# Cash Flow Discounting A NEW CASH FLOW DISCOUNTING METHOD

Chaamjamal, Thailand, 2012

EQUATIONS ARE MORE LEGIBLE IN THE SOURCE DOCUMENT [LINK]

1. ABSTRACT. Here we show that 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 step-wise model of compounding, by contrast, is a redundant innovation that is more complicated than the generalized model and limited in scope.
2. 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 (Jackson, 2008). The more bacteria there are, the greater are the number undergoing reproduction and therefore the greater the rate of growth. In finance we understand the dynamics of this behavior as “compounding”. The difference between bacterial systems and financial systems is that 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.
3. The effect of randomness is that interest payment and compounding occur at small increments of time throughout the year. In contrast, most financial contracts specify discrete interest payments that are made only on certain dates of the calendar in chunks. 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 (Lewin, 1970). It is this arithmetical innovation of the 17th century that has preserved for us the mathematics of discrete step-wise compounding we teach in business schools today as the “time value of money” or the discounted cash flow method of valuation (Brealey, 2000). 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 step-wise 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 dynamics of exponential growth.
4. Both in teaching finance and in carrying out valuation of securities in practice the exponential model offers an advantage over the step-wise compounding model. Contrary to the conventional wisdom that the exponential model only applies to securities with continuous compounding (Bodie, 2000), we find that the exponential model is a simple and robust tool that may be used for valuation of a a wide range of real financial contracts including continuously compounded instruments. Yet, 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.
5. Suppose we start with a fixed sum of \$w0 and allow it to grow exponentially to wt in t years at a continuously compounded rate of c% per year. We can compute zt, the value today of one dollar received t years from now according to Equations 4. Other useful forms of the equation are shown below. The derivation of the exponential growth model may be found on Wikipedia (Wikipedia, 2014).
Equation 1 wt = w0ect
Equation 2 w0 = wte-ct
Equation 3 zt = w0/wt
Equation 4 zt= e-ct
Equation 5 1/zt = ect
Equation 6 c*t = ln(1/zt)
6. Equations are more legible in the source document [LINK]
That we would pay zt dollars today for each dollar to be delivered t years from now if we expect our investment to grow at c percent per year is the foundation relationship for valuation in all of finance. We now show how this relationship is used to price all forms of financial contracts covered in college textbooks (Bodie, 2000).
7. Flow annuities. In the exponential model of compounding, cash flow streams may be thought of as a continuous flow of funds like water flowing in river. In most capital budgeting applications, sales, and cash flows often occur throughout the year in the form of a flow. A flow annuity is a cash flow stream that flows at a constant rate. If the constant rate is \$f per year then, applying equation 2, the discounted value at time zero of an incremental amount of flow that will occur during a small increment of time = dt years, t years from now may be written as
Equation 7 dw0 = e-ctfdt
We can compute the value of w0 as the sum of all of these incremental values by integration.
Equation 8 wo = f*(1- e-ct)/c
Expressing w0/f as vt and e-ct as zt we may write
Equation 9 vt = (1- zt)/c
That is, we would pay vt dollars today for continuous flow payments of \$1 per year over a contract period of t years. Equations 4 and 9 may now be used as the basic building blocks for the pricing of all patterned streams of cash flows in any combination of lumps and flows. The future value of this annuity at the end of the annuity period is
Equation 10 wt = vt*ect = (ect – 1)/c
8. Chunky annuities. 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 there exists an equivalent “flow annuity” with value equal to that of the chunky annuity.
Equation 11 f = c/(ec/m-1)
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/(ec/m-1) per year. The proof for this equality is provided in the Appendix. Substituting this value into the flow equation we get the value today of a contract that pays \$1 per payment m times a year for t years.
Equation 12 ut = (1- zt)/(ec/m-1)
Here we define ut as the value of discrete stepwise payments of \$1 to be paid m times a year for t years. This relationship may be used 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 4 and 11 as B0 = zt + ut*(period coupon rate).
9. 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:
Equation 13 u = 1/(ec/m-1)
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 a perpetuity because 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.
10. Growing flow annuity. Consider a flow annuity in which the annuity paid per year grows at a constant rate of g% per year continuously throughout the year. That is, the annuity flow rate p at time t is given by
Equation 14 pt= p0egt
Substituting this relationship for annuity flow we may re-write the differential equation for flow annuities as
Equation 15 dw0 = p0*egt * e–ct * dt = p0* e–(c-g)t * dt
Integration yields
Equation 16 vt = (1-e-(c-g)t)/(c-g)
Here vt is the value of a flow annuity with an initial rate at year zero of \$1 per year that grows at g% per year. The valuation is similar to equation 9 but with (c-g) rather than c as the discount rate. It should be noted that stepwise growth rates encountered in textbooks must be converted to continuous growth as we shall see.
11. Growing chunky annuities: If the growing annuity is chunky we can transform it to the “equivalent flow annuity” according to Equation 11
Equation 17 f = (c-g)/( e(c-g)/m)
Therefore the value of a chunky annuity in which the payment amount grows at g% per year is given by
Equation 18 ut = (1 – e-(c-g)t)/( e(c-g)/m)
We would pay ut dollars today to receive \$1 in the 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
Equation 19 u = 1/(e(c-g)/m-1)
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 willing to pay u dollars today for the common stock.
12. Step-wise compounding. The rate of return c in equations 1 through 17 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 most 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 interest amount is instantaneously added and capitalized. 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 simple and well known (Seitz, 2005). 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
Equation 20 c = n*ln(1+r)
Equation 21 r = ec/n – 1
The derivation of this relationship is shown in the Appendix. For any conventional stepwise rate of return quoted as a periodic rate r or as APR = n*r, an equivalent continuous rate c may be inferred using equation 18 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 growth rate quoted in textbooks normally assume a step growth at the end of the growth period normally one year. The g parameter of growth in our model is a continuous growth rate. The conversion between stepwise growth rate h and continuous growth rate g follows the same format.
Equation 22 g = n*ln(1+h)
Equation 23 h = eg/n -1
13. Equations are more legible in the source document [LINK]
14. PROOFS
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 = e-ct = 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 ec = (1+r)n. Take the natural logarithm of both sides and the result is c = n*ln(1+r).
15. SOME TEXTBOOK EXAMPLES: Equations more legible in source document [LINK]
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-5*0.10 = 0.60653066. The value today is 0.60653066*1000 = \$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 = n*ln(1+r) = 1*ln(1+0.10517) = 0.10
zt = e-ct = e-20*0.10 = 0.135335
vt = (1-zt)/c = 0.8646647/0.10 = 8.646647 = w0
wt = w0ect = 8.646647*e0.10*20 = 8.646647*7.389056099 = 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 = n*ln(1+r) = 1*ln(1+0.10517) = 0.10
zt = e-ct = e-20*0.10 = 0.135335
vt = (1-zt)/c = 0.8646647/0.10 = 8.646647
The value of the project is 8.64667*100,000 + 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 = ect = e0.10*5 = 1.64872. You will have \$1,648.72
2 Annual Percentage Yield
A NEW DISCOUNTING MODEL FOR TEACHING FINANCE, JAMAL MUNSHI, 2014 8
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 = n*ln(1+r) = 1*ln(1+0.10) = 0.09531018
wt = ect = e20* 0.09531018 = 6.727499949. You will have \$672.75
Example 6: A financial contract offering a lump sum payment of \$4,661 twenty years from now is priced at \$1,000 today. What rate of return will you earn if you invest in this instrument?
ect = 4661/1000 = 4.661
c*t = ln(4.661) = 1.539230017
c = 1.539230017/20 = 0.076961501 (continuous compounding)
APY = ec/1 -1 = e0.076961501 – 1 = 0.08 (annual compounding)
Example 7: An investment of \$1,000 is expected to earn 8% APY. How long will it take for this investment to grow to \$4,661?
c = n*ln(1+r) = 1*ln(1+0.08) = 0.076961501
ect = 4661/1000 = 4.661
c*t = ln(4.661) = 1.539230017
t = 1.5392300/0.076961501 = 20 years
Example 8: A financial contract offers \$20,000, t=5 years from now. The offer price is \$10,000 and our required rate of return is 10% API. Is this a good investment ?
c = n*ln(1+r) = 1*ln(1+0.10) = 0.09531018
zt = e-ct = e-5*0.09531018 = 0.620921323
The value of this investment is 0.620921323*20,000 = 12418.42646
Since the value exceeds the offer price it is a good investment.
Example 9: You invest \$1,000 at the end of each year for 40 years into a retirement fund that offers a rate of return of 10% API. How much will you have at the end of 40 years?
c = n*ln(1+r) = 1*ln(1+0.10) = 0.0953101
wt = (ect -1)/c = (e0.0953101*40 -1)/0.0953101 = 464.3695799 for a flow annuity
equivalent flow annuity = c/(ec-1) = 0.953101
future value for chunky annuity = 0.953101*464.36957 = 442.5911015*1000
Example 10: An investment offers 10% APY. How much should we invest at the end of each year to have \$1 million at the end of 5 years?
c = n*ln(1+r) = 1*ln(1+0.10) = 0.0953101
the future value of the flow annuity of \$1 at the end of 5 years is (ect – 1)/c = (e0.0953101*5 – 1)/0.0953101 = 6.405506749
the future value of the chunky annuity is 6.405506749*c/(ec/m – 1) = 6.1051
The needed payment is 1,000,000/6.1051 = 163797.4808.
A NEW DISCOUNTING MODEL FOR TEACHING FINANCE, JAMAL MUNSHI, 2014 9
Example 11: If we invest \$1,000 at the end of each year for 10 years and receive \$19,337, what rate of return did we earn?
We do this by trial and error as we need c to compute c. First we guess a value of c, say 10%. Now we can compute the equivalent flow annuity EFA = c/(ec/m -1) = c/(ec -1) since m=1 period per year. Next we compute the target ratio of the future value to the equivalent flow annuity payment as 19.337/EFA. And finally the actual ratio of FV/EFA = (ect -1)/c = (e10*t -1)/c since t=10 years. Now e we compare our 2 values of FV and guess again until the two values are close enough. For example:
Guess c=10%: c/(ec -1) = c/(ec -1)= 0.950833 and so 19.337/0.950833 = 20.3369 and (e10*t -1)/c = 17.1828. Since 17.1828 ≠20.3369, we must guess again
Guess c=12%: c/(ec -1) = c/(ec -1)= 0..9412 and so 19.337/0..9412 = 20.545 and (e10*t -1)/c = 19.334. Since 19.334 ≠20.545, we must guess again
Finally at a guess of 13.1%, c/(ec -1) = c/(ec -1)= 0.0.93596 and so 19.337/0..93596 = 20.66 and
(e10*t -1)/c = 20.65. And we decide to accept that guess.
Converting the continuous compounding rate to annual compounding we get
r = ec/n – 1 = ec -1 = 14%
Example 12: A “\$20 million” lottery winner receives \$1 million immediately and then \$1 million at the end of each of the following 19 years. At 10% API, what is the present value of the prize?
c = n*ln(1+r) = 1*ln(1+0.10) = 0.0953101
zt = e-ct = e-19*0.09531 = 0.163507991
ut = (1-zt)/(ec/m -1) = 0.836492009/0.1 = 8.364920091
Example 13: A business opportunity requires an initial investment of \$475,000. the useful life is 10 years. If we receive equal net cash flows at the end of each of the next 10 years, what must these cash flows be for us to realize a 10% rate of return?
c = n*ln(1+r) = 1*ln(1+0.10) = 0.09531
zt = e-ct = e-10*0.09531 = 0.385543289
ut = (1-zt)/(ec/m -1) = 0.614456711/0.1 = 6.14456711
Constant cash flow required at the end of each year = 475,000/6.144579 = 77303.91
Example 14: A business opportunity offers \$1,000 at the end of each year from year 4 to year 20. If our required rate of return is 10% APY, what is the present value of this investment?
First compute the value at the end of year 3.
c = n*ln(1+r) = 1*ln(1+0.10) = 0.09531
zt = e-ct = e-17*0.09531 = 0.197844669
ut = (1-zt)/(ec/m -1) = 0.802155331/0.1 = 8.02155331 three years from now
The value today is 8.021568443* e-3*0.0953 = 6.026723098 * 1,000 = 6,026.7231
A NEW DISCOUNTING MODEL FOR TEACHING FINANCE, JAMAL MUNSHI, 2014 10
Example 15: A business opportunity offers an annuity of \$25,000 from years 2 through 10 and an additional lump sum of \$80,000 in year 10. It requires initial investments of \$100,000 in year zero and \$50,000 in year one. All cash flows are received and investments made at the end of the year. If our required rate of return is 10% APY, what is the present value of this investment?
The value of the annuity at the end of year 1:
c = n*ln(1+r) = 1*ln(1+0.10) = 0.09531
zt = e-ct = e-9*0.09531 = 0.424097618
ut = (1-zt)/(ec/m -1) = 0.575902382/0.1 = 5.75902382 at the end of year 1
It’s value today is 5.759034682*e-0.09531 = 5.235486076 times 25000 = 130887.1519
The value of 80,000 received 10 years from now = 80000*e-10*0.09531 = 30843.46312
The value of -50000 1 year from now is -50000*e-0.09531 = -45454.54545
The value of the project is -100000-45454.54545+30843.46312+130887.1519 = 16276.06957
Example 16: Find the present value of an investment that includes (1) -40,000 investment in year 0,
(2) -6000 investment at the end of year 5, (3) +50,000 cash flow in year 7, and (4) an annuity of cash flows of 8,000 in years 1 through 7. The required rate of return is 10%
c = n*ln(1+r) = 1*ln(1+0.10) = 0.09531
z5 = e-0.09531*5 = 0.620921323
z7 = e-0.09531*7 = 0.513158118
u7 = (1-z7)/(ec/m -1) = 0.486841882/0.1 = 4.86841882
Present value = 4.868428004*8000 +0.513158118*50000 – 0.620921323*6000-40000 = 20879.80199
Example 17: A credit union pays 10% APR3 on savings compounded semiannually. If you deposit \$100 today what will be your balance 3 years from now?
r=5% per period, n=2 compounding periods per year
c = n*ln(1+r) = 2*ln(1+0.05) = 0.097580328
1/z3 = e0.097580328*3 = 1.340095641
Example 18: A financial contract offers \$10,000, 10 years from now. Your required rate of return is 16% APR compounded quarterly. What is the equilibrium price of this contract?
r=4% per period, n=4 compounding periods per year
c = n*ln(1+r) = 4*ln(1+0.04) = 0.156882853
z10 = e-0.156882853*10 = 0.208289045
3 Annual percentage rate = periodic rate x the number of periods per year
A NEW DISCOUNTING MODEL FOR TEACHING FINANCE, JAMAL MUNSHI, 2014 11
Example 19: A loan requires monthly payments of \$460 for 20 years. The interest rate is 8% APR. What is the present value of this loan?
r=.08/12 = 0.006667 per period, n=12 compounding periods per year, m=12 payments per year
c = n*ln(1+r) = 12*ln(1+0.0066667) = 0.07973491
z20 = e-0.07973491*20 = 0.202969776
u20 = (1-z20)/ (e0.07973491/12 -1) = (1- 0.202969776)/ 0.07973491 = 9.996000798
EFA = The equivalent flow annuity = c/(ec/m -1) = 0.07973491/(e0.07973491/12 -1) = 11.9601767
Value of the annuity = 9.996000798*11.9601767 = 119.5539358 per dollar of payment
460*119.5539358 = 54994.81047
Example 20: A firm invested \$33.33333 million per year into a new project continuously over a period of 3 years. What is the value of this investment at the end of year 3 if the required rate of return is 10% APY. (This example is a case of a flow annuity that is normally approximated with chunky annuities in conventional textbook examples. Here is the flow annuity solution.)
c = n*ln(1+r) = 1*ln(1+0.10) = 0.09531
wt = (e0.09531*3 -1)/c = (1.331-1)/ 0.09531 = 3.472877977
Example 21: A preferred stock pays an annual dividend of \$7.80. What is the equilibrium price of this stock if our required rate of return is 7.46%?
c = n*ln(1+r) = 1*ln(1+0.0746) = 0.071948499
u = 1/(ec/m -1) = 1/(e0.071948499/1 -1) = 13.40482574
Example 22: A common stock is expected to pay dividends of \$4 a share at the end of the first year. Thereafter dividends are expected to grow in annual steps at a rate of 5% per year forever. The required rate of return is 10% APY. Compute the value of this stock.
c = n*ln(1+r) = 1*ln(1+0.10) = 0.09531
g = n*ln (1+h) = 1* ln(1+.05) = 0.04879
c-g = 0.09531 – 0.04879 = 0.04652
u = 1/(e(c-g)/m -1) = 1/(e(0.04652)/1 -1) = 21.00000722
price today = 21.00000722 per dollar of dividends D0
D0 = 4/1.05 = 3.80952381
Share price = \$ 3.80952381*21.00000722 = 80.00002751
Example 23: A corporate bond with 2 years to maturity makes interest payments of \$40 every 6 months. If investors’ required rate of return is 6% APR from this investment what is the equilibrium price of the bond?
m=2 payments per year
n=2 compounding periods per year
r = 3% per period
c = n*ln(1+r) = 2*ln(1+0.03) = 0.059117604
z2 = e-2c = e-2*.059117604 = 0.888487048, and u2 = (1-z2)/ (ec/m -1) = (1-0.888487048)/ (e0.059117604/2 -1) = 3.717098403. Bond value = 1000*0.888487048 + 40*3.717098403 = 1037.170984
A NEW DISCOUNTING MODEL FOR TEACHING FINANCE, JAMAL MUNSHI, 2014 12
Example 24: A certain acquisition offers managers cash flows of \$700 million at the end of each year forever. The cost of the acquisition is 5,100 million. What is the net present value of the target firm if (a) they do not expect cash flows to grow and (b) if they expect cash flows to grow at 4% per year. The required rate of return is 15% API.
c=ln(1.15) = 0.139761942, m=1
(a) No growth: u=1/(e0.139761942 -1) = 6.666666667
value = 6.666666667*700 = 4666.666667
NPV = 4666.666667 – 5100 = -433.33
(b) Growth = 4% per year g = ln(1.04) = 0.039220713
year 0 pmt = 700*e-0.039220713 = 673.0769231
c-g = 0.139761942 – 0.039220713 = 0.100541229
u = 1/(e0.100541229 -1) = 9.454545476
value = 673.0769231* 9.454545476 = 6363.636378
NPV = 6363.636378- 5100 = 1263.63
16. Equations more legible in source document [LINK]
17. 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 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.
18. EQUATIONS ARE MORE LEGIBLE IN THE SOURCE DOCUMENT [LINK]
19. REFERENCES:
Bodie, K. a. (2000). Essentials of Investments. New York: McGraw Hill.
Brealey, M. a. (2000). Fundamentals of corporate finance. New York: Irwin Mcgraw Hill.
Gapenski, B. a. (1996). Intermediate financial management, 5th Edition. New York: Dryden Press.
Jackson, R. (2008). Wine science principles and applications. New York: Elsevier.
Lewin, C. G. (1970). An Early Book on Compound Interest – Richard Witt’s Arithmeticall Questions. Lewin, C G (1970). “An Early Book on CoJournal of the Institute of Actuaries 96 (1): 121–132.
Taggart, R. (1996). Quantitative analysis for investment management. Saddle River, NJ: Prentice Hal.
Varian, H. (1987). The arbitrage principle in financial economics. Journal of Economic Perspectives .

EQUATIONS ARE MORE LEGIBLE IN THE SOURCE DOCUMENT [LINK]