Question

Suppose you just won the state lottery, and you have a choice between receiving $2,700,000 today or a 20-year annuity of $250,000, with the first payment coming one year from today. Assuming both choices have the same present value, what rate of return is built into the annuity? Disregard taxes.

Answer 1

Present value of annutiy and one time payment is same as $2700000

Yearly annuity (P)= 250000

Number of annnuity (n) = 20

Present value of annuity = P*(1-(1/(1+i)^20))/i

We will calculate i by trial and error method.

Assume is is 6%

Present value of annutiy = 250000*(1-(1/(1+6%)^20))/6%

2867480.305

Assume i is 7%

Present value of annuity = 250000*(1-(1/(1+7%)^20))/7%

2648503.561

Actual value 2700000 is in between above values. So we will calcuate rate of return (i) by trial and error method.

interpolation formula = lower i +( uper i - lower i)*(lower i value - Actual i value)/(lower i value- uper i value)

6%+( (7%-6%)*(2867480.305-2700000)/(2867480.305-2648503.561))

=0.0676483147 or 6.76%

Note : financial calculator or excel function = rate(number of periods, annuity, -Present value)

=rate(20,250000, - 2700000)

=6.75%

So exact rate of return implied is 6.75%