Question

An initial deposit of $10000 is made into an account now, and a second deposit of $18000 is made into the same account after 15 years. During the first 10 years, the account earns interest quoted at a nominal quarterly discount rate of d %. After 10 years have passed, any money in the account earns a nominal semi-annual interest rate of 4.6%. At the end of 30 years, the total value of the account will be $86440. What is the value of d (to the nearest 0.1%)?

Answer 1

Formula for compound interest:

A = P x (1 + r/m) ^mt

A = Future value of investment

P = Principal

r = Rate of interest

m = No. of compounding in a year

t = No. of years

Future value of $ 10,000 deposited today + Future value of $ 18,000 deposited after 15 years = Total fund value at the end of 30 years

$ 10,000 x (1+d/4) ^4x10 x (1+ 0.046/2) ^20x2+ $ 18,000 x (1+ 0.046/2) ^15x2 = $ 86,440

$ 10,000 x (1+d/4) ⁴⁰ x (1+ 0.023) ⁴⁰+ $ 18,000 x (1+ 0.023) ³⁰ = $ 86,440

$ 10,000 x (1+d/4) ⁴⁰ x (1+ 0.023) ⁴⁰+ $ 18,000 x (1+ 0.023) ³⁰ = $ 86,440

$ 10,000 x (1+d/4) ⁴⁰ x (1.023) ⁴⁰+ $ 18,000 x (1.023) ³⁰ = $ 86,440

$ 10,000 x (1+d/4) ⁴⁰ x 2.483278043 + $ 18,000 x 1.9781946 = $ 86,440

$ 10,000 x (1+d/4) ⁴⁰ x 2.483278043 + $ 35,607.50275 = $ 86,440

$ 10,000 x (1+d/4) ⁴⁰ x 2.483278043 = $ 86,440 - $ 35,607.50275

$ 10,000 x (1+d/4) ⁴⁰ x 2.483278043 = $ 50,832.497253

$ 10,000 x (1+d/4) ⁴⁰ = $ 50,832.497253/2.483278043

$ 10,000 x (1+d/4) ⁴⁰ = $ 20,469.91774

(1+d/4) ⁴⁰ = $ 20,469.91774/$ 10,000

(1+d/4) ⁴⁰ = 2.046991774

1+d/4 = (2.046991774)^1/40

1+d/4 = 1.018070615

d/4= 1.018070615 – 1

d/4 = 0.018070615

d = 0.018070615 x 4 = 0.07228246 or 7.2 %