A loan of $4000 is to be repaid over a period of 8 years. During the first years, exactly half of the loan principal is to be repaid (along with accumulated compound interest) by a uniform series of payments A1 dollar per year. The other half of the loan principal is to be repaid over four years with accumulated interest by a uniform series of A2 dollar per year. If interest rate=9% per year what are A1 and A2?
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since A1 is just half the total yearly payment (annuity) without any interest for first 4 years.
Hence for 4 years total money to be repaid=2000$ , A1 is for 1 year so A1=2000/4=500$.
interest for these 4 years = 4000$ * (1+0.09)^4 - 4000 =1646.32$
since 2000$ has been repaid, total amount to repay now = 2000$+1646.32$=3646.32$
This has to be paid in the form of uniform annuity, here comes the present value concept.
The present value of annuity has to be determined for the annuities you are paying in 5th,6,7,8th years, that present value(pv) is equal to 3646.32$.
pv= A2(1/ (1+0.09) + 1/ (1+0.09)^2+1/ (1+0.09)^3+1/ (1+0.09)^4)=3646.32$.
A2* 3.23971 = 3646.32$
A2 = 1125.50$
and we got A1=500$