Question

A rich aunt has promised you $2,000 one year from today. In addition, each year after that, she has promised you a payment (on the anniversary of the last payment) that is 7% larger than the last payment. She will continue to show this generosity for 20 years, giving a total of 20 payments. If the interest rate is 7%, what is her promise worth today? The present value is $. (Round to the nearest cent.)

Answer 1

The present value of the promise from the rich aunt is approximately $29,100.59 today.

Step-by-step explanation:

To calculate the present value of the promised payments, we need to discount each payment back to the present using the interest rate. Let's calculate the present value step by step:

1. The payment promised one year from today is $2,000. To find its present value, we need to discount it by dividing it by (1 + interest rate) raised to the power of the number of years (in this case, 1).
2. For the subsequent payments, we use the formula: Payment / (1 + interest rate)^t, where t is the number of years from the time of the last payment. For example, for the second payment, we have a payment of $2,000 * (1 + 0.07) = $2,140. The present value of the second payment is then $2,140 / (1 + 0.07) = $2,000.
3. We repeat this process for each subsequent payment until we reach the 20th payment.
4. Finally, we sum up all the present values of the payments to find the total present value of the promise.

After performing these calculations, the promise from the rich aunt is worth approximately $29,100.59 today.

Answer 2

To calculate the present value of the 20 payments your aunt promised, since the growth rate equals the discount rate, you simply multiply the first payment by the number of periods which sums up to $40,000.

Step-by-step explanation:

The question involves calculating the present value of a series of payments that are growing at a constant rate, given an interest rate. This is solved using the formula for the present value of a growing annuity. Given the first payment of $2,000, an annual growth rate of 7%, and an interest or discount rate also at 7%, we can find the present value.

To find the present value of this annuity, the formula is:
PV = Pmt x ((1 - (1 + g)^(-n)) / (r - g)),
where Pmt is the first payment, g is the growth rate, r is the discount rate, and n is the number of payments.

The present value of the annuity is given by:
PV = $2,000 x ((1 - (1 + 0.07)^(-20)) / (0.07 - 0.07)),
which simplifies to PV = $2,000 x 20, as the growth rate and discount rate are equal, hence the denominator becoming 0 would result in a division by zero which is not feasible. Instead, we simply multiply the first payment by the number of periods as the present value of each payment in such a scenario would equal the payment itself.

Therefore, the present value of your aunt's promise is:
PV = $2,000 x 20 = $40,000.