Question

present value of an annuity. winners of the georgia lotto drawing are given the choice of receiving the winning amount divided equally over 20 years or as a lump - sum cash option amount. the cash option amount is determined by discounting the annual winning payment at 5% over 20 years. this week the lottery is worth $20 million to a single winner. what would the cash option payout be? the cash option payout would be $ (round to the nearest cent.) (use your financial calculator or you may use the financial tables in appendix c in computing your answer, and round to the nearest cent.)

Explanation:

Step1: Recall present - value of an ordinary annuity formula

The formula for the present - value of an ordinary annuity is $PV = A\times\frac{1-(1 + r)^{-n}}{r}$, where $PV$ is the present value, $A$ is the annual payment, $r$ is the interest rate per period, and $n$ is the number of periods. Here, the total winning amount is divided equally over 20 years. If the total winning amount is assumed to be paid out as an annuity, and we want to find the present value of this annuity to get the cash - option payout. Let's assume the total winning amount is $A_{total}=20000000$. The annual payment $A=\frac{20000000}{20}=1000000$, the interest rate $r = 0.05$, and the number of years $n = 20$.

Step2: Substitute values into the formula

$PV=1000000\times\frac{1-(1 + 0.05)^{-20}}{0.05}$. First, calculate $(1 + 0.05)^{-20}$. Using the formula $x^{-n}=\frac{1}{x^{n}}$, we have $(1 + 0.05)^{-20}=\frac{1}{(1.05)^{20}}$. $(1.05)^{20}\approx2.653297705$. So, $(1 + 0.05)^{-20}\approx\frac{1}{2.653297705}\approx0.376889483$. Then, $1-(1 + 0.05)^{-20}=1 - 0.376889483 = 0.623110517$. $\frac{1-(1 + 0.05)^{-20}}{0.05}=\frac{0.623110517}{0.05}=12.46221034$.

Step3: Calculate the present value

$PV = 1000000\times12.46221034=12462210.34$.

Answer:

$12462210.34$