91Ó°ÊÓ

Find the following values, using the equations, and then work the problems using a financial calculator or the tables to check your answers. Disregard rounding differences. (Hint: If you are using a financial calculator, you can enter the known values, and then press the appropriate key to find the unknown variable. Then, without clearing the TVM register, you can "override" the variable that changes by simply entering a new value for it and then pressing the key for the unknown variable to obtain the second answer. This procedure can be used in parts b and \(\mathrm{d}\), and in many other situations, to see how changes in input variables affect the output variable.) Assume that compounding/discounting occurs once a year. a. An initial \(\$ 500\) compounded for 1 year at 6 percent. b. An initial \(\$ 500\) compounded for 2 years at 6 percent. c. The present value of \(\$ 500\) due in 1 year at a discount rate of 6 percent. d. The present value of \(\$ 500\) due in 2 years at a discount rate of 6 percent.

Step by step solution

01

Understanding the Problem

We need to find the future values of an initial amount after compounding interest and the present values given a future amount and a discount rate. We use the compound interest formula for future values and the present value formula for present values.

02

Compounding for 1 Year

For part (a), we use the future value formula: \( FV = PV \times (1 + r)^n \), where \( PV \) is the present value \( \\(500 \), \( r = 0.06 \), and \( n = 1 \) year. Thus, \( FV = 500 \times (1 + 0.06)^1 = 500 \times 1.06 = \\)530 \).

03

Compounding for 2 Years

For part (b), again use the future value formula \( FV = PV \times (1 + r)^n \), with \( n = 2 \) years. Then, \( FV = 500 \times (1 + 0.06)^2 = 500 \times 1.1236 = \$561.80 \).

04

Calculating Present Value for 1 Year

For part (c), use the present value formula: \( PV = \frac{FV}{(1 + r)^n} \), where \( FV = \\(500 \), \( r = 0.06 \), and \( n = 1 \). Thus, \( PV = \frac{500}{1.06} \approx \\)471.70 \).

05

Calculating Present Value for 2 Years

For part (d), use the same present value formula but with \( n = 2 \). Then, \( PV = \frac{500}{(1 + 0.06)^2} = \frac{500}{1.1236} \approx \$444.54 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Future Value Calculation

When we talk about future value in finance, we're referring to the amount an investment is expected to grow to over a period of time, given a specific interest rate.
This concept is crucial for understanding how money can grow over time due to compounding interest.
The formula we use to calculate the future value is:\[ \text{FV} = \text{PV} \times (1 + r)^n \]Where:

• \( \text{FV} \) represents the future value
• \( \text{PV} \) is the present value, or initial amount invested
• \( r \) is the interest rate
• \( n \) is the number of years the money is invested for

Let's apply this to part (a) of our example, where we start with \(500 at an interest rate of 6% for 1 year:
\[ \text{FV} = 500 \times (1 + 0.06)^1 = 530 \]This means that at the end of one year, with compounded interest, our initial \)500 grows to $530.
This formula not only helps in calculating growth for one year but is also valid for any number of years.

Present Value Calculation

Present value calculations are used to determine the current worth of an amount due in the future.
This is done by discounting the future amount using a specified interest rate, often referred to as the discount rate.
The present value formula is:\[ \text{PV} = \frac{\text{FV}}{(1 + r)^n} \]Where:

• \( \text{PV} \) is the present value
• \( \text{FV} \) is the future value or the amount expected in the future
• \( r \) is the discount rate
• \( n \) is the number of years until the future value is received

For example, in part (c) of our problem, we want to know what \(500 due in 1 year is worth today at a 6% discount rate.
\[ \text{PV} = \frac{500}{1.06} \approx 471.70 \]This means that \)500 received a year from now is equivalent to about $471.70 today at a 6% interest rate.
Understanding present value helps in making informed decisions about investments and comparing cash flows occurring at different times.

Discount Rate

The discount rate is fundamental in both calculating present value and understanding how the value of money changes over time.
It reflects the time value of money—helping to assess the desirability of an investment or the cost of borrowing money.
Consider it the reverse of the interest rate used to grow money; instead of pushing the value up (as with future value), the discount rate pulls future value down to its present value.
Factors that influence the choice of a discount rate include:

• Market interest rates
• Risk level of the investment
• Inflation expectations

In practical applications, if we know the future value, the discount rate helps us determine how much we would need to invest today to achieve that future amount.
As shown in part (d), determining present values for future cash flows is essential for investment appraisals and comparing options that involve time-differentiated cash flows.

Financial Calculator Usage

Using a financial calculator can simplify the process of calculating future and present values.
These calculators are specifically designed to handle time value of money (TVM) problems efficiently and accurately.
Here's how to use it for our exercise:

• Input the known variables into the calculator: present value (PV), interest rate (r), and number of periods (n).
• Press the appropriate key to find the unknown variable—either future value (FV) or present value (PV).
• Without clearing the calculator's memory, easily adjust any of these variables to see how changes affect the result. This is useful for sensitivity analysis, to understand how variations in interest rates or investment duration impact future or present values.

By practicing these steps, students can quickly verify their calculations and deepen their understanding of how financial models operate.
As demonstrated in the original exercise, the flexibility of financial calculators showcases their practicality and powerful capability in financial analysis.