Loan and annuity (prospective methods)

Question

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.

Answer 1

You made a mistake on your third line. It should be

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

Answer 2

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.