A debt of $9000$ is to be amortized with $12$ equal semiannual payments. If the interest rate is $6\%$, compounded semiannually, what is the size of each payment? Round your answer to the nearest cent.

1. Problem Description

A debt of $9000$ is to be amortized with $12$ equal semiannual payments. If the interest rate is $6\%$, compounded semiannually, what is the size of each payment? Round your answer to the nearest cent.

2. Solution Steps

The present value of an ordinary annuity formula is:

$PV = PMT \times \frac{1 - (1+i)^{-n}}{i}$,

where

$PV$ is the present value of the annuity,

$PMT$ is the periodic payment,

$i$ is the interest rate per period, and

$n$ is the number of periods.

In this case, the present value $PV$ is the debt, which is $9000$.

The number of payments is $n=12$.

The annual interest rate is $6\%$, so the semiannual interest rate is $i = \frac{6\%}{2} = 3\% = 0.03$.

We need to find the payment $PMT$.

$9000 = PMT \times \frac{1 - (1+0.03)^{-12}}{0.03}$

$9000 = PMT \times \frac{1 - (1.03)^{-12}}{0.03}$

$9000 = PMT \times \frac{1 - 0.70138012}{0.03}$

$9000 = PMT \times \frac{0.29861988}{0.03}$

$9000 = PMT \times 9.953996$

$PMT = \frac{9000}{9.953996}$

$PMT = 904.158752$

Rounding to the nearest cent, the payment is $904.16$.

3. Final Answer

$904.16$