1 Communications in Mathematical Finance, vol. 5, no. 2, 2016, 1-11 ISSN: (pint), X (online) Scienpess Ltd, 2016 Compound Inteest Doubling Time Rule: Extensions and Examples fom Antiquities Saad Taha Baki 1 Abstact Compound inteest calculations ae used in most financial tansactions concening loans and investments. Of special inteest, is calculating the time it takes a pincipal to double at a cetain compound inteest ate. This aticle stats by discussing the famous Rule of 70 (o 72) that gives a simple estimate of the doubling time unde compound inteest. The Rule of 70 is then extended to estimate the time fo a pincipal to gow to a highe fold (tiple, quaduple, etc.) unde compounding. Finally, this aticle shows that doubling time (o to highe folds) calculations wee caied out in antiquity as evidenced by many excavated ancient cuneifom texts fom Mesopotamia (c BCE). Mathematics Subject Classification: 91G30; 01A17 Keywods: Babylonian mathematics; cuneifom; m-fold time ules; ule of 70 1 College of Business Administation, Alabama State Univesity, Montgomey, AL. USA. Aticle Info: Received: May 12, Revised: June 14, Published online : Septembe 1, 2016.
2 2 Compound Inteest Doubling Time Rule 1 Intoduction Inteest beaing accounts can gow by eaning inteest accoding to the simple inteest method o to the compound inteest method. In the simple inteest method, the inteest is paid on the initial investment (the pincipal) only, not on the inteest eaned ove time. In the compound inteest method, the inteest is paid on the pincipal and on the inteest accued ove time. Thus, compound inteest is called inteest on the inteest. While compound inteest is used in most banking and financial tansactions such as in saving accounts and loan epayments, simple inteest is used in some shot-tem financial tansactions. Although compound inteest is cuently a diving foce in most moden economies, it is by no means a moden invention. Cuneifom text tablets fom the Sumeian peiod (ca BCE) and fom the Old Babylonian peiod (ca BCE) showed concete evidence of compound inteest calculations fo tansactions between individuals and between city states. Some of those ancient examples demonstate calculations of the time (doubling time) it takes a pincipal to double (o gow to highe folds) unde compounded inteest. The doubling time of a pincipal is still of inteest in ou moden days. The pesent-day ule of 70 states that it will take 70 yeas fo a pincipal to double at an annual compound inteest ate of (as a decimal). Thus at 5% annual inteest ate, a pincipal doubles in 70 = 14 yeas In this aticle, we discuss the exact and appoximate doubling time ules unde compound inteest. Then we deive exact and appoximate mathematical extensions fo calculating the times fo a pincipal to tiple, quaduple, quintuple, etc., unde compound inteest. Two inteesting examples fom antiquities ae pesented to demonstate the ealy histoy of compound inteest calculations.
3 Saad Taha Baki 3 2 Peliminay Notes Fo the benefit of the eade, we give the basic teminology and fomulae egading pesent-day mathematics of compound inteest: P = the initial pincipal (o pesent value) that is invested, = annual inteest ate expessed in decimal fom. In pecent fom, 100% is called the annual pecentage ate (APR). n = fequency of compounding peiods pe yea. Fo example, n = 1, 2, 4, 12, 52, and 365 coesponds to annual, semi-annual, quately, monthly, weekly, and daily compounding, espectively. Also n = 1/2, 1/3, 1/4, 1/5, and 1/10 coesponds to compounding evey two yeas (biannual), evey thee yeas (tiennial), evey fou yeas (quadennial), evey five yeas (quinquennial), and evey 10 yeas (decennial). An ancient histoy example of compounding once evey five yeas is pesented in this pape. t = time of investment in yeas, F = futue value of the pincipal P, and I = total accued inteest. The basic compound inteest fomulae ae F = P 1+ n nt I = F P, (2) Fo continuous compounding (i.e., when n ), the basic fomula becomes F = P lim 1 + n n nt = P limn 1 + t n n = P[e ] t = Pe t. (3) The simple inteest method has two main fomulae: F = P(1 + t) and = Pt. (1)
4 4 Compound Inteest Doubling Time Rule 3 The m-fold Exact Time Rule The question at hand is How long does it take a pincipal to gow m-folds at an annual inteest ate of and a compounding fequency of n times pe yea? Solution: Substitute F = mp in fomula (1) to obtain mp = P 1 + n nt m = 1 + n nt = nt ln 1 + n nt = ln 1+ n in compounding peiods. In numbe of yeas, the m-fold exact time compounding fomula is t = nln 1+ n in yeas. (4) Fo continuous compounding, the m-fold exact time fomula can be deived fom (3) as follows: t = F = Pe t mp = Pe t m = e t = t ln e = t in yeas. (5) Note that the value of m can be equal to 1.5 if one equies the time fo a pincipal to gow by 50%. Of special inteest is when m = 2, which leads to the doubling exact time fomula t = ln 2 n ln 1+ n in yeas (6) Fo continuous compounding, the doubling exact time fomula becomes t = ln 2 in yeas. (7)
5 Saad Taha Baki 5 4 The m-fold Appoximate Time Rules Instead of using the doubling exact time in fomula (6) that equies logaithmic calculations, pesent-day financies use a simple ule (the ule of 70) to appoximate the equied doubling time unde compound inteest. The ule of 70 states that at an annual inteest ate of (in decimal fom), a pincipal doubles in yeas. Thus, at a 7% APR, money doubles in = 10 yeas. Sometimes people pefe using 72 instead of 70 because 72 has moe divisos than does 70. The 72 double-you-money ule becomes We will extend the ule of 70 (o 72) to ules that give the appoximate time fo a pincipal to gow m-folds (tiple, quaduple, etc.). Suppose that a pincipal P is invested at an annual inteest ate of (in decimal) compounded at a fequency of n > 0. It is equied to detemine the appoximate time fo the pincipal to gow m-folds, i.e., becomes mp, whee m 1. Recall the m-fold exact time fomula in (4), =. nln 1+ n Using the Maclauin's seies expansion (see any calculus book), we get ( / n) ( / n) 2 3 ln 1 + = ( / n) +, fo ( / n) < 1. n 2 3 ( ) ( ) ( ) 2 3 Because / n is small, we can ignoe tems involving / n, / n, etc. Thus, ln 1 + ( / n). n Theefoe, fomula (4) becomes
6 6 Compound Inteest Doubling Time Rule t =. n ( / n) n ln 1+ n The m-fold appoximate time fomula becomes t, in yeas. (8) When m=2, we get the famous appoximate double time fomula (the Rule of 70) as follows: t ln 2 = t = , o t 0.70, which leads to in yeas. (9) Note 1: The m-fold appoximate time in fomulae (8) and (9) ae indiffeent to the compounding fequency, n. Note 2: The m-fold t appoximate and the exact time fomulae fo continuous compounding ae identical: t = in yeas. Table 1 shows extensions of the Rule of 70 to m-fold appoximate time ules. Table 1: The m-fold appoximate compound inteest time ules in yeas, whee is the annual inteest ate. m m-fold appoximate time ules in yeas = 0.02 yeas = 0.05 yeas = 0.10 yeas = 0.20 yeas / 40/100 ule of / 70/100 ule of / 92/100 ule of / 110/100 ule of / 140/100 ule of / 160/100 ule of / 180/100 ule of / 195/100 ule of
7 Saad Taha Baki / 208/100 ule of / 220/100 ule of / 230/100 ule of Compound Inteest Time Calculations in Antiquities Mesopotamia (Ancient Iaq in southwest Asia) is the land whee the Sumeian, Akkadian, Babylonian, and Assyian ancient civilizations had flouished and made significant impints in the human histoy, especially in mathematics. Excavated cuneifom witings evealed that Mesopotamians handled poblems involving aithmetic and geometic pogessions, squae and cube oots of integes, quadatic and simultaneous equations, Pythagoas theoem applications, aeas and volumes of geometical figues, and compound inteest calculations. Refeences on Mesopotamian (o Babylonian) mathematics include Baqi [1, witten in Aabic], Neugebaue and Sachs , Nuegebaue , and Robson . Boye  is a geneal book on the histoy of mathematics. Roux  gives a shot histoy of Mesopotamia. In this aticle, we discuss two Mesopotamian atifacts that involve compound inteest tansactions; specifically doubling time calculations. Mesopotamians used exponential tables and thei inveses (logaithmic tables) to handle compound inteest calculations. Annual inteest ates of 10% up to 33% on loans wee common in Mesopotamia in ealy peiods. Usually, acheologists assign alphanumeic identifications to the cuneifom atifacts they excavate. Fo example, AO stands fo Antiquite` Oientales in the Louve in Pais and VAT stands fo Vodeasiatische Abteilung, Tontafeln in Staatliche Museen in Belin. In Section 5.1, we discuss cuneifom tablet AO 6770 that calculates the time fo a pincipal to double at 20% APR compounded annually. In Section 5.2, we discuss cuneifom tablet VAT 8528 that calculates the time fo a pincipal to gow
8 8 Compound Inteest Doubling Time Rule 64 folds when the fequency of compounding occus once evey five yeas (quinquennial). 5.1 Tablet AO 6770 (c BCE, in the Louve in Pais) In effect, Tablet AO 6770 asks the question: How long does it take a pincipal to double at 20% APR compounded annually? We will discuss the pesent-day, the Rule of 70, and the Mesopotamian solutions to the above question, in that ode. The Pesent-day Exact Solution: Substitute = 20% = 0.20 and n = 1 in the exact doubling time fomula in (6) to obtain ln 2 t = n ln 1 + n = ln 2 ln 2 ln = 1 ln(1.20) = = 3.80 yeas Thus, the pesent-day exact answe is 3.80 yeas, which is equivalent to 3 yeas, 9 months and 18 days. The Rule of 70 Appoximate Solution: Substitute = 0.20 in the ule of 70, fomula (9), to obtain t = 70 = 70 = 3.5 yeas Thus, the ule of 70 answe is: 3 yeas and 6 months. The Mesopotamian Solution: The Mesopotamian solution in Tablet AO 6770 educed the question to that of finding the time t in the exponential equation 2 = 6 5 t. This is equivalent to ou pesent-day fomula F = P 1 + n nt, in which = 2P, n = 1, and = 0.20 = 1 5. As evidenced by excavations, the Mesopotamians had calculated tables of powes of vaious numbes. By seaching though the powe tables of 6, they 5
9 Saad Taha Baki 9 found that 2 = 6 5 t leads to values of t falling between 3 and 4. Though intepolation between t = 3 and t = 4, thei answe was 3 yeas and 9 4 months. 9 This Mesopotamian answe is equivalent to 3 yeas, 9 months, 13 days and one-thid of a day. It can be seen that the Mesopotamian answe of 3 yeas, 9 months, 13 days and one-thid of a day is vey close the pesent-day exact answe of 3 yeas, 9 months and 18 days. Fo futhe eading on tablet AO 6770, see Cutis  and Muoi . 5.2 VAT 8528 (c BCE, in Belin) Tablet VAT 8528 deals with finding the time equied fo a pincipal invested at 20% APR to gow 64 folds when compounding occus once evey five yeas. It asks the question: If you lend one mina of silve at an annual inteest ate of 12/60 of a mina pe yea, how long does it take to be epaid as 64 minas? In VAT 8528, the Mesopotamian compounded inteest by capitalizing the inteest only when the outstanding pincipal doubled. In VAT 8528 the annual inteest ate is given as 12 = 20%. The basic simple inteest scenaio fomula is F = 60 P(1 + t). Substitute F = 2P, = 0.20 to obtain 2 = ( t). Solving, we see that doubling the standing pincipal occus evey t = 5 yeas. Pesent-day Exact Solution: Accounting fo the fact that compounding occus once evey five yeas, the question in VAT 8528 becomes: How long will it take a pincipal to gow 64-folds if compounding occus once evey five yeas at a 20% APR? Substituting m = 64, = 0.20, n = 1/5 in fomula (4), we obtain t = ln 64 = = 5 ln 64 = 5(4.1588) nln 1+ 1 n 5 ln ln = 30 yeas.
10 10 Compound Inteest Doubling Time Rule The m-fold Appoximate Time Rule: Substitute m = 64 and = 0.20 in fomula (4), the m-fold appoximate time ule to obtain t = ln 64 = = 20.8 yeas This appoximation of 20.8 yeas is vey fa off the exact answe of 30 yeas. The eason fo this goss discepancy is that the appoximation ln 1 + n n equies < 1. In ou case, = = 1.0 and ln 1 + = ln 2 = n n ae quite diffeent. The time appoximation ules may become invalid when the compounding fequency is less than once yea. The Mesopotamian Solution: The Mesopotamian solution in Tablet VAT 8528 involves educing the question to that of solving the exponential equation 2 t 5 = 64. One can aive at this equation by substituting F = 64, P = 1, = 0.2, and n = 1 5 in fomula (1) to obtain: F = P 1 + n nt t 64 = = (2) t 5. Then the Mesopotamians seached thei powe tables to look up the powe of 2 such that 2 t 5 = 64. Knowing that 2 2 = 4, 2 3 = 8, 2 4 = 16, 2 5 = 32, and 2 6 = 64, leads to t 5 = 6. Thus the solution, t = 30 yeas. Again, the Mesopotamian solution of 30 yeas coincides with the pesent-day exact time solution. Fo futhe details on VAT 8528 (and a simila tablet VAT 8521), see Neugebaue  and Muoi .
11 Saad Taha Baki 11 Refeences  T. Baqi, A Bief Histoy of the Sciences and Knowledge in the Ancient and Aabic-Islamic Civilizations (witten in Aabic), Ceatespace. USA,  O. Neugebaue and A. Sachs, Mathematical Cuneifom Texts, Ameican Oiental Society, Seies 29. New Haven,  O. Nuegebaue, The Exact Sciences in Antiquity, Second edition, Dove Publication Inc., New Yok,  E. Robson, Mathematics in Ancient Iaq: A Social Histoy, Pinceton Univesity Pess, Pinceton,  C. B. Boye, A Histoy of Mathematics, John Wiley and Sons, New Yok,  G. Roux, Ancient Iaq, Penguin Goup, New Yok,  L. J. Cutis, Concepts of the Exponential Law Pio to 1900, Ame. J. Physics, 46(9), (1978),  K. Muoi, A new Intepetation of Babylonian Mathematical Text AO 6770 (in Japanese), Kagakusi Kenkyu, 162, (1987),  O. Neugebaue, Mathematische Keilschift-Texfe, 3, ( ).  K. Muoi, Inteest Calculation in Babylonian Mathematics: New Intepetations of VAT 8521 and VAT 8528, Histoia Scientiaum, 39, (1990),