## Systematic Risk versus Firm Specific Risk

The success of a portfolio selection rule depends on the quality of the input list, that is, the estimates of expected security returns and the covariance matrix. In the long run, efficient portfolios will beat portfolios with less reliable input lists and consequently inferior reward-to-risk trade-offs.

Suppose your security analysts can thoroughly analyze 50 stocks. This means that your input list will include the following:

n = 50 estimates of expected returns n = 50 estimates of variances (n2 - n)/2 = 1,225 estimates of covariances 1,325 estimates

This is a formidable task, particularly in light of the fact that a 50-security portfolio is relatively small. Doubling n to 100 will nearly quadruple the number of estimates to 5,150. If n = 3,000, roughly the number of NYSE stocks, we need more than 4.5 million estimates.

Another difficulty in applying the Markowitz model to portfolio optimization is that errors in the assessment or estimation of correlation coefficients can lead to nonsensical results. This can happen because some sets of correlation coefficients are mutually inconsistent, as the following example demonstrates:1

Standard |
Correlation Matrix | |||

Asset |
Deviation (%) |
A |
B |
C |

A |
20 |
1.00 |
0.90 |
0.90 |

B |
20 |
0.90 |
1.00 |
0.00 |

C |
20 |
0.90 |
0.00 |
1.00 |

Suppose that you construct a portfolio with weights: -1.00; 1.00; 1.00, for assets A; B; C, respectively, and calculate the portfolio variance. You will find that the portfolio variance appears to be negative (-200). This of course is not possible because portfolio variances cannot be negative: we conclude that the inputs in the estimated correlation matrix must be mutually inconsistent. Of course, true correlation coefficients are always consistent.2 But we do not know these true correlations and can only estimate them with some imprecision. Unfortunately, it is difficult to determine whether a correlation matrix is inconsistent, providing another motivation to seek a model that is easier to implement.

Covariances between security returns tend to be positive because the same economic forces affect the fortunes of many firms. Some examples of common economic factors are business cycles, interest rates, technological changes, and cost of labor and raw materials. All these (interrelated) factors affect almost all firms. Thus unexpected changes in these variables cause, simultaneously, unexpected changes in the rates of return on the entire stock market.

1 We are grateful to Andrew Kaplin and Ravi Jagannathan, Kellogg Graduate School of Management, Northwestern University, for this example.

2 The mathematical term for a correlation matrix that cannot generate negative portfolio variance is "positive definite."

### 294 PART III Equilibrium in Capital Markets

Suppose that we summarize all relevant economic factors by one macroeconomic indicator and assume that it moves the security market as a whole. We further assume that, beyond this common effect, all remaining uncertainty in stock returns is firm specific; that is, there is no other source of correlation between securities. Firm-specific events would include new inventions, deaths of key employees, and other factors that affect the fortune of the individual firm without affecting the broad economy in a measurable way.

We can summarize the distinction between macroeconomic and firm-specific factors by writing the holding-period return on security i as ri = E(r) + mi + e,- (10.1)

where E(r) is the expected return on the security as of the beginning of the holding period, mi is the impact of unanticipated macro events on the security's return during the period, and ei is the impact of unanticipated firm-specific events. Both mi and ei have zero expected values because each represents the impact of unanticipated events, which by definition must average out to zero.

We can gain further insight by recognizing that different firms have different sensitivities to macroeconomic events. Thus if we denote the unanticipated components of the macro factor by F, and denote the responsiveness of security i to macroevents by beta, (3,-, then the macro component of security i is mi = (F, and then equation 10.1 becomes3

Equation 10.2 is known as a single-factor model for stock returns. It is easy to imagine that a more realistic decomposition of security returns would require more than one factor in equation 10.2. We treat this issue later in the chapter. For now, let us examine the simple case with only one macro factor.

Of course, a factor model is of little use without specifying a way to measure the factor that is posited to affect security returns. One reasonable approach is to assert that the rate of return on a broad index of securities such as the S&P 500 is a valid proxy for the common macro factor. This approach leads to an equation similar to the factor model, which is called a single-index model because it uses the market index to proxy for the common or systematic factor.

According to the index model, we can separate the actual or realized rate of return on a security into macro (systematic) and micro (firm-specific) components in a manner similar to that in equation 10.2. We write the rate of return on each security as a sum of three components:

Symbol | ||

1. |
The stock's expected return if the market is neutral, that is, | |

if the market's excess return, rM - rf, is zero | ||

2. |
The component of return due to movements in the overall | |

market; (3, is the security's responsiveness to market | ||

movements |
3/(M - f) | |

3. |
The unexpected component due to unexpected events that | |

are relevant only to this security (firm specific) |
e/ |

The holding period excess return on the stock can be stated as ri - rf = «i + (i(rM - rf) + ei

3 You may wonder why we choose the notation |3 for the responsiveness coefficient because |3 already has been defined in Chapter 9 in the context of the CAPM. The choice is deliberate, however. Our reasoning will be obvious shortly.

CHAPTER 10 Single-Index and Multifactor Models 295

Let us denote excess returns over the risk-free rate by capital R and rewrite this equation as

We write the index model in terms of excess returns over rf rather than in terms of total returns because the level of the stock market return represents the state of the macro economy only to the extent that it exceeds or falls short of the rate of return on risk-free T-bills. For example, in the 1950s, when T-bills were yielding only 1% or 2%, a return of 8% or 9% on the stock market would be considered good news. In contrast, in the early 1980s, when bills were yielding over 10%, that same 8% or 9% would signal disappointing macroeco-nomic news.4

Equation 10.3 says that each security has two sources of risk: market or systematic risk, attributable to its sensitivity to macroeconomic factors as reflected in RM, and firm-specific risk, as reflected in e. If we denote the variance of the excess return on the market, RM, as oM, then we can break the variance of the rate of return on each stock into two components:

Symbol | |

1. The variance attributable to the uncertainty of the common | |

macroeconomic factor |
P2 |

2. The variance attributable to firm-specific uncertainty |
<j2(ei) |

The covariance between RM and ei is zero because ei is defined as firm specific, that is, independent of movements in the market. Hence the variance of the rate of return on security i equals the sum of the variances due to the common and the firm-specific components:

What about the covariance between the rates of return on two stocks? This may be written:

But since ai and aj are constants, their covariance with any variable is zero. Further, the firm-specific terms (e;, ej) are assumed uncorrelated with the market and with each other. Therefore, the only source of covariance in the returns between the two stocks derives from their common dependence on the common factor, RM. In other words, the covariance between stocks is due to the fact that the returns on each depend in part on economywide conditions. Thus,

These calculations show that if we have n estimates of the expected excess returns, E(R;) n estimates of the sensitivity coefficients, ( n estimates of the firm-specific variances, o2(e;) 1 estimate for the variance of the (common) macroeconomic factor, oM, then these (3n + 1) estimates will enable us to prepare the input list for this single-index security universe. Thus for a 50-security portfolio we will need 151 estimates rather than

4 Practitioners often use a "modified" index model that is similar to equation 10.3 but that uses total rather than excess returns. This practice is most common when daily data are used. In this case the rate of return on bills is on the order of only about .02% per day, so total and excess returns are almost indistinguishable.

296 PART III Equilibrium in Capital Markets

1,325; for the entire New York Stock Exchange, about 3,000 securities, we will need 9,001 estimates rather than approximately 4.5 million!

It is easy to see why the index model is such a useful abstraction. For large universes of securities, the number of estimates required for the Markowitz procedure using the index model is only a small fraction of what otherwise would he needed.

Another advantage is less obvious but equally important. The index model abstraction is crucial for specialization of effort in security analysis. If a covariance term had to be calculated directly for each security pair, then security analysts could not specialize by industry. For example, if one group were to specialize in the computer industry and another in the auto industry, who would have the common background to estimate the covariance between IBM and GM? Neither group would have the deep understanding of other industries necessary to make an informed judgment of co-movements among industries. In contrast, the index model suggests a simple way to compute covariances. Covariances among securities are due to the influence of the single common factor, represented by the market index return, and can be easily estimated using equation 10.4.

The simplification derived from the index model assumption is, however, not without cost. The "cost" of the model lies in the restrictions it places on the structure of asset return uncertainty. The classification of uncertainty into a simple dichotomy—macro versus micro risk—oversimplifies sources of real-world uncertainty and misses some important sources of dependence in stock returns. For example, this dichotomy rules out industry events, events that may affect many firms within an industry without substantially affecting the broad macroeconomy.

Statistical analysis shows that relative to a single index, the firm-specific components of some firms are correlated. Examples are the nonmarket components of stocks in a single industry, such as computer stocks or auto stocks. At the same time, statistical significance does not always correspond to economic significance. Economically speaking, the question that is more relevant to the assumption of a single-index model is whether portfolios constructed using covariances that are estimated on the basis of the single-factor or singleindex assumption are significantly different from, and less efficient than, portfolios constructed using covariances that are estimated directly for each pair of stocks. We explore this issue further in Chapter 28 on active portfolio management.

Suppose that the index model for stocks A and B is estimated with the following results:

Ra = 1.0% + .9Rm + eA Rb = -2.0% + 1.1Rm + eB ctm = 20% a(eA) = 30% a(eB) = 10%

Find the standard deviation of each stock and the covariance between them.

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