How does the interest rate affect the present value of an annuity due payment?

The interest rate has a significant impact on the present value of an annuity due payment. An annuity due refers to a series of equal cash flows or payments received or made at the beginning of each period for a specified number of periods. The present value of an annuity due payment is the current value of all future cash flows discounted at a specific interest rate.

When the interest rate increases, the present value of an annuity due payment decreases. This is because a higher interest rate implies a higher discount rate, which reduces the value of future cash flows. As a result, the present value of each payment received at the beginning of each period is lower.

Conversely, when the interest rate decreases, the present value of an annuity due payment increases. A lower interest rate means a lower discount rate, which increases the value of future cash flows. Consequently, the present value of each payment received at the beginning of each period is higher.

To illustrate this relationship, consider an example. Let's assume an annuity due payment of $1,000 per year for five years, with an interest rate of 5%. Using the formula for the present value of an annuity due, we can calculate the present value as follows:

PV = PMT * [(1 - (1 + r)^(-n)) / r]

Where PV is the present value, PMT is the annuity payment, r is the interest rate, and n is the number of periods.

Using the given values, the present value would be:

PV = $1,000 * [(1 - (1 + 0.05)^(-5)) / 0.05]
PV = $1,000 * [(1 - 1.27628) / 0.05]
PV = $1,000 * (-0.27628 / 0.05)
PV = $1,000 * (-5.5256)
PV = -$5,525.6

Now, let's assume the interest rate increases to 8%. Using the same formula, the present value would be:

PV = $1,000 * [(1 - (1 + 0.08)^(-5)) / 0.08]
PV = $1,000 * [(1 - 1.46933) / 0.08]
PV = $1,000 * (-0.46933 / 0.08)
PV = $1,000 * (-5.8666)
PV = -$5,866.6

As you can see, when the interest rate increased from 5% to 8%, the present value of the annuity due payment decreased from -$5,525.6 to -$5,866.6. This demonstrates the inverse relationship between the interest rate and the present value of an annuity due payment.

In conclusion, the interest rate has a direct impact on the present value of an annuity due payment. An increase in the interest rate leads to a decrease in the present value, while a decrease in the interest rate results in an increase in the present value.