# Loan and annuity (prospective methods)

The question is a loan of $10,000 is to be repaid over 10 years by level annual repayment of capital and interest. The interest rate to be charged on the capital outstanding will be 6% per annum for the first 4 years and 8% per annum thereafter i)How much of the loan is outstanding at the beginning of the 3rd Year. I have already know the answer is$8345.53. But i got the wrong answer (8107.66) when i using the prospective methods. So i am so confuse about which part am i wrong in the prospective methods. Thank you so much.

$$v^2_{0.06}a_{\overline{6|}0.08}+a_{\overline{2|}0.06}$$

• Thank you so much. Finally i can find the correct answer. – alan.cheung Apr 6 '15 at 15:10

Let $P=10{,}000$, let $r=.06$, and let $A=1403.14$. What remains to be repaid after the first payment, at the end of the first year, is

$$(1+r)P-A$$

What remains to be repaid after the second payment, at the end of the second year, is

$$(1+r)((1+r)P-A)-A$$

I would say this is also the outstanding loan at the beginning of the third year, unless your loan shark tacks on the year's interest up front.

• But if i would like to do it in the prospective methods, i notice that it still have 8 payments need to pay at the beginning of 3rd year. For the 8 payments, which interest should i use? Is it 6 payments is 0.08 ( Year 5 to Year 10 ) and 2 payment is 0.06 ( Year 3 to Year 4). I am so confusing about it coz my answer is wrong when I do this methods – alan.cheung Apr 6 '15 at 10:43
• @alan.cheung, if you edit your question to show how you calculated the level annual payment to be 1043.14, it might be more clear what you are confused about. (I did not check that 1043.14 is the correct amount.) As I read the problem, you have two more payments at six percent and six payments at eight percent. – Barry Cipra Apr 6 '15 at 12:24
• I have uploaded the photo which is my steps. i know that prospective methods should be equal to retrospective methods, but i dun know which step am i wrong in the prospective methods, coz the correct answer is \$8345.53. Thank you – alan.cheung Apr 6 '15 at 12:35
• I'm afraid I have to admit defeat. I don't recognize the notation or terminology you are using, so I don't see how you managed to write the result of your calculation (R=1043.14) in the very first line. Perhaps there will be someone who does. – Barry Cipra Apr 6 '15 at 12:57
• For what it's worth, I think the mistake lies in your calculation of R=1403.14. I think it's too small a level payment. (It might be the right answer if you make four payments at six percent and TEN payments at eight percent.) – Barry Cipra Apr 6 '15 at 13:09