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I'm having trouble with this problem:

A loan is being repaid with level annual payments based on an annual effective interest rate of $8\%$. If the outstanding balance immediately after the $10$th payment is $1000$, calculate the amount of interest in the $11$th payment

My attempt:

Since $B_{10} = 1000$, then $I_{11} = i*B_{10} = 0.08*1000 = 80$, but I feel like this answer is too easy and I'm missing something

user270494
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1 Answers1

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You're right because $B_t=L\,a_{\overline{n-t}|i}$ and $I_{t+1}=iB_t$.

alexjo
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  • OK, but if it's being repaid with level annual payments, then the next payment would be $80 as would all future payments, even though the balance is decreasing (unless it's an "in perpetuity" loan). We probably do need the total number of payments to work this out though. –  Mar 31 '16 at 14:14
  • NO! Each payment $P_t=P$ (constant) at time $t$ is the sum of the principal repayed $C_t$ and interest payed $I_t$. So the next payment won't be 80 (because we have to sum the principal repayment)! and we don't need the total number of payments because at each $t$ we have $I_{t+1}=iB_t$. – alexjo Mar 31 '16 at 14:28
  • I might be misunderstanding your answer. What do you think each payment amount would be, if not 80? –  Mar 31 '16 at 14:36
  • We cannot calculate each payment because we don't know the number $n$ of payments to find $P=\frac{L}{a_{\overline{n}|i}}$. But doesn' matter because the question is related to the interest payed a time $t+1$ knowing the debt at time $t$ and obviously we pay the interest $iB_t$. So if $I_{t+1}=80$ is the interest a time $t+1$, the payment will be $P_{t+1}\equiv P=C_{t+1}+I_{t+1}=C_{t+1}+80$. – alexjo Mar 31 '16 at 14:56
  • OK, to me, level annual payments means each payment must be the same amount, like in a mortgage or other amortization situation. –  Mar 31 '16 at 14:57
  • Yes, each payment! not each interest payed! So in the amortization schedule you have a principal repayed part $C_t$ increasing and a interest payed part $I_t$ decrreasing but the sum is constant $P=C_t+I_t$. – alexjo Mar 31 '16 at 15:01
  • You're right, I'm wrong. I thought amortization would have an effect on annual interest, but it turns out it doesn't. You always pay the same rate of interest on the outstanding balance. –  Mar 31 '16 at 15:11