A debt of $500 due in four years and $1000 due on six years is to be repaid by a single payment three years from now. If the interest rate is 4.4% compounded quarterly, how much is the payment

To determine the single payment required to repay the two debts, we need to compute the present values (PVs) of the two debts at the time of the payment (three years from now), taking into account the interest rate of 4.4% compounded quarterly.

### Step 1: Formula for present value with compound interest
The formula for the present value is given by:

\[
PV = \frac{A}{(1 + i)^n}
\]

where:
- $ A $ = amount due in the future
- $ i $ = interest rate per period
- $ n $ = number of periods from the current time to the payment time

The interest rate is compounded quarterly, so we need to adjust both the rate and the time to match quarterly periods.

### Step 2: Adjusting the interest rate
The nominal annual interest rate is 4.4%, so the quarterly interest rate $ i $ is:

\[
i = \frac{4.4\%}{4} = \frac{0.044}{4} = 0.011
\]

### Step 3: Time periods for each debt
The payment is to be made 3 years from now. We need to calculate how many quarters will pass from the payment time to the time the debts are due:
- For the $500 due in 4 years (i.e., 1 year after the payment), the time difference is 1 year = 4 quarters.
- For the $1000 due in 6 years (i.e., 3 years after the payment), the time difference is 3 years = 12 quarters.

### Step 4: Calculate present values
Now, calculate the present values of both debts at the time of the single payment.

#### For the $500 debt:
\[
PV_1 = \frac{500}{(1 + 0.011)^4} = \frac{500}{(1.011)^4}
\]

#### For the $1000 debt:
\[
PV_2 = \frac{1000}{(1 + 0.011)^{12}} = \frac{1000}{(1.011)^{12}}
\]

### Step 5: Total payment
The total payment is the sum of the present values of the two debts:

\[
\text{Total payment} = PV_1 + PV_2
\]

Let me calculate that for you.The single payment required to repay the debts is approximately **$1,355.57**, three years from now.

Would you like further details or have any questions? Here are some related questions to consider:

1. How would the payment change if the interest rate were compounded annually instead of quarterly?
2. What would the payment be if it were made 2 years from now instead of 3 years?
3. How does compounding frequency affect the total amount repaid in general?
4. How can this problem be approached with continuous compounding interest?
5. How would the solution change if the debt amounts were repaid in multiple payments instead of a single one?

**Tip:** When dealing with compounded interest, always ensure that the interest rate and time periods are in compatible units (e.g., both quarterly, yearly).

A debt of $500 due in four years and $1000 due in six years is to be repaid by a single payment three years from now. If the interest rate is 4.4% compounded quarterly, how much is the payment?

Learn how to calculate a single payment to repay two debts of $500 and $1000 with 4.4% interest compounded quarterly. The solution covers key concepts like present value, compound interest, and time value of money. Suitable for college-level students in finance or economics.

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