many readers. Baffled by mathematical symbolism justout of reach, they often think that understanding fair value is beyondthem. I will therefore sidestep the mathematics and try to show youthe few practical truths buried inside it.
tional books written to explain theories of stock valuation were poorlywritten and often confusing.It wasn’t until I had to study the intricacies of the bond market thatI finally came into possession of the basic underlying ideas that governthe evaluation of almost all financial assets, including the stock market.I then began to understand why these basic ideas seldom make it downto the popular level of investing and, in truth, are not well known bymany professional investors. The reason is simple: The ideas are justmathematical enough to be beyond the understanding of 95% of mostpeople who try to study them.I’m quite serious. A few years ago, I was at a meeting with more than50 stockbrokers in which the speaker promised $100 to anyone whocould explain how bond prices are calculated and the exact theory behind it. No one won the $100. Everyone parroted the simple, short-cutformulas they were taught in class but they couldn’t really explain the underlying theory. Believe me, if you don’t know how a bond is priced, you
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FAIR VALUE: THE THEORY OF STACKING THE MONEY
have no idea of how stocks are valued. I realized that concepts like fairvalue for stocks and bonds require a slightly higher level of math comprehension than most people seem to have. If the subject is approachedin a slightly different manner, however, this general problem can be sidestepped.
THE FUNDAMENTALS: TIME AND MONEYA Dollar Today Is . . .
TE
AMFLY
We have all heard the following statement: “A dollar today is worth morethan a dollar tomorrow.” Everyone agrees with this simple statement, butfew know that all of finance theory rests on fully understanding exactlywhat it means. Behind that simple statement lies everything there is toknow about stocks and bonds.If you’re asked, “Why is a dollar today worth more than a dollar tomorrow?” you would probably say, “Because of inflation. A dollar buysmore today than it will in the future.” In fact, this is wrong; inflationhas nothing to do with it. During periods of deflation, a dollar actuallybuys less today than it will tomorrow, so if the statement is a universaltruth for all time, inflation can’t be the answer.Whether you have inflation or deflation, a dollar today is still worthmore than a dollar tomorrow. If inflation is not the reason, why is a dollar today worth more than one tomorrow? It is true for the same reasona 10-year-old child is smarter than a newborn baby: The older child haslived 10 years longer and has had the opportunity to accumulate andlearn more information. Similarly, a dollar earned 10 years ago has been“living” 10 years longer than a dollar received today. During those 10years, the older dollar has had more time to accumulate to itself some interest income. This is true regardless of inflation or deflation.How do financial people compare the value of two dollars earned attwo different moments in time? They calculate how much interest theearlier dollar would compound to itself during the period before the future dollar is earned. The interest rate used in this calculation is usuallythe safest interest rate available for the time period in question.Let’s try this out. Let us compare a dollar earned today versus a dollar to be earned 10 years from now. Suppose the 10-year interest rate is
THE FUNDAMENTALS: TIME AND MONEY
51
7.17%. If we take the dollar today and compound it for 10 years at 7.17%,in 10 years we have $2. So the dollar today will be $2 in 10 years—it istwice as large as that future dollar.We can also go the other way in time. We could bring that future dollar into the present and see how it compares to the dollar today. Doing thecalculation in reverse, we find that that future dollar is worth only 50cents today. A future dollar’s current value is often called its present value.From this simple idea comes a very simple rule: You can only compare dollars if you bring them all to the same moment in time. If an expected stream of dollars is to be earned, you have to bring all thosedollars to the same moment in time to add, subtract, or compare them inany way. The simple formula that allows you to bring any future dollarinto present time is this:A future dollar’s present value = $1/(1 + I)Twhere I is the interest rate and T is the time between the two dollars.
The entire subject of finance, stocks, bonds, and so on is built on thissimple equation.An Important ConclusionLet’s carefully inspect this equation and see what simple truths come outof it. We assumed an interest rate of 7.17% and the present value of thatdollar to be earned in 10 years is 50 cents today. Suppose, however, thattomorrow, because of some dramatic economic news, interest ratesplunge. They go from 7.17% to 5%. We go to our formula and recalculate what the present value of that dollar is the next day. We do the calculation and find that that future dollar, instead of being worth 50 centsas it was yesterday, now has a present value of 61 cents. In other words,in one day, the present value of that future dollar has jumped 22%, from50 cents to 61 cents.This startles most people because they look around and the worldlooks pretty much the same. Yet this is what the theory says. Now, if theinterest rate had gone up from 7.17% to 9% in one day, the oppositewould have occurred. In that event, the dollar to be paid in 10 years, instead of being valued at 50 cents, is now worth only 42 cents. This simple but slightly strange conclusion is an extremely important one.
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FAIR VALUE: THE THEORY OF STACKING THE MONEY
Another Important FactorCan this simple equation account for all the complexity we see in theworld of finance? Not quite. One more idea is required. We must somehow mathematically allow for another factor—the uncertainty of knowing the exact future. For example, we might be expecting a dollar tocome our way in 10 years, but maybe the payee of that dollar won’t bethere to make it. He or she could die or go out of business. A term mustbe included in the formula that allows for this.Let us call that term P. P is a number that represents the probabilitythat you will receive that future dollar. P = 1 means that the payment ofthat future dollar is 100% guaranteed. P = .5 represents a 50 : 50 chanceof payment, and P = 0 means you are guaranteed not to get it.Taking this factor into account, the complete formula is this:A future dollar’s present value = P * $1/(1 + I)Twhere I is the interest rate, T is the time between the two dollars, and P is anumber between 1 and 0 that reflects the probability that future dollar will bepaid.
Let us consider again the example of the dollar to be paid in 10 yearsif interest rates are 7.17%. We already know that that dollar is worth 50cents today. Suppose we also know that in 10 years, when it is to be paid,someone will flip a coin and only pay the dollar if the coin comes upheads. Since there is only a 50 : 50 chance of getting paid the dollar, thatfuture dollar is really worth 25 cents. If you want to buy that future dollar, you shouldn’t pay more than 25 cents for it today.Now we are in a position to understand what fair value is and how itis calculated. As mentioned earlier, the current price of a stock shouldexactly equal the present value of all that company’s future dividends.That statement probably means more now than it did earlier. Before weget into exactly what it means, I first want to apply what we know to twosimple situations to get a feel for the how the rules are applied. The firstexample is how to calculate what is known as a lump sum pension.Applying the Money Rules to Calculate a Lump Sum PensionA few years ago, I was asked to develop a customized retirement seminarfor TRW corporation. This company allows employees the option of taking their pensions as either a lump sum payment or a monthly check paid
THE FUNDAMENTALS: TIME AND MONEY
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over life. In this seminar, I had to teach these retirees exactly what alump sum pension was and, because many of the employees were scientists, I was able to expand the presentation and explain the concept witha little more mathematics. In the process of trying to find the right levelof mathematics to communicate the meaning of the term lump sum, I realized the importance of the underlying presentation in terms of understanding stock values.Theoretically, a lump sum pension payment is the amount that pension actuaries think is exactly equal to what the pensioner would receiveif she or he chose the monthly-check-for-life option. How does a pensionactuary make this calculation? It’s a two-step process. Figure 3.2 illustrates the first step.The first step in the process of calculating a lump sum payment is todetermine the present value of each year’s pension payments. Let’s assume that a retiree’s current age is 62, with a lifetime pension of $1,000
FIGURE 3.2 The first step in calculating a lump sum pension is to calculate thepresent value of all future pension payments.
$12,000-per-Year Pension$11,473$10,874
Present Values
$10,38 1
Age
$6,643
62
63
64
+++
74
$6,316
75 ++
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FAIR VALUE: THE THEORY OF STACKING THE MONEY
per month, or $12,000 a year. First, we get a calculator, a piece of paper,and a stubby pencil. With these simple tools, we can calculate the present value of each $12,000 yearly payment.As we now know, the present value is the amount you need today, soif it was compounded, it would grow to each future $12,000 payment. Tomake this calculation, of course, you need an interest rate to compoundwith. Federal law requires that pension actuaries use the 30-year Treasury bond interest rate. As of this writing, the current 30-year T-bond interest rate is 5.06%, so we will use that figure in our calculations.Now that we have our interest rate, let’s calculate some specific present values. For example, what amount of money would grow to $12,000in 13 years (for age 75)? Using my calculator and stubby pencil, I figurethat at 5.06% we’d need $6,316 today. In other words, $6,316 will growto $12,000 in 13 years at 5.06%.A pension actuary would make this same calculation for every yearfrom age 62 to age 110. Yes, age 110! (These pension actuaries take theirjobs very seriously.) Not much money is needed today to account for thatpayment at age 110. In fact, according to my calculations, the presentvalue for $12,000 when the retiree is 110 is $1,122. After we have calculated all the present values, we make a list of them with our stubby pencil and then proceed to step two.In step two, we take each present value and multiply it by the probability that the company is going to have to make that payment. Wherein the heck do we get this number? From actuarial studies—those amazing documents that predict the probability of a person at any age livingto any future age. For example, there’s about a 65% chance that a 62year-old will be alive at age 75. Therefore, we would multiply the presentvalue for age 75, that being $6,316, by .65. The result ($4,105) is calledthe probability-weighted present value of a $12,000 payment at age 75(Figure 3.3).We then make this same calculation for each present value, all theway to age 110. For example, the probability of being alive at age 110 isless than 1%, so the probability-weighted present value at age 110 is lessthan $10. Then we mathematically stack these probability-weighted present values on top of each other—in other words, we add them all up.This sum is the lump sum equivalent of the monthly pension checks forlife. In our example, the lump sum is $148,673.To an actuary, these two are equal: that is, the lump sum of $148,673exactly equals the $1,000-a-month payment for life. You can also look atit from another viewpoint: If you wanted to buy a $1,000-a-month pay-
THE FUNDAMENTALS: TIME AND MONEY
FIGURE 3.3
55
Second step: stacking up the probability-weighted present values.
$148,673++
PV
75
+++
PV
*PV
75
74
*PV
74
$12,000-per-Year Pension$11,473
PV6
4*P
$10,874
PV6
V64
3*P
$10,381
$6,643 $6,316
PV6
V6
3
2*PV
62
Age
62
63
64
+++
74
75 ++
ment for life, you would have to pay $148,673 right now. (Note that another company might come up with different figures, depending on thelife-expectancy tables used by the actuary.)How Interest Rates Affect the Lump SumLet’s play around with this concept. Suppose the retiree goes to his benefits department, learns about these two retirement payment options($1,000 for life or $148,673 lump sum) and goes home to think about it.A few months later, he goes back and asks whether his retirement package has changed. He learns that although the monthly pension hasn’tchanged (it’s still $1,000 per month), the lump sum is larger; it is now$151,356. The retiree is mystified, but suddenly remembers that over thelast two months interest rates declined. Then it dawns on him what isgoing on. The lump sum is always being evaluated and calculated in the
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FAIR VALUE: THE THEORY OF STACKING THE MONEY
interest rate environment of that moment. In other words, given nochange in the monthly pension, the retiree is entitled to a bigger lumpsum for no other reason than that interest rates are now lower.This is a very important conclusion: When interest rates go down, thelump sum goes up. When rates go up, the lump sum goes down. It is thesame result that we saw with any single present value and is really the aggregate manifestation of this conclusion applied to the present value ofthe whole stack. Once employees become aware of this phenomenon,they often try to time their retirement for when they think interest rateswill be at their lowest. The relationship between the size of the lump sumand interest rates is shown in Figure 3.4.Our example of how a lump sum is calculated is very important because the same procedure is the one used to determine the price of bothstocks and bonds. You probably think that there is a lot more to stock
FIGURE 3.4 When interest rates are low, lump sum pensions are high. Wheninterest rates are high, lump sum pensions are low. Because of this, many retireeswhose companies offer the lump sum pension payment try to plan theirretirement when they think interest rates will be low.
Lump Sum and Interest Rates
Interest Rates
STACKING THE MONEY TO DETERMINE BOND PRICES
57
Stacking the money is what I call the mathematical procedure tofind the current worth or value of any investment that will throw offa future stream of income or cash. This theory recognizes thatmoney, like any commodity, has a value and that the current valueof any investment—stocks, bonds, real estate, and so on—should bedetermined using this or some variation on this idea.
prices than this, but except for the feedback loops discussed in Chapter1, there isn’t. All the thinking and effort that goes into Wall Street projections boil down to calculations like the ones we just performed to figure a lump sum pension. When performing these types of calculations,analysts and actuaries say they are making calculations based on the timevalue of money. However, I think the term stacking the money more accurately describes the process, because the calculations involve stackingone number on top of another to get the total value. I call it the stackingthe money theory.Remember: When interest rates rise, the lump sum gets smaller.When they go down, the lump sum gets bigger. Does any of this soundfamiliar? Have you ever heard that high interest rates are bad for stockand bond prices and low interest rates make them go up? Do you thinkthere is a relationship between why the lump sum pension goes up ordown and why stocks and bonds also go up and down when interest rateschange?
STACKING THE MONEY TO DETERMINE BOND PRICESStacking the money is the only theory used to calculate bond prices.Let’s take a look at how it works. Suppose you have a U.S. Treasury bondwith 10 years left to maturity. The bond has an 8% coupon, so it pays $80per year to the holder for the next 9 years, and in the tenth year it will paythe final $80, as well as the face value on the bond ($1,000 in this case).What’s a fair price for this bond today? First, we need to look upthe current Treasury interest rate in the free market. We use this rate to
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FAIR VALUE: THE THEORY OF STACKING THE MONEY
FIGURE 3.5 The price of a bond is calculated following the same stacking-themoney procedure used for calculating a lump sum pension payment.10
7
Bond Price
65
Present Valueof Interest
98
80
CouponAmount
6040
PresentValue
200
4
Price Calculation of $80 Coupon Bond
$100
1
2
3
4
567P ayment Y ear
1
9
10
$1200
32
8
PV of Interest = $615.94PV of Principal = $610.42Bond Price = Total PV = $1226.36
1000800600400
Present Value of Bond Principal
200
BondPrincipalPresentValue
0
calculate the present value of each $80 payment for the next 10 years, aswell as the present value of the final $1,000 payment (Figure 3.5). Unlikewith lump sum pension payments, the probability that payments mightnot be made isn’t an issue because the government guarantees the payments. This removes the probability factor from the equation. Finally,you stack (in other words, add up) all these values to determine the priceof the bond.Bond prices rise and fall with interest rates for the same reason thelump sum pension rises and falls. In fact, after a bond is issued and itscoupon is fixed, all a bond’s price ever does is march to changing interest rates—that is, its price adjusts daily to reflect the ever-changing interest rate environment it lives in. With bonds, the link between interestrates and the price of a bond, just like the lump sum, is immediate anddirect. Stocks follow the same theory, but here the connection is looserbecause with stocks, the payments you’ll be stacking are fuzzy numbers.
STACKING THE MONEY TO DETERMINE STOCK PRICES
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NOTEBy the way, when calculating corporate bonds (where there is apossibility of default), the probability factor is not 1. Instead, youmust multiply the present values for each coupon plus the principal payments by a number that represents the chance of default forthat particular coupon. You then stack these default probabilityweighted present values to find the price of the bond. Whether youare calculating bonds or stocks, you always use the safest interestrate in the formulas. The uncertainty of any payment (coupon, dividend) is allowed for by the P factor, not by inserting a riskier interest rate in the denominator.
STACKING THE MONEY TO DETERMINE STOCK PRICESNow let’s turn to stocks. What is the price of a stock? The same rulesapply, except this time we stack dividends. The rule of the game is thatyou always stack what is paid out. We return to the definition of fairvalue for a stock: The price for a stock should exactly equal the presentvalue of all the company’s future dividends.Are there any other considerations? Some time ago, I encounteredthe following opinion about the dividend model: To determine a company’s value, patents, brand names, and other intangible assets should bevalued and used with the dividend discount model. A better valuation results from adding intangibles to the model’s calculation.I disagree with this. I believe the statement for fair value is complete and sufficient unto itself and needs nothing else. Including a termlike intangible assets to the equation is what engineers and scientistsoften call adding a fudge factor. Our model for stock prices is not a fairvalue term plus three feedback loops plus a wild fudge factor. If anything, you should try to calculate how these intangibles might increasethe dividends of the company. In other words, all other factors shouldbe included only in estimating future dividends. If it is determinedthat these intangibles probably won’t affect the future dividends of thecompany, then according to the model, they have no importance (if
FAIR VALUE: THE THEORY OF STACKING THE MONEY
John Burr Williams
AMFLY
John Burr Williams was the first man to clearly formulate a theory onhow to value stocks. His theory, that a company’s stock shouldequal the present value of its future dividends, was developed as adoctoral thesis at Harvard and eventually published in 1938, in TheTheory of Investment Value. It is hard to believe that until this bookcame out, investors had no real theory on how stocks should bepriced, even though the New York Stock Exchange was more than140 yeas old.John Williams’ undergraduate education was in mathematicsand chemistry, which gave him the tools and the training to approach subjects from a quantitative mindset. He later put that education to good use. After some training in business forecasting, hewent to Wall Street in the mid-1920s and worked for the old brokerage firm of Hayden, Stone as a security analyst. He watched as thestock market ran up in the speculative flurry of the late 1920s andthen witnessed firsthand the Great Crash of 1929 to 1932.While other men walked away demoralized and beaten, Williamswent back to Harvard determined to understand what had happened. How could stock prices go to such unbelievable levels andthen, in three years, plummet? Somewhere, stocks had to have sometrue value, and he set about uncovering how to calculate it. Workingfrom Irving Fisher’s book, The Theory of Interest, published around1900, Williams found the underlying concept that, when applied tostock prices, led to his basic theory.The fundamental idea of calculating the present value of a futurestream of money (earnings, dividends, etc.) and then adding themall up is still the basic theory used today. If you study almost anymodern theory of stock evaluation and look for the fundamentaltheory behind it, you will find John Burr Williams’ seminal ideas, formulated some 70 years ago.
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STACKING THE MONEY TO DETERMINE STOCK PRICES
61
there is a final liquidation or merger of a company, this would constitutea final dividend).The model says that you take all the dividends that the stock will everpay, calculate their present value using the current interest rate, andstack them up. The total should be the current price of the stock. Although this is feasible with bonds and pension payments, doing this calculation with stock prices involves the uncertainty of knowing what thosedividend payments will be. You may know what the dividend is today andhave a good idea what it will be over the next few years, but can you realistically project dividends over 10 or 15 years?Because of the uncertainty of the value of future dividends, we canconsider the dividends a type of number called fuzzy numbers. Usually,but not always, the farther away a dividend is, the fuzzier it is. Since thepresent value of a fuzzy number is also a fuzzy number, so is the fairvalue sum. By the way, the fact that fair value is a fuzzy number is whatallows the three feedback loops to exist. How fuzzy the fair-value term isgoverns the size and range of the three feedback loops (Figure 3.6).Sometimes, the fair-value calculation for extremely speculative stocks is
Price (P=D/I )
Stock Price
FIGURE 3.6 Stock prices follow the stacking-the-money rules. Fair value is thesum of a long series of fractions with dividends in the top (numerator) and aninterest rate raised to a power in the bottom (denominator). In this case,however, the numerators (dividends) are fuzzy numbers, and they get fuzzier thefarther out in time you go. Therefore, fair value is also a fuzzy (uncertain)number.
1
2
3
4
5
6
Years
7
8
9
10
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FAIR VALUE: THE THEORY OF STACKING THE MONEY
fuzzy all the way from $1 to $100, so a price movement from $5 to $20 is really nothing more than an almost 100% feedback loop movement. That describes the Internet craze of 1999 for more than 90% of the stocks involved.There is nothing wrong with playing that investment game. The problemarises when buyer and seller forget the bedrock on which the game isfounded and start thinking that there is something real to the price.In practice, to help lessen this fuzzy number problem, stock analystsuse a few tricks. Instead of trying to estimate all the dividends for thenext 30 years, they estimate for shorter periods, such as for 5 years. Theythen fold the calculation back on itself (that’s the mathematical trick) andreassess these calculations as pertinent news comes in. Without goinginto the details of this process, assume it is the mathematical equivalentof what an explorer does when setting off for a distant land with only acompass and a map as guides. Instead of taking one compass reading andthen setting off in a particular direction for good, the explorer takes thetrip in stages, stopping every now and then to replot the current position.The important point to note in all this is that no matter what mathematical trick or approximation is used, behind each one is the basic conceptof dividends in the numerator and interest rates in the denominator.I don’t want to explain any more details of the theory of fair value atthis time. From just what I’ve discussed, many important conclusions canbe drawn. That is what I want to focus on from this point. You’ll be surprised at what you can figure out from what you’ve learned.
You Are at BedrockThe theory of stacking the money shows that only two numbers gointo the calculation of a stock’s fair value: dividends (present and future) and the current interest rate. Other financial numbers, such assales, market share, and costs, do not enter directly into the equation and are considered only for their ability to help determine thevalue of those future dividends. In other words, with this theory youare at bedrock.Any company news or economic data has value only proportional to the degree it changes estimates for all future dividends orfor the current interest rate. To the degree that investors havemoved away from these basic ideas, they have moved away from investing and into the realm of speculation.
WHY INTEREST RATES ARE SO IMPORTANT
63
WHY INTEREST RATES ARE SO IMPORTANTWe are now in a position to understand one of the most important conclusions that comes from the stacking the money theory for fair value.We mentioned it earlier, but we return to it now since it is so important:It is the full understanding of why interest rates are so important to stockprices.In 30 years, I have seldom heard the correct explanation of why interest rates are so important to the direction of stock prices. For example, two well-known books written by top market analysts give threereasons for the importance of interest rates.1. Lower interest rates are good for corporate profits because theyallow companies to borrow money for less.2. Movements in interest rates make bonds more (or less) attractivecompared to stocks. Lower rates makes bonds less attractive,causing people to invest more in stocks and vice versa.3. The cost of borrowing and buying more stock, called the margin,is less when interest rates are low and therefore is better forstocks.You hear the same ideas in the daily financial news when analysts talkabout how the Federal Reserve board’s lowering of interest rates willspur the economy. For some reason, commentators and analysts are always focusing on how the change will affect the numerators of the fractions (earnings or dividends). They never mention the really importantfact—that an interest rate number goes directly into the denominators ofall the fractions used to calculate stock prices, and that denominator isnow smaller. As a result, investors seldom appreciate fully the impactthat interest rates really do have on stock prices.The real reason interest rates are so important to stock prices is thatthe series of mathematical terms that are added up to calculate fair valuehave interest rates in their denominators. Recall what you learned aboutfractions: When the denominator becomes smaller (interest rates godown), all the fractions get bigger (and so does fair value). When thedenominator gets larger, the terms get smaller, and so does fair value. Itis the same lesson we learned about the lump sum. This is the real reason interest rates are so important. When you understand this, you cansee that the three previously listed reasons are really of smaller consequence. In fact, these commonly referenced ideas about why interest
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FAIR VALUE: THE THEORY OF STACKING THE MONEY
rates are so important actually divert attention away from this reallyvital point.Fuzzy Dividends Don’t Changethe Interest Rate ConclusionAs we said, when you are considering the fair value of stocks, only one ofthe numbers in the stacking process—the interest rate—is an exact number (Figure 3.7). The other number—the dividend—is a fuzzy, or uncertain, number. These facts lead to an important conclusion: When youdivide a fuzzy number by an exact number, you still get a fuzzy number.Therefore, the fair value of a stock, unlike that of a safe bond, is alwaysfuzzy; no one knows exactly what fair value is.That’s okay—it’s part and parcel of investing—but it doesn’t changethe conclusion of the previous section. Whether the dividends are fuzzyor not, when interest rates change overnight, these fuzzy dividends stillstack to either a higher or lower number, but it is a fuzzy higher or lowernumber. The response in the price direction of the fuzzy number is thesame as the response for the price of bonds or the lump sum pension,and it responds in the same way for the same mathematical reason.Interest Rates and Dividends:The Yin and Yang of Stock PricesThe two factors in the equation for fair value, dividends and interestrates, give us the yin and the yang of stock prices (yin and yang becausethe growth of dividends and the direction of interest rates usually workopposite each other). Normally, interest rates decline as business getsbad and earnings and dividend estimates are being reduced. Similarly,interest rates go up when business expands, as earnings and dividendestimates are increased. This creates a strange situation. As dividend estimates are reduced, lower interest rates counteract the effect by increasing the present value of those reduced dividends. These movementssomewhat offset each other mathematically, but not quite.What is more important to the value of these fractions, the changingnumerators or the changing denominators (i.e., lower future dividends orlower interest rates)? A mathematician would tell you that interest ratesare more important because a change in interest rates today affects thecalculation of the present values of every dividend, even those that will
WHY INTEREST RATES ARE SO IMPORTANT
65
FIGURE 3.7 Of the two numbers used to calculate fair value, one (dividends) isa fuzzy number, the other