Problem 103 - Essential Calculus

Question

In Exercises \(103-106,\) use the formula for the \(n\) th partial sum of a geometric series $$\sum_{i=0}^{n-1} a r^{i}=\frac{a\left(1-r^{n}\right)}{1-r}$$ The winner of a \(\$ 1,000,000\) sweepstakes will be paid \(\$ 50,000\) per year for 20 years. The money earns \(6 \%\) interest per year. The present value of the winnings is \(\sum_{n=1}^{20} 50,000\left(\frac{1}{1.06}\right)^{n}\) Compute the present value and interpret its meaning.

Step-by-Step Solution

Answer

The present value of the winnings is the actual worth of the winnings today when the potential earnings from interest over the 20 years are considered. This amount is less than \$1,000,000 due to the time value of money concept.

Step 1: Identify the terms

In the given geometric series, the first term \(a= \$50,000\), the ratio \(r=\frac{1}{1.06}\) and the number of terms \(n=20\).

Step 2: Apply the formula for nth partial sum of a geometric series

The formula given is \(\sum_{i=0}^{n-1} a r^{i}=\frac{a\left(1-r^{n}\right)}{1-r}\). Substituting the values of \(a, r, n\), we get the present value of the winnings as \(\$50,000\left(\frac{1-\left(\frac{1}{1.06}\right)^{20}}{1-\frac{1}{1.06}}\right)\).

Step 3: Calculate the present value

Use a calculator to compute the value of the above expression. The result is the actual winnings in today's dollars, which is less than $1,000,000 due to the effect of interest.

Step 4: Interpret the meaning

The result of the calculation is the present value of the \(\$1,000,000\) sweepstakes. This is the actual worth of the winnings today if the effect of earning interest over the 20 years is considered.

Key Concepts

Present Value

The concept of **present value** is fundamental in finance and helps us understand the worth of a sum of money at a specific point in time. It considers how future cash flows, such as the payments from a sweepstakes, are affected by factors like interest rates. When you calculate the present value, you determine how much the future cash flows are worth right now.
The formula to calculate present value in a geometric series setting is:

• The sum of future cash flows.
• Discounted at a specific interest rate.
• Over a certain period.

The present value essentially helps investors and individuals like the sweepstakes winner understand the current value of future money given the rate of interest. It can highlight how the total of all those payments, when adjusted for interest, is less than the nominal value spread over many years.

Partial Sum

The term **partial sum** refers to a key idea in series and sequences, especially when dealing with geometric series. In this context, the partial sum is used to calculate the present value by combining a set series of payments, like annuity installments, into one value.
In our exercise, the formula for the nth partial sum is applied:

• \( S_n = \frac{a(1 - r^n)}{1 - r} \)
• Where "a" is the first term, "r" is the common ratio, and "n" is the number of terms.

For the winner receiving $50,000 for 20 years, the partial sum allows us to calculate how these payments accumulate, factoring in decreasing value over time due to the interest rate. This sum will yield the present value by giving us the worth of these payments today.

Interest Rate

An **interest rate** represents the cost of borrowing money or the return on investment for saving. In the context of the sweepstakes, it’s the rate used to calculate how each payment devalues over time. Here, a 6% interest rate indicates how each of the yearly $50,000 payments is worth less if received today rather than in the future.
Interest rates are critical because they can significantly affect the total present value calculation:

• Higher rates lead to a lower present value.
• Lower rates increase the present value.

Understanding interest rates helps to better grasp why the present value of $50,000 received over 20 years differs from their face value, and the real cost or benefit they represent over time.

Sweepstakes Payment

A **sweepstakes payment** refers to the method by which a winner receives their prize, usually in periodic installments. For example, instead of a one-time payment of $1,000,000, the winner gets $50,000 every year for 20 years. This is designed to provide steady income over a longer period.
The payment structure allows:

• Cash flow management over an extended time.
• Possibly reduced tax burden by spreading income.

However, understanding the present value of such payments, using the geometric series and interest rate, allows recipients to know the real worth of this income stream today. They see how the total collectible amount compares to a lump-sum alternative if offered.