Chapter 6, Problem 52QP

Summary Introduction

To calculate: The value of annuity for five years from the present, the value for three years from present and the current value

Introduction:

The future sum of money that worth today is described by the present value. The formula of the present value of annuity is helpful to find the future periodic payment values for the given time.

Answer to Problem 52QP

The value of annuity for five years from the present is $41,721.62, the value for three years from the present is $35,571.70, and the current value is $28,003.99.

Explanation of Solution

Given information:

The five-year annuity of $7,100 for ten semiannual payments will start nine years from now. The first payment is 9.5 years from now. Assuming the discount rate to be 8% that is compounded monthly.

Time line of the sales:

Note: The cash flows in the given information are semiannual, so it is necessary to find the effective semiannual rate. The annual percentage rate is 8%.

Formula to calculate the monthly rate with the annual percentage rate:

$\text{Monthly rate}=\frac{\text{Annual percentage rate}}{\text{Number of months in a year}}$

Compute the monthly rate with the annual percentage rate:

$\begin{array}{c}\text{Monthly rate}=\frac{\text{Annual percentage rate}}{\text{Number of months in a year}}\\ =\frac{0.08}{12}\\ =0.0067\end{array}$

Hence, the monthly rate is 0.0067

Formula to calculate the effective semiannual rate:

$\text{Effective semiannual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^6-1\right)$

Compute the effective semiannual rate:

$\begin{array}{c}\text{Effective semiannual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^6-1\right)\\ ={\left(1+0.0067\right)}^6-1\\ =1.04067-1\\ =0.04067\end{array}$

Note: To calculate the effective semiannual rate, the time period is assumed to be six months. The APR is the annual percentage rate. The monthly rate for the annual percentage rate is calculated above.

Hence, the effective semiannual rate is 0.0406 or 4.06%.

Formula to calculate the present value annuity:

$\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}$

Note: C denotes the annual cash flow, r denotes the rate of exchange, and t denotes the period.

Compute the present value annuity at year 9:

$\begin{array}{c}\text{Present value of annuity at year 9}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ =\$7,100\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.04067\right)}^{10}}\right)\right]}{0.04067}\right\}\\ =\$7,100\left\{\frac{\left[1-\left(\frac{1}{{\left(1.04067\right)}^{10}}\right)\right]}{0.04067}\right\}\\ =\$7,100\left\{\frac{\left[1-\left(\frac{1}{1.489808167}\right)\right]}{0.04067}\right\}\end{array}$

$\begin{array}{c}=\$7,100\left\{\frac{\left[1-0.671227358\right]}{0.04067}\right\}\\ =\$7,100\left\{\frac{0.328772641}{0.04067}\right\}\\ =\$7,100\times 8.083910524\\ =\$57,395.02\end{array}$

Note: This is the value for the first period of six months previous to the first payment, thus it is the value at the year nine. Therefore, the value at different periods asked in the question utilizes this value of nine years from now.

Hence, the present value annuity at year nine is $57,395.02

Formula to calculate the present value:

$\text{Present value}=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}$

Note: r denotes the rate of discount and t denotes the number of years.

Compute the present value at year 5:

$\begin{array}{c}\text{Present value at year five }=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\frac{\$57,395.02}{{\left(1+0.04067\right)}^8}\\ =\frac{\$57,395.02}{{\left(1.04067\right)}^8}\\ =\frac{\$57,395.02}{1.375638369}\end{array}$

$=\$41,721.62$

Note: The present value for the fifth year can also be calculated using the effective annual rate, the present values for the remaining years can also be calculated using the effective annual rate.

Hence, the value of annuity at 5 year is $41,721.62

Formula to calculate the effective annual rate:

$\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)$

Compute the effective annual rate:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\\ ={\left(1+0.0067\right)}^{12}-1\\ =1.0830-1\\ =0.0830\end{array}$

Hence, the effective annual rate 0.0830 or 8.30%

Formula to calculate the present value:

$\text{Present value}=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}$

Note: r denotes the rate of discount and t denotes the number of years.

Compute the present value at year 5:

$\begin{array}{c}\text{Present value at year five }=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\frac{\$57,395.02}{{\left(1+0.0830\right)}^4}\\ =\frac{\$57,395.02}{{\left(1.0830\right)}^4}\\ =\$41,721.62\end{array}$

Hence, the value of annuity at 5 year is $41,721.62

The value of annuity for the other years is calculated as follows:

$\begin{array}{c}\text{Present value at year three }=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\frac{\$57,395.02}{{\left(1+0.04067\right)}^{12}}\\ =\frac{\$57,395.02}{{\left(1.04067\right)}^{12}}\\ =\$35,571.70\end{array}$

$\begin{array}{c}\text{Present value at year three }=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\frac{\$57,395.02}{{\left(1+0.0830\right)}^6}\\ =\frac{\$57,395.02}{{\left(1.0830\right)}^6}\\ =\$35,571.70\end{array}$

Note: The present value at year 3 is calculated using the calculated r values

Hence, the value of year three is $35,571.70

$\begin{array}{c}\text{Present value at year 0 }=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\frac{\$57,395.02}{{\left(1+0.04067\right)}^{18}}\\ =\frac{\$57,395.02}{{\left(1.04067\right)}^{18}}\\ =\$28,003.99\end{array}$

$\begin{array}{c}\text{Present value at year 0 }=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\frac{\$57,395.02}{{\left(1+0.0830\right)}^9}\\ =\frac{\$57,395.02}{{\left(1.0830\right)}^9}\\ =\$28,003.99\end{array}$

Note: The present value at year 0 is calculated using the calculated r values

Hence, the current value is $28,003.99.