16.2 William Sharpe’s SI Model

  • What is SI model used for? Extension of the CER model to capture the stylized fact of a common component in the returns of many assets.
    • Provides an explanation as to why assets are correlated with each other: share an exposure to a common source.
    • Provides a simplification of the covariance matrix of a large number of asset returns.
    • Provides additional intuition about risk reduction through diversification.
  • Example of a factor model for asset returns. Factor models are used heavily in academic theories of asset pricing and in industry for explaining asset returns, portfolio construction and risk analysis. Sharpe’s SI model is the most widely used.
  • Motivate by showing positive correlation between individual asset returns and market index (sp500)
  • Mostly used for simple returns; but can be used for cc returns for risk analysis.

Let \(R_{it}\) denote the simple return on asset \(i\) over the investment horizon between times \(t-1\) and \(t\). Let \(R_{Mt}\) denote the simple return on a well diversified market index portfolio, such as the S&P 500 index. The single index (SI) model for \(R_{it}\) has the form:96 \[\begin{eqnarray} R_{it} & = & \alpha_{i}+\beta_{i}R_{Mt}+\varepsilon_{it},\tag{16.1}\\ R_{Mt} & \sim & iid\,N(0,\sigma_{M}^{2}),\tag{16.2}\\ \epsilon_{it} & \sim & \mathrm{GWN}\,(0,\sigma_{\epsilon,i}^{2}),\tag{16.3}\\ \mathrm{cov}(R_{Mt},\epsilon_{is}) & = & 0,\,\mathrm{f}\mathrm{or\,all}\,t\,\mathrm{and}\,s,\tag{16.4}\\ \mathrm{cov}(\epsilon_{it},\epsilon_{js}) & = & 0,\,\mathrm{for\,all}\,i\neq j\,\mathrm{and\,all}\,t\,\mathrm{and}\,s.\tag{16.5} \end{eqnarray}\] The SI model (16.1) - (16.5) assumes that all individual asset returns, \(R_{it},\) are covariance stationary and are a linear function of the market index return, \(R_{Mt},\) and an independent error term, \(\epsilon_{it}.\) The SI model is an important extension of the regression form of the CER model. In the SI model individual asset returns are explained by two distinct sources: (1) a common (to all assets) market-wide source \(R_{Mt}\); (2) and asset specific source \(\epsilon_{it}\).

16.2.1 Economic interpretation of the SI model

16.2.1.1 Interpretation of \(\beta_{i}\)

First, consider the interpretation of \(\beta_{i}\) in (16.1). The coefficient \(\beta_{i}\) is called the asset’s market exposure or market “beta”{Market beta}. It represents the slope coefficient in the linear relationship between \(R_{it}\) and \(R_{Mt}\). Because it is assumed that \(R_{Mt}\) and \(\epsilon_{it}\) are independent, \(\beta_{i}\) can be interpreted as the partial derivative \[ \frac{\partial R_{it}}{\partial R_{Mt}}=\frac{\partial}{\partial R_{Mt}}(\alpha_{i}+\beta_{i}R_{Mt}+\varepsilon_{it})=\beta_{i}. \] Then, for small changes in \(R_{Mt}\) denoted \(\Delta R_{Mt}\) and holding \(\varepsilon_{it}\) fixed, we have the approximation \[ \frac{\Delta R_{it}}{\Delta R_{Mt}}\approx\beta_{i}\Rightarrow\Delta R_{it}\approx\beta_{i}\times\Delta R_{Mt}, \] Hence, for a given change in the market return \(\beta_{i}\) determines the magnitude of the response of asset \(i's\) return. The larger (smaller) is \(\beta_{i}\) the larger (smaller) is the response of asset \(i\) to the movement in the market return.

The coefficient \(\beta_{i}\) in (16.1) has another interpretation that is directly related to portfolio risk budgeting. In particular, from chapter 14 \[\begin{equation} \beta_{i}=\frac{\mathrm{cov}(R_{it},R_{Mt})}{\mathrm{var}(R_{Mt})}=\frac{\sigma_{iM}}{\sigma_{M}^{2}},\tag{16.6} \end{equation}\] and \[ \mathrm{MC}\mathrm{R}_{i}^{\sigma_{M}}=\beta_{i}\sigma_{M}. \] Here, we see that \(\beta_{i}\) is the “beta” of asset \(i\) with respect to the market portfolio and so it is proportional to the marginal contribution of asset \(i\) to the volatility of the market portfolio. Therefore, assets with large (small) values of \(\beta_{i}\) have large (small) contributions to the volatility of the market portfolio. In this regard, \(\beta_{i}\) can be thought of as a measure of portfolio risk. Increasing allocations to assets with high (low) \(\beta_{i}\) values will increase (decrease) portfolio risk (as measured by portfolio volatility). In particular, when \(\beta_{i}=1\) asset \(i's\) percent contribution to the volatility of the market portfolio is its allocation weight. When \(\beta_{i}>1\) asset \(i's\) percent contribution to the volatility of the market portfolio is greater than its allocation weight, and when \(\beta_{i}<1\) asset \(i's\) percent contribution to the volatility of the market portfolio is less than its allocation weight.

The derivation of (16.6) is straightforward. Using (16.1) we can write \[\begin{align*} \mathrm{cov}(R_{it},R_{Mt}) & =\mathrm{cov}(\alpha_{i}+\beta_{i}R_{Mt}+\varepsilon_{it},R_{Mt})\\ & =\mathrm{cov}(\beta_{i}R_{Mt},R_{Mt})+\mathrm{cov}(\varepsilon_{it},R_{Mt})\\ & =\beta_{i}\mathrm{var}(R_{Mt})\text{ }\ (\text{since }\mathrm{cov}(\varepsilon_{it},R_{Mt})=0)\\ & \Rightarrow\beta_{i}=\frac{\mathrm{cov}(R_{it},R_{Mt})}{\mathrm{var}(R_{Mt})}. \end{align*}\]

16.2.1.2 Interpretation of \(R_{Mt}\) and \(\epsilon_{it}\)

To aid in the interpretation of \(R_{Mt}\) and \(\epsilon_{it}\) in (16.1) , re-write (16.3) as \[\begin{equation} \varepsilon_{it}=R_{it}-\alpha_{i}-\beta_{i}R_{Mt}.\tag{16.7} \end{equation}\] In (16.3), we see that \(\epsilon_{it}\) is the difference between asset \(i's\) return, \(R_{it},\) and the portion of asset \(i's\) return that is explained by the market return, \(\beta_{i}R_{Mt}\), and the intercept, \(\alpha_{i}.\) We can think of \(R_{Mt}\) as capturing “market-wide” news at time \(t\) that is common to all assets, and \(\beta_{i}\) captures the sensitivity or exposure of asset \(i\) to this market-wide news. An example of market-wide news is the release of information by the government about the national unemployment rate. If the news is good then we might expect \(R_{Mt}\) to increase because general business conditions are good. Different assets will respond differently to this news and this differential impact is captured by \(\beta_{i}.\) Assets with positive values of \(\beta_{i}\) will see their returns increase because of good market-wide news, and assets with negative values of \(\beta_{i}\) will see their returns decrease because of this good news. Hence, the magnitude and direction of correlation between asset returns can be partially explained by their exposures to common market-wide news. Because \(\epsilon_{it}\) is assumed to be independent of \(R_{Mt}\) and of \(\epsilon_{jt}\), we can think of \(\epsilon_{it}\) as capturing specific news to asset \(i\) that is unrelated to market news or to specific news to any other asset \(j\). Examples of asset-specific news are company earnings reports and reports about corporate restructuring (e.g., CEO resigning). Hence, specific news for asset \(i\) only effects the return on asset \(i\) and not the return on any other asset \(j\).

The CER model does not distinguish between overall market and asset specific news and so allows the unexpected news shocks \(\epsilon_{it}\) to be correlated across assets. Some news is common to all assets but some is specific to a given asset and the CER model error includes both types of news. In this regard, the CER model for an asset’s return is not a special case of the SI model when \(\beta_{i}=0\) because the SI model assumes that \(\epsilon_{it}\) is uncorrelated across assets.

16.2.2 Statistical properties of returns in the SI model

In this sub-section we present and derive the statistical properties of returns in the SI model (16.1) - (16.5). We will derive two types of statistical properties: unconditional and conditional. The unconditional properties are based on unconditional or marginal distribution of returns. The conditional properties are based on the distribution of returns conditional on the value of the market return.

16.2.2.1 Unconditional properties

The unconditional properties of returns in the SI model (16.1) - (16.5) are: \[\begin{eqnarray} E[R_{it}] & = & \mu_{i}=\alpha_{i}+\beta_{i}\mu_{M},\tag{16.8}\\ \mathrm{var}(R_{it}) & = & \sigma_{i}^{2}=\beta_{i}^{2}\sigma_{M}^{2}+\sigma_{\epsilon,i}^{2},\tag{16.9}\\ \mathrm{cov}(R_{it},R_{jt}) & = & \sigma_{ij}=\beta_{i}\beta_{j}\sigma_{M}^{2},\tag{16.10}\\ \mathrm{cor}(R_{it},R_{jt}) & = & \rho_{ij}=\frac{\beta_{i}\beta_{j}\sigma_{M}^{2}}{\sqrt{(\beta_{i}^{2}\sigma_{M}^{2}+\sigma_{\epsilon,i}^{2})(\beta_{j}^{2}\sigma_{M}^{2}+\sigma_{\epsilon,j}^{2})}},\tag{16.11}\\ R_{it} & \sim & iid\,N(\mu_{i},\sigma_{i}^{2})=N(\alpha_{i}+\beta_{i}\mu_{M},\beta_{i}^{2}\sigma_{M}^{2}+\sigma_{\epsilon,i}^{2})\tag{16.12} \end{eqnarray}\] The derivations of these properties are straightforward and are left as end-of-chapter exercises.

Property (16.8) shows that \[\begin{equation} \alpha_{i}=\mu_{i}-\beta_{i}\mu_{M}.\tag{16.13} \end{equation}\] Hence, the intercept term \(\alpha_{i}\) can be interpreted as the average return on asset \(i\) that is in excess of the average return due to the market.

From property (16.9), an asset’s return variance is additively decomposed into two independent components: \[\begin{eqnarray} \mathrm{var}(R_{it}) & = & \sigma_{i}^{2}=\beta_{i}^{2}\sigma_{M}^{2}+\sigma_{\epsilon,i}^{2}\tag{16.14}\\ \mathrm{(total\,asset\,}i\,\mathrm{variance} & = & \mathrm{market\,variance}+\mathrm{asset\,specific\,variance)}\nonumber \end{eqnarray}\] Here, \(\beta_{i}^{2}\sigma_{M}^{2}\) is the contribution of the market index return \(R_{Mt}\) to the total variance of asset \(i\), and \(\sigma_{\epsilon,i}^{2}\) is the contribution of the asset specific component \(\epsilon_{it}\) to the total return variance, respectively. If we divide both sides of (16.14) by \(\sigma_{i}^{2}\) we get \[\begin{eqnarray*} 1 & = & \frac{\beta_{i}^{2}\sigma_{M}^{2}}{\sigma_{i}^{2}}+\frac{\sigma_{\epsilon,i}^{2}}{\sigma_{i}^{2}}\\ & = & R+(1-R^{2}), \end{eqnarray*}\] where \[\begin{equation} R^{2}=\frac{\beta_{i}^{2}\sigma_{M}^{2}}{\sigma_{i}^{2}}\tag{16.15} \end{equation}\] is the proportion of asset \(i's\) variance that is explained by the variability of the market return \(R_{Mt}\) and \[\begin{equation} 1-R^{2}=\frac{\sigma_{\epsilon,i}^{2}}{\sigma_{i}^{2}}\tag{16.16} \end{equation}\] is the proportion of asset \(i's\) variance that is explained by the variability of the asset specific component \(\epsilon_{it}.\)

  • Sharpe’s rule of thumb. A typical stock has \(R^{2}=0.30\). That is, 30% of an asset’s variability is explained by the market movements. - \(R^{2}\) can be interpreted as the fraction of risk that is non-diversifiable, \(1-R^{2}\) gives the fraction of risk that is diversifiable. Come back to this point after discussing the SI model and portfolios below.
  • This is an example of factor risk budgeting

Properties (16.10) and (16.11) show how assets are correlated in the SI model. In particular,

  • \(\sigma_{ij}=0\) if \(\beta_{i}=0\) or \(\beta_{j}=0\) or both. Assets \(i\) and \(j\) are uncorrelated if asset \(i\) or asset \(j\) or both do not respond to market news.
  • \(\sigma_{ij}>0\) if \(\beta_{i},\beta_{j}>0\) or \(\beta_{i},\beta_{j}<0\) . Assets \(i\) and \(j\) are positively correlated if both assets respond to market news in the same direction.
  • \(\sigma_{ij}<0\) if \(\beta_{i}>0\) and \(\beta_{j}<0\) or if \(\beta_{i}<0\) and \(\beta_{j}>0\). Assets \(i\) and \(j\) are negatively correlated if they respond to market news in opposite directions.

From (16.11), assets \(i\) and \(j\) are perfectly correlated (\(\rho_{ij}=\pm1)\) only if \(\sigma_{\epsilon,i}=\sigma_{\epsilon,j}=0\).

Property (16.12) shows that the distribution of asset returns in the SI model is normal with mean and variance given by () and (), respectively.

In summary, the unconditional properties of returns in the SI model are similar to the properties of returns in the CER model: Returns are covariance stationary with constant means, variances, and covariances. Returns on different assets can be contemporaneously correlated and all asset returns uncorrelated over time. The SI model puts more structure on the expected returns, variances and covariances than the CER model and this allows for a deeper understanding of the behavior of asset returns.

16.2.2.2 Conditional properties

The properties of returns in the SI model (16.1) - (16.5) conditional on \(R_{Mt}=r_{Mt}\) are: \[\begin{eqnarray} E[R_{it}|R_{Mt} & = & r_{Mt}]=\alpha_{i}+\beta_{i}r_{Mt},\\ \mathrm{var}(R_{it}|R_{Mt} & = & r_{Mt}]=\sigma_{\epsilon,i}^{2},\\ \mathrm{cov}(R_{it},R_{jt}|R_{Mt} & = & r_{Mt}]=0,\\ \mathrm{cor}(R_{it},R_{jt}|R_{Mt} & = & r_{Mt}]=0,\\ R_{it}|R_{Mt} & \sim & iid\,N(\alpha_{i}+\beta_{i}r_{Mt},\sigma_{\epsilon,i}^{2}). \end{eqnarray}\]

Recall, conditioning on a random variable means we observe its value. In the SI model, once we observe the market return two important things happen: (1) an asset’s return variance reduces to its asset specific variance; and (2) asset returns become uncorrelated.

16.2.3 SI model and portfolios

A nice feature of the SI model for asset returns is that it also holds for a portfolio of asset returns. This property follows because asset returns are a linear function of the market return. To illustrate, consider a two asset portfolio with investment weights \(x_{1}\)and \(x_{2}\) where each asset return is explained by the SI model: \[\begin{eqnarray*} R_{1t} & = & \alpha_{1}+\beta_{1}R_{Mt}+\epsilon_{1t},\\ R_{2t} & = & \alpha_{2}+\beta_{2}R_{Mt}+\epsilon_{2t}. \end{eqnarray*}\] Then the portfolio return is \[\begin{align*} R_{p,t} & =x_{1}R_{1t}+x_{2}R_{2t}\\ & =x_{1}(\alpha_{1}+\beta_{1}R_{Mt}+\varepsilon_{1t})+x_{2}(\alpha_{2}+\beta_{2}R_{Mt}+\varepsilon_{2t})\\ & =\left(x_{1}\alpha_{1}+x_{2}\alpha_{2}\right)+\left(x_{1}\beta_{1}+x_{2}\beta_{2}\right)R_{Mt}+\left(x_{1}\varepsilon_{1t}+x_{2}\varepsilon_{2t}\right)\\ & =\alpha_{p}+\beta_{p}R_{Mt}+\varepsilon_{p,t} \end{align*}\] where \(\alpha_{p}=x_{1}\alpha_{1}+x_{2}\alpha_{2}\), \(\beta_{p}=x_{1}\beta_{1}+x_{2}\beta_{2}\), and \(\varepsilon_{p,t}=x_{1}\varepsilon_{1t}+x_{2}\varepsilon_{2t}\).

16.2.3.1 SI model and large portfolios

Consider an equally weighted portfolio of \(N\) assets, where \(N\) is a large number (e.g. \(N=500\)) whose returns are described by the SI model. Here, \(x_{i}=1/N\) for \(i=1,\ldots,N\). Then the portfolio return is \[\begin{align*} R_{p,t} & =\sum_{i=1}^{N}x_{i}R_{it}\\ & =\sum_{i=1}^{N}x_{i}\left(\alpha_{i}+\beta_{i}R_{Mt}+\varepsilon_{it}\right)\\ & =\sum_{i=1}^{N}x_{i}\alpha_{i}+\left(\sum_{i=1}^{N}x_{i}\beta_{i}\right)R_{Mt}+\sum_{i=1}^{N}x_{i}\varepsilon_{it}\\ & =\frac{1}{N}\sum_{i=1}^{N}\alpha_{i}+\left(\frac{1}{N}\sum_{i=1}^{N}\beta_{i}\right)R_{Mt}+\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{it}\\ & =\bar{\alpha}+\bar{\beta}R_{Mt}+\bar{\varepsilon}_{t}, \end{align*}\] where \(\bar{\alpha}=\frac{1}{N}\sum_{i=1}^{N}\alpha_{i}\), \(\bar{\beta}=\frac{1}{N}\sum_{i=1}^{N}\beta_{i}\) and \(\bar{\varepsilon}_{t}=\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{it}\). Now, \[ \mathrm{var}(\bar{\varepsilon}_{t})=\mathrm{var}\left(\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{it}\right)=\frac{1}{N^{2}}\sum_{i=1}^{N}\mathrm{var}(\varepsilon_{it})=\frac{1}{N}\left(\frac{1}{N}\sum_{i=1}^{N}\sigma_{\epsilon,i}^{2}\right)=\frac{1}{N}\bar{\sigma}^{2} \] where \(\bar{\sigma}^{2}=\frac{1}{N}\sum_{i=1}^{N}\sigma_{\epsilon,i}^{2}\) is the average of the asset specific variances. For large \(N\), \(\frac{1}{N}\bar{\sigma}^{2}\approx0\) and we have the Law of Large Numbers result \[ \bar{\varepsilon}_{t}=\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{it}\approx E[\varepsilon_{it}]=0. \] As a result, in a large equally weighted portfolio we have the following:

  • \(R_{p,t}\approx\bar{\alpha}+\bar{\beta}R_{Mt}:\) all non-market asset-specific risk is diversified away and only market risk remains.
  • \(\mathrm{var}(R_{p,t})=\bar{\beta}^{2}\mathrm{var}(R_{Mt})\Rightarrow\mathrm{SD}(R_{p,t})=|\bar{\beta}|\times\mathrm{SD}(R_{Mt}):\) portfolio volatility is proportional to market volatility where the factor of proportionality is the absolute value of portfolio beta.
  • \(R^{2}\approx1:\) Approximately 100% of portfolio variance is due to market variance.
  • \(\bar{\beta}\approx1\). A large equally weighted portfolio resembles the market portfolio (e.g., as proxied by the S&P 500 index) and so the beta of a well diversified portfolio will be close to the beta of the market portfolio which is one by definition.97

These results help us to understand the type of risk that gets diversified away and the type of risk that remains when we form diversified portfolios. Asset specific risk, which is uncorrelated across assets, gets diversified away whereas market risk, which is common to all assets, does not get diversified away.

  • (Relate to average covariance calculation from portfolio theory chapter).
  • Related to asset R2 discussed earlier. R2 of an asset shows the portion of risk that cannot be diversified away when forming portfolios.

16.2.4 The SI model in matrix notation

  • Need to emphasize that the SI model covariance matrix is always positive definite. This is an important result because it allows for the mean-variance analysis of very large portfolios.

For \(i=1,\ldots,N\) assets, stacking (16.1) gives the SI model in matrix notation \[ \left(\begin{array}{c} R_{1t}\\ \vdots\\ R_{Nt} \end{array}\right)=\left(\begin{array}{c} \alpha_{1}\\ \vdots\\ \alpha_{N} \end{array}\right)+\left(\begin{array}{c} \beta_{1}\\ \vdots\\ \beta_{N} \end{array}\right)R_{Mt}+\left(\begin{array}{c} \epsilon_{1t}\\ \vdots\\ \epsilon_{Nt} \end{array}\right), \] or \[\begin{equation} \mathbf{R}_{t}=\alpha+\beta R_{Mt}+\epsilon_{t}.\tag{16.17} \end{equation}\] The unconditional statistical properties of returns (16.8), (16.9), (16.10) and (16.12) can be re-expressed using matrix notation as follows: \[\begin{eqnarray} E[\mathbf{R}_{t}] & = & \mu=\alpha+\beta\mu_{M},\tag{16.18}\\ \mathrm{var}(\mathbf{R}_{t}) & = & \Sigma=\sigma_{M}^{2}\beta \beta ^{\prime}+\mathbf{D},\tag{16.19}\\ \mathbf{R}_{t} & \sim & iid\,N(\mu,\Sigma)=N(\alpha+\beta\mu_{M},\sigma_{M}^{2}\beta \beta ^{\prime}+\mathbf{D}),\tag{16.20} \end{eqnarray}\] where \[\begin{eqnarray} \mathbf{D} & = & \mathrm{var}(\epsilon_{t})=\left(\begin{array}{cccc} \sigma_{\epsilon,1}^{2} & 0 & \cdots & 0\\ 0 & \sigma_{\epsilon,2}^{2} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & \sigma_{\epsilon,N}^{2} \end{array}\right)\\ & = & \mathrm{diag}(\sigma_{\epsilon,1}^{2},\sigma_{\epsilon,1}^{2},\ldots,\sigma_{\epsilon,N}^{2}).\nonumber \end{eqnarray}\]

The derivation of the SI model covariance matrix (16.19) is \[\begin{eqnarray*} \mathrm{var}(\mathbf{R}_{t}) & = & \Sigma=\beta \mathrm{var}(R_{Mt})\beta ^{\prime}+\mathrm{var}(\epsilon_{t})\\ & = & \sigma_{M}^{2}\beta \beta ^{\prime}+\mathbf{D}, \end{eqnarray*}\] which uses the assumption that the market return \(R_{Mt}\) is uncorrelated will all asset specific error terms in \(\epsilon_{t}\).

It is useful to examine the SI covariance matrix (16.19) for a three asset portfolio. In this case, we have \[\begin{align*} R_{it} & =\alpha_{i}+\beta_{i}R_{Mt}+\varepsilon_{it},\text{ }i=1,2,3\\ \sigma_{i}^{2} & =\mathrm{var}(R_{it})=\beta_{i}^{2}\sigma_{M}^{2}+\sigma_{\varepsilon,i}^{2}\\ \sigma_{ij} & =\mathrm{cov}(R_{it},R_{jt})=\sigma_{M}^{2}\beta_{i}\beta_{j} \end{align*}\] The \(3\times3\) covariance matrix is \[\begin{align*} \Sigma & =\left(\begin{array}{ccc} \sigma_{1}^{2} & \sigma_{12} & \sigma_{13}\\ \sigma_{12} & \sigma_{2}^{2} & \sigma_{23}\\ \sigma_{13} & \sigma_{23} & \sigma_{3}^{2} \end{array}\right)\\ & =\left(\begin{array}{ccc} \beta_{1}^{2}\sigma_{M}^{2}+\sigma_{\varepsilon,1}^{2} & \sigma_{M}^{2}\beta_{1}\beta_{2} & \sigma_{M}^{2}\beta_{1}\beta_{3}\\ \sigma_{M}^{2}\beta_{1}\beta_{2} & \beta_{2}^{2}\sigma_{M}^{2}+\sigma_{\varepsilon,2}^{2} & \sigma_{M}^{2}\beta_{2}\beta_{3}\\ \sigma_{M}^{2}\beta_{1}\beta_{3} & \sigma_{M}^{2}\beta_{2}\beta_{3} & \beta_{3}^{2}\sigma_{M}^{2}+\sigma_{\varepsilon,3}^{2} \end{array}\right)\\ & =\sigma_{M}^{2}\left(\begin{array}{ccc} \beta_{1}^{2} & \beta_{1}\beta_{2} & \beta_{1}\beta_{3}\\ \beta_{1}\beta_{2} & \beta_{2}^{2} & \beta_{2}\beta_{3}\\ \beta_{1}\beta_{3} & \beta_{2}\beta_{3} & \beta_{3}^{2} \end{array}\right)+\left(\begin{array}{ccc} \sigma_{\varepsilon,1}^{2} & 0 & 0\\ 0 & \sigma_{\varepsilon,2}^{2} & 0\\ 0 & 0 & \sigma_{\varepsilon,3}^{2} \end{array}\right). \end{align*}\] The first matrix shows the return variance and covariance contributions due to the market returns, and the second matrix shows the contributions due to the asset specific errors. Define \(\beta =(\beta_{1},\beta_{2},\beta_{3})^{\prime}.\) Then \[\begin{eqnarray*} \sigma_{M}^{2}\beta \beta ^{\prime} & = & \sigma_{M}^{2}\left(\begin{array}{c} \beta_{1}\\ \beta_{2}\\ \beta_{3} \end{array}\right)\left(\begin{array}{ccc} \beta_{1} & \beta_{2} & \beta_{3}\end{array}\right)=\sigma_{M}^{2}\left(\begin{array}{ccc} \beta_{1}^{2} & \beta_{1}\beta_{2} & \beta_{1}\beta_{3}\\ \beta_{1}\beta_{2} & \beta_{2}^{2} & \beta_{2}\beta_{3}\\ \beta_{1}\beta_{3} & \beta_{2}\beta_{3} & \beta_{3}^{2} \end{array}\right),\\ \mathbf{D} & = & \mathrm{diag}(\sigma_{\varepsilon,1}^{2},\sigma_{\varepsilon,2}^{2},\sigma_{\varepsilon,3}^{2})=\left(\begin{array}{ccc} \sigma_{\varepsilon,1}^{2} & 0 & 0\\ 0 & \sigma_{\varepsilon,2}^{2} & 0\\ 0 & 0 & \sigma_{\varepsilon,3}^{2} \end{array}\right), \end{eqnarray*}\] and so \[ \Sigma=\sigma_{M}^{2}\beta \beta ^{\prime}+\mathbf{D}. \]

The matrix form of the SI model (16.18) - (16.20) is useful for portfolio analysis. For example, consider a portfolio with \(N\times1\) weight vector \(\mathbf{x}=(x_{1},\ldots,x_{N})^{\prime}.\) Using (16.17), the SI model for the portfolio return \(R_{p,t}=\mathbf{x}^{\prime}R_{t}\) is \[\begin{eqnarray*} R_{p,t} & = & \mathbf{x}^{\prime}(\alpha+\beta R_{Mt}+\epsilon_{t})\\ & = & \mathbf{x}^{\prime}\alpha+\mathbf{x}^{\prime}\beta R_{Mt}+\mathbf{x}^{\prime}\epsilon_{t}\\ & = & \alpha_{p}+\beta_{p}R_{Mt}+\epsilon_{p,t}, \end{eqnarray*}\] where \(\alpha_{p}=\mathbf{x}^{\prime}\alpha\), \(\beta_{p}=\mathbf{x}^{\prime}\beta\) and \(\epsilon_{p,t}=\mathbf{x}^{\prime}\epsilon_{t}\).

  1. The single index model is also called the* market model* or the single factor model.↩︎

  2. The S&P 500 index is a value weighted index and so is not equal to an equally weighted portfolio of the stocks in the S&P 500 index. However, since most stocks in the S&P 500 index have large market capitalizations the values weights are not too different from equal weights.↩︎