Mm B0mm BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF Beni'g W. Sage 1891 i,/^l.^Ai f/:^/f^z. Cornell University Library HG229 .N88 Statistical studies in the New Yoric iripne olin 3 1924 032 510 962 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31 92403251 0962 Statistical Studies in the New York Money-Market PUBLICATIONS OF THE DEPARTMENT OF THE SOCIAL SCIENCES YALE UNIVERSITY STATISTICAL STUDIES IN THE NEW YORK MONEY-MARKET PRECEDED BY A BRIEF ANALYSIS UNDER THE THEORY OF MONEY AND CREDIT WITH STATISTICAL TABLES, DIAGRAMS AND FOLDING CHART BY JOHN PEASE NORTON, Ph.D. PUBLISHED FOR THE Department of tbe Social Sciences, Kale laniversitie BY THE MACMILLAN COMPANY New York 1902 Copyright, 1902 by Yale UmvERSixv THE TUTTLE, MOREHOUSE & TAYLOR COMPANY, NEW HAVEN, CONN. PREFACE In the following pages an attempt is made to apply the mathematical methods of interpolation and correlation to the financial statistics of discount rates and banking items as pub- lished weekly by the financial journals. In studying correlation the methods of Professor Karl Pearson have been of very great use. Although Professor Pearson's writings have been largely in connection with the biological problems of evolution, his statistical methods have been found to apply satisfactorily to the problems presented under the theory of money and credit. I have tried as far as possible to subordinate the mathematical side of the work, and when for clearness it was necessary to introduce some mathe- matical formulas I have tried to state the logic of the formu- las also in words and to express the meaning of the tables graphically by charts. I am indebted to Professor Sumner and Professor Schwab for very material assistance; and to my mother for help in revising the proof and in verifying the statistical tables. yale University, New Haven, April 27, 1901. CONTENTS PART I PAGE Brief Analysis under the Theory of Money and Credit i PART II Statistical Studies in the New York Money-Market 13 CHAPTER I The Statistics of the New York Associated Banks 15 CHAPTER II The Growth Element 24 CHAPTER III The Percentage Deviations 35 CHAPTER IV Periodicity in the Reserve Deviations 45 CHAPTER V Periodicity in the Loan Deviations 62 CHAPTER VI Correlation 68 VI CHAPTER VII Correlation between the Call Discount Rate and the Ratio of Reserves to Deposits 78 CHAPTER VIII Correlation between the Call Discount Rate and the Percentage Devia- tions of the Total Reserves 88 CHAPTER IX Correlation between the Reserve and Loan Periods 92 CHAPTER X The Crisis , 98 CHAPTER XI Summary loi Folding Chart facing p. 104 Index 105 PART I BRIEF ANALYSIS UNDER THE THEORY OF MONEY AND CREDIT NEW YORK MONEY-MARKET PART I Brief Analysis under the Theory of Money AND Credit § I. In any short time on a limited habitat, there are two elements in exchange, commodities and the media of exchange. Commodities consist of goods and labor. Media of exchange consist of money and credit. With the two elements commodities and media of exchange there are four permutations, — (i) Commodities against Com- modities, (2) Commodities against Media of Exchange, (3) Media of Exchange against Commodities, (4) Media of Exchange against Media of Exchange. In the end, men simply wish to exchange commodities. Obviously this reduces to three cases : — * Case I. Commodities against Commodities, i. e. Barter. Case II. Commodities against the Media of Exchange, i. e. Exchange. The principal Media of Exchange are two, Money and Banking Devices. Case III. Media of Exchange against Media of Exchange, i. e. Money Changing and Banking. §2. If we mark off a limited area and limited time, we may throw the three cases into one equation representing the societary circulation. We may use letters in the follow- ing senses to represent the various factors. C — commodities ; Cc — when bartered ; Cmd — when exchanged against the media of exchange. Md — media of exchange. Pmd — prices reckoned in the media of exchange. * Cf. Newcomb, Principles, Chapter XV. Pc — prices reckoned in other commodities. D — rate of discount on " futures " in the media of exchange (i. e. loans or " paper") bought by the banks. Mdf — " futures" in the media of exchange.* Map — " spot" media of exchange. Mdc — stock of money proper in circulation. Mdb — stock of money proper in the total reserves of the banks. Vc — rate of turnover of money in circulation during the time (T). Vb — rate of turnover of money in bank-vaults in indirectly effecting exchange. We may now write the following equality as a first approx- imation : — f barter-|-exchange — banking expenditures+receipts= barter-|- SPcCc+ 2P„,dC„,d - 2(i-D)Mdf + 2Mdp =2PcCe+ money paid in exchange-|-banking devices paid in exchange SMdoVoT + SMdbVbT. Inasmuch as the part of business transactions effected by barter is relatively small, we may, to simplify the work, neglect both of the 2PcCc terms. We then have the follow- ing equation : — 2Pn,dCn,d-S(i-D)Ma(+SMdp = 2MdcVeT+2MdbVbT. This is the general equation of the societary circulation developed by Professor Newcomb and other writers in a slightly different form. President Hadley,;]; in treating of the monetary aspect of this subject, has already brought into common use in his text-book the formula, RxM = PxT, where "the amount of money in the country" is "repre- * Cf . Schwab, Unpublished Lectures. f Cf. Fisher, The R6le of Capital in Economic Theory, Economic Journal, December, i8g7. "Gifts, bequests, charity, taxes, etc.," as well as failure of banking expenditures to exactly equal receipts during a short time makes this statement a rough approximation. X Economics, p. 196. -3— sented by M and its rapidity of circulation by R." The price level P is a percentage of the price level of a preceding year "treated as unity." Transactions of the given year, "esti- mated at the prices of the ' standard year,' are represented by T." § 3. In the United States the basis of the media of exchange is gold coin, as now defined by the New Currency Act of March 1900. At a given time, there is a certain amount of gold coin in the country which does not fluctuate very widely from day to day. It may, of course, during longer periods change in amount owing to exports, loss, use by arts, imports, production and coinage. Increase by (i) Imports; Decrease by (i) Exports; (2) Coinage. (2) Loss; (3) Use by Arts. The stock of gold may be said to be located in three places in the country, in the (i) U. S. Treasury; (2) Banks; (3) Elsewhere. All this is perfectly obvious and is stated simply to call atten- tion to facts which later will be used. With this apology, let us look at a sample U. S. Treasury statement* showing the amount of money in the United States on July I, 1899. Table No. i. Gold coin, including bullion in Treasury, . . . Standard silver dollars, including bullion in Treasury, Subsidiary silver, Gold certificates Silver certificates, Treasury notes, act July 14, 1890, United States notes, Currency certificates, act June 8, 1872, .... National-bank notes, Total, $2,745 General stock coined or issued, in millions. Proportionate part. $963 564 35!« 20 75 3 34 406 I 15 94 3 347 12 21 I 241 9 100^ Report of the Secretary of the Treasury for 1899, Table M. — 4— According to this statement, gold constitutes 35^ of the stock of money, silver coin and bullion 20^, silver certificates 1$%, United States notes 12^, bank notes 9^, and other forms 9%. The distribution of this fund between the U. S. Treasury, the total reserves of the national banks, reserves of all other non-national banks and circulation among the people is roughly gauged by Table No. 2. Table No. 2. In millions. Proportionate part. In Treasury (Sept. i, 1899) National Banks (Sept. 7, 1899), All other banks $931 466 210 1,228 33^ 16 8 Circulation, 43 Total, $2,835 100^ All these figures are in error owing to the methods of con- jecture employed by the U. S. Treasury officials, but as we study further we shall find that it is not total amounts we care for, but changes on the margin. The really important problem to solve is the relation existing between prices, volume of transactions and media of exchange. To predict prices is the ultimately useful goal. Increasing transactions, it is said, make more exchange-work for the standard money and its dependent instruments of credit, enhancing in some degree the value of money, and consequently causing prices to fall. All this is very general and lacks the quantitative test. § 4. Without knowing the correct figures, we may, never- theless, write out the equation of the relations. In this section, let us designate the various factors by the following letters : — Letter. In Treasury. In Banks. Media of Exchange. National. Non-National. In Circulation. Gold Silver Silver certificates, . . United States notes, . Bank notes, . . . Other forms, . . . G S c N B L Gt s. Ct Nt Bt Lt Gt Sb Cb Nb Bb Lb Up B^ G„ Sc c„ No Bo Lo The amount of money piled up in the Treasury, which can do no direct exchange-work as long as it lies in the vaults, is represented by the following expression : — (Gt+St+Ct+Nt+Bt+Lt). The gold, however, may be said to do indirectly the exchange- work of all fiduciary money and instruments dependent on it. In the vaults of the national banks, we may represent the same as follows: — (Gb+Sb+Cb+Nb+Bb+U). The holdings of currency by all other banks may be tokened by use of the subscript (p) : — (Gp+Sp+Cp+Np+Bp+Lp). Elsewhere circulating among the people, we have the remaining amount: — (Ge+Se+Ce+Nc+Bc+Lc). The sum of these four expressions is, of course, the total amount of currency in the country. § 5. So far we have been dealing with quantities of cur- rency in existence at a time. The next step is to throw into exact form the theoretical exchange-work done by the media of exchange, as represented by the above expressions. The exchange-work accomplished during a time is found by multi- plying the quantities of different kinds of currency by their respective "numbers of times of turnover" during the given time. I. Money in Circulation. During the day, the various kinds of money pass from hand to hand in exchange for goods at different velocities. By velocity we mean the number of times the dollar exchanges for goods per unit of time. The exchange-work done for a short time, during which we may consider the sum of money in circulation relatively constant, is given by the following expression : — T(GoVg+ScV,-fCoVe+N.V„ + BeVb+LeV,), where (V) represents the average velocity* of the kind of money whose subscript it bears, and (T) the time. It is probable that the government will, in course of time, institute investigations to determine the velocities of differ- ent kinds of money within limits. Such investigations could hardly fail to have extremely fruitful results. For much misunderstanding in monetary discussions arises in exactly this lack of statistical data. It follows that : — T(GeVg+SeVs+CcVc+NeVn+BeVt+LeVi) _ T(Gc+Se+Ce+Ne+Be+Le) where V denotes the weighted average velocity of the whole stock of money in circulation for the time (T). We may now write the exchange-work accomplished by money in passing from hand to hand thus : — Eo=TV(Gc+Se+Ce+Ne+Bo+Le). II. Money in bank vaults. The next step is to develop a like equation for the exchange-work of money in banks. f On the books of the banks are all the records of all trans- actions effected by checks and drafts. That a check or draft be honored, a deposit must stand on the books of the bank subject to the order of the drawer. As the days pass, the deposits of yesterday are checked out, and new deposits are entered. The weekly statements of the New York Asso- ciated Banks furnish a concrete example. In the figure below, let OT be the time line, OD the aver- age specie holdings for the week January i to January 8, DC the average legal tender holdings. It follows that OD+DC, i. e. OC, equals the total reserves. AC represents the aver- age deposits. As the days pass, old deposits are checked out, and new deposits are entered. There were AC deposits on January I. On January 3, there were PL deposits of the old AC * Cf. Fisher, The R6le of Capital in Economic Theory, Economic Journal, l8g7, pp. 520. t Cf. § 2. -7— deposits left, and LK represents new deposits. On January 7, all the old deposits are cancelled, and NM new deposits stand on the books. T = CM = time of turnover of the deposits AC. The reciprocal i/T — U, the rate of turnover of each dollar of the deposits. It follows that T tan a= CM XNM/CM = TXD/T = exchange- work done by the deposits AC. In general, if D be the deposits at a time, U the rate of turnover,* and T a short time during which the deposits and rate of turnover vary little in amount, the exchange-work done is given by the expression TxDxU. The equation of societary circulation is generally given as E = T(MxV + DxU),t where M represents money, V velocity of money, D deposits and U rate of turnover of deposits. This expression did not * a. Pierre Des Essars, LaVitesse de la Circulation de la Monnaie, Jour. Soc. Statistique, Paris, April, 1895. Relation of V to crises is developed. •f Cf. Newcomb, Principles; Fisher, Unpublished Lectures; Gaines, Unpublished Lectures. seem to me to readily lend itself to statistical investigation. In fact, interest attaches to the relation existing between deposits and the total reserves of the banks. The Act of January 1875 prescribes that banks shall hold a minimum reserve against deposits. In the reserve cities, the reserves must be 25^ of the deposits; for all other banks IS%. A bank in a reserve city may deposit 1/2 of its required reserve in any central reserve city bank, and any bank out- side of the reserve cities may deposit 3/5 of its required reserves with other banks in the central reserve cities. Consequently, with a given amount of total reserves, we may write the equations by which the maximum deposits with those reserves may be reached. By R designate total reserves. The reserves of the country banks will be Ri, of the reserve city banks R^, and of the cen- tral reserve city banks R^, deposits D^, D^, D3. For Country Banks, deposits are forbidden to exceed the ratio 100/15 to reserves. It follows that Maximum 0^=100/15 R^. But 3/S of R, may consist of deposits in the central reserve cities. Consequently Maximum Di= 100/15 x 5/2 Rm,= 16.667 Rmi. Rmi stands for the lawful money reserve. For the Reserve City Banks, the ratio is 100/25. Thus Maximum 0^=100/25 R,. Here 1/2 of Rj may be deposited with the central reserve banks. Therefore Maximum D,= 100/25x2/1 Rm,=8 Rm^. In the same way for the Central Reserve Banks, Maximum 03=100/25=4 R^,. By adding together Max. D,4-Max. D,+Max. D„ — 9— we obtain the maximum lawful deposits attainable by the National Banking System. Substituting the above values we have Maximum 0=16.667 Rm,+8 Rma4-4 Rm,. This then is the maximum towards which the deposits of the National Banking System are all the time tending. The nearer bank presidents can approximate this condition, the greater are their profits. If business presses hard on the maximum, discount rates rise. As business lags far behind the maximum, discount rates fall that the loan account may increase and so deposits increase. If we take the Comptroller's return for October 2, 1890, we find by the equation : — Maximum 0=16.667 Rm.+S Rm,+4 Rm, = $1,172+1614+1369 = $2,154.8 millions. Now the actual deposits were $1,758.7 millions, or 82^ of the possible lawful maximum. It is never safe to approach too closely to the maximum, for the funds in country banks, reserve city banks and central reserve banks are all the time flowing from one to the other. At certain seasons, there is a strong ebb in one direction, and, at other seasons, opposite currents. (This will be shown later by statistical charts.) And so a margin must be left against these flows as well as against a possible increased demand of money for currency circulation. To illustrate the variations, four recent returns of the Comptroller are reduced to the above percentage of the maximum. December i, 1898 — 94^ September 7, 1899 — 80^ February 4, 1899 — 72^ September 5, 1900 — 83^ These percentages measure the credit strain to which the banking system is at a time subjected. The lawgivers pre- scribed 100^ as the safety limit for the stress. If we let Kj, K„ and K, stand for the percentage propor- tion the actual deposits D„ D^, and D^ bear to the maximum D„ MAXIMUM Dj and maximum D3, we may write for the whole country Maximum D = D/K. But we know D= 16.667 K,R:n,-|-8 K,R„i,-f4 K^R^s as the more exact expression. Further, let U stand for the weighted average rate of turn- over of deposits for the whole country, and U„ U„ U3, the velocities for country, reserve city, and central reserve city banks respectively. We may then write the familiar expres- sion D X U in the following form : — DxU= 16.667 K,U,Rm,+8 K,U,R„,,+4 K3U3Rra,. This gives the exchange-work equation. It stands for the exchange-work done per unit of time by the specie and legal tenders lying in the vaults of the national banks throughovxt the country. III. The exchange- work equation for the remaining State Banks, Private Banks and Trust Companies may be worked out in a manner analogous to the above. If we let Up represent the average weighted velocity of deposits. Dp deposits, Kp the percentage proportion as above between actual and maximum deposits, Rmp the total specie and legal tender holdings, and Cp the coefficient of maximum deposits to specie and legal tender holdings as fixed by the laws of the various states, the general equation of exchange-work may be written : — DpUp^ CpKpUpRmp. We may now write the complete equation of exchange- work (E), done by money directly by passing from hand to hand against goods, and by money indirectly as the reserve in national and other banks. The letters are used in pre- cisely the same senses as above. The equation comes by addition of the expressions already derived. E = T< V„,(G^+S„.+C„,+N„.+R„+U)+i6.667 K.U,R„,+ 8 K,U,R.,+4 K3U3Rm3+CpKpUpRp> — II — This expression represents the media of exchange side, i. e. the right hand side of the general equation of the societary circulation developed in § 2. § 6. Let us consider for a moment the statistical data bear- ing upon these items. The reports of the Treasurer of the United States contain an estimate of the amount of money of all kinds in circulation on the last day of each month. These estimates are con- veniently collected in the Commercial Year-book of the Journal of Commerce. Kj, K„ K3 are easily found from the Reports of the Comp- troller of the Currency. Rm,, Rm^, Rms are likewise given in the same report. Vm we do not know save for a limited set of observations.* We have indirect methods of judging its limitations, which, however, have been little worked. Cp, Kp, and Up are given in no one place. The Comp- troller gives estimates of Rmp. Uj, Uj, and U^ we have no way of finding directlyf at the present time. A limited range of statistics exists, col- lected by Dr. Gaines for some New York banks. There is also a series compiled for the Bank of France.:]: As a sub- stitute. Dr. Gaines has suggested the ratio of clearings to average deposits. The Commercial and Financial Chronicle publishes weekly the clearings for the whole country. These ratios for the three classes of national banks we will designate by W„ W„ W3. W, = CyD„ W,=C,/D„ W,=C3/D3. *"Ahundred such returns among students at Yale University indicated an average velocity of forty-five times a year, making the average length of time a dollar rests in one man's hands about eight days." (Fisher, cited above, pp. 520.) f There is no more deserving a question for investigation by the Government in the whole subject of money and banking than the determination of this data with the aid of the banks. The question is not divorced from practical affairs; for it is so intimately connected with crises and price levels that exact knowledge will render safeguards possible. X Pierre des Essars also gives annual averages for the Banks of France, Ger- many, Belgium, Portugal, Spain, Italy and Greece. 12 Five times a year, the national banks make returns of Di, Dj, and D,. In the case of D,, however, the central reserve banks publish in addition weekly averages. This determines W, weekly. The ratio W, is smaller than U^ by the amount of checks that cancel each other in the banks and fail to pass through the Clearing-house. It is probably unsafe to say that Wj and U, vary together in fixed proportion, but there is no doubt that they vary in the same directions. § 7. We have now a complete statement of the monetary factors of the equation of the societary circulation. In a laboratory of economic statistics, we may imagine on file records of each one of these items for a long series of years. We might then write the exchange-work equation across the top of a wide sheet of paper and in columns below each letter fill in the appropriate value for each week. Such a statement would be but a systematic arrangement of facts. The interest of science demands more than an arrangement of facts. The predominating interest of all science is in discovering corre- lations, laws, which derived from past phenomena may be relied upon with some degree of probability to predict future phenomena. Social utilities are served by relations found to exist between the motions of the heavenly bodies and the dimen- sion time. The tides, time of sunset and sunrise, eclipses, all may be predicted for months and years. Time in economics is of no less interest in the business world. In a search for correlations it is wise at the start to divide the work into four classes of problems : — (i) Proportional relations at a time between the items of the equation showing the relative importance of each. (2) Deviations in items at one time which are followed later by deviations in other items, thus showing anticipatory correlation. (3) Changes in a single column coincident with the passage of a long series of years. (4) Periodic changes in one column with the seasons of the year. PART II STATISTICAL STUDIES IN THE NEW YORK MONEY-MARKET CHAPTER I THE STATISTICS OF THE NEW YORK ASSOCIATED BANKS § 8. Theory and statistics are the two legs of economic science. Some theory in the past without statistics has notoriously gone lame. This lameness in the past had a justifiable palliation. There were no statistics to be had. To-day this lack of accurate statistics is rapidly vanishing. Trade journals, governments, states, cities, corporations, commissions — public, semi-public and private — are all throw- ing out daily and weekly masses of undigested statistics that are appalling. In the past statisticians have too often been content to study such arrays of statistics by an average. Yearly aver- ages and often averages for five and ten-year periods have been taken. The results of such conglomerations of many tendencies into a single index have been often small. An average is easy, often useful, but many times a senseless thing. In many cases, it must be confessed, the average tells very little. In the future, it will perhaps fall back to its place as only one of the several decisive quantities of the frequency curve of the array. The great fault of a series of averages is the lack of con- tinuity. This becomes glaring when quinquennial and bien- nial, or often annual, averages are used. In many investiga- tions quarterly and monthly averages are not sufficiently continuous. The week is much nearer the ideal unit ; for it combines practicability of handling with an approximate con- tinuity. Yet in problems of correlating variations in stock quotations and the call discount rate at the stock exchange in times of serious disturbance to credit, a twenty or thirty minute average is necessary for results. Such misuse of indices has brought down that sarcasm of Bernard Moses, — "We have plenty of statisticians, but no — 16— statistics." The reverse is perhaps nearer the truth. There is an abundance of excellent statistics, but a narrow dispersion of statistical method as it exists in other branches of science. To make light of our statistics is to cut huge lumps from beneath the ground-work of economic science. § 9. In discussing the problems arising in a contemplated perfect table of statistics covering many years and prepared as suggested in Part I, we divided the problems for conveni- ence into four classes. Two of these classes of problems consisted in correlating changes in different items, (i) at the same time, (ii) at different times, the one giving immediate and the other anticipatory correlation. The other two prob- lems were concerned with changes in one item coincident (i) with the passage of a long lapse of time, and (ii) with recurrent or periodic time. Unfortunately no such perfect table exists. The investi- gator must needs attempt the compiling of a part. The complete expression for the exchange-work (E), it will be remembered, was [T< V„(G„.+S^+C„+N„+R„,+L,„)+ 1 6. 667 K,U,R,..+ 8 K,U,R,.,+K3U3R„,,+CpKpUpRp>]. In the following investigation the factor Rmj was split up. Rms stands for the total reserves of the central reserve city banks. It consists of the reserves of the Associated Banks of New York, Chicago and St. Louis. Inasmuch as the returns for New York are by far the most important, consti- tuting the largest portion of the factor Rmj, it seemed best to make a beginning with that series. It will be convenient to consider first the single item and later the problems of corre- lation. § 10. The weekly statements* of the New York Associated Banks go back for over forty years. The method of prepar- ing the statement is the so-called " system of averages." The * "Weekly bank statements were not made by any banks until Aug. 6, 1853, when the New York banks began the custom, and others gradually followed." Sumner, History of American Currency, pp. 175-6. —17— typical bank which is a member of the Clearing House Asso- ciation, prepares a table showing for each day of the week the amount of loans, deposits, specie and legal tenders held within its vaults. On the last day of the week, the bank adds up the six values for each item and divides the totals by six. If a holiday intervenes, the divisor is of course five. The averages of each bank for the several items are reported to the Clearing House and there the weekly state- ments of the New York Associated Banks are compiled. Thus the items in the statement for loans or deposits are the sums of the amounts of loans or deposits reported by the several banks. The statement as issued by the Clearing House gives in addition the changes, positive or negative, with respect to the preceding week in each item. The bank statement is often the "feature" in the Saturday market and, when it appears at about 11.30 A. M., frequently becomes the cause of an advance or of a decline in quotations, as the statement seems favorable or unfavorable to the oper- ators. The theory is that smaller reserves (i. e. shrinkage in the sum of specie and legal tenders) lower the ratio of reserves to deposits, which by law shall not sink below 25^. The bankers' method of guarding against a too rapidly shrinking tendency in the total reserves is to raise discount rates. Higher discount rates tend necessarily to make the burden of carrying stocks on a margin greater and conse- quently "make" for lower prices. Inasmuch as, on the whole, increasing loans tend to increase deposits, all the items affect directly or indirectly the ratio. The ways in which these items are correlated together and with the discount rates on call loans will appear later. It is enough here to note: — (i) That the statistics are well prepared and represent the banking conditions at one of the most important financial centers of the world ; (2) That the weekly average affords very satisfactory con- tinuity ; m o fa* o Z K < H O H O > O Z 1 O^ ■'to ■^ M O'O iH cnO 'JSNtJ-OOOcOW -^co C^ N m co o^ g; CO M O r^r>mj>-oo o^co cno ■^'d-o cT'O u-»vo -^d ooo -^r^cno oo O CO + OOHMMh-iwOOC^OOOOOOOOiHWWcnc^'t'^'Tr MH«N o OO en O vio « coo w r^u-iTtM or^OOcoi>Ti-c*TO r^trimrl-'O 0^ OO l-l 1 ■ThwvOO cor-r-'N O inr-'-i OOw r^'^OmO O c<-)w O O l> CO 1 c^iocOO o or^co 00«m0 ON rJ-r-^r-u-i-^l-M H O OOcO WNCO-^unHOOOOWMC^MC^NCJMWWWWNMNlH en CO M o (r)Hc-( cqenTfTi--J^c^NMHMjHOiMNWNNWNCnNMMOO C4 CO ^■g,^ ss'S ^^>8 s ? a s> a"!?. ? 2 S"^-^ s s s s s s CO 1 o CO M 1 22^225 5^22222^222222222222 CO 00 1 O i^NvO r^c*-u>.o woo or>-\nou-)W -"tO 0"->v0 mn « Oco CO CO CO M oSSg 1 22555K5S22222222SS52222&22 CO 00 + 2 S S "S S H" S 2 2 2 2 2 ^" ^ ^ 2 ^" " ^ ^ ^" ^ ^ CO oo M 1 S2E'S"E'S?&2H^H"22222222222222 OO CO M 1 S'SS§-|3^'g,S|,^^S;^S||^^?S,S,S.S^'5> CO 00 M C4 P u M m M OTj-r-cow H o OM rhO '-wco O « r^Om OOOOOmmmOO OOOO OoO cO CO oO 00 O r* l> t^QO oo CO CO 1 oococococoooco r-r^oovDOO r^t--r^r^c30oocooococo a^ a^ CO CO 00 -r^r^i>-r^r^i>r--coooooooco r^i>.cocoooco H OD OO + vO M cncnrJ-riH cncooo n h q coO m mOTj-oooo Tl-\nTj-iriTi- r^co oo CO CO 00 CO r--vO o t^ r^ r^ t-^ r^oo ooco o^o^o^i^o^o^o^o^ O CO oo 1 £^o 1 w inOOOOM u^c^ O r^vO tJ-tJ-^tI-O h inco « c<^utO i> ^ooooovo r^t^r>r>- t^vo ■ovO'O^O'O'O r^t-* r^co oo oo oo oo CO I d^t^w cH^oo cnoi^oo c<^o mococo «co w o N Thcnt^ •Avd JduipnodsdxiOQ t^Thi-ico tJ-wco lo-^Moo miico mw OO, coO r^cnO t-^ri-iH MNd MWW MMW l-l««^IH«« MW« 1 1 1 1 IIS -3133 AV iH N en'^m-O r-'CO OO w « cotJ-uiO r^oo oo w n eo^+u-iyD MMWMMMHWMMCTMNNCTNN —19— 1 1 OHir)Oc*^«'^0>cncncOHTtOOu^i^'ONooinr~--t^ON m cn'^Tj-iriir)mTr'*'»r)u-i'<;t-TtcoNNiHHMMiHM«NMM -n OO o M c^ tH \0 M r^co I-^ N H in cOOO OtncncnONcnTtMvOcONin o cnencONNN C^ 0) « HH OO OiO^OiON OOO COCOOOnOn O^ N en o -■^cnwMNciNMO ovco coo^a^o•HMMOMl-ll-(MH(^^ cn O CO m 1 cnrhONcnMco t^ONO"^r~^o^« oco o>r->t-i o>i-i rhr-* o^oo en o^ «r) OiO^OvQ o ONONa^ON o>co r^j>-r^yD^o t-^oo i>cooooococooo r^co ON 00 "is a' 1 k. ed o 1 xncoco«aNcnOTt'«oOMvooO"^t^ON'-oO'^oOMTj' o Tt--^ThTi-cncnNN«NMWNcncnNWWNcn-cococococooooo r^r^^o^o inTtTj-rttninminTt-m-Tj-'^ tJ- 00 1-1 M 1 Onm hqo t1-m -^cnNco r^r^r^r^oo w « w o w ■^r^mii-»« en MNMI-tlHlH)HtHHOOOOOOlHlHrHl-ll-llHONr^l>l> t^- en CO yn Tt-sO w ONvO O >0 N vO u") tj- c^ Onco oor^TfNNOcooocn r^ OnOnOnOM^i>0000 OnOnO w N « enrj-mo^co 0\ O O O O O HMMMMMIHIHI-flHHtHMClN W OO M "'i'§a^ MOO w T^O H r-^enOMTiOco u-»-^ooococo l~^Ooo O Ococo t^co inTj-inininmTi-Ti-encncnMW nwm m mwhwnnw w m m H ON OO 1 r-.u-iM M M ONOMn-^O O ONvn'^co NOr^w tJ-O^c; 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A ^ 1 6 & o ■5139 AV ■Boijscan^ r-co oOm n en- i»g I or Log Y=A+B Log X Y=A + B Log X + C Log' X Y=A-|-BC^ Y=A+BLog(X-C) Log Y=A + B Log (X-C) Y=A + BC'''s^ Y=A + B(CX + D)« Y=(A + BX)« Log Y=A-I-BX« or Y=io "'+''^°' Hyperbola Straight line Straight line Parabola Parabolic curve Logarithmic curve Logarithmic curve Exponential curve No name Logarithmic curve Logarithmic curve No name No name No name No name No name —26— too great an influence in the determination of the probable variation. § 15. After a consideration of the practical difficulties as well as the needs of theory, it seemed best to use as a growth axis the geometrical curve* of simplest type ¥=6^ (Stein- hauser's form 10). This form is familiar in the algebraic formula for compound interest, A=P(i+R)', where A is the amount after T years, P the principal and R the rate. This form is probably not the best fit theoretically, but it is a good fit approximately, as we shall see, and to increase the number of constants was practically an added burden in arithmetical computation that was out of the ques- tion. Moreover, the convenience of the formula would be greatly lessened by greater complication and the gain would not be, perhaps, after all very apparent for our purposes. So having once assumed the geometrical form Y = BC as the growth axis, it is simply necessary to determine the con- stants B and C of the form Y = BC'' for the three items of the bank statements, the total reserves, the loans and the deposits. § 16. Although the method of interpolating the geometrical curve is quite widely known in economics, for the sake of completeness it may be well to sketch roughly the procedure. The geometrical form Y = BC'' when written in logarithms Log Y = Log B+X Log C, becomes an arithmetical form, i. e. if plotted (Diagram No. 2 on the Chart), the logarithmic curve becomes a straight line. Now it is obvious that, having collected the statistics of the three bank items for the years 1879— 1900, we know the amount of each item for every week. If we represent by (Y) the amount of the item for a week which is (X) years from the first week of 1879, we may say that the observational values of (Y) and (X) in the above equation are known for every week of the twenty-two years. We do not know the * The writer tried tentatively several modified forms by graphic processes. —2 7— values of the constants (B) and (C). Graphically (Diagram No. 2 on folding chart), Log B represents the point at which the straight line starts. Log C is the increase at the end of one year. To find these two unknown constants B and C, the following tables are prepared. I. II. Log Y = Log B + X Log C (I) (2) 2 Log Y = N Log B + Log CSX X Log Y = X Log B + X^ Log C (3) (4) IX ■ Log Y = Log B 2X + Log CSX^ In column (i) all observations for the twenty-two years are entered weekly. In column (2) is entered the time of the entry in column (i), counting from January 7, 1879. Columns (i) and (2) are multiplied through by the successive values of X, giving columns (3) and (4). Inasmuch as X = o for the record of January i, 1879, the first records vanish in (3) and (4), and so the first series of equations contains one more equation than the second series. The entries in each column are added and the sums form the two equations written at the foot of the tables. The solution of these two equations determines the two unknowns, Log B and Log C. The values of the constants B and C are, of course, deter- mined by the logarithms and may now be inserted in the form Y=BC^. This formula renders it possible to calculate for any week the value as given by the form. These values plotted form the steadily rising geometrical curve. This, in brief, is the statistical mechanism of the growth axis. § 17. In practical work, it is impossible to use every record. To do so would involve an unbearable amount of arithmetical computation. For instance, in each case we should have to add four columns of over 1 100 numbers of four or five figures in addition to over 2200 multiplications of an average of seven figures by three. The work might be lessened by the use of multiplication tables and the adding machine, but the end would not justify the means. It is enough to take readings at equal intervals, say monthly or bi-monthly. In some cases it may be well to —28— underweight years out of the ordinary as crises, but on the whole the impartial method of equal intervals is preferable ; for too much work should not be put upon the growth axes, which are after all but the scaffoldings of the polygons. § 1 8. To illustrate the method of interpolation more fully, the Growth Axis of the Total Reserves is in point. In the form Y=BG', Y equals the value of the total reserve, Rt, for any week, T, and may be written Rt. X represents the interval of time since January i, 1879, fo'' ^^7 week. B stands for the first value, Ro, and C stands for (i+r), i- e. one plus the annual rate of increase. Writing the form Y = BC^ in these new letters, we obtain the formula Rt = Ro(i+r)'. Writing logarithms, we have Log kt=Log Ro-f-T Log (i+r). The two unknowns are, of course, B and C, i. e. Log Ro and Log (i+r). In words, it is necessary to know at what point to start the growth axis, and how rapidly to make it rise. In the following work 136 records were taken. Construct- ing the two tables as stated in § 16, the two following equa- tions were obtained on adding the tour columns.* 276.378877= 136 Log Ro-f 1,216.2 Log (i-|-r) 2,599.298623=1,216.2 Log Ro-|-i6, 196.8 Log (i-|-r) Solving, Log(i-|-r)=o.o233i7 Log Ro =1.823682 The logarithmic equation of the growth axis is then Log Ro=i.823682-|-o.o233i7 t. Dispensing with logarithms, we have approximate!)- Ro = 66.63 (1.055)'. § 19. Following the method of interpolation used in the determination of the growth axis of the reserve polygon, * In actual work, it should be stated, there are several short cuts which con- siderably modify the burden of calculation. -29- the equation of the Growth Axis of the Deposits was found to be Y = 279.8 (1.0397)', or expressed in logarithms, Log ¥=2.446987+0.016926 t. This is based on 1 5 5 records at more or less even intervals with a slight weighting. The equation of the Growth Axis of the Loans is based on 88 records. The equation is Lt=264.7 (1.042)*, or expressed in logarithms, Log Lt=2.4227i7-j-o.oi7742 t. § 20. These equations enable us to work out the values of the growth axes for any week of the twenty-two years. In the following table (No. 5) are the values of the growth axes of the reserves, loans and deposits on the first day of January in each year. The values in this table, when plotted on the Chart, form the "three steadily ascending smooth curves" mentioned in § 11. The lowest smooth curve is the growth axis of the reserves. Next above is the growth axis of the loans. The highest smooth curve represents the growth ele- ment of the deposits. Table No. 5. Growth Axes Values for January i of each year, in millions of dollars. Year. Reserves. Deposits. Loans. Year. Reserves. Deposits. Loans. 1879 67 280 265 189I 127 447 432 1880 70 291 276 1892 134 465 450 1881 74 303 287 1893 141 483 469 1882 78 315 299 1894 149 502 489 1883 83 327 312 1895 157 522 509 1884 87 340 325 1896 166 543 530 1885 92 354 338 1897 175 565 552 1886 97 368 352 1898 185 587 575 1887 102 382 367 1899 195 610 599 1888 108 398 382 1900 206 635 624 1889 114 413 398 1901 217 660 650 1890 120 430 415 1902 229 686 677 —so— §2 1. If now we turn to the Chart and consider how well each geometrical growth axis fits its respective polygon, the question arises what is meant by a ' good fit? ' The popular notion would perhaps be that the polygon should balance as evenly as possible on its growth axis. From the notion of an even balance the mathematical dogma has been developed that the minimizing of the squares of the deviations constitutes the criterion of a ' good fit.' In an ordinary arithmetical form — an example would be the expansion of mercury under heat — the deviations are in amplitude no greater with a higher temperature than with a lower one. In such a case the higher deviations exert the same influence in the determination of the probable variation as do the lower deviations, and in such cases of a constant law of deviation the quantitative test of least squares is of real value. But in the cases of an increasing or of a decreas- ing amplitude of deviations, this quantitative index is unrep- resentative and so far as I can see a very bad test. In this case of increasing funds in which the deviations become in amplitude constantly more violent, a very difier- ent notion of an even balance is involved. For in a con- tinually increasing amplitude the amplitude might come, at last, to equal the whole fund so that the final squares would exert so vast an influence upon the sum of the squares, that the squares of the last few deviations would equal the sum of the squares of all the deviations that had gone before. In this extreme case, the meaning of the least square test is entirely lost. It might seem best to interpolate the deviations without regard to signs and to obtain a ratio between the fund and the amplitude by a comparison of the growth axes of the fund and of the deviations. In this way, we might reduce all the deviations by this changing ratio to the fund and apply the least square test to these proportionate deviations. Such a method brings up a rather complex theory in probabilities. The difficulties in practical work would be also very great owing to the large amount of tentative arithmetical calcu- lation. —31— Out of the original, popular notion of an even balance, another interpretation may be developed. Abandon all idea of a quantitative test of measured deviations, and rely upon a qualitative test. Practically what is wanted is a growth axis which shall all the time be crossed and recrossed by the polygon. The form, then, which shows a greater number of crossings with a higher evenness of distribution is the better form. This is a convenient method for the statistician. All that is required, in an example of this sort, is a table showing the number of crossings per year. Calculate the average number of yearly crossings with the probable varia- tion, and this range becomes the test of fitness. The greater the average and the smaller the probable variation, the better is the form. It is not a decisive criterion; for sometimes the advantage between a slightly lower average with narrower range and a higher average with wider range would be diffi- cult to decide. In such a case, simplicity of form should be the deciding factor for the statistician. Table No. 6 records the number of times during each year that the reserve, loan and deposit polygons crossed their respective growth axes. Table No. 6. Year. Reserves. Loans. Deposits. Year. Reserves. Loans. Deposits. 1879 6 5 1891 I 1880 4 2 2 1892 I 2 2 1881 4 4 1893 3 3 1882 5 2 4 1894 1883 5 4 1895 3 2 1884 2 I 2 1896 1885 5 1897 5 3 5 1886 2 2 3 1898 2 2 1887 2 2 2 1899 I 1888 3 2 4 1900 5 1889 1890 6 2 8 2 4 Average 2.8 1-5 1.9 Range 1.2 I.O 1.2 Thus, during the twenty-two years, the reserve polygon crosses the growth axis 62 times, or on an average of 2.8 crossings per year. The deposit polygon crosses 42 times, —32— or on an average of 1.9 times yearly, and the loan polygon makes 33 crossings, with an average of 1.5 crossings yearly. The probable annual range of the reserve polygon with respect to the number of crossings is 1.6 to 4.0, of the deposits 0.7 to 3.1, and of the loans 0.5 to 2.5. The method of calculating the probable range is doubtless familiar. It is briefly described at a later point.* The commonly accepted meaning is that in any year the chances are even that the number of crossings will lie within this range. §22. One or two interesting mathematical properties of these growth axes may be remarked. The ratios of annual increase are for the reserve axis 5.5^, the deposit axis 4.0^, and the loan axis 4.2^, approximately. Now, owing to these varying rates, the ratios existing between the growth axes for any date must change. Thus, for the first week in 1879, the ratio of reserve axis to deposit axis is 23.8^ and of reserve axis to loan axis 25.2^. Twenty-two years later, the ratio of reserve axis to deposit axis is 32.0^, and the ratio of reserve axis to loan axis is 32.5^. During the same years, the ratio of the loan axis to the deposit axis has increased from 94.6^ to 98.2^. Now these changes to a considerable extent represent the influences at work changing the relative amounts of the vari- ous funds. The rise of the ratio of reserve axis to loan axis from 25.2^ to 32. $fc is probably largely due to the diminished relative use of bank-note circulation based on bonds. A larger relative reserve in money is now required to take the place of the relatively diminished use of bank-notes. The same tendency accounts for the increased ratio of the loan axis to the deposit axis. The ratio advances from 94.6^ to 98.2^. But here too much stress should not be laid; for the growth axis and the loan axis are gradually converging. If we solve the logarithmic equation, Log Lt=Log Dt 2. 422717+0. 017742 t=2.446987-l-o. 016926 t, to discover when the axes will be equal, we shall find that * Section 33. —33— they cross about 1910. Consequently, before results should be drawn from such interpolation, it would be advisable to construct the axes with greater care. For our purposes, however, as scaffolding, they are sufficiently accurate. The summary of the facts of the growth axes, as they are, will be found in Table No. 7. Table No. 7. Reserves. Loans. Deposits. Equation, .... Log. Equation, . Ratio 1879, Ratio 1900, . . . Crossings (1879-1900), Average Crossings, Probable Variation, 66.6(1.055)' 1.823682+0.023317 t R/D 23.8j^ 32-0^ 62 2.8 1.2 264.7(1.042)' 2.422617+0.017742 t L/D 94.6^ 98.2^ 33 1.5 i.o 279.8(1.040)' 2.446987+0.016926 t R/L 25.2^ 32.5% 42 1.9 1.2 §23. The mathematical determination of these several growth axes naturally leads up to the question, ' To what use can they be put? ' It will be remembered that in a preliminary analysis, three main classes of elemental influences were assumed to be at work governing the values of the funds at any time. The three influences we called (i) the growth element, (2) the periodic elements, and (3) the dynamic elements. The growth elements have now been studied and mathematical laws derived to express their motion. It has also been previously remarked that the three poly- gons are continually crossing and recrossing their respective growth axes. At this point, we may bring in a new working assumption, — that the fluctuations or deviations from the growth axes during these years are due to other influences at work than the growth element. It remains to discover whether the results justify the use of this assumption. It is thus that we are enabled to eliminate the element of growth. A study of the folding Chart discovers two facts : (i) That the growth axes afford an excellent standard from which to measure the deviations of the polygons at any date ; (2) That these deviations increase in amplitude as the values of the funds swell. 3 —34— The growth axis as a standard from which to measure deviations is analogous to the "ideal" surface of the ocean in another science. Nor is the analogy lost in (2) — that the deviations increase with the increasing value of the funds. For the waves that ruffle the surface of a fish globe are in amplitude much smaller than the swell of the Atlantic Ocean. Whether the deviations are fairly represented by a per- centage proportion, will be considered later. In the follow- ing pages I have assumed a percentage proportion and have reduced the actual deviations to a percentage basis. With all the criticism that exists in regard to governmental statistics, they nevertheless remain of very great value. For although the final sums, after forty years of additions and subtractions of four and five estimated amounts annually, is subject, undoubtedly, to very great error, still it often becomes difficult to conceal the tendencies in the changes from year to year. It is with these changes that interesting problems in eco- nomics are connected, not with gross sums. Like the sailor or captain on the ocean, we are interested, not in the depth below of the ocean-bed, — for we are sure that the ocean-bed will remain during the little while that we are sailing over it, — but we do fear the height of the waves upon the surface. Even the captain measures the height of the waves, not from the ocean-bed, but from the "ideal" surface. So with us,- — we will measure the financial waves from an "ideal" stand- ard, which for convenience we have called the growth axis. Nor are these waves unimportant. It was some chemist who said, " Never throw away your residues. Look in these for your results." This practice of studying the residues is perhaps most common in the business world. A glance through the tables of trade and financial statistics compiled by our leading journals will convince the reader how familiar this method is outside the text-books. The mill owner and the speculator are constantly watching the net changes, the differences, and these become the motives of their actions. The economist may profitably study these differences for the verification of his laws. CHAPTER III THE PERCENTAGE DEVIATIONS § 24. The Percentage Deviation for any week may be defined as the ratio of the algebraic difference of the actual fund and the growth axis for the given week to the growth axis. Let MN (Diagram No. 3 on the Chart) be the growth axis and PAQ the polygon. If AB is the actual deviation, BC the growth axis for the actual fund AC, then AB/BC is the percentage deviation.* As the polygon lies above or below the growth axis, the sign is positive or negative. It is thus possible, given the requisite data, to construct a table of percentage deviations. In such a table the element of growth and the increasing amplitude of deviations (§21) are eliminated from the statistics. These percentage devia- tions may then be plotted upon a horizontal axis, which is the growth axis reduced to a straight line.f The rising polygon, then, becomes undulatory in motion, similar in type to curves of fluctuations of the daily average tempera- ture, the foreign exchange rate, etc. It is necessary to reduce the actual polygons to an undulatory motion to sat- isfactorily study correlation with other undulatory curves. § 25. Following the method laid down in the last section, I have calculated the weekly percentage deviations of reserves, loans and deposits for the twenty-two years. The weekly percentage deviations of the total reserves (which for brevity we will call, hereafter, reserve deviations), may be consulted in Table No. %.% The weekly percentage deviations of the * The percentage deviation (D) for the geometrical growth axis maybe algebra- ically expressed as follows: — AC-BC ^ Rti-Rtg _ Rti-Ro(i-l-r)t " BC Rtg " R„(i-l-r)» ■ Rtg represents the value of the growth axis for the time (t). f Cf. Mercator's Chart for a parallel projection. X In Tables Nos. 8 and 9 fractions of \% are expressed as the nearest whole percentage. In the actual calculation, however, of all following statistical tables, fractions were expressed as the nearest tenths of if,. -36- loans (i. e. the loan deviations), appear in Table No. 9. The deposit deviations are omitted for precisely the same reason that was given for the omission of the actual deposits. Reserve, loan and deposit deviations are represented in the large Chart. The heavy straight line, in the lower half of the Chart, stands for the growth axes of reserves, loans and deposits reduced to a straight line. This line is zero value for the percentage deviations, which are plotted from this line as origin. § 26. In this Chart are represented the really important movements in these financial statistics. The motion of the growth element is slow and gradual. Its effect is scarcely felt. But in the deviations are the movements which are forever puzzling financiers, and upon whose often apparently eccentric movements great fortunes are made or wrecked, panics are bred and crises precipitated. These deviations do not, it is hardly necessary to say, produce such serious calamities as crises ; but they are the barometers of the state of that conglomeration of many tendencies in the societary circulation, working for good or for ill, that are in themselves prosperity or depression. Indeed, they may be made to form a measure of the severity of crises or of the affluence of periods of prosperity, as we shall later see. §27. Before commenting upon certain features suggested to the eye by the movements of the three polygons of the Chart, it is, perhaps, at this point the best place to explain what is meant by the weekly changes in the percentage devia- tions. The changes in the percentage deviations are simply the increases (or decreases) of the consequent week over the antecedent week. There are several advantages in working with the first differences instead of with the percentage deviations. Among these are the following: — (i) The changes are smaller in amount and consequently much less arithmetical work is involved in calculating aver- ages. (2) The changes are easily found directly from the weekly report, and afford a convenient method of continuing a table —37— of percentage deviations. The actual change as reported by the papers is of course the distance AE in Diagram* No. 4. Now the change between the percentage deviations of the /AC A'C'\ two weeks would be (oc~B^')" ^"* BC=[(i+ry''=X r,>r^n T. ^ „ u AC A'C AC-A'C'(i+r)'." B'C']. It follows that g^-3^^^^-p^..= BC ' Thus the change (C) (it may be shownf by a few steps), is given by the formula ^_ R,-R.(i+ry/- ^- r;^ ' where R, is the actual reserve for the second week, R^ for the first week, r the annual rate of increase of the growth axis, and Rg^ the value of the growth axis for the second week. For the growth axis of the reserves, {i-\-ry''°' equals the constant (1.0013). Often, for practical purposes, it is sufficiently accurate to divide the actual change as reported in the statement by the value of the growth axis for that date. But if it is planned to carry along a table of the percentage deviations by adding and subtracting the changes according to the signs, it is better to follow this rule. To find the weekly change in the reserve deviations, sub- tract the product of the actual value for the previous week multiplied by the constant 1.0013 from the value for the present week and divide by the value of the growth axis for the present week. Add the change to the percentage devia- tion for the previous week, — or subtract if the change is negative. The result is the value of the percentage devia- tion for the present week. The values of the growth axis are, of course, found from a previously prepared table. * See folding Chart. t AC-A'C (i+r)'^''' ^ R,-Rg,-(Ri-Rgi)(i4-r)'^'^' ^ R,-R.(l+r)'^'''-Rg,+Rgi(i+r)'^°'' ^R, -R.(i-hr)'/" Rg2 Rg2 For Rgj=Rg,(i + r)'/« -38- « O > Q O 1 1 « \n O uir-oooo inc^oo \rt rj- r^ixi m il-vOco r^t^ON« w w OnOn + + + + + + + + + + + + + + -f- + + + + + + + + + + + ON CO coNr^M «NMM ONi>.Ti'TtNi-(CJ-cof^co r>.cnoo u-)r^o Ooo onm w h Tht^w h -^0^0 w MMMMMM MtH HMMMMNMMMCOcn + + + + + + + + + + + -l--l--l--l--l-4- + -h-l--l--h + + + + QO + cnONco'O M o cn-^c^Nooco r^vO r^o^ONinmo r^r-'co o w <>■ ON 00 ■*00 TtN « ^O « eo'^o r^cooooovo cnminTj-eno ■^O m TTi 1 1 1 iTTTTTTTTTTTTTTTTTTT m On C30 H I H U100OO ooinp-jM wt-M « cnwco Tl-O cno c'^-^inci « o + + + + + + + + + 1 iT T T T 1 1 + + + + + + + + oo M 1 + + + + + + + + + + + -h + + + -i--i--i--i- + + + -h + + + cn ON OO M o cnoo cno H mo ^cocooo r^iricno^o M Tt-o M t^ tn t^ o Ti + iiTTTTTTTTTTTTTiTTTTr a* OO iih^ ■wOnO cnN(N Hl^OMvo^-co cooo oo co t:*- ^t « w 1 + + + + + + + + + + + 1 1 1 + + + + + + + + + + + 00 OO CO °sag, rt*iH t^ONONr^mN N cnrhu^o r^u^w h ^ONTt-mw Ti-r-.r^i>. I+ + + + + + + +I 1 1 1 1 1 I+ + 4- + + + + + + + OO CO 1 \nONH mrteno »J^'N HI o cno \ninincniriovoo r-.i>.i^ONON + + + + + + + + + + 1 1 1 1 1 1 1 II 1 1 M II o CO OO + O c^ cn^o UTO N tncni-tsOoo rnco Onco m-^mmmcnOoOCO u-> N cncncncncnenw « « w ++++++++++++++++++++++++-H- CO OO + r-- c^ N -^vo w O\oo r^r-.r-.r>tnx-^ON« -^ xnca o m m rj-T^o ut ^mininm\n^Tt'r}-'^rh'<1"Thrh'^mvnin \n\0 \0 O vO vO vO nO ++++++++++++++++++++++++++ OO OO M m-Tl-Ocnrtoor^t^Tj-ocoOcONNMciinHOTt-iHONcnO'* + + + + + + + + + + + + + + + II 1 1 1 1 1 1 11 1 en CO OO M 1 cncnThvO cnpi w inoo m cntn o mcnw ONcncncncninvnr^r^ 1 + + + + + 1 1 1 1 1 1 1 1 1 1 1 1 II + + + + + + M OO 00 M OoocqNO'^wr^HON\nir)coOcn'^r>t~>u-iM«MO'Oi^c^ + + + + +I Ml 1 1 1T1+ + + + +I 1 +4- + M CO CO M 1 c^ONNoic^eiooo Onqo ^OnO cnMco cnco inoNco rru^Tfu-ien MMMH MMC<«(NNiHO-^i-< HWM WHCI HMN W«« MWW MHW s' 1 1 1 1 i -= IBOijaoinjsj H M en Tj-ino r-^oo OnO w « cn-^ino r-^oo o^O hi w cn^fu-io ^HlH^H^-ll-^MMMMMC^NclNMe^C^ —39— w 1 CO o N 'i-u-)mM cninu-)« OMnMco u-i>0\0 tJ-o O mcoo r»o ++++++++++++++++++++++++++ OO H ++ + +++ + + + + + 1 1 1 1 1 iT T 1 1 1 1 1 1 CO OO IH + covO O OO r^miTivO w mu-»0 cnii^ o^oo Oco tH M M N en m U-) +++++++++ 1 1 1 +++++++++++++ 00 « + CO M en N ooo o O r^ en w w^vD cooocnMcnM o« cnMMeno ++++++++++ 1 1 1 1 1 1 1 1 1 ++++ 1 '»0 enenO OxW'Oco c>OOCM^»nvD r^r>-\£)oo M inn Ooo m in MI-IW«MM ooo Oooco M ■^■^Tj-Tj-'d-o r^r^m'Ocoo n c< O O ^^TJ-'^'^en'^encn'^encnenencnentnencncnenw h m w w ++++++++++++++++++++++++++ en CO H o inmenr^u->r^'>*M f^tJ-co «o «0 h r^tn o enOMOvOOco o encncncnTh-^-^Thenenc»NiHM wh-i««encncncnri- 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 +++++++++++ CO CT^O M ri-o r^Ti-H tnooo O M Tl-OO inr^r^o -rh t1-vO vO r^o ++++++++I 1 iTTTTTTTTTTTTTTT M ox CO + M r^ r^ r- ci c» M eno vor^owcomc^NOcnooooooiHOM^ TiiiiiTTTTTTTTTTTTTTiiii + 00 1 eniH«c»OHO'*en cn^ a»r^oo vo o O O ■^'^■^■^r-HiniHoo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 CO CO H 1 1 i+i Mil iTTTTTTTTTTTTTTT 00 CO CO + + + + + + + + + + + + + + + + + + + + + 1 1 1 1 1 CO CO 1 vO inir>r^OW Thm^ioO ThO O O w inr^OO W CI COM O o iMiiTTTTTTTTTTTiiiTTTrrri CO OO C4 + r^oooo^ c» c^ u-iooo r-^r^u^ooo O^co o>o en tJ-o o O O a> +++++ iiiiiiiiiriiiiiiiirTi 00 CO + CO OcOcOO enenOO N ONO encioO MCO T^inO u-ienO -rJ-« tn vD r^vo oyDOO^ou^iJ^'d-'^-d--*encn« « m ci m m e^ w« n + + + + + + + + + + + + + + + + + H- + + + + + + + + OO CO IH 1 iH cd O 1 'tag 1 O « r>«o OH « M ooo^ovDvO oenenN «o H rho r^r^vovo t-iiH«iH«co u^OnO Ocovr>M O h >-i hi o + + + + + + + 1 1 1 1 1 1 1 T T M 1 III CI 00 OO H o oenoo en-^iH w c^ oco w tJ-w o om moo mco cnco u^mco WH( MMHIM MMHIMHIHI +++++++ 1 II II II 11 M II M 11 II OO CO M m ( "^cooooo M inco « H cnenTh-<:t"00 O u->h c» cnenO OO OOO «NNO»C^HI HI HI + + + + + + + + + + + + 1 1 1 1 1 1 1 1 1 1 1 1 II O OO CO M mH o -, N +£|°+ uio cnwco r^N M o O Ot^oo o i>-r-.oo r-.incneno ot-mm «e^«C»HIMHIHHIHHI HI MM ++++++++++++++++++++ 1 1 1 1 1 1 00 1 M ^ cnoo enc» t-^M w -^rt-O cnO tn m t^o coOHMM-^d-Ocn M MMMMMMMMMMMM MM + + + + + + 1 1 1 1 1 1 1 1 1 II 1 1 1 1 1 1 1 II Suipuodssjjoo OO *ri CI O m W a^\D « OO cnO r^T*-McO tJ-mcO »r»c» OO en O HICIC? MWCI MNen HNN WMC* Mc^cn ^ S° Hh ^ > 6 "^ ^ ^ o O intn -40- O X O Z O !>: a- <; H 8 M 1 Onco onO n inr^o^O w o f^oo O w w c* ■^Th'i-»n\0 l--cOco t-^ ++++++++++++++++++++++++++ CO o O^O O 1-1 du->OC0 O- 0^00 OnOnOM-^'O u->cO r^vO (ne'CO o^ ++++++++++++++++++++++++++ CO M + •OM200 O^O HM H OM-^incncOM 0«MNNOi-icO'cor-~0 TTTTTTTTTi iTTTTTTTTTTi i i i i M o u a i 1-1 rt O 1 MWWMOOOWOO^O^OOOOOOOWWWNWNNM HMMMMMMIHM MHt-fMMI-tlHMMHMMHMM 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 M 1 CO M 1 cn-^'^rj-'^xninvo vninmoo r^r^r^r^O u^Tj-cncncnw >-* m 1 1 1 M 1 1 1 1 M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 CO M 1 ■^tmTj-vTi'^MiHOiHMOOO O^CO r^ r^\0 ^OvOvOnOOOOO TTTTTTTTTTTTTi i i i i i i i i i i i i CO M Ovommc^M c^ m Tfvo r^ o^ o^co o^o O o o^n cncnfriTt-mrh iiiiiiiiiiiiiiiTTTiTTTTTTT N CO M M H M o N cnmoco o>o>cocococococococo r^r^co o^oo co co o* CO w + M H M ocoi>-r-~i>-r^ r^o vO iri inso r-*co oo O'ihc^nconnm TTTTiiiMiiiiiiiiiMTTTTTT 8- CO M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 M 1 1 1 1 1 1 1 CO CO M 1 N COM c^ O « c* w cn'i-vr)u-iinTi-T;j-T:j-cn'oo r-Tt--C7' Onco O WHMW h-ll-IMIHtHMW + + + + + + + + + + ++ + + + + + + + ++ + + + + + O CO CO w O O H C'l'Tj-'^'^Oi^Oin'^Tj-cow OO Ot-i coco« h h h co +++++++++++++ 1 1 1 1 1 +++ On CO 1 w CO 01 c^ O Onco CO r-^r^co O-O ci Tr^f'^t-t oo 'd-'^'^u^mvo TTTTTi 1 1 1 1 1 iTTTTTTTi i i i i i i puB Tiiaopj Suipaodsajioo r^Tj-Mco tj-moo in-ri-t-Hco xniHco ma o^\o coo r^coo r-^Tti-t l-HCqC^ HMM MMOi MC^C^ M«M WHN ^ ? 4 ^ ^ § 9 •3193AV 1 M w cnTj-mo t^co o-O w n co-^^mo r^co o^O '-' w co-^mo 1 HlHIHtHl-ltHMWtHM(M«N• l-l iz 1 o r Ovo irnOvOO i>co r^oo oco r^r^in«:h(T)cnM w cnri-Ti-N m en NNMMWNNC- r^ a^'-^ N -^vOvOO r^oo c^oo CO + 10lOTt-'^tOC)HMO«NMMOOC*MOl-'Mrl-TfvO'0\00 1 1 1 1 1 1 M ++++ 1 1 ++++++++ CO TTTTTTTTTTTTTTTTTTTTTTTTTT in CO M 1 M N cncnw w C4 w M 1-1 o '-' cncnTj-'^u-)\n»£) r^t~-.r^r^inoo o 1 M 1 1 1 M 1 1 1 1 11 1 1 1 1 1 1 1 1 1 M CO 1 CnCi-)Cn-*TtCOCrjCriNCI«IHMMMMIHMMMW«00«in 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 II CO M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 CO + + + + + + + + + + + + II 1 1 1 1 1 II 1 1 1 + mwNCIMi-iihihOO Ooo O O O^ O Ooo 0^ Oco r^ l^O vn c<^ TTTTTTTTTTiiiTmmmiiiiii 8^ CO w 1 ^iniDOvO xnvo r^co r>-cocOcovO mmo r^r-C7>0 m o h o m iiiiiiiiiiiiiiiiiiiiTTTTTT co CO 1 + + + + + + + 1 1 1 1 1 1 1 1 1 1 1 1 1 1 CO CO CO M cnc^(ntnenc4i-iHOOOMi-(OHOOOiHHi-tM«enen« 1 1 1 1 1 1 1 1 II + 1 1 1 1 1 1 1 1 CO CO M 1 Tj-Tt-inmiDvo i^r^oooooooo a^oo r^r-~co r->r^i>.r^r^r^ooco t^ 1 1 M 1 1 1 II 1 1 1 M 1 1 1 1 1 1 1 1 1 I II CO + HI c» w N M N r^r^cooo t^t>o uroo^o r^voo \r> -ri- -rf^n^o <:> 1 1 1 1 1 II 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 in oo CO M + M H M c^ O O Oco r^^D sO\0 \nir>-^c<^c>» cocncncnrJ-rtTl-rt'^ TTTTTiiiiiiiiiMiiiiiiiiii CO CO M O c^^cocncncOTl-c*-)T:}-cnc*^c*^cnc*^c*^cocnc*lrJ-p-)'«d-u^»r>Ttcocn TTTTTTTTTTTTTTTTTTTTTTTTTT CO oo 1 +++++++++++++++++ +++++ CO CO M o r^ocoosoooo^cor--*OvD'^WMOHcoe0 c^ OO cnO r^-^wco T^wco idm CT»vO en O i^ bo •% ^ > 6 -^ 3 g' U O o M c^ fri-d-m^r^co oo w m tOTi-vno t^oo oo w w M « M cncncncOcncOtncocncnTj-Tt'^'^^^'^'^'<4-'^uim\n —42— §28. To return to the Chart showing the reserve, loan and deposit deviations, several general features may be at once remarked: — (i) With respect to violence of fluctuation, the reserve deviations surpass both in extent of deviation and in number of fluctuations. The deposit deviations are second in these respects, and the loan deviations third. Although the ex- planation will come out more fully later, the great deviating activity in the reserves is due to the deposit of currency in the New York banks by the other national banks of the United States as well as by the trust companies. The sur- plus cash of banks outside New York will draw interest if deposited in New York. When a use arises for this surplus cash, the banks in the West and South withdraw their bal- ances, producing a descending fluctuation. This is one important cause. Another is found in the field of Foreign Exchange, — in the export and import of specie. The deposits are second in this respect, because they stand on one side of the balance sheet offsetting reserves and loans. Fluctuations in reserves consequently produce fluctuations in deposits. The loan deviations are most lethargic owing to the fact that loans exist in response to business demands that are of a constant nature. Time is specified in the majority of the loan contracts, and this contract-time makes for greater stability. (2) The leader in the up and down movement with respect to time is unquestionably the reserve. It begins ascending and begins descending in advance of the deposits and the loans. The deposits follow closely, and the loans follow shortly afterwards. There seems on the average about a three weeks interval between the turning points of the reserve and loan deviations. This is, of course, in conso- nance with banking theory, that expanding credits follow closely expanding reserves, and contracting reserves are fol- lowed by contracting credit. In this connection is the inter- esting study of pool activity in the flotation of new securi- ties. —43— (3) The position of these lines with respect to the hori- zontal axis is an index of business prosperity. The lean years are the years in which the loan line is far below the horizontal axis with (as we shall see) interest low, and the fat years vice versa. For when more men can borrow more money at higher interest, then the profits of business are greater. (4) This chart suggests strikingly the similitude of crises ; but as this subject belongs under the dynamic element, dis- cussion will be deferred. (5) For purposes of comparison, there exists a real value in such a series of statistics. The removal of the growth element admits of comparison of the position of the fund at any time with respect to its position at any other time, by means of simple indices. Such a method would, it seems to me, be of very great value if applied to the clearing statistics as published weekly by the Commercial and Financial Chronicle. The percentage of increase or of decrease should be calculated by the method of percentage deviations from a growth axis, and not, as now, from the same week of the previous year. The latter method does not admit of valid comparison; for a 100^ increase after a year of great stagnation may be in reality far less than a 50^ decrease from a year of great activity. The results of the two systems may be seen by the following table for the week ending July i in three years. Year. Actual Value. Growth Axis. Percentage Deviation. Change in Percent. Dev. Chronicle Change. 1887 1888 1889 40 90 50 50 60 71 — 20)? + S0% -29^ -1-70^ -79^ + 125^ - 44^ How the rational change is distorted appears when we com- pare -\-70% with -|-I25^ or —7g% with —44^. In addition to the fallacy involved in the divisor, there arises another serious one. In some statistics there is no more reason for using as a basis of comparison the same week a year ago than for using any other week in past time ; —44— for there often occurs no annual period, but a cycle of several years duration which is of great statistical importance. In such a case the rationale of the divisor entirely vanishes. It seems probable that, could growth axes come into general use, the statistics published by the journals would be of much greater service to statistician, economist and business man. For they could then be conveniently used to study correla- tion with other sets of statistics, like the rate of foreign exchange or different varieties of the interest rate, all of which have an undulatory rhythm. CHAPTER IV PERIODICITY IN THE RESERVE DEVIATIONS § 29. In the preceding section, the growth element has been our main concern. We now take up the subject of periodic- ity. Periodicity is a very ancient notion in economics. It has remained a notion in economics more than in many of the other sciences. A farmer's boy reading his almanac frames a concept of periodicity, if he has not already derived this notion from his own experience. In the almanac, he finds the lunar period, the solar period, the period of the tides and the period of the temperature. The almanac maker, however, has carried the notion beyond the hazy stages of common experience to the stage of statistical measurements and, finally, has derived tables so accurate as to admit of prediction. Why this should not become eventually possible in economics has never yet been clearly shown. Such a result, however, can come only through the statistical half of economics. It should be remembered that time in itself has no causal connection with periodic movements in economic statistics, inasmuch as time is not a force. The causes of periodicity lie in the economic forces at work, but without knowing the forces — if sufficient periodicity exists — we may be able to derive the period. Men could predict the time of high and low tide from simple observations connecting the phenomena with the time of day centuries before they ever arrived at a the- ory which connected the tides with the attractive power of the moon, or even guessed at a dynamical theory of the tides. In the following sections, an attempt has been made to derive the periods of the reserves and loans from the sta- tistics, and then to consider, though meagrely, the causes at work producing the periodicity. § 30. Before commencing a study of the statistics of the reserves, a word should be said in regard to the period-year. -46- The period-year is the ideal numerical year. The first week ends on January 7. Since there are fifty-two weeks in the year plus a fraction of a week, the weekly statements pub- lished on Saturday do not fall for the first week on January 7 every year. It becomes necessary, therefore, to define as the first week, the weeks ending January 4, 5, 6, 7, 8, 9 and 10. The records are thus evenly divided. This method is used throughout the following pages. §31. From a table of the weekly changes in the reserve deviations (described in § 27), Table No. 10 has been com- piled. In column I, are the numbers of years out of the Table No. 10. I. n. I. II. 'S •rf 11 i i III. IV. V. 1^ s i ^ III. IV. V. iz; 1 u Mix 27 1 Blx I 23 -1-22 -l-i.oo -I-I.06 — O.IO 12 10 -H 2 -H .09 -H .10 -HI. 56 2 22 -1-22 -hi. 00 -I-I.06 -1-0.96 28 19 2 -H18 -H • 78 + .83 -H2.39 3 20 2 -M8 + .82 -1- .87 -I-1.83 29 II 9 -H 2 -H .09 + .10 +2.49 4 20 2 -I-18 + .82 + .87 +2.70 30 6 16 — 10 — •45 — .42 -H2.07 5 14 8 -h 6 + ■ 27 + .29 -1-2.99 31 6 15 - 9 — •41 — •39 -H1.68 6 6 16 — 10 — .45 — •42 H-2.57 32 4 18 — 14 — .64 — .60 -Hi.oB 7 4 18 -14 — .64 — .60 -HI. 97 33 7 15 - 8 — ■ 36 — • 34 -HO. 78 8 2 20 -18 — .82 — • 77 -1-1.20 34 2 20 —18 — .82 — • 77 -Ho.oi 9 I 21 —20 — .91 — • 85 -fo.35 35 6 16 — 10 — •45 — .42 —0.41 10 2 20 -18 — .82 — ■ 77 — 0.42 36 8 13 - 5 — .23 — .22 —0.63 II 4 18 -14 — .64 — .60 — 1.02 37 4 17 -13 — ■59 — • 56 — 1.19 12 5 17 — 12 — .09 — .08 — 1. 10 38 8 13 - 5 — • 23 — .22 — 1.41 13 7 15 - 8 — .36 — •34 —1.44 39 7 15 - 8 — .36 — ■34 — 1^75 14 6 16 — 10 — .45 — • 42 —1.86 40 9 13 - 4 — .18 — ■ 17 — 1.92 15 18 4 + 14 -1- .64 -1- .68 —1. 18 41 9 12 — 3 — • 14 — ■13 — 2^15 16 18 4 -H14 -1- .64 + • 64 — 0.50 42 12 9 -H 2 + • 14 -H • 15 — 2.00 17 15 7 + 8 + .36 + • 38 — 0.12 43 12 8 + 4 + .18 -H .19 —1.79 18 13 9 -1- 4 + .18 + .19 -1-0.07 44 7 15 - 8 — .36 — • 34 —2.13 19 14 8 + 6 + • 27 + .29 +0.42 45 7 15 - 8 — ■ 36 — • 34 —2.47 20 14 8 -t- 6 -1- .27 + .29 4-0.65 46 17 5 -H12 + ■ 55 -H • 58 —1.89 21 15 7 -t- 8 — .36 — • 38 -HI. 03 47 12 9 -H 3 + • 14 -H ■ 15 — 2.00 22 12 9 -1- 3 + .09 + .10 -HI.13 48 14 8 -H 6 + .27 -H .29 -1.85 23 13 7 -(- 6 + • 27 + .29 -Hi.42 49 7 15 - 8 — .36 — • 34 —1.79 24 16 6 -hio + • 45 + .48 -Hi.90 50 II 9 -H 2 + .09 + .10 —1.69 25 10 II - 6 — .05 — •05 -HI. 85 51 13 9 -H 4 + .18 + .19 —1.50 26 6 1 15 - 9 — .41 1- •39 -Hi.46 52 15 7 -H 8 + .36 -H • 38 — 1. 12 twenty-two years of our study, which have shown for each week of the period-year increases and decreases in the reserve deviations. Thus during the first week every year has shown Diag-ram No. — 4;— an increase. In the fifth week, increases have occurred in fourteen years and decreases in eight years. Column I is represented in diagram No. 5. Diagram No. 5 is a" suggestion " picture of the weekly changes in the reserves during twenty-two years. The length of the dotted line above the horizontal axis mea- sures for that week of the period-year the number of years out of the twenty-two years which have shown increases in the reserve deviations. The heavy black lines in the same way denote the number of years which have shown decreases. By joining the upper extremities of the dotted lines and the lower extremities of the black lines, we plainly distinguish a band-like movement which is worthy of further investigation. A cursory examination shows heavy increases during Jan- uary, heavy decreases during February and March, a majority of increases for the months of April, May, June and July, with the exception of a sharp decrease for the weeks contain- ing the dates of July i and July 4. By August, the chart shows decreases in the majority and this tendency continues until late into the fall. By December i, however, increases again predominate. To illustrate this movement in another way, column II (Table No. 10) is formed by subtracting for each week the decreases from the increases and in prefixing a negative sign when the decreases exceed the increases. Column II is graphically represented in Diagram No. 5 by the circled dots. The circled dots record the excess number of years showing increases or decreases for each week of the period- year. By joining the dots the same periodic movement may be discerned. § 32. Although this diagram shows the weekly changes in the fund, it does not show the continuing movement of the fund itself throughout the year. To get an idea of this period with a minimum of arithmetical work, the follow- ing method was devised: — (i) Divide through the excess number of years showing increases or decreases for each week (column II) by the total —48— number of years (in this case 22). This operation results in a percentage table (column III). (2) Find in this percentage table (column III) the sum of the positive quantities and the sum of the negative quantities. Subtract the latter from the former and distribute this differ- ence over the fifty-two records by subtracting algebraically the quotient of this difference divided by the sum of the two sums for each unit of advance, and adding the same quotient for each unit of decline. In this corrected table, the sum of the advances equals the sum of the declines. Thus the differ- ence of the two sums is 1.29. The quotient 1.29/2 1.87 is equivalent to 0.059 fo'" each unit of fluctuation. The cor- rected figures, obtained by adding the correction (o.c6 X each unit of advance) to each advance, and subtracting a like product from each decline without regard to signs, are in column IV. (3) Add the record for each consequent week to the alge- braic sum of all the antecedent weeks, counting advances positive and declines negative. This results in a table of the relative position of the fund at any week with reference to all other weeks. (4) To balance the curve on a horizontal axis, find the average of all the records. By subtracting this average from each record, the table of the annual relative period is obtained. Thus the successive algebraic sums are found from column IV and the average of these sums, 1.16, is subtracted from each of the fifty-two records. The result- ing period (column V) is represented by the dotted line in Diagram No. 8. A glance at the movement of the fund displayed by the dotted line suggests a well defined period in the total reserves. It is not a quantitative period, for the amount of deviation does not enter. It is rather a period of the degree of occurrence of the position of the fund with the week of the period-year. The value of this method lies in its arithmetical convenience in tentative work. It is useful as a prospecting method. The suggestion is, however, one of degree. A quantitative expression is required. —49- §33- Quantitative indices may be obtained by averaging for twenty-two years the weekly changes of the reserve deviations for each week of the period-year. These aver- ages of the weekly changes may be consulted in Table No. II (22 year column). As the sign is positive or negative, the change is an increase or a decrease. Table No. ii. Average Changes in the Reserve Deviations for each week of the Period- Year^ with Probable Errors. I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 i6 17 i8 22 years. +4. +5 +0 2, — 2, — 2 2, 2, I, O, o, I. -I-I. H-2. +1. +0. 686±, 323 ± 105 ± .oi8± 082 ± .300± 141 ± .986± 9I9± 374± i36±, 836±, 723 ±, 323 ±, 823 ±, 2i8±. 091 ±. 809 ±, 18 years. +4. +5. +2. — O, — O, — I, —3. — 2, — 2, — I. — I. — O, 1. +3. H-2. +0. +0. 783±.4 6i7±-3 428 ±.4 533±-3 628±.4 805 ±.4 3II±.5 I05±.4 8i8±.3 322±.4 6o6±.4 ooo±.4 723±.3 027X.5 i44±-4 o83±.4 822±.3 539±-6 22 years. +2. +1. — o, +0. — o, -I-I. — o, — 0, +0, +2. +0. — o, ■2, '2. I. — I, 068 ±. oi8±. 023±. 723 ±, 373±. 623 ±, I36±, 759±. 245 ±, 241 ±, oi8±. 7i4±. 5i8±, 364 ±, 050 ±, 673 ±, 555±. 18 years. +1. +1 +1. +0, +1. +1. +0, — o, +0. +2. — o, o, — I, ■2, — 2. 2 — 2 994±.5 778±.5 ioo±.3 4oo±.5 278±.3 ■550 ±.4 I56±.3 933 ±-4 333±.5 422±.5 094 ±.4 878±.3 622±.4 I33±.4 061 ±.5 056X.2 I33±.5 22 years. — I I. — I — o. — o. +0. +0. — I. — o +2 +0. — o. — o. +0. 4-0. 4-0. ii8±, 304 ±, 068 ±, 541 ±. 327±, 359±. 800 ±, 800 ±, 100 ±, 823 ±, 6i3±, 886±, 323±, i±, 359±. 3I3±. 500 ±, 18 years. I, I, — O. o, — o, — o, 4-0. 4-0. — i, ■ — o, 4-1. 4-0. — o, — o, 4-0. 4-0. +0. 639±.4 322±.3 56i±.6 7o6±.7 26i±.4 644±.5 461 ±.6 76i±.5 6ii±.4 6ii±.4 622±.3 i33±-4 o89±.4 144 ±.4 7ii±.3 678±.3 5ii±.4 Diagram No. 6 affords a graphic representation of the average changes in the reserve deviations for each week of the period-year on the twenty-two year basis. The length of the vertical, dotted line measures the average change for that particular week, and as the line extends upwards or downwards the change is an increase or a decrease. The communities of changes all in one direction or the other at different seasons of the year are significant. Before going into details, however, it may be well to inquire whether these averages are the most representative that can be derived from this statistical array. It is possible that extraordinary events have occurred in some years, so extraordinary as to quite distort the average or the normal. The Chart showing the actual course of the reserve devia- 4 —50— tions during the twenty-two years, will be remembered. During 1893 occurred a crisis which in its severity was unequalled in all the last quarter of the century. The year 1894 was full of governmental bond operations and the after- effects of the crisis. So extreme were the deviations from the axis that more or less distortion must necessarily result if such years are included in the averages. To remedy this distortion, the averages in the second column, Table No. 11, were calculated. The sole difference between the two series is the omission of the years 1879, 1893, 1894 and 1900 from the latter. The eighteen year averages are represented in Diagram No. 6 by the heavy vertical lines. On the whole, the differences between the two series are not striking. How are we to decide which of the two sets of averages is more representative of the normal movement? This question we can settle without being obliged to resort to general impressions. It is only necessary to calculate the probable variations of the averages in the two series, com- pare the two sets of probable variations, and to select as the more representative the series having the smaller probable variation. The method of calculating the probable variation is doubt- less familiar. The expression of the probable variation is 'hV>' approximately ±2/3^/ , where v is the deviation of any one of the numbers entering in to form the average from the average, n the number of records forming the average.* To illustrate briefly, the average change for the second week in the eighteen year series is -1-5.617. The numbers entering the average are eighteen in number. The differences between the successive numbers and the average are given in the column under v. The squares of the v's follow and on adding the column we have 54.83, which is the 2v' of the expression *(v) (v=) -0.6 0.36 + 1.2 1.44 -1-2. 1 4.41 —0.2 0.04 -f3-7 13.69 -0.6 0.36 -I-3-7 13.69 —0.1 o.oi — i.o 1. 00 -0.6 0.36 -2.3 5.29 2v'= = 54-83 ±2/34/ .■.54.83-5-17 = 3.22. The square root of 3.22 r n— I multiplied by 2/3 is 1.2 or the probable variation of the statis- tical array. Biagmm No.lo Ig vears —Si- Table No. 12 contains the two series of probable variations. In Diagram No. 6 the broken, dotted line (in the lower part) represents the twenty-two year series of probable variations, and the broken, heavy line the eighteen year series. The Table No. 12. Probable Variations of the eighteen year and twenty-two year Average Changes in the Reserve Deviations. *A ^ j>a M V S V A i i 1 V a 1 u 1 1 t— 1 V q u ^ H 1 ca U «« I 14 5 + 9 14 II 10 + I 27 13 8 + 5 40 10 12 — 2 2 6 15 - 9 15 7 14 - 7 28 6 15 — 9 41 10 II — I 3 17 5 + 12 16 8 14 - 6 29 5 14 - 9 42 4 17 -13 4 13 9 + 4 17 9 12 - 3 30 7 13 - 6 43 7 15 — 8 5 18 2 + 16 18 15 6 + 9 31 17 4 + 13 44 II II 6 18 3 + 15 19 10 II — I 32 16 5 + 11 45 4 II - 7 7 16 6 + 10 20 9 II — 2 33 13 9 + 4 46 7 14 - 7 8 10 10 21 8 14 — 6 34 12 10 + 2 47 10 12 — 2 9 15 5 + 10 22 14 7 + 7 35 6 16 — 10 48 13 8 + 5 10 II 10 + I 23 15 5 + 10 36 13 9 + 4 49 10 II — I II 9 12 - 3 24 12 9 + 3 37 9 13 - 4 50 8 14 - 6 12 6 16 — 10 25 12 6 + 6 38 5 IS — 10 51 14 7 + 7 13 5 15 — 10 26 16 4 + 12 39 5 16 — II 52 13 7 + 6 sented in Diagram No. 9. The dotted lines above the horizontal axis measure the years with increases for each one of the fifty-two weeks, and the heavy lines the de- creases. Two movements are apparent: — (i) an annual movement resembling, in some degree, the period of the reserves; (2) an irregular monthly movement in which the rise falls on the week containing the first day of the month. This is shown by the saw-tooth, upper edge. The points of the teeth fall, as a rule, on the weeks containing the first days of the month. Just before the first days of the months, business men borrow money to meet obligations then com- ing due. ;^i n ^ IOi:'i;; 8:;:;!i a; ; ; ; : .'■It- 18 5.^ ■J'^ 31 iaoramNo.Q. It 3 4- i> T, If I! I 1 ^ m J'f ii II ;;; n ; ;m ; * #11* ft|| t" ji* sc ^3 31. f, » I/JW I3S :?« ; I, , ij '1 1, '1 4Iv '•'»' ■i1^^&iii:iH?ii'iiii^>^iiikiii^H'^!i'^^ i'li '1 is"'- i| 55 37 33. ■5^ ^^ '5> ,^5 ^8 sy 3) 15 Hi. fc TliaQ'rQffl NQ.iQ. -63- §41. The average weekly changes in the loan deviations, calculated both on the basis of twenty-two years and eighteen years, appear in Table No. 15. In the eighteen year aver- ages, the years 1879, 1893, 1894 and 1900 are omitted, as in the reserve averages. Table No. 15. Average Weekly Changes in the Loan Deviations on the basis of twenty-two years and eighteen years, with Probable Errors. •s i i % S2 years. 18 years. & 22 years. 18 years. 1 22 years. 18 years. I -1-0.445 ±0.1 +0.467 ±0.1 19 — 0.109 ±0.1 — 0.267±0.2 36 — 0.105 ±0.1 — 0.094 ±0.1 2 — o.327±o.i — 0.261 ±0.1 20 —0.045 ±0.1 — 0.lo6±o.2 37 — 0.250±0.1 — 0.328 ±0.1 3 -|-o.509±o.i +o.438±o.l 21 — 0.l77±o.2 — o.294±o.2 38 — 0.382 ±0.1 — 0.500 ±0.1 4 +0.264 ±0.1 +0.536 ±0.1 22 +0.459 ±0.1 +o.5ii±o.2 39 — o.39i±o.i — o.359±o.i 5 -f-I.208±O.I +1.033 ±0.1 23 +0.364 ±0.1 -i-o.4ii±o.2 40 +0.055 ±0.1 — 0.045 ± O.I 6 +I.082±0.1 +0.800 ±0.1 24 +o.i73±o.l +0.278 ±0.1 41 — 0. 191 ±0.1 — 0.2II±0.2 7 +0.632 ±0.1 +0.6l6±0.l 25 +0.300 ±0.1 +0.411 ±0.1 42 — 0.632 ±0.1 —0.523 ±0.1 8 +0.068 ±0.1 +0.083 ±0.1 26 +o.555±o.i +0.672 ±0.1 43 — o.i73±o.i 0.222±0.1 9 +o.o77±o.3 +0.435 ±0.2 27 +0.414 ±0.1 +0.241 ±0.1 44 +0.064 ±0.1 — 0.000 ±0.1 10 — 0.218 ±0.2 — o.i78±o.i 28 — 0.2I3±O.I — 0.6ll±0.2 45 — 0.468 ±0.1 — o.639±o.i II —0.291 ±0.1 — o.i78±o.i 29 —0.477 ±0.2 — o.i5o±o.i 46 — o.i77±o.i — o.i36±o.i 12 —0.595 ±0.1 — o.237±o.i 30 — o.oi8±o.i +0.033 ±0.1 47 — 0.018 ±0.2 — 0.295 ±0.2 13 —0.364 ±0.1 — 0.306 ±0.1 31 +o.5i4±o.i +0.483 ±0.1 48 +0.027±0.2 —0.073 ±0.2 14 — 0.1l8±0.2 +0.100 ±0.2 32 +0.386 ±0.1 +0.267 ±0.1 49 +0.141 ±0.1 +0.050 ±0.1 15 —0.350 ±0.1 —0.383 ±0.2 33 —0.045 ±0.1 —0.033 ±0.1 50 — 0.045 ±0.1 +0.032 ±0.2 16 — 0.291 ±0.2 — 0.400 ±0.2 34 — o.i36±o.2 — 0.250 ±0.1 51 +o.i86±o.i +0.432 ±0.1 17 — o.i73±o.i — o.237±o.i 35 — o.736±o.3 — o.3ii±o.i 52 -f-o.259±o.i +0.288 ±0.1 18 +0.664 ±0.1 +0.500 ±0.1 The eighteen year and twenty-two year series are drawn in one Diagram (No. 10) for purposes of comparison. The heavy lines stand for the eighteen year averages and the dotted lines for the twenty -two year series. Both sets of averages are subject to one well defined tendency. Increases are the order from the week ending January 2 1 st to the week ending March 4th, decreases from March nth to May 27th with the strong exception of the week containing May ist. Increases then follow up to the end of the first week of July. Three weeks of decreases ensue followed by two weeks of increases for the first two weeks of August, and from the middle of August a general declining tendency continues to the end of November with slight increases for October ist and November ist. From -64— December ist the loans increase with the exception of the recession for December 15th. This is the general movement of the year. It is interesting also to note how plainly the first days of each month produce their effects, often com- pletely counter-balancing the prevailing tendency. §42. The differences between the two sets of averages are not striking. In order to select the more representative series, the probable variations (Table No. 16) were calculated Table No. 16. Probable Variations of the eighteen year and twenty-two year Average Changes in the Loan Deviations. ii i i J4 s H u Ji 1 18 years. 22 years. 18 years. 22 years. V 18 years. 22 years. 18 years. 22 years. I 0.7 0.7 14 0.7 0.9 27 0.6 0.7 40 0.6 0.6 2 0.4 0.4 15 0.7 0.7 28 0.7 0.8 41 0.7 0.7 3 0.5 0.5 16 0.7 0.9 29 0.6 0.5 42 0.6 0.5 4 0.5 0.5 17 0.4 0.4 30 0.6 0.5 43 0.6 0.6 5 0.5 0.5 18 0.6 0.7 31 0.6 0.6 44 0.6 0.5 6 0.6 0.7 19 0.7 0.7 32 0.5 0.5 45 0.5 0.5 7 0.5 0.4 20 0.8 1.0 33 0.5 0.5 46 0.5 0.4 8 0.5 0.5 21 I.O 0.9 34 0.6 I.I 47 0.7 0.9 9 0.7 1.2 22 0.8 0.7 35 0.6 1-3 48 0.9 0.6 10 0.9 0.8 23 0.7 0.7 36 0.5 0.5 49 0.4 0.5 II 0.6 0.6 24 0.6 0.6 37 0.6 0.6 50 0.7 0.7 12 0.5 0.5 25 0.5 0.5 38 0.6 0.6 51 0.6 0.7 13 0.4 0.5 26 0.4 0.5 39 0.5 0.5 52 0.6 0.6 for each week, as in the case of the reserves. The dotted line in the lower half of Diagram No. 10 joins the ordinates of the twenty-two year variations, the heavy line the eighteen year series. It is obvious that the heavy axis (the average of the eighteen year variations) is lower than the dotted axis (the average of the twenty-two year variations). It follows that the eighteen year changes are the more representative, on the average, by about o. i. At several points the broken heavy line advances above the axis. These oscillations show that the forces producing them vary widely between different years. They are con- nected with loans on foreign exchange, excess of exports or imports and, of course, with the cotton and grain movements. -65- §43- The band movement of the probable variations of the loans is represented by Diagram No. ii. The rectangles are the limits within which the chances are even that the changes will lie for the various weeks. These ranges are percent- agely much smaller than the ranges found for the reserves. In other words, the ratio of the amplitude of period to growth axis is less in loans than in reserves. § 44. Before constructing the annual period of the loans, it is necessary to correct the eighteen year averages by sub- tracting 0.074 from each unit of advance and adding 0.074 to each uni*. of decline. From the corrected averages (Table No. 17, first column), the loan period is obtained in exactly the same manner as in the reserves. The loan period (Table Table No. 17. Annual Period of the Loans. ■ss ■ss •ss 1 11 p i 14 > V 8^ rtt) n i 1 3i I +0.432 —1.340 +0.093 +0.770 27 +0.223 +1.757 40 — 0.049 —0.394 2 —0.280 — 1.620 15 — 0.41 1 +0.359 28 —0.656 --1.051 41 — 0.226 — 0.620 3 — 0.406 — 1. 214 lb —0.430 — 0.071 29 — 0.161 +0.940 42 —0.562 —1. 182 4 - -0.515 — 0.699 17 —0.255 — 0.326 30 —0.031 +0.971 43 —0.238 — 1.420 ■; - -0.957 -J-0.258 18 +0.463 +0.137 31 +0.448 — 1.419 44 0.000 — 1.420 6 --0.741 +0.999 19 —0.287 —0.150 32 +0-247 +1.666 45 —0.686 — 2.106 7 - -0.570 +1.569 20 — 0.114 +0.264 33 —0.035 +1.631 46 —0.144 — 2.250 8 - -0.076 +1.645 21 — 0.316 —0.580 34 — 0.269 +1.632 47 —0.317 —2.567 Q - -0.403 +2.048 22 +0.473 — 0.107 35 —0.333 +1.029 48 — 0.078 —2.645 10 — 0.302 +1.764 23 +0.381 +0.274 36 O.IOI +0.929 49 +0.046 —2.569 n — 0.I9I +1-555 24 +0.257 +0.531 37 —0.352 +0-577 50 +0.030 —2.569 12 —0.549 +1.006 25 +0.381 +0.912 38 —0-537 +0.030 51 -I-0.400 — 2.169 13 —0.329 +0.677 26 +0.622 +1-534 39 -0-385 —0-345 52 +0.267 — i.g02 No. 17) is represented by Diagram No. 12. To show the peculiar correspondence between the loan and the reserve periods, the latter is represented by the dotted line. In the loan period, the upper extremes occur about the weeks ending March 4th and July 8th, and again August I3th-i9th. The low points come during the weeks ending January 14th, April 29th and May 27th. A considerable recession occurs from July 8th to July 29th. The low point of the year falls about December 2d. 5 —66— ■ §45- The quarterly period of the loans (Table No. i8) is represented by Diagram No. 13 (A). The average weekly changes for the quarterly period were derived from the weekly changes of the annual period, which are simply divided into four parts. The indices for each of the thirteen weeks showing the quarterly movement of the fund are obtained as in the former periods by correcting the changes and subtracting the quarterly average. Table No. 18. Quarterly and Monthly Periods of the Loans. Week. Quarterly Period. Week. Quarterly Period. Day of Month. Montlily Period. I 2 3 4 5 6 7 +0.103 —0.289 -0.472 —0.481 +0.030 +0.216 +0.292 8 9 10 II 12 13 +0.215 +0.165 +0.203 +0.083 —0.061 —0.049 5 12 19 26 30 +0.126 +0.073 +0.022 -0.045 -0.074 The period is only suggestive, for the truly representative period should be based on the deviations from the interpo- lated form* which should best fit the annual period. As a suggestion period, it is significant. The first month shows a declining tendency, with the minimum about the end of the fourth week. During the next two weeks there is a rapid rise of about 0.8. This is in anticipation of the mid-quarterly settling date. The next weeks show a declin- ing tendency with a slight advance for the first week of the third month. During the eleventh and twelfth weeks, there are rapid declines. The thirteenth week shows an advance in anticipation of the next quarter. This is roughly true to the tendency. The indices are, however, more suggestive than quantitative. The 0.8 in- crease during the fifth, sixth and seventh weeks amounts at the present time to only about six and one-half millions of * A form of the sine curve (Poisson) would, perhaps, be statistically convenient. Dia?tai Nal3. -67- dollars in money. How this period may be made quantita- tively representative will appear later (§ 68). § 46. The monthly period may be suggested roughly from five dates. The dates are the 5th, 12th, 19th, 26th and 30th. The position of the fund at each date (Table No. 18) is meas- ured from an average in the manner previously explained. Like the quarterly period, the monthly period (Diagram No. 1 3 (B)) is suggestive rather than quantitatively representative. It consists, however, of the averages of a large number of weeks and is, it is safe to say, true to the tendency. It is plain that the maximum occurs in the early part of the month. The average for the first date, which is the 5th of the month, is the largest, and a declining tendency ensues up to the last record, which is about the 30th of the month. There is an advance between the 30th and the 5th of 0.3, or at the present value of the growth axis of the loans, of about two and one-half millions of dollars. The chart is interesting in that it displays the habit that business men have formed of borrowing money at the first of the month and paying off gradually up to the close. This is, in distinction from the annual period of the reserves, not dependent upon the facts of nature, but upon the customs of men acting with reference to an artificial division of time. CHAPTER VI CORRELATION § 47. Before commencing the subject of correlation, it is necessary to introduce two additional statistical tables. Table No. 19 contains the average weekly discount rate on call loans at the New York Stock Exchange from 1885 to 1900. These averages from 1887 to 1900 have been col- lected from the files of the Financial Review. The two earlier years, 1885 and 1886, have been collected from the editorials of the Commercial and Financial Chronicle. This series of the Chronicle is probably the most reliable array of call rate statistics in existence. There exists another series of bank call discounts, which during the earlier years may be found printed on the weekly statements of the New York Clearing House. These statistics are, however, thor- oughly unreliable, and the practice of publishing such a weekly rate has now for a good many years been abandoned by the Clearing House. The call discount rate at the Stock Exchange is probably of all barometers the most sensitive to the immediate monej^ market. The rate at the Stock Exchange is even more sensi- tive than the call rates at the banks. The reason for this lies in the fact that "bank loans are not called so soon as the loans on the Exchange and hence always command a slightly higher rate."* The result inevitably forces the quickest changes upon the rate at the Stock Exchange and makes it the barometer of the money market at the time rather than an average rate governing an anticipated period of thirty to sixty days. The range of the rate at the Stock Exchange is very wide. The rate has at times fallen as low as three-quarters of one per cent, for several days at a time. At the banks, on the other hand, the practice is, when money becomes superfluous, * Wall Street Journal, January 30, 1901. -69- Table No. 19. Weekly Average Call Discount Rate at the Stock Exchange. 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 V Date —5 — 3 i -4 -5 + 1 — I —2 -3 -4 -6 — I —2 -4 + 2 — I — 2 I J any. 7 *lX *3 5 5 5>^ 20 5 2X 5 1/, iX 7 2 3X 3X 6 2 14 *I 2 5 4 3 6 4>^ 3 4/2 iX 5 iX 2X 2X 5 3 21 *i^ *i^ 5 4 2;^ 5 4 2% 3>^ i>^ 4X iX 2 2X 33/ 4 28 *i>^ 2 4^ 3 2X 4 3 2 3 i>^ 4 iX iX 2.x iX 5 Feby. 4 *I i^ 4 3 2 3>^ 3 2 2 f 4 17^ !X 2-X 2X 6 II #y i^ 4 2/2 2>^ 4 3 2 2/. iX 4X iX iX 2X 2X 7 18 *I^ *i^ 4 2/2 2 4 3 2 3>^ IX 4 iX iXs 2X 2X 8 9 March 25 4 *I 2>^ 3>^ 3^ 2X 2>^ 2 2X 4 5 2y. 3 2 2 4 6 I^ 3¥ iX ?x 23/ 5X 5X 5X 2% 10 li *I 2X 4 3 2yi 4>^ 3 2 15 2 3X iX 2 2X 2X II 18 *I 2 4 2/2 2>^ 4/2 2>4 2 9 I>^ 2X 3X iX 2X 4 5 12 25 *I 2>^ 4>^ 2% 2>^ 4 3 2 4 I>^ 2X 2'}4 4 2X 5 t/2 13 April I ^I 3 5 2% 3>^ 4 3 2 5 ^y 3X iX 2% 5 14 8 *I 3 6 2>^ 4>^ 4J5^ 3 2 5 i/i 2X 3X iX 2X 7 3X 15 15 *I 2^ 6I4: 2>4 3/2 4>^ 3 2 4/2 i/i 2>4 3X iX 3X 4X 3X 16 22 *I *2 6 2 3 4 3>^ 2 5 i/s 2X 3 iX 3 4X 3 17 18 May 29 6 *I *2X 2 4>^ 5 2X 2 23/ 2X 4 4>^ 2 2 \ HI :f 2X :| f 4 4X 2^ 19 13 *I 2X sy 2 2>^ 5 4 I>^ 4 iXs 2X iX 2X 4 2X 20 20 *I 2 5 I^ 2 5 4/2 I>^ 3 \/s iX 3 iX iX 3X 2 21 27 *I I^ 5 I>^ 2 5 25^ p IJ< 2^ iX IJ^ 3 2 22 June 3 •»j 2^ 5 1;^ 2;^ 4/2 \yi. I^ 2/2 iX 2 iX iXs 5X "iX 23 10 *I *2>^ 4 I^ 2>^ 5 4 iX A/2 iX iX iX iX 2X iX 24 17 *I 2X 5 I>^ 2^ 4>^ 3^ if 7 IX IX iX iX 2X iX 25 26 July 24 I *I *I 2 2 7 10 2 3>^ f ^y. if :« 2X iX ;i % i % 27 8 *lX 3 6 I>^ 3^4 5 3 IX 8 2X iX iX 5 iX 28 15 2 5 i;^ 3>^ 5 2>^ 2>^ 8 iX IX iX iX 5 iX 29 22 2 4>^ I>^ 3 4>^ 2 2 5 iX 2X iX iX 4 iX 30 29 *I^ 2 4>^ I>^ 2X' 4>^ 2 I^ 10 lK vi iH iX 3X iX 31 Aug. 5 I>^ 2X 4>^ I>^ 2^ 4 fK I^ 9 iYs iX i\ ^y 4~ i-X 32 12 l¥ 2^ 4>^ I>^ 3 5>4 I^ I>^^ 5 IX 3X iX iX 3X iX 33 19 iX 6 5 ^¥2 3>^ 10 2 I>^ 4 5 iX iX 2X iX 34 26 I>^ nyi 5/2 I>^ 5 25 ^ I^ 5 42^ iX 2 2X iX 35 Sept. 2 I>^ bU 5/2 i;^ 4^ 6 ^% 2' 4% r 6^ 1/ 2-X 3X iX 36 9 I>^ ty^ *5X 2 3>^ 6 3>^ 3/2 4 6X iX 3X 5 iX 37 16 *iX iH 6 *2>^ 3>^ 10 4 4 4 iX 5X 2X 3X 6 iX 38 23 *IX s'X 5 2 4>^ 7 3 4>^ 3 2 6 3X 4X 6X iX 39 30 I>^ 6 5 2 5>^ 4 6 4 iX 4^ 3 3X 8 % 40 Oct. 7 IX *6X 5 3 8 4 5 ■O^ 3" 2r ? 2^ 12" 41 14 *IX 7 4>^ 2>^ 7 4>^ 5>^ 5 2>^ 2X 6X 2X 2X 7 3X 42 21 IX 6>^ 4 2 8 4/2 \y2 6 2 2X 6X 2X 2 5X 3X 43 28 ^% 6X 3>^ 2 8 6 6 2 2^8 .F^ 2 13/ 6 4X 44 Nov. 4 *3 5 3/2 2 8 6 3% Y/2 2 2X 2"" iX iff 4% 45 II 2;^ 7 *4X 2^ 5^ 6 5>^ 6 2 2X 15 iX 2 9 9 46 i8 2>^ 5;5^ 4/2 2X 5>^ 8 4>^ 5X i>^ 2* 4 iX 2X 7 4 47 25 2^ 5>^ 5 2>^ 6 8 i 1 l>^. I^ 34^ ?x 2J^ ^ 34 48 Dec. 2 *1^ 8 *5X 2>^ 6 6 VA 4/2 iK h/s 2- 2l^ I^ 2X 6 3X 49 9 *2X 7 4^ 2J^ 6 6 3 V/2 iX 1/8 2X 2X iX 2X 7 4X 50 16 2 7 5 3 6 6 3 aU iX iX 2 IX 3X 2X5 7 5X 51 23 *2>^ 25 4>^ 4 6 4 2^ 7 I 1/ 4 iX 3X 2X 25 5X 52 30 *2>^ 6 5 *5 7 4 2. '^ . ^ i^ 10 2 3% 2J< 5X 5X 5>^ 5>^ 2 — 70— to make "a uniform rate of 3^ for all periods. Banks do not like to go below 3^, because they reckon the cost of money to them, including reserve requirements, at about 2 and j4%-"* The upper limit of the call rate at the Stock Exchange is around 200^. A rate of 186^ occurred during the panic of December 1899, and has been registered at several other such disturbances in credit. In the earlier years the average was estimated from daily fluctuations for some dates. These dates are marked in the table by asterisks. § 48. The second additional table (No. 20) contains the weekly ratios of reserves to deposits. These ratios are published in the Clearing House weekly statements. The table was compiled from the files of the Financial Review for the years 1885 to 1900. § 49. Of the several classes of problems outlined in Part I (§7), we have already considered "changes in one column" of the media of exchange expression ' ' coincident with the passage of time " under the head of Growth Axes. " Changes in one column coincident with recurrent periods of time," we have treated to some extent under the head of Periodicity. We now come to the third division, ^^correlation between items in different columns." This correlation from the standpoint of time, may be either immediate or anticipatory . The nature of this class of problems is best seen from a concrete example. Let us take the movements of three banking items on the Chart. These items are the average weekly discount rate on call loans as reported by the Com- mercial and Financial Chronicle since 1885 (Table No. 19), the ratios of reserves to deposits (Table No. 20), and finally the array of reserve deviations (Table No. 8). The Chart is properly suggestive. The heavy black line represents the reserve deviations, the dotted line the call loan rate and the smooth black line the ratio of reserves to deposits, with the straight line beneath it the 25^ limit required by law. * Wall Street Journal, January 30, 1901. -71- Table No. 20. — Ratios of Reserves to Deposits ( i88s-igoo). • — in vO t-^ CO C^ M M c^ >* m >o r^ CO 00 00 00 00 00 O- 0^ C3^ o> a> o^ o> oo 00 CO CO CO CO 00 CO CO CO CO CO CO QO CO 4 M M H M M H M tH H M M 1 1 -4 36.7 -5 31-4 +1 29.0 27.9 — 2 26.8 -3 25-4 -4 27.2 28.7 27.0 — I 41.2 — 2 31-5 -3 -4 29.0 -5 31-3 + 1 28.3 27.8 — I I Jan. 7 26.5 2 14 38.6 32.4 30.0 29.2 28.5 26.5 28.4 29.0 28.3 42.5 32.5 30-5 33.0 28.6 28.4 27.2 3 21 39-5 33-5 30.0 30.6 29-3 26.9 29.8 30.0 29.7 43-9 33.1 31-6 34-4 29-3 2g.i 28.1 4 28 39-8 34-0 30.8 30.4 29.7 28.5 30.9 31.6 29.7 44-9 33-2 32.7 35.0 29.9 29-5 28.7 5 Feb. 4 40.3 34.3 30.6 30.9 29.4 28.4 30.8 32.0 28.8 45.2 31.7 33.1 34.6 29.7 29.2 28.8 6 II 40.5 34-0 33-3 30.2 28.3 27.3 29.9 31.5 28.5 41-3 30.2 33.2 33.3 29.4 28.9 28.4 7 18 39-5 33-7 29.0 29.6 28.9 26.7 29.4 31-5 27.8 39.1 31-3 32.5 34.1 28.5 28.8 27.9 8 25 39-1 33-3 32.8 29.0 28.6 25.9 28.7 30.8 27.0 39-0 30.6 30.9 32.8 28.1 28.3 27.3 9 Mar. 4 39-0 31.6 27.4 28.4 27.8 25.6 28.3 30.0 26.5 39-3 30.3 30.0 35.0 27.9 27.7 26.6 10 II 38.4 30.9 27.1 28.0 26.9 25.1 27.6 29.0 26.0 39-2 29.3 29.7 34.6 28.3 27-5 25.6 II 18 38.2 30.2 26.6 27.7 26.8 25.3 27.4 28.0 26.4 39-4 28.4 29.4 34-1 29.1 27.1 25-3 12 25 38.5 29.4 26.7 27.5 26.5 25.8 27.2 28.0 27.1 39-5 27.8 28.9 33-4 29.9 27.0 25.7 13 April I 38.6 28.4 26.2 27.4 26.5 26.1 27.0 28.4 27.4 40.3 27.7 28.7 33.5 30.3 26.7 26.2 14 8 38.4 28.2 26.1 27.3 25-3 25.4 26.5 28.7 27.1 39-6 27.8 28.5 33-4 30.2 26.7 25.9 15 15 38.7 29.2 26.2 27.9 26.4 25-1 26.3 28.1 27.5 39-3 28.0 28.7 33.1 30.6 27.2 26.3 16 22 38.1 29.9 26.8 28.8 27.7 25.3 26.5 28.0 28.3 39-4 28.9 28.8 33-3 31.6 27.7 26.7 17 29 39-9 28.6 26.9 29-3 28.0 25.8 26.7 28.7 27.8 39-6 29.9 29.2 33.6 31.8 27.9 27.0 18 May 6 40.4 28.2 26.2 29.7 27.2 25.8 26.8 28.7 28.0 39-3 30.2 29.6 33.5 31-7 27.1 26.8 19 13 40.3 28.0 34-5 30.7 27.0 25.4 26.2 27.8 29.1 38.9 30.8 29.1 34.5 32.1 28.0 26.7 20 20 40.7 28.4 26.2 32.1 28.2 25.5 26.2 28.0 30.6 38.7 31.8 28.8 32.8 32.6 28.8 26.9 21 27 41.5 28.8 26.6 32.2 28.4 25.9 26.3 28.7 30.8 38.5 32.1 29.4 33.0 32.7 29.9 27.1 22 June 3 41.8 28.8 26.2 3I.6 28.3 26.2 26.9 30.0 29.8 38.6 32.3 29.4 33.1 32.4 29.8 27.2 23 10 41.5 28.3 26.2 31-9 27-3 26.0 26.7 29.4 28.4 38.4 32.0 28.7 32.4 32.4 29.3 27.0 24 17 42.1 29.0 26.2 32.0 27.4 26.7 27.7 29.3 27.1 38.4 31.7 29.1 33.1 33.1 28.3 26.9 25 24 41.8 29.2 25-9 31.8 27.1 26.5 29.1 29.2 26.4 38.3 31-4 29.0 33.2 33.4 27.8 26.7 26 July I 41.8 28.9 26.0 31.6 26.5 26.6 29.6 29.0 25-3 38.0 31.0 29.1 33.2 33-3 26.6 26.9 27 8 41.2 27.9 25.6 30.9 26.1 25.9 28.8 28.4 23-7 37.4 30.6 29.1 31.8 31.9 25.5 26.7 28 15 41.7 28.5 27.2 31-6 26.5 26.5 28.6 27.9 23-9 37-5 30.9 29.4 31.8 31-5 26.2 27.2 29 22 41.6 28.6 27.3 31.4 26.6 26.4 29.5 29.0 24.4 37.6 31.8 29.3 32.5 30.8 26.4 27.7 30 29 41.7 28.8 27.2 31-6 26.6 26.5 29.8 29.4 23.9 37-3 32.4 28.7 32.6 30.6 26.2 28.1 31 Aug. 5 41.9 28.4 26.9 31.5 26.9 27.2 29.8 29.6 21.2 36.9 32.1 28.6 32.3 30.3 25.9 28.2 32 12 40.9 27.3 26.4 30.8 26.6 25.3 29.6 28.6 20.6 36.5 31-9 27.9 31-5 29.3 26.7 28.1 33 19 40.3 27.0 26.2 30.2 25-1 24.8 29-3 28.0 21.7 36.6 32.1 27.0 31.2 28.8 26.8 27-3 34 26 39-9 26.9 26.4 30.1 25.5 24.4 28.5 27.4 23.1 36.4 31.5 27.0 31.3 27.9 26.4 27.6 35 Sept. 2 39-5 26.3 26.5 29.1 26.1 24.9 28.2 26.9 24.6 36.2 31.8 26.9 31.2 27.0 26.1 27.9 36 9 38.3 27.0 31.9 27.9 26.9 24.6 27.3 26.5 25.8 35-0 3I.O 26.8 31-9 26.0 25.3 27.8 37 16 37.8 27.2 26.1 27.5 26.2 24.1 27.2 26.0 27.8 35-2 29.7 26.9 29.2 25.6 25.0 27-3 38 23 37-2 27.2 26.7 27.9 25.5 25.5 26.9 26.0 29.6 35.2 29.0 27.3 28.1 26.1 25.4 26.8 39 30 39-3 27.6 27.6 28.6 25.2 28.5 26.0 26.1 31-2 35.4 2g.i 28.2 27.6 27.2 25.2 26.4 40 Oct. 7 35.8 26.7 27-3 27.8 24.6 27.8 25.8 26.0 32.1 35.1 28.0 28.6 27.5 27.6 25.0 25.2 41 14 35.4 27.6 27.3 27.8 24.8 25.8 26.6 25.4 33-4 35-3 27.6 28.4 27.2 27.7 25.3 25.5 42 21 34-4 26.3 27.6 29.0 25.2 24.0 27.2 25.1 35.1 35.5 27.9 27.6 27.3 28.1 25.2 25-3 43 28 33-1 26.6 28.3 28.8 25-3 25.0 28.0 25-5 36.3 35.7 28.2 28.3 28.7 28.4 25-4 25.7 44 Nov. 4 32.4 26.7 27.7 28.3 25-3 25.2 28.0 25-9 36.5 35.6 28.3 29.9 28.9 27.5 24.9 25.6 45 II 31-7 26.6 27.7 27.8 24.6 24.4 26.2 25.6 37-7 35-4 28.7 28.4 28.4 26.9 24.6 25-5 46 18 32.0 27.2 27.1 27.8 25.1 24.8 27.3 25.6 39.1 35-6 28.8 30.2 28.4 27.1 24.9 25.9 47 25 32.2 27.8 28.4 28.0 25.4 24.1 28.2 26.0 39-9 36.2 28.7 32.6 28.6 27.4 25.9 26.4 48 Dec. 2 32.0 27-5 26.7 27-5 25-5 25.1 28.0 26.5 40.6 34-0 28.5 33.2 28.4 27.1 26.1 26.2 49 9 32.0 26.7 26.8 26.8 25.2 24.4 28.2 26.4 40.5 30.8 28.9 31-4 28.3 27.1 25-9 25.6 50 16 32.6 26.1 27-3 27.4 25-7 25.2 28.5 26.2 40.4 31-9 28.5 3I.I 27.8 27.1 25.9 25.7 51 23 32.1 26.1 27.6 26.8 25.8 26.2 29.1 26.2 40.6 31.1 28.3 31-3 27-3 27.4 26.2 26.1 52 30 30.3 27.1 28.4 27.4 26.6 25.5 27.0 29-3 26.4 26.5 41.0 31-4 28.1 31-5 26.9 25.0 29.8 26.5 26.3 —72 — Glancing at the continuous movement of these lines in relation to each other, three points may be remarked : — (i) That the reserve deviations and call discounts vary inversely ; (2) That the ratio of reserves to deposits and call discounts vary inversely ; (3) That the inverse variation increases in amplitude as the ratio of reserves to deposits approaches the legal ratio of 25^. These co-variations of two or more items are properly correlations. The Chart suggests that correlation exists. The suggestion is simply one of fact and not quantitative. Hence the following questions at once arise. To what degree are these variations correlated? Is one correlation greater than another? If we knew next week's value for one item, what value could we predict for the other and within what limits? All such questions the diagrams do not answer. A method is needed to work out the quantitative measure of correlation. Such a method has been invented for use in biology by Prof. Karl Pearson of London. It has not been used to any extent in economics.* Consequently a brief exposition of the method may not be out of place. § 50. In constructing the coefficient of correlation, two quan- tities must be calculated for each set of statistics. These quantities are the mean and Xh& standard deviation. The last two quantities are very important constants for the frequency curve of an}' array of statistics. Hence it is important to know at the ovitset exactly what is meant by these several terms. Concrete examples of frequency curves are the distribu- tion of the number of weeks from 1885 to 1900 according to the ratio of reserves to deposits, the distribution of bond- dollars according to the actual rate of interest realized at the market price, or the distribution of the number of weeks from 1885 to 1900 according to the call discount rate. * A few exceptions to this statement may be noted, as Yule, on Pauperism (Roy. Stat. Soc, 1899, Vol. 62, pp. 249-286). -73- Table No. 21. Frequency of Bond-Dollars^ to Actual Yield, in Millions. Actual Bond Actual Bond Actual Bond Actual Bond Rate of Dollar Rate of DoUar Rate of Dollar Rate of Dollar Interest. Frequency. Interest. Frequency. Interest. Frequency. Interest. Frequency, 2.85-3.04 13 4.05-4.24 1088 5.25-5.44 26 6.45-6.64 3.05-3.24 131 4.25-4.44 444 5.45-5-64 2 6.65-6.84 3-25-3-44 183 4.45-4.64 3" 5.65-5-84 14 6.85-7.04 4 3.45-3-64 190 4.65-4.84 251 5.85-6.04 2 total 4204 3-65-3-84 540 4-85-5.04 96 6.05-6.24 mean rate 4.0736 3.85-4.04 881 5.05-5.24 27 6.25-6.44 I S. D. 0.5990 Thus in Diagram No. 14 (A) we lay off upon the horizontal axis at equal intervals the numbers or ratios 20^, 21%, 22^, .... 45^, which are the ratios of reserves to deposits. At each ratio of reserves to deposits, perpendiculars are erected proportionate to the number of weeks with that particular ratio during the years 1885 to 1900. The outline joining the tops of these perpendiculars is the so-called frequency polygon. In the bond-dollar frequency array (Dia- gram No. 14 B), the base line represents the actual rate of interest and the ordinates bond-dollars. In Diagram No. 14 (C), the abscissas signify discount rates and the ordinates number of weeks. Every such frequency polygon has a number of constants, which are of very great importance in the characterisation of the statistics. Among these con- stants, the most important are the mean, the mode and the standard deviation. The mean is what is commonly called in economics the weighted average. Its formula is '— where x is the abscissa quantity, y the ordinate quantity and n the number of records. In physics, the mean is known as the abscissa of the centre of gravity. Upon the Chart, the circle marks the height of the ordinate to the mean, and where the line cuts the horizontal axis lies the mean of the ratios of reserve to * The above table is compiled for bonds listed on the New York Stock Exchange about November i, 1901. Data for the actual yield were gathered from the Wall Street Journal. The array represents about $4,200,000,000 of capital invested in railroads, which were paying interest regularly at the time. —74— deposits (29.81 15). The mode* represents the highest peak of the polygon, i. e. the class of greatest frequency. In this case it is about 27. Some methodf of appreciating how the ' ' deviations are dis- tributed along the range" is our next consideration. The diagram suggests to the eye a certain representation, but it is again only suggestive, not a measurement. The notion of a "numerical value of the variation" may be appreciated from the following example in physics. If a bar hung with weights (Diagram No. 14, D) be set "rotating on the given rough pivot at a certain speed, fric- tion will bring it to rest in a given time." " Now the greater the concentration of weights about the pivot, the sooner the bar comes to rest ; the further out from the pivot the weights are, the longer it takes to come to rest. In other words, the time the bar takes to come to rest is a m.ea- sure of the concentration or scattering of the weights along the range.'' ' ' Now physicists tell us that this time is proportional to the square of a certain quantity termed the spin or swing radius, and which I will denote by the Greek letter (o")." " o- is then shown to be the mean of the squares of all the individual deviations, and in our quantitative study of evolu- tion ''^i/<''s. Cf. Pearson and Filon, Phil. Trans. A, Vol. 191, 1898, pp. 229-311 Probable Errors, 0.67449 a/ Vn. 0.67449 "^Z ^2n. 0.67449 (1 — ''°)/ ^"■ 0.67449 <^iA2|/ -— —84— tions, coefficients of correlation and coefficients of regression for the total number of weeks, i. e. N = 780, and for the two divisions in which N = 488 and N = 292 respectively. Table No. 25. N^78o weeks. N=488 weeks. N=2g2 weeks. (M) Means. Ratios. Discounts. 2g.8ii5±o.io2i 3.ii20±o.o6i4 27.2869±o.o4i5 4.2234±o.o8i3 34.il30±o.i523 i.7038±o.o298 (a) Standard Deviations. Ratios. Discounts. 4.2285±0.0722 2.5429±cS.0434 i.389i±o.o297 2.7288±o.o583 3.8566±o.io77 o.7556±o.o2ii {r) Coefficients of Correlation. Ratios and Discounts. o.523i±o.oi42 o.5920±o.oi97 o.5997±o.o253 (R) Coefficients of Regression. Discounts on Ratios. o.3i46±o.oi24 i.i6i7±o.o572 o.ii75±o.oo62 § 60. The regression lines are plotted in Diagram No. 1 5 . In the case of the total number of weeks (N = 780), it is obvious that the regression line (DE) on the linear basis is not a good fit. The line of regression (slope R = 0.3146) is con- siderably higher in the middle and lower at the extremities than the actual records. To remedy this, the total number of weeks was split into two parts. The major number of weeks (N = 488) fell below the mean of the ratio of reserves to deposits. The slope (R) of the regression line (F E) equals 1.1617. This line is a thoroughly satisfactory fit. The slope (G H) for the upper portion (N = 292, R = 0.1 175) is slight, and the relation is comparatively of small importance. Now theoretically the best fit for the whole system is a curve. But practically since the majority of the cases fall below the mean of the ratios of reserves to deposits 29.8115, and since it is in these cases that the regression is of real importance, simplicity of expression favors the straight line.* * Pearson says in a somewhat similar study, — " Now the reader has only to look at our regression diagrams, in particular at that for brethren, to assure himself no curve will serve for practical purposes substantially better than the straight line. . . All this is independent of any theory of frequency distribution, and the vanishing of (r) with the correlation simply flows from the fundamental problem that the chance of a combined event is the product of two independent probabili- —85- §6i. What, in words, is the meaning of a coefficient of regression of 1.1617 in the case of call discounts on the ratio of .reserves to deposits? It means simply that with every point of fall in the ratio of reserves to deposits call discounts tend to rise approximately one and one-sixth points. This relation is as authentic and as well justified by its appropriate facts and within its appropriate limits as the physical law of the expansion of mercury with the application of heat. It enables us to write the regression equation of call discounts on the ratio of reserves to deposits in a mathematical formula. In the biological language* of Prof. Pearson, — "if we have a first organ of magnitude m, = M^-\-x, then the most probable value of the second organ (i. e. the mean value of its array) is M^^M^-\-j/, where j/=;irr o-yo-j. " Thus the value of m, is given by m^='M.,-\-r a J a., {sn^ — M.^. Such an equation is termed a regression equation, and R^i (i. e. r aja^ is termed the coefficient of regression. In words : the probable deviation of a second organ from its mean is the product of the coefficient of regression into the observed deviation of the first organ from its mean. When the regression is perfect, i. e. m^-=M.^, the coefficient of regres- sion, or the correlation r, must vanish. When the correlation is perfect, or r=i, then the regression is least, or m^ differs most from M,." Pearson's expression, m^-^yi^-^r trja^ (»«, — M,), in the present case may be written »«d= Md-|-rO-d/ffr (»«r — Mr). Substituting the values given for the letters in Table No. 25, we have 0=4.2234— (i. 161 7) (R— 27.2896) = 35.927 — 1. 1617R. Thus if we knew beforehand that the probable value of the ties. . . . Our conclusions in this paper are deduced from the above value of (r) and from the slope of the regression line, and they involve no further assump- tion than the approximately linearity of the regression curves." (Roy. Soc. Proc, 1899, Vol, 65, 295.) * Grammar of Science, p. 401. —86— ratio of reserves to deposits would be 25.0^, we could sub- stitute the value R=:2 5.o and solve for (D), which in this case would be about Tfo. This equation simply states mathematically the rule, pre- viously stated in words, that, whenever the ratio of reserves to deposits is below 29. 5^^, for every point of fall in the ratio, call discounts tend to rise approximately i and 1/6 points. § 62 . A practical consideration of the above results may suggest several questions. Of what use, it may be said, beyond satisfying an academic curiosity, is this relation between the call discount rate and the ratio of reserves to deposits? It is an immediate correlation and not anticipa- tory. How, it may be said, not knowing the ratio, can we predict the discount rate? The bearing of all this is seen at once when it is remem- bered that the ratio of reserves to deposits is but the quotient of these two quantities. The reserves form the numerator of the fraction, the deposits the denominator. Consequently, knowing the average periodicity of the reserves, we can within limits predict the influence of the numerator on the ratio. The periodicity of the deposits has not yet been determined. But inasmuch as the deposits are, to a very large extent, the offset to the loans, and vary with the loans, knowing the periodicity of the loans enables us to study the influence of the loans upon the denominator of the ratio. Thus we may hope that, with a further development of the dynamic element, we may be able in time to predict the ratio within close limits, and ultimately the call discount rate. As correlations among these items and other items in the equation of the societary circulation and in the foreign exchange field* are developed, the limits of error may be greatly reduced. Indeed, it requires only time and a suffi cient number of investigations to reduce all this field of finance to an exact science. §63. A consideration of the means and of the standard * See § 70. —87- deviations in Table No. 25 brings out a striking antithesis. It will be noted that the high mean discount rate 4.2234 occurs with the low mean ratio 27.2869, and that the high mean ratio 34. 1 1 30 occurs with the low average discount 1.7038. In the same manner, variability as measured by the standard deviation is subject to a similar antithesis. The small standard deviation of the ratios occurs with a high standard deviation for call discounts. This of course testifies to the urgent use the banks make of the discount rate as a weapon against falling reserves. The coefficient of variation has been defined by Pearson to be the ratio of the standard deviation to the mean multiplied by 100. The following table shows how variability changes with the different classifications and also how distinctly the bond-rate is differentiated as a non-fluctuating type. Coefficients of Variation. Bond-rates. Weeks. Call-rates. Ratios R/D. 15 780 292 488 60 44 33 14 II 5 § 64. As a business development of such statistical theory and investigation there would seem to be room for under- writing companies, whose chief business should be the selling of ' puts ' and ' calls ' on the discount rates for different periods of time. Such work, whether done by independent under- writing corporations or by banking houses, would be of very great avail in the flotation of new securities and in the opera- tion of the pools which are so necessary to the existence of a great financial center. Such a means of insuring the interest rate would do much to lessen the destructive effects of panics. For it would tend, like all insurance, to spread the losses and would do much to prevent the domino-series of failures, which attend on every such disturbance to credit. CHAPTER VIII CORRELATION BETWEEN THE CALL DISCOUNT RATE AND THE PERCENTAGE DEVIATIONS OF THE TOTAL RESERVES §65. An a priori reason for correlation between the call discount rate and the reserve deviations flows from the part the reserves play in the numerator and, indirectly, in the denominator of the ratio of reserves to deposits. Now, inasmuch as the correlation between the ratio of reserves to deposits and the discount rate was slight when the ratio stood above 30^, it seemed best to deal only with correlation between the discount rate and the reserve devia- tions when the ratio stood below 30^. The following table (No. 26) is the correlation table of the above items under the given condition. Table No. 26. Reserve Deviations Rate. % 1 ro 1 8, 1 1 % 1 10 1 1 1 + + + + + I.O . I i.S 4 12 14 10 4 2 3 4 2 3 24 II 5 3 5 I 2 I II 10 5 5 2 3 2 6 7 I 2 I I 6 I 2 I I 3 I 2 I I 2 2 4 4 2 10 5 7 5 12 15 8 13 5 4 2 9 II 9 6 12 15 II 4 2 3 6 I I 5 9 6 9 4 8 10 6 8 I 5 2 I I I 13 15 7 10 7 3 2.5 3-0 3-5 4.0 4.5 5.0 5.5 6.0 7 I I I I I I 2 I 2 6 3 I 2 4 I I I I I 6.5 7.0 8.0 I I 9.0 10. 2 12.0 15.0 20.0 I 25.0 2 2 § 66. The regression averages of the call discount rate on the reserve deviations (Table No. 27) are drawn in Diagram No. 16. -89- Table No. 27. Rcgressioii Averages of Call Discounts on Reserve Deviations. Reserve Average Reserve Average Deviations. Frequency. Discount Rate. Deviations. Frequency. Discount Rate. -45 3 5.67 57 3.15 -40 I 5.00 + 5 56 3-03 -35 4 7.00 + 10 57 2.47 -30 9 7- 50 + 15 28 2.78 -25 19 8.34 + 20 20 2.65 — 20 22 5.18 + 25 16 2.84 -15 99 4-44 + 30 9 2.67 — 10 91 4.14 + 35 2 2.75 - 5 77 5-09 The heavy line joining the solid dots is the regression polygon. The light line connecting the circled dots joins the averages, which can hardly be considered representative owing to their low frequency. The regression line is sensibly skew, and follows the general form of the regression polygon of the discount rate on the ratio of reserves to deposits. The two points previously remarked (§ 56) may be repeated. (i) As the reserve falls, discount rates rise. (2) The rate of increase is not uniform, but increases as reserves fall. As in the previous case, a curve is the better fit, and, as soon as the method of moments is published, these averages will be fitted with an appropriate curve. Provisionally, how- ever, the regression averages have been fitted on the linear basis. The following table (No. 28) contains the various con- stants of correlation. Table No. 28. Reserve Deviations. Call Discounts. Mean, . .... — 3-li9±3-986 4.l23±o.077 Standard Deviation, I3.947±2.8i8 2.6gi±o.o54 CoeiEcient of Correlation, r=— 0.3707 ±0.0246 Coefficient of Regression, Call Discounts on Reserve Deviations, R=— 0.0715 ±0.0052 Even on a linear basis the coefficient of regression amounts to 0.0715 with a probable error of 0.0052, or 7^. Diagram — 9° — No. 1 6 shows by the slope of the heavy line (AB) the coeffi- cient of regression 0.0715. This shows the basis underlying that close scrutiny of the movement of the total reserves by the operators and traders of the Stock Exchange. By experience they know that fall- ing reserves, other things equal, tend to cause the discount rate to rise. In this light, the statement is favorable or unfavorable to advancing prices. Inasmuch as the correlation is truly skew, it is hardl}'^ worth while to pay but a passing notice to the regression equation which follows: = 3.900—0.0715 r. D is the call discount rate for a given reserve deviation r. To illustrate the form of the skew curve I have drawn (Diagram No. 16) the curve (M N) by freehand, which may stand for the regression curve on the skew basis. Now this curve (when it shall be expressed in a mathematical form) will perhaps be among the first supply curves derived from actual statistics. By supply curve, I mean the conception of Cour- not. Estimates have been made of demand and supply curves from the times of Gregory King down to Marshall, but these have been simply estimates and for purposes of illustration. It is interesting to glance back across the space of over sixty years to the pithy sentences of Cournot.* ' ' Observations must therefore be depended on for furnishing the means of drawing up between proper limits a table of the corresponding values of D and P." (D = annual demand and P = average price); "after which, by the well known methods of interpolation or by graphic processes, an empiric formula or a curve can be made to represent the function in question ; and the solution of problems can be pushed as far as numer- ical applications." Here we have the supply)- curve of interest, or better, the * Researches into the Mathematical Principles of the Theory of Wealth, pp. 48. f If the view is held that demand and supply curves are statistically impossible, then the above interpolation is a correlation, and nothing more. Indeed, Cournot's vievir, that demand curves for commodities may be derived from statis- —91 — rate-price curve of the supply of call capital. Thus we may write T> = (S). By the letter (D) we mean the discount rate and by the letter (S) the supply of capital. As soon as the mathematical formula of the curve is determined, we may substitute the exact, algebraic expression for the indefinite function ^(S). By the above method the discount rate is quantitatively determined, on the supply side, by the static curve* of the supply of call capital on the New York Stock Exchange. The supply of call capital is, of course, measured in percent- age deviations from the growth axis of the total reserves rather than in absolute figures. In the same manner, the static curve of the demand for capital on the New York Stock Exchange, as represented by the percentage deviations of the loans, may be derived. The general form is represented in Diagram No. 17 by AB. The regression curve of discounts on reserves is represented by DC. The two curves are so drawn that the three means coincide, and the two ranges are extended to the same distances from the means on the figure. Now, if R is the reserve deviation at a time and L the loan deviation, the lines RF and LE represent the two rates of discount as determined by the two curves. We may, therefore, suppose that, so far as these two factors are concerned, the discount rate will lie between E' and F. Thus we have two means of prediction, the second correcting the first. This is, of course, the case of three correlated variables. Corrections, however, are neces- sary on account of the fact that loan deviations are affected by changes in time as well as call loans. This correlation between the reserve deviations and call discounts suggests an annual period in the call discounts to correspond with the period in the reserves. On this basis, the extremes of the discount period would seem to be about i}{% and 5^, with the correlation, of course, inverse. tics, is practically untenable. But the conclusion that it is impossible to derive demand and supply curves for capital at a price, whether call loans, railroad bonds or mortgages, by no means follows. * Cf. Professor J. B. Clark's distinctions between static and dynamic con- ditions. CHAPTER IX CORRELATION BETWEEN THE RESERVE AND THE LOAN PERIODS § 67. So far we have been considering the correlation as immediate. A change in one item, we found, was accom- panied by a change in the second item at the same time. We will now take up a case of anticipatory correlation. In Diagram No. 12, the periods of reserve and loan devia- tions are represented. This diagram suggests very strongly that the loan period follows the reserve period after an interval. The differences between the two periods come out in this diagram. Since the total amplitude of the reserves is 19^ of the growth axis or approximately 1/5 of the growth fund, the amplitude of the loan period is 4-7% or approximately 2/9 of the growth fund. Expressed as 19^ and 4.7^, the latter is about 1/4 of the former and 4: i makes a convenient scale for plotting. The absolute scale in millions of dollars at the present time is easily found. The amplitudes are proportionate 4 : i . The growth funds are proportionate 1:3. Consequently reducing the scale of the plot of the loans to 1:3, we should have a comparison in actual figures. For January 7, 1901, the million dollar amplitude of the reserves is roughly forty millions of dollars, and for the loans about thirty millions of dollars. It follows that the dynamic element of the loans is of far greater importance than the dynamic element of the reserves compared with their relative periodicities. The tradition of probably higher money rates for the last weeks of December is also expressed in this chart. The rise in the loans during the last two weeks of December is much more rapid than in the reserves. The result is, the ratio of reserves to deposits is diminished, and high discount rates are the result. II- ■ 10 10 30 10 Dia&raYn \No./ 19. -1 Vi —93— Roughly the loan period is the shadow of the reserve period with a three or four weeks interval. To show the wave motion which these two funds are undergoing with the seasons, I have drawn the periods for two years in succes- sion. In Diagram No. i8, the heavy line represents the reserve period, the line broken by solid dots the loan period and the dotted line broken by circles the period of the tem- perature. § 68. If the loan period may be truly thought of as the shadow of the reserve period, it remains to discover how great is the interval. Does the loan index follow the reserve index of last week, two weeks ago, or a week still more remote? The amount of the interval may be discovered by calculat- ing the coefficients of correlation for lapses of one, two, three and four weeks. The correct interval is determined by the one having the highest coefficient of correlation. Table No. 29 contains the constants of correlation for the different inter- vals. Table No. 2g. Coefficients of Correlation (immediate and anticipatory ) of Reserve and Loan Periods. Correlatioa after Coefficients. Immediate Anticipatory r=o.489±o.07i r=o.6i5±o.o58 ?-=0.872±0.022 ?-=o.g58±o.oo8 )-=o.9i4±o.o54 1 week. 2 weeks. 3 weeks. 4 weeks. By Diagram No. 19, it is apparent that the highest corre- lation is reached after a lapse of three weeks, but that even to the fourth week there is a large degree of correlation. The existence of correlation after every interval is due to the continuous movement of the periods in one direction or the other for a considerable number of weeks. It is this that causes the almost perfect correlation (0.9575) after the three- week interval to flow over to the other cases. A consideration of the form of the curve in Diagram No. 19 as well as of the changes that take place in the probable —94— errors of the several coefficients would seem to substantiate the statement, that the highest correlation between the two periods is reached after an interval of nineteen to twenty days. This high correlation between the loan and the reserve periods throws a new light upon the meaning of the loan period. The loan period is not properly a period of the loans sui generis, but an ebb wave in the loans which follows after three weeks and is caused by the tide wave in the reserves. In other words, the loan period is really the shadow of the reserve period, and it is due, in a very large part, not prima- rily to the yearly period of time, but to a correlation (not yet worked out) which exists between the percentage deviations of the loans and the reserves. This correlation has appar- ently an interval of approximately three weeks. This knowledge makes it clear that the genuine annual period of the loans should be derived by averaging the loan deviations, after corrections have been made for this corre- lation with the reserves. We should then have the loan material vibrating for reasons of its own, and not through the transmitted vibration of the reserves. The reason for a correlation between the reserve and loan deviations is not hard to find in banking theory. Increasing reserves, we have seen, are correlated with decreasing call discounts. Now a falling discount rate is correlated with increasing loans. Consequently ensues the correlation between reserve and loan deviations after an interval of sufficient duration for the working out of these two correlations. Such an interval is common in physical phenomena. The regression equation of the indices of the loan period on the indices of the reserve period after a three weeks interval is Lt+3=o.2203 Rt. § 69. If the reader will carefully follow out the movements of the lines in Diagram No. 18, a peculiar correlation between the temperature and reserve periods will be discovered. When the line of the temperature period is below the axis. —95— the line of the reserve period is in inverse correlation. When the temperature line is above the axis, the reserve line is in direct correlation. Nor is this species of correlation irrational. The sowing time comes before and about the point at which the line of the temperature crosses the axis on the rise, and the harvest time falls before and about the point at which the temperature line crosses the axis on the descent. In consequence, there is this shuttle-cock correlation. The reserve fluctuation resembles a drop hammer. The drop hammer is released just before the blow is desired, and, when once the blow is struck, the hammer is raised to await the next release. We may think of the reserves as released at about the time of the turn of direction in the temperature. The blow is struck before the temperature has risen half its course, and the reserves rise to await the turn in the temper- ature. On the turn of the temperature for the descent, the reserves again fall. The blow is struck before the temperature has much more than crossed the axis, and the reserves again rise to await the next turn in the temperature. § 70. Among other correlations, which are believed to exist for reasons and upon which work* is being carried on are the following : — Correlation between the reserve and loan deviations, imme- diate and anticipatory. Correlation between the loan deviations and the call dis- count rate. Correlation between loan deviations and the rate of foreign exchange (immediate and anticipatory). Correlation between call discounts at New York and the rate of foreign exchange, as a case under the correlation of the call discount rate at London, Paris, Berlin and New York, and the rate of foreign exchange at each place. * [If various statistical agencies would spend less time in working out annual averages and expend the economised clerk-hire in publishing tables of contin- uous daily or weekly figures, much labor would be saved the investigator by eliminating the toilsome task of collecting the statistics from the files of periodi- cals.] -96- Correlation between call discounts and sixty-day paper at New York. Correlation between the rate of foreign exchange and the sixty-day paper rate at New York. Correlation between the percentage deviations of the in- terior clearings and the reserve deviations, immediate and anticipatory. §71. To compare the values of the coefficients of correla- tion, found in the last three chapters in our financial studies, with the values ascertained in several biological investiga- tions, I submit Table No. 30, compiled from some of Pear- son's Memoirs.* Table No. 30. Comparison of Correlation Coefficients in Biology "with those derived in the Preceding Pages. (A) PearsoH^s Table of Correlation Coefficients in Jlfan, Length and breadth of skull, . . o.2g to 0.49 Breadth and height of skull, ... . . . o.iotoo.34 Length and capacity of skull, 0.50 to 0.89 Weight and length (babies), . . . .... 0.62 to 0.64 Weight and stature (adults)', .... 0.50 to 0.72 Right and left femur bone, . . . 0.96 Strength of pull and stature, . . 0.22 to 0.30 Strength of pull and weight, . . 0.34 to 0.54 Age at death of fathers and sons, . . : . . 0.09 to 0.13 fBJ Correlation Coefficients in the Preceding Pages. Ratio of reserves to deposits and discount rate (780 weeks), o.52±o.oi Ratio of reserves to deposits and discount rate (488 weeks), 0.59 ±0.02 Ratios of reserves to deposits and discount rates (292 weeks), o.6o±o.oi Reserve deviations and discount rate, .... . 0.37 ±0.02 Reserve and loan periods (immediate), . . . o.49±o.07 Reserve and loan periods (after one week), . ... o.62±o.o6 Reserve and loan periods (after two weeks), . . . o.87±o.o2 Reserve and loan periods (after three weeks), . . . o.96±o.oi Reserve and loan periods (after four weeks), . . cgiico; It is apparent by the above table that the highest degree of correlation in the biological table, the correlation between * Before the table of the biological coefficients of correlation (taken from Grammar of Science, p. 402) is the following statement: "I close this section with a table of a few coefficients of correlation in man, so that the reader may have some idea of the extent to which characters and organs in one type of life are correlated." —97— the right and left femur bones, is equalled by the correlation which we have found to exist between the reserve and loan periods. Professor Pearson closes the chapters on evolution with these words.* "The reader who has followed the author through the somewhat difficult quantitative discussions of this and the previous chapters, will probably arise from the perusal with the conviction that biology is almost as exact as any branch of physical science." If the biologist can point to these correlations as satis- factory scientific laws, it is hardly more than fair to grant the same privilege to the economist. In short, it seems reasonable to conclude that economics may become almost as exact as biology. * Grammar of Science, p. 500. CHAPTER X THE CRISIS §72. The dynamic elements, as already noted, cover the movements of the funds which do not come under the statistical laws of the growth axis or periodicity. These dynamical movements were classified as, case I, a cycle, i. e. an uncertain period of gradual change; case II, a catastrophe, i. e. a crisis with violent change; case III, minor dynamical changes. Now it should be noted that although we have not been able to derive simple laws to express the dynamical move- ments correlate with continuous or recurrent time, never- theless we should by no means despair of correlating these cases with movements in other items. But inasmuch as this field cannot be advantageously worked until the previous elements have been studied, it will be sufficient to point out at this stage a theoretical point of some practical interest. A study of the disturbances in credit during the years 1884, 1890, 1893 and 1899 discloses a sequence of four phe- nomena : (i) a rapid fall in reserve deviations ; (ii) culminating with high discount rates ; (iii) a fall in the loan deviations ; (iv) a readjustment of the above items with a rapid rise in reserve deviations to great proportions. In the extent of these movements and in the level from which the movement starts, we have, criteria for the definition of crises and panics and a means of comparing these disturb- ances in credit, as measured by banking barometers. Thus we may define a panic, in general, as a less violent fall in the reserves from a higher level of shorter duration, and a crisis as a more violent fall from a lower level of longer duration. § 73. As an illustration, I have prepared Diagram No. 20, showing the movement of the dynamic elements of the reserves during the years 1893 and 1899. The one year contains the crisis of 1893 and the latter year the panic of —99— December i8th. The dynamic indices are the differences between the percentage deviations and the periodic indices. In comparing these two disturbances the two points already mentioned should be noted : (i) the level from which the disturbance starts, and (ii) the rate of decrease per week. The lower the level and the swifter the decrease, the more severe is the catastrophe. In the history of crises and panics, writers have called naturally sudden decreases from high levels panics, and violent and prolonged decreases from low levels crises. The crisis of 1893 starts from the low level of approxi- mately — 15. The panic of December 18th, or the prelimi- nary liquidation, started from a level of about +25. Black Wednesday, July 26, 1893, occurred after a fall of forty points in six weeks, or an average fall of over six points per week. The panic of December i8th occurred after a fall of about twenty-five points, distributed over twenty-seven weeks, or an average weekly fall of a little less than one point per week. Diagram No. 2 1 (folding Chart) represents the reserve deviations for the crises of 1884-5 ^.nd 1893-4 on the same scale. The similarity is apparent. The sharp break causes discount rates to rise violently. Prices fall and failures ensue. The volume of business transactions dimin- ishes and the currency which is no longer needed throughout the country to effect exchange accumulates at New York. By these two criteria taken in connection with reserve and loan deviations and call and sixty-day paper rates, a very concrete measure may be made for determining the severity of such catastrophes. A striking physical parallel is the barometer. In forecast- ing weather, the observer takes note of the level as well as of the sudden change. The barometric reading is a measure- ment of the atmospheric pressure, agreed upon by scientists. So the expansion of mercury is taken as a measure of heat. To compare heat by subjective feeling is the same as the comparison of two crises by men who have lived through both, and then state their opinions of the relative severities. lOO Here is a possible bank test — not perfect, for other causes enter in. But it is like the thermometer. It was 90° in the shade yesterday, is 80° to-day. The subjective experience, however, may profoundly differ on account of varying humidity, wind or other modifying influence. But as the thermometer is useful, so the bank test of crises may become of use. Even taking the reserve deviations alone, the level and the rate of fall show a decided difference between 1893 and 1899. 1893 represents the emaciation of adversity. 1899 stands for the "indigestion of prosperity." CHAPTER XI SUMMARY In Part I an attempt was made to state clearly the lead- ing factors of the societary circulation and to express as an approximation the relations of these factors in the equation of exchange-work. These factors may be thought of as parts of the delicate and complicated machinery which is con- stantly in operation in a modern society with the object of increasing the trade and diminishing the friction. The fac- tors of the exchange-work equation, i. e. the frame-work of the machinery, are nearly all represented by letters in the equation, every letter having a certain numerical value for every moment of time. A priori analysis brings us as far as this equation of the relations of these factors. This serves as a systematic arrangement of the facts in preparation for the statistical analysis. The statistical analysis corresponds to the test tubes and mortars of the chemist, by use of which the component parts of a compound mixture are isolated and measured. At about this point, it is well to suspend theoriz- ing and to commence collecting the facts. This is an onerous though necessary process ; for each factor has a different value in each succeeding moment. The writer — as an illustration of some of the methods which should be applied to each and every letter — ^collected from the files of journals the statistics of several of the letters weekly for a series of years. These factors were the loans, deposits, reserves, ratios of reserves to deposits and call discount rates of the New York banks. These items were then subjected to statistical analysis to determine how each set of factors varied with the continuous lapse of time, with the recurrence of intervals of time, as the year, the quarter and the month, and with the movements (immediate and anticipatory) in other sets of statistics. As far as possible, these results have been graphically repre- 102 sented, but they have been in all cases numerically expressed and the probable errors have been generally attached. Conclusions from the statistical analysis may be summarized briefly as follows : (i). As a first approximation, the exchange-work equation affords a starting point for this as well as future investiga- tions. In the present stage of economic science, such in- vestigations of the facts will, it seems to the writer, be far more fruitful than further eclectic research of authorities. The motto of the Biometrika may well be taken over bodily into economics : "Nihil est quod non nuinerandum." (2). Indices for measuring the successive credit strains to which the national banking system of the United States is subjected, are one of the fruits of the equation of exchange- work. These percentages are believed to bear important relations to the movements of prices, interest rates, the foreign exchanges, and the import and export of specie. (3). The use of the growth axis to represent solely the element of growth, rather than all of the statistical move- ments, is believed to have many applications to all kinds of continuous statistics, both economic and social, that are sub- ject to rates of increase. By thus eliminating the influence of growth, the percentage deviations can be prepared for the application of Pearson's correlation methods, which have produced so many strikingly original results in evolution. (4). In the interpolation of the growth axis, a qualitative test of fitness seemed on the whole truer to the representation of the curve as a scaffolding for the statistics, than the com- monly accepted test of least squares. (S). The percentage deviations represent those fluctuations which affect human interests most vitally. They are more evident ; thus the reserve deviations travel up and down on an average of twenty points while the growth axis is advanc- ing one point. The removal of the growth element reduces the deviations to an undulatory type. (6). The annual period of the reserve deviations, long known to bankers by experience but hitherto unmeasured, — 103— is clearly revealed. The normal, annual period consists of a double fluctuation in which the distance between the extremes is approximately i8^ of the growth axis; thus, during a year, the reserve deviations travel up and down the yearly period a distance which is between 60 and 70^ of the growth axis. (7). An annual period in the loans, corresponding to and believed to be the result* of the period in the reserves, has been isolated and measured. The loan period is, however, relatively less important than the period of the reserves; thus, in the course of a year, the loan deviations travel up and down the normal period a distance which is between 15 and 20^ of the growth axis. (8). The statistical analysis has also suggested quarterly and monthly periods in the loans, but the indices are believed to be suggestive rather than quantitatively representative. (9). The specialization of loans into various types, with the result that all the funds of a people are constantly productive, is illustrated by the contrast between call and bond loans both with respect to duration, actual rate of interest and variabil- ity of rate. (10). The importance of the ratio of reserves to deposits as a criterion of the money-market, long recognized by the operators, is numerically demonstrated. During fifteen years, the coefficient of correlation has stood above 0.50 with a probable error of o.oi. Further, call discounts tend to rise much more rapidly as the ratio decreases absolutely. (i i). The importance of the bank statement is numerically emphasized by the high correlation between the reserve deviations and the call discount rate. It is believed that there is, further, an anticipatory correlation of some import- ance between reserve deviations and call rates. (12). The importance to the public of the weekly bank statements, as a barometer of credit conditions, is quite * As far as possible, mention of cause and effect has hitherto been excluded ; for the primitive fallacy, post hoc ergo propter hoc, is nowhere more dangerous than in work of this nature. — 104 — obvious ; because the recent rapid growth of trust companies brings about the possibility of the concealment of loans and thus causes the bank statements to be no longer representa- tive for this item in the money-market, force is lent to the movement for weekly statements by the trust companies. (13). The almost perfect correlation between the reserve period and the loan period is interpreted as cause and effect. The shuttle-cock correlation between the reserve period and the temperature period emphasizes the importance of climate even on occupations so far removed from the soil as banking. It is supposable that this banking period would undergo con- siderable modification for a climate where the crops were raised continuously. (14). The peculiar likeness of panic to panic and crisis to crisis, even during the few years covered by this study, suggests a field for very careful gathering of facts on an extensive scale. The crisis is so fraught with vicissitudes to men that it seems worth while to seek knowledge from the past in the hope of finding adequate safeguards. (15). The high correlations among economic phenomena, immediate and anticipatory, — a few instances are presented in the preceding pages and many more are in process of verification but as yet unpublished — suggest strongly that foresight in business may in the future be vastly increased. Indeed, the phenomena of economics lend themselves far more readily to the possibility of prediction than do the phenomena of meteorology. Such prediction, however, can come only through the finer methods of statistical analysis and through concerted action in the assembling of the facts. Chart Showing the Movements of the Weekly Averages of the Deviations of the Total Reserves, Loans and Deposits ( In general, every line is broken at intervals of five weeks by a solid clot or circle, counting from the yearly verticals. Tlie fift\-sccond week falls upon the vertical. In the original drawing, four spines were placed between each dot or circle, and by counting dots and spines the exact record for any week can be located. An attempt was made to draw the various lines accurately on cross section paper, and in the reduction tlie cross section lines were eliminated to prevent confusion. In the space between the ver- ticals for the years 1885 and 1886, all ihe lines are plainly marked. Thk Total R!;sf.rV£s are represented b}- the heavy line joining solid (lots, marked R in 1879. The smooth line G,j is the growth axis of tlie reserves. The scale is in millions of dollars, and the scale numbers are placed on the right hand side of the yearly verticals abo\-e and below the growth axis under the letter R. The vertical scale of the reserves is twice Ihe scale of the loans and deposits. The Loans are represented by the line joining circles, marked L in 1879. Tlie growth axis of the loans is marked Gj . The scale, in millions of dollars, Total Reserves, Loans and Deposits of *u^ New York Associated Banks ( I J879-I900), Ratios of Reserve to Deposits (^^854 900), and the Discount is the same for loans and deposits. Th ^ ale numbers arc placed along the left hand side of the )'early verticals untlei' t'^'- ^'^"''^'^' ^' The I;)e1'OSITS are represented by the fi"tted line joining solid dots, marked D in 1879. The highest of the «''>\\''' ^^''^^' ^"' ^^ g'l'f^wth axis for the dei»osits. The scale of the denn-its i^ *'^^' same as the scale of the loans. TiFE Fi:i;CENTA(;E Devlvtions of THr T"''-^'- Ri-^i'-i-'-^'i'''^ ''ii's ^c:prcsented by the licav\' line joining solid dots, marked "iicser\c Deviations" in 1885. The percentage deviations of reserves, loan.'; and depi)sits fluctuate above and below the lowest straight line, which is pnrallel to and next above the base line of the chart. This straight line is zeri) value for tlic percentage devia- tions of reserves, loans and deposits and stands for the three growth axes reduced to one straight Hue. The same scale is used in plotting the per- centage deviations of reserves, loans and deposits. The scale numbers are placed to the right of the yearly verticals, -|~ lO, -f 20, ^ 30, etc., when above and — 10, — 20, — 30, etc., when below the zero axis. By this device it becomes possible to study the nio\cmci!ts of rising curves in their 879-1900), wi Rate on Call L correlations wit measures of the The Perci heavy line joini The Pei^lci dotted line join TtiE Ratk line unbroken This line is gen which is the 2 are placed to t straight line is The Call by solid dots < placed half wa) Financial the base line at ^i'^'-' RESERVES (R) <-— LORNS Ct.) — DEPOSITS RATIOS R/£> CALL DISCOUNT RATE^Dr) fl Banks (J 8794900), with their Respective Growth Axes; also Percentage Discount Rate on Call Loans at the Stock Exchange (1885-1900). correlations with movements of undulatory curves, and furtiier to obtain measures of the correlations. The Perckntage Deyiatioxs ov the Loans arc represented by the heavy line joining circles. The FERtEXTAGE Devl-vtions oe the Derosits arc represented by the dotted line joining- solid dots. The Ratios of Reserves to Deposefs are represented by the heavy bne unbroken by solid dots or circles, marked "Ratios R/D ' in 1885. This line is generally above the straight line marked "Lawful Limit R/D, which is the 2^^'e minimum reserve required by hwv. The scale numbers are placed to the left of the yearly verticals under the letters R/T). The straight line is the 25^ value. The Caei, Discount Rates are represented by a dotted line, unbroken by solid dots or circles, plainly marked in 1885. The scale numbers are placed half way between the )-carly verticals luider the letter D^^. Financial E\ents, as crises, panics, bond sales, etc., are marked along the base line at the dates of occurrence. dots, axis f tlie inted 1885. ' and base cvia • axes per- 3 are when this their \iT RATE(l)r) Chart Showing the Movements of the Weekly Averages of the Deviations of the Total Reserves, Loans and Deposits ( In general, every line is broken at intervals of five weeks by a solid dot or circle, counting from the yearly verticals. The fifty-second week falls upon the vertical. In the original drawing, four spines were placed between each dot or circle, and by counting dots and spines the exact record for any week can be located. An attempt was made to draw the various lines accurately on cross section paper, and in the reduction the cross section Unes were eliminated to prevent confusion. In the space between the ver- ticals for the years 1885 and 1886, all the lines are plainly marked. The Total Reserves are represented by the heavy line joining solid dots, marked R in 1879. The smooth line G^ is the growth axis^of the reserves. The scale is in millions of dollars, and the 'scale numbers are placed on the right hand side of the yearly verticals above and below the growth axis under the letter R. The vertical scale of the reserves is twice the scale of the loans and deposits. The Loans are represented by the line joining circles, marked L in 1879. The growth axis of the loans is marked G,,. The scale, in uiillions of dollars,' Total Reserves, Loans and Deposits ^^ New York Associated Banks ( I J879-I900), Ratios of Reserve to Depc^^s (^3854900), and the Discount is the same for loans and deposits. Xh nle niunbcrs are placed along the left hand side of the yearly verticals u' /,'' the letter L. liiE De!>()SITS are represented by the dotted line joining solid dots, marked D in ,879. The highest of L o-rno- the 1 dots, th axis of the 3sentcd n 1885. ive and le base devia ■ :h axes le per- ers a)"e , when iy this n their correlations with momiebts of undulatory curves, and further to obtain measures of the corre»i(j^g_ The PKRCEXTAGtyi-jviATioNs OF THE LoANS are represented by the heavy line joining '^"^s. The PERCENTA(i^|feEViATiONS OF THE Deposffs are represented by the dotted line joining soM d,ots. The Ratios ok ©serves to Deposffs are represented by the heavy line unbroken by soil,, dots or circles, marked "Ratios R/D " in 1885. This line is generally i)ove the straight line marked "Lawful Limit R/D," which is the 25^; aii"&uin reserve required by law. The scale numbers are placed to the l^ftw tjhe yearly verticals under the letters R/D. The straight line is the 2 5wvailue The Call DiscoMt Rates are represented by a dotted line, unbroken by solid dots or circM plainly marked in 1885. The scale numbers are placed half way betwBn the yearly verticals under the letter Dj^. Financial EviiNil' as crises, panics, bond sales, etc., are marked along the base line at tlie dUes of occurrence. JNT RATE(l)r) i INDEX Analysis : meaning of element, 23 ; classification of elements, 23 ; gen- eral problem, 12. Average: value and limits, 15. Banking : place in scheme, i ; Act of Jan., 1875, 8. Bank Notes : amount in U. S., 3-4. Banks : state banks, 10 ; private banks, 10 ; present system of small banks, 59 ; branch-banks, 59 ; lend- ing indirectly on real estate, 60. Bank Statements, see N. Y. Associ- ated Banks. Barter, i. Bonds : actual rates, 73 ; duration, 79. Call Discount Rate : table (No. ig) of weekly averages, 1885-igoo, 69 ; barometer of money market, 68 ; range, 68 ; practice of banks in call loans, 70 ; sensitiveness, 79 ; com- parison with interest rates on bonds, 79 ; correlation with ratio of reserves to deposits (Table No. 23), 80 ; re- gression averages of call discounts on ratios of reserves to deposits (Table No. 24 and Diagrams Nos. 15 and 16), 81-2 ; regression equa- tion of call discounts on ratio of reserves to deposits, 85-6 ; meaning of regressive equation, 86 ; correla- tion between reserve deviations and call discount rates (Table No. 26), 88 ; regression averages (Table No. 27), 8g ; constants (Table No. 28), 89 ; regression equation, 90 ; annual period, gi ; determination by two correlations, gi ; reason for high rates in December, 92. Circulation, see Societary Circula- tion : meaning, 2. Clark, J. B.: on static conditions, 91. Commercial and Financial Chroni- cle : clearings, 11 ; importance of percentage deviations in clearing statistics, 43 ; call discount rates, 68. Commercial Year-book ; for amounts of money in U. S., 11. Comptroller of Currency : reports, II ; returns by banks, 12. Correlation : immediate and antici- patory, 16; meaning and illustration, 70-72 ; Pearson's measure of corre- lation,' 72 ; discount rates and ratios of reserves to deposits, 78-87 ; measure, 80-81; coefficient,' 82; formulas, 83 ; call discount rates and reserves deviations, 88-91 ; re- serve and loan periods, 92-4 ; notion of anticipatory correlation, 93 ; pe- riods of reserves, loans arid the temperature, 94-5 ; shuttle-cock type, 95 ; correlations believed to exist, 95-6 ; comparison of correlation coefficients in biology and econom- ics, 96. CoURNOT, AuGUSTiN : On foreign ex- change, 61 ; on determination of demand curves, go. Credit : in effecting exchange, i. Crisis : meaning, 23 ; definitions of crises and panics, 98-100 ; sequence of phenomena in crises, 98 ; com- parison of crises of 1884 and 1893, 98 ; comparison of crisis of 1893 and panic of 1899, 99. Currency : see Media of Exchange. Cycle: definition, 23. Deposits : rate of turnover (Diagram No. i), 7 ; relation of lawful maxi- mum deposits to reserves, 8-10 ; percentages, g. -io6- DiscoUNT, see Call Discount Rate, 2. Dynamic Elements : meaning and classification, 23. Economics : an exact science, 97. Edgeworth : interpolation, 24. EssARS, Pierre Des : rate of turnover of deposits, 7 ; statistics of turnover of deposits, 11. Exchange, see Media of Exchange : elements of, i ; media of, i. Exchange-work : notion, 5 ; equa- tions, 5-8, 10 ; money in circulation, 5 ; money in bank-vaults, 6, 7. FiLON AND Pearson : errors of corre- lation, 83. Financial Review : source of call discount rates and ratios of reserves to deposits, 68, 70. Fisher, Irving: on societary circula- tion, 2 ; velocity of money, 6 ; velocity of money and credit, 7 ; student statistics of velocity of money, 11. Foreign Exchange Market : effect on reserves, 56 ; need of mathemati- cal theory, 60. Fourth of July : Effect on reserves, 56. Frequency Curves : ratios of reserves to deposits, 73 ; bond-dollars (Table No. 21), 73 ; discount rates, 73 ; constants, 73-74 ; Pearson's method of analysis, 76-7 ; types, 76. Gaines, John M. : velocity of money and credit, 7 ; statistics of rates of turnover of deposits, 11. Geometrical Curve, see Interpolation. Gold : as standard, 3 ; amount, 3 ; changes in amount, 3 ; where lo- cated, 3. Growth Axes : equation of total re- serves, 28 ; equation of deposits, 29 ; equation of loans, 29 ; annual values for reserves, loans and de- posits (Table No. 5), 29 ; use as a standard, 33 and 34 ; assumption that axis accounts for growth, 33. Growth Element : meaning, 23 ; rep- resented by a geometrical form, 24. Hadley, Arthur T. . on societary circulation, 2-3. Harvest : effect on reserve period, 56. Interest : actual rate on bond-dollars, 73 ; sensitiveness of call and bond rates, 79. Interpolation : geometrical curve, 24 ; Steinhauser's table, 25 ; method of least squares, 25 ; method of moments, 24, 81 ; method of inter- polating geometrical curve, 23-28 ; tests of fitness, 30, 75 ; average yearly number of crossings with range (Table No. 6), 31 ; proportional relations among three growth axes, 32 ; summary of growth axes (Table No. 7), 33 ; Pearson's skew curves, 73, 75-6. King, Gregory : demand curves, 90. Loans : Table No. 4 (folding chart, 20 ; polygon, 22 ; Table (No. 14) showing increases and decreases, 52 ; period- icity, 62 ; probable variations (Table No. 16 and Diagram No. 11), 64; cor- rections applied, 65 ; annual period (Table No. 17 and Diagram No. I2), 65 ; quarterly period (Table No. 18 and Diagram No. 13), 66 ; monthly period (Table No. 18 and Diagram No. 13), 67 ; correlation of reserve and loan periods, 92-4. Marshall : demand curves, 90. Mean : definition, 73. Media of Exchange : definition, i ; varieties in U. S., 3 ; relative pro- portions, 3 ; table of letters, 4 ; dis- tribution in U. S., 4; rate of turn- over, 6-7 ; equation of exchange- work, II. — 107 — Merriman : Pearson's criticism of the- ory of least squares, 76. Money, see Media of Exchange : rate of turnover, 2 ; velocity, 5 ; in circu- lation, 5 ; in bank vaults, 6. Money-changing, i. National Bank of U. S.: a necessity, 57 ; advocated by Mr. Stickney, 57. Newcomb, Simon : equation of socie- tary circulation, 2 ; turnover of money, 7. New Currency Act ; gold basis, 3. New York Associated Banks : week- ly statements, 6, 15-17 ; statistics of total reserves and loans (Tables Nos. 3 and 4 and folding chart), 18-21. New York Clearing House : early statistics of bank call rates unrelia- ble, 68 ; ratios of reserves to depos- its, 70. Out-of-town Banks : influence on money-market, 56 and 58 ; induce- ment to deposit funds at New York, 57-8. Pareto, Vilfredo : income curve, 24. Pearson, Karl : standard deviation, 74-5 ; frequency curves, 73, 75-6 ; theory of least squares, 76; measure of fitness, 77 ; measure of correla- tion, 72,80-81; method of moments, 24, 81; coefficient of correlation, 82; formulas of constants of correlation, 83 ; regression equation, 85 ; coeffi- cient of variation, 87. Percentage Deviations, see Reserves and Loans : gross deviations, 33 ; importance of deviations in eco- nomic science, 34, 36; definition, 35; tables (Nos. 8 and 9) of reserve and loan deviations, 38-41 ; weekly changes, 36, 37 ; general features of the polygons of deviations, 42; com- parisons of rational change with chronicle change, 43 ; convenience in problems of correlation, 43-4. Periodicity • meaning, 23, 45 ; of re- serve deviations, 45 ; occurrence period, 46-48, 54 ; average weekly changes of reserve deviations, 49 ; prediction, 52 ; correction, 53 ; an- nual period of reserves, 53 ; mean- ing of reserve periodicity, 55-56, 59-60. Period-year : definition, 45-46. Probable Variation : definition, 50 ; reserves, 51 ; meaning, 51-2; loans, 64. Pools: connection with reserves, 6r. Prices : letters, 1-2 ; price-level, 3 : connected with volume of transac- tions, 3-4. Ratios of Reserves to Deposits : weekly ratios (1885-igoo), 71 ; corre- lation with discount rates (Table No. 23), 80 ; regression averages of call discounts on ratios (Table No. 24), 81-82. Reserves : provision for minimum re- serves, 8 ; Table No. 3 (folding chart), 18 ; polygon, 22 ; table (No. 10) showing weekly increases and decreases, 46 ; occurrence period, 48 ; average weekly changes, 49 ; selection of the more representative series by the probable variation, 50 ; prediction, 52 ; annual period, 53- 60 ; correlation with temperature, 58 and 95 ; correlation with discount rates, 88 ; regression averages, 89 ; regression equation, 90 ; anticipa- tory correlation of reserve and loan periods, 92-4. Schwab, John C. : " futures " in the media of exchange, 2. Silver: amount in U. S., 3-4; certifi- cates, 3-4. SociETARY Circulation : cases, i ; first approximation, 2; general equa- tion for media of exchange side, 11. Sowing, effect on reserve deviations, 55- — io8 — Standard Deviation : definition, 74. Steinhauser, Anton: interpolation forms, 25. Stickney, a. B.: central bank, 57. Sumner, William G. : origin of bank statements, 16, 22 ; deposits by coun- try banks at New York in the fifties, 55 ; the banking period, 59. Supply Curve : Cournot's notion, 90; data for an algebraic equation of a supply curve, go. Temperature : cause of periodicity in reserves, 58 ; period of temperature, 58 ; correlation of reserve, loan and temperature periods, 94-5. Total Reserves, see Reserves : mean- ing, 2. Trust Companies: exchange- work, 10 ; influence on reserves and loans, 56, 60 and 104. U. S. Notes : amount, 3-4. U. S. Treasury : statement, 3 ; in- fluence on money-market, 56-7 ; bond purchases, 57. Wall Street Journal: trust com- panies, 60 ; call loans, 68 ; data for actual rates of interest on bond-dol- lars, 73. Yule, G. Udny : association of attri- butes in statistics, 54 ; pauperism, 72. ??tr^