Question

Susan wishes to purchase a laptop computer. She is offered the following payment options:

Option 1 : $0 down, $432 in 1 year, $300 in 2 years

Option 2 : $82.56 down, $250 in 1 year, $400 in 2 years

Determine the range of the interest rate i for which the present value of Option 1 is less than the present value of Option 2.

A) 4.2%< i <16.3%

B) i <4.2% or i >16.3%

C) 3.5%< i <10.2%

D) i <3.5% or i >10.2%

E) i >4.2%

Answer 1

Range of interest rate

Let I be the interest rate.

NPV of option 1,NPV1=432/(1+i)+300/(1+i)²

NPV of option 2,NPV2=82.56+250/(1+i)+400/(1+i)²

We have to get the range of values of i for which NPV1<NPV2

That means, 432/(1+i)+300/(1+i)² < 82.56+250/(1+i)+400/(1+i)²

Multiply through out with (1+i)² ,we get

432*(1+i)+300<82.56*(1+i)² +250(1+i)+400

Transposing, 82.56*(1+i)² +250(1+i)+400 >432*(1+i)+300

82.56*(1+i)² +250(1+i)+400-432*(1+i)-300>0

82.56(1+2i+i²)+250+250i+400-432-432i-300>0

82.56+165.12i+82.56i² +250+250i+400-432-432i-300>0

82.56i² -16.88i+0.56>0

We may use the equation to find roots of a quadratic equation of model ax²+bx+c=0,

Roots will be –b+?b²-4ac/2a,-b-?b²-4ac/2a,where a=82.56,b=-16.88 and c=0.56

Roots ={--16.88+?16.88*16.88-4*82.56*0.56}/2*82.56 and

={--16.88-?16.88*16.88-4*82.56*0.56}/2*82.56

=(16.88+10)/165.12 and ( 16.88-10/165.12)

=0.162791 and 0.041667 .

Thus i-0.162791>0 or i>0.162791 or i<0.041667

for which NPV1<NPV2

Approximately range of i is <4.2 or i>16.3