Chapter 7 - Asset Pricing Models Flashcards
Two asset pricing models (equilibrium models)
- Capital asset pricing model
2. Arbitrage pricing theory
Assumptions of the CAMP (8)
- Investors make their decisions purely on the basis of expected return and variance. So all expected returns, variances and covariances of assets are known.
- Investors are non-satiated and risk-averse.
- There are no taxes or transaction costs.
- Assets may be held in any amounts.
- All investors make the same assumptions about the expected returns, variances and covariances of assets.
(Notes) Investors have the same estimates of the expected returns, standard deviations and covariances of securities over the one-period horizon. - All investors measure returns consistently (eg in the same currency or in the same real/nominal terms)
- The market is 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.
- All investors may lend or borrow any amounts of a risk-free asset at the same risk-free rate r.
Notes:
- All investors have the same one-period horizon.
Results of the CAMP (4)
- All investors have the same efficient frontier of risky assets. (Because investors have homogeneous expectations)
- The efficient frontier collapses to the straight line in E - σ space which passes through the risk-free rate of return on the E axis and is tangential to the efficient frontier for risky securities. (Because all investors are subject to the same risk-free rate of interest)
- All investors hold a combination of the risk free asset and M, the portfolio of risky assets at the point where the straight line through the risk-free return touches the original efficient frontier.
- M is the market portfolio - it consists of all assets held in proportion to their market capitalisation. The proportion of a particular investor’s portfolio consisting of the market portfolio will be determined by their risk-return preference.
Separation theorem
The separation theorem suggests that the investor’s choice of portfolio of risky assets is independent of their utility function.
The fact that the optimal combination of risky assets for an investor can be determined without any knowledge of their preferences towards risk and return (or their liabilities) is often known as the separation theorem.
Market price of risk
(Em - r) / σm
Capital market line
The straight line denoting the new efficient frontier is called the capital market line. Its equation is:
Ep - r = (Em - r) * σp / σm
where
Ep : is the expected return of any portfolio on the efficient frontier
σp : is the standard deviation of the return on portfolio P
Em : is the expected return on the market portfolio
σm : is the standard deviation of the return on the market portfolio
r : is the risk-free rate of return
Thus the expected return on any efficient portfolio is a linear function of its standard deviation
Security market line
An equation relating the expected return on any asset to the return on the market:
Ei - r = βi (Em - r)
where
Ei : is the expected return on security i
r : is the return on the risk-free asset
Em : is the expected return on the market portfolio
βi : is the beta factor if security i defined as cov[Ri,Rm]/Vm
Since the beta of a portfolio is the weighted sum of the betas of its consituent securities, the security market line equation applies to portfolios as well as to individual securities.
Main limitations of basic CAMP (2)
- Most of the assumptions are unrealistic.
- Empirical studies do not provide strong support for the model.
- There are basic problems in testing the model since, in theory, account has to be taken of the entire investment universe open to investors, not just capital markets.
Attempts to develop the theory of CAPM
- Models have been developed whici allow for decisions over multiple periods and for the optimisation of consumption over time.
- Other versions of the basic CAPM have been produced which allow for taxes and inflation, and also for the situation where there is no riskless asset.
- A model has been developed which allows for groups of investors in different countries, each of which considers their domestic currency to be risk-free.
Assumptions of APT
APT arequires that the returns on any stock be linearly related to a set of indices as follows:
Ri = ai + bi,1 I1 + bi,2 I2 +…+ bi,L IL + ci
where
Ri : is the return on security i
ai and ci : are the constant and random parts respectively of the component of return unique to security i
I1 ,…, IL : are the returns on a set of L indices
bi,k : is the sensitivity of security i to index k
We also have:
E[ci] = 0
E[cicj] = 0 for all i <> j
E[ci(Ij - E[Ij])] = 0 for all stocks and indices
This is exactly the same as the multi-index model for returns on individual securities.
Contribution of APT
The contribution of APT is to describe how we can go from a multi-index model for individual security returns to an equilibrium market model.
The general result of APT
All securities and portfolios have expected returns described by the L-dimensional hyperplane:
Ei = λ0 + λ1 bi,1 + λ2 bi,2 + … + λL bi,L
Strengths and weaknesses of APT
Strengths
- The principal strength of the APT approach is that it is based on the no-arbitrage conditions
- Another important characteristic is that it is extremely general.
Weaknesses:
- The fact that it is extremely general is also a weakness
- Although it allows to describe equilibrium in terms of any multi-index model, it gives no evidence as to what might be an appropriate multi-index model.
- APT tells us nothing about the size or the signs of the λ’s.
Uses of CAPM
- The model can be used to price assets, where this could be financial securities or other assets such as capital projects.
- If the beta of an asset can be estimated from past data, then the security market line can be used to estimate the prospective return that the asset should offer given its systematic risk. This return can then be used to discount projected future cash flows and so price the asset.
Uses of APT
- The APT can be used for both passive and active portfolio management.
- For example, it may be desired to track a share index but without holding all of the constituent shares in the index. In this instance, APT could be used to find a smaller sample of shares with the same sensitivities to the same factors as the index.
- More generally, by using APT to estimate the exposure of a portfolio to different risk factors, the extend of that exposure can be managed as required.
- Finally, it can be used to estimate the expected return on a financial security given its exposure to the various risk factors modelled. This return can then be used to discount projected future cash flows and so price the security amd determine if it appears to be under-valued or over-valued.
