Question

Four people share an annuity which makes payments at the end of each year. Person A gets the first ten payments of P. Person B gets the next 10 payments of 2P. Person C gets the next ten payments of 3P and person D gets the final ten payments of 4P. The present value of C's share is one-third the present value of A's share.

a)What is the ratio of the present value of D's share to B's share?

b) If P = $10,000, what is the present value of the entire annuity?

c) Assuming A's payments are each $10,000 how much would each of D's payments need to be in order for the present value of D's payments actually have to be equal to the present value of A's payments?

Answer 1

(a) The present value of an ordinary annuity is given by

PV = P[1-(1+r)^-n]/r

PV_A = P[1 - (1+r)^-10]/r

PV_B = (2P[1 - (1+r)^-10]/r)/(1+r)¹⁰

PV_C = (3P[1 - (1+r)^-10]/r)/(1+r)²⁰

PV_D = (4P[1 - (1+r)^-10]/r)/(1+r)³⁰

Given PV_C = PV_A/3

=> (3P[1 - (1+r)^-10]/r)/(1+r)²⁰ = (P[1 - (1+r)^-10]/r) /3

=> (1+r)²⁰ = 9

=> r = 0.1161 or 11.61%

PV_D/PV_B = (4P[1 - (1+r)^-10]/r)/(1+r)³⁰ / (2P[1 - (1+r)^-10]/r)/(1+r)¹⁰ = 2/(1+r)²⁰ = 0.222

(b) PV = PV_A + PV_B + PV_C + PV_D

= P[1 - (1+r)^-10]/r + (2P[1 - (1+r)^-10]/r)/(1+r)¹⁰ + (3P[1 - (1+r)^-10]/r)/(1+r)²⁰ +  (4P[1 - (1+r)^-10]/r)/(1+r)³⁰

= 10000[1 - (1+0.1161)^-10]/0.1161 + (20000[1 - (1+0.1161)^-10]/0.1161)/(1+0.1161)¹⁰ + (30000[1 - (1+0.1161)^-10]/0.1161)/(1+0.1161)²⁰ +  (40000[1 - (1+0.1161)^-10]/0.1161)/(1+0.1161)³⁰

= $123358.84

(c) Let Ds payment be X

=> PV of A's payment = PV of D's payment

=> P[1 - (1+r)^-10]/r = (X[1 - (1+r)^-10]/r)/(1+r)³⁰

=> X = P(1+r)³⁰ = 10000*(1+0.1161)³⁰ = 269831.87

Hence, for PV of D to be equal to A, the annual payment for D should be $269831.87