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1 ACTS 408 Instructor: Natala A. Humphreys SOLUTION TO HOMEWOR 4 Secton 7: Annutes whose payments follow a geometrc progresson. Secton 8: Annutes whose payments follow an arthmetc progresson. Problem Suppose you buy a perpetuty-due wth varyng annual payments. The frst 5 payments are constant and equal to. Startng the sxth payment, the payments start to ncrease so that each year s payment s % larger than the prevous years payment. At an annual effectve nterest rate of 7%, the perpetuty has a present value of 05. Calculate, gven < 7. A). B). C).5 D).7 E).9 Soluton. 0 4 ( + ) 5 ( + ) 6 Ths perpetuty conssts of two cash flows: () Annuty-due of for 4 years () Annuty whose payments follow a geometrc progresson startng wth payment of at the begnnng of the 5th year Hence, P V = ä = = 05 = = = % B Problem The prce of a stock s \$ per share. Annual dvdends are pad at the end of each year forever; the frst dvdend s \$ and the expected growth rate for the dvdends s % per year. The annual effectve nterest rate s 5%. Calculate. Soluton. = = 0.0 = Problem Morrs makes a seres of payments at the end of each year for 9 years. The frst payment s. Each subsequent payment through the tenth year ncreases by 5% from the prevous payment. After the tenth payment, each payment decreases by 5% from the prevous payment. Calculate the present value of these payments usng an annual effectve rate of 7%. Soluton We splt our cash flow nto two cash flows:

2 ACTS 408. AU 04. SOLUTION TO HOMEWOR 4. () Annuty whose payments follow a geometrc progresson startng wth payment of at tme, wth multpler q =.05 for 0 years; () Annuty whose payments follow a geometrc progresson startng wth payment of.05 8 at tme, wth multpler q =.05 for 9 years, dscounted for 0 years. The present value of the frst cash flow s: P V = ( = Let us now calculate the present value of the second cash flow. Let =.05 8, q = /.05, v = /.07. Then, Hence, P V = v 0 ( v + qv + q v + + q n v n = v 0 v( + qv + (qv) + + (qv) n ) ) = ( = v 0 v ) (qv)n, qv P V = (.07) ( Alternatvely, ) 9 ) 0 = (.07) = P V = P V + P V = = Agan, We splt our cash flow nto two cash flows: () Annuty whose payments follow a geometrc progresson startng wth payment of at tme, wth multpler q =.05 for 0 years; () Annuty whose payments follow a geometrc progresson startng wth payment of at tme, wth multpler q = 0.95 for 9 years, dscounted for 0 years. The present value of the frst cash flow s: P V = ( = Let us now calculate the present value of the second cash flow. Let = , q = /.05, v = /.07. Then, Hence, P V = v 0 ( v + qv + q v + + q n v n = v 0 v( + qv + (qv) + + (qv) n ) ) = ( = v 0 v ) (qv)n, qv P V = (.07) ( 0.95 ) = ) 0 P V = P V + P V = = Problem 4 Common stock S pays a dvdend of 5 at the end of the frst year, wth each subsequent annual dvdend beng 4% greater than the precedng one. Mary purchases the stock at a theoretcal prce Copyrght Natala A. Humphreys, 04 Page of 5

3 ACTS 408. AU 04. SOLUTION TO HOMEWOR 4. to earn an expected annual effectve yeld of 8%. Immedately after recevng the th dvdend, Mary sells the stock for a prce of P. Her annual effectve yeld over the -year perod was 6.75%. Calculate P. Soluton. The ntal prce of the stock s Therefore, 5 ) P ( S = = = P = 875 P = Problem 5 Vernon buys a 0-year decreasng annuty-mmedate wth annual payments of 0, 9, 8,,. On the same date, Elzabeth buys a perpetuty-mmedate wth annual payments. For the frst years, payments are,,,,. Thereafter, payments reman constant at. At an annual effectve nterest rate of, both annutes have a present value of. Calculate. A) 7 B) 8 C) 88 D) 9 E) 98 Soluton. Vernon: Elzabeth: P V V = (Da) 0 = 0 a = 0 0 P V E = (Ia) 0 + ( + ) 0 ä0 0v0 = + v0 Equatng these two values, we obtan: 0 a 0 = ä 0 + v 0. Recall that: ä n = a n + v n. Hence, = ä0 + v0 ä 0 = a 0 + v 0 0 a 0 = a 0 + v 0 + v 0 a 0 = 9 a 0 = 4.5 = 5.5% = = ä0 + v0 = B = Problem 6 You are gven two seres of payments. Seres A s a perpetuty wth payments of at the end of each of the frst years, at the end of each of the next years, at the end of each of the next years, and so on. Seres B s a perpetuty wth payments of at the end of each of the frst years, at the end of each of the next years, at the end of each of the next years, and so on. The present values of the two seres of payments are equal. Calculate. A) B) d C) a a D) a ä E) s s Soluton. Seres A: Copyrght Natala A. Humphreys, 04 Page of 5

4 ACTS 408. AU 04. SOLUTION TO HOMEWOR P V A = (Ia) j + ( + ) (Ia) j, where j = (0.5) 0.5 (Ia) j = j + j = + j ( + ) ( + ) j = (( + ) = ) ( + ) ( + j) 0.5 = + j = ( + ) = ( + )( + + ) = ( + ) ( + ) ( + ) P V A = ( + ) (Ia) j = ( + ) ( + ) = ( + ) Seres B can be thought as a payment of s, s, s, at the end of years, 6, 9, etc Therefore, ( P V B = s j + ) j, where j = ( ), ( + j) = +, + j = ( + ), j = ( + ) s = ( + ), j + j = + j j = P V B = ( + ) ( + ) (( + ) ) ( + ) ( + ) (( + ) = ) (( + ) ) Snce the present values of the two seres of payments are equal P V A = P V B ( + ) ( + ) = ( + ) (( + ) ) ( + ) = + ( + ) = ( + ) ( + )( + ) ( + ) ( + ) = ( + ) = (( + ) )( + ) = ( + ) (( + ) )( + ) = s s ( + ) = (( + ) )( + ) = (( + ) )v (( + ) )( + )v = v (( + ) )v = v v = a C a Problem 7 At an annual effectve nterest rate of, the present value of a perpetuty-mmedate startng wth a payment of 500 n the frst year and ncreasng by 0 each year thereafter s Calculate. A).5% B) 5.0% C) 7.5% D) 40.0% E) 4.5% Soluton. Ths perpetuty can be represented as the sum of two perpetutes: a level perpetutymmedate of 500, plus an ncreasng perpetuty wth the frst payment of 0 at the end of the second year ncreasng by 0 n each subsequent year: = Copyrght Natala A. Humphreys, 04 Page 4 of 5

5 ACTS 408. AU 04. SOLUTION TO HOMEWOR 4. Therefore, Problem 8 P V = ( + ) 0 (Ia) = ( + + ) = = = = = 0 D = = 90. = , = = 0.75 = 7.5% C Payments of are made at the end of each month for a year. These payments earn nterest at a nomnal rate of j% convertble monthly. The nterest s mmedately renvested at a nomnal rate of % convertble monthly. At the end of the year, the accumulated value of the payments and the renvested nterest s Calculate j. Soluton. Interest payments are: I = j I I 4I 4 I + I (Is) % = I s % 68.50I = I =.4 j = = 0.68 = 95.55, s % =.685 Problem 9 Payments of are made at the begnnng of each year for 5 years. These payments earn nterest at the end of each year at an annual effectve rate of 8%. The nterest s mmedately renvested at an annual effectve rate of 5%. At the end of 5 years, the accumulated value of the 5 payments and the renvested nterest s Calculate. A) 47 B) 5 C) 57 D) 6 E) 67 Soluton. I I 4 4I 4 5 5I 5 I = 0.08, 5 + I (Is) 5 5% = s6 5% = 4000 = A 0.05 = 4000, s 6 5% =.6575 Problem 0 Mke receves a cash flow of today, 00 n two years, and n four years. The present value of ths cash flow s 78 at an annual effectve rate of nterest. Calculate. A).9% B).9% C) 4.9% D) 5.9% E) 6.9% Soluton. Copyrght Natala A. Humphreys, 04 Page 5 of 5

6 ACTS 408. AU 04. SOLUTION TO HOMEWOR v + v 4 = 78, x = v x + x + =.78 ( + x) =.78, x = v = =.9% A Copyrght Natala A. Humphreys, 04 Page 6 of 5

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