Question

Sally's rich aunt wants to deposit a sum of money, P, in a bank account that earns 5% APY and will provide Sally a payment of $20,000 every 10 years forever. How much money, P, should Sally's aunt deposit today?

Answer 1

She will have to deposit P such that Net present value(NPV) of all future Value is equal to the P i.e. NPV = P.

PV = A/(1 + r)ⁿ

where A = amount after n years, r = interest rate = 5% = 0.05 and n = time period

Hence, NPV = 20,000/(1 + 0.05)¹⁰ + 20,000/(1 + 0.05)²⁰ + 20,000/(1 + 0.05)³⁰ + ------------------- infinity

=> NPV = 20,000(1/(1 + 0.05)¹⁰ + 1/(1 + 0.05)²⁰ + 1/(1 + 0.05)³⁰ +------------------+ infinity)

According to infinite series formula

a + ar + ar² --------------------infinity = a/(1 - r) if 1 > r > -1

Here a = 1/(1 + 0.05)¹⁰ , r = (1/(1 + 0.05)²⁰)/(1/(1 + 0.05)¹⁰) = 1/(1 + 0.05)¹⁰

=> NPV = 20,000((1/(1 + 0.05)¹⁰)/(1 - (1/(1 + 0.05)¹⁰))) = 31801.83

=> P = NPV = 31801.83

Hence, Amount that she should deposit today is P = $31801.83