Question

The present value of a dollar declines as (LO5) a) the interest rate declines and the number of years you wait for your money declines b) the interest rate rises and the number of years you wait for your money rises c) the interest rate declines and the number of years you wait for your money rises d) the interest rate rises and the number of years you wait for your money declines

Short answer

The present value of a dollar declines as \(\textbf{the interest rate rises and the number of years you wait for your money rises}\), as given in Option b).

Step-by-step solution

Option a)

In this option, both the interest rate and the number of years you wait for your money decline. First, let's analyze the effect of a decline in the interest rate. If the interest rate decreases, the denominator of the present value formula will decrease, leading to an increase in the present value. Secondly, a decrease in the number of years will reduce the denominator's power. Thus, both effects will lead to an increase in the present value. So, Option a) will not yield a decline in the present value of a dollar.

Option b)

In this option, both the interest rate and the number of years you wait for your money rise. First, let's analyze the effect of a rise in the interest rate. If the interest rate increases, the denominator of the present value formula will increase, resulting in a decrease in the present value. Secondly, an increase in the number of years will raise the denominator's power. Both effects will lead to a decline in the present value. So, Option b) will yield a decline in the present value of a dollar.

Option c)

In this option, the interest rate declines while the number of years you wait for your money rises. First, let's analyze the effect of a decline in the interest rate. If the interest rate decreases, the denominator of the present value formula will decrease, leading to an increase in the present value. Secondly, an increase in the number of years will raise the denominator's power. These two effects are opposite in their impact on present value, and therefore, it is not clear whether the present value will increase or decrease. So, Option c) cannot be concluded to cause a decline in the present value of a dollar.

Option d)

In this option, the interest rate rises while the number of years you wait for your money declines. First, let's analyze the effect of a rise in the interest rate. If the interest rate increases, the denominator of the present value formula will increase, resulting in a decrease in the present value. Secondly, a decrease in the number of years will reduce the denominator's power. These two effects are opposite in their impact on present value, and therefore, it is not clear whether the present value will increase or decrease. So, Option d) cannot be concluded to cause a decline in the present value of a dollar. Based on the analysis of the given options, we can conclude that the present value of a dollar declines as "the interest rate rises and the number of years you wait for your money rises," as given in Option b).

Key concepts

Interest Rate

The interest rate is a critical component in the study of finance, acting as the cost of borrowing money or the reward for saving and investing it. When you deposit money in a savings account, you earn interest because the bank is effectively paying you for the privilege of using your funds. Conversely, when you take out a loan, you pay interest for accessing funds now rather than later.

Mathematically, the interest rate is often expressed as a percentage and is used to calculate the interest payment or receivable over a certain period. For instance, an annual interest rate of 5% implies that you will earn \(5 per year for a \)100 investment. In financial calculations, the interest rate is a pivotal factor that affects the present value of future cash flows, influencing how much future money is worth in today's terms.

Time Value of Money

The time value of money is a financial concept that expresses a fundamental truth: a dollar in hand today is worth more than a dollar promised in the future. This is due to the potential earning capacity of money; if you have it now, you can invest it and earn interest. As a result, we discount future money to reflect its present value.

In equations, we use the formula \( PV = \frac{FV}{(1 + r)^n} \), where \(PV\) represents the present value, \(FV\) stands for future value, \(r\) is the interest rate, and \(n\) is the number of periods. As either the interest rate increases or the time until receipt of the money lengthens, the present value diminishes because of higher opportunity costs and greater uncertainty. Conversely, lowering the interest rate or reducing the time frame elevates the present value.

Financial Mathematics

Financial mathematics is the application of mathematical methods to solve problems related to finance. This branch of applied mathematics is essential for understanding concepts like present value, compound interest, annuities, and amortization schedules. It encapsulates a variety of tools to help interpret the financial marketplace and make informed economic decisions.

Performing financial calculations often involves determining the present or future value of cash flows, assessing risk, and analyzing investment strategies. Financial mathematics relies heavily on exponential functions, probability theory, and stochastic processes to model and predict the behavior of financial markets. The accuracy of these calculations can be the difference between profitable decisions and costly mistakes in personal and corporate finance. For students, mastering financial mathematics is crucial to not only understanding how money grows over time but also to optimizing financial outcomes in a real-world setting.