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.
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%
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