Arhat Virdi
Graph Description: This is a Mean-Variance diagram, illustrating the concept of the Sharpe Ratio. The horizontal x-axis represents risk, measured by standard deviation ($\sigma_p$), and the vertical y-axis represents expected return ($E(r_p)$).
The graph shows a scatter of points representing individual risky assets and a C-shaped curve known as the efficient frontier. A straight line, labeled $CAL_T$ (Capital Allocation Line), starts at the risk-free rate ($r_f$) on the vertical axis and extends upwards, becoming tangent to the efficient frontier at a single point. This point is labeled the "Tangency portfolio". The slope of this line is the Sharpe Ratio, which represents the maximum possible risk-adjusted return. An investor finds the optimal risky portfolio by maximizing this ratio.
An asset's ratio of risk premium to risk is called its Sharpe Ratio.
$$ \frac{E(r_{T})-r_{f}}{\sigma(r_{T})} $$
The Sharpe Ratio is the slope of the Capital Allocation Line (CAL).
The optimal risky portfolio is the one that maximizes the Sharpe Ratio.
$$ Max_{\{W_{i}\}}(\frac{E(r_{T})-r_{f}}{\sigma(r_{T})}) \quad \text{subject to:} \quad \sum_{i=1}^{N}w_{i}=1 $$
Graph Description: This graph illustrates the Two-Fund Separation Theorem using the Capital Allocation Line (CAL). The axes represent expected return ($E(r_p)$) versus risk ($\sigma$). The CAL is shown as a straight line starting from the risk-free rate ($r_f$) and tangent to the efficient frontier of risky assets at point T.
The theorem states that all efficient portfolios are simply combinations of these two funds: the risk-free asset and the single tangency portfolio of risky assets.
All efficient portfolios combine the risk-free asset and the tangency portfolio.
An investor's specific risk aversion determines their location on the CAL.
Graph Description: This diagram shows how an individual investor chooses their optimal complete portfolio. The x-axis is Standard Deviation (%) and the y-axis is Expected Return (%).
The graph features the efficient frontier of risky assets ("Opportunity Set of Risky Assets") and the Capital Allocation Line (CAL(P)), which is tangent to it at point P ("Optimal Risky Portfolio"). The risk-free rate is shown at 5% on the y-axis. An "Indifference Curve" is also drawn, which is convex to the origin. This curve represents all combinations of risk and return that provide the investor with the same level of utility. The investor's "Optimal Complete Portfolio" is at point C, which is the point where their indifference curve is tangent to the Capital Allocation Line.
To find the optimal location, we solve the utility maximization problem:
$$ max_{w_{P}}(E(r_{C})-\frac{\gamma}{2}{\sigma_{C}}^{2})=max_{w_{P}}(r_{f}+w_{P}(E(r_{P})-r_{f})-\frac{\gamma}{2}w_{P}^{2}\sigma_{P}^{2}) $$
This yields the optimal weight in the risky portfolio, $w_P^*$:
$$ {w_{P}}^{*}=\frac{E(r_{P})-E(r_{f})}{\gamma\sigma_{T}^{2}} $$
The CAPM assumptions imply that all investors agree on the shape of the efficient frontier and the composition of the tangency portfolio. Therefore, every investor holds the same optimal portfolio of risky assets. They only differ in how much they allocate between this tangency portfolio and the risk-free asset.
In equilibrium, the aggregate demand for risky assets must equal the total supply.
For the market to be in equilibrium, demand must equal supply. This means the tangency portfolio must be the market portfolio. If the weight of a stock in the tangency portfolio were different from its weight in the market portfolio, its price would adjust until the weights are equal.
For the market portfolio (M) to be the optimal risky portfolio, the return-to-risk ratio (RRR) of every individual risky asset *i* must be the same and equal to that of the market portfolio.
$$ \frac{E(r_{i})-r_{f}}{\frac{\sigma_{i,M}}{\sigma_{M}}}=\frac{E(r_{M})-r_{f}}{\sigma_{M}} $$
Rearranging this gives the main CAPM equation:
$$ E(r_{i})=r_{f}+\beta_{i}(E(r_{M})-r_{f}) $$
where $\beta_{i}=\frac{cov(r_{i},r_{M})}{\sigma_{M}^{2}}$ and $E(r_{M})-r_{f}$ is the market risk premium.
Graph Description: This graph shows the Security Market Line (SML), a core concept of the CAPM. The horizontal x-axis represents Beta ($\beta$), the measure of systematic risk. The vertical y-axis represents Expected Return ($E(r)$).
The SML is a straight, upward-sloping line that graphically represents the CAPM equation. It starts at the risk-free rate ($r_f$) on the y-axis, which corresponds to a Beta of 0. The line passes through a point labeled "Market portfolio," which by definition has a Beta of 1.
According to CAPM, every security in the market should lie on this line. The second version of the graph shows a "Security X" plotted above the SML. This represents a security that is considered underpriced because its expected return is higher than what CAPM predicts for its level of systematic risk.
If CAPM holds, there is a linear relationship between an asset's expected return $E(r_i)$ and its beta $\beta_i$, which is depicted by the SML.
Image Description: This slide presents two graphs side-by-side to contrast the Capital Market Line (CML) and the Security Market Line (SML).
The left graph shows the CML. The axes are Expected Return versus Total Volatility (standard deviation). It displays the curved efficient frontier of risky assets. The CML is a straight line starting from the risk-free rate and tangent to this frontier at the Market Portfolio. Individual stocks like IBM and Disney are plotted to the right of the CML, indicating they are inefficient portfolios on their own as they have too much diversifiable risk for their level of return.
The right graph shows the SML. The axes are Expected Return versus Beta. The SML is a straight upward-sloping line representing the risk-return relationship predicted by CAPM. In this graph, all the individual stocks that were scattered in the CML plot now fall directly onto the SML. This illustrates the key idea: while individual assets are not "efficient" in terms of total risk, their expected returns are considered to be in equilibrium based on their non-diversifiable, systematic risk, which is measured by Beta.
The key prediction of CAPM is that an asset's expected return is determined by its systematic risk (beta). We can test this with a two-stage approach.
Stage 1: Estimate Betas
For each asset *i*, we run a time-series regression of its excess returns against the market's excess returns to estimate its beta ($\beta_i$).
$$ r_{i,t}=\alpha+\beta_{i}(r_{m,t}-r_{f,t})+\epsilon_{i,t} $$
This is an application of Ordinary Least Squares (OLS) regression, where we find the line that best fits the data by minimizing the sum of the squared errors. The formula for beta from this regression is mathematically identical to its theoretical definition: $\hat{\beta}=\frac{cov(y,x)}{\sigma_{x}^{2}}$.
Stage 2: Test the SML relationship
We then run a cross-sectional regression using the estimated betas from Stage 1 to explain the average returns across all assets.
$$ r_{i}=a+b\beta_{i}+cX_{i}+\epsilon_{i} $$
Here, $X_i$ represents other characteristics that might predict returns (like firm size). According to CAPM, we should find that:
Image Description: This slide shows three graphs that empirically test the Security Market Line (SML).
The first graph plots Average Monthly Return against Estimated Beta. The theoretical SML is a straight red line rising from the origin. The data points, which are portfolios sorted by beta, are represented by black dots. A dashed blue line, the "Empirical SML," is fitted to these points. This empirical line is clearly flatter than the theoretical SML and has a higher intercept, suggesting that low-beta stocks earn more than predicted by CAPM, while high-beta stocks earn less.
The second and third graphs compare the SML across two time periods. The axes are Average risk premium (%) versus Portfolio beta. In the 1931-1965 period, the data points align very closely with the theoretical SML. In the more recent 1966-2020 period, the relationship is much weaker, with the points forming a significantly flatter line than predicted.
The evidence suggests:
Image Description: This set of slides illustrates the concept of "alpha" and the "betting against beta" anomaly.
The first graph is a standard SML plot (Expected Return vs. Beta). It shows that for a stock with a beta of 1.2, the SML predicts a return of 15.6%. However, the stock's actual expected return is plotted at 17%. The vertical distance between the actual return and the SML is labeled as $\alpha_i$, representing the stock's abnormal return.
The second graph is a bar chart of monthly alpha for U.S. equities sorted into 10 portfolios by beta (P1 is low-beta, P10 is high-beta). The alphas systematically decrease as beta increases. The low-beta portfolio has a high positive alpha, while the high-beta portfolio has a negative alpha.
The third slide shows four smaller bar charts demonstrating that this "betting against beta" phenomenon (low-beta assets outperforming and high-beta assets underperforming) holds across various international markets, including Commodities, Equity Indices, Country Bonds, and Foreign Exchange.
A related way to test CAPM is to look for "alpha" ($\alpha_i$). If CAPM holds, after controlling for market exposure, an asset's alpha should be zero.
$$ E(r_{i})-r_{f}=\alpha_{i}+\beta_{i}(E(r_{M})-r_{f})+\epsilon_{i} $$
However, empirical evidence shows a "betting against beta" pattern: low-beta stocks tend to have positive alphas, while high-beta stocks have negative alphas. This departure from CAPM is a common theme observed across many different asset classes and markets.
Beyond beta, several other firm characteristics have been found to predict stock returns.
Image Description: Three charts illustrate factors that predict stock returns beyond what CAPM suggests.
The first chart shows the "Small-Firm Effect." It's a bar chart of Annual return (%) versus Size decile (1=small, 10=large). There is a clear downward trend, with the smallest firms (decile 1) showing the highest average returns (18.4%) and the largest firms (decile 10) showing the lowest (11.6%).
The second chart shows the "Book-to-Market" (or value) effect. It's a bar chart of Annual return (%) versus Book-to-market decile (1=low, 10=high). The trend is generally upward, with high book-to-market "value" stocks (decile 10) earning higher returns (16.7%) than low book-to-market "growth" stocks.
The third chart is a line graph tracking the cumulative value of $1 invested in 1927 in two strategies: "Small minus big" (long small stocks, short large stocks) and "High minus low book-to-market" (long value stocks, short growth stocks). Both lines show a strong upward trend over the long term, indicating these have historically been profitable strategies.
The existence of these persistent patterns raises questions about whether they are compatible with market efficiency.
CAPM's failures may stem from its strong assumptions.
How restrictive are these assumptions?
Relaxing some of these assumptions may help explain the empirical evidence.
Beyond the theoretical assumptions, there are several practical problems in testing and implementing CAPM.
The table shows expected returns for several stocks calculated using the CAPM formula $E(r_i) = r_f + \beta_i(r_m - r_f)$, with $r_f = 0.2\%$ and $r_M = 7.2\%$ (based on Feb 2009 data).
| Stock | Beta ($\beta$) | Expected Return |
|---|---|---|
| Amazon | 2.16 | 15.4% |
| Ford | 1.75 | 12.6% |
| Dell | 1.41 | 10.2% |
| Starbucks | 1.16 | 8.4% |
| Boeing | 1.14 | 8.3% |
| Disney | 0.96 | 7.0% |
| Newmont | 0.63 | 4.7% |
| Exxon Mobil | 0.55 | 4.2% |
| Johnson & Johnson | 0.50 | 3.8% |
| Campbell Soup | 0.30 | 2.4% |
The calculated expected return for a stock can be used as the opportunity cost of capital for a project with similar risk.
A primary use of CAPM is to find the appropriate risk-adjusted discount rate, *r*, for the Dividend Discount Model:
$$ P_{t}=\sum_{j=1}^{\infty}\frac{E[CF_{t+j}]}{(1+r)^{j}} $$
For any project or asset, you can estimate its beta and then use CAPM to find the "fair return" or cost of capital, *r*. The intuition is that a stock portfolio with the same beta represents an alternative investment opportunity for investors.
Image Description: A horizontal bar chart shows the results of a survey on firms' "Debt-Equity Ratio Policies." The policies range from "Very Strict Target" to "No Target Ratio." The longest bars correspond to "Flexible Target" (about 38%) and "Somewhat Tight Target/Range" (about 34%), indicating that while most firms aim for a target capital structure, they allow for flexibility.
When valuing an entire firm (which is financed by both debt and equity), the discount rate should reflect the risk of the firm's total assets, not just its equity.
If a firm maintains a fixed debt-to-equity ratio, its overall discount rate ($r_{firm}$) is the Weighted Average Cost of Capital (WACC).
$$ r_{firm}=r_{E}\frac{E}{V}+r_{D}\frac{D}{V} $$
Where $r_E$ is the cost of equity, $r_D$ is the cost of debt, and E, D, and V are the market values of equity, debt, and the total firm, respectively.
In practice, firms often have flexible targets for their debt-equity ratios, meaning the WACC formula is an approximation.
A project should be accepted if its Net Present Value (NPV) is greater than or equal to zero.
$$ NPV=\sum_{t=0}^{T}\frac{E(CF_{t})}{(1+r)^{t}}\ge0 $$
When calculating NPV, it is crucial to only consider incremental cash flows—the changes in cash flow that result directly from the project. This includes opportunity costs and indirect effects, but excludes sunk costs.
There are three main approaches to valuation: