Suppose you just won the state lottery, and you have a choice between receiving $2,550,000 today or a 20-year annuity of $250,000, with the first payment coming one year from today.What rate of return is built into the annuity? Disregard taxes.
A) 7.12%
B) 7.49%
C) 7.87%
D) 8.26%
E) 8.67%

Question

Quizplus · Accepted Answer

To compare the two options, we need to find the present value of the annuity payments. Using the formula for the present value of an annuity, we get:

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

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

Substituting PMT = $250,000, n = 20, and PV = $2,550,000, we can solve for r:

$2,550,000 = $250,000 x [1 - (1+r)^-20] / r

Multiplying both sides by r and rearranging, we get:

$2,550,000r = $250,000[1 - (1+r)^-20]

Expanding the brackets and simplifying, we get:

$2,550,000r = $250,000 - $250,000(1+r)^-20

$2,550,000r + $250,000(1+r)^-20 = $250,000

$2,550,000r + $250,000/(1+r)^20 = $250,000

$2,550,000r = $250,000 - $250,000/(1+r)^20

$2,550,000r = $250,000[(1+r)^20 -1]/(1+r)^20

$2,550,000r(1+r)^20 = $250,000[(1+r)^20 -1]

Dividing both sides by (1+r)^20 and simplifying, we get:

r = 7.49%

Therefore, the rate of return built into the annuity is 7.49%. As this is greater than the expected rate of return on other investments, it is better to choose the annuity option.

Suppose You Just Won the State Lottery, and You Have