Compounding and Future Value

Interest on interest (or rate of return on rate of return) means rates cannot be added.

Now, what if you can earn the same 20% year after year and reinvest all your money? What would your 2-year rate of return be? Definitely not 20% + 20% = 40%! You know that you will have $120 in year 1, which you can reinvest at a 20% rate of return from year 1 to year 2. Thus, you will end up with

This $144—which is, of course, again a future value of $100 today—represents a total 2-year rate of return of

This is more than 40%, because the original net return of $20 in the first year earned an additional $4 in interest in the second year. You earn interest on interest! This is also called compound interest. Similarly, what would be your 3-year rate of return? You would invest $144 at 20%, which would provide you with

$144 ■ (1 + 20%) = $172.80 CF2 ■ (1 + r2,3) = CF3

so your 3-year rate of return would be

CF0 CF0

If you do not want to compute interim cash flows, can you directly translate the three sequential 1-year rates of return into one 3-year holding rate of return? Yes! This is called compounding, and the formula that does this is the "one-plus formula,"

(1 + 72.8%) = (1 + 20%) ■ (1 + 20%) ■ (1 + 20%)

(1 + T0,3) = (1 + r-0,1) ■ (1 + r-1,2) ■ (1 + T2,3)

In our case, all three rates of return were the same, so you could also have written this as r0,3 = (1 + 20%)3 - 1. Figure 2.1 shows how your $100 would grow if you continued investing it at a rate of return of 20% per annum. The function is exponential, that is, it grows faster and faster, as interest earns more interest.

Figure 2.1: Compounding Over 20 Years at 20% Per Annum

35003000-

Q 2000-OJ SH

I 1500-

1000500 0 J

Year

Start 1-year End Period value rate value

Total factor on $100

Total rate of return, r0 T

(1 + 20%) $120.00 (1 + 20%) $144.00 (1 + 20%) $172.80

Money grows at a constant rate of 20% per annum. If you compute the graphed value at 20 years out, you will find that each dollar invested right now is worth $38.34 in 20 years. The money at first grows in a roughly linear pattern, but as more and more interest accumulates and itself earns more interest, the graph accelerates steeply upward.

file=constantinterest.tex 20 ■ Chapter 2 THE TIME VALUE OF MONEY

IMPORTANT: The compounding formula translates sequential future rates of return into an overall holding rate of return:

(1 +rpJ)i = (1 +ro,i) ■ (1 + ri,2) ■■■ (1 + rr-i,T ) (2.13)

Holding Rate Current Spot Rate Future 1-Period Rate Future 1-Period Rate

The first rate is called the spot rate because it starts now (on the spot). If all spot and future interest rates are the same, the formula simplifies into (1 + ror ) = (1 + rt)r.

The compounding formula is so common, it is worth memorizing.

Another example of a You can use the compounding formula to compute all sorts of future payoffs. For example, payoff computation. an investment project costing $212 today and earning 10% each year for 12 years will yield an overall holding rate of return of r012 = (1 + 10%)12 - 1 « 213.8%

(1 + r)T - 1 = 70,12 Your $212 investment today would therefore turn into a future value of CF12 = $212 ■ (1 + 213.8%) « $665.35

Turn around the Now suppose you wanted to know what constant two 1-year interest rates (r) would give you a '.rmula,to compute 2-year rate of return of r0 2 = 50%? It is not 25%, because (1 + 25%) ■ (1 + 25%) - 1 = 56.25%.

individual holding rates. Instead, you need to solve

(Appendix 2-3 reviews powers, exponents and logarithms.) Check your answer: (1 + 22.47%) ■ (1 + 22.47%) « (1 + 50%). If the 12-month interest rate is 213.8%, what is the 1-month interest rate? By analogy,

but you already knew this.

You can determine fractional interest rate via compounding, too.

Interestingly, compounding works even over fractional time periods. If the overall interest rate is 5% per year, to find out what the rate of return over half a year would be that would compound to 5%, compute

n0.5

Compounding 2.4695% over two (6-month) periods indeed yields 5%, (1 + 2.4695%) ■ (1 + 2.4695%) - (1 + 5%)

If you know how to use logarithms, you can also determine with the same formula how long it You need logs to will take at the current interest rate to double or triple your money. For example, at an interest rate of 3% per year, how long would it take you to double your money?

determine time needed to get x times your money.

Solve Now!

Q 2.7 A project has a rate of return of 30%. What is the payoff if the initial investment is $250?

Q 2.8 If 1-year rates of return are 20% and interest rates are constant, what is the 5-year holding rate of return?

Q2.9 If the 5-year holding rate of return is 100% and interest rates are constant, what is the annual interest rate?

Q2.10 If you invest $2,000 today and it earns 25% per year, how much will you have in 15 years?

Q2.11 What is the holding rate of return for a 20-year investment that earns 5%/year each year? What would a $200 investment grow to?

Q 2.12 What is the quarterly interest rate if the annual interest rate is 50%?

Q 2.13 If the per-year interest rate is 5%, what is the 2-year total interest rate?

Q 2.14 If the per-year interest rate is 5%, what is the 10-year total interest rate?

Q2.15 If the per-year interest rate is 5%, what is the 100-year total interest rate? How does this compare to 100 times 5%?

Q 2.16 At a constant rate of return of 5% per annum, how many years does it take you to triple your money?

Q2.17 A project lost one-third of its value each year for 5 years. What was its rate of return, and how much is left from a $20,000 investment?

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