CAPM And Factor Models

CAPM

• The CAPM model considers the economic concept of equilibrium
• Focused on the analysis of singular assets with the mean-variance approach
• The assumptions for the model is as follows:

Assumptions

Assumptions Cont

Individual Optimisation

• Individuals are faced with the problem of maximising utility, which wants high expected return with low standard deviation

\[ \max_{\mathbf{w}}U(r_f+\mathbf{w}^\top(\mathbf{z}-r_f\mathbf{1}), \mathbf{w^\top\Sigma w})\]

• Under CAPM the solution to the this optimisation problem is:

\[ \mathbf{w} = -\frac{U_1(.)}{2U_2(.)}\mathbf{\Sigma}^{-1}(\mathbf{z}-r_f\mathbf{1}),\quad U_1 = \frac{\partial U}{\partial\mu},U_2 = \frac{\partial U}{\partial\sigma}\] * Noting that the tangency portfolio, which is the market portfolio under equilibrium, has the weights:

\[ \mathbf{w} = \frac{1}{A-Br_f}\mathbf{\Sigma}^{-1}(\mathbf{z}-r_f\mathbf{1})\] * CAPM considers the aggregation of individual optimisation, therefore everyone will purchase the same fund of risky assets and the aggregate supply = aggregate demand for each of the assets in the risky portfolio. The resulting fund must be the market portfolio, a summation of all assets. + No mean-variance analysis is needed, the market determines the optimal portfolio for us

Capital Market Line

• The capital market line denotes the efficient portfolios when you consider the risk free asset. The intersection with the minimum variance frontier is at the tangency portfolio.

Capital Market Line

Security Market Line

• The security market line is the link between an asset’s returns and the determined market risk \(\beta\).

Derivation

• Noting that the covariance between any two portfolios \(i,j\) is calculated:

\[ \mathbf{w^\top_i\Sigma w_j}\]

• The vector of covariance of assets with the market portfolio (noting the market portfolio is the tangency portfolio in equilibrium, \(\mathbf{w^\top 1} = 1\)):

\[ \begin{split}\mathbf{\sigma_{*,M}} &= \mathbf{\Sigma w_M}\\ &= \mathbf{\Sigma\Sigma^{-1}}\frac{1}{B-Ar_f}(\mathbf{z}-r_f\mathbf{1})\\&= \frac{1}{B-Ar_f}(\mathbf{z}-r_f\mathbf{1})\end{split}\] * The variance of the market portfolio is:

\[ \begin{split} \sigma^2_M &= \mathbf{w_M^\top\Sigma w_M}\\ &= \frac{1}{B-Ar_f}\mathbf{w^\top_M}(\mathbf{z}-r_f\mathbf{1})\\&=\frac{1}{B-Ar_f}(\mathbf{w^\top_Mz}-r_f\mathbf{w^\top_M 1})\\&=\frac{1}{B-Ar_f}(z_M-r_f)\end{split}\]

• Equating these two equations together, we get the CAPM equation:

\[\begin{split} (\mathbf{z}-r_f\mathbf{1}) &= \frac{\sigma_{*,M}}{\sigma_M^2}(z_m-r_f)\\&=\mathbf{\beta}(z_m-r_f)\end{split}\] * This can be used to find the SML for a single asset:

\[ z_i = r_f +\beta_i(z_m-r_f)\] * A diagram of the Security Market Line:

Security Market Line

Pricing of Assets

• To get the actual price from the CAPM expected return; consider the random return on an asset. Note the future price is a random variable \(X\).

\[ r = \frac{X-p}{p},\quad\therefore\frac{\mathbb{E}[X]-p}{p} = r_f + \beta_i(\mathbb{E}[r_M]-r_f)\]

\[ p=\frac{\mathbb{E}[X]}{1+r_f + \beta_i(\mathbb{E}[r_M]-r_f)} = \frac{\mathbb{E}[X]}{1 + z_i}\]

Certainty Equivalent Form

• We can get the discounted certainty equivalent of the random payoff \(X\), noting:

\[ \beta = \frac{Cov(\frac{X}{p}-1, r_M)}{\sigma_M^2} = \frac{Cov(X,r_M)}{p\sigma^2_M}\]

• Subbing this into the original price equation we get the certainty equivalent form discounted:

\[ p = \frac{1}{1+r_f}\left[\mathbb{E}[X]-\frac{Cov(X,r_m)(\mathbb{E}[r_M]-r_f)}{\sigma^2_M}\right]\]

Decomposition of Risk

• Consider a random ‘noise’ variable that denotes the difference of actual and expected returns under CAPM

\[ \xi_i\equiv r_i-r_f-\beta_i(r_M-r_f)\] * Returns when considering this noise:

\[ r_i = r_f +\beta(r_m-r_f)+\xi_i\] * This noise variable has 0 mean and is uncorrelated with market return (to prove this write out \(Cov(\xi_i,r_M)\) in full form):

\[ \mathbb{E}[\xi_i] = 0,\quad Cov(\xi_i,r_m) = 0\] * Therefore we can decompose the variance:

\[ \sigma_i^2 = Cov(r_i, r_i) = Cov(\beta_ir_M+\xi_i,\beta_ir_M+\xi_i) = \beta_i^2\sigma_M^2 + \sigma^2_{\xi_i}\] ### For a portfolio

• Average return of portfolio:

\[ \mu_P = \sum^N_{i=1}w_i\mu_i\]

• Variance of portfolio

\[ Var(r_p) = Cov(r_p,r_p) = \beta_P^2\sigma^2_M + \sigma^2_{\xi_P},\quad\beta_P = \sum^N_{i=1}w_i\beta_i,\quad \sigma^2_{\xi_{i}}=\sum^N_{i=1}w^2_i\sigma^2_{\xi_i} \]

• Diversification, considering the limit of portfolio variance \(\sigma_P\), and an equally weighted portfolio \(w_i = \frac{1}{n}\forall i\)

\[ \lim_{N\rightarrow\infty}[\beta_P^2\sigma^2_M + \sigma^2_{\xi_P}] = \beta^2_P\sigma^2_M+\lim_{N\rightarrow\infty}\sum^N_{i=1}\frac{1}{N^2}\sigma^2_{\xi_i} = \beta^2_P\sigma^2_M\]

Limitations

• Limitations of the CAPM Model:
  • Assumptions can be unrealistic
  • Empirical support is not very strong
  • Hard to define the market portfolio as it contains all assets
• Alternatives:
  • No riskless asset
  • Different borrowing and lending rates
• However CAPM gives a good stepping stone
  • Systematic risk is what matters
  • Linear relationship between systematic risk and return over long period

Factor Models

• Factor models address some of the problems with the mean-variance approach
  • Computational complexity
    • \(3N+2\) parameters estimated rather than \(2N+\frac{N(N-1)}{2}\)
  • Instability and unreliability of estimates
    • Firm today and its future return may have significantly different character to its past
  • Common Risk Factors
    • Links the randomness of asset returns to a few common risk factors

Single Factor Model

• Single factor model (SFM) assumes that the asset returns are correlated with a single common risk factor, typically a proxy for the market portfolio.
  • Factor model attempts to capture major economic forces that systematically move prices of all securities
  • Any aspect of a security’s return unexplained by the factor model is assumed to the unique and firm specific
• Single factor model assumes that the random return for a stock \(i, r_i\) is related to the random outcome factor \(f\).
  • \(\alpha_i\) and \(\beta_i\) are stock specific factors

\[ r_i = \alpha_i + \beta_1f +\epsilon_i,\quad\epsilon_i\sim N(0,\sigma^2)\]

Assumptions

• \((r_i, f)\) are jointly normally distributed

• \(\mathbb{E}[\epsilon_i]=0\)

• \(Var(\epsilon_i) = \sigma^2_{\epsilon_i}\)

• \(Cov(\epsilon_i,f) = 0\)

• \(Cov(\epsilon_i,\epsilon_j) = 0\)

• \(\epsilon\sim N\)

• Unbiased expected return:

\[ \mathbb{E}[r_i] = \alpha_i+\beta_i\mu_f\]

• Variance:

\[ \sigma^2_i = \beta_i^2\sigma^2_f + \sigma^2_{\epsilon_i}\]

• Covariance:

\[ \sigma_{i,j} = \beta_i\beta_j\sigma^2_f \]

Portfolios Under SFM

• Return of the portfolio:

\[r_P = \alpha_p+\beta_pf+\epsilon_P,\quad \alpha_P=\mathbf{w^\top\alpha},\quad\beta_P=\mathbf{w^\top\beta},\quad\epsilon_P=\mathbf{w^\top\epsilon}\]

• Return is weighted sum of returns, with weights equating to 1:

\[ r_p = \mathbf{\mathbf{w^\top r}},\quad\mathbf{w^\top 1} = 1\]

• These results imply that:

\[ \mathbb{E}[r_P] = \alpha_P+\beta_P\mu_f,\quad\sigma^2_P=\beta^2_P\sigma^2_P+\sigma^2_{\epsilon_P}\]

\[ \sigma_{\epsilon_P} = \sum^N_{i=1}w^2_i\sigma^2_{\epsilon_i}\]

• The diversification of the portfolio is measured by:

\[ R^2_P = \frac{\beta^2_P\sigma^2_f}{\sigma^2_P} = \frac{\text{Systematic}}{\text{Total}}\]

• The parameters of a single factor model are often estimated using least squares linear regression:

\[ \hat{r}_{i,t} = \hat{\alpha}_i+\hat{\beta}_if_t\] * Note that the relative country’s index is often used as a proxy for the market portfolio.

Multi Factor Models

• The single factor model can be extended using multiple common risk factors

\[ r_i = \alpha_i+\beta_{i,1}f_1+\beta_{i,2}f_2+...+\beta_{i,k}f_k+\epsilon_i\]

• In matrix form:

\[ \mathbf{r}_i = \mathbf{f}\gamma_i+\epsilon\] * Where \(\gamma\) is estimated through least squares (same as beta a typical least squares regression):

\[ \mathbf{\gamma} = \mathbf{(f^\top f)^{-1}f^\top r}\]

Arbitrage Pricing Theory

• Replacement of many assumptions of the CAPM model with ‘no arbitrage’.
  • Law of one price arguments
• Assumption of the university of securities \(N\) being large, and therefore portfolios are diversified
• Returns follow a particular factor model

Simple Arbitrage Pricing Theory

• Arbitrage is a risk free profit now or in the future with no downside risk or upfront cost.
• APT works under much weaker assumptions compared to CAPM
  • Based on the LOOP, two items with the same risk must have the same price
• APT assumes that when returns are certain, investors prefer greater returns

Assumptions of Single Factor APT Model:

• All asset returns satisfy the following single factor model:

\[ r_i = \alpha_i + \beta_{1,i}I_1+\epsilon_i,\quad\mathbb{E}[I_1]=\mu_{I_1},\quad Var(I_1)=\sigma^2_{I_1}>0\]

• \(\mathbb{E}[\epsilon_i]=0\)

• \(Cov(\epsilon_i,\epsilon_j) = 0\)

• \(Cov(\epsilon_{i},I) = 0\)

• Number of securities \(N\) is very large

• No restrictions on short selling

• Define \(f_1 = \frac{I_1-\mu_1}{\sigma_{I_1}}\) such that \(\mathbb{E}[f_1] = 0,\quad \sigma_{f_1}=1\)

• Each asset can be treated as a diversified asset:

• Portfolio P is considered diversified if weights satisfy the relation:

  \[ |w_i|<\frac{A}{N}, \quad A>0\]

  • Supposing the unsystematic risk of individual assets is bounded by \(\sigma^2_{\epsilon_i}<B,\quad B>0\)

  \[ \sigma^2_{\epsilon_P} = \sum^N_{i=1}w_i^2\sigma^2_{\epsilon_i}\leq\sum^N_{i=1}\left(\frac{A}{N}\right)^2B=\frac{A^2B}{N}\approx 0\] \[ r_P = a_P+b_{1,i}f_1\]

• Considering the assumption that all assets can be treated as a diversified portfolio, the return equation becomes:

  • Noting that the variance of \(f_1 = 1\) as it is a standardized factor

\[ r_i = a_i+b_{1,i}f_1,\quad\sigma_i = b_{1,i}\]

• The APT states that \(a_i\) can be written as a combination of a risk free return \(\lambda_0\) and a factor price \(\lambda_1\):
  • The factor price is the rate at which the market rewards risk associated with undiversified risk factor.
  • \(\lambda_0\) and \(\lambda_1\) are common among assets.

\[\mathbb{E}[r_i] = a_i = \lambda_0+b_{1,i}\lambda_1\]

\[ \lambda_0 = r_f\]

\[ \lambda_1 = \frac{a_i-\lambda_0}{b_{1,i}} \]

Derivation

• Consider a portfolio of 2 assets (remember that assets can be treated as diversified portfolios themselves):

\[ \begin{split} r_P &= wr_i + (1-w)r_j \\ &= a_p+b_{1,P}f_1\\&=[wa_i+(1-w)a_i]+[wb_i+(1-w)b_i]f_1\end{split}\]

• If we select w such that \(b_P = 0\), the portfolio would be risk free as it becomes a constant. The weights that satisfy this is:

\[ w = \frac{-b_{1,j}}{b_{1,j}-b_{1,i}} = \frac{b_{1,j}}{b_{1,i}-b_{1,j}}\]

• The no arbitrage condition implies that all risk free portfolios should have the same risk free rate of return \(\lambda_0 = r_f\)

\[ \lambda_0 = \frac{a_ib_{1,j}}{b_{1,j}-b_{1,i}} + \frac{a_jb_{1,i}}{b_{1,i}-b_{1,j}}\]

• We can rearrange this to define \(\lambda_1\)

\[ \lambda_1 = \frac{a_i-\lambda_0}{b_{1,i}} = \frac{a_j-\lambda_0}{b_{1,j}} \]

• Rearranging this creates an expression where \(a_i\) and \(b_{1,i}\) are related under no arbitrage condition.

\[ \mathbb{E}[r_i] = a_i = \lambda_0 + b_{1,i}\lambda_1\]

• This can also be written as the factor price:
  • The rate at which the market rewards risk associated with undiversifiable risk factor. This is common across all assets

\[ \lambda_1 = \frac{\mathbb{E}[r_i]-\lambda_0}{\sigma_i}\]

• If there is an arbitrage opportunity, buy the undervalued asset and sell the overvalued.

Multiple Factor Simple APT

• Added Assumptions:
  • \(Cov(\epsilon_i,I_j) = 0\)
  • \(Cov(I_i,I_j) = 0\)
  • Define \(f_k = \frac{I_k-\mu_k}{\sigma_k}\)
• Returns therefore follow the multiple factor model:

\[ r_i = a_i+b_{1,i}f_1+b_{2,i}f_2 + \epsilon_i\]

• Single factor APT can be generalised to multiple factor APT:
  • \(\lambda_1, \lambda_2\) are common factor prices

\[ \mathbb{E}[r_i] = a_i = \lambda_0 + \lambda_1b_{1,i} + \lambda_2b_{2,i},\quad r_f=\lambda_0\]

• To solve equations set up simultaneous equations for \(a_i\) as \(\mathbb{E}[r_i] = a_i\)

Comparison of APT to CAPM

• Similarities:
  • Expected return is a linear function of the risk free rate and risk premium/s
  • Both imply broad diversification
  • The two can be the same under the case where the single risk factor is the market risk premium
• Main differences:
  • CAPM is based on utility theory, and in particular on mean-variance individuals
  • CAPM uses equilibrium
  • APT is based on arbitrage
  • APT says nothing about optimality of the market portfolio