Risky Portfolios and the CAPM

Investment Theory

Lorenzo Naranjo

Fall 2024

The Investment Opportunity Set

A Portfolio of Two Risky Assets

• Suppose we have two risky assets A and B. The return of a portfolio P in which we invest 1 - w in A and w in B is r_{P} = (1 - w) r_{A} + w r_{B}. \tag{1}
• The expected return of the portfolio is \mu_{P} = (1 - w) \mu_{A} + w \mu_{B}, \tag{2} whereas its variance can be computed as \begin{aligned} \sigma_{P}^{2} & = \operatorname{V}((1 - w) r_{A} + w r_{B}) \\ & = (1 - w)^{2} \sigma_{A}^{2} + w^{2} \sigma_{B}^{2} + 2 w (1 - w) \sigma_{A, B}. \end{aligned} \tag{3}
• Here \sigma_{A, B} denotes the covariance of returns between A and B.

Example 1 (A Portfolio with Two Risky Assets) Consider two risky assets A and B for which you have the following information.

Asset	Expected Return	Standard Deviation
A	10%	20%
B	15%	35%

The correlation between the assets returns is 0.4. If you invest 40% in A and 60% in B, your portfolio will have an expected return of \mu = 0.4 \times 0.10 + 0.6 \times 0.15 = 13.0\%. Since the covariance of returns is \sigma_{AB} = 0.20 \times 0.35 \times 0.4 = 0.028, the standard deviation of the portfolio returns is \sigma = \sqrt{0.4^{2} \times 0.20^{2} + 0.6^{2} \times 0.35^{2} + 2 \times 0.4 \times 0.6 \times 0.028} = 25.29\%.

Short-Selling a Stock

• In financial markets, it is possible to borrow an asset and then sell it.
  • We call this transaction short-selling the asset.
  • Typically, there is no specific date when the asset must be paid back, but must be paid back as soon as the lender requires it.
• In this class, we assume that short-selling is allowed.
• Short selling implies that w can be greater than one, in which case we borrow A to overinvest in B, or less than zero, in which case we borrow B to overinvest in A.

The Investment Opportunity Set

• The investment opportunity set generated by the two risky assets is an hyperbola.

Figure 1: The figure shows the investment opportunity set generated by two risky assets A and B.

The Minimum Variance Portfolio (MVP)

• As shown in the previous plot, there is a portfolio that has the minimum variance among all portfolios between A and B.
• To find its composition, we can use standard optimization techniques: \begin{aligned} \frac{d}{dw} \sigma_{P}^{2} & = \frac{d}{dw} (1 - w)^{2} \sigma_{A}^{2} + w^{2} \sigma_{B}^{2} + 2 w (1 - w) \sigma_{A, B} \\ & = - 2 (1 - w) \sigma_{A}^{2} + 2 w \sigma_{B}^{2} + 2 (1 - 2 w) \sigma_{A, B} = 0. \end{aligned} \tag{4}
• Thus, - (1 - w) \sigma_{A}^{2} + w \sigma_{B}^{2} + (1 - 2 w) \sigma_{A, B} = 0. or w_{MV} = \frac{\sigma_{A}^{2} - \sigma_{A, B}}{\sigma_{A}^{2} + \sigma_{B}^{2} - 2 \sigma_{A, B}}. \tag{5}

Property 1 (The Minimum Variance Portfolio) Given two risky assets A and B, there is a portfolio that has the minimum variance among all possible portfolios that can be built with A and B. The weights of the minimum variance portfolio are given by \begin{aligned} w_{A} & = \frac{\sigma_{B}^{2} - \sigma_{AB}}{\sigma_{A}^{2} + \sigma_{B}^{2} - 2 \sigma_{AB}}, \\ w_{B} & = \frac{\sigma_{A}^{2} - \sigma_{AB}}{\sigma_{A}^{2} + \sigma_{B}^{2} - 2 \sigma_{AB}}. \end{aligned} In the previous expression, \sigma_{A} and \sigma_{B} denote the standard deviation or volatility of returns. The term \sigma_{AB} denotes the covariance of A and B and is equal to \sigma_{AB} = \sigma_{A} \sigma_{B} \rho_{AB}, where \rho_{AB} is the correlation between the returns of A and B.

Example 2 (Minimum Variance Portfolio) Using the data of Example 1, we find that the weights of the minimum variance portfolio are given by \begin{aligned} w_{A} & = \frac{0.35^{2} - 0.028}{0.20^{2} + 0.35^{2} - 2 \times 0.028} = 88.73\% \\ w_{B} & = \frac{0.20^{2} - 0.028}{0.20^{2} + 0.35^{2} - 2 \times 0.028} = 11.27\% \end{aligned} Thus, the expected return and volatility of the the minimum variance portfolio are \begin{aligned} \mu & = 0.8873 \times 0.10 + 0.1127 \times 0.15 = 10.56\%, \\ \sigma & = \sqrt{0.8873^{2} \times 0.20^{2} + 0.1127^{2} \times 0.35^{2} + 2 \times 0.8873 \times 0.1127 \times 0.028} \\ & = 19.66\%. \end{aligned}

Location of the MVP

• The MVP is the point farthest to the left of the investment opportunity set.

Figure 2: The figure shows the minimum variance portfolio obtained from combining two risky assets.

Properties of the Minimum Variance Portfolio

• By definition, any other portfolio on the investment set has a variance greater than the MVP \sigma_{P}^{2} = \sigma_{MV}^{2} + \sigma_{Z}^{2} where r_{Z} = r_{A} - r_{MV} is a zero-cost portfolio orthogonal to r_{MV}, i.e., \operatorname{Cov}(r_{Z}, r_{MV}) = 0.
• Thus, \operatorname{Cov}(r_{P}, r_{MV}) = \sigma_{MV}^{2}.
• The covariance of the MVP with any other portfolio is always the same and equal to its own variance!
  • The correlation of the MVP with any other portfolio is always positive.

Example 3 (Computing a Correlation with the MVP) Consider an investment opportunity set where the MVP has a standard deviation of returns of 20%. An asset A has a standard deviation of returns equal to 30%. Thus, \sigma_{A} \sigma_{MV} \rho_{A, MV} = \sigma_{A, MV} = \sigma_{MV}^{2}, or \rho_{A, MV} = \frac{\sigma_{MV}}{\sigma_{A}} = \frac{1}{3}.

An Equally-Weighted Portfolio

• Say that we have N securities and we invest the same amount in each, so the return of this portfolio that we call P is r_{P} = \frac{1}{N} \sum_{i = 1}^{N} r_{i}.
• The variance of P is given by \sigma_{P}^{2} = \frac{1}{N^{2}} \sum_{i = 1}^{N} \sum_{j = 1}^{N} \operatorname{Cov}(r_{i}, r_{j}). \tag{6}

Average Variance and Covariance

• The average variance of the securities is \text{Avg. Variance} = \frac{1}{N} \sum_{i = 1}^{N} \operatorname{V}(r_{i})
• The average covariance of the securities is \text{Avg. Covariance} = \frac{1}{N (N - 1)} \sum_{i = 1}^{N} \sum_{\substack{j = 1 \\ i \neq j}}^{N} \operatorname{Cov}(r_{i}, r_{j})
• We can write (6) as \sigma_{P}^{2} = \frac{1}{N} (\text{Average Variance}) + \frac{N - 1}{N} (\text{Average Covariance}). \tag{7}

Understanding Diversification

• As N increases in (7), we have that \sigma_{P}^{2} \xrightarrow[N \rightarrow \infty]{} \text{Average Covariance}.
• Therefore, there is so much diversification that you can achieve given that assets comove with each other.
  • This is why we had a financial crisis in 2008 with subprime mortgages!
  • There was an underlying covariance structure in the default assumptions that many financial institutions disregarded.
• The minimum variance portfolio cannot eliminate all the risk, even though it looks for weights that are the ones that achieve the minimum variance among all portfolios.

Adding a Risk-Free Asset

• Adding a risk-free asset allows us to generate a CAL for each risky portfolio in our original investment opportunity set.

Figure 3: The figure shows the capital allocation lines of portfolios A, B, and C that are members of the same investment opportunity set of risky assets.

The Tangency Portfolio

• There is one portfolio Q that achieves the highest Sharpe ratio.

Figure 4: The figure shows the capital allocation line of the tangency portfolio.

The Efficient Frontier

• Investors like portfolios with the highest Sharpe ratio, therefore, will choose a portfolio located in the CAL of the tangency portfolio.
• We call this CAL the efficient frontier.
• All portfolios in this CAL are efficient portfolios since they all have the highest Sharpe ratio.
  • Any two portfolios in this line are enough to generate all the other efficient portfolios.
• Any other portfolios in this economy will have lower Sharpe ratios than the tangency portfolio.
  • To determine if a portfolio is efficient or not, just compute its Sharpe ratio and compare it to the Sharpe ratio of an efficient portfolio.

Example 4 (Testing for Efficiency) You know that the tangency portfolio has an expected return of 15% with a standard deviation of returns of 20%. The risk-free rate is 5% per year. You have the following information of two risky assets.

Asset	Expected Return	Standard Deviation
A	10%	20%
B	25%	40%

Are these portfolios efficient? The Sharpe ratio of the tangency portfolio is (15 - 5) / 20 = 0.5. The Sharpe ratio of A is (10 - 5) / 20 = 0.25 < 0.5, whereas the Sharpe ratio of B is (25 - 5) / 40 = 0.5. Thus only B is an efficient portfolio.

Example 5 (Constructing an Efficient Portfolio) In Example 4, note that by investing 50\% in the risk-free asset and 50\% in the tangency portfolio, we obtain a portfolio with the same expected return as A but with a lower standard deviation of 0.5 \times 0.2 = 10\%.

An investor who wants to achieve a 10% expected return would prefer to invest in this efficient portfolio rather than A. Less risk is always better!

The Capital Asset Pricing Model

The Main Idea

• The previous analysis shows that all investors with utility functions U = \mu - \frac{1}{2} A \sigma^{2} should invest in a portfolio composed of the tangency portfolio and the risk-free asset.
  • Aggregate borrowing should match aggregate lending.
  • All investors hold the tangency portfolio by investing different amounts in it.
• Overall, they own the same portfolio of risky assets in the same proportions.
• The tangency portfolio must be the market portfolio.
  • We also say that the market portfolio is an efficient portfolio.

Mean-Variance Investors Like Efficient Portfolios

Figure 5: The figure shows the investment opportunity set available to an investor. The upper line determines the efficient frontier whereas the lower line is the set of inefficient portfolios. For a given portfolio A, there is a portfolio A' that has the same expected return as A but the lowest standard deviation. Both A' and M are efficient portfolios.

Implications of the CAPM

• In Figure 5, portfolio A' is efficient, and therefore a combination of the market portfolio with the risk-free asset. r_{A'} = (1 - \beta) r_{f} + \beta r_{M}.
• The residual \varepsilon = r_{A} - r_{A'} by construction has mean zero, implying that \operatorname{E}(r_{A}) = (1 - \beta) r_{f} + \beta \operatorname{E}(r_{M}).
• The residual is also orthogonal to its projection r_{A'}, 0 = \operatorname{Cov}(\varepsilon, r_{A'}) = \operatorname{Cov}(r_{A} - r_{A'}, r_{A'}) = \beta \operatorname{Cov}(r_{A}, r_{M}) - \beta^{2} \operatorname{V}(r_{M}), or \beta = \frac{\operatorname{Cov}(r_{A}, r_{M})}{\operatorname{V}(r_{M})}.

Decomposing Returns

• The CAPM implies that we can decompose returns into a systematic and a firm-specific part, r_{A} = (1 - \beta) r_{f} + \beta_{A} r_{M} + \varepsilon_{A}
• The residual \varepsilon_{A} is orthogonal to the risk of the market and therefore is specific to the asset.
  • Firm-specific or idiosyncratic risk is diversifiable.
• The term \beta_{A} r_{M} captures the systematic risk of the asset.
  • The higher the \beta of the asset the more exposure the asset has to market risk.
  • Market risk is non-diversifiable.
• Note that if the CAPM is not true, we could replace M by the tangency portfolio Q and the analysis holds!

Property 2 (The Capital Asset Pricing Model) The returns of any asset can be decomposed into a systematic component which characterizes the non-diversifiable risk, and a firm-specific or idiosyncratic component containing the risk that can be diversified. Thus, r_{A} = (1 - \beta) r_{f} + \beta_{A} r_{M} + \varepsilon_{A}, \tag{8} where \beta = \frac{\operatorname{Cov}(r_{A}, r_{M})}{\operatorname{V}(r_{M})}. Since the risk in \varepsilon_{A} is not priced, the expected return of the asset depends on how the asset returns covary with the market risk, \operatorname{E}(r_{A}) = (1 - \beta) r_{f} + \beta \operatorname{E}(r_{M}). \tag{9}

Example 6 (Computing an Expected Return) If the risk-free rate is 5%, \beta_{DELL} = 1.3, and \operatorname{E}(r_{M}) = 14\%, then the CAPM predicts that: \operatorname{E}(r_{DELL}) = 0.05 + 1.3 \times (0.14 - 0.05) = 16.7\%. Dell stock must have an expected annual return of 16.7%.

Note that the CAPM is a prediction, and tells us how much the price of the stock should increase on average next year.

Example 7 (Computing a Stock Beta) Suppose that you know that the correlation between stock A and the market is 0.6. If the standard deviation of A returns is 40% per year, and the standard deviation of the market is 20% per year, the beta of A is \begin{aligned} \beta_{A} & = \frac{\operatorname{Cov}(r_{A}, r_{M})}{\operatorname{V}(r_{M})} = \frac{\sigma_{A} \sigma_{M} \rho_{A,M}}{\sigma_{M}^{2}} \\ & = \frac{\sigma_{A} \rho_{A,M}}{\sigma_{M}} = \frac{0.4 \times 0.6}{0.2} = 1.2. \end{aligned}

Violations of the CAPM

• What if a security has an expected return higher or lower than the CAPM predicts?
• One possibility is that markets are inefficient and the stock is mispriced.
  • Some investors can profit from this since the rest of the markets seems not to care.
• Another possibility is that investors not only care about mean and variance when choosing portfolios, but consider other attributes of the returns.
  • In this case the market portfolio will not be efficient and the analysis does not follow.
• Conclusion: it is not possible to disentangle both hypothesis using data.
  • In finance we always face the problem of a dual hypothesis test!

Variance Decomposition

• The return decomposition in equation (8) allows us to decompose the variance of any asset into two components. \sigma_{A}^{2} = \beta_{A}^{2} \sigma_{M}^{2} + \sigma^{2}(\varepsilon_{A}). \tag{10}
• The term \sigma_{A}^{2} measures the total variance of the asset.
• The total variance can then be decomposed into a systematic variance given by \beta_{A}^{2} \sigma_{M}^{2}, and a firm-specific variance equal to \sigma^{2}(\varepsilon_{A}).

Example 8 (Computing the Residual Risk) Suppose that stock A has a standard deviation of returns of 40% per year and a beta of 1.1 with the market. The standard deviation of the market returns is 20%.

This means that the residual variance is \sigma^{2}(\varepsilon_{A}) = \sigma_{A}^{2} - \beta_{A,M}^{2} \sigma_{M}^{2} = 0.1116. Therefore, the standard deviation of the firm-specific risk is \sqrt{0.1116} = 33.41\%.