In This Chapter
Understanding the time value of money
Realizing the power of compounding and the Rule of 72 Seeing the power of buying cheap
Examining the devastating effects of underperformance Determining what future returns are worth today Watching for hidden pitfalls in large numbers Factoring in inflation, taxes, interest, and risk
I
t’s hard to imagine the words “short” and “painless” being used in the same sentence as “math,” but, this chapter attempts to prove that math — at least the math you need for investing — can indeed be both short and relatively painless. We explore a handful of fundamental math concepts in value investing, keeping it simple and practical and focusing on how the concepts are applied.You won’t find any statistics, stochastics, or oscillators — just some harm- less algebra and arithmetic and basic principles that value investors employ daily. No fancy Ivy League portfolio-theory higher-math stuff that you may have heard about, or seen, or even studied in school, because that doesn’t really apply to value investing. And you won’t come across formulas with little Greek symbols, either. As Warren Buffett once said, “If calculus were required, I’d have to go back to delivering papers.”
The tools and underlying principles covered in this chapter are thoroughly understood and employed by value investing masters. They will become part of your investing vocabulary, just as knowing the taste and strength of garlic is part of your cooking vocabulary. Knowing the chemical makeup and con- centration of the allyl propyl disulfide in garlic is hardly important for cook- ing. Similarly, the math technicalities themselves aren’t so important either, leaving us to ponder whether we should have used the word “math” in this chapter in the first place.
Lesson 1: Time Value of Money
First, we explore a cornerstone principle in all of business and finance:
A dollar today isn’t worth the same amount as a dollar yesterday, nor is it worth the same amount as a dollar tomorrow. The bottom line is, invested money appreciates with time — how and how much is examined here.
Note there will be exponents in this discussion, but keep the faith — the calculations are easy, and today’s calculators and spreadsheets easily perform them.
If you are already familiar with time value of money basics, you can probably skip Lessons 1 through 3 in this chapter.
Money and time: An interesting story
Suppose that you have a $10 bill in your pocket. What’s it worth today? Ten bucks. You’re right, but keep reading.
Now suppose that the $10 is in a bank account or some other investment vehicle that pays a return. This return can be a fixed payment in return for the financial institution’s use of the money, known as interest. Or it can be a return in some other form, say, a profit generated through the use of the
$10 in a business, increasing the value of the business.
Either way, if you leave these dividends alone and don’t withdraw them, they become part of the investment. At a 10 percent rate of return, the $10 becomes $11 in the first year, and $11 is invested in the interest or profit- generating asset for the second year. With $11 invested, assuming that the interest or profitability stays the same, you reap the greater rewards due an
$11 investment. In this case, the reward is now $1.10, not $1. The investment is now worth $12.10 ($10 + $1 + $1.10). Now $12.10 is invested, and the return is $1.21. And so forth. Both the investment and the incremental dollar return grow over time. Each year’s golden eggs become part of the next year’s goose, which then lays still more golden eggs.
Present and future value
So what is the investment worth? It depends. The investment is worth $10 today, as we said earlier. That ten bucks is known as the present value.Pretty simple so far, right? Now what about the future? If the investment grows over time, the total value will include the initial $10 plus all returns generated during that time. This is known as future value.
Invested money grows and compounds. In other words, there is growth on the original investment, plus return and growth on returns already earned. A snowball rolling downhill is a good analogy. As the ball gets bigger, it picks up
ever-larger amounts of snow. How much? Compounding formulas, which are driven by rate of return and the amount of time, supply the answer.
Investment returns in the future: When it isn’t yours yet
Suppose that someone promises to pay you $10 five years from now. Are you $10 wealthier? In five years you are, but what about now? Well, the truth is, if you look in the mirror today, you can’t say that you’re worth $10 more.
The reality: To have $10 in the future, you only need to put some fraction of that $10 in the bank today. The exact fraction depends on the same factors that drive future value: rate of return and time. At 10 percent, you would need to deposit only $6.21 today to have $10 five years from now. Same formula, but this time, the approach is from the opposite direction. Instead of asking,
“What is my $10 worth in five years?” you ask, “What would I need today to have $10 in five years?”
The magic compounding formula
The fundamental time-value-of-money, or compounding, formula provides an indispensable foundation for value investors. A word of advice: It’s just as important to understand the formula, the dynamics, and the factors that drive or have the most influence on the result as it is to memorize the for- mula to do lots of math problems. Furthermore, just as it takes more than garlic to cook, you need a lot more than this formula to select stocks and be successful. Here’s the formula:
FV = PV ×(1 + i)n where . . .
FVis future value PVis present value
iis the interest rate, or rate of return nis the number of years invested
Now, to take apart the formula: The future value is a function of the present value, expanded or compounded by the interest rate over time. To calculate a return for one year, simply take PV and multiply by 1 (to preserve the original value) plus i(to increment by the interest rate or rate of return). The result is future value.
To calculate the return for more than one year, it gets more interesting.
Multiply PV by (1 + i) factored by the number of years, so 5 years is (PV) × (1 + i) ×(1 + i) ×(1 + i) ×(1 + i) ×(1 + i). Each (1 + i) indicates another year of compoundinginterest. The exponent is mathematical shorthand for such sequential multiplications The FV of $10 invested at 10 percent over 5 years is
FV = $10 ×(1 + .10)5 or
$10 ×(1.61), or $16.10
Fine, we can calculate future value. But what about present value? What if you want to figure out what interest rate would give you $16.10 on a $10 investment if held for 5 years? In other words, what if you want to work back- ward? You can transpose the formula algebraically to calculate PV, i,and even n:
1) PV = FV ÷ (1 + i)n
2) i = ((FV ÷ PV )(1/n))-1 or the nth rootof (FV/PV) 3) n is trickier — it involves logarithms!
Why Lesson 1 is important
Time value of money helps you estimate or determine the future value of an investment held over time. The importance of time value of money calcula- tions doesn’t stop there. Some value investing techniques call for discounting, or calculating the present value of future income streams. Time value calcula- tions are an important ingredient in measuring the value of investments and comparing them to alternatives. And you see later in this chapter and in those that follow how compounding becomes a main engine powering the value investing concept.