Panos Parpas

1. Asset Pricing Models Panos Parpas 381 Computational Finance Imperial College London

2. Problem Types in Investment Science Determining correct, arbitrage free price of an asset: price of a bond, a stock the best action in an investment situation: how to find the best portfolio – how to devise the optimal strategy for managing an investment Single period Markowitz model

3. Topics Covered The Capital Asset Pricing Model (CAPM) Single and Multi Factor Models CAPM as a Factor Model The Arbitrage Pricing Theory (APT)

4. M-V model investor chooses portfolios on the efficient frontier – deciding if given portfolio is on efficient frontier or not no guarantee that a portfolio that was efficient ex ante will be efficient ex post statistical considerations regarding time period over which to estimate & which assets to include are non-trivial not mention implications of m-v optimisation on asset pricing CAPM describes MV portfolios and provides asset pricing

5. CAPM: Capital Asset Pricing Model developed by Sharpe, Lintner and Mossin single period asset pricing model determines correct price of a risky asset within the mean-variance framework highlights the difference between systematic & specific risk

6. Assumptions All investors are mean variance optimisers –portfolios on efficient frontier plan their investments over a single period of time use the same probability distribution of asset returns: the same mean, variance, & covariance of asset returns borrow and lend at the risk free rate are price-takers: investors’ purchases & sales do NOT influence price of an asset There is no transaction costs and taxes

7. Market Portfolio Everyone purchases single fund of risky asset, borrows (lends) at risk-free rate. Form a portfolio that is a mix of risk free asset and single risky fund Mix of the risky asset with risk free asset will vary across individuals according to their individual tastes for risk Seek to avoid risk – have high percentage of the risk free asset in their portfolio More aggressive to risk – have a high percentage of the risky asset What is the fund that everyone purchases? This fund is Market Portfolio and defined as summation of all assets – total invested wealth on risky assets An asset weight in market portfolio is the proportion of that asset’s total capital value to total market capital value – capitalization weights

8. The Capital Market Line (CML) Consider single efficient fund of risky assets (market portfolio) and a risk free asset (a bond matures at the end of investment horizon): If a risk free asset does not exist, investor would take positions at various points on the efficient frontier. Otherwise, efficient set consists of straight line called CML. Pricing Line: prices are adjusted so that efficient assets fall on this line CML describes all possible mean-variance efficient portfolios that are a combination of the risk free asset and market portfolio Investors take positions on CML by • buying risk free asset (between M and rf) or • selling risk free asset(beyond point M) and • holding the same portfolio of risky assets

9. The Capital Market Line Equation describes all portfolios on CML CML relates the expected rate of return of an efficient portfolio to its standard deviation The slope the CML is called the price of RISK! How much expected rate of return of a portfolio must increase if the risk of the portfolio increases by one unit? Expected Value of market rate of return Standard Deviation of market rate of return

10. The Pricing Model How does the expected rate of return of an individual asset relate to its individual risk? If the market portfolio M is efficient, then the expected return of an asset i satisfies The beta of an asset (risk premium):

11. The Pricing Model expected excess rate of return of an asset is proportional to the expected excess rate of return of the market portfolio: proportional factor is the beta of asset. Amount that rate of return is expected to exceed risk free rate is proportional the amount that market portfolio return is expected to exceed risk free rate describes relationship between risk and expected return of asset

12. Beta of an Asset beta of an asset measures the risk of the asset with respect to the market portfolio M. high beta assets earn higher average return in equilibrium because of beta of market portfolio: average risk of all assets

13. The Beta of Portfolio If the betas of the individual assets are known, then the beta of the portfolio is This can be shown by using rate of return of the portfolio covariance

14. Systematic and Specific Risk CAPM divides total risk of holding risky assets into two parts: systematic (risk of holding the market portfolio) and specific risk Consider the random rate of return of an asset i: Take expected value and the correlation of the rate of return with rM The total risk of holding risky asset i is

15. Summary: CAPM The capital market line: expected rate of return of an efficient portfolio to its standard deviation The pricing model: expected rate of return of an individual asset to its risk The risk of holding an asset i is s = b s + s 2 2 2 2 total risk s risk ystematic specific risk

16. Beta of the Market Average risk of all assets is 1 (beta of the market portfolio) Beta of market portfolio is used as a reference point to measure risk of other assets. Assets or portfolios with betas greater than 1 are above average risk: tend to move more than market. Example: If risk free rate is 5% per year and market rises by 10 %, then assets with a beta of 2 will tend to increase by 15%. If market falls by 10%, then assets with a beta of 2 will tend to fall by 25% on average. Assets or portfolios with betas less than 1 are of below average risk: tend to move less than market. Capital Market Line Security Market Line M M

17. CAPM as a Pricing Formula CAPM is a pricing model. standard CAPM formula only holds expected rates of return suppose an asset is purchased at price P and later sold at price S. rate of return is substituted in CAPM formula

18. Discounting Formula in CAPM

19. Single-Factor Model Consider n assets with rates of return ri for i=1,2,…,n and one factor f which is a random quantity such as inflation, interest rate Assume that the rates of return and single factor are linearly related. Errors have zero mean are uncorrelated with the factor are uncorrelated with each other Factor Loadings Intercept Error

20. Multi-Factor Model Single factor model is extended to have more than one factor. For two factors f1 and f2 the model can be written as For k number factors

21. How to Select Factors? Factors are external to securities: consumer price index, unemployment rate Factors are extracted from known information about security returns: the rate of return on the market portfolio Firm characteristics: price earning ratio, dividend payout ratio How to select factors: It is part science and part art! Statistical approach – principal component analysis Economical approach – its beta, inflation rate, interest rate, industrial production etc.

22. The CAPM as a Factor Model Special case of a single-factor model f = rM

23. The CAPM as a Factor Model: Example • Single factor model equation defines a linear fit to data • Imagine several independent observations of the rate of return and factor • Straight line defined by single factor model equation is fitted through these points such that average value of errors is zero. • Error is measured by the vertical distance from a point to the line

24. Arbitrage: “The law of one price” Arbitrage relies on a fundamental principle of finance : the law of one price • says– security must have the same price regardless of the means of creating that security. • implies – if the payoff of a security can be synthetically created by a package of other securities, the price of the package and the price of the security whose payoff replicates must be equal.

25. Arbitrage – Example How can you produce an arbitrage opportunity involving securities A, B,C? Replicating Portfolio: combine securities A and B in such a way that replicate the payoffs of security C in either state Let wA and wB be proportions of security A and B in portfolio

26. Example Continued • Payoff of the portfolio • Create a portfolio consisting of A and B that will reproduce the payoff of C regardless of the state that occurs one year from now. • Solving equation system, weights are foundwB = 0.6 and wA = 0.4 An arbitrage opportunity will exist if the cost of this portfolio is different than the cost of security C. • Cost of the portfolio is 0.4 x £70 + 0.6 x £60 = £64 - price of security C is £80. The “synthetic” security is cheap relative to security C.

27. Example – Continued • Riskless arbitrage profit is obtained by “buying A and B” in these proportions and “shorting” security C. • Suppose you have £1m capital to construct this arbitrage portfolio. • Investing £400k in A £400k  £70 = 5714 shares • Investing £600k in B £600k  £60 = 10,000 shares • Shorting £1m in C £1m  £80 = 12,500 shares The outcome of forming an arbitrage portfolio of £1m

28. The Arbitrage Pricing Theory CAPM is criticised for two assumptions: The investors are mean-variance optimizers The model is single-period Stephen Ross developed an alternative model based purely on arbitrage arguments Published Paper: “The Arbitrage Pricing Theory of Capital Asset Pricing”, Journal of Economic Theory, Dec 1976.

29. APT versus CAPM APT is a more general approach to asset pricing than CAPM. CAPM considers variances and covariance's as possible measures of risk while APT allows for a number of risk factors. APT postulates that a security’s expected return is influenced by a variety of factors, as opposed to just the single market index of CAPM APT in contrast states that return on a security is linearly related to “factors”. APT does not specify what factors are, but assumes that the relationship between security returns and factors is linear.

30. Simple Version of APT Consider a single factor model. Assume that the model holds exactly; no error The uncertainty comes from the factor f APT says that ai and bi are related if there is no arbitrage

31. Derivation of APT Choose another asset j such that Form a portfolio from asset i and j with weights of w and (1-w) Choose w so that the coefficient of factor is zero; so

32. Derivation of APT ai and bi are not independent

33. Arbitrage Pricing Formula Once constants are known, the expected rate of return of an asset i is determined by the factor loading. The expected rate of return of asset i CAPM?

34. CAPM as a consequence of APT The factor is the rate of return on the market APT is identical to the CAPM with