# load repayment with increasing annuity

Question is : A loan of $10,000 is to be repaid in ten years by payments at the end of each year. The payments grow by 3% per year, so if the first payment is P, then the second payment is 1.03P and the third payment is (1.03)^2P. Compute the first payment, on the basis of an interest rate of 9% p.a I have no idea how to do it. become so confused. what I have got now is the sum of ten payments is P+1.03P+1.03^2P+...+1.03^9P and the value of load after 10 years is 1000*1.09^10 ## 1 Answer The present value of the ten annual payments must equal the principal. So, at an effective annual interest rate of$i = 0.09$, the present value discount factor is$v = (1+i)^{-1} = 1/1.09and the present value of the payments is \begin{align*} 10000 &= Pv + 1.03Pv^2 + (1.03)^2 Pv^3 + \cdots + (1.03)^9 P v^{10} \\ &= Pv(1 + (1.03v) + (1.03v)^2 + \cdots + (1.03v)^9) \\ &= Pv \cdot \frac{1 - (1.03v)^{10}}{1 - 1.03v}. \end{align*} Therefore, the level paymentP\$ is...?

• it does make sense except that whether 10000 load will increase after 10 years @ 0.09 rate p.a or not? – tinglingling Mar 16 '15 at 3:50