Question

You have two options to repay a loan. You can repay $6,000 now and $5,940 in one year; or you can repay $12,000 in 6 months. Find the annual effective interest rate(s) i at which both options have the same present value.

Answer 1

Answer:

We will consider positive interest rate which is i=0.21 or i=21%

Explanation:

The formula for Future value is:

$FV=PV(1+i)^n$

The present value will become:

$PV=FV(1+i)^{-n}$

where:

n is the number of years

Since the condition is same present value,so the given data form the equation:

$6000+5940(1+i)^{-1}=12000(1+i)^{-1/2}$

Divide above equation by $(1+i)^{-1}$

$6000(1+i)+5940=12000(1+i)^{1/2}$

Let $z=(1+i)^{1/2}$. Above equation will become:

$6000z^2+5940=12000z$

Rearranging above equation:

$5940-12000z+6000z^2=0$

Solving the quadratic equation:

z=1.1,    z=0.9

Let $z=(1+i)^{1/2}$ will become:

$z=(1+i)^{1/2}\\\\z^2=1+i$

$i=z^2-1$

For z=1.1

$i=(1.1)^2-1\\i=0.21$

For z=0.9

$i=(0.9)^2-1\\i=-0.19$

we will consider positive interest rate which is i=0.21 or i=21%