The winner of a lottery chooses to receive annual payments of $160,000 at the end of each year for 25 years. If the current interest rate is 5.9%, find the present value (in dollars) of the payments. (Round your answer to the nearest cent. See Example 5 in this section.)
The correct answer and explanation is:
Let’s solve this step-by-step.
Problem Summary:
A lottery winner receives $160,000 at the end of each year for 25 years.
The current interest rate (discount rate) is 5.9% per year.
We want to find the present value (PV) of these payments.
What is Present Value?
The present value is the current worth of a stream of future payments, discounted back to the present using the interest rate. It tells us how much the series of future payments is worth right now.
Since the payments are the same every year, this is an ordinary annuity problem.
Present Value Formula for an Ordinary Annuity:
PV=P×1−(1+r)−nrPV = P \times \frac{1 – (1 + r)^{-n}}{r}
Where:
- P=160,000P = 160,000 (annual payment)
- r=0.059r = 0.059 (annual interest rate as a decimal)
- n=25n = 25 (number of years/payments)
Step 1: Calculate the present value factor
1−(1+r)−nr=1−(1+0.059)−250.059\frac{1 – (1 + r)^{-n}}{r} = \frac{1 – (1 + 0.059)^{-25}}{0.059}
First, calculate (1+0.059)−25(1 + 0.059)^{-25}: 1+0.059=1.0591 + 0.059 = 1.059 1.059−25=11.059251.059^{-25} = \frac{1}{1.059^{25}}
Calculate 1.059251.059^{25}:
Using a calculator: 1.05925≈4.2561.059^{25} \approx 4.256
So, 1.059−25=14.256≈0.2351.059^{-25} = \frac{1}{4.256} \approx 0.235
Now substitute back: 1−0.2350.059=0.7650.059≈12.97\frac{1 – 0.235}{0.059} = \frac{0.765}{0.059} \approx 12.97
Step 2: Calculate the present value
PV=160,000×12.97=2,075,200PV = 160,000 \times 12.97 = 2,075,200
Final answer:
$2,075,200.00\boxed{\$2,075,200.00}
Explanation:
The present value calculation discounts each of the $160,000 payments back to today’s value using the 5.9% interest rate. Because the payments continue for 25 years, we use the annuity formula to sum the discounted value of all payments. This shows that receiving $160,000 yearly for 25 years is equivalent to receiving about $2,075,200 as a lump sum today, assuming the money could be invested at a 5.9% annual rate.
This present value is useful for comparing options: for instance, if the lottery winner could take the entire amount now instead of annual payments, they’d expect to get around $2.075 million to be financially equivalent. If they could invest at a rate higher than 5.9%, the lump sum might be lower, and vice versa.