We
learned in the previous chapter how to construct portfolios and how to
evaluate their returns. We now step back and examine the returns on
individual securities more closely. By doing this we shall find that the
portfolios inherit and alter the properties of the securities they
comprise.
To be concrete, let us consider the return on the equity of a company called Flyers. What will determine this equity's return in, say, the coming month?
The return on any equity traded in a financial market consists of two parts. First, the normal or expected return from the equity is the part of the return that shareholders in the market predict or expect. It depends on all of the information shareholders have that bears on the company, and it uses all of our understanding of what will influence the share price in the next month.
The second part is the uncertain or risky return on the equity. This is the portion that comes from information that will be revealed within the month. The list of such information is endless, but here are some examples: p. 300
where R is the actual total return in the month, is the expected part of the return, and U stands for the unexpected part of the return.
We must exercise some care in studying the effect of these or other news items on the return. For example, the government might give us GNP or unemployment figures for this month, but how much of that is new information for shareholders? Surely, at the beginning of the month, shareholders will have some idea or forecast of what the monthly GNP will be. The expectations of shareholders should be factored into the expected part of the return, , as of the beginning of the month. On the other hand, insofar as the announcement by the government is a surprise, and to the extent to which it influences the return on the shares, it will be part of U, the unanticipated part of the return.
As an example, suppose shareholders in the market had forecast that the GNP increase this month would be 0.5 per cent. If GNP influences our company’s share price, this forecast will be part of the information shareholders use to form the expectation, , of the monthly return. If the actual announcement this month is exactly 0.5 per cent, the same as the forecast, then the shareholders learned nothing new, and the announcement is not news. It is like hearing a rumour about a friend when you knew it all along.
On the other hand, suppose the government announced that the actual GNP increase during the year was 1.5 per cent. Now shareholders have learned something  that the increase is one percentage point higher than they had forecast. This difference between the actual result and the forecast, one percentage point in this example, is sometimes called the innovation or surprise.
Any announcement can be broken into two parts, the anticipated or expected part and the surprise or innovation:
Announcement = Expected part + Surprise
The expected part of any announcement is part of the information the market uses to form the expectation, , of the return on the equity. The surprise is the news that influences the unanticipated return on the equity, U.
For example, to open the chapter, we compared Yahoo!, Apple Computer, and ASML. In Yahoo!’s case, even though the company’s revenue and profits jumped substantially, neither number met investor expectations. In Apple’s case, in addition to the good earnings, the company announced that sales in the next quarter would be lower than previously expected. And for ASML, even though sales and profits were below the previous year, both numbers exceeded expectations.
When we speak of news, then, we refer to the surprise part of any announcement and not the portion that the market has expected and therefore has already discounted.
p. 301There are important differences, though, among various sources of risk. Look at our previous list of news stories. Some of these stories are directed specifically at Flyers, and some are more general. Which of the news items are of specific importance to Flyers?
Announcements about interest rates or GNP are clearly important for nearly all companies, whereas the news about Flyers’ chairman, its research, its sales, or the affairs of a rival company are of specific interest to Flyers. We shall divide these two types of announcement and the resulting risk, then, into two components: a systematic portion, called systematic risk; and the remainder, which we call specific or unsystematic risk. The following definitions describe the difference:
In contrast, the announcement of a small oil strike by a company may affect that company alone or a few other companies. Certainly, it is unlikely to have an effect on the world oil market. To stress that such information is unsystematic and affects only some specific companies, we sometimes call it an idiosyncratic risk.
This permits us to break down the risk of Flyers’ equity into its two components: the systematic and the unsystematic. As is traditional, we shall use the Greek epsilon, e, to represent the unsystematic risk and write
where we have used the letter m to stand for the systematic risk. Sometimes systematic risk is referred to as market risk. This emphasizes the fact that m influences all assets in the market to some extent.
The important point about the way we have broken the total risk, U, into its two components, m and e, is that e, because it is specific to the company, is unrelated to the specific risk of most other companies. For example, the unsystematic risk on Flyers’ equity, e_{F}, is unrelated to the unsystematic risk of a company, such as Vodafone, in another industry, e_{V}. The risk that Flyers’ equity will go up or down because of a discovery by its research team  or its failure to discover something  probably is unrelated to any of the specific uncertainties that affect Vodafone’s equity.
Using the terms of the previous chapter, this means that the unsystematic risks of Flyers’ equity and Vodafone’s equity are unrelated to each other, or are uncorrelated. In the symbols of statistics:
To be concrete, let us consider the return on the equity of a company called Flyers. What will determine this equity's return in, say, the coming month?
The return on any equity traded in a financial market consists of two parts. First, the normal or expected return from the equity is the part of the return that shareholders in the market predict or expect. It depends on all of the information shareholders have that bears on the company, and it uses all of our understanding of what will influence the share price in the next month.
The second part is the uncertain or risky return on the equity. This is the portion that comes from information that will be revealed within the month. The list of such information is endless, but here are some examples: p. 300
 News about Flyers’ research
 Government figures released for the gross national product (GNP)
 Results of the latest arms control talks
 Discovery that a rival’s product has been tampered with
 News that Flyers’ sales figures are higher than expected
 A sudden drop in interest rates
 The unexpected retirement of Flyers’ founder and chairman
where R is the actual total return in the month, is the expected part of the return, and U stands for the unexpected part of the return.
We must exercise some care in studying the effect of these or other news items on the return. For example, the government might give us GNP or unemployment figures for this month, but how much of that is new information for shareholders? Surely, at the beginning of the month, shareholders will have some idea or forecast of what the monthly GNP will be. The expectations of shareholders should be factored into the expected part of the return, , as of the beginning of the month. On the other hand, insofar as the announcement by the government is a surprise, and to the extent to which it influences the return on the shares, it will be part of U, the unanticipated part of the return.
As an example, suppose shareholders in the market had forecast that the GNP increase this month would be 0.5 per cent. If GNP influences our company’s share price, this forecast will be part of the information shareholders use to form the expectation, , of the monthly return. If the actual announcement this month is exactly 0.5 per cent, the same as the forecast, then the shareholders learned nothing new, and the announcement is not news. It is like hearing a rumour about a friend when you knew it all along.
On the other hand, suppose the government announced that the actual GNP increase during the year was 1.5 per cent. Now shareholders have learned something  that the increase is one percentage point higher than they had forecast. This difference between the actual result and the forecast, one percentage point in this example, is sometimes called the innovation or surprise.
Any announcement can be broken into two parts, the anticipated or expected part and the surprise or innovation:
Announcement = Expected part + Surprise
The expected part of any announcement is part of the information the market uses to form the expectation, , of the return on the equity. The surprise is the news that influences the unanticipated return on the equity, U.
For example, to open the chapter, we compared Yahoo!, Apple Computer, and ASML. In Yahoo!’s case, even though the company’s revenue and profits jumped substantially, neither number met investor expectations. In Apple’s case, in addition to the good earnings, the company announced that sales in the next quarter would be lower than previously expected. And for ASML, even though sales and profits were below the previous year, both numbers exceeded expectations.
When we speak of news, then, we refer to the surprise part of any announcement and not the portion that the market has expected and therefore has already discounted.
11.2 Risk: Systematic and Unsystematic
The unanticipated part of the return  that portion resulting from surprises  is the true risk of any investment. After all, if we got what we had expected, there would be no risk and no uncertainty.p. 301There are important differences, though, among various sources of risk. Look at our previous list of news stories. Some of these stories are directed specifically at Flyers, and some are more general. Which of the news items are of specific importance to Flyers?
Announcements about interest rates or GNP are clearly important for nearly all companies, whereas the news about Flyers’ chairman, its research, its sales, or the affairs of a rival company are of specific interest to Flyers. We shall divide these two types of announcement and the resulting risk, then, into two components: a systematic portion, called systematic risk; and the remainder, which we call specific or unsystematic risk. The following definitions describe the difference:
 A systematic risk is any risk that affects a large number of assets, each to a greater or lesser degree.
 An unsystematic risk is a risk that specifically affects a single asset or a small group of assets.^{1}
In contrast, the announcement of a small oil strike by a company may affect that company alone or a few other companies. Certainly, it is unlikely to have an effect on the world oil market. To stress that such information is unsystematic and affects only some specific companies, we sometimes call it an idiosyncratic risk.
This permits us to break down the risk of Flyers’ equity into its two components: the systematic and the unsystematic. As is traditional, we shall use the Greek epsilon, e, to represent the unsystematic risk and write
where we have used the letter m to stand for the systematic risk. Sometimes systematic risk is referred to as market risk. This emphasizes the fact that m influences all assets in the market to some extent.
The important point about the way we have broken the total risk, U, into its two components, m and e, is that e, because it is specific to the company, is unrelated to the specific risk of most other companies. For example, the unsystematic risk on Flyers’ equity, e_{F}, is unrelated to the unsystematic risk of a company, such as Vodafone, in another industry, e_{V}. The risk that Flyers’ equity will go up or down because of a discovery by its research team  or its failure to discover something  probably is unrelated to any of the specific uncertainties that affect Vodafone’s equity.
Using the terms of the previous chapter, this means that the unsystematic risks of Flyers’ equity and Vodafone’s equity are unrelated to each other, or are uncorrelated. In the symbols of statistics:
11.3 Systematic Risk and Betas
The
fact that the unsystematic parts of the returns on two companies are
unrelated to each other does not mean that the systematic portions are
unrelated. On the contrary, because both companies are influenced by the
same systematic risks, their total returns will also be related.
For example, a surprise about inflation will influence almost all companies to some extent. How sensitive is Flyers’ share price return to unanticipated changes in inflation? If Flyers’ share price tends to go up on news that inflation is exceeding expectations, we would say that it is positively related to inflation. If the share price goes down when inflation exceeds expectations and up when inflation falls short of expectations, it is negatively related. In the unusual case where an equity’s return is uncorrelated with inflation surprises, inflation has no effect on it.
p. 302We capture the influence of a systematic risk by using the beta coefficient. The beta coefficient, β, tells us the response of the equity’s return to a systematic risk. In the previous chapter beta measured the responsiveness of a security’s return to a specific risk factor, the return on the market portfolio. We used this type of responsiveness to develop the capital asset pricing model. Because we now consider many types of systematic risk, our current work can be viewed as a generalization of our work in the previous chapter.
If a company’s share price return is positively related to the risk of inflation, it has a positive inflation beta; if it is negatively related to inflation, its inflation beta is negative; and if it is uncorrelated with inflation, its inflation beta is zero.
It’s not hard to imagine some equities with positive inflation betas and others with negative inflation betas. The equity of a company owning gold mines will probably have a positive inflation beta, because an unanticipated rise in inflation is usually associated with an increase in gold prices. On the other hand, an automobile company facing stiff foreign competition might find that an increase in inflation means that the wages it pays are higher, but that it cannot raise its prices to cover the increase. This profit squeeze, as the company’s expenses rise faster than its revenues, would give its equity a negative inflation beta.
Some structure is useful at this point. Suppose we have identified three systematic risks on which we want to focus. We may believe that these three are sufficient to describe the systematic risks that influence share price returns. Three likely candidates are inflation, GNP, and interest rates. Thus every equity will have a beta associated with each of these systematic risks: an inflation beta, a GNP beta, and an interest rate beta. We can write the return on the equity, then, in the following form:
where we have used the symbol βI to denote the stock’s inflation beta, β_{GNP} for its GNP beta, and β_{r} to stand for its interest rate beta. In the equation, F stands for a surprise, whether it be in inflation, GNP, or interest rates.
Let us go through an example to see how the surprises and the expected return add up to produce the total return, R, on a given security. To make it more familiar, suppose that the return is over a horizon of a year and not just a month. Suppose that at the beginning of the year inflation is forecast to be 5 per cent for the year, GNP is forecast to increase by 2 per cent, and interest rates are expected not to change. Suppose the security we are looking at has the following betas:
The magnitude of the beta describes how great an impact a systematic risk has on a security’s returns. A beta of +1 indicates that the security’s return rises and falls one for one with the systematic factor. This means, in our example, that because the security has a GNP beta of 1, it experiences a 1 per cent increase in return for every 1 per cent surprise increase in GNP. If its GNP beta were 2, it would fall by 2 per cent when there was an unanticipated increase of 1 per cent in GNP, and it would rise by 2 per cent if GNP experienced a surprise 1 per cent decline.
Let us suppose that during the year the following events occur: inflation rises by 7 per cent, GNP rises by only 1 per cent, and interest rates fall by 2 per cent. Suppose we learn some good news about the company, perhaps that it is succeeding quickly with some new business strategy, and that this unanticipated development contributes 5 per cent to its return. In other words:
Let us assemble all of this information to find what return the security had during the year.
p. 303First we must determine what news or surprises took place in the systematic factors. From our information we know that
Expected inflation = 5%
Expected GNP change = 2%
and
Expected change in interest rates = 0%
This means that the market had discounted these changes, and the surprises will be the difference between what actually takes place and these expectations:
FI = Surprise in inflation
= Actual inflation − Expected inflation
= 7% − 5%
= 2%
Similarly,
F_{GNP} = Surprise in GNP
= Actual GNP − Expected GNP
= 1% − 2%
= −1%
and
F_{r} = Surprise in change in interest rates
= Actual change − Expected change
= −2% − 0%
= −2%
The total effect of the systematic risks on the security return, then, is
Combining this with the unsystematic risk portion, the total risky portion of the return on the security is
Last, if the expected return on the security for the year was, say, 4 per cent, the total return from all three components will be
The model we have been looking at is called a factor model, and the systematic sources of risk, designated F, are called the factors. To be perfectly formal, a kfactor model is a model where each security’s return is generated by
p. 304where ε is specific to a particular security, and is uncorrelated with the ε term for other securities. In our preceding example we had a threefactor model. We used inflation, GNP, and the change in interest rates as examples of systematic sources of risk, or factors. Researchers have not settled on what is the correct set of factors. Like so many other questions, this might be one of those matters that is never laid to rest.
In practice, researchers frequently use different models for returns. They do not use all of the economic factors we used previously as examples; instead they use an index of stock market returns  such as the FTSE 100 or DAX  in addition to returns on arbitrage portfolios representing factors that have been identified as being important from earlier research. Using the singlefactor model we can write returns like this:
Where there is only one factor (such as the returns on the FTSE 100 or DAX index), we do not need to put a subscript on the beta. In this form (with minor modifications) the factor model is called a market model. This term is employed because the index that is used for the factor is an index of returns on the whole market. The market model is written as
where R_{M} is the return on the market portfolio.^{2} The single β is called the beta coefficient.
In the past 20 years practice has changed quite significantly in the area of factor models because of two seminal research articles, published by Eugene Fama and Kenneth French in 1993 and Mark Carhart in 1997. Both papers recognized that the market index alone cannot fully explain the variation in asset returns. Fama and French introduced the threefactor model using two new factors, HML and SMB. HML stands for ’High Minus Low Book to Market Equity’ and represents the return on an arbitrage portfolio that is long (a positive investment) in high book to market equity companies and short (a negative investment or borrowing) in low book to market equity companies. An arbitrage portfolio has a zero net investment because the positive weights completely cancel out the negative weights on assets. Similarly, SML means ’Small Minus Big Companies’, and corresponds to an arbitrage portfolio that is long in small companies and short in big companies. Carhart (1997) added another factor that represented a momentum effect that is measured by the return on an arbitrage portfolio that is long in the bestperforming equities of the previous year and short in the worstperforming equities of the previous year. Algebraically, the fourfactor model is expressed as
where there are four factors representing the market risk premium, high minus low B/M, small minus big, and momentum arbitrage portfolios respectively. The FamaFrench threefactor model is simply the fourfactor model without the momentum factor.
We shall create portfolios from a list of N securities, and use a onefactor model to capture the systematic risk. The ith security in the list will therefore have returns
where we have subscripted the variables to indicate that they relate to the ith security. Notice that the factor F is not subscripted. The factor that represents systematic risk could be a surprise in GNP, or we could use the market model and let the difference between the market return and what we expect that return to be, be the factor. In either case, the factor applies to all of the securities in the portfolio.
p. 305The β_{i} is subscripted because it represents the unique way the factor influences the ith security. To recapitulate our discussion of factor models, if β_{i} is zero, the returns on the ith security are
In words, the ith security’s returns are unaffected by the factor F if β_{i} is zero. If β_{i} is positive, positive changes in the factor raise the ith security’s returns, and negative changes lower them. Conversely, if β_{i} is negative, its returns and the factor move in opposite directions.
Figure 11.1 illustrates the relationship between a security’s excess returns, , and the factor F for different betas, where β_{i} ≫⃒ 0. The lines in Fig. 11.1 plot Eq. (11.1) on the assumption that there has been no unsystematic risk: that is, ε_{i} = 0. Because we are assuming positive betas, the lines slope upwards, indicating that the return rises with F. Notice that if the factor is zero (F = 0), the line passes through zero on the yaxis.
Now let us see what happens when we create equity portfolios where each equity follows a onefactor model. Let Xi_{ }be the proportion of equity i in the portfolio. That is, if an individual with a portfolio of €100 wants €20 in Axa, we say X_{AXA} = 20%. Because the Xs represent the proportions of wealth we are investing in each of the equities, we know that they must add up to 100 per cent or 1:
We know that the portfolio return is the weighted average of the returns on the individual assets in the portfolio. Algebraically, this can be written as follows:
p. 306We saw from Eq. (11.1) that each asset, in turn, is determined by both the factor F and the unsystematic risk of ε_{i}. Thus by substituting Eq. (11.1) for each R_{i} in Eq. (11.2), we have
Equation (11.3) shows us that the return on a portfolio is determined by three sets of parameters:
Weighted average of expected returns:
Weighted average of betas ×F:
Weighted average of unsystematic risks:
This rather imposing equation is actually straightforward. The first row is the weighted average of each security’s expected return. The items in the parentheses of the second row represent the weighted average of each security’s beta. This weighted average is, in turn, multiplied by the factor F. The third row represents a weighted average of the unsystematic risks of the individual securities.
Where does uncertainty appear in Eq. (11.4)? There is no uncertainty in the first row because only the expected value of each security’s return appears there. Uncertainty in the second row is reflected by only one item, F. That is, while we know that the expected value of F is zero, we do not know what its value will be over a particular period. Uncertainty in the third row is reflected by each unsystematic risk, ε_{i}.
As it turns out, something unusual occurs to Eq. (11.4): the third row actually disappears in a large portfolio. To see this, consider a gambler who divides £1,000 by betting on red over many spins of the roulette wheel. For example, he may participate in 1,000 spins, betting £1 at a time. Though we do not know ahead of time whether a particular spin will yield red or black, we can be confident that red will win about 50 per cent of the time. Ignoring the house take, the investor can be expected to end up with just about his original £1,000.
Though we are concerned with securities, not roulette wheels, the same principle applies. Each security has its own unsystematic risk, where the surprise for one security is unrelated to the surprise of another security. By investing a small amount in each security, we bring the weighted average of the unsystematic risks close to zero in a large portfolio.^{4}
p. 307Although the third row completely vanishes in a large portfolio, nothing unusual occurs in either row 1 or row 2. Row 1 remains a weighted average of the expected returns on the individual securities as securities are added to the portfolio. Because there is no uncertainty at all in the first row, there is no way for diversification to cause this row to vanish. The terms inside the parentheses of the second row remain a weighted average of the betas. They do not vanish, either, when securities are added. Because the factor F is unaffected when securities are added to the portfolios, the second row does not vanish.
Why does the third row vanish while the second row does not, though both rows reflect uncertainty? The key is that there are many unsystematic risks in row 3. Because these risks are independent of each other, the effect of diversification becomes stronger as we add more assets to the portfolio. The resulting portfolio becomes less and less risky, and the return becomes more certain. However, the systematic risk, F, affects all securities because it is outside the parentheses in row 2. Because we cannot avoid this factor by investing in many securities, diversification does not occur in this row.
We begin by noting that the relevant risk in large and welldiversified portfolios is all systematic, because unsystematic risk is diversified away. An implication is that when a welldiversified shareholder considers changing her holdings of a particular security, she can ignore its unsystematic risk.
Notice that we are not claiming that equities, like portfolios, have no unsystematic risk. Nor are we saying that the unsystematic risk of an equity will not affect its returns. Shares do have unsystematic risk, and their actual returns do depend on the unsystematic risk. Because this risk washes out in a welldiversified portfolio, however, shareholders can ignore this unsystematic risk when they consider whether to add an equity to their portfolio. Therefore, if shareholders are ignoring the unsystematic risk, only the systematic risk of a security can be related to its expected return.
This relationship is illustrated in the security market line of Fig. 11.3. Points P, C, A, and L all lie on the line emanating from the riskfree rate of 10 per cent. The points representing each of these four assets can be created by combinations of the riskfree rate and any of the other three assets. For example, because A has a beta of 2.0 and P has a beta of 1.0, a portfolio of 50 per cent in asset A and 50 per cent in the riskless rate has the same beta as asset P. The riskfree rate is 10 per cent and the expected return on security A is 35 per cent, implying that the combination’s return of 22.5 per cent [(10% + 35%)/2] is identical to security P’s expected return. Because security P has both the same beta and the same expected return as a combination of the riskless asset and security A, an individual is equally inclined to add a small amount of security P and to add a small amount of this combination to her portfolio. However, the unsystematic risk of security P need not be equal to the unsystematic risk of the combination of security A and the riskfree rate, because unsystematic risk is diversified away in a large portfolio.
Of course, the potential combinations of points on the security market line are endless. We can duplicate P by combinations of the riskfree rate and either C or L (or both of them). We can duplicate C (or A or L) by borrowing at the riskfree rate to invest in P. The infinite number of points on the security market line that are not labelled can be used as well.
p. 309Now consider security B. Because its expected return is below the line, no investor would hold it. Instead, the investor would prefer security P, a combination of security A and the riskless asset, or some other combination. Thus security B’s price is too high. Its price will fall in a competitive market, forcing its expected return back up to the line in equilibrium.
The preceding discussion allows us to provide an equation for the security market line of Fig. 11.3. We know that a line can be described algebraically from two points. It is perhaps easiest to focus on the riskfree rate and asset P, because the riskfree rate has a beta of 0 and P has a beta of 1.
Because we know that the return on any zerobeta asset is R_{F} and the expected return on asset P is , it can easily be shown that
In Eq. (11.5), can be thought of as the expected return on any security or portfolio lying on the security market line. β is the beta of that security or portfolio.
A large, diversified portfolio has no unsystematic risk, because the unsystematic risks of the individual securities are diversified away. Assuming enough securities so that the market portfolio is fully diversified, and assuming that no security has a disproportionate market share, this portfolio is fully diversified and contains no unsystematic risk.^{6} In other words, the market portfolio is perfectly correlated with the single factor, implying that the market portfolio is really a scaledup or scaleddown version of the factor. After scaling properly, we can treat the market portfolio as the factor itself.
The market portfolio, like every security or portfolio, lies on the security market line. When the market portfolio is the factor, the beta of the market portfolio is 1 by definition. This is shown in Fig. 11.4. (We deleted the securities and the specific expected returns from Fig. 11.3 for clarity; the two graphs are otherwise identical.) With the market portfolio as the factor, Eq. (11.5) becomes
p. 310where is the expected return on the market. This equation shows that the expected return on any asset, , is linearly related to the security’s beta. The equation is identical to that of the CAPM, which we developed in the previous chapter.
For example, a surprise about inflation will influence almost all companies to some extent. How sensitive is Flyers’ share price return to unanticipated changes in inflation? If Flyers’ share price tends to go up on news that inflation is exceeding expectations, we would say that it is positively related to inflation. If the share price goes down when inflation exceeds expectations and up when inflation falls short of expectations, it is negatively related. In the unusual case where an equity’s return is uncorrelated with inflation surprises, inflation has no effect on it.
p. 302We capture the influence of a systematic risk by using the beta coefficient. The beta coefficient, β, tells us the response of the equity’s return to a systematic risk. In the previous chapter beta measured the responsiveness of a security’s return to a specific risk factor, the return on the market portfolio. We used this type of responsiveness to develop the capital asset pricing model. Because we now consider many types of systematic risk, our current work can be viewed as a generalization of our work in the previous chapter.
If a company’s share price return is positively related to the risk of inflation, it has a positive inflation beta; if it is negatively related to inflation, its inflation beta is negative; and if it is uncorrelated with inflation, its inflation beta is zero.
It’s not hard to imagine some equities with positive inflation betas and others with negative inflation betas. The equity of a company owning gold mines will probably have a positive inflation beta, because an unanticipated rise in inflation is usually associated with an increase in gold prices. On the other hand, an automobile company facing stiff foreign competition might find that an increase in inflation means that the wages it pays are higher, but that it cannot raise its prices to cover the increase. This profit squeeze, as the company’s expenses rise faster than its revenues, would give its equity a negative inflation beta.
Some structure is useful at this point. Suppose we have identified three systematic risks on which we want to focus. We may believe that these three are sufficient to describe the systematic risks that influence share price returns. Three likely candidates are inflation, GNP, and interest rates. Thus every equity will have a beta associated with each of these systematic risks: an inflation beta, a GNP beta, and an interest rate beta. We can write the return on the equity, then, in the following form:
where we have used the symbol βI to denote the stock’s inflation beta, β_{GNP} for its GNP beta, and β_{r} to stand for its interest rate beta. In the equation, F stands for a surprise, whether it be in inflation, GNP, or interest rates.
Let us go through an example to see how the surprises and the expected return add up to produce the total return, R, on a given security. To make it more familiar, suppose that the return is over a horizon of a year and not just a month. Suppose that at the beginning of the year inflation is forecast to be 5 per cent for the year, GNP is forecast to increase by 2 per cent, and interest rates are expected not to change. Suppose the security we are looking at has the following betas:
The magnitude of the beta describes how great an impact a systematic risk has on a security’s returns. A beta of +1 indicates that the security’s return rises and falls one for one with the systematic factor. This means, in our example, that because the security has a GNP beta of 1, it experiences a 1 per cent increase in return for every 1 per cent surprise increase in GNP. If its GNP beta were 2, it would fall by 2 per cent when there was an unanticipated increase of 1 per cent in GNP, and it would rise by 2 per cent if GNP experienced a surprise 1 per cent decline.
Let us suppose that during the year the following events occur: inflation rises by 7 per cent, GNP rises by only 1 per cent, and interest rates fall by 2 per cent. Suppose we learn some good news about the company, perhaps that it is succeeding quickly with some new business strategy, and that this unanticipated development contributes 5 per cent to its return. In other words:
Let us assemble all of this information to find what return the security had during the year.
p. 303First we must determine what news or surprises took place in the systematic factors. From our information we know that
Expected inflation = 5%
Expected GNP change = 2%
and
Expected change in interest rates = 0%
This means that the market had discounted these changes, and the surprises will be the difference between what actually takes place and these expectations:
FI = Surprise in inflation
= Actual inflation − Expected inflation
= 7% − 5%
= 2%
Similarly,
F_{GNP} = Surprise in GNP
= Actual GNP − Expected GNP
= 1% − 2%
= −1%
and
F_{r} = Surprise in change in interest rates
= Actual change − Expected change
= −2% − 0%
= −2%
The total effect of the systematic risks on the security return, then, is
Combining this with the unsystematic risk portion, the total risky portion of the return on the security is
Last, if the expected return on the security for the year was, say, 4 per cent, the total return from all three components will be
The model we have been looking at is called a factor model, and the systematic sources of risk, designated F, are called the factors. To be perfectly formal, a kfactor model is a model where each security’s return is generated by
p. 304where ε is specific to a particular security, and is uncorrelated with the ε term for other securities. In our preceding example we had a threefactor model. We used inflation, GNP, and the change in interest rates as examples of systematic sources of risk, or factors. Researchers have not settled on what is the correct set of factors. Like so many other questions, this might be one of those matters that is never laid to rest.
In practice, researchers frequently use different models for returns. They do not use all of the economic factors we used previously as examples; instead they use an index of stock market returns  such as the FTSE 100 or DAX  in addition to returns on arbitrage portfolios representing factors that have been identified as being important from earlier research. Using the singlefactor model we can write returns like this:
Where there is only one factor (such as the returns on the FTSE 100 or DAX index), we do not need to put a subscript on the beta. In this form (with minor modifications) the factor model is called a market model. This term is employed because the index that is used for the factor is an index of returns on the whole market. The market model is written as
where R_{M} is the return on the market portfolio.^{2} The single β is called the beta coefficient.
In the past 20 years practice has changed quite significantly in the area of factor models because of two seminal research articles, published by Eugene Fama and Kenneth French in 1993 and Mark Carhart in 1997. Both papers recognized that the market index alone cannot fully explain the variation in asset returns. Fama and French introduced the threefactor model using two new factors, HML and SMB. HML stands for ’High Minus Low Book to Market Equity’ and represents the return on an arbitrage portfolio that is long (a positive investment) in high book to market equity companies and short (a negative investment or borrowing) in low book to market equity companies. An arbitrage portfolio has a zero net investment because the positive weights completely cancel out the negative weights on assets. Similarly, SML means ’Small Minus Big Companies’, and corresponds to an arbitrage portfolio that is long in small companies and short in big companies. Carhart (1997) added another factor that represented a momentum effect that is measured by the return on an arbitrage portfolio that is long in the bestperforming equities of the previous year and short in the worstperforming equities of the previous year. Algebraically, the fourfactor model is expressed as
where there are four factors representing the market risk premium, high minus low B/M, small minus big, and momentum arbitrage portfolios respectively. The FamaFrench threefactor model is simply the fourfactor model without the momentum factor.
11.4 Portfolios and Factor Models
Now let us see what happens to portfolios of equities when each follows a onefactor model. For purposes of discussion we shall take the coming onemonth period and examine returns. We could have used a day, or a year, or any other period. If the period represents the time between decisions, however, we would rather it be short than long, and a month is a reasonable time frame to use.We shall create portfolios from a list of N securities, and use a onefactor model to capture the systematic risk. The ith security in the list will therefore have returns
where we have subscripted the variables to indicate that they relate to the ith security. Notice that the factor F is not subscripted. The factor that represents systematic risk could be a surprise in GNP, or we could use the market model and let the difference between the market return and what we expect that return to be, be the factor. In either case, the factor applies to all of the securities in the portfolio.
p. 305The β_{i} is subscripted because it represents the unique way the factor influences the ith security. To recapitulate our discussion of factor models, if β_{i} is zero, the returns on the ith security are
In words, the ith security’s returns are unaffected by the factor F if β_{i} is zero. If β_{i} is positive, positive changes in the factor raise the ith security’s returns, and negative changes lower them. Conversely, if β_{i} is negative, its returns and the factor move in opposite directions.
Figure 11.1 illustrates the relationship between a security’s excess returns, , and the factor F for different betas, where β_{i} ≫⃒ 0. The lines in Fig. 11.1 plot Eq. (11.1) on the assumption that there has been no unsystematic risk: that is, ε_{i} = 0. Because we are assuming positive betas, the lines slope upwards, indicating that the return rises with F. Notice that if the factor is zero (F = 0), the line passes through zero on the yaxis.

Now let us see what happens when we create equity portfolios where each equity follows a onefactor model. Let Xi_{ }be the proportion of equity i in the portfolio. That is, if an individual with a portfolio of €100 wants €20 in Axa, we say X_{AXA} = 20%. Because the Xs represent the proportions of wealth we are investing in each of the equities, we know that they must add up to 100 per cent or 1:
We know that the portfolio return is the weighted average of the returns on the individual assets in the portfolio. Algebraically, this can be written as follows:
p. 306We saw from Eq. (11.1) that each asset, in turn, is determined by both the factor F and the unsystematic risk of ε_{i}. Thus by substituting Eq. (11.1) for each R_{i} in Eq. (11.2), we have
Equation (11.3) shows us that the return on a portfolio is determined by three sets of parameters:
 The expected return on each individual security,
 The beta of each security multiplied by the factor F
 The unsystematic risk of each individual security, ε_{i}
Weighted average of expected returns:
Weighted average of betas ×F:
Weighted average of unsystematic risks:
This rather imposing equation is actually straightforward. The first row is the weighted average of each security’s expected return. The items in the parentheses of the second row represent the weighted average of each security’s beta. This weighted average is, in turn, multiplied by the factor F. The third row represents a weighted average of the unsystematic risks of the individual securities.
Where does uncertainty appear in Eq. (11.4)? There is no uncertainty in the first row because only the expected value of each security’s return appears there. Uncertainty in the second row is reflected by only one item, F. That is, while we know that the expected value of F is zero, we do not know what its value will be over a particular period. Uncertainty in the third row is reflected by each unsystematic risk, ε_{i}.
Portfolios and Diversification
In the previous sections of this chapter we expressed the return on a single security in terms of our factor model. Portfolios were treated next. Because investors generally hold diversified portfolios, we now want to know what Eq. (11.4) looks like in a large or diversified portfolio.^{3}As it turns out, something unusual occurs to Eq. (11.4): the third row actually disappears in a large portfolio. To see this, consider a gambler who divides £1,000 by betting on red over many spins of the roulette wheel. For example, he may participate in 1,000 spins, betting £1 at a time. Though we do not know ahead of time whether a particular spin will yield red or black, we can be confident that red will win about 50 per cent of the time. Ignoring the house take, the investor can be expected to end up with just about his original £1,000.
Though we are concerned with securities, not roulette wheels, the same principle applies. Each security has its own unsystematic risk, where the surprise for one security is unrelated to the surprise of another security. By investing a small amount in each security, we bring the weighted average of the unsystematic risks close to zero in a large portfolio.^{4}
p. 307Although the third row completely vanishes in a large portfolio, nothing unusual occurs in either row 1 or row 2. Row 1 remains a weighted average of the expected returns on the individual securities as securities are added to the portfolio. Because there is no uncertainty at all in the first row, there is no way for diversification to cause this row to vanish. The terms inside the parentheses of the second row remain a weighted average of the betas. They do not vanish, either, when securities are added. Because the factor F is unaffected when securities are added to the portfolios, the second row does not vanish.
Why does the third row vanish while the second row does not, though both rows reflect uncertainty? The key is that there are many unsystematic risks in row 3. Because these risks are independent of each other, the effect of diversification becomes stronger as we add more assets to the portfolio. The resulting portfolio becomes less and less risky, and the return becomes more certain. However, the systematic risk, F, affects all securities because it is outside the parentheses in row 2. Because we cannot avoid this factor by investing in many securities, diversification does not occur in this row.
EXAMPLE 11.1  Diversification and Unsystematic Risk  
The
preceding material can be further explained by the following example.
We keep our onefactor model here but make three specific assumptions:
Return on Walter V. Bagehot’s portfolio: We mentioned before that as N increases without limit, row 3 of Eq. (11.4) becomes equal to zero.^{5} Thus the return to Walter Bagehot’s portfolio when the number of securities is very large is The key to diversification is exhibited in Eq. (11.4’). The unsystematic risk of row 3 vanishes while the systematic risk of row 2 remains. This is illustrated in Fig. 11.2. Systematic risk, captured by variation in the factor F, is not reduced through diversification. Conversely, unsystematic risk diminishes as securities are added, vanishing as the number of securities becomes infinite. Our result is analogous to the diversification example of the previous chapter. In that chapter, we said that undiversifiable or systematic risk arises from positive covariances between securities. In this chapter, we say that systematic risk arises from a common factor F. Because a common factor causes positive covariances, the arguments of the two chapters are parallel.

11.5 Betas and Expected Returns
The Linear Relationship
p. 308We have argued many times that the expected return on a security compensates for its risk. In the previous chapter we showed that market beta (the standardized covariance of the security’s returns with those of the market) was the appropriate measure of risk under the assumptions of homogeneous expectations and riskless borrowing and lending. The capital asset pricing model, which posited these assumptions, implied that the expected return on a security was positively (and linearly) related to its beta. We shall find a similar relationship between risk and return in the onefactor model of this chapter.We begin by noting that the relevant risk in large and welldiversified portfolios is all systematic, because unsystematic risk is diversified away. An implication is that when a welldiversified shareholder considers changing her holdings of a particular security, she can ignore its unsystematic risk.
Notice that we are not claiming that equities, like portfolios, have no unsystematic risk. Nor are we saying that the unsystematic risk of an equity will not affect its returns. Shares do have unsystematic risk, and their actual returns do depend on the unsystematic risk. Because this risk washes out in a welldiversified portfolio, however, shareholders can ignore this unsystematic risk when they consider whether to add an equity to their portfolio. Therefore, if shareholders are ignoring the unsystematic risk, only the systematic risk of a security can be related to its expected return.
This relationship is illustrated in the security market line of Fig. 11.3. Points P, C, A, and L all lie on the line emanating from the riskfree rate of 10 per cent. The points representing each of these four assets can be created by combinations of the riskfree rate and any of the other three assets. For example, because A has a beta of 2.0 and P has a beta of 1.0, a portfolio of 50 per cent in asset A and 50 per cent in the riskless rate has the same beta as asset P. The riskfree rate is 10 per cent and the expected return on security A is 35 per cent, implying that the combination’s return of 22.5 per cent [(10% + 35%)/2] is identical to security P’s expected return. Because security P has both the same beta and the same expected return as a combination of the riskless asset and security A, an individual is equally inclined to add a small amount of security P and to add a small amount of this combination to her portfolio. However, the unsystematic risk of security P need not be equal to the unsystematic risk of the combination of security A and the riskfree rate, because unsystematic risk is diversified away in a large portfolio.

Of course, the potential combinations of points on the security market line are endless. We can duplicate P by combinations of the riskfree rate and either C or L (or both of them). We can duplicate C (or A or L) by borrowing at the riskfree rate to invest in P. The infinite number of points on the security market line that are not labelled can be used as well.
p. 309Now consider security B. Because its expected return is below the line, no investor would hold it. Instead, the investor would prefer security P, a combination of security A and the riskless asset, or some other combination. Thus security B’s price is too high. Its price will fall in a competitive market, forcing its expected return back up to the line in equilibrium.
The preceding discussion allows us to provide an equation for the security market line of Fig. 11.3. We know that a line can be described algebraically from two points. It is perhaps easiest to focus on the riskfree rate and asset P, because the riskfree rate has a beta of 0 and P has a beta of 1.
Because we know that the return on any zerobeta asset is R_{F} and the expected return on asset P is , it can easily be shown that
In Eq. (11.5), can be thought of as the expected return on any security or portfolio lying on the security market line. β is the beta of that security or portfolio.
The Market Portfolio and the Single Factor
In the CAPM the beta of a security measures its responsiveness to movements in the market portfolio. In the onefactor model of the arbitrage pricing theory (APT) the beta of a security measures its responsiveness to the factor. We now relate the market portfolio to the single factor.A large, diversified portfolio has no unsystematic risk, because the unsystematic risks of the individual securities are diversified away. Assuming enough securities so that the market portfolio is fully diversified, and assuming that no security has a disproportionate market share, this portfolio is fully diversified and contains no unsystematic risk.^{6} In other words, the market portfolio is perfectly correlated with the single factor, implying that the market portfolio is really a scaledup or scaleddown version of the factor. After scaling properly, we can treat the market portfolio as the factor itself.
The market portfolio, like every security or portfolio, lies on the security market line. When the market portfolio is the factor, the beta of the market portfolio is 1 by definition. This is shown in Fig. 11.4. (We deleted the securities and the specific expected returns from Fig. 11.3 for clarity; the two graphs are otherwise identical.) With the market portfolio as the factor, Eq. (11.5) becomes

p. 310where is the expected return on the market. This equation shows that the expected return on any asset, , is linearly related to the security’s beta. The equation is identical to that of the CAPM, which we developed in the previous chapter.
11.6 The Capital Asset Pricing Model and the Arbitrage Pricing Theory
The
CAPM and the APT are alternative models of risk and return. It is worth
while to consider the differences between the two models, both in terms
of pedagogy and in terms of application.
However, the APT has an offsetting advantage. The model adds factors until the unsystematic risk of any security is uncorrelated with the unsystematic risk of every other security. Under this formulation, it is easily shown that (a) unsystematic risk steadily falls (and ultimately vanishes) as the number of securities in the portfolio increases, but (b) the systematic risks do not decrease. This result was also shown in the CAPM, though the intuition was cloudier because the unsystematic risks could be correlated across securities.
p. 311In this equation, β stands for the security’s beta with respect to the first factor, g stands for the security’s beta with respect to the second factor, and so on. The equation states that the security’s expected return is related to the security’s factor betas. The intuition in Eq. (11.6) is straightforward. Each factor represents risk that cannot be diversified away. The higher a security’s beta with regard to a particular factor, the higher is the risk that the security bears. In a rational world, the expected return on the security should compensate for this risk. Equation (11.6) states that the expected return is a summation of the base expected return plus the compensation for each type of risk that the security bears.
As an example, consider the hypothetical coefficients for the fourfactor model for British Land Company plc, the UK property developer. Assume that the expected monthly return on any equity, S, can be described as
Suppose British Land Company had the following betas: β = 1.1, γ = 2, δ = 3, η = 0.1. The expected monthly return on that security would be
Assuming that British Land Company is unlevered, and that one of the firm’s projects has risk equivalent to that of the firm, this value of 0.01778 (i.e. 1.78%) can be used as the monthly discount rate for the project. (Because annual data are often supplied for capital budgeting purposes, the annual rate of 0.2355 [= (1.01778)^{12} 1] might be used instead.)
Because many factors appear on the right side of Eq. (11.6), the fourfactor formulation has the potential to measure expected returns more accurately than does the CAPM. However, as we mentioned earlier, we cannot easily determine whether these factors are appropriate. The factors in the preceding study were included because they were found to explain a significant proportion of returns for US companies. They were not derived from theory, and they may not be particularly appropriate for European, Middle Eastern or African companies.
By contrast, the use of the market index in the CAPM formulation is implied by the theory of the previous chapter. We suggested in earlier chapters that market indices (such as the FTSE 100 and DJ Euro Stoxx 50) mirror stock market movements quite well.
Differences in Pedagogy
We feel that the CAPM has at least one strong advantage from the student’s point of view. The derivation of the CAPM necessarily brings the reader through a discussion of efficient sets. This treatment  beginning with the case of two risky assets, moving to the case of many risky assets, and finishing when a riskless asset is added to the many risky ones  is of great intuitive value. This sort of presentation is not as easily accomplished with the APT.However, the APT has an offsetting advantage. The model adds factors until the unsystematic risk of any security is uncorrelated with the unsystematic risk of every other security. Under this formulation, it is easily shown that (a) unsystematic risk steadily falls (and ultimately vanishes) as the number of securities in the portfolio increases, but (b) the systematic risks do not decrease. This result was also shown in the CAPM, though the intuition was cloudier because the unsystematic risks could be correlated across securities.
Differences in Application
One advantage of the APT is that it can handle multiple factors while the CAPM ignores them. Although the bulk of our presentation in this chapter focused on the onefactor model, a multifactor model, such as the Carhart (1997) fourfactor model, is probably more reflective of reality. That is, we must abstract from many marketwide and industrywide factors before the unsystematic risk of one security becomes uncorrelated with the unsystematic risks of other securities. Under the fourfactor model, the relationship between risk and return can be expressed asp. 311In this equation, β stands for the security’s beta with respect to the first factor, g stands for the security’s beta with respect to the second factor, and so on. The equation states that the security’s expected return is related to the security’s factor betas. The intuition in Eq. (11.6) is straightforward. Each factor represents risk that cannot be diversified away. The higher a security’s beta with regard to a particular factor, the higher is the risk that the security bears. In a rational world, the expected return on the security should compensate for this risk. Equation (11.6) states that the expected return is a summation of the base expected return plus the compensation for each type of risk that the security bears.
As an example, consider the hypothetical coefficients for the fourfactor model for British Land Company plc, the UK property developer. Assume that the expected monthly return on any equity, S, can be described as
Suppose British Land Company had the following betas: β = 1.1, γ = 2, δ = 3, η = 0.1. The expected monthly return on that security would be
Assuming that British Land Company is unlevered, and that one of the firm’s projects has risk equivalent to that of the firm, this value of 0.01778 (i.e. 1.78%) can be used as the monthly discount rate for the project. (Because annual data are often supplied for capital budgeting purposes, the annual rate of 0.2355 [= (1.01778)^{12} 1] might be used instead.)
Because many factors appear on the right side of Eq. (11.6), the fourfactor formulation has the potential to measure expected returns more accurately than does the CAPM. However, as we mentioned earlier, we cannot easily determine whether these factors are appropriate. The factors in the preceding study were included because they were found to explain a significant proportion of returns for US companies. They were not derived from theory, and they may not be particularly appropriate for European, Middle Eastern or African companies.
By contrast, the use of the market index in the CAPM formulation is implied by the theory of the previous chapter. We suggested in earlier chapters that market indices (such as the FTSE 100 and DJ Euro Stoxx 50) mirror stock market movements quite well.
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