Question

$1,000 is invested into a fund at a nominal rate of interest X convertible quarterly during the two years. The fund earns interest at a nominal rate of discount X convertible quarterly thereafter. The total fund value at the end of the 4th year is $1,616.31.

1) Show the fund value at the end of the 2nd year in terms of X.

2) Calculate X.

Answer 1

The nominal rate of interest X convertible quarterly, what this means that X/4 is the rate of interest quarterly as the payments occur quarterly.

So the fund value at the end of 2 nd year would be the initial amount invested compounded over (4 quarters*2) = 8 periods at an interest rate of X/4.

So, in terms of X at the end of 2 years

Final Value= Present Value*(1+periodc interest rate)^(number of periods)

So fund value at the end of second year = $ -------------------(1)

Next up, the fund earns interest at a nominal rate of discount X.Here we need to convert first the nominal discount to the periodic discount which is much simpler, given it is quarterly

So quarterly rate of discount is X/4.

Now , deduce the quarterly interest rate based on quarterly discount

So,

Thus the quarterly interest rate for tne next two years is ---------------------(2)

Given that the value of the fund at the end of the 4 th year, we can decompose the value at the end of the 4th year as

Final Value at Year 4 = Final Value at Year 2 *(1+ periodic interest rate)^8----------------------(3_

From (2)

So

(3) becomes

= = 1616.31

So

=1.06186

Solving by adding 1 on both sides and transposing

X= 0.12 or 12% , that is the nominal rate of interest.