Questions: Find the present value for the following future amount. 9810 at 2.5% compounded semiannually for 11 years The present value is (Do not round until the final answer. Then round to the nearest cent as needed.)

Find the present value for the following future amount. 9810 at 2.5% compounded semiannually for 11 years

The present value is
(Do not round until the final answer. Then round to the nearest cent as needed.)

Transcript text: Question 5, 13.1.37 points Points: 0 of 1 Find the present value for the following future amount. $\$ 9810$ at $2.5 \%$ compounded semiannually for 11 years The present value is $\$$ $\square$ (Do not round until the final answer. Then round to the nearest cent as needed.)

Solution

Solution Steps

To find the present value of a future amount compounded semiannually, we use the present value formula for compound interest:

\[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \]

where:

$ PV $ is the present value,
$ FV $ is the future value (\$9810 in this case),
$ r $ is the annual interest rate (2.5% or 0.025),
$ n $ is the number of compounding periods per year (2 for semiannual),
$ t $ is the number of years (11 years).

Step 1: Identify the Given Values

We are given the future value $ FV = 9810 $, the annual interest rate $ r = 0.025 $, the number of compounding periods per year $ n = 2 $ (since it is compounded semiannually), and the time in years $ t = 11 $.

Step 2: Use the Present Value Formula

To find the present value, we use the formula for compound interest:

\[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \]

Substituting the given values into the formula:

\[ PV = \frac{9810}{(1 + \frac{0.025}{2})^{2 \times 11}} \]

Step 3: Calculate the Present Value

First, calculate the base of the exponent:

\[ 1 + \frac{0.025}{2} = 1.0125 \]

Next, calculate the exponent:

\[ 2 \times 11 = 22 \]

Now, raise the base to the power of the exponent:

\[ 1.0125^{22} \approx 1.314 \]