Lesson 4 Annuities: The Mathematics of Regular Payments


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1 Lesson 4 Annuities: The Mathematics of Regular Payments Introduction An annuity is a sequence of equal, periodic payments where each payment receives compound interest. One example of an annuity is a Christmas club at a bank. Here you make deposits each week or month to build up a Christmas fund. Another example is the payments you make on a car loan. In order to understand the behavior of annuities we need to investigate the mathematical idea of a sequence and, in particular, a geometric sequence. Geometric Sequences A sequence is a collection of numbers in a particular order. The individual numbers are called the terms of the sequence. Example 1 1, 5, 9, 13, Can you see how the terms were generated? Example 2 3, 1, 1/3, 1/9, Can you see how the terms were generated? Example 3 1, 1, 2, 3, 5, 8, 13, Can you see how the terms were generated? This is a famous sequence called the Fibonacci sequence. In our study of annuities we will need a particular type of sequence called a geometric sequence. A geometric sequence starts with an initial term P and from then on every term in the sequence is obtained by multiplying the preceding term in the sequence by the same constant c. The number c is called the common ratio of the geometric sequence. (We will see why in a minute.) Here are some examples. Example 4 5, 10, 20, 40, 80, Initial term = Note: case), a common ratio. Common ratio =. The ratio of any term to the preceding one is constant (2 in this Example 5 27, 9, 3, 1, 1/3, Initial term = Common ratio = Example 6 27, 9, 3, 1, 1/3, Initial term = Common ratio =
2 Notation: A geometric sequence with initial term P and common ratio c can be written as P, cp, c 2 P, c 3 P, c 4 P, We will label the terms using a common letter in this case G for geometric. We will use subscripts to indicate which term of the sequence we are referring to. We will start with 0 rather than 1 so that the subscript matches the power of c in the term. G 0 = P, G 1 = c 1 P, G 2 = c 2 P, G 3 = c 3 P, Summary For a geometric sequence G 0, G 1, G 2, we have 1) G 0 = P and G N = c G N 1 (called a recursive formula) 2) G N = c N P (called an explicit formula) Geometric sequences play an important role in the world of finance. Consider the following example. Example 7 Consider a geometric sequence with initial term P = 5000 and common ration c = Then the first few terms of this sequence are : G 0 = 5000 G 1 = (1.06)5000 = 5300 G 2 = (1.06) = 5618 G 3 = (1.06) = , etc. Notice that if we put dollar signs in front of these numbers, we get the principal and the future values over the first 3 years of an investment with a principal of $5000 and an interest rate of 6% compounded annually! Let s see if we can generalize this example. Suppose that we have a principal P and an interest rate per period of i. Then the balances in the account at the end of each compounding period are the terms of a geometric sequence with initial term P and common ratio (1 + i)! P, P(1 + i), P(1 + i) 2, P(1 + i) 3, In discussing how much money will accumulate in an annuity (like a Christmas club) we will need to add up all the terms in such a geometric sequence (and there may be lots of them!). Thus, it would be great if we had a simple way to add up all the terms in a geometric sequence. Geometric Sequence Sum Formula Problem: Find the sum P + cp + c 2 P + c 3 P + + c N 1 P. To find the sum we are going to use a clever strategy that will eliminate most of the terms that we need to add up! Step 1 Multiply each term in given sum above by c: c(p + cp + c 2 P + c 3 P + + c N 1 P) = cp + c 2 P + c 3 P + + c N P
3 Step 2 Take the result from Step 1 and then subtract the original sum from it. (cp + c 2 P + c 3 P + + c N 1 P + c N P) ( P + cp + c 2 P + c 3 P + + c N 1 P) = c N P P = P(c N 1) Notice that almost all the terms have canceled out, appearing once with a plus sign and once with a minus sign! If we denote the original sum that we wanted to find by S (for sum!), then in Step 1 we constructed cs while is Step 2 we found cs S = S(c 1). Thus we have S(c 1) = P(c N 1) or, solving for S we get: S = or P + cp + c 2 P + c 3 P + + c N 1 P =. Notes: 1. The formula fails for c = 1. Why? (What is the sum of first N terms if c = 1?) 2. If you think of the left hand side as the sum of the first N terms of a geometric sequence and then think of the exponent of the right hand side as being one more than the highest power of c on the left hand side (or the number of terms being added ), then you have a simple way to remember the formula.
4 Future Value of an Annuity Let s begin by looking at examples of the 2 most common types of annuity: ordinary annuity and annuity due. Example 1. On August 10 Sherah joins a Christmas club at her bank. She will make $200 deposits on the first of each of the next 3 months and on December 1 will be able to withdraw her money for shopping. Assume that the interest rate will be 6% compounded monthly. How much will be in her account on December 1? Solution: To do this problem we have to track the future value of each payment separately and then combine the results. The September 1 deposit will earn interest for 3 periods (months) where the rate per period is so FV Sept = 200(1.005) 3 = $ The October 1 deposit will earn interest for 2 periods (months) so FV Oct = 200(1.005) 2 = $ The November 1 deposit will earn interest for only 1 period so FV Nov = 200(1.005) 1 = $ Thus, the future value of the annuity is given by the sum of these 3 future values: FV = $ $ $201 = $ Sherah earned $6.03 in interest. Before we do our next example, let s look at some vocabulary associated with annuities. 1) The payment period of an annuity is the time between payments. 2) The term of an annuity is the time from the beginning of the first payment period to the end of the last payment period. (Sherah s annuity had a term of 3 months.) 3) An annuity is said to have expired at the end of its term. 4) An annuity is called simple if its compounding period is the same as its payment period. All of our annuities will be simple annuities. 5) An annuity due is an annuity for which each payment is due at the beginning of each payment period. 6) An ordinary annuity is an annuity for which each payment is due at the end of its payment period. Sherah s annuity is an example of an annuity due. The next example is an ordinary annuity. Example 2. Dan joins a Christmas club for September, October and November with interest at 6% compounded monthly. He makes $200 payments at the end of each month. How much will be in his account for Christmas shopping on December 1? Solution: We again think of the future value of each payment separately and then combine the results.
5 We know that i =.005 again and so we have the following future values. FV Sept = 200(1.005) 2 = $202.01, FV Oct = 200(1.005) 1 = $ and FV Nov = 200(1.005) 0 or just $200. The total for Dan is $603.01, and he earned $3.01 interest. Question: Why did Sherah earn more interest than Dan? Answer: Because each payment in her annuity earned interest for 1 more month. Let s compare the two annuities sidebyside to see exactly how they are related. Sept. Oct. Nov. FV Sherah = $ = 200(1.005) (1.005) (1.005) 1 FV Dan = $ = 200(1.005) (1.005) Notice that if we multiply each term in Dan s future value by 1.005, (1 + i), we will get Sherah s future value exactly; that is FV Dan (1.005) = FV Sherah. This shows us a general principle about the relationship between the future of an ordinary annuity and an annuity due, assuming that the interest rate and the terms are the same. In general, FV Ordinary (1 + i) = FV Due. We can convert an ordinary annuity to an annuity due by leaving all funds accumulated in the account for one additional period. If Dan leaves his $ in his account for the month of December he will have (1.005) = $ which is the same as Sherah s future value! Example 3. If you deposit $100 every 6 months into an ordinary annuity paying 6% compounded semiannually, how much money would you have after a term of 3 years? Solution: Let s do a timeline and track the future value of each payment, adding them up at the end years periods deposits FV 1 = 100(1.03) 5 = $115.93, FV 2 = 100(1.03) 4 = $112.55, FV 3 = 100(1.03) 3 = $109.27, FV 4 = 100(1.03) 2 = $106.09, FV 5 = 100(1.03) 1 = $ and FV 6 = $100 for a total of $ What would the future value have been if we had used an annuity due with same interest rate? Answer: (1.03) = $
6 Since there were only 6 payments it was fairly easy to look at each future value separately and then add up the results. But what if there were 200 payments? It would be nice to have a fast, convenient way to add up the future values of all those payments, and, in fact, we already have the tools we need: the formula for the sum of a geometric sequence! We will denote our periodic payments by R and we will assume that our account is earning interest at a rate of i for a term of n periods (that is, there will be n payments). We construct the following time line n3 n2 n1 n periods R R R R R R R R The last payment will earn no interest so its future value is just R. The next to last payment will earn interest for one period and so its future value will be R(1 + i) 1. The second to last payment will earn interest for two periods and so its future value will be R(1 + i) 2. We continue working backwards this way and we find that the second payment will earn interest for all but the first 2 periods, that is it will interest for n 2 periods and so will have a future value of R(1 + i) n2. Similarly, the first payment will earn interest for n 1 periods (all but the first period) and will have a future value of R(1 + i) n1. To find the future value F of the annuity, we now need to add up all the future values of the separate payments. F = R + R(1 + i) 1 + R(1 + i) 2 + R(1 + i) R(1 + i) n2 + R(1 + i) n1. Notice that this is a geometric sequence with first term R and with the common ratio c = 1 + i! Thus we can plug into our sum formula for a geometric sequence to find our future value formula. This is our formula for the future value of an ordinary annuity. For an annuity due, we need only multiply this result by 1 + i to get the correct future value. F due = F ordinary (1 + i) Let s look at some examples. Example 4 Verify the result of Example 3 by using the future value formula for an ordinary annuity. Solution: In Example 3 we had R = $100, i =.03 (6% compounded semiannually) and n = 6 (semiannual payments for 3 years). Thus, F = Example 5 What is the future value of an ordinary annuity at the end of 5 years if $100 per month is deposited in an account earning 9% compounded monthly? What would this future value be for an annuity due?
7 Solution: Here R = 100, n = kt = 12(5) = 60 periods, and i =.09/12 = Substituting into our formula for the future value of an ordinary annuity gives. For an annuity due, we only have to multiply the answer we just got by 1 + i = : (1.0075) = $ Example 6 Jim puts $200 per month into an ordinary annuity paying 8.75% compounded monthly for 35 years. Find the future value of his annuity. What would the future value be as an annuity due? Solution: Here R = 200, r =.0875, k = 12 and t = 35. Thus i =.0875/12 and n = 12(35) = 420 periods in 35 years. Using our formula,. For an annuity due, we simply multiply the future value for the ordinary annuity by 1 + i = /12: F Due = 552,539.96( /12) = $556, Sinking Funds An account that is established to accumulate funds for a future obligation is called a sinking fund. If regular, periodic payments are made to this fund, we have an annuity. Example 7. A couple sets up a sinking fund for their new baby s education. How much should they have deducted from their biweekly (every two weeks) paycheck to have $30,000 in 18 years at 9.25% interest? Assume an ordinary annuity. Solution: Here F = 30000, i = r/k =.0925/26 and n = kt = 26(18) = 468 total periods, and we want to find R, the periodic payment. Filling in the values in our formula for the future value of an ordinary annuity, we have. Solving for R gives R = or $25 to the nearest penny. Example 8. A company estimates that it will need $10,000 in 5 years to replace a piece of equipment. A sinking fund is established using an ordinary annuity with monthly payments earning interest at 6% compounded monthly. What should be payments be? Solution: We have F = 10,000, i = r/k =.06/12 =.005, and n = kt = 12(5) = 60. Substituting into our formula for the future value of an ordinary annuity gives Present Value of an Annuity. Solving for R gives R = $ The present value of an annuity is the lump sum that can be deposited at the beginning of the annuity s term at the same interest rate and compounding period, and which would yield the same future value as
8 the annuity at the end of its term. We sometimes say that the lump sum will buy or generate the annuity. *We are looking for a present value P so that F compound = F annuity. Example 9. What lump sum would the company in Example 8 have to deposit now at 6% compounded monthly in order to have the same amount after 5 years as their annuity will give them? Solution: We know from Example 8 that their annuity will accumulate the $10,000 they will need to replace their piece of equipment. So now we have a standard present value problem for compound interest. We know F = $10,000, r =.06, k = 12 and t = 5. Thus i = r/k =.06/12 =.005, and n = kt = 60. Substituting these values in our formula gives = P(1.005) 60 or P = Example 10. Find the present value of an ordinary annuity that has $200 monthly payments for 25 years if the account receives 10 ½ % interest (compounded monthly). Solution: To find the present value P, we need to solve the equation (*) for P. We have i =.105/12, n = 12(25) = 300 and R = $200. Substituting and solving for P gives P = $21, Note: We can use (*) to find a formula for P by dividing both sides by (1 + i) n : Example 11. What is the present value of an annuity that has $200 monthly payments for 5 years with a rate of 6% (compounded monthly)? Solution: We know that R = $200, i = r/k =.06/12 =.005 and n = kt = 12(5) = 60. We will substitute into the formula above.
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