Derive CAPM

The Capital Asset Pricing Model (CAPM) is a foundational model in finance that describes the relationship between an asset’s expected return and its risk, as measured by beta:

\[E(\tilde r_i) = r_f + \beta_i (E(\tilde r_m) - r_f)\]

where:

• $E(\tilde r_i)$ is the expected return of asset $i$.
• $r_f$ is the risk-free rate.
• $\beta_i$ is the sensitivity of the asset’s return to the market return.
• $E(\tilde r_m)$ is the expected return of the market.
• $E(\tilde r_m) - r_f$ is the market risk premium.

As usual, we use a tilde, as in $\tilde r$, to denote a random variable, and $r$ to denote a deterministic variable. For example, $\tilde r_i$ is the random return of asset $i$, and $r_i$ is the deterministic return of asset $i$.

The CAPM is built on the idea that investors demand higher returns for taking on additional risk. In this section, we derive the CAPM using both the mean-variance analysis framework and an alternative approach.

[!IMPORTANT]
Before reading this section, you should be familiar with utility theory and mean-variance analysis. This section focuses on the derivation of CAPM; you may skip it if you are not interested in the mathematical details.

Derivation of CAPM from Mean-Variance Analysis

Recall that in mean-variance analysis, if the two-fund separation holds, any desirable portfolio can be expressed as a combination of the risk-free asset and the market portfolio. Thus, any portfolio can be written as:

\[\begin{align*} E[\tilde r_j] &= r_f + \beta_{jm} (E[\tilde r_m] - r_f) \\ \beta_{jm} &= \frac{Cov(\tilde r_j, \tilde r_m)}{Var(\tilde r_m)} \end{align*}\]

For more details, see the Market Portfolio and Security Market Line section, and the broader Mean-Variance Analysis discussion.

With this, we have derived the CAPM from the mean-variance framework. The key assumptions required are:

• Investors are rational and risk-averse (i.e., have increasing, concave utility functions).
• Only mean and variance matter for utility optimization, which holds if:
  • Returns are normally distributed, or
  • Investors have quadratic utility functions.
  • See From Optimized Utility to Mean-Variance Analysis for more.
• The two-fund separation holds (see Market Portfolio and Security Market Line).
• The market is in equilibrium.

This is an equilibrium approach: assuming all investors are rational, risk-averse, and have the same information, the market reaches equilibrium, allowing us to derive the CAPM. However, these are strong assumptions!

Alternative Derivation of CAPM

We can also derive the CAPM directly from utility maximization, without relying on mean-variance analysis. The full utility maximization problem is discussed in Utility Theory; here, we briefly outline the process.

Assume:

• $r_f$ is the risk-free rate.
• There are $N$ risky assets with returns $r_1, r_2, \ldots, r_j, \ldots, r_N$, which are jointly normally distributed.
• $\tilde W_i$ is the optimally invested wealth for investor $i$.

The first-order condition (FOC) for utility maximization is:

\[\begin{align*} E[u_i^\prime(\tilde W_i)(\tilde r_j - r_f)] = 0, \quad \forall i, j \\ \text{where} \quad \tilde W_i = W_0^i [(1 + r_f) + \sum_{j=1}^N w_{ij}(\tilde r_j - r_f)]. \end{align*}\]

Using the definition of covariance, the FOC can be rewritten as:

\[\begin{align*} E[u_i^\prime(\tilde W_i)(\tilde r_j - r_f)] &= Cov(u_i^\prime(\tilde W_i), \tilde r_j - r_f) + E[u_i^\prime(\tilde W_i)]E[\tilde r_j - r_f] = 0, \\ E[u_i^\prime(\tilde W_i)]E[\tilde r_j - r_f] &= -Cov(u_i^\prime(\tilde W_i), \tilde r_j - r_f) \\ &= -Cov(u_i^\prime(\tilde W_i), \tilde r_j) + Cov(u_i^\prime(\tilde W_i), r_f) \\ &= -Cov(u_i^\prime(\tilde W_i), \tilde r_j) + 0 \\ &= -Cov(u_i^\prime(\tilde W_i), \tilde r_j). \end{align*}\]

By Stein’s Lemma: If $\tilde x$ and $\tilde y$ are bivariate normal, then $cov[g(\tilde x), \tilde y] = E[g^\prime(\tilde x) \tilde y]$, where $g$ is a differentiable function of $\tilde x$.

Thus:

\[E[u_i^\prime(\tilde W_i)]E[\tilde r_j - r_f] = -E[u_i^{\prime\prime}(\tilde W_i)]Cov(\tilde W_i, \tilde r_j).\]

So:

\[\begin{align*} Cov(\tilde W_i, \tilde r_j) &= \theta_{i}^{-1} E[\tilde r_j - r_f], \\ \text{where} \quad \theta_{i} &= - \frac{E[u_i^{\prime\prime}(\tilde W_i)]}{E[u_i^{\prime}(\tilde W_i)]}. \end{align*}\]

Here, $\theta_{i}$ is the absolute risk aversion of investor $i$, measuring how much risk the investor is willing to take. Higher $\theta_{i}$ means greater risk aversion. See Absolute and Relative Risk Aversion for more.

Aggregating across all investors gives the market allocation:

\[\begin{align*} Cov(\tilde W, \tilde r_j) = (\sum_{i=1}^N \theta_{i}^{-1}) E[\tilde r_j - r_f], \\ E[\tilde r_j - r_f] = (\sum_{i=1}^N \theta_{i}^{-1})^{-1} Cov(\tilde W, \tilde r_j). \end{align*}\]

Meanwhile, world wealth is $\tilde W = W_0 [1 + \lambda \tilde r_m + (1 - \lambda) r_f]$, where $\lambda$ is the weight of risky assets in world wealth. Combining the equations, we get:

\[\begin{align} E[\tilde r_j - r_f] = W_0 \lambda (\sum_{i=1}^N \theta_{i}^{-1})^{-1} Cov(\tilde r_m, \tilde r_j). \end{align}\]

Setting the market portfolio as portfolio $j$: \(\begin{align} E[\tilde r_m - r_f] = W_0 \lambda (\sum_{i=1}^N \theta_{i}^{-1})^{-1} Cov(\tilde r_m, \tilde r_m) = W_0 \lambda (\sum_{i=1}^N \theta_{i}^{-1})^{-1} Var(\tilde r_m). \end{align}\)

Dividing (2) by (1) yields:

\[E[\tilde r_j - r_f] = \frac{Cov(\tilde r_m, \tilde r_j)}{Var(\tilde r_m)} E[\tilde r_m - r_f].\]

This is the CAPM! We have derived the CAPM directly from utility maximization, without relying on mean-variance analysis.

Interpretation of terms:

• $(\sum_{i=1}^N \theta_{i}^{-1})^{-1}$ is the market’s absolute risk aversion, i.e., the harmonic mean of all investors’ absolute risk aversion.
• $W_0 (\sum_{i=1}^N \theta_{i}^{-1})^{-1}$ can be interpreted as the aggregate relative risk aversion of the market.

Summary of assumptions:

• Investors are rational and risk-averse (increasing, concave utility functions).
• Risky assets are jointly normally distributed (required for Stein’s Lemma).