1. Computational Finance 1/34 Panos Parpas Asset Pricing Models 381 Computational Finance Imperial College London

2. Computational Finance 2/34 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. Computational Finance 3/34 Topics Covered   The Capital Asset Pricing Model (CAPM)   Single and Multi Factor Models   CAPM as a Factor Model   The Arbitrage Pricing Theory (APT)

4. Computational Finance 4/34 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. Computational Finance 5/34 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. Computational Finance 6/34 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. Computational Finance 7/34 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. Computational Finance 8/34 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 r f ) or – selling risk free asset (beyond point M) and – holding the same portfolio of risky assets

9. Computational Finance 9/34 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. Computational Finance 10/34 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. Computational Finance 11/34 .  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 The Pricing Model

12. Computational Finance 12/34 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. Computational Finance 13/34 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. Computational Finance 14/34 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 r M   The total risk of holding risky asset i is

15. Computational Finance 15/34 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 risk specific 2 risk ystematic 22 risk total 2 s 

16. Computational Finance 16/34 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 LineSecurity Market Line MM

17. Computational Finance 17/34 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. Computational Finance 18/34 Discounting Formula in CAPM

19. Computational Finance 19/34 Single-Factor Model   Consider n assets with rates of return r i 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 1. have zero mean 2. are uncorrelated with the factor 3. are uncorrelated with each other Intercept Factor Loadings Error

20. Computational Finance 20/34 Multi-Factor Model   Single factor model is extended to have more than one factor.   For two factors f 1 and f 2 the model can be written as   For k number factors

21. Computational Finance 21/34 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 –  Statistical approach – principal component analysis  Economical approach –  Economical approach – its beta, inflation rate, interest rate, industrial production etc.

22. Computational Finance 22/34 The CAPM as a Factor Model   Special case of a single-factor model f = r M

23. Computational Finance 23/34 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. Computational Finance 24/34 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. Computational Finance 25/34 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 w A and w B be proportions of security A and B in portfolio SecurityPrice Payoff in State 1 Payoff in State 2 A£70£50£100 B£60£30£120 C£80£38£112

26. Computational Finance 26/34 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 found w B = 0.6 and w A = 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. Computational Finance 27/34 Example – Continued SecurityInvestmentState 1State 2 A-4000005714 x 50 = 2857005714 x 100 = 571400 B-60000010000 x 30 = 30000010000 x 120 =1200000 C100000012500 x 38 = -47500012500 x 112 = -1400000 Total£0£110,700£371,400 The outcome of forming an arbitrage portfolio of £1m  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

28. Computational Finance 28/34 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. Computational Finance 29/34 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. Computational Finance 30/34 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 a i and b i are related if there is no arbitrage

31. Computational Finance 31/34 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. Computational Finance 32/34 Derivation of APT a i and b i are not independent

33. Computational Finance 33/34 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. Computational Finance 34/34 CAPM as a consequence of APT   The factor is the rate of return on the market   APT is identical to the CAPM with