A woman wins the lottery worth $8.74 million. The woman receives an immediate check for $641,000, and she plans to receive 24 annual checks of $337,458, with the first check coming exactly one year from now. If a buyer offers to purchase the 24 remaining checks for $6.72 million to be paid immediately, what is the buyer's annual interest rate? State your answer as a percentage with decimals. In other words, enter 8.3 not 0.083.
Answer
As buyers offers to purchase the 24 remaining checks for 6.72 million, he must have calculated the present value of the checks and found that Net Present Value of these checks = 6.721 million
Formula:
Present Value of Annuity(Periodic Payments) is given by:
NPV = (P/i)(1 - 1/(1 + i)n)
Here, P = Periodic Payment = 337458
n = time = 24
i = interest rate to calculate
NPV = (337458/i)(1 - 1/(1 + i)24)
This NPV must equal to 6.72 million
=> (337458/i)(1 - 1/(1 + i)24) = 6.72 million
=> (1 - 1/(1 + i)24) = 19.91i
Let t = i + 1
=> (1 - 1/t24) = 19.913(t - 1)
=> 20.913 = 19.913t + 1/t24
=> 19.913t25 - 20.913t24 + 1 = 0
Solving above equation we get :
t = 1.0155
=> r = t - 1 = 0.0155 ~ 1.55%
Hence Buyers annual interest rate = 1.55% ( 1.6% approx)
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