Question

The future value of an anjuity at the end of ten years is 80000, the payments are made at the end of each year, and the interest rate is 11% compounded semiannually. What isbthe size of each annual payment?

Answer 1

Here,

Given Information

Future Value of annuity at end of 10 years =80,000

Interest rate =11% compounded semi-annually

Time Period =10 years

Using the Effective Annual rate formula

$EAR=(1+\frac{r}{m})^m-1$

Where, EAR=Effective Annual rate

r=Annual nominal rate of interest

m=Number of compounding periods in a year

$EAR=(1+\frac{0.11} {2} ) ^2 −1 \\EAR=(1+0.055) ^2 −1 \\EAR=(1.055) ^2 −1\\EAR= 1.113025−1\\ EAR=0.113025 \space or \space 11.3025\%$

Now,

Effective Annual Rate =11.3025%

Using the Future Value of annuity formula

Future Value of Annuity $=P[\frac{(1+r) ^n −1} r ]$

Where P=Periodic Payment 

r=rate per period 

n=number of periods 

$80000=P[\frac{(1+0.113025)^ {10} −1}{ 0.113025}\\80000\times 0.113025= P[(1.113025) ^{10} −1]\\ 9042=P[2.9177574906−1]\\ 9042=P[1.9177574906]\\ \frac{9042}{1.9177574906}=P\\ 4,714.8818577530 \space or\space 4,714.88 =P$

The Size of each annual payment is 4,714.88