A debt payment of ​$6500 is due in 60 months. If money is worth 9.9​% p.a. compounded semi dash annually​, find the equivalent payments at the times shown below.
​(a) now
​(b) in 30 months
​(c) in 60 months
​(d) in 72 months
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Part 1
​(a) The equivalent debt payment is ​$

enter your response here.
​(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as​ needed.)

To find the equivalent debt payments at different times given the future value of $6500 due in 60 months, with an interest rate of 9.9% per annum compounded semi-annually, we can use the present value formula for compound interest:

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

Where:
- $ PV $ is the present value (the equivalent debt payment now or at a specified time)
- $ FV $ is the future value ($6500)
- $ r $ is the annual interest rate (9.9% or 0.099)
- $ n $ is the number of times the interest is compounded per year (2 for semi-annually)
- $ t $ is the time in years

First, we need to convert the interest rate to match the semi-annual compounding:

\[ \text{Semi-annual rate} = \frac{0.099}{2} = 0.0495 \]

Now we can proceed with the calculations for each part.

### (a) Equivalent payment now
\[ t = \frac{60}{12} = 5 \text{ years} \]
\[ PV = \frac{6500}{(1 + 0.0495)^{2 \cdot 5}} \]

Let's calculate the denominator first:
\[ (1 + 0.0495)^{10} \approx 1.628894 \]

Now calculate $ PV $:
\[ PV = \frac{6500}{1.628894} \approx 3990.93 \]

### (b) Equivalent payment in 30 months
\[ t = \frac{30}{12} = 2.5 \text{ years} \]
\[ PV = \frac{6500}{(1 + 0.0495)^{2 \cdot 2.5}} \]

Calculate the denominator:
\[ (1 + 0.0495)^{5} \approx 1.273318 \]

Now calculate $ PV $:
\[ PV = \frac{6500}{1.273318} \approx 5104.69 \]

### (c) Equivalent payment in 60 months
The equivalent payment in 60 months is simply the future value itself, as it is due at that time:
\[ PV = 6500 \]

### (d) Equivalent payment in 72 months
\[ t = \frac{72 - 60}{12} = 1 \text{ year after the due date} \]
Since this payment is beyond the due date, we need to calculate the future value of the $6500 one year after it is due.

\[ FV = 6500 \times (1 + 0.0495)^{2 \cdot 1} \]

Calculate the denominator:
\[ (1 + 0.0495)^{2} \approx 1.101761 \]

Now calculate $ FV $:
\[ FV = 6500 \times 1.101761 \approx 7161.45 \]

So the equivalent payments are:
- (a) Now: $3990.93
- (b) In 30 months: $5104.69
- (c) In 60 months: $6500
- (d) In 72 months: $7161.45

Would you like more details or have any questions?

Here are 8 related questions to further explore the topic:

1. How does the frequency of compounding affect the present value of a future sum?
2. What would be the equivalent payment if the interest rate were compounded quarterly?
3. How would the payments change if the interest rate was 12% p.a. compounded semi-annually?
4. What is the difference between nominal and effective interest rates?
5. How do you calculate the present value of an annuity?
6. What is the impact of inflation on debt repayments?
7. How do you determine the future value of a series of payments (annuity)?
8. What are the implications of making early payments on a loan?

**Tip:** Always round intermediate values to at least six decimal places to ensure accuracy in your final results.

Learn how to calculate equivalent debt payments at different times using compound interest with a detailed step-by-step solution. This problem involves determining present values for future payments at various intervals, including immediate and delayed payments.

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