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. the total 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 completing your answer, and round to the nearest cent.)

Explanation:

Step1: Identify the 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, we assume the lottery winning amount of $20$ million is paid out evenly over 20 years. Let's assume the annual payment $A$ is calculated as if the total winning of $20000000$ is divided evenly over 20 years, so $A=\frac{20000000}{20}=1000000$, $r = 0.05$ (5% interest rate), and $n = 20$.

Step2: Calculate the present - value factor

First, calculate $(1 + r)^{-n}=(1 + 0.05)^{-20}$. Using the formula $x^{-y}=\frac{1}{x^{y}}$, we have $(1.05)^{-20}=\frac{1}{(1.05)^{20}}$. $(1.05)^{20}\approx2.653297705$, so $(1.05)^{-20}\approx0.376889483$. Then $1-(1 + r)^{-n}=1 - 0.376889483 = 0.623110517$.

Step3: Calculate the present - value of the annuity

$\frac{1-(1 + r)^{-n}}{r}=\frac{0.623110517}{0.05}=12.46221034$. And $PV=A\times\frac{1-(1 + r)^{-n}}{r}=1000000\times12.46221034 = 12462210.34$.

Answer:

$12462210.34$