Problem 14 - Accounting

Question

Determine the present value of $400,000 to be received in three years, using an interest rate of 10%, compounded annually, as follows: a. By successive divisions. (Round to the nearest dollar.) b. By using the present value table in Exhibit 4.

Step-by-Step Solution

Answer

Present value is approximately $300,525 using successive divisions and $300,520 using the table.

Step 1: Understand the Present Value Formula

The present value (PV) formula is used to find the current worth of a sum of money to be received in the future, given an interest rate. The formula is: \[ PV = \frac{FV}{(1 + r)^n} \] where \( FV \) is the future value, \( r \) is the interest rate, and \( n \) is the number of periods.

Step 2: Identify the Variables

In the given problem, \( FV = 400,000 \), \( r = 0.10 \) (10% interest), and \( n = 3 \) years. We will use these values to solve for the present value using the formula.

Step 3: Calculate Using Successive Divisions

First, calculate \((1 + r)^n\):\[ (1.10)^3 = 1.1 \times 1.1 \times 1.1 = 1.331 \]Now, divide the future value by this factor:\[ PV = \frac{400,000}{1.331} \approx 300,525 \]So, the present value is approximately $300,525.

Step 4: Use the Present Value Table

Refer to the present value table in Exhibit 4 (which lists values of \(\frac{1}{(1 + r)^n}\) for different \( r \) and \( n \)) to find the present value factor for 10% and 3 years. The table gives approximately 0.7513 for these parameters. Multiply this factor by the future value:\[ PV = 400,000 \times 0.7513 = 300,520 \]Hence, the present value is $300,520 according to the table.

Key Concepts

Time Value of Money

The concept of the Time Value of Money (TVM) is a fundamental principle in finance. It proposes that a sum of money in the present is worth more than the same sum in the future. This happens because money today can be invested to earn a return, leading to more money tomorrow. Essentially, having money now gives you the opportunity to grow wealth through investment, leveraging interest to increase your sum over time.

• Money earned today can be reinvested to earn more, compounding its value.
• Inflation erodes the purchasing power of money over time.
• Risk factors can impact the future value of money.

Understanding TVM is crucial because it helps people make informed decisions about investing, saving, and spending. It underlies the processes of discounting, which you might use to calculate present value, and compounding, which determines future value.

Interest Rate

Interest rates are the cost of borrowing money or the reward for saving, usually expressed as a percentage of the principal amount per year. In finance, they can significantly impact the present and future value of money.
An interest rate determines how much money can earn over time. The higher the rate, the more money you can potentially earn or be charged when borrowing. There are different types of interest rates:

• Simple Interest: Earned or paid on the original principal alone.
• Compound Interest: Accumulates on both the initial principal and the interest from previous periods.

Changes in interest rates can affect economic activity, influencing how companies invest or consumers spend and save. Knowing how to calculate its impact is essential for making sound investment and financing decisions.

Future Value

Future Value (FV) refers to the amount of money an investment will grow to over a period of time at a specific interest rate. It's a forward-looking measure that allows you to predict how much your investments today will be worth in the future.
By using the future value formula, you can calculate how much money you will have after interest is applied over a certain number of periods. For instance, investing with compound interest increases the future value of money more significantly compared to simple interest. Here's how it works with compound interest:

• Calculate compound interest using the formula: \[ FV = PV \times (1 + r)^n \] where \( PV \) is the present value, \( r \) is the interest rate, and \( n \) is the number of periods.

Being able to calculate future value is particularly important for setting financial goals and retirement savings, as it gives an idea of the growth potential of current investments.

Present Value Table

A Present Value Table, often used in time value of money calculations, helps simplify the calculation of present value by providing the present value factors. These factors can quickly indicate what a future sum is worth in today's terms given a certain interest rate and period.
Using the formula for present value: \[ PV = \frac{FV}{(1 + r)^n} \] The present value table takes the hassle out of calculating \((1 + r)^n\). Instead, it directly provides the factor which can be multiplied with the future value to arrive at the present value. Here's how it works:

• Locate the interest rate across the top of the table.
• Find the corresponding period along the side of the table.
• The intersection point gives you the present value factor.

This table is a handy tool for investors, accountants, and financial analysts, as it provides a quick and accurate way to convert future cash inflows to their present value, aiding in better financial planning and analysis.