Capital Asset Pricing Model
Is a set of predictions concerning equilibrium expected returns on risky assets.
In equilibrium the assets are priced such that all of them are included in the market portfolio.
- it is sensitive to market risk
- there is no alpha
CAPM argues that the market portfolio is the optimal risky portfolios.
It claims that investors are compensated by taking systematic risk only, since idiosyncratic risk is diversified in a portfolio.
CAPM predicts a linear relationship between a stock’s expected return and its beta.
According to CAPM, two stocks have different expected return only when their betas are different.
If CAPM holds, securities should lie on the SML, and investors should not be able to forecast alphas accurately.
Role of Markowitz in CAPM
Extension of Markowitz portfolio selection model.
If we assume that all investors hold the same view about all securities and optimise their portfolios according to Markowitz.
- All investors are expected to hold the same optimal risky portfolio.
- This optimal risky portfolio becomes the market portfolio (same for everyone)
- Through the price adjustment process this market portfolio will contain all investable assets in the market.
Suppose all investors optimised their portfolios like Markowitz.
- each investor uses an input list, consisting of expected returns and covariance matrix
- to draw an efficient frontier employing all available risky assets
- identifies an efficient risky portfolio, P, by drawing the tangent CAL to the frontier.
As a result, each investor holds securities in a investible universe with weights arrived at by the Markowitz optimisation process.
CAPM Aim
CAPM asks what would happen if all investors shared an identical investable universe and used the same input list to draw their efficient portfolios.
This means that their efficient frontiers would be identical. Facing the same risk-free rate, they would then identify tangent,CAL and naturally all arrive at the same risky portfolio, P.
All investors therefore would choose the same set of weights for each risky asset. This brings the question of What must be these weights?
What must be these weights?
A key insight of CAPM is this, because the market portfolio is if the aggregation of all these identical risky portfolios, it too will have the same weights.
Therefore, if all investors choose the same risky portfolio, it must be the market portfolio. This market portfolio is the value-weighted portfolio of all assets in the investable universe.
Therefore, the CAL based on each investors optimal risky portfolio will be the same as the CML.
What is the market portfolio, M?
When we sum over, or aggregate, the portfolios of all individual investors, lending and borrowing will cancel out (because each lender has a corresponding borrower), and the value of the aggregate risky portfolio will equal the entire wealth of the economy.
Do all Investors hold the market portfolio?
The proportion of each stock in this portfolio equals the market value of the stock divided by the sum of the market value of all stocks.
Proportion of any stock will be the same in each investors risky portfolio.As a result, the optimal risky portfolio of all investors is simply a share of the market portfolio
When all investors avoid a stock, the demand is 0 and the price falls. Then as stock gets progressively cheaper, it
becomes ever more attractive and other stocks look relatively less attractive. Ultimately, stock reaches a price where it is attractive enough to include in the optimal stock portfolio.
Such a price adjustment process guarantees that all stocks will be included in the optimal portfolio.
Mutual Fund Theorem
The passive strategy of investing in a market-index portfolio is efficient. For this
reason, we sometimes call this result a mutual fund theorem.
It is another incarnation of the separation property.
The practical significance of the mutual fund theorem is that a passive investor may view the market index as a reasonable first approximation to an efficient risky portfolio.
If the passive strategy is efficient, then attempts to beat it simply generate trading and research costs with no offsetting benefit, and ultimately inferior results.
Expected returns on individual securities
CAPM, all investors use the same input list, that is, the same estimates of expected returns, variances, and covariances.
To calculate the variance of the market portfolio, we use the bordered covariance matrix with the market portfolio
weights
Beta
The beta of a security is the appropriate measure of its risk because beta is proportional to the risk the security contributes to the optimal risky portfolio.
Expected return–beta relationship as a reward–risk equation.
Risk-averse investors measure the risk of the optimal risky portfolio by its variance.
Security Analysis
This analysis suggests that the starting point of portfolio management can be a passive market-index portfolio.
The portfolio manager will then increase the weights of securities with positive alphas and decrease the weights of securities with negative alphas.
Hurdle Rate (IRR)
Capital Budgeting
For a firm considering a new project, the CAPM can provide the required rate of return that the project needs to yield, based on its beta, to be acceptable to investors.
Managers can use the CAPM to obtain this
cutoff internal rate of return (IRR), or “hurdle rate” for the project.
Key implications of the CAPM
- The market portfolio is efficient
2. The risk premium on a risky asset is proportional to its beta.
CAPM and Single Index Model
Suppose instead that they all face a market where excess stock returns, Ri, are normally distributed and driven by one systematic factor.
Each firm-specific, zero-mean residual, e i , is uncorrelated across stocks and uncorrelated with the market factor, R M .
Residuals represent diversifiable, nonsystematic, or unique risk. The total risk of a stock is then just the sum of the variance of the systematic component,
biRM , and the variance of ei.
The single-index model suggests that the market index can explain some of the returns on a stock.
CAPM suggests that, in equilibrium, the expected risk premium on a stock is proportional to its amount of systematic risk (beta).
How do investors choose stocks?
Investors have two considerations when forming their portfolios:
First, they can diversify nonsystematic risk. Since the residuals are uncorrelated, residual risk, becomes ever smaller as diversification reduces portfolio weights.
Second, by choosing stocks with positive alpha, or taking short positions in negative-alpha stocks, the risk premium on Q
can be increased. Alpha is the returns on the stock.
Due to this investors pursue positive alpha stocks, short negative alpha stocks. Due to this prices of the positive alpha stocks will rise and the prices of the negative alpha stocks will fall. This will continue until alpha values are driven to zero. At this point, investors will be content to minimise risk by completely eliminating unique risk, that is bu holding the market portfolio. When all the alphas have zero alphas, the market portfolio is the optimal risky portfolio.
Deriving the CAPM.
Markowitz
- Start from mean-variance portfolio selection.
- Evaluate mean, variance, covariance of all available assets.
- Form the minimum-variance frontier
- As we can borrow and lend at the risk-free rate, we are able to identify one portfolio on the frontier that yields us the best risk-return trade-off.
Thus, we get the optimal risky portfolio.
With homogenous expectations, the investors face the same lending and borrowing rate, therefore have the same optimal risky portfolio.
CAL
- graphs risk premium of only efficient portfolios consisting of market and Rf, goes northwest to make a better decision.
- as a function of standard deviation
- every investor’s utility lie on this line
Investors will hold a combination of the risk free asset and the optimal risky portfolio, their portfolios will lie on the CAL.
CML
Two mutual fund theorem:
All investors will hold combinations of only two portfolios, the market portfolio and the risk-free asset. And the line that links the risk-free asset to the market portfolio is called the CML. In equilibrium the optimal portfolio is the market portfolio.
How are individual assets priced?
- This line can be used to price or describe the return of all efficient portfolios.
- All efficient portfolios lie of CML. These portfolios are a combination of risk-free asset and the market portfolio.
- all other inefficient portfolio lie below it.
- all efficient portfolios lie on CML & SML
As all investors hold the well diversified market portfolio, when they purchase assets they become parts of the diversified portfolio. In this case, the idiosyncratic risks are diversified away when they are in the market portfolio. This investors do not care about idiosyncratic risk and only about systemic risk which is measured by beta.
The only dimensions of security that need be of concern are expected return and beta.
SML
Is used to price all assets based on their systemic risk.
- In equilibrium all assets lie on the SML.
- Prices everything efficient & inefficient
- Contains idiosyncratic risk
- Beta = 0 for all risk free asset
- Beta = 1 for market portfolio
- CAPM is used in the money-management industry.
Underpriced : plot above the SML. It will provide an expected return in excess of their fair return stipulated by the SML.
Security’s risk premium is directly proportional to both the beta and the risk premium of the market portfolio; that
is, the risk premium equals b [ E (rM)- rf ]
- graphs individual asset risk premiums as a function of asset risk.
- for individual assets, sd is not a relevant measure of risk. Rather it is the contribution of the asset to the portfolio’s variance, which is measured by the asset’s beta
Only systematic risk is priced
CAPM suggests that the differences in expected returns of securities arise
from the differences in the betas.
Provides a benchmark for the evaluation of investment performance. Given risk (beta) of an investment, the SML provides the required rate of return necessary to compensate investors for risk as well as the time value of money.
Investors receive a risk premium only on the amount of systematic risk they bear.
- The amount is the security’s beta times the market risk premium.
- There is no risk premium on firm-specific risk, since it is diversified away in a large portfolio at no cost.
CAPM does not day
That high beta yield high returns each period. It says that over many periods, high-beta stocks on average should yield high returns, than their low-beta counterparts.
The price adjustment process
Under CAPM, all assets should lie on the SML in equilibrium.
When the asset lies above or below the SML, this creates an arbitrage opportunity that can be exploited by investors. This competition among investors in pursuit of arbitrage profit ensures that the asset quickly moves back on to the SML, this is the price adjustment process
Example: Price Adjustment
Investor finds security below/above the SML
E has an expected return less than required by its risk level. Thus investors can take advantage of this mispricing using the following trading strategy.
- Sell Security E - $100
- Use the proceed to buy portfolio C - $100
- The expected profit on E - Er* $100 = -8
- The expected profit on C - Er* $100 = 11
The expected profit of the arbitrage portfolio which is short E and long C is
(-8-11=3)
Portfolio C and the asset E have same bat, but offers different expected return. Which is why investors sell E and buy C to form the arbitrage portfolio.
In this case the beta’s cancel out to be 0. The arbitrage portfolio requires no investment, has no systemic risk, but positive expected profit.
If everyone shorts, the current P falls. Therefore the denominator falls until the ER is 11% and reached equilibrium. Then it matches the expected return on c, the portfolio that has the same beta as E. The arbitrage opportunity disappears and E lies on the SML.
Alpha
The difference between the fair and Er of the stock is called the stock’s Alpha.
Alpha, considered the active return on an investment, gauges the performance of an investment against a market index or benchmark that is considered to represent the market’s movement as a whole.
SML is the graphical representation of expected return-beta relationship, “fairly price” assets plot exactly on the SML. The Er is commensurate with their risk.
Under CAPM, we assume that all investors have homogenous expectations regarding return and risk of all assets. But this is not in reality.
One might say that security analysis is about uncovering securities with non-zero alphas. A portfolio manager can start with a market-index portfolio, and then
- Increase the weights of stocks with positive alphas
- Decrease the weights of stocks with negative alphas
- Short sell, downward pressure, eliminates alpha
Assumptions about the CAPM.
- Individual Investors
a. Investors are rational, mean-variance optimisers. Maximising return, minimising risk. Utility only depends on risk & return.
b. Their planning horizon is a single period
c. Investors have homogenous expectations - Market Structure
a. All assets are publicly held and trade on public exchanges, short positions are allowed and investors can borrow or lend at a common risk-free rate. Normal people don’t have access to Rf rate and there are several regulations against shorting.
b. All information is publicly available, in reality there are also private info available.
c. No taxes
d. No transaction costs. There is room in CAPM to make predictions on net of asset cost.
