Question

A deposit of X is made into a fund which pays an rate of 12 for 8 years. At the same time, 2X is deposited into another fund which pays a continuous compounded rate (force of interest) of o for 8 years From the end of the 4th year until the end of the 8th year, both funds the same dollar amount of interest. Calculate d (A) 7.2% (B) 7.5% |(C) 7.8% (D) 8.1% (E) 8.4% annual effective discount earn

Answer 1

The future value of the deposits at the end of the fourth year are:

for the X-deposit

X(1 0.12)

and $2Xe^{4\delta }$ for the 2X-deposit

The future value of the deposits at the end of the eight year are :

for the X-deposit

X(1 0.12)

and $2Xe^{8\delta }$ for the 2X-deposit

Therefore the interest earned from the end ot the 4th year to the end of the 8th year is

for the X-deposit

X(1 0.12) X(1-0.12)

and     for the 2X-deposit

2Xe2Xe

Since both fund earn the same amount of interest we have

X (1 0.12) X(1 0.12)=2Xe-2Xe

(1 0.12)-(1 - 0.12) 2e2e 8

( (1-0.12)(1 0.12)-4) e-e5 8

We need to solve for \delta, so let's use the following substitution

40 y= e

Then we have

y2-y((1-0.12)8( 0.12)-4)

$y^2-y-0.55654367=0$

So that

$y=\frac{1\pm\sqrt{(-1)^2-4\cdot(- 0.55654367)}}{2}=\frac{1\pm1.796155}{2}$

Since y can't be negative then

$y=\frac{1+1.796155}{2}=1.398077$

therefore

1.398077

$\delta=\frac{1}{4}\text{ln}(1.398077)= 0.08377$

So the answer is

(E)8.4%