The Time Value of Money: 동영상 및 연습문제

The time value of money is the concept that a dollar received today is worth more than a dollar received in the future. This is because money available now can be invested to earn interest or a rate of return, increasing its value over time. For example, if you receive \$1 today and invest it at a 3% interest rate, in one year it will grow to \$1.03. Therefore, receiving money now allows you to generate additional income, making it more valuable than the same amount received later. This principle is crucial in business decisions, especially in capital budgeting, where companies compare current cash outflows to future cash inflows.

To calculate the future value (FV) of an investment, you use the formula: $\mathrm{FV}=\mathrm{PV}\cdot (1+r)n^{}$, where $\mathrm{PV}$ is the present value or initial amount, $r$ is the interest rate (expressed as a decimal), and $n$ is the number of periods. For example, if you invest \$100,000 at a 5% interest rate for 10 years, the future value is calculated as $100000\cdot (1+0.05){10}^{}$, which equals approximately \$162,889.46. This means your investment will grow to that amount after 10 years.

Present value (PV) is the current worth of a future amount of money, discounted at a specific interest rate. It answers the question: how much is a future sum worth today? The formula to calculate present value is: $\mathrm{PV}=\frac{\mathrm{FV}}{{(}^1}$, where $\mathrm{FV}$ is the future value, $r$ is the interest rate, and $n$ is the number of periods. For example, if you expect to receive \$162,889 in 10 years and the interest rate is 5%, the present value is $\frac{162889}{{(}^1}$, which equals \$100,000. This means \$162,889 in 10 years is equivalent to \$100,000 today.

A present value table simplifies the calculation of present value by providing factors based on the interest rate and number of periods. Instead of using the formula with exponents, you find the factor where the interest rate column and the number of periods row intersect. For example, for a 5% interest rate and 10 periods, the factor might be 0.614. To find the present value of a future amount, multiply the future value by this factor. If the future value is \$162,889, then the present value is \$162,889 × 0.614 = \$100,000. This method reduces errors and speeds up calculations, especially when dealing with multiple periods and rates. However, present value tables are typically only available for present value calculations, so future value calculations still require the formula.

Present value is more commonly used in capital budgeting because it allows businesses to assess the value of future cash inflows and outflows in today's dollars. Since investments involve spending money now (outflows) to receive money later (inflows), converting future amounts to present value helps compare and evaluate these cash flows on a consistent basis. This approach accounts for the time value of money, ensuring that earlier cash flows, which are more valuable, are properly weighted. While future value shows what current money will grow to, present value helps determine whether an investment's future returns justify the initial cost, making it essential for decision-making in capital budgeting.