Question

Your great aunt left you $15,000 when she died. You can invest the money to earn 6.5% per year. If you spend $2,100 per year out of this inheritance, how long will the money last? Try to use an appropriate annuity formula, and make sure to clearly state any assumptions you make.

Answer 1

Answer:

The money will last for approximately 10 years.

Explanation:

Assuming the withdrawal to spend $2,100 is made at the end of each year, the relevant formula to use is therefore the formula for calculating the present value (PV) of an ordinary annuity as follows:

PV = P * [{1 - [1 / (1 + r)]^n} / r] …………………………………. (1)

Where;

PV = Present value of the inheritance left by your great aunt = $15,000

P = yearly withdrawal = $2,100

r = interest rate = 6.5%, or 0.065

n = number of years the money will last = ?

Substitute the values into equation (1) and solve for n as follows:

15,000 = 2,100 * [{1 - [1 / (1 + 0.065)]^n} / 0.065]

15,000 / 2,100 = {1 - [1 / 1.065]^n} / 0.065

7.14285714285714 = [1 - 0.938967136150235^n] / 0.065

7.14285714285714 * 0.065 = 1 - 0.938967136150235^n

0.464285714285714 = 1 - 0.938967136150235^n

0.938967136150235^n = 1 - 0.464285714285714

0.938967136150235^n = 0.535714285714286

Loglinearize both sides, we have:

n * log 0.938967136150235 = log 0.535714285714286

n = log 0.535714285714286 / log 0.938967136150235

n = -0.271066772286538 / -0.0273496077747564

n = 9.9112 years, or approximately 10 years

Therefore, the money will last for approximately 10 years.