Question

question 25
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consider an investment with a nominal interest rate of 9% compounded quarterly for 23 years, resulting in a future value of r823,400. calculate the annual payment:

Explanation:

Step1: Calculate the effective - annual interest rate

The nominal interest rate $r = 9\%=0.09$ and the number of compounding periods per year $m = 4$. The effective - annual interest rate $i$ is calculated using the formula $i=(1+\frac{r}{m})^m - 1$.
$i=(1+\frac{0.09}{4})^4 - 1=(1 + 0.0225)^4-1=1.0225^4-1\approx1.093083 - 1=0.093083$.

Step2: Calculate the number of periods

The investment is for $n = 23$ years. Since we want to find the annual payment, the number of periods $n = 23$.

Step3: Use the future - value of an ordinary annuity formula

The future - value of an ordinary annuity formula is $FVA = A\times\frac{(1 + i)^n-1}{i}$, where $FVA$ is the future value of the annuity, $A$ is the annual payment, $i$ is the interest rate per period, and $n$ is the number of periods.
We know that $FVA = 823400$, $i = 0.093083$, and $n = 23$. We need to solve for $A$.
Rearranging the formula for $A$, we get $A=\frac{FVA\times i}{(1 + i)^n-1}$.
$(1 + 0.093083)^{23}=1.093083^{23}\approx7.53777$.
$A=\frac{823400\times0.093083}{7.53777 - 1}=\frac{76644.4422}{6.53777}\approx11723.94$.

Answer:

11723.94