The mathematics of continuous compounding is simpler than that for discrete stepwise compounding; and contrary to conventional wisdom its application is not limited to continuously discounted financial instruments but rather it is a versatile and robust model that may be used for valuation of all financial contracts normally encountered. Further, this model offers a more effective tool for teaching discounting to students of finance as it consists of a simple, consistent, and coherent set of equations that apply without modification to the complete range of applications normally covered in college courses. The discrete stepwise model of compounding, by contrast, is a redundant innovation that is more complicated and it is often a barrier to learning.

INTRODUCTION

Exponential growth is a natural phenomenon and is understood colloquially as the “snowball effect”. It occurs whenever the instantaneous rate of growth is directly proportional to accumulation. For example, bacterial population in fermenting wine behaves in this manner. The more bacteria there are, the greater are the number undergoing reproduction and therefore the greater the rate of growth. In finance we understand this behavior as “compounding”. <

But mathematical models of bacterial systems may not apply directly to financial systems because bacterial reproduction – or “compounding” – occurs at random but financial compounding is synchronized with the calendar. The bacterial system is akin to a portfolio of many zero coupon bonds that mature at random while the financial system is similar to a bacterial population in which the reproductive activities of the bacteria are synchronized. It is this distinction that has prevented financial analysts from applying the relatively simple mathematical models of exponential growth in natural systems to the same phenomenon in financial systems.

The effect of randomness is that interest payment and compounding occurs at small increments of time throughout the year; but financial contracts specify discrete interest payments only on certain dates of the calendar. Although this behavior is easily accommodated mathematically within the exponential model for natural systems, the calculus of financial compounding has been developed on an entirely different premise. It is this development that has produced the mathematics of discrete stepwise compounding we teach in business schools today as the “time value of money” or the discounted cash flow method of valuation.

In this paper we present an alternative to conventional discounting mathematics in finance by making a synchronization correction to the exponential mathematics for natural systems. In comparing this model of compounding to the conventional stepwise mathematics in finance textbooks we find that the textbook approach is a redundant innovation that adds unnecessary complexity and limitations to modeling and understanding the compounding process in finance.

We conclude that both in teaching finance and in carrying out valuation of securities in practice the exponential model offers an advantage over the stepwise compounding model. Contrary to the conventional wisdom that the exponential model only applies to securities with continuous compounding we find that the exponential model with a “synchronization correction” is a simple and robust tool that may be used for valuation of a real financial contracts including continuously compounded instruments. Most finance textbooks do not present the exponential model at all. Those that do, present it as a model that is limited to the treatment of financial contracts that are continuously compounded. We show that this is not the case.

THE BASICS OF EXPONENTIAL GROWTH

Lump Sum Contracts

Suppose we start with a fixed sum of $Wo and allow it to grow exponentially to Wt in t years at a continuously compounded rate of c% per year. Exponential growth occurs when the instantaneous rate of growth is not constant but is proportional to the current level of funds accumulated at any given time t. Algebraically we may state that

Various algebraic forms of this equation are used in practice. For example, Wt/Wo = e^ct may be used to explicitly describe Wt, the value to which an investment of Wo dollars today would grow in t years if the rate of return is c percent per year. In fact, all valuation relationships in finance are derived from equation 1. Two useful forms are shown below.

1/z = e^ct, and ln(1/z) = ct

Flow Annuity Contracts

In the exponential model of compounding, cash flow streams may be thought of as a continuous flow of funds. An annuity is a cash flow stream that flows at a constant rate.If the constant rate is $d per year then, applying equation 2, the discounted value at time zero of an incremental amount of flow during dt years may be written as

Wo is then computed as the sum of all of these incremental values by integration.

Expressing Wo/d as v(t) and e^-ct as z(t) we may write

That is, we would pay v dollars today for continuous flow payments of $1 per year over a contract period of t years. Equations 1 and 2 may now be used as the basic building blocks for the pricing of all patterned streams of cash flows.

Stepwise Annuity Contracts

Suppose that an annuity pays $1 per payment m times a year in chunks at equal intervals rather than as a continuous flow. It can be shown that an equivalent “flow annuity” exists and its value may be derived from equation 1 as

That is, discrete stepwise payments of $1 per period m times a year has the same wealth effect as a continuous flow annuity of $c/(e^c/m-1) per year. The proof for this equality is provided in the Appendix. Substituting this value into the flow equation we get

Here we define u(t) as the value of discrete stepwise payments of $1 to be paid m times a year for t years. We may now use this relationship to price all patterned “chunky” cash flows that may be modeled as discrete stepwise annuities.

For example, coupon bond contracts may be priced as a complex security that consists of an ordinary annuity and a zero coupon bond and the bond price as a fraction of face value may be written in terms of equations 1 and 4 as

Perpetuities are annuities in which t is not bounded. Since z approaches zero as t goes to infinity, the value of perpetuity may be expressed as

That is, we would pay u dollars today for each dollar to be delivered m times a year forever. Preferred stocks may be valued as perpetuity as they are like the annuity portion of bonds but the contract does not expire and there is no balloon payment pending at any time in the future.

Contracts with Increasing Annuity Payments

Consider a flow annuity in which the annuity paid per year grows at a constant rate of g% per year. That is, the annuity at time t is given by

Substituting this relationship for d we may re-write the differential equation for flow annuities as

Collecting terms we may re-write this relationship as

Integration yields

Here v_t is the value of a flow annuity with an initial rate of $1 per year but growing at g% per year. The valuation is similar to equation 2 but with (c-g) rather than c as the discount rate. The “equivalent flow annuity” term may then be written as

Therefore the value of a chunky annuity in which the payment amount grows at g% per year is given by

We would pay u dollars today to receive $1 next period and further payments m times a year growing at g% per year for a contract term of t years.

Common stock valuation differs from preferred stock valuation by virtue of the investor’s ability to participate in the growth of the firm. Its valuation therefore represents perpetuity of constant growth dividends that may be expressed as

The equation states that if the next dividend payment is expected to be $1, and if there are m dividend payments per year, and if dividends grow at g percent per year continuously forever, then we would pay u dollars today for the common stock.

STEPWISE COMPOUNDING

The rate of return c in equations 1 through 5 may be interpreted as a “continuously compounded” rate; that is, growth of funds due to interest earnings occurs at every interval of time however small. In real financial contracts, for various spurious reasons, this is not the case. Each contract arbitrarily specifies a “step” size in time during which growth is not permitted to occur. The typical step size is one month, three months, six months, or one year. At the end of each step, an accumulated growth amount is instantaneously added. For our model of exponential growth to be directly applicable to these types of financial contracts, it must be faithful to this start-stop nature of compounding.

A simple method of modeling the stepwise growth behavior is to develop and apply a generalized relationship between the stepwise rate of return r and the continuous rate of return c so that for any quoted stepwise rate an equivalent continuous rate may be computed. Once the equivalent continuous rate is available the continuous exponential model may be used for valuation purposes.

In fact the relationship between c and r is a simple one. In general, if there are n steps per year, and the rate of return is r per step then the relationship between r and c may be written as

For any conventional stepwise rate of return quoted as r or as APR = n*r, an equivalent continuous rate c may be inferred using equation 6 and the continuous compounding model shown in Table 1 may be applied directly for all valuation purposes. No additional complexity is necessary to accommodate stepwise compounding.

THE PROPOSED DISCOUNTING MODEL

Based on the arguments presented above we propose a wholly new approach to compounding models in finance. We demonstrate that the new model is easier to use than conventional discrete compounding and that it is general in scope since it applies directly to all forms of financial contracts including those that use flow annuities and continuous compounding.

Summary of the Proposed Model

The working equations of the proposed model are summarized in Table 1 and the notation used in the model is summarized in Table 2. The model is compact and simple and yet versatile in its application to real financial contracts. It requires the additional rate conversion step in all cases and this step is not always necessary in conventional stepwise compounding models but there is a significant payoff in coherence and comprehension.

Lump Sum Contracts z_t = e^-ct
Annuity Contracts u_t = (1- z_t)/(e^c/m-1
Growing Annuities u_t = (1 - e^-(c-g)t )/( e^(c-g)/m)
Perpetuities u = 1/(e^c/m-1
Growing Perpetuities u = 1/(e^(c-g)/m-1)
Rate Conversion c = n*ln(1+r) and r = e^c/n-1

c = continuously compounded rate of return in % per year
r = periodically compounded rate of return in % per period
n = compounding periods per year
m = periodic payments per year
t = time in years
z(t) = value of a dollar to be paid t years from now
u(t) = value of an annuity of $1 per period m times a year for t years
v(t) = value of a flow annuity of $1 per year for t years

By way of contrast, the conventional and therefore the textbook approach to stepwise compounding is altogether different. It begins by re-writing equation 1 for stepwise growth as W(step x) = W(step x-1)*(1+r). The resulting equations are more difficult to apply in practice particularly when m and n are not both equal to one year or to each other or when the financial contract specifies continuous compounding or continuous flow annuities

The continuous exponential compounding model presented in Table 1 is simple and robust. In these equations we see that m and n are independent of each other and any combination is possible without increasing the complexity of the model and without the need for case specific equations. Values of m and n that are less than one are also easily accommodated. The resultant model is completely general in scope and the equations are easily rendered explicit in a number of different variables that are often the object of financial analysis. We now demonstrate the simplicity and lucidity of the model in pricing an assortment of financial instruments.

Application of the Proposed Discounting Model

Example 1: Lump Sum Contracts

We wish to price a T-Bill that pays $10,000 in 70 days and yields 5% per year with annual compounding. First we compute c = ln(1.05) = 0.04879 and t in years as t = 70/365 = 0.19178 years. We can now evaluate Wo = 10,000*e^{-0.04879*0.19178} = $9906.87

But what if the compounding period were, say, 90 days instead of one year? There are 365/90 or 4.0555 of these periods per year so n=4.0555. If the quoted 5% rate is the APR then the 90-day rate is 5%/4.0555 or 0.012329 and the equivalent continuous rate is c = 4.0555*ln(1.012329) or 0.04969. The T-Bill pricing may therefore be computed as 10,000*e^{-.04969*0.19178} or $9905.16

Alternately, we might want to know what rate of return we would earn if the T-Bill were quoted at $9900. The rate is c = ln(9900/10000)/t = 0.0524. The equivalent 90-day compounded rate is r = e^0.0524/4.055-1 or 0.013 and so the corresponding APR quote would be 4.055*0.013 or 5.274% and the APY quote is given by r = e^.0524-1 or 5.38%.

A $20,000 car loan is to be paid over a period of 5 years in 60 equal monthly payments. The interest on the loan is quoted as 9% APY. The value of c is ln(1.09) or 8.6177%, t is 5 years, and m is 12; so we may compute e^-ct = 0.64993 and e^c/m = 1.007207. Wo/d = 0.35007/0.007207 or 48.5736. Therefore the payment is 20,000/48.5736 or 411.7462 or $411.75.

Alternately we might compute how much the buyer would have in the bank in five years if she saves 411.7462 per month for 5 years and earns 9% APY. Wt = 20,000/0.64993 or $30,772.54.

Suppose that the buyer wishes to pay weekly at a rate of $100 per week. How long will it take to pay off the car at the same rate of interest and without changing the compounding period? We set Wo/d to 20000/100 or 200 and m to 52. Since c*t must equal 0.377445 and so t must be 4.6212 years or 240.3 weeks.

This example is used to illustrate the utility of this model in the valuation of cash flows with complex compounding and payment streams. Suppose that a consumer with a $10,000 credit card balance makes weekly purchases of $100 and makes monthly payments of $500. The issuer charges 14% APR compounded every 25 days. What is the balance on the account after 3 years? First, we compute the 25-day period rate as 0.14*25/365 = 0.009589. Setting n=365/25=14.6 we find c = 14.6*ln1.009589 = 0.13933.

With the value of c in hand we can now set up the credit card problem as a sum of two annuities and one lump sum. At t=3 years, the lump sum of $10,000 is worth 10,000*e^3*0.13933 = $15,189. The purchases form an annuity of $100 with m=52 and its value at t=3 years is 100*(e^3*0.13933 –1)/ (e^0.1393/52 –1) or $19,341. The monthly payments at m=12 are worth 500*(e^3*0.13933 –1)/ (e^0.1393/12 –1) or $22,221. The year 3 balance is 15,189+19,341-22,221 or $12,309.

To maintain her $10,000 balance, the consumer’s monthly payments must be 500*(22,221+2309)/22,221 or $551.96. For a zero balance she must pay 500*34530/22221 = $776.97. The solution is simple and clear in the exponential model but not so in the stepwise textbook model.

A Treasury makes coupon payments of $4 every six months and yields 5% APY. It has 10.75 years to maturity at which time a redemption value of $100 will be paid in addition to the coupon payment. First we note that the 5% APY corresponds to a c-value of ln(1.05) or 0.04879. The corresponding z and u values are z = e^{-10.75*0.04879} = 0.591855, u = (1-z)/(e^0.04879/2 –1) = 16.5274

Since the coupon rate per period is 4% the price of the bond is Bo = 0.591855 + 0.04*16.5274 = 1.25295 times $100 or $125.395.

Bond yield computations require an iterative procedure. Suppose that the bond above is available for $120. Since the coupon rate is 4% twice a year, the bond price will be $100 at c= 2*ln1.04 or 7.844%. Interpolating between 7.844% and 4.879%, our first guess is 4.879+0.1172*(125.2951-120) = 5.5 %. At c=5.5% we compute bond price = 120.755. Our second guess is 5.5% + 0.1172*0.755 = 5.588% and the computed bond price is 119.98.

Suppose that the bond in Example 4 makes the 8% annual coupon payment as a flow annuity. In that case we compute v = (1-z)/0.04879 = 8.36534. The bond price is 0.591855 + 8.36534*0.08 or 1.26108 times face value or $126.108.

A young couple wishes to take out a $300,000 home mortgage at an APY of 7.5%. Payments are to be made monthly for 30 years. We compute n=1, c = ln1.075 or 0.07232, m=12, z= 0.114221, e^c/m = 1.00604486, and u = 146.534. Therefore the payment is 300,000/146.534 = 2047.303 per month.

Suppose they agree to increase the payment by say 2.5% per year. The equivalent continuous rate of growth in payments is g = 0.0246926 and g-c=0.047627 and u = 0.760406/0.0039768 = 191.211. The first year payments will be 300,000/191.211 = 1568.947 per month. In the 30^th year, when the couple’s income is expected to be higher, their payments will be 1568.947*e^0.0246926*30 = 3290.97per month.

A common stock pays dividends of $1 every quarter and we expect these dividends to grow at a rate of 1.5% per quarter. Our required rate of return from this stock is 16% APY. To price this stock we first compute c = ln(1.16) or 14.842% and the continuous growth rate in dividends as g=4*ln(1.015) = 5.955%. The stock price is = 1/(exp(0.08887/4)-1) = $44.51. At zero growth rate and a perpetuity of $1/quarter dividends, the price would be 1/exp(0.14842/4 –1) = $26.45

When the yield curve is not flat coupon bond pricing must be consistent with the strip market to reach a state of equilibrium where no arbitrage is possible between the coupon market and the strip market (Varian 1987). To check for equilibrium, each coupon must be priced according to the yield curve. Equation 3 does not apply in this case and Equation 1 must be used repeatedly for each payment in the cash flow stream.

Suppose a Treasury with a face value of $100 matures in 5 years and makes annual coupon payments of $10. The yield curve is upward sloping. Strip maturing in [1, 2, 3, 4, 5] years yield [2, 3, 4, 5, 6]% APY. Equation 2 must be applied to each payment. First we compute c = [0.0198, 0.02956, 0.03922, 0.04879, 0.05827] and use Equation 2 to compute the value of $1 delivered under each of the 5 conditions as z = [0.9804, 0.9426, 0.8890, 0.8227, 0.74725]. For payments of $[10, 10, 10, 10, 110], equilibrium price of the bond is 118.5445.

The equilibrium values of interest rate futures quotations are determined by the yield curve because any divergence between these values provides arbitrage opportunities (Taggart 1996). Suppose that the yield curve is given by strip yields of [c1, c2, c3]% for terms of [t1, t2, t3] years and interest rate futures are quoted at continuously compounded rate of f% for term = (t3-t2) years for strips to be delivered t2 years from now. We set up the arbitrage as shown below:

Spot leg: invest $1 in t3 strips for t3 years for W(t3) = $ e^c3*t3
Futures leg: go short $1 in t2 strips for t2 years and at the same time sell (t3-t2) year futures for delivery t2 years from now for an obligation in year 3 of W(t3) = $ e^c2*t2 * e^f*(t3-t2)

Here f is the continuously compounded interest rate on the futures contract. We find the equilibrium value of f by setting arbitrage profits to zero and solving for f.

e^c2*t2 * e^f*(t3-t2) = e^c3*t3
c2*t2 + f*(t3-t2) = c3*t3
f = (c3*t3 – c2*t2)/(t3-t2)

As a numerical example consider the yield curve given by r = [4.0, 4.5, 5.0]% APY for term = [1, 2, 5] years respectively. The equilibrium futures quotation f for a 3-year strip to be delivered 2 years from now may be computed as follows:

The equivalent continuous rates are given by ln(1+r) as c = [0.03922, 0.04402, 0.04879] and so f = (0.24395 – 0.088034)/3 = 0.05197. The equivalent APY quote is e^c/n –1 = e^0.05197 - 1 = 0.05334

When learning the principles of NPV analysis for capital budgeting purposes, the student is faced with a stream of cash flows that do not follow a given pattern. No shortcut algebraic solution exists in these cases. These cash flows may be priced one at a time by repeatedly applying Equation 1. We show in the example below that both NPV and IRR may be easily computed using the valuation model presented in this paper.

A project with a life of 5 years requires an initial investment of $100 and offers net cash flows of $[20, 25, 25, 30, 30] in years [1, 2, 3, 4, 5]. Our term-dependent required rate of return from projects of this nature are [10, 10, 11, 11, 12]% APY. To evaluate this project we first compute c = [0.0953, 0.0953, 0.1044, 0.1044, 0.1133]. The corresponding z-values are [0.9091, 0.82646, 0.7311, 0.6586, 0.5675]. The value of the cash flows is therefore = 18.02+20.66+18.278+19.758+17.025 = 93.74. The NPV is -6.26.

Alternately, one may compute the value of the cash flows at time t = Wt = [20, 25, 25, 30, 30] * [e^4*0.1044, e^3*0.1044,e^2*0.0953, e^0.0953*1,1] = 30.366+34.196+30.25+33+30 = 157.8117.

The computation of IRR requires an iterative procedure. As a first guess we may use c = ln(157.8117/100)/5 = 0.09123. The corresponding 1/z-values are [1.4404, 1.3148, 1.200, 1.0955, 1.00]. The new Wt value is 154.543 and our next guess is c=ln(154.543/100)/5 = 0.08706. The corresponding z-values are [0.91662, 0.84018, 0.77014, 0.70593, 0.64707]. The computed investment amount is 99.18. The target is 100.

The exponential model accommodates negative interest rates in a simple and symmetrical manner. For example if c is a negative 5%, t is 10 years, and m is 12, then the value of z is e^-(-0.05)*10 = e^0.5 = 1.648, that is, you must invest $1.648 today to receive $1 ten years from now.

The value of u is (1-1.648)/(e^-0.05/12) = 0.648/-0.004158 = 156.844, that is we must pay $156.844 today to receive 10*12 = 120 monthly payments of $1. At zero interest the value of the annuity would be m*t or $120.

SUMMARY AND CONCLUSIONS

Because financial contracting involves stepwise compounding and discrete cash flow streams it is thought that the exponential model does not apply because it subsumes continuous compounding. Conventional wisdom is that an entirely new approach to valuation is necessary.

We show in this paper that this is not so because for any financial contract that offers a stepwise rate of return r, there exists an equivalent continuous rate of return c, and for any discrete annuity there exists an equivalent flow annuity; and there are simple relationships between the continuous compounding parameters and the discrete compounding parameters.

The continuous compounding model is not limited to pricing continuously compounded securities but in fact the model is more general in scope than the discrete model. We show that the continuous exponential model is robust and simple and is easily applied to all forms of financial contracting for valuation purposes.

The simple and consistent set of equations presented here apply without modification under all combinations of payment and compounding periods and may be used to price a wide variety of financial contracts.

REFERENCES

Beninga, Simon, Numerical techniques in finance, MIT Press, Cambridge, MA, 1989

Black, Fisher and Myron Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, May-June, 1973

Brealey, Myers, and Marcus, Fundamentals of corporate finance, 3rd edition, Irwin McGraw Hill, 2000

Brigham and Gapenski, Intermediate financial management, 5th Edition, Dryden Press, 1996

Finnerty, John, An overview of corporate securities innovation, Continental Bank Journal of Applied Corporate Finance, Winter 1992

Rubenstein, Mark, Derivative assets analysis, Journal of Economic Perspectives, Fall 1987

Sundaresan, Suresh, Fixed income markets and their derivatives, South Western College Publishing, Cincinnati, Ohio, 1997

Taggart, Robert Jr., Quantitative analysis for investment management, Prentice Hall, Upper Saddle River, NJ, 1996

Varian, Hal, The arbitrage principle in financial economics, Journal of Economic Perspectives, Fall 1987

APPENDIX

The Equivalent Flow Annuity

Consider a chunky annuity that pays $1 per period m times a year at equal intervals of time. After the first period, 1/m years have passed and $1 has been received. Using equation 1 we compute the value of this payment as z = e^-ct = e^-c/m. Using equation 2 we may price a flow annuity of EFA dollars per year for 1/m years as v = EFA*(1-e^-c/m)/c

For equal wealth we set z = v and solve for EFA = c/(1-e^-c/m).

Equate the value of Wt at any value of t say t=1 with n compounding periods per year and a period rate of r.

e^c = (1+r)ⁿ

Take the natural logarithm of both sides

c = n*ln(1+r)

As x approaches zero from the positive side, i.e., as x becomes a smaller and smaller positive number, the function e^x approaches 1+x; and as x becomes infinitely large e^x also goes to infinity and therefore e^-x goes to zero. Applying these limits to equation 2 and 3 we find that
Limit (c->zero): z -> 1+ct -> 1
Limit (c->infinity): z -> 0
Limit (c->zero): u -> (1-1+ct)/(1+c/m –1) = ct/(c/m) = mt
Limit (c->infinity): u -> (1-0)/(infinity) = 0

That is, as long as interest rates have positive values, z is bounded by 0 and 1 and u is bounded by 0 and m*t. These results are consistent with our intuition.