Question

Question text You have just won the lottery. The Lottery Corporation offers you two options: 1. a lump-sum payment of $400,000, or 2. an annuity of $800,000 to be received in equal installments over the next 20 years. At what interest rate, r, will you be indifferent between the two options?

Answer 1

Use present value equation.
If present value of annuity equals 400,000, then you are indifferent.


`PV = A(v + v^2 + ...v^(20)) \\ \\ PV = A(v(1-v^(20)))/(1-v)`

where v = 1/(1+r) and A = annual payment = 800,000/20 = 40,000

PV = 400,000
After subbing in the numbers, the equation is:

`10 = (v(1-v^(20)))/(1-v) \\ \\ 10 - 10v = v - v^(21) \\ \\ v^(21) - 11v+10 = 0 \\ \\ v = 0.928034 \\ \\ (1)/(1+r) = 0.928034 \\ \\ r = 0.0775`

Therefore at interest rate of about 7.75% you are indifferent.