Actuarial Science: Capital Asset Pricing Model

The construction of general equilibrium models allows us to determine the relevant measure of risk of any asset and the relationship between expected return and risk for any asset when markets are in equilibrium.

CAPM is the simplest form of an equilibrium model. The CAPM model is a straight line plot of the excess returns of the stock compared to the excess returns of the benchmark. The capital asset pricing model tells us about the relationship between risk and returns in the security market as a whole, assuming that investors act in accordance with mean-variance portfolio theory and that the market is in equilibrium. Mean-variance portfolio theory, sometimes called modern portfolio theory (MPT), specifies a method for an investor to construct a portfolio that gives the maximum return for a specified risk or the minimum risk for a specified return. It uses just the expected return and variance of the asset.

The standard form of capital asset pricing model was developed by Sharpe, Lintner, and Mossin, hence it is often referred to as the Sharpe-Lintner-Mossin form of CAPM.

Assumptions of the Model The real world is sufficiently complex that to understand it and construct models of how it works, one must assume away those complexities that are thought to have an only small effect on its behavior As the physicist builds models of the movement of matter in a frictionless environment, the economist builds models where there are no institutional frictions to the movement of stock prices.

  • All investors have the same one-period horizon. 

  • All investors can borrow or lend unlimited amounts at the same risk-free rate. 

  • The markets for risky assets are perfect. Information is freely and instantly available to all investors and no investor believes that they can affect the price of a security by their own actions. 

  • Investors have the same estimates of the expected returns, standard deviations and covariance of securities over the one-period horizon. 

  • All investors measure in the same ‘currency’ e.g. pounds or dollars or in ‘real’ or ‘money’ terms. 

  • All assets are marketable.

There are a few assumptions that underlie many financial models like mean-variance portfolio theory, they are:

  • all expected returns, variances and covariance of pairs of assets are known
  • investors make their decisions purely on the basis of expected return and variance
  • investors are non-satiated
  • investors are risk-averse
  • there is a fixed single-step time period
  • there are no taxes or transaction costs
  • assets may be held in any amounts, (with short-selling, infinitely divisible holdings, no maximum investment limits)

The capital market line

The straight line denoting the new efficient frontier is called the capital market line. Its equation is:

Where,

All investors will end up with portfolios somewhere along the capital market line, and all efficient portfolioswould lie along the capital market line. However, not all securities or portfolios lie along the capital market line.

(Expected return) = (Price of time) + (Price of risk) * (Amount of risk)

The market price of risk: It is the extra return that can be gained by increasing the level of risk (standard deviation) on an efficient portfolio by one unit. It is calculated using the formula,

Security Market Line

The Security Market Line

It is also possible to develop an equation relating the expected return on an asset to the return on the market:

Where,

Risk of any stock can be divided into systematic and unsystematic risk. Beta is the index of systematic risk. Security Market Line validates that systematic risk is the only important ingredient in determining expected returns and that non-systematic risk plays no role.

SML states that the expected return on any security is the riskless rate of interest plus the market price of risk times the amount of risk in the security of the portfolio.

Problems with the Model

Most of the assumptions underlying the CAPM violate conditions in the real world. Although the CAPM may describe equilibrium returns on the macro level, it certainly is not descriptive of micro (individual investor) behavior. For example, most individuals and many institutions hold portfolios of risky assets that do not resemble the market portfolio. the CAPM assumes several real-world influences away, it does not provide us with a mechanism for studying the impact of those influences on capital market equilibrium or on individual decision making.

Only by recognizing the presence of these influences can their impact be investigated. For example, if we assume personal taxes do not exist, there is no way the equilibrium model can be used to study the effects of taxes. All models are wrong but that doesn’t mean they cannot serve any purpose.

Implications of the model

Under the assumptions of the CAPM, we say the only portfolio of risky assets that any investor will own is the market portfolio. Market portfolio is a portfolio in which the fraction invested in any asset is equal to the market value of that asset divided by the market value of all risky assets.

The impact of a security on the risk of the market portfolio is given by

SML clearly shows that return is a linearly increasing function of risk. Furthermore, it is the only market risk that affects return. The investor receives no added return for bearing diversifiable risk.

Even if the standard CAPM model explains the behavior of security returns, it obviously does not explain the behavior of individual investors.

Non-Standard Form of CAPM

The standard form of CAPM can be modified to incorporate more realistic assumptions about each of the following influences:

  • Short sales disallowed: the same CAPM relationship is derived here.
  • Riskless lending and borrowing: It seems much more realistic to assume that investors can lend unlimited sums of money at the riskless rate but cannot borrow at a riskless rate.
  • Personal taxes: The general equilibrium pricing equation for all assets and portfolios, given differential taxes on income and capital gains, is
  • Nonmarketable assets:
  • Heterogeneous expectations
  • Non-price-taking behavior
  • Multi-Period Models
  • Multi-Beta CAPM
  • Inflation model: It is simplest form of multi-Beta model.
  • Zero beta CAPM

We have modified one assumption at a time. The derivations and explanations of these models have been detailed in the reference book mentioned below, for curious readers.

Despite having shortfalls, CAPM is widely used to price assets. If the beta of a stock can be estimated from the past data, then the model can be used to find the prospective return that the asset should offer. This return can then be used to discount projected future cash flows and so price the security and determine if it appears to be under-valued or over-valued.

References: Modern portfolio theory and investment analysis by Elton and Gruber

Data Used in the above example can be downloaded from here

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