Question

After shopping for a car Amelia ended up borrowing $14300 from her grandparents at 7% per year compounded annually with repayment at the end of 5 years. Her Grandparents asked her to develop some alternative repayment options.
If Amelia's TVOM is 10%, what is the present worth for Amelia of each of the following 3 alternatives?
1) Interest only at the end of each year and principal at the end of the fifth year. $____
2) Equal annual payments. $ ____
3) Pay the principal and interest in one lump sum after 5 years? $ ____

If Amelia's TVOM is 4%, what is the present worth for Amelia of each of the following 3 alternatives?
1) Interest only at the end of each year and principal at the end of the fifth year. $____
2) Equal annual payments. $____
3) Pay the principal and interest in one lump sum after 5 years? $____

Answer 1

Amount borrowed = $14300

annual interest rate = 7% = 0.07

loan period = 5 years

1) when TVOM = 10%

a)  Interest only at the end of each year and principal at the end of the fifth year

Present value =( 7% * 14300)* PVIFA( 5 years, 10%) + 14300/(1.1)⁵

PVIFA( 5 years, 10%) = present value interest rate factor of annuity= ((1.1)⁵ - 1 )/((1.1)⁵ * 0.1)

= (1.61051-1)/(1.61051*0.1) = 0.61051/0.161051 = 3.790786

Present value = 1001*3.790786 + 8879.17492 = $12673.75171

b)

when Equal annual payments are made

the equal annual payments = loan amount /PVIFA

PVIFA = present value interest rate factor

= ((1.10)⁵-1)/((1.10)⁵*0.10) = 3.790787

Equal annual payments = 14300/3.790787 = 3772.304

PV of equal annual payments = equal annual payment * PVIFA = 14300

((c) Pay the principal and interest in one lump sum after 5 years

Interest for year 1,I1 = 0.07*14300 = 1001

interest for year 2,I2 = 0.07*(14300 + I1) = 1071.07

interest for year 3,I3 = 0.07*(14300 + I1 +I2) = 1146.045

interest for year 4,I4 = 0.07*(14300 + I1 +I2 + I3) = 1226.268

interest for year 5,I5 = 0.07*(14300 + I1 +I2+ I3 + I4) = 1312.107

Total interest paid = I1 + I2 +I3+I4+I5 = 1001 + 1071.07 + 1146.045 + 1226.268 + 1312.107 = 5756.49

Total amount paid at the end of 5th year = total interest + principal = 5756.49 + 14300 = 20056.49

Present value of this amount = total amount paid/(1.10)⁵ = 20056.49/(1.10)⁵ = $12453.5

2) when TVOM = 4%

a)  Interest only at the end of each year and principal at the end of the fifth year

Present value =( 7% * 14300)* PVIFA( 5 years, 4%) + 14300/(1.04)⁵

PVIFA( 5 years, 4%) = present value interest rate factor of annuity= ((1.04)⁵ - 1 )/((1.04)⁵ * 0.04)

= 4.4518228

Present value = 1001*4.4518228 + 11753.55765 = $16,209.83227

b)

when Equal annual payments are made

the present value will be equal to $14300

c) Pay the principal and interest in one lump sum after 5 years

total interest that will be paid after 5 years = I1 + I2 + I3 + I4 + I5 = 5756.49

Total lump sum payment for loan made after 5 years = total interest + loan amount = 5756.49 + 14300 = $20056.49

Present value = Total lump sum payment for loan made after 5 years/(1.04)⁵ = 20056.49/1.2166529 = $16484.97259