Chapter 3, Problem 67RE

Solution

To find: the present value PV of a loan.

$28,794.06

Given information:

It is given that, “the loan is with an annual interest rate r = 5.5% and periodic payments r = $550 for a term of t = 5 years, with payments made and interest charged 12 times per year.”

The data given is:

  $r=5.5\%,R=\$550,t=5$

Formula Used:

Present Value of an Annuity

The present value PV of an annuity consisting of n equal payments of R dollars earning an interest rate i per period (payment interval) is:

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

Explanation:

Here, the interest is credited 12 times a year, so the interest rate is:

  $\begin{array}{c}i=\frac{r}{k}\\ =\frac{0.055}{12}\end{array}$

The number of payments is:

  $\begin{array}{c}n=k\cdot t\\ =12\cdot 5\\ =60\end{array}$

Substitute these values in the above formula,

  $\begin{array}{c}PV=R\frac{1-{\left(1+i\right)}^{-n}}{i}\\ =550\cdot \frac{1-{\left(1+\frac{0.055}{12}\right)}^{-60}}{\frac{0.055}{12}}\\ \approx \$28,794.06\end{array}$

The present value is $28,794.06.