Lottery Winnings In most states, the winnings of...

Lottery Winnings In most states, the winnings of million-dollar lottery jackpots are divided into equal payments given annually for 20 years. (In Colorado, the results are distributed over 25 years.) This means that the present value of the jackpot is worth less than the stated prize, with the actual value determined by the interest rate at which the money could be invested. Source: The New York Times Magazine. a. Find the present value of a $\$ 1$ million lottery jackpot distributed in equal annual payments over 20 years, using an inter- est rate of 5$\%$ . b. Find the present value of a $\$ 1$ million lottery jackpot distributed in equal annual payments over 20 years, using an interest rate of 9$\% .$ c. Calculate the answer for part a using the 25 -year distribution time in Colorado. d. Calculate the answer for part b using the 25-year distribution time in Colorado.

Lottery Winnings In most states, the winnings of million-dollar lottery jackpots are divided into equal payments given annually for 20 years. (In Colorado, the results are distributed over 25 years.) This means that the present value of the jackpot is worth less than the stated prize, with the actual value determined by the interest rate at which the money could be invested. Source: The New York Times
Magazine.
a. Find the present value of a $\$ 1$ million lottery jackpot distributed in equal annual payments over 20 years, using an inter- est rate of 5$\%$ .
b. Find the present value of a $\$ 1$ million lottery jackpot distributed in equal annual payments over 20 years, using an interest rate of 9$\% .$
c. Calculate the answer for part a using the 25 -year distribution time in Colorado.
d. Calculate the answer for part b using the 25-year distribution time in Colorado.

Key Concepts

Discount Rate

The discount rate, which is equivalent to the interest rate in these calculations, is used to determine the present value of future cash flows. It reflects the opportunity cost of capital and the risk involved in deferring receipt of money.

Annuity

An annuity refers to a series of equal payments made at regular intervals over a specified period. The concept is crucial for understanding how periodic payments, such as lottery winnings, are valued in today's terms.

Time Value of Money

This is the fundamental financial principle stating that a dollar today is worth more than a dollar in the future due to its potential earning capacity. It underpins the need to discount future cash flows to determine their current value.

Present Value

Present value is the current worth of a future sum of money or stream of cash flows given a specific rate of return. It is calculated by discounting future payments back to their value today using an appropriate interest rate.

Transcript

00:01 Okay, so for part a, we're playing the lottery, and we just won, and we're getting paid out over, let's see, equal annual payments over 20 years, we're getting paid a million dollars.

00:13 So it wants us to find the present value.

00:16 So our yearly payment is going to be a million dollars divided about 20.

00:23 So a million dollars divided by 20, i believe is, well, let's just make sure.

00:31 A million dollars divided about 20, yes, is 50 ,000.

00:35 So our present value is going to be 50 ,000 times one minus our, we need to know our interest rate.

00:45 So for part a, so it's finding to distribute equal annual payments over 20 years using an interest rate of 5%.

00:53 So by 5%, so we're going to be 1 minus 1 .05 to the negative 20th power divided by 0 .05.

01:08 Okay, so we plug that into a calculator.

01:15 50 ,000 times 1 minus 1 .05 to the negative 20th power divided by 0 .05.

01:28 And we're going to get the present value of this winnings as $623 ,110 and $52 .2 .2.

01:49 Part b says find the present value of a million dollar lottery jackpot to straight and equal payments over 20 years using 9%.

02:01 So again, everything's the same.

02:04 Only we're going to change the 0 .05 to 1 .09 to the negative 20th over 0 .09.

02:17 Okay, it should be easy to just go back in your calculator and change those 1 .05s to 1 .09s.

02:25 Okay, so this amount is only going to be worth $456 ,427...

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