*********************************************************** DATA FOR DON ALLEN'S JULY/AUGUST 1998 REVIEW ARTICLE "HOW CLOSELY DO BANKS MANAGE VAULT CASH?" Pages 43-54 *********************************************************** FIGURE 1 (Left Panel) Vault Cash as a Percent of Demand Deposits "SOURCE: Friedman & Schwartz ""A Monetary History of the United States, 1867-1960.""" DD Deman Deposits Adjusted "Table A-1 Currency Held by the Public and Deposits, Seasonally Adjusted," Billions of Dollars (page 713) VC "Vault Cash, Bank Reserves" "Table A-2 Bank Reserves, Seasonally Adjusted" Billions of Dollars (page 739) RATIO YEAR VC DD VCDD 1930 0.883 21.113 4.182257377 1931 0.819 17.29 4.736842105 1932 0.716 15.511 4.616078912 1933 0.731 14.92 4.899463807 1934 0.813 18.215 4.463354378 1935 0.87 22.153 3.927233332 1936 0.952 25.386 3.750098479 1937 0.894 23.565 3.793761935 1938 1.101 26.131 4.213386399 1939 1.207 29.797 4.050743363 1940 1.257 34.633 3.629486328 1941 1.364 38.615 3.532306099 1942 1.359 48.861 2.781359366 1943 1.5 61.263 2.448459919 1944 1.7 67.443 2.520647065 1945 1.9 76.216 2.492914874 1946 2.1 81.3 2.58302583 1947 1.903 85.8 2.217948718 1948 2.042 84.9 2.405182568 1949 2.013 85 2.368235294 1950 2.165 90.2 2.400221729 1951 2.283 95.8 2.38308977 1952 2.397 99.1 2.41876892 1953 2.454 100.4 2.444223108 1954 2.563 104.4 2.454980843 1955 2.63 106.8 2.462546816 1955 2.702 108.2 2.497227357 1957 2.732 107.2 2.548507463 1958 2.815 112.2 2.508912656 1959 2.982 112.6 2.648312611 1960 3.128 111.4 2.807899461 FMD "Money Stock: Demand Deposits (SA, Bil.$)" FMOT "Money Stock: Other Checkable Deposits (SA, Bil.$)" FARVTN "Total Vault Cash (NSA, Mil.$)" FIGURE 1 (Right Panel) Vault Cash as a Percent of Demand Deposits (BIL $) Ratio FMD FMOT fmdfmot FARVTN FARVTN VCDD 1959 110.8 0 110.8 2408 2.408 2.173285199 1960 111.6 0 111.6 2581 2.581 2.312724014 1961 115.5 0 115.5 2864 2.864 2.47965368 1962 117.1 0 117.1 3137 3.137 2.678906917 1963 120.6 0.1 120.7 3451 3.451 2.85915493 1964 125.8 0.1 125.9 3617 3.617 2.872915012 1965 131.3 0.1 131.4 3936 3.936 2.99543379 1966 133.4 0.1 133.5 4235 4.235 3.172284644 1967 142.5 0.1 142.6 4521 4.521 3.170406732 1968 153.6 0.1 153.7 4737 4.737 3.081977879 1969 157.3 0.2 157.5 4959 4.959 3.148571429 1970 164.7 0.1 164.8 5340 5.34 3.240291262 1971 175.1 0.2 175.3 5675 5.675 3.237307473 1972 191.6 0.2 191.8 6098 6.098 3.179353493 1973 200.3 0.3 200.6 6635 6.635 3.307577268 1974 205.1 0.4 205.5 7179 7.179 3.493430657 1975 211.6 0.9 212.5 7772 7.772 3.657411765 1976 221.6 2.7 224.3 8548 8.548 3.810967454 1977 236.8 4.2 241 9352 9.352 3.880497925 1978 250.6 8.5 259.1 10331 10.331 3.987263605 1979 257.7 16.8 274.5 11344 11.344 4.132604736 1980 261.5 28.1 289.6 18149 18.149 6.26691989 1981 231.4 78.7 310.1 19538 19.538 6.30054821 1982 234 104.1 338.1 20392 20.392 6.031351671 1983 238.4 132.1 370.5 20755 20.755 5.601889339 1984 243.7 147.4 391.1 22193 22.193 5.674507799 1985 266.6 179.8 446.4 23336 23.336 5.227598566 1986 302.1 235.6 537.7 24538 24.538 4.563511252 1987 286.8 259.5 546.3 26676 26.676 4.883031301 1988 286.8 280.9 567.7 28204 28.204 4.968116963 1989 279.3 285.3 564.6 29827 29.827 5.282855119 1990 277.4 293.9 571.3 31789 31.789 5.564326974 1991 289.6 332.5 622.1 32509 32.509 5.225687189 1992 339.5 384.4 723.9 34541 34.541 4.771515403 1993 385.2 414.5 799.7 36818 36.818 4.603976491 1994 384 403.9 787.9 40378 40.378 5.124762026 1995 391 356.4 747.4 42094 42.094 5.6320578 1996 403.6 275.9 679.5 44379 44.379 6.531125828 1997 397.1 245.2 642.3 43970 43.97 6.845710727 FIGURE 2 Eighth District TRR Total Required Reserves ($1000) WVC2 Toal Vault Cash ($1000) TRR WVC2 890109 1876385 1151039 890123 1821589 1089913.5 890206 1749798 1008931 890220 1748422 967185.5 890306 1744315 975995.5 890320 1763856 958931.5 890403 1739209 1008438 890417 1823174 977524 890501 1768185 1041861.5 890515 1748354 981816.5 890529 1681784 1010699.5 890612 1749407 985097 890626 1711120 1003583.5 890710 1744557 997390 890724 1725580 1047122.5 890807 1716324 973171.5 890821 1727535 1000294.5 890904 1721690 1003516 890918 1774031 1022989.5 891002 1687229 1022881 891016 1757276 979453 891030 1692520 1045335 891113 1756592 979928.5 891127 1748047 1051442.5 891211 1803157 1063156.5 891225 1823466 1061916 900108 1901072 1165771.5 900122 1841003 1103203.5 900205 1744339 1015321.5 900219 1749576 974117.5 900305 1755878 984169 900319 1781835 968701 900402 1754037 995829 900416 1861660 963902.5 900430 1796269 1046921.5 900514 1777885 979337 900528 1730177 1014566.5 900611 1810658 1002233.5 900625 1777531 1029361.5 900709 1811750 1027194.5 900723 1807619 1060780.5 900806 1792859 1013373 900820 1804258 1016462.5 900903 1771728 1039698 900917 1843670 1040938.5 901001 1774601 1058829.5 901015 1829532 1011420.5 901029 1760195 1063928.5 901112 1821356 987285 901126 1829119 1070285 901210 1867089 1092812 901224 1709856 1079071.5 910107 1574348 1172377.5 910121 1540771 1092058 910204 1461822 1033563.5 910218 1471915 992585 910304 1478991 1026241.5 910318 1517429 1007935 910401 1500989 1045435 910415 1594006 1033961.5 910429 1531440 1089756 910513 1543445 1018925 910527 1510523 1057638 910610 1561876 1063231.5 910624 1562277 1084513.5 910708 1589405 1067964 910722 1581781 1106992.5 910805 1566190 1045346 910819 1604685 1044860 910902 1568450 1083705 910916 1654833 1074697.5 910930 1591014 1105538 911014 1661727 1037361.5 911028 1645623 1114374 911111 1704886 1054249.5 911125 1708634 1128270.5 911209 1761465 1091132.5 911223 1802671 1110797 920106 1836561 1217255 920120 1836628 1143664.5 920203 1767753 1093106.5 920217 1823948 1019122 920302 1808346 1131757.5 920316 1911324 1056793 920330 1858975 1120973 920413 1702105 1073195 920427 1674772 1154244 920511 1662177 1090385 920525 1646303 1123685.5 920608 1664407 1138495 920622 1685306 1130027.5 920706 1663803 1091295 920720 1693684 1150502.5 920803 1667283 1134262 920817 1752866 1103877 920831 1697339 1165039 920914 1799805 1104856.5 920928 1749394 1170297.5 921012 1806755 1096743.5 921026 1791022 1159159 921109 1837032 1117720.5 921123 1877267 1194866 921207 1895115 1193448 921221 1922810 1201899.5 930104 1902603 1243242 930118 1939779 1259318 930201 1811108 1235291.5 930215 1818149 1129252.5 930301 1802793 1248628.5 930315 1871609 1155163.5 930329 1828601 1187317 930412 1933647 1122522.5 930426 1918070 1216863.5 930510 1909464 1157247 930524 1912384 1199692 930607 1936753 1187297.5 930621 1976520 1208135.5 930705 1947784 1201114 930719 2030863 1260239.5 930802 1946510 1256621 930816 2032102 1199662.5 930830 1961423 1282079.5 930913 2050462 1222808.5 930927 2002757 1274568 931011 2083046 1206999 931025 2076516 1282247.5 931108 2116376 1257015 931122 2169842 1317557 931206 2169255 1343702 931220 2233223 1319789 940103 2191319 1381712 940117 2243153 1371113 940131 2105966 1361102 940214 2110603 1249399.5 940228 2089457 1363169 940314 2150545 1257296.5 940328 2102980 1302556.5 940411 2217376 1274307.5 940425 2196897 1367727 940509 2136119 1321717 940523 2103911 1385442.5 940606 2113361 1391976.5 940620 2172999 1404391.5 940704 2100157 1391150 940718 2160391 1433813.5 940801 2076012 1425119.5 940815 2130594 1379356 940829 2060761 1443774.5 940912 2146130 1395671.5 940926 2070910 1410444.5 941010 2088875 1350702 941024 2098569 1421762 941107 2098280 1391750 941121 2135750 1431134 941205 2117050 1461572 941219 2171808 1423821 950102 2153654 1486290.5 950116 2192599 1527424.5 950130 2034471 1489926.5 950213 2033785 1352593 950227 1987491 1421738.5 950313 2033721 1354605.5 950327 1983482 1399331 950410 2048637 1358366 950424 2100578 1441444 950508 2007636 1411573.5 950522 2009830 1460394 950605 2009902 1466087 950619 2065947 1452735 950703 1995898 1476860 950717 2105186 1501820 950731 2029197 1503517.5 950814 2026591 1417817 950828 1958728 1484399.5 950911 2019979 1438301 950925 1964162 1498259 951009 1868592 1425463.5 951023 1880526 1486061 951106 1889629 1435994 951120 1927166 1449150 951204 1908353 1515994.5 951218 1923539 1466021 960101 1837643 1523493.5 960115 1808067 1567293.5 960129 1751314 1518489 960212 1739542 1382138 960226 1719493 1451773.5 960311 1767229 1393670.5 960325 1764285 1464517 960408 1763129 1451629.5 960422 1802381 1518145 960506 1686433 1466645.5 960520 1655269 1481847.5 960603 1617588 1537912.5 960617 1648243 1486505.5 960701 1569967 1532127 960715 1571046 1539830.5 960729 1494975 1528080.5 960812 1499097 1460171.5 960826 1481104 1528277.5 960909 1515262 1511863 960923 1529101 1551398.5 961007 1473179 1517291 961021 1487941 1586096 961104 1498197 1567340.5 961118 1499710 1556672 961202 1498264 1624506 961216 1520874 1611114.5 961230 1535191 1652581.5 Figure 3 Surplus Vault Cash SURPLUS VAULT CASH (SVC) BILLIONS OF DOLLARS SVC = (Total vault cash - Applied vault cash)/1000 Year SVC 1959 3.1805 59 2.9496 59 2.9017 59 2.9755 59 2.9415 59 3.0618 59 3.0903 59 3.0372 59 3.14 59 3.1235 59 3.1503 59 3.1938 60 3.0767 60 2.9062 60 2.8918 60 2.8849 60 2.8637 60 2.9581 60 2.927 60 2.8549 60 2.4353 60 2.3889 60 2.1277 60 1.0913 61 1.0592 61 1.0418 61 1.056 61 1.0395 61 1.0756 61 1.1443 61 1.0795 61 1.1244 61 1.0933 61 1.0666 61 1.1194 61 1.1373 62 1.1408 62 1.0859 62 1.0947 62 1.077 62 1.1456 62 1.1455 62 1.1351 62 1.1825 62 1.125 62 1.1402 62 1.1882 62 1.1805 63 1.2364 63 1.1499 63 1.1289 63 1.1712 63 1.2205 63 1.1912 63 1.2342 63 1.2365 63 1.1968 63 1.2438 63 1.1998 63 1.2993 64 1.332 64 1.2033 64 1.1756 64 1.2717 64 1.2069 64 1.2405 64 1.3053 64 1.231 64 1.3127 64 1.3254 64 1.2876 64 1.4449 65 1.3986 65 1.3513 65 1.3415 65 1.395 65 1.3075 65 1.3728 65 1.4033 65 1.3546 65 1.4372 65 1.3944 65 1.4204 65 1.5478 66 1.48 66 1.456 66 1.4859 66 1.4632 66 1.4292 66 1.5366 66 1.4803 66 1.498 66 1.5546 66 1.456 66 1.5271 66 1.6059 67 1.5739 67 1.5391 67 1.5916 67 1.5326 67 1.5824 67 1.6813 67 1.6038 67 1.6765 67 1.6276 67 1.5791 67 1.6507 67 1.6457 68 1.7833 68 1.6648 68 1.5989 68 1.6648 68 1.7203 68 1.6993 68 1.7609 68 1.7525 68 1.7019 68 1.7587 68 1.6934 68 2.0161 69 1.684 69 1.6311 69 1.6548 69 1.8104 69 1.6769 69 1.8765 69 1.8595 69 1.739 69 1.8707 69 1.8104 69 1.7875 69 2.1125 70 1.8004 70 1.7449 70 1.8008 70 1.9037 70 1.8165 70 2.0296 70 1.9524 70 1.9374 70 2.0783 70 1.9673 70 2.0565 70 2.2135 71 2.0201 71 1.963 71 2.0629 71 2.1183 71 2.0239 71 2.2581 71 2.1883 71 2.2035 71 2.253 71 2.1631 71 2.3326 71 2.4396 72 2.3143 72 2.1811 72 2.272 72 2.4208 72 2.442 72 2.5432 72 2.5697 72 2.5841 72 2.4662 72 2.5856 72 2.6569 72 2.7088 73 2.7139 73 2.4365 73 2.5084 73 2.7119 73 2.7446 73 2.8225 73 2.9023 73 2.8249 73 2.8116 73 2.8791 73 2.8649 73 3.1128 74 2.9739 74 2.7766 74 2.7589 74 3.1033 74 2.9853 74 3.1664 74 3.2792 74 3.1109 74 3.2252 74 3.2262 74 3.1198 74 3.604 75 3.1517 75 3.0379 75 3.0686 75 3.4296 75 3.1834 75 3.4863 75 3.6586 75 3.3672 75 3.5892 75 3.472 75 3.4891 75 4.0198 76 3.5459 76 3.3674 76 3.512 76 3.8098 76 3.4815 76 3.9471 76 3.6896 76 3.7841 76 3.8837 76 3.7405 76 4.0252 76 4.1454 77 3.9794 77 3.6104 77 4.0877 77 4.0953 77 3.9279 77 4.2402 77 4.2339 77 4.4105 77 4.2811 77 4.2654 77 4.4959 77 4.3877 78 4.7329 78 4.1756 78 4.3091 78 4.5376 78 4.6249 78 4.6917 78 4.8504 78 4.7395 78 4.8702 78 4.945 78 5.1056 78 5.0805 79 5.1746 79 4.6994 79 4.8041 79 5.2403 79 5.0277 79 5.2508 79 5.3796 79 5.2759 79 5.3865 79 5.477 79 5.5774 79 5.7537 80 5.6734 80 5.2818 80 5.4124 80 5.7571 80 5.4298 80 5.7319 80 5.879 80 5.7887 80 6.009 80 5.9295 80 5.0828 80 5.3814 81 4.9526 81 4.4018 81 4.2594 81 4.6798 81 4.3316 81 4.5775 81 4.5803 81 4.5828 81 4.1245 81 3.7978 81 4.0714 81 4.0419 82 4.1216 82 3.8159 82 3.6931 82 3.906 82 3.6381 82 4.2302 82 3.9577 82 4.0084 82 3.5166 82 3.2495 82 3.6472 82 3.6521 83 3.9901 83 3.5095 83 3.7397 83 3.767 83 3.7539 83 4.0699 83 3.6488 83 3.9711 83 3.6125 83 3.5033 83 3.9698 83 3.683 84 4.0347 84 3.0739 84 3.7976 84 4.1975 84 4.3562 84 4.1953 84 4.7873 84 3.9481 84 4.4068 84 3.8817 84 4.0629 84 4.3179 85 4.9953 85 2.4718 85 3.3803 85 3.9353 85 3.7549 85 3.8497 85 4.1969 85 3.2741 85 4.0924 85 3.4041 85 3.4854 85 4.6582 86 4.381 86 1.724 86 2.8656 86 3.5895 86 3.2911 86 3.5799 86 4.0187 86 3.0582 86 3.7288 86 2.8168 86 3.2958 86 3.9808 87 3.9803 87 1.4601 87 2.6494 87 3.2802 87 3.1639 87 3.3361 87 3.1832 87 3.1681 87 3.6907 87 2.3369 87 3.4403 87 3.635 88 4.2915 88 1.3693 88 2.2688 88 3.1049 88 2.6891 88 3.6929 88 3.1858 88 2.9517 88 3.443 88 2.6924 88 3.6444 88 3.8227 89 4.7058 89 0.80691 89 2.5958 89 3.2513 89 3.3263 89 3.7411 89 3.8021 89 3.151 89 3.7025 89 2.9298 89 3.6644 89 4.1692 90 4.8973 90 0.6223 90 2.4072 90 3.3208 90 3.1467 90 3.5829 90 3.6955 90 2.9044 90 3.8025 90 3.0152 90 3.8901 90 4.6319 91 5.921 91 2.8368 91 4.0213 91 4.7872 91 4.6832 91 5.1026 91 5.3723 91 4.1139 91 5.2668 91 4.2456 91 4.7573 91 5.8611 92 5.0725 92 1.8811 92 3.5901 92 4.3539 92 4.6937 92 4.6773 92 4.3664 92 3.9901 92 5.0975 92 3.7625 92 5.2916 92 4.411 93 4.4325 93 3.7923 93 3.8656 93 4.401 93 3.942 93 4.6268 93 4.3135 93 4.2559 93 4.5403 93 4.1536 93 4.6634 93 4.2202 94 4.6083 94 3.8798 94 3.9742 94 4.4087 94 4.4762 94 4.9044 94 4.5773 94 4.736 94 4.6762 94 4.641 94 5.4553 94 4.7054 95 5.3578 95 3.7309 95 4.2894 95 4.8775 95 5.1584 95 4.9523 95 5.3032 95 5.1064 95 5.0772 95 5.24 95 6.0404 95 5.9929 96 7.0836 96 5.0255 96 5.0555 96 5.7928 96 6.121 96 5.8346 96 7.1939 96 6.7621 Program for Table 3: Sample parameters for equation 6 "A Mathematica Program to Determine [S,s]" Clear; (*** Define the probability Distribution ***) " f[x_,mu_,sigma_]:=((1/((2 Pi sigma^2)^0.5)) E^((-(x-mu)^2)/(2 sigma^2)))" " F[y_,mu_,sigma_]:=Simplify[Integrate[f[x,mu,sigma],{x,-Infinity,y}]]" (* Define L1=penalty for lost sales *) " L1[Istar_]:=p Simplify[ Integrate[(x-Istar) f[x,mu,sigma],{x,Istar,Infinity}]]" (* Define the storage cost for inventory *) " L2[Istar_]:= r Simplify[ Integrate[(Istar-x) f[x,mu,sigma],{x,0,Istar}]]" (* Total (expected) cost of inventory is sum of L1 & L2 *) L[Istar_]:= L1[Istar] + L2[Istar] (* Define variable cost as quadratic polynomial *) VC[Istar_]:= anought + aone*Istar + atwo*Istar^2 TC[Istar_]:= VC[Istar]+L[Istar] "(*Initialize parameter vector: mu=mean of demand,sigma=standard deviation of " "demand, p=price, r=storage cost, anought,aone,atwo=coefficients of" variable cost... atwo=0 implies no quadratic term*) mu=0.3;sigma=5;p=0.1;r=.02;anought=.0025;aone=0;atwo=-0.0; "vec:={mu,sigma,p,r,anought,aone,atwo};" periods=10; "param=Table[0,{periods}];" "Band=Table[0,{periods},{2}];" " Smax=Istar /. FindMinimum[L[Istar],{Istar,1}] [[2]];" " smin=s /. FindRoot[VC[s]-anought+L[s]==VC[Smax]+L[Smax],{s,0}] [[1]];" (* A loop to compute changes in S and s for growth in mu (or any other parameter)*) k=1; "Band[[k]]={Smax,smin};" " (Label[begin]; Smax=Istar /. FindMinimum[L[Istar],{Istar,1}] [[2]];" " smin=s /. FindRoot[VC[s]-anought+L[s]==VC[Smax]+L[Smax],{s,0}] [[1]];" k=k+1; "Band[[k]]={Smax,smin};" vec; smin; "Plot[L[Istar],{Istar,0,12}];" "param[[k]]={vec,Smax,smin};" mu=1.02 mu; " If[ k < periods, Goto[begin]]);" "plotS=ListPlot[Transpose[Band][[1]],PlotJoined -> True];" "plots=ListPlot[Transpose[Band][[2]], PlotJoined -> True];" "Show[plotS,plots];" Pr int[param];