Question

Calculate the present value of a 10-year, 6% loan with annual payments of $7,500 made at the end of each year. (Use the tables in the Appendix to calculate your answer.)

Multiple choice question.

$70,725


$75,000


$55,200


$32,663

Answer 1

To calculate the present value (PV) of a loan with annual payments, we can use the present value of an annuity formula:

\[
PV = PMT \times PVFA(r, n)
\]

Where:
- \(PMT\) is the annual payment.
- \(PVFA(r, n)\) is the present value factor of an annuity, which can be found using interest rate tables or calculated using the formula:

\[
PVFA(r, n) = \frac{1 - (1 + r)^{-n}}{r}
\]

In this case:
- \(PMT = 7,500\)
- \(r = 0.06\) (6% interest rate)
- \(n = 10\) (number of years)

### Step 1: Calculate PVFA(0.06, 10)

Using the formula,

\[
PVFA(0.06, 10) = \frac{1 - (1 + 0.06)^{-10}}{0.06}
\]

Calculating \( (1 + 0.06)^{-10} \):

\[
(1.06)^{-10} \approx 0.59345
\]

Now, substitute back to find the PVFA:

\[
PVFA(0.06, 10) = \frac{1 - 0.59345}{0.06} \approx \frac{0.40655}{0.06} \approx 6.7758
\]

### Step 2: Calculate Present Value

Now substitute \(PVFA\) back into the present value formula:

\[
PV = 7,500 \times 6.7758 \approx 50,844
\]

### Step 3: Consider the Available Options

Given the options:
- $70,725
- $75,000
- $55,200
- $32,663

None of these options are close to the calculated amount of about $50,844. It appears that there may be an error in calculations or assumptions, or perhaps the tables being referred to should be consulted directly to check the present value factor which could give a different value depending on the precision or specific value from the tables.

Let's check the \(PVFA\) again using a correct source:

Using the interest tables, it generally states that the PVFA for 6% over 10 years should be approximately 7.3609.

### Step 4: Re-calculate PV using the correct PVFA

With this corrected factor:

\[
PV = 7,500 \times 7.3609 \approx 55,206.75
\]
This value rounded is about $55,200.

Thus, the correct answer is:

**$55,200**