Mathematics and Finance Summer School at CMI

Chennai Mathematical Institute

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Capital Asset Pricing Model

• In order to measure the performance of a particular security or fund against that of benchmark portfolio, the `security characteristic lines’ (SCL) are very useful.

• The SCL equation is \[ \mathbb{E}(R_i)-R_F=\alpha_i+\beta_i(\mathbb{E}(R_i)-R_F) \] where \(\alpha_i\) is active return.

• Suppose there are \(N\) risky assets with return \(R_1,R_2,...,R_N\) with weights \(\omega_1,\omega_2,...,\omega_N\).

• The return of the portfolio is \[ R_P=\omega_1 R_1+\omega_2 R_2+...+\omega_N R_N. \]

• Let \(R_M\) be the return of the market index. According to SML, \[ \mathbb{E}(R_i)=R_F+\alpha_i+\beta_i\mathbb{E}(R_M-R_F), \] which means, \[ \mathbb{E}(R_P)=R_F+\Big(\sum_{i=1}^N\omega_i \alpha_i\Big)+\Big(\sum_{i=1}^N\omega_i \beta_i\Big)\mathbb{E}(R_M-R_F), \] i.e., \[ \mathbb{E}(R_P)=R_F+\alpha_P+\beta_P\mathbb{E}(R_M-R_F), \] where \(\alpha_P=\Big(\sum_{i=1}^N\omega_i \alpha_i\Big)\) and \(\beta_P=\Big(\sum_{i=1}^N\omega_i \beta_i\Big)\).

• If this is indeed the model then expected price of the portfolio is \[ \mathbb{E}(P_t|P_{t-1})=P_{t-1}\exp\{R_F+\alpha_P+\beta_P\mathbb{E}(R_M-R_F)\}, \] where \(R_P=\log(P_t)-\log(P_{t-1})\)

Factor Model

• Factor models generalize the CAPM by allowing more factors than simply the market risk and the unexplained unique risk of each asset.

• The model is defined as, \[ R_i-R_F=\beta_{0i}+\beta_{1i}x_{1}+...+\beta_{pi}x_{p}+\epsilon_i, \] where \(x_1,x_2,...,x_p\) are the \(p\) factors.

• Typically, there are two major objectives.

• If the objective is to predict the return; any factor which has a good predictive power could be used in the model. A model with good predictive power means higher chance of a good return on investment.

• For example, experts in behavioral finance and data mining experts try to measure the market sentiment with good predictive power and they use the market sentiment as a factor in the model.

• However, if the objective is to analyze the relationship between the return and factors, one should try to find the theoretical reason for including a factor in the model.

Factor Models in Matrix Notation

• We express the factor model in the matrix notation as follows: \[ r_t=B x_t+\epsilon_t \] where \[r_t=\Bigg[\begin{array}{c} r_{t1}\\ \vdots\\ r_{tN} \end{array}\Bigg]=\Bigg[\begin{array}{c} R_{t1}-R_F\\ \vdots\\ R_{tN}-R_F \end{array}\Bigg],\] is the vector of excess return over risk-free rate on day \(t\) for \(N\) many assets, \[ B=\Bigg[\begin{array}{ccc} \beta_{11}&...&\beta_{1p}\\ \vdots&...&\vdots\\ \beta_{N1}&...&\beta_{Np}\\ \end{array}\Bigg]_{N \times p}, \] is the asset’s sensitivity matrix of factors, \[ x_t=\Bigg[\begin{array}{c} x_{t1}\\ \vdots\\ x_{tN} \end{array}\Bigg] \] is the vector of risk factors.

• The portfolio return of day \(t\) is \[ r_{Pt}=\omega^Tr_t=\sum_{i=1}^N\omega_i r_{ti}, \] where \(\omega^T=\{\omega_1,...,\omega_N\}\) is the weight vector, such that \(\sum_{i=1}^N\omega_i=1\) and \(\omega_i\) is the weight of the \(i^{th}\) asset in the portfolio.

• This can be expressed as \[ r_{Pt}=\omega^TBx_t+\omega^T\epsilon_t. \]

• The expected portfolio return is \[ \mathbb{E}(r_{P})=\omega^TBx=\beta x, \] where \(\omega^TB=\beta\) is the portfolio’s sensitivity to risk factor \(x\).

• The portfolio’s covariance matrix is \[ \begin{eqnarray*} D(r_P)&=&\omega^T B \Sigma_x B^T \omega + \omega^T \Sigma_{\epsilon} \omega,\\ &=& \omega^T\Big[B \Sigma_x B^T+\Sigma_{\epsilon}\Big]\omega,\\ &=& \omega^T\Sigma_P\omega, \end{eqnarray*} \] where \(\Sigma_{\epsilon}\) is the diagonal matrix whose each diagonal elements are the unexplained volatility of each asset, \(\Sigma_x\) is the covariance matrix of risk factor, \(B\Sigma_x B^T\) is the volatility that due to risk factor.

Benefit of Diversification

• You can express the unexplained part of the portfolio volatility \(\omega^T\Sigma_{\epsilon}\omega\) as \[\begin{eqnarray*} \omega^T\Sigma_{\epsilon}\omega=(\omega_1,\hdots,\omega_N)^T \Bigg[\begin{array}{ccc} \sigma_1^2& \hdots&0\\ \vdots&\hdots&\vdots\\ 0 &\hdots&\sigma_N^2 \end{array}\Bigg]\Bigg(\begin{array}{c} \omega_1\\ \hdots\\ \omega_N \end{array}\Bigg)=\sum_{i=1}^N\omega_i^2\sigma_i^2. \end{eqnarray*}\]

• If \(\omega_i=\frac{1}{N},~~\forall~i\), then we \[ \omega^T\Sigma_{\epsilon}\omega=\frac{1}{N^2}\sum_{i=1}^N\sigma_i^2\leq \frac{\sigma_M^2}{N}, \] where \(\sigma_M^2=\max(\sigma_1^2,...,\sigma_N^2)<\infty\) is the maximum of all unexplained variances, which is finite.

• Therefore \[ \omega^T\Sigma_{\epsilon} \omega \rightarrow 0, \text{ as }N\rightarrow \infty. \]

data<-read.csv("stock_treasury.csv")
# Risk Free Rate is in percentage and annualised. 
# So the following conversion is required.
Rf<-data$UST_Yr_1/(100*250)
plot(ts(Rf),ylab="US Treasury 1 Year Yield")

n<-nrow(data)

## Compute log-return
ln_rt_snp500<-diff(log(data$SnP500))-Rf[2:n]
ln_rt_ibm<-diff(log(data$IBM_AdjClose))-Rf[2:n]
ln_rt_apple<-diff(log(data$Apple_AdjClose))-Rf[2:n]
ln_rt_msft<-diff(log(data$MSFT_AdjClose))-Rf[2:n]
ln_rt_intel<-diff(log(data$Intel_AdjClose))-Rf[2:n]

## log-return of the portfolio
ln_r <- cbind(ln_rt_ibm,ln_rt_apple,ln_rt_msft,ln_rt_intel)
head(ln_r)

##         ln_rt_ibm   ln_rt_apple  ln_rt_msft  ln_rt_intel
## [1,] -0.015870443 -2.858644e-02 -0.00924877 -0.011350577
## [2,] -0.021811910  8.413819e-05 -0.01479610 -0.018822896
## [3,] -0.006567005  1.391479e-02  0.01261529  0.020748139
## [4,]  0.021492855  3.769331e-02  0.02898454  0.018420655
## [5,]  0.004337269  1.063127e-03 -0.00844949  0.001897247
## [6,] -0.016930825 -2.495693e-02 -0.01258907 -0.004369651

• Portfolio Allocation:

Company	IBM	Apple	Microsoft	Intel
Weights	20%	30%	25%	25%
——–	——	——-	———–	——-

w = c(0.2,0.3,0.25,0.25)
ln_rt_portf = ln_r%*%w

capm_ibm<-lm(ln_rt_ibm~ln_rt_snp500)
capm_ibm_analysis<-coefficients(summary(capm_ibm))
capm_ibm_analysis <- round(capm_ibm_analysis,digits = 5)
rownames(capm_ibm_analysis)<-c("alpha","beta")
## Result of capm using lm() for IBM
capm_ibm_analysis

##       Estimate Std. Error  t value Pr(>|t|)
## alpha -0.00050    0.00058 -0.86605   0.3873
## beta   1.01601    0.05863 17.32801   0.0000

plot(ln_rt_snp500,ln_rt_ibm,xlab="S&P 500",ylab="ibm")
abline(capm_ibm,col="blue")
grid(col="red")
## Adjusted R-Squareed
summary(capm_ibm)$adj.r.squared

## [1] 0.5468333

rse<-summary(capm_ibm)$sigma
al <-capm_ibm$coefficients[1]-2*rse
b <- capm_ibm$coefficients[2]
abline(a= al,b= b,col=3,lty=2)
au <-capm_ibm$coefficients[1]+2*rse
abline(a=au,b=b,col=3,lty=2)

## CAPM on Microsoft
capm_msft<-lm(ln_rt_msft~ln_rt_snp500)
## Adjusted R-Squareed
summary(capm_msft)$adj.r.squared

## [1] 0.4736175

## CAPM on Apple
capm_msft<-lm(ln_rt_msft~ln_rt_snp500)
## Adjusted R-Squareed
summary(capm_msft)$adj.r.squared

## [1] 0.4736175

## CAPM on Intel
capm_intel<-lm(ln_rt_intel~ln_rt_snp500)
## Adjusted R-Squareed
summary(capm_intel)$adj.r.squared

## [1] 0.4085883

• CAPM on Portfolio

capm_portf<-lm(ln_rt_portf~ln_rt_snp500)
capm_portf_analysis<-round(coefficients(summary(capm_portf)),digit=5)
rownames(capm_portf_analysis)<-c("alpha","beta")
## Result of capm using lm() for Microsoft
 capm_portf_analysis

##       Estimate Std. Error  t value Pr(>|t|)
## alpha  0.00008    0.00044  0.17406  0.86196
## beta   1.11029    0.04434 25.04207  0.00000

## Adjusted R-Squareed
 summary(capm_portf)$adj.r.squared

## [1] 0.7162813