Chapter 4, Problem 48QP

Summary Introduction

To compute: The present value of annuity.

Purchasing Power:

It refers to the ability of the currency to buy something. Purchasing power decreases with increase in inflation and increase with decrease in inflation.

Explanation of Solution

Given,

Annuity is of 5 years.

Semi annual payments are of $6,175.

Discount rate is 11%.

Semiannual rate is 5.054%.

Formula to calculate present value of annuity,

$PVA=\text{Semiannual}\text{payment}\left[\frac{1-{\left(\frac{1}{1+\text{semiannual}\text{rate}}\right)}^{10}}{\text{semiannual}\text{rate}}\right]$

Where,

• PVA is present value of annuity.

Substitute $6,175 for semiannual payment, and 5.054% for semi annual rate,

$\begin{array}{c}PVA=\$6,175\left[\frac{1-{\left(\frac{1}{1.055}\right)}^{10}}{0.055}\right]\\ =\$6,175\left[\frac{1-0.5850}{0.055}\right]\\ =\$6,175\left[\frac{0.4150}{0.055}\right]\\ =\$46,592.84\end{array}$

Hence, the present value of annuity is $46,592.84.

Effective annual rate is 11.48%.

Given, number of years is 4.

Formula to calculate present value of annuity for 5 years,

$PVA=\frac{\text{Total}PVA}{{\left(1+\text{Effective}\text{annual}\text{rate}\right)}^{\text{Number}\text{of}\text{year}}}$

Where,

• PVA means present value of annuity.

Substitute $46,592.84 for total PVA, 5.5% for effective annual rate and 4 for number of years.

$\begin{array}{c}PVA=\frac{\$46,592.84}{{\left(1+0.055\right)}^4}\\ =\frac{\$46,592.84}{\left(1.2388\right)}\\ =\$37,610.52\end{array}$

Hence, the present value of annuity of 5 years is $37,610.52.

Compute present value of annuity for 3 years.

Number of years that will be taken is 6.

Formula to calculate present value of annuity for 3 years,

$PVA=\frac{\text{Total}PVA}{{\left(1+\text{Effective}\text{annual}\text{rate}\right)}^{\text{Number}\text{of}\text{year}}}$

Where,

• PVA means present value of annuity.

Substitute $46,592.84 for total PVA, 5.5% for effective annual rate and 6 for number of years.

$\begin{array}{c}PVA=\frac{\$46,592.84}{{\left(1+0.055\right)}^6}\\ =\frac{\$46,592.84}{\left(1.3788\right)}\\ =\$33,791.26\end{array}$

Hence, the present value of annuity of 3 years is $33,791.26.

Compute present value of annuity for current year.

Number of years that will be taken is 9.

Formula to calculate present value of annuity for current year,

$PVA=\frac{\text{Total}PVA}{{\left(1+\text{Effective}\text{annual}\text{rate}\right)}^{\text{Number}\text{of}\text{year}}}$

Where,

• PVA is present value of annuity.

Substitute $47,154.54 for total PVA, 5.5% for effective annual rate and 9 for number of years.

$\begin{array}{c}PVA=\frac{\$46,592.84}{{\left(1+0.055\right)}^9}\\ =\frac{\$46,592.84}{\left(1.6190\right)}\\ =\$28,778.78\end{array}$

Hence, the present value of annuity of 3 years is $28,778.78.

Working Note:

Computation of monthly rate,

$\begin{array}{c}\text{Monthly}\text{rate}=\frac{\text{Rate}}{\text{Number}\text{of}\text{months}}\\ =\frac{0.11}{12}\\ =0.0091\end{array}$

Hence, the monthly rate is 0.0091.

Computation of semi annual rate,

$\begin{array}{c}\text{Semiannual}\text{rate}={\left(1+\text{Monthly}\text{rate}\right)}^6-1\\ ={\left(1+0.0091\right)}^6-1\\ =1.055-1\\ =5.5\%\end{array}$

Hence, the semiannual rate is 5.5%.

Computation of early annual rate,

$\begin{array}{c}\text{Effective}\text{annual}\text{rate}={\left(1+0.0091\right)}^{12}-1\\ ={\left(1.0091\right)}^{12}-1\\ =1.1148-1\\ =0.1148\end{array}$

Hence, the effective annual rate is 11.48%.

Hence, the present value annuity for 5 years, 3 years and currently is $37,610.52, $33,791.26 and $28,778.78.