=== Summary of Section 9.2 === | The essential concept of chapter 9 is that loaned money | grows with time. This leads to the notion of present | value. To compare money at different times you have to | multiply or divide by the appropriate growth factor. | This concept is known as the "time value of money" (see | "http://en.wikipedia.org/wiki/Time_value_of_money"), and it | is the central concept of everyday mathematical finance. The | notion of present value requires that you specify an interest | rate and period. (That is, you must specify how the money grows | with time.) Present Value: | In general, the *present value* of an amount of money A which is | to be paid at a time T in the future is the amount of money P | that would grow to A if we invested it now. To find this amount | we have to divide A by the investment growth factor. For example: assuming 25% annual yield, the present value of $100 to be paid one year from now is the amount P such that $100 = P*1.25, Solving for P, we see that the present value of $100 to be paid after one year of 25% annual interest is the final amount divided by the growth factor: P = $100/1.25 = $80. The present value of an amount A to be paid after n periods of time with an interest rate per period k is the principal P such that A = P*(1+k)^n. Solving for P shows again that the present value is the final amount divided by the growth factor: P = A/(1+k)^n. Annuities. | An *annuity* is a scheduled series of regular equal payments. | For an ordinary annuity (annuity-immediate) the payments are | made at the *end* of each payment period (typically a month). | To find the present value of a series of future payments we | find the present value of each payment and add them up. Let Y = size of payments, let n = number of payments, let k = interest rate per payment period, let V_m = present value of payment m, let V = present value of all the payments, and let a = (1+k) = growth factor per payment period. The present value of the m-th payment is the amount V_m that will grow to become Y in m payment periods: Y = V_m*a^m; To solve for the present value we divide by the growth factor: V_m = Y/a^m. The present value of all the payments is V = V_1 + V_2 + ... + V_n = Y/a + Y/a^2 + ... + Y/a^n. To find this sum we first need to study geometric series. Sum of a Geometric Series Let S by the sum of the first n powers of a growth factor a: S = 1 + a + a^2 + ... + a^n This is called a geometric series. The trick to find S is to multiply by a and subtract the original equation: a*S = a + a^2 + ... + a^n + a^(n+1) - [ S = 1 + a + a^2 + ... + a^n ] When you subtract these two equations, all the terms on the right cancel except the first and the last, so you get S*(a-1) = a^(n+1) - 1. Solving for S gives: S = (a^(n+1) - 1)/(a-1). The formula on page 426 is incorrect. (It has "n-1" instead of "n+1".) Fix it. Observe that the sum S = 1 + a + a^2 + ... + a^(n-1) is given by the same formula, with n replaced by n-1: S = (a^n - 1)/(a-1). Present Value of an annuity (continued). Now resume computing the present value of an annuity. Recall V = Y/a + Y/a^2 + ... + Y/a^n. Factoring out Y gives: V = Y*(1/a + 1/a^2 + ... + 1/a^n). Multiply by a^n. Get: a^n*V = Y*(a^n/a + a^n/a^2 + ... + a^n/a^n). Recall the general property of division of powers: a^p/a^q = a^(p-q). So we get: a^n*V = Y*(a^(n-1) + a^(n-2) + ... + 1). We see a geometric series on the right. Using the formula for the sum of a geometric series, a^n*V = Y*(a^n-1)/(a-1) Solving for V, we get V = Y*(1 - 1/a^n)/(a-1). Recall that a = 1+k. Making this substitution gives the formula on page 428 of the book for the present value of an annuity with payments Y, n payment dates, and an interest rate per period k: V = Y*(1 - 1/(1+k)^n)/k. For example, the present value of an annuity that pays $100 every month for 12 months relative with respect to an interest rate of 1% per month is V = $100*(1 - 1/1.01^12)/.01 = $1125.507747348463. Amount of an annuity. | The amount of an annuity is the "future value" of | the payments at the end of the last payment period: | Again, to determine future value an interest | rate k and payment period t must be specified. If the | payments are put in an account accumulating interest | at rate k and investment period t, the money acccumulated | in the account at the end of the last payment period | is the amount of the annuity. In particular, using the same variable definitions that we used for the present value of a series of payments, A = Y*a^(n-1) + Y*a^(n-2) + ... + Y*a^2 + Y*a + Y = Y*(a^n-1)/(a-1) = Y*((1+k)^n-1)/k, in agreement with equation 9.6 on page 427. For example, the amount of an annuity that pays $100 every month for 12 months relative with respect to an interest rate of 1% per month is A = Y*((1+k)^n-1)/k, = $100*(1.01^12-1)/.01, = $1268.250301319698. Recall that we previously calculated the present value of this annuity to be V = $1125.507747348463. Observe that the total of all the payments is $1200. Notice that the present value of the total payments is less than the total payments, and the amount is more. That should make sense to you. | In general the present value of the annuity is the | present value of the amount of the annuity. Let's check that. We need that A/(1+k)^n = V. You can see that this is true in the formulas. For our specific case, we have that the growth factor is (1+k)^n = 1.01^12 = 1.12682503013197, and $1268.250301319698/1.12682503013197 = $1125.507747348463, as needed.