Question

Bethany deposits $1500 into an account today and then $1000 into the account 10 years later. Assume that interest is credited to the account at a nominal annual interest rate of i^(6), convertible every two months, for the first 5 years, and at a nominal discount rate of 10%, compounded quarterly, thereafter. The accumulated balance in the fund after the 20 years is $10,000. Find i^(6)

Answer 1

Future value = present value*(1+r)^n

Where r = rate of interest per period

n = number of periods

Let interest rate = x ( 2 months interest rate)

So after 5 years (30 periods) balance = 1500*(1+x)^30

For the remaining 15 years (60 periods) interest rate = 10% compunded quarterly (10%/4 = 2.5%)

After 20 years balance

= 1500*(1+x)^30 * (1+2.5%)^60............(i)

Addition of 1000 after 10 years

Future value = 1000*(1+2.5%)^40 = 2685.06......(ii)

(i) + (ii) = 10,000

1500*(1+x)^30 *(1+2.5%)^60 + 2685.06 = 10000

1500*(1+x)^30 * 4.40 = 7314.94

(1+x)^30 = 7314.94 / 6599.68

x = (7314.94 / 6599.68)^(1/30) - 1

x = 0.3436%

i = 0.3436% * 6 = 2.06%

answer is 2.06% (Rounded to two decimals)