Question

You are offered the choice of two payment streams: (a) \(\$ 150\) paid one year from now and \(\$ 150\) paid two years from now; (b) \(\$ 130\) paid one year from now and \(\$ 160\) paid two years from now. Which payment stream would you prefer if the interest rate is 5 percent? If it is 15 percent?

Short answer

The preferred payment stream will depend on the results of the present value calculations and can vary by the interest rate. Perform the calculations to find the exact answers.

Step-by-step solution

Calculate Present Values for 5% interest rate

First, calculate the present value of each payment stream assuming a 5% interest rate. Use the formula for calculating present value: \(P = \frac{F}{(1+r)^n}\), where P is the present value, F is the future value (the money to be received in the future), r is the interest rate, and n is the number of periods. For stream (a) the present value is \(P_{a} = \frac{150}{(1+0.05)^1} + \frac{150}{(1+0.05)^2}\) For stream (b) the present value is \(P_{b} = \frac{130}{(1+0.05)^1} + \frac{160}{(1+0.05)^2}\) Calculate these values.

Calculate Present Values for 15% interest rate

Next, calculate the present value of each payment stream assuming a 15% interest rate. The formula remains the same, only the rate (r) changes. For stream (a) the present value is \(P_{a} = \frac{150}{(1+0.15)^1} + \frac{150}{(1+0.15)^2}\) For stream (b) the present value is \(P_{b} = \frac{130}{(1+0.15)^1} + \frac{160}{(1+0.15)^2}\) Calculate these values.

Comparison and Decision

Once the present values are calculated, compare them to see which is higher. If the present value of stream (a) is higher than stream (b) under a particular interest rate, then that is the preferred payment stream. Conversely, if the present value of stream (b) is higher than that of stream (a), then stream (b) is the preferred payment stream. Repeat this step for both interest rates

Key concepts

Present Value

The concept of present value is fundamental in the world of finance, as it allows you to determine the worth of cash to be received in the future when compared to receiving the same amount of cash today. Imagine you have a time machine, but instead of traveling through time, you're moving money. If you bring money from the future to today, it will not have the same value—it might be less. What causes this difference? It's because of the potential earning capacity of that money, such as the interest it could earn if deposited in a bank.

For example, in the given exercise, payments are to be received at different times in the future. To find out which payment stream is more valuable to you today, you'd calculate the present value of each payment. This process is like discounting the future money, taking into account the 'time value of money'. It helps to answer questions like: 'Would you rather have \(100 today or a year from now?' Considering most circumstances, having \)100 today is usually better because you can invest it and earn interest over that year.

Interest Rate

The interest rate, often denoted as 'r' in financial formulas, is the price of money. It refers to the percentage charged on borrowed money or earned through investment. It represents the opportunity cost of using money for a specific purpose instead of investing it elsewhere. The interest rate is also a measure of the risk involved – higher rates often signal higher risk.

In the textbook exercise, the interest rate is crucial because it affects the present value of future payments. A higher interest rate would decrease the present value since it indicates that money today could potentially earn more if invested. To illustrate, a 5 percent interest rate would discount future money less than a 15 percent interest rate, because at 15 percent, the money is theoretically able to earn more over time. Therefore, when comparing payment streams, adjusting for the interest rate is necessary to make a fair comparison.

Future Value

In contrast to present value, the future value represents the amount of money that an investment will grow to over a period of time at a given interest rate. It's like taking cash today and sending it to the future to see what it will become. If you deposit \(100 in a savings account today, the future value is how much you'll have in that account at some future date, after accounting for the interest it accumulates over the investment period.

Future value is important in our exercise because we start with known future values (\)150 and \(130 in one year, \)150 and $160 in two years) and try to determine their value in today's terms. This comparison helps to judge which payment stream is preferable based on the future values and how they relate to present values when we apply a specific interest rate.

Present Value Formula

The present value formula is a powerful tool for converting future cash flows into their current monetary worth. The formula looks like this:
\[\begin{equation}P = \frac{F}{(1+r)^n}\end{equation}\]
Where

• P represents the present value,
• F is the future value or the amount of money to be received in the future,
• r is the interest rate per period, and
• n is the number of periods until the payment is received.

To apply this formula, as we saw in the exercise, you divide the future amount by the product of one plus the interest rate raised to the power of the number of periods. When you do this for each future payment and add them up, you get the total present value of that payment stream.

It's important to adjust the rate and periods correctly. For example, if the interest rate is annual and payments are also annual, then you use the rate directly. However, if you were dealing with monthly payments, you'd need to adjust the interest rate accordingly and redefine 'n' to reflect the number of months instead of years. This formula is the bedrock of time value of money calculations and is widely used in finance to determine the best investment or loan repayment options.