Ming borrows X for 10 years at an annual effective interest rate of 8%. If he pays the principal and accumulated interest in one lump sum at the end of 10 years, he would pay 468.05 more in interest than if he repaid the loan with 10 level payments at the end of each year. Calculate X.

X = $700

Explanation:

the future value of X = X · (1 + 8%)¹⁰ = 2.158925X

X = annual payment · 6.7101 (PV annuity factor, 8%, 10 periods)

annual payment = X / 6.7101

2.158925X = 10 annual payments + 468.05

2.158925X = 10X/6.7101 + 468.05

2.158925X = 1.490291X + 468.05

0.668634X = 468.05

X = 468.05 / 0.668634 = $700

if you payback the loan in one lump sum at the end of 10 years, you will pay = $700 x 2.158925 = $1,511.25

or you could make 10 annual payments = $700 / 6.7101 = $104.32, in total you would pay $1,043.20

the difference between both = $1,511.25 - $1,043.20 = $468.05

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