A rich aunt has promised you $4 000 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 3% 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 4%, what is her promise worth today?
Question
A rich aunt has promised you $4 000 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 3% 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 4%, what is her promise worth today?
Solution
To solve this problem, we need to use the formula for the present value of a growing annuity. The formula is:
PV = Pmt * [(1 - (1 + g / r) ^ -n) / (r - g)]
Where:
- PV is the present value of the annuity
- Pmt is the first payment
- g is the growth rate of the payments
- r is the interest rate
- n is the number of payments
In this case:
- Pmt = $4,000
- g = 3% = 0.03
- r = 4% = 0.04
- n = 20
Substituting these values into the formula, we get:
PV = 4000 * [(1 - (1 + 0.03 / 0.04) ^ -20) / (0.04 - 0.03)]
Solving this equation will give you the present value of the aunt's promise.
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