college level interest problem?

Note: There will be small differences between my answers and the answers you may get from your book or teacher, due to intermediate rounding. I am rounding all results to 2 decimal places.

Let I(t) = interest paid in period t
B(t) = unpaid balance in period t
i = effective interest rate per payment period

Recall the formula

I(t) = B(t-1) * i
Rearranging terms, we get i = I(t) / B(t-1)

Therefore,
i = I(1) / B(0)
= 10 / 1000
= .01

So, i = .01

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a) finding the payment

I will assume that each payment is the same.

Recall the formula:

L = x * [1 - (1+i)^-n] / i, where

L = amount of loan
x = amount of level periodic payment
i = effective interest rate per payment period
n = number of payments

Then

1000 = x * (1 - 1.01^-20) / .01
x = 1000 * .01 / (1 - 1.01^-20)
x = 55.42

Therefore, the level payment is 55.42

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b) finding the unpaid balance in the 15th period

A forward-looking formula (based on the future) for the balance at time t is

B(t) = present value of remaining payments

Therefore,
B(15) = 55.42 * (1 - 1.01^-5) / .01
= 268.98

Therefore, the unpaid balance in the 15th period is 268.98.

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c) finding the interest paid in the 15th period


Let K = amount of level payment
i = effective interest rate per payment period
n = number of payments

Then

I(t) = K * [1 - (1+i)^-(n-t+1)]

Therefore,

I(15) = 55.42 * [1 - 1.01^-(20-15+1)]
= 55.42 * (1 - 1.01^-6)
= 3.21
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d) finding the balance reduction (aka the principal repaid) in the 15th period

P(t) = K(t) - I(t)
= K - I(t)

Therefore,

P(15) = 55.42 - 3.21
= 52.21

Therefore, the balance reduction in the 15th period is 52.21.