# Using CAPM to derive the following

Background Information:

Say there are $s = 1,\ldots,S$ possible future outcomes (states) with known probabilities $\pi_s > 0$, $\sum_{s=1}^{S}\pi_s = 1$. Define the expected payoff as $\mathbb{E}_\pi\left[X\right] = \sum_s \pi_s x_s = \mu_x$

Question:

The CAPM is often used to compute fair market prices of assets. There are two commonly used pricing formulae derived from the CAPM for this purpose. One solves for the price as the discounted risk-adjusted expected payoff

$$P = \frac{\mathbb{E}\left[\tilde{X} - \Pi\right]}{(1+r_f)}$$

and the other solves for the price as the expected payoff discounted by a risk adjusted discount rate $$P = \frac{\mathbb{E}\left[\tilde{X}\right]}{(1+r_f + \pi)}$$ Derive both of these expression from the CAPM and identify the risk-adjustment returns.

Thoughts: I am not familiar with the notation above, and referring to the CAPM equation online I am not sure how to use that and derive the latter. Any thoughts or suggestions should help.

• Hint: What does CAPM say? It says securities have to be priced to give a certain required return $r=\cdots$. What does $P = \frac{\mathbb{E}\left[\tilde{X}\right]}{(1+r_f + \pi)}$ say? It says the security price is it's future expected value discounted at the rate $r_f+\pi$. So you can identify $r$ from CAPM and $r_f+\pi$, i.e. set them equal. That will show what $\pi$ is that will make this work. Similarly for the other equation you should be able to figure out what $\Pi$ is assuming the CAPM holds. – Alex C Feb 3 '17 at 0:11
• @AlexC still a bit lost, could you provide an answer? – user26356 Feb 3 '17 at 16:05

The standard formula for Capital Asset Pricing Model is:

\begin{equation} \bar{r} = r_f + \beta \cdot ( \bar{r_m} - r_f) \quad (1) \end{equation} in which: \begin{equation} \bar{r} \textit{ - expected return of an asset} \end{equation} \begin{equation} \ r_f \textit{ - risk-free rate} \end{equation} \begin{equation} \beta \textit{ - beta of an asset} \end{equation} \begin{equation} \bar{r_m} \textit{ - expected market return} \end{equation}

Expected return of an asset is the simple formula of expected price of an asset and a current price of an asset. For example, if the current price is 2 and the expected price is 3, then expected return is 50%.

\begin{equation} \bar{r} = \frac{E( \tilde{X} ) - P}{ P } \quad (2) \end{equation}

In other words the price is discounted expected price:

\begin{equation} \ P = \frac{E( \tilde{X} )}{ 1 + \bar{r} } \quad (3) \end{equation}

Combining the equations (1) and (3) we have your second equation:

\begin{equation} \ P = \frac{E( \tilde{X} )}{ 1 + r_f + \beta \cdot ( \bar{r_m} - r_f) } \quad (4) \end{equation}

The formula for an asset beta is:

\begin{equation} \beta = \frac{Cov( \bar{r}, \bar{r_m} )}{ \sigma_m^2} \quad (5) \end{equation}

After a little modification of the equation (2) we can use it in the equation (5).

\begin{equation} \beta = \frac{Cov( \frac{E( \tilde{X} )}{ P } - 1, \bar{r_m} )}{ \sigma_m^2} = \frac{Cov( E( \tilde{X} ), \bar{r_m} )}{ \sigma_m^2 \cdot P} \quad (6) \end{equation}

Going back to the equation (1)

\begin{equation} \bar{r} = r_f + \frac{Cov( E( \tilde{X} ), \bar{r_m} )}{ \sigma_m^2 \cdot P} \cdot ( \bar{r_m} - r_f) \quad (7) \end{equation}

After a little modification of the equation (3) and combining it with (7) we have:

\begin{equation} \frac{E( \tilde{X} )}{P} = 1 + r_f + \frac{Cov( E( \tilde{X} ), \bar{r_m} )}{ \sigma_m^2 \cdot P} \cdot ( \bar{r_m} - r_f) \quad (8) \end{equation}

After multiplying by P, and ordering the elements of the equation we have:

\begin{equation} \ P= \frac{E( \tilde{X} ) - \frac{ ( \bar{r_m} - r_f)}{ \sigma_m^2} \cdot Cov( E( \tilde{X} ), \bar{r_m} )}{1 + r_f} \quad (9) \end{equation}

Maybe you noticed that in the expression there is a market price of risk formula (how much market is willing to pay for the risk). We write it as lambda

\begin{equation} \lambda = \frac{ ( \bar{r_m} - r_f)}{ \sigma_m^2} \quad (10) \end{equation}

\begin{equation} \ P= \frac{E( \tilde{X} ) - \lambda \cdot Cov( E( \tilde{X} ), \bar{r_m} )}{1 + r_f} \quad (11) \end{equation}

Both (9) and (11) are your first formula written differently. We call them certainty-equivalent form of CAPM. It represents the price as the expression of present value of a risk-adjusted payoff. If the asset is uncorrelated with the market, then the result would be:

\begin{equation} \ P = \frac{E( \tilde{X} )}{ 1 + r_f} \quad (12) \end{equation}

Note that exactly the same result would occur, for your second formula.

If the asset is positively correlated with the market, then the adjustment expression (13) makes price lower. If the asset is negatively correlated with the market, then the price is higher. The value of the expression depends obviously also on the value market price of risk.

\begin{equation} \ - \lambda \cdot Cov( E( \tilde{X} ), \bar{r_m} ) \quad (13) \end{equation}