LINK TO SOURCE AND FULL TEXT: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2417143

ABSTRACT

The mathematics of continuous compounding is not limited to the valuation of continuously compounded financial instruments and flow annuities, but rather it is a versatile and robust model that may be used for valuation of all financial contracts normally encountered. It consists of a simple, consistent, coherent and conceptually appealing set of equations that apply without modification to the complete range of applications. The discrete stepwise model of compounding, by contrast, is a redundant innovation that is more complicated than the generalized model and limited in scope.

FINDINGS 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 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. It is taken for granted that sufficient computational machinery and software exist so that in practice it will not be necessary to grind out the algebra presented in this paper or in textbooks. It is proposed only that the algebra presented here is superior to the textbook approach for the purpose of teaching the concept of exponential growth in finance.

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. We can compute the value of this payment as zt = ect = e-c/m. We now price a flow annuity of EFA dollars per year for 1/m years as vt = EFA(1-e-)/c. For equal wealth we set zt = vt to get EFA = c/(1-e -c/m). The Equivalent Continuous Rate. Equate the value of wt at any value of t say t=1 with n compounding periods per year and a period rate of to get e c = (1+r)n . Take the natural logarithm of both sides and the result is c = nln(1+r).

SOME TEXTBOOK EXAMPLES

EXAMPLE 1:

Your deferred compensation plan will pay $1,000 in 5 years. What is the value of this plan today if your required rate of return is 10% per year with continuous compounding?

zt = e-ct = e-50.10 = 0.60653066. The value today is 0.606530661000 = $606.53

EXAMPLE 2:

You decide to invest $10,000 a year for 20 years in a bond mutual fund with payments spread continuously over the year. The fund pays 10.5175% APY2 . How much will your contributions be worth after 20 years? What lump sum could you invest today to have the same amount in 20 years?

c = nln(1+r) = 1ln(1+0.10517) = 0.10 zt = e-ct = e-200.10 = 0.135335 vt = (1-zt)/c = 0.8646647/0.10 = 8.646647 = w0 wt = w0e ct = 8.646647e0.1020 = 8.6466477.389056099 = 63.89056099

The present value is $86,466.47 and the future value is $638,905.61

EXAMPLE 3:

The Searcy Hat Company is planning an expansion that is expected to provide cash flows of $100,000 per year, generated continuously throughout the year as a flow annuity, for 20 years. At the end of the 20-year period, the investment can be liquidated for $200,000. The required rate of return is 10.517% APY. What is the value of this project today?

c = nln(1+r) = 1ln(1+0.10517) = 0.10 zt = e-ct = e-200.10 = 0.135335 vt = (1-zt)/c = 0.8646647/0.10 = 8.646647 The value of the project is 8.64667100,000 + 0.135335*200,000 = $891,734

EXAMPLE 4:

You invest $1,000 today in a fund that pays 10% per year continuously compounded.
How much will you have at the end of 5 years?
1/zt = e, ct = e0.10*5 = 1.64872. You will have $1,648.72

EXAMPLE 5:

Montgomery National Bank pays 10% interest on savings compounded annually. If you invest $100 today, how much will you have 20 years from now?
c = nln(1+r) = 1ln(1+0.10) = 0.09531018, wt = ect = e20* 0.09531018 = 6.727499949.

You will have $672.75

MORE EXAMPLES