An initial deposit of $10000 is made into an account now, and a second deposit o
ID: 2809156 • Letter: A
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%)? 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%)?Explanation / Answer
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 %