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4.1. 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 10$ 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!

4.2. Future Value of Annuities

Annuities are level streams of payments. Each payment is the same amount, and occurs at a regular interval. Sometimes, one may be curious to learn how much a recurring stream of payments will grow to after a number of periods.

4.3. Future Value of an Annuity Due

An annuity due (also known as an annuity in advance) involves a level stream of payments, with the payments being made at the beginning of each time period. For instance, perhaps you plan on saving for retirement by investing $5,000 at the beginning of each year for the next 5 years. If the annual interest rate is 10% per year, how much will you accumulate by the end of the 5-year period? The following graphic shows how each of the five individual payments would grow, and the accumulated total would reach $33,578:

Although the graphic provides a useful explanatory tool, it is a bit cumbersome to implement. The same conclusion can be reached by reference to a FUTURE VALUE OF AN ANNUITY DUE TABLE. Examine the table linked at the website to find the value of 6.71561 (10% column/5-period row). Multiplying the $5,000 annual payment by this factor yields $33,578 ($5,000 x 6.71561). This means that the $25,000 paid in will have grown to $33,578; perhaps Albert Einstein was right!

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