Frank Cronin must make a balloon payment ​$20 comma 000 in 6 years. Find the present value of the payment if it includes annual interest of 7.7​% compounded quarterly.
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Part 1
The present value is ​$
  
enter your response here.
​(Round to the appropriate​ cent.)


To find the present value of a future payment, we use the **present value formula** for compound interest:

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

Where:
- $ PV $ is the present value (what we are solving for),
- $ FV = 20,000 $ is the future value or the balloon payment,
- $ r = 0.077 $ is the annual interest rate (7.7%),
- $ n = 4 $ because the interest is compounded quarterly,
- $ t = 6 $ is the number of years.

### Step-by-step solution:

1. **Substitute the values** into the formula:
\[
PV = \dfrac{20,000}{(1 + \frac{0.077}{4})^{4 \times 6}}
\]
2. **Simplify the expression** inside the parentheses:
\[
PV = \dfrac{20,000}{(1 + 0.01925)^{24}} = \dfrac{20,000}{(1.01925)^{24}}
\]
3. **Calculate** $ (1.01925)^{24} $:
\[
(1.01925)^{24} \approx 1.569132
\]
4. **Divide** $ 20,000 $ by $ 1.569132 $:
\[
PV \approx \dfrac{20,000}{1.569132} \approx 12,749.65
\]

So, the present value is approximately **$12,749.65**.

Would you like more details or have any questions?

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Here are 5 related questions:
1. What is the formula for continuously compounded interest?
2. How does compounding frequency affect the present value?
3. If the interest rate changes, how does it affect the present value?
4. What is the difference between nominal and effective interest rates?
5. How would the present value change if compounding occurred monthly instead of quarterly?

**Tip:** When working with compound interest, always double-check the compounding frequency to avoid calculation errors.

Frank Cronin must make a balloon payment of $20,000 in 6 years. Find the present value of the payment if it includes annual interest of 7.7% compounded quarterly. The present value is $ (Round to the appropriate cent).

Find the present value of a $20,000 balloon payment due in 6 years with 7.7% interest compounded quarterly. Learn how to apply the present value formula to calculate it, including detailed steps and final value rounded to the nearest cent.

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