Solve the following problem.
n equals 39​; iequals 0.023​; PMT equals $ 190​; PV equals question mark
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Part 1
PV equals $

enter your response here ​(Round to two decimal​ places.)

To solve for the present value (PV) given the number of periods ($ n $), interest rate per period ($ i $), and periodic payment ($ PMT $), we use the formula for the present value of an ordinary annuity:

\[ PV = PMT \times \left( \frac{1 - (1 + i)^{-n}}{i} \right) \]

Given:
- $ n = 39 $
- $ i = 0.023 $
- $ PMT = 190 $

We can substitute these values into the formula.

First, compute $ (1 + i)^{-n} $:

\[ (1 + 0.023)^{-39} \]

Next, calculate $ 1 - (1 + 0.023)^{-39} $:

\[ 1 - (1.023)^{-39} \]

Then, divide this result by $ i $:

\[ \frac{1 - (1.023)^{-39}}{0.023} \]

Finally, multiply by $ PMT $:

\[ PV = 190 \times \left( \frac{1 - (1.023)^{-39}}{0.023} \right) \]

Let’s compute these step by step.

1. Compute $ (1 + 0.023)^{-39} $:

\[ (1.023)^{-39} \approx 0.405381 \]

2. Compute $ 1 - (1.023)^{-39} $:

\[ 1 - 0.405381 = 0.594619 \]

3. Divide by $ i $:

\[ \frac{0.594619}{0.023} \approx 25.85213 \]

4. Multiply by $ PMT $:

\[ 190 \times 25.85213 \approx 4911.90 \]

Therefore, the present value $ PV $ is approximately \$4911.90.

Do you want details or have any questions?

Here are some related questions:

1. How does the interest rate affect the present value of an annuity?
2. What is the difference between the present value and future value of an annuity?
3. How does the number of periods impact the present value calculation?
4. Can the present value formula be applied to any type of annuity?
5. What are some practical applications of calculating the present value of an annuity?
6. How would the present value change if the payments were made at the beginning of each period (annuity due)?
7. What is the effect of compounding frequency on the present value?
8. How can we adjust the present value formula for continuous compounding?

**Tip:** Always double-check your calculations, especially when dealing with exponential functions and financial formulas, to ensure accuracy.

Learn how to calculate the present value (PV) of an annuity with n = 39 periods, interest rate i = 0.023 per period, and periodic payment PMT = $190. Step-by-step solution provided.

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