Question

A deposit of X is made into a fund which pays an annual effective interest rate of 8% for 5 years. At the same time, 2X is deposited into another fund which pays an annual effective rate of discount of d for 3 years. The amounts of interest earned over the 10 past years are equal for both funds. Calculate d.

Answer 1

For the deposit of X:

Final value = X (1.08) ^ 10

Therefore, the interest = [X (1.08) ^ 10] – X

                                    = 2.158925X – X

                                    = 1.158925X

For the deposit of 2X:

Final value = 2X × (1/(1-d)^10)

                    = 2X (1/(1-d)^10)

Therefore, the interest = 2X (1/(1-d)^10) – 2X

                                    = 2X {(1/(1-d)^10) -1}

Now, as per the condition, both these interests are equal

1.158925X = 2X {(1/(1-d)^10) -1}

By solving,

{(1/(1-d)^10) -1} = 1.158925X / 2X

(1/(1-d)^10) -1 = 1.158925 / 2

(1/(1-d)^10) = 0.579462 + 1

1 / (1 – d)^10 = 1.579462

1 = 1.579462 (1 – d)^10

1/1.579462 = (1 – d)^10

0.633126 = (1 – d)^10

0.633126 ^ (1/10) = (1 – d) ^ (10/10)

0.633126 ^ 0.1 = 1 – d

0.955320 = 1 – d

1 - 0.955320 = d

d = 0.04467 (Answer)