Chapter 12.3, Problem 91AYU

To determine

To calculate: The value of TSA be after the ${80}^{th}$ deposits ($20$ years) when Don contributes $\$500$ at the end of each quarter to a tax-sheltered annuity with the per annum rate of return is assumed to be $5\%$ compounded quarterly.

Answer to Problem 91AYU

Solution:

The value of TSA be after the ${80}^{th}$ deposit is $\$68059.40$

Explanation of Solution

Given information:

Don contributes $\$500$ at the end of each quarter to a tax-sheltered annuity with the per annum rate of return is assumed to be $5\%$ compounded quarterly.

Formula used:

Theorem of amount of an annuity:

Suppose that $P$ is the deposit in dollars made at the end of each payment period for an annuity paying $i$ percent interest per payment period. The amount $A$ of the annuity after $n$ deposits is

$A=P\frac{{\left(1+i\right)}^n-1}{i}$

Calculation:

Don contributes $\$500$ at the end of each quarter to a tax-sheltered annuity, so $P=\$500$

Per annum rate of return is $5\%$

Per quarter rate of return is $\frac{5\%}{4}=0.0125$

The number of deposits $n=80$

By using the theorem of amount of an annuity, The value of money after ${80}^{th}$ deposits $=500\frac{{\left(1+0.0125\right)}^{80}-1}{0.0125}$

$=500\frac{{\left(1.0125\right)}^{80}-1}{0.0125}$

$=500\frac{2.70148494-1}{0.0125}$

$=500\frac{1.70148494}{0.0125}=68059.40$

Therefore, the value of TSA be after the ${80}^{th}$ deposit is $\$68059.40$