# Compound Interest and Present Value

You have heard the expression that "time is money." In capital budgeting this concept is measured and brought to bear on the decision process. The fundamental idea is that a dollar received today is worth more than a dollar to be received in the future. This result occurs because a dollar in hand can be invested to generate additional returns; such would not be the case with a dollar received in the future.

In the context of capital budgeting, assume two alternative investments have the same upfront cost. Investment Alpha returns $100 per year for each of the next five years. Investment Beta returns $50 per year for each of the next 10 years. Based solely on this information, you should conclude that Alpha is preferred to Beta. Although the total cash returns are the same, the time value of money is better for Alpha than Beta. With Alpha, the money is returned sooner, allowing for enhanced reinvestment opportunities. Of course, very few capital expenditure choices are as clear cut as Alpha and Beta. Therefore, accountants rely on precise mathematical techniques to quantify the time value of money.

## Compound Interest

The starting point for understanding the time value of money is to develop an appreciation for compound interest. "The most powerful force in the universe is compound interest." The preceding quote is often attributed to Albert Einstein, the same chap who unlocked many of the secrets of nuclear energy. While it is not clear that he actually held compound interest in such high regard, it is clear that understanding the forces of compound interest is a powerful tool. Very simply, money can be invested to earn money. In this context, consider that when you spend a dollar on a soft drink, you are actually foregoing 100 per year for the rest of your life (assuming a 10% interest rate). And, as you will soon see, that annual dime of savings builds to much more because of interest that is earned on the interest! This is the almost magical power of compound interest.

Compound interest calculations can be used to compute the amount to which an investment will grow in the future. Compound interest is also called future value. If you invest $1 for one year, at 10% interest per year, how much will you have at the end of the year? The answer, of course, is $1.10. This is calculated by multiplying the $1 by 10% ($1 X 10% = $0.10) and adding the $0.10 to the original dollar. And, if the resulting $1.10 is invested for another year at 10%, how much will you have? The answer is $1.21. That is, $1.10 X 110%. This process will continue, year after year.

The annual interest each year is larger than the year before because of "compounding." Compounding simply means that your investment is growing with accumulated interest, and you are earning interest on previously accrued interest that becomes part of your total investment pool. This formula expresses the basic mathematics of compound interest:

Where "i" is the interest rate per period and "n" is the number of periods

So, how much would $1 grow to in 25 years at 10% interest? The answer can be determined by taking 1.10 to the 25th power [(1.10)25], and the answer is $10.83. Future value tables provide predetermined values for a variety of such computations (such a table is found at the FUTURE VALUE OF $1 link on the companion website). To experiment with the future value table, determine how much $1 would grow to in 10 periods at 5% per period. The answer to this question is $1.63, and can be found by reference to the value in the "5% column/10-period row." If the original investment was $5,000 (instead of $1), the investment would grow to $8,144.45 ($5,000 X 1.62889). In using the tables, be sure to note that the interest rate is the rate per period. The "period" might be years, quarters, months, etc. It all depends on how frequently interest is to be compounded. For instance, a 12% annual interest rate, with monthly compounding for two years, would require you to refer to the 1% column (12% annual rate equates to a monthly rate of 1%) and 24-period row (two years equates to 24 months). If the same investment involved annual compounding, then you would refer to the 12% column and 2-period row. The frequency of compounding makes a difference in the amount accumulated - for the given example, monthly compounding returns 1.26973, while annual compounding returns only 1.25440!