Risk and required return according to the CAPM
A brief digression
The size and value premiums
The three-factor model: Overview
The three-factor model: Implementation
The three-factor model: Application
The big picture
Excel section
Challenge section
T
he CAPM we discussed in the previous chapter argues that the only variable important for the estimation of required returns on equity is systematic risk measured by beta. That’s a very strong statement. Particularly when the empirical evidence seems to point to other variables that are clearly correlated to returns. Two of these variables are a company’s market capitalization and book-to-market ratio, which can be articulated (together with beta) into the increasingly popular three-factor model.Risk and required return according to the CAPM
When it comes down to popularity, the CAPM beats the competition hands down: over 80% of practitioners claim to use the CAPM when calculating a company’s required (or expected) return on equity. Does that make it the ‘best’
model? Not necessarily. It is by far the most popular, though which one is the best (whatever that means, anyway) remains an open question.
The difference across models used to estimate required returns on equity largely stems from the way each proposes to estimate the risk premium. Recall that the required return on any stock i, E(Ri), can be thought of as the sum of the risk-free rate (Rf) and the stock’s risk premium (RPi); that is, E(Ri) = Rf+ RPi. Recall, also, that the CAPM essentially argues that this risk premium can be calculated as the product of the market risk premium, MRP= E(RM) – Rf, where E(RM) is the required return on the market portfolio, and the stock’s beta (ßi), that is, RPi= {E(RM) – Rf} · ßi= MRP· ßi.
The previous arguments collapse into the central message of the CAPM, which says that the required return on equity on any stock ican be expressed as
E(Ri) = Rf+ {E(RM) – Rf} · ßi= Rf+ MRP · ßi (8.1)
or, similarly, as
E(Ri) – Rf= {E(RM) – Rf} · ßi= MRP · ßi (8.2)
where beta captures systematic risk, the onlyrelevant source of risk according to this model.
A brief digression
Now, thatis a strong statement! Think about it. Total risk (volatility), currency risk, bankruptcy risk, and as many others sources of risk as you can imagine . . . they don’t matter. They provide no information about the required or expected return on stocks. If this sounds strange, it’s simply because it is.
And yet, a die-hard supporter of the CAPM would justify this strong argument in many ways, two of which we’ll consider briefly. The first line of defense would be theoretical. He would claim that, unlike the vast majority of its contenders, the CAPM is solidly grounded in theory. More precisely, he would argue that in a model in which investors behave optimally, beta must be the only relevant source of risk. In other words, this ‘strange’ prediction of the CAPM is not an assumption but the resultof a model of optimal behavior.
Very little can be argued against this line of reasoning. It is indeed true that the CAPM is supported by a solid theoretical background and that it results from a model of optimal investor behavior. And it is also true that the vast majority of its competitors are models in which the variables used to determine the required return on equity come from guesses or plausible stories, or worse, from the result of trying one variable after another until something that correlates with returns is found. And yet it could be argued that a theory is only as good as its predictions, and if the evidence does not support these predictions, then the theory should be discarded.
That takes us straight into the second line of defense, which is empirical. But this is a tricky one; we could fill a room with studies that test the validity of the CAPM, in different countries, over different periods of time, and with different methodologies. The problem is that there is a huge amount of evidence on both sides of the fence. Both those who defend the CAPM and those who defend alternative models could point to a vast amount of evidence supporting their position. As a result, the evidence doesn’t go a long way toward clearly supporting either the CAPM or one of its contenders.
The size and value premiums
And yet, at least some empirical evidence is surprisingly consistent. Data for different countries and over different time periods shows a consistent negative relationship between market capitalization and returns. In other words, the evidence seems to clearly show that small companies tend to deliver higher
returns than large companies. This empirical regularity is usually known as the size effect.
Similarly, data for different countries and over different time periods show a consistent positive relationship between book-to-market ratios (BtM) and returns. In other words, companies with high BtMtend to deliver higher returns than those with low BtM. This ratio, recall, is a measure of cheapness in the sense that high and low BtMindicate cheap and expensive stocks (relative to book value), respectively. So the evidence seems to clearly show that cheap (also called value) stocks tend to outperform expensive (also called growth) stocks.
This empirical regularity is usually known as the value effect.
Now, however clear the evidence may be, there doesn’t seem to be any good theoretical reason for the size and value premiums. In other words, no model of optimal investor behavior converges into a result in which stock returns depend on size and value. Some may not consider this a problem; they would claim that as long as we can isolate the variables that explain differences in returns, we should use them to determine expected returns. Yet others would argue that without a good theory behind, there is no point in using a model for this purpose. You can pick your side on this debate.
If you think a bit about it, though, at first glance these two risk premiums seem to make sense. Small companies are probably less diversified and less able to withstand negative shocks than large companies. And as for cheap companies, well, there must be a reason why they’re cheap! Put simply, it’s not very difficult to come up with some plausible story to explain why small stocks and cheap stocks are riskier than large stocks and expensive stocks, and therefore why they should deliver higher returns.
But those are just stories. Perhaps a better alternative is attempting to link empirically size and value to obvious sources of risk. The evidence on this seems to point to the fact that small companies and cheap companies are less profitable (have lower earnings or cash flow relative to book value) than large companies and expensive companies. In other words, small companies and cheap companies are distressed because of their poor profitability, and are therefore perceived as riskier by investors.
The CAPM argues that stocks with high systematic (market) risk should outperform those with low systematic risk. Complementary evidence shows that small stocks outperform large stocks, and that cheap (value) stocks outperform expensive (growth) stocks. Put all this together and we get the result that stock returns are affected by a market premium, a size premium, and a value premium. And that is, precisely, the message from the three-factor model.
The three-factor model: Overview
Estimating required returns from the three-factor model is just a tiny bit more difficult than doing it with the CAPM. That is simply because we need some additional data and we have to estimate two additional beta coefficients. Other than that, as we’ll soon see, the model poses no difficult obstacles to practitioners.
According to the three-factor model, the required return on stock ifollows from the expression
E(Ri) = Rf+ MRP · ßi+ SMB · ßiS+ HML · ßiV (8.3)
where SMB (small minus big) and HML (high minus low) denote the size and value premiums, respectively, and ßiSand ßiVdenote the sensitivities (betas) of stock i with respect to the size factor and the value factor, respectively. Let’s think a bit about these magnitudes.
Recall that MRP, the market risk premium, seeks to capture the additional compensation required by investors for investing in risky assets as opposed to investing in risk-free assets. Recall, also, that it is measured by the average historical difference between the return of the market portfolio (some widely accepted benchmark index of stocks) and the risk-free rate. And recall, finally, that ßimeasures the sensitivity of the returns of stock ito changes in the market risk premium (or, simply, to changes in the returns of the market).
Similarly, SMB, the size premium, seeks to capture the additional com- pensation required by investors for investing in small companies as opposed to investing in large companies. It is measured as the average historical difference between the returns of a portfolio of small stocks and those of a portfolio of large stocks. And the beta associated with this factor, usually called the size beta (ßiS), measures the sensitivity of the returns of stock i to changes in the size premium, or, simply, the exposure of company ito size risk.
Finally, HML, the value premium, seeks to capture the additional com- pensation required by investors for investing in cheap stocks as opposed to investing in expensive stocks. It is measured as the average historical difference between the returns of a portfolio of stocks with high BtM and those of a portfolio of stocks with low BtM. And the beta associated with this factor, usually called the value beta (ßiV), measures the sensitivity of the returns of
stock ito changes in the value premium, or, simply, the exposure of company i to value risk.
Note that just as we stressed in the previous chapter about MRP, neither SMBnor HMLin equation (8.3) has an isubscript. That means that the average size and value premiums, as well as Rfand MRP, are independent of the stock we’re considering. Note, on the other hand, that the size beta (ßiS) and the value beta (ßiV), as well as ßi, all have a subscript i, indicating that they are specific to the company we’re considering.
The three-factor model: Implementation
The three-factor model, just like the CAPM, is silent about several practical issues. What is a portfolio of small stocks? And one of large stocks? What is a portfolio of cheap (value) stocks? And one of expensive (growth) stocks? Should we estimate betas out of daily, weekly, monthly, or annual data? Over what period of time? Again, we’ll find ourselves looking at the convergence among practitioners for guidance.
Let’s start with what we already know from the previous chapter regarding the estimation of the CAPM. We need a risk-free rate that we approximate with the yield on 10-year Treasury notes (or with the yield on a bond of maturity equal to the average maturity of a company’s projects). We need a market portfolio that we approximate with a widely accepted benchmark of stocks (such as the S&P500 in the US). We need a market risk premium that we calculate as the average historical difference between the returns on the benchmark of stocks and a long-term risk free rate; the time period is as long as the data allows and the average can be either arithmetic or geometric. And we need a company’s beta, which we estimate with respect to the benchmark of stocks using monthly returns over a five-year period.
Because the three-factor model essentially adds two factors to the CAPM, we’ll focus now on what we need to estimate the size and value premiums. But a quick comment first. The three-factor model was proposed by professors Eugene Fama and Kenneth French in a series of articles published in the 1990s;
that’s why you may occasionally find this model referred to as the Fama–
French three-factor model. In the ‘Data Library’ of Ken French’s web page (http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/) you will find a wealth of information about this model, as well as data to implement it. For that reason, we’ll focus here on the essentials; if you want to get into details, do visit that web page.
Let’s start with the estimation of SMBand HML. To estimate SMBwe need to calculate the average historical difference between the returns of a portfolio of small stocks and those of a portfolio of large stocks. The formation and rebalancing of each of these two portfolios is less than trivial but you don’t have to worry about it. In Ken French’s web page you’ll find annual returns for the SMB portfolio from 1927 on. The third column of Table 8.1 displays these returns for the 1994–2003 period only. Note that, on average since 1927, small stocks outperformed large stocks by almost 4 percentage points a year.
TABLE 8.1
Year MRP SMB HML
(%) (%) (%)
1994 –4.1 0.4 –0.1
1995 31.0 –6.9 –3.5
1996 16.3 –1.9 0.2
1997 26.1 –3.7 11.1
1998 19.4 –23.3 –15.0
1999 20.2 11.7 –39.4
2000 –16.7 –5.7 21.4
2001 –14.8 28.4 27.3
2002 –22.9 4.4 3.7
2003 30.7 28.1 15.1
AM(1927–2003) 8.5% 3.9% 4.4%
GM(1927–2003) 6.4% 2.9% 3.4%
The estimation of HMLis similar. We need to calculate the average historical difference between the returns of a portfolio of stocks with high BtM and those of a portfolio of stocks with low BtM. Again, the formation and rebalancing of each of these two portfolios is less than trivial, but you don’t have to worry about it. In Ken French’s web page you’ll find annual returns for the HML portfolio from 1927 on. The fourth column of Table 8.1 displays these returns for the 1994–2003 period. Note that, on average since 1927, cheap (value) stocks outperformed expensive (growth) stocks by almost 4.5 per- centage points a year.
We have already discussed the estimation of MRP in detail in the previous chapter (and briefly at the beginning of this chapter), so one final quick comment now. The returns of this portfolio available from Ken French’s web page are calculated a bit differently from how we discussed it (and from standard practice). Nothing you should worry about from a practical point of view. The
second column of Table 8.1 displays the annual returns of the MRPportfolio for the 1994–2003 period, and the (arithmetic and geometric) average since 1927.
The three betas we need to implement the three-factor model are estimated jointly by running a time-series regression between the risk premium of stock i, RPi= Ri– Rf, and the three portfolios that capture the market, size, and value premiums (MRP, SMB, and HML), that is,
Rit– Rft= ß0+ ß1· MRPt+ ß2· SMBt+ ß3· HMLt+ ut (8.4)
where ß0, ß1, ß2, and ß3are coefficients to be estimated, uis an error term, and t indexes time. Note that ß1is the usual beta with respect to the market, ß2is the size beta (ßiS), and ß3is the value beta (ßiV).
This regression is typically estimated using monthly returns during a five-year period. Monthly returns for the MRP, SMB, and HML portfolios are available from Ken French’s web page. It is not unusual in practice to run this regression with the returns of stock i(Rit) as the dependent variable, instead of with the risk premium of stock ias in equation (8.4). In theory, the estimates of the betas should be the same either way. In practice, however, because Rfvaries a bit over time, the two sets of estimates may be slightly different.
The three-factor model: Application
Let’s now put everything together and estimate required returns on equity from the three-factor model. And let’s do it, as in the previous chapter, for the 30 stocks of the Dow. For the risk-free rate we’ll use, also as in the previous chapter, the yield on the 10-year US Treasury note, which at the end of 2003 was 4.3%.
To estimate MRP, we’ll depart slightly from the last chapter. Instead of using a market risk premium of 5.5% as we did before, we’ll now use 6.4%, which is the (geometric) average MRPsince 1927 as calculated by Fama and French (see the last line of Table 8.1). To estimate SMB and HML we’ll also use the portfolios calculated by Fama and French. And as the last line of Table 8.1 shows, from 1927 on, the (geometric) average SMB and HML are 2.9% and 3.4%, respectively.
We then have all the estimates common to all stocks that we need to implement the three-factor model. In other words, we will estimate required returns from the expression
E(Ri) = 0.043 + 0.064 · ßi+ 0.029 · ßiS+ 0.034 · ßiV (8.5)
Note that equation (8.5) is the same as (8.3) but with specific estimates for Rf, MRP, SMB, and HML. All we need now to use this model, then, are estimates for the three betas of any of stock of our interest.
We’ll estimate the three betas for each of the 30 stocks of the Dow using equation (8.4), five years of monthly returns (January 1999–December 2003), and the three portfolios provided by Fama and French. These betas are shown in the second, third, and fourth columns of Table 8.2.
TABLE 8.2
Company ßi ßiS ßiV CAPM 3FM Diff
(%) (%) (%)
3M 0.6 –0.2 0.2 8.2 8.3 0.1
Alcoa 1.8 0.1 0.4 15.8 17.5 1.7
Altria 0.3 0.0 0.6 6.3 8.3 2.0
American Express 1.2 –0.7 0.2 11.8 10.6 –1.2
American Intl. 0.8 –0.9 0.1 9.7 7.3 –2.4
Boeing 0.8 0.1 0.7 9.1 11.5 2.4
Caterpillar 1.1 –0.4 0.7 11.2 12.3 1.2
Citigroup 1.4 –0.5 0.3 13.4 13.1 –0.3
Coca-Cola 0.3 –0.1 0.4 6.3 7.3 1.0
DuPont 1.0 –0.4 0.3 10.5 10.2 –0.4
Exxon Mobil 0.4 –0.1 0.3 7.1 7.8 0.8
General Electric 1.1 –0.7 –0.2 11.1 8.3 –2.9
General Motors 1.3 0.0 0.5 12.6 14.5 1.9
Hewlett-Packard 1.7 1.0 0.0 15.0 17.9 2.9
Home Depot 1.3 0.0 –0.1 12.9 12.6 –0.3
Honeywell 1.4 –0.5 0.7 13.1 13.9 0.8
Intel 2.0 0.3 0.2 16.8 18.3 1.5
IBM 1.4 0.3 0.6 13.3 16.0 2.8
Johnson & Johnson 0.3 –0.7 0.0 6.0 4.0 –2.0
JP Morgan Chase 1.8 0.0 0.6 15.6 17.5 1.9
McDonald’s 0.8 –0.4 0.3 9.1 9.1 0.0
Merck 0.3 –1.2 0.1 6.5 3.4 –3.1
Microsoft 1.5 0.1 –0.2 14.2 13.9 –0.3
Pfizer 0.4 –0.7 0.1 6.7 4.9 –1.8
Procter & Gamble –0.1 –0.1 0.1 3.7 3.9 0.1
SBC Comm. 0.8 –0.6 0.5 9.7 9.5 –0.2
United Tech. 1.1 –0.3 0.2 11.1 10.8 –0.4
Verizon 1.0 –0.6 0.4 10.5 10.2 –0.3
Wal-Mart 0.7 –0.6 –0.1 9.0 7.1 –2.0
Walt Disney 1.0 0.0 0.4 11.0 12.5 1.6
Average 1.0 –0.3 0.3 10.6% 10.7% 0.2%
Note from the outset that although the monthly MRP we used to estimate betas in this chapter is different from the monthly MRPwe used in the previous chapter, the betas with respect to the market risk premium (ßi) are virtually identical. (Compare, company by company, the betas in the second column of Table 8.2 with those in the second and fifth columns of Table 7.3.)
The fifth column from Table 8.2 shows the required return on equity estimated from the CAPM using only the first two terms of the right-hand side of equation (8.5), that is, E(Ri) = 0.043 + 0.064 · ßi. If we compare, company by company, the required returns on equity from the CAPM shown in Tables 8.2 and 7.3, the difference between them is not substantial. In fact, the average difference across all 30 companies of 0.9% (10.6% in this chapter versus 9.7% in the previous chapter) follows almost exclusively from the higher MRP we’re using in this chapter (6.4% here versus 5.5% in the previous chapter).
The sixth column of Table 8.2 shows the required return on equity of the 30 companies of the Dow estimated with the three-factor model using equation (7.5). Note that, company by company, the difference between these numbers and those generated by the CAPM is in general not large. In fact, as the last column shows, on average, both models yield almost exactly the same required return on equity (10.6% the CAPM and 10.7% the three-factor model). Note, however, that positive and negative differences tend to cancel out in the average. Still, even if we take the average of the absolute value of the differences, we find that it is 1.3%.
Could this explain, at least partially, the popularity of the CAPM? Note that the CAPM is widely taught in business schools, is easy to understand, and easy to implement. Most alternative models are rarely taught at business schools, are more demanding in terms of data collection, their intuition is not always clear, and they are more difficult to implement. Is the additional trouble worth the extra cost of implementation? Well, if the differences in required returns between the CAPM and alternative models are around 1%, probably not. After all, that is usually quite a bit less than the difference between using a short-term and a long-term risk-free rate when we implement the CAPM.
The big picture
The CAPM makes the strong statement that the only variable that should have an impact on the required or expected return of a stock is the stock’s beta.
However, evidence both from the US and from international markets seems to quite clearly show that size and value do matter. That is, small stocks tend to