We turn to discuss the future value of an ordinary annuity. In short, it is the sum of the compound amounts of all payments (we will visualise it shortly). Let us consider the following situation.
You will deposit money in your savings account n times. Each time, you are depositing Ddollars. Suppose the periodic rate is denoted by r. The deposit will start at the end of
85 3.8 Annuities: what is the value of your home loan?
Figure 3.8 A time line 2.
Period 1, and will occur at the end of each period until the end of Period n. Compounding takes place at the end of each period as we are looking at an ordinary annuity.
The future value of the annuity S can be written as follows:
S = D + D(1 + r) + D(1 + r)2+ D(1 + r)3+ · · · + D(1 + r)n−2+ D(1 + r)n−1. (3.52) Again the use of a time line helps you visualise this equation. In Figure 3.8 we describe how all the repayments are discounted (forward) to their future values. We start with D deposited at the end of Period n, which needs no discounting. D deposited at the end of Period n− 1 is discounted forward one period to D(1 + r), D deposited at the end of Period n− 2 is discounted forward two periods to D(1 + r)2, and so on. The RHS of Equation (3.52) is merely the collection of these values discounted forward.
Again, S is the geometric series of n terms with the initial value D and the common ratio (1+ r). Using the geometric series formula, you should be able to simplify S.
Question Simplify Equation (3.52).
Exercise 3.20 Simplifying Equation (3.52).
So the future value S of an ordinary annuity of D dollars per payment period for n periods at the interest rate of r per period can be written as:
S = D ·(1+ r)n− 1
r . (3.53)
Here’s an exercise for you.
Question Find the future value of an annuity of $50 at the end of every three months for 3 years at an interest rate of 8 per cent compounded quarterly. What is the compound interest? You may round your answer to two decimal places.
Solution
S = 50 ·(1+ 0.02)12− 1 0.02
≈ 670.60 (dollars).
Compound interest: 670.60− 12 × 50 = 70.60.
Exercise 3.21 The future value of an ordinary annuity.
3.9 Perpetuity
Financial assets that yield regular payments for an infinite number of periods are called perpetuities. An example of a perpetuity is a particular type of a bond. A bond is a promise by the issuer to pay the holder a fixed sum (a redemption value) at a specified maturity date and to make interest payments (coupon interest payments) at regular intervals. So a bond that has no maturity date or redemption value and which pays coupon interest forever is a perpetuity (we will look at this in the example shortly).
In fact, a perpetuity can be considered as an ordinary annuity with an infinite duration.
Despite this, we can show that the present value of a perpetuity approaches a certain value, R
r. To check this, we use the present value formula as in Equation (3.51). The present value of a perpetuity can be obtained by taking the limit, lim
n→∞A:
n→∞lim A= R lim
n→∞
1− (1 + r)−n r
= R r lim
n→∞
1− (1 + r)−n
= R r −R
r lim
n→∞(1+ r)−n
= R r −R
r lim
n→∞
1 (1+ r)n.
The second term of the RHS approaches zero because (1+ r) is greater than unity (the denominator approaches infinity and the numerator is constant at unity, so the ratio approaches zero). Therefore, the present value of a perpetuity approachesR
r .
87 3.10 Additional exercises
Question Suppose the government guarantees that all holders of a bond will be paid
$100 at the end of each quarter forever. If interest of 8 per cent is compounded quarterly, what are the future and the present values of this perpetuity?
Solution
P V = 100
0.02 = 5000.
Exercise 3.22 The present value of a perpetuity.
3.10 Additional exercises
1. (Limits) Find the following limits where possible.
(1) lim
2. (Summations) Express the following sums using a summation operator.
(1) 1+ 3 + 5 + 7 + 9.
(2) 1+ 4 + 9 + 16 + 25 + 36 + · · · + n2. (3) 2x+ 4x2+ 8x3+ 16x4+ 32x5. (4) ai1b1j + ai2b2j+ · · · + ainbnj.
3. (The geometric series) Find the sum to infinity of the following series. [Hint. Do it in two steps. Step 1: describe the sum assuming there are n terms. Step 2: think what will happen to this sum when n approaches infinity, i.e. n→ ∞.]
(1) 243, 81, 27, 9, 3, 1, . . .
Table 3.3. Net cash flows.
Project Constant
End of year Cash flow PV at 5% PV at 10%
1 50
2 50
3 50
4 50
5 50
Project Increasing
End of year Cash flow PV at 5% PV at 10%
1 0
2 20
3 50
4 100
5 100
4. (The effective rate) What is the effective rate of interest? Denote the principal by P, the nominal rate by i, and the number of compounding periods per year by n.
Carefully derive the effective rate re. If $2000 is accumulated to $6000 over a period of 6 years in an account where interest is compounded daily, how can we obtain the effective rate? You may use the following information at the end of your calculation:
316 ≈ 1.20.
5. (The effective rate and the nominal rate) If a major credit-card company has a finance charge of 1 per cent per month on the outstanding debt, obtain the nominal rate compounded monthly. Also, obtain the effective rate. You may use the following information at the end of your calculation: 1.0112≈ 1.13.
6. (Doubling the money) How many years will it take for our money to double if interest is compounded continuously at a nominal rate of 2 per cent? Express your answer using the natural logarithm. Calculate it using the information that ln 2≈ 0.7.
How many years will it take for our money to double if interest is compounded annually at the same nominal rate? Express it using the natural logarithm. Do you expect this duration to be shorter than the one you obtained under continuous compounding?
7. (Net present value) A firm has two investment projects, Constant and Increasing.
Cash flows that these projects will generate are given in Table 3.3. Calculate the sum of the present value of cash flows for each project if interest is compounded annually at (a) 5 per cent, and (b) 10 per cent. Suppose both projects require the same amount of initial investment. Give advice to this firm as to which of these projects is better to undertake under (a) and (b). Carefully explain your findings. In calculating, you may round numbers to two decimal places for each of the present values of cash flows.
89 3.10 Additional exercises
8. (Compound interest) The population growth rate in Fefmland for the 5 years between 1999 and 2003 had been r1 per annum. Because of a change in policy the population growth rate of Fefmland was r2 per annum for the 5 years between 2004 and 2008. Express the average population growth rate per annum over the 10-year period between 1999 and 2008 in terms of r1 and r2. Here, the average population growth rate per annum, denoted by g, for the 10-year period is defined by the following equation: P1998(1+ g)10 = P2008, where P1998 and P2008 are the population of Fefmland in the beginning of 1999 and the end of 2008, respectively.
9. (Future value of an annuity) A company called Generous Insurance sells an education policy to parents with new babies. If the nominal rate of 8 per cent is compounded quarterly, what is the size of the payments that must be made by parents at the end of each quarter (so the first payment is made 3 months after the child is born), if they wish to receive $15 000 when their child turns 18? What will the size of the payments be if those must be made at the beginning of each quarter?
10. (Perpetuity with increasing payments) Consider the situation where Nasty Bank offers new-home buyers the option of making payments of continually increasing size at the end of each period. That is, instead of payments of R, customers will make payments of:
R, R(1+ g), R(1 + g)2, R(1+ g)3, . . . (IP) Here, g is an increase in the size of the payment from one period to the next. For example, if the payment increases by 1 per cent from one period to the next, then g = 0.01. Assume that the periodic rate is r > 0 and r = g.
Obtain the present value of a perpetuity when we have payments of an increasing size as in (IP). [Hint. Recall we obtained the PV of an annuity in the main text when payments are constant (Equations (3.50) and (3.51)). Using the same approach, you need to derive the PV of an annuity when payments are increasing. Think what Equation (3.50) will become if payments are increasing. And then, using the geometric series formula, you can simplify the sum similar to Equation (3.51), which should involve R, r, g and n. Then, take the limit of it when n approaches infinity.] Does it depend on the relative size of r and g? What will the present value be if g is zero? Explain.