Nathan was supposed to make a payment of $2,000 in 2 years and another payment for $500 in 4 years to Loon Company as part of a payment plan.
Instead, she is trying to reach an agreement with the company where she would pay an upfront amount now, and an amount of $800 in 5 years. Assume that money is worth 3.18% compounded quarterly.
a. Calculate the equivalent value of the $2,000 payment and the $500 payment today.

To calculate the equivalent value of the $2,000 and $500 payments today, we need to discount these future payments to their present values using the given interest rate, which is 3.18% compounded quarterly.

### Step 1: Determine the quarterly interest rate.
The nominal annual interest rate is 3.18%, and since it is compounded quarterly, we divide this rate by 4 to get the quarterly interest rate:

\[
\text{Quarterly interest rate} = \frac{3.18\%}{4} = \frac{0.0318}{4} = 0.00795 \text{ or } 0.795\%
\]

### Step 2: Calculate the present value of the $2,000 payment due in 2 years.
The number of compounding periods over 2 years is:

\[
n_1 = 2 \times 4 = 8 \text{ quarters}
\]

The present value $ PV_1 $ of the $2,000 payment is calculated using the formula:

\[
PV_1 = \frac{2000}{(1 + 0.00795)^{8}}
\]

### Step 3: Calculate the present value of the $500 payment due in 4 years.
The number of compounding periods over 4 years is:

\[
n_2 = 4 \times 4 = 16 \text{ quarters}
\]

The present value $ PV_2 $ of the $500 payment is calculated using the formula:

\[
PV_2 = \frac{500}{(1 + 0.00795)^{16}}
\]

### Step 4: Add the present values to get the total equivalent value today.

\[
PV_{\text{total}} = PV_1 + PV_2
\]

Let's calculate these values.The present values of the payments are as follows:

- The present value of the $2,000 payment due in 2 years is approximately **$1,877.23**.
- The present value of the $500 payment due in 4 years is approximately **$440.50**.

Therefore, the total equivalent value of these payments today is approximately **$2,317.73**.

Would you like to delve into any specific details or have any questions? Here are five related questions you might consider:

1. How would the present values change if the interest rate was compounded annually instead of quarterly?
2. What would be the effect on the present value if the payments were deferred by an additional year?
3. How would you calculate the equivalent value if the interest rate was different for each payment period?
4. What is the present value of a $1,000 payment due in 3 years under the same conditions?
5. How does changing the payment plan to monthly compounding affect the total present value?

**Tip:** When calculating present value, the more frequently the interest is compounded, the lower the present value will be for future payments.

Learn how to calculate the present value of future payments using compound interest at 3.18% compounded quarterly. This problem involves determining the present values of payments due in 2 and 4 years, emphasizing the time value of money and compound interest concepts. Suitable for undergraduate students and financial analysts.

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