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CAPITAL ASSET PRICING MODEL (CAPM) AND ARBITRAGE PRICING THEORY (APT) OPPORTUNITY SET WITH MANY RISKY ASSETS AND 1 RISK FREE ASSET The opportunity set with N risky assets and 1 risk-free asset case can be easily extended from the N risky only assets case: Any point on (and inside) the opportunity set can be combined with the risk free asset The combinations will lie on a straight line. Example: for any 1>a>0 invested in Rf and (1-a) in A, the combination will lie on the straight line ABRf. The efficient set now becomes RfMA However, a risk-free asset is equivalent to having the ability to borrow and lend at the rate Rf. The efficient set can be extended (through short-selling the Rf asset to become a linear set with M at a tangent to curve AMBCD THE CAPITAL MARKET LINE Similar to Fisher Separation, we get Two-Fund Separation All investors will seek a combination of Rf and M to maximize their utility. Recall: Fisher separation – All investors agreed there was one optimal production point determined by the market rate of interest, regardless of their personal preferences. Personal preferences only determined present vs future consumption choice, not the production choice. Two-Fund Separation – All investors agree that the two relevant assets to be holding are the risk free asset and the portfolio corresponding to point M. Personal preferences determine only the weights assigned to Rf vs M. The line RfMC is called the Capital Market Line (CML) THE CAPITAL MARKET LINE Once again, we see the introduction of a capital market (ability to borrow & lend) improves the aggregate welfare Utility for some improves, and no one is worse off (Pareto optimality) In equilibrium, M will be the market portfolio (why ) All assets must be held at market clearing prices in equilibrium i.e. excess demand to what is already held in the market portfolio must be zero Slope of the CML will be (E(Rm) – Rf)/σ(Rm) The fact that all investors hold a combination of only 2 assets – the risk free asset and the market portfolio – is a very useful result We don’t have to worry about computing the efficient set for the entire combination of risky assets (the full curve MB). Just the knowledge of the market portfolio will suffice. PORTFOLIO DIVERSIFICATION AND INDIVIDUAL ASSET RISK Recall: Portfolio Variance calculation involves N variance terms and N(N-1) covariance terms Assume an equally weighted portfolio i.e. wi = 1/N The portfolio variance can be separated into variance and covariance terms = 12∑ σ + =1 12 ∑ ∑ σ=1 ≠ =1 Suppose the largest individual asset variance is L and the average covariance is σ Then 1 2 ∑ σ ≤=1 /N and as N becomes large the limit approaches 0 1 2 ∑ ∑ σ =1 ≠ =1 = (N-1)/N * σ which in the limit approaches σ PORTFOLIO DIVERSIFICATION AND INDIVIDUAL ASSET RISK As we increase the size of a portfolio (in terms of number of assets, N), the covariance terms become relatively more important – and in the limit are the only contributor to portfolio volatility or risk Diversification reduces the impact of the variance terms Typically, for a portfolio of stocks the first 10-15 stocks have the largest impact Contribution of a single asset to portfolio risk: Evaluate the change in portfolio variance w.r.t change in asset’s weight () = 2wiσi2 + 2∑ σ=1≠ Again, suppose wi = 1/N In the limit, the marginal contribution of a single asset to portfolio risk is a function of its average covariance with the other assets in the portfolio The above insight was crucial to the development of the CAPM While variance is a good measure of portfolio risk, covariance is the right measure of risk for an individual security THE CAPITAL ASSET PRICING MODEL (CAPM) Based on similar suppositions we have encountered previously.. Main assumptions of the CAPM are: 1. Investors are risk-averse and maximize expected utilty 2. Asset returns have joint normal distributions (and there is agreement amongst investors i.e. homogenous beliefs) 3. Existence of a risk free asset and capital markets that allows for borrowing and lending at this rate This also implies frictionless markets. In addition no market imperfections such as taxes, or regulations such as restrictions on short selling are assumed Note: Above assumptions imply the existence of a Capital Market Line and a linear efficient set THE CAPITAL ASSET PRICING MODEL (CAPM) We will also accept the claim that the Market Portfolio is an efficient portfolio in equilibrium (i.e. lies on the efficient set) Proof of the market portfolio being efficient is quite involved but can be inferred from other observations As long as investors have homogenous beliefs, there will be one efficient set or frontier All individuals will select an efficient portfolio which lies on the efficient set (anything else will provide a lower return for the same level of risk – or same return for a higher level of risk) Since the market portfolio is the sum of all individual holdings, and the individual holdings are themselves efficient, the market portfolio must also be efficient. The implication of the market portfolio being efficient is that any demand for an asset in excess of what it already contributes to the market portfolio will be zero. Prices of all assets must adjust until they are all held by individuals. THE CAPITAL ASSET PRICING MODEL (CAPM) Derivation: In equilibrium, weight of each asset in the Market portfolio is: wi = Market Value (MV) of asset/ MV of all assets Suppose a portfolio consisting a% invested in asset I and (100-a)% in the market portfolio. (Note: The opportunity set from any combination of I and M will lie on the curve IMI’). We will have: E( P) = a E( i) + (1-a) E( m) σ( P) = [a2σi2 + (1-a)2σm2 + 2a(1-a)σim] Compute the marginal return and volatility contribution of asset I to the portfolio: () = E( i) – E( m) σ() = [a 2σi2 + (1-a)2σm2 + 2a(1-a)σim]- * [2aσi2 – 2σm2 + 2aσm2 + 2σim – 4aσim] But in equilibrium, the market portfolio already has the value weight wi and excess demand a must be zero () |a=0 = E( i) – E( m) σ() |a=0 = [σm 2 ]- * [-2σm2 + 2σim] = σim σm 2 σm THE CAPITAL ASSET PRICING MODEL (CAPM) So the slope of the risk-return trade off (curve IMI’) evaluated at M is: E( )/ σ( )/ |a=0 = E( i) E( m)(σim σm2)/σm But the slope at point M must also equal the slope of the Capital Market Line E( m) Rf σm = E( i) E( m)(σim σm2)/σm Rearranging: E( i)= Rf + [E( m) Rf] σim σm2 The above equation represents the CAPM. The term σim σm2 is the Beta risk measure of the asset By default, beta of the market portfolio is 1 The linear plot of beta risk vs return is called the Security Market Line PROPERTIES OF THE CAPM In equilibrium, all assets must be priced such that they lie on the Security Market Line. Even assets that don’t lie on the efficient frontier must still lie on the SML Total risk of an asset can be decomposed into two components Total Risk = Systematic Risk + Unsystematic Risk Only the Systematic portion of total risk will garner a risk premium (i.e. will get priced) The unsystematic risk can be diversified away (costlessly) The realized return for asset i can be expressed as: i = ai + bi m + I Beta can be estimated (albeit a na ve measure) by regressing asset returns on market returns – since the regression coefficient, b, is Cov(Rm,Ri)/Var(Rm) The beta risk measure is linearly additive i.e. a% investment in asset X and b% investment in asset Y will have a combined beta of aβx + bβy βp= E{[aX + bY aE(X) bE(Y)][Rm E(Rm)]} VAR(Rm) = aβx + bβy This is a very useful property – betas of individual assets are sufficient to compute the portfolio beta. PROPERTIES OF THE CAPM Exercise: Which assets below are potentially mispriced What is unique about Stock D above TESTS OF THE CAPM (EMPIRICAL) Researchers have tested the model since its development – and found contradicting or non- conclusive “evidence” Main tests have been along the

lines of: Whether the model is linear Whether factors other than beta are needed to explain returns Testing the CAPM is not straight forward – various econometric challenges to overcome Example: Distinction between ex-ante and ex-post forms of the model Rjt = E(Rjt ) + βjδmt + εjt , where δmt = Rmt – E(Rmt); εjt is a random error term; βj = COV((Rjt, Rmt)/VAR(Rmt ) Substituting E(Rjt ) from CAPM: Rjt = Rft + E(Rmt – Rft) βj + βj(Rmt – E(Rmt)) + εjt = Rft + (Rmt – Rft)βj + εjt , or Rjt – Rft = (Rmt – Rft)βj + εjt TESTS OF THE CAPM (EMPIRICAL) Rjt – Rft = (Rmt – Rft)βj + εjt Empirical tests are usually based on running regressions using the above specification. Most studies look for the following: The intercept term of the regression should be zero (or not significantly different from zero) Beta should be the only significant factor explaining returns. If other factors (such as dividend yield, P/E ratios, firm size, or non-linear terms such as beta-squared are included in the specification, they should turn out insignificant i.e. have no explanatory power) When the estimation is done over a long period of time, the rate of return on the market portfolio should be greater than the risk-free rate (though this is not directly a test of CAPM – more of a ‘sanity check’) Main findings have been: The intercept term is found to be different from zero usually Most studies find that using other factors improves the model (i.e. they have some explanatory power) but beta dominates Over long periods of time excess market returns are positive There has been much debate however whether methodological issues are the main reason for the observed deviations! There seems to be some agreement that beta indeed captures some level of risk and hence the model is useful (and used widely in the industry) THE ARBITRAGE PRICING THEORY (APT) APT is similar to CAPM but more general: Rate of return on an asset is a linear function of k factors: i = E( i) + bi1 1+ bi2 2 + + bik k+ I Note: the factors, Fi, are mean zero factors common to all assets These factors can be interpreted to be macro economic factors in some cases (interest rates, inflation, etc) – but not always The CAPM can be seen as a special case of APT where covariance with the market is the only risk factor ARBITRAGE PRICING THEORY (APT) Required Conditions for APT: Perfectly Competitive and Frictionless Markets Homogenous beliefs (i.e. everyone agrees on the factors and the linear specification) Number of assets, n, be much larger than number of factors, m Noise term, i, be independent of all factors and noise terms for other assets (idiosyncratic) General idea behind APT All portfolios that can be formed using i) no wealth, and ii) have no risk, must earn no return on average (Arbitrage portfolios) i) How to form a portfolio with no wealth ;0 n iw 1 i ARBITRAGE PRICING THEORY (APT) ii) How to form a portfolio with no risk Eliminate systematic and idiosyncratic risk nnnn i n i ip RwR ~~~~ ~ 1 ~ Idiosyncratic risk can be eliminated by keeping wi small i.e. wi ≈1/n i i ikik i ii i ii i i ewFbwFbwREw 11 11 11 …)( and n is large Systematic risk can be eliminated by choosing wi such that for each factor k the weighted sum of systematic factor loadings is zero 0)( ~~ i n ip REwR (Why = 0 ) 1 i ARBITRAGE PRICING THEORY (APT) What does all this mean The expected return vector can be represented as a linear combination of a constant and the coefficient vectors. Why Because of the orthogonality conditions 01. 1 n i iw 0 1 ik n i ibw 0)( ~ 1 i n i i REw In other words : If th i i kl t ith t R R (Wh ) ikkiii bbbRE …)( 22110 ~ ere s a r s ess asse , w re urn f 0 = f y So the expected return can be represented in excess returns form: bbbRRE )( ~ This is a very intuitive result: Excess return is a function of the factor loadings (bik) and the factor risk premiums ( k) ikkiifi …2211 ARBITRAGE PRICING THEORY (APT) Arbitrage Pricing Line Risk premium, , can be defined as the excess expected return of a portfolio that has a unit response to the kth factor and zero response to the other factors. PRACTICE PROBLEMS FOR THE MID-TERM PRACTICE PROBLEMS FOR THE MID-TERM PRACTICE PROBLEMS FOR THE MID-TERM CODING ASSIGNMENT Implementing a multifactor Gaussian Copula Model (An APT based Approach for Evaluating Structured Credit Securities) STRUCTURED CREDIT RISK FACTORS Primary Risk Drivers Default probability Performance would be directly affected by quality of underlying asset pool Correlation (Dependence) Determines the likelihood of joint defaults Recovery Determines level of loss in instances of defaults Why does correlation matter Correlation determines the shape of the (collateral) loss distribution Shape of the loss distribution determines the risk (hence fair value / price) of tranches. Change in loss distribution Change in the probability of experiencing a given loss level Increase / decrease in correlation increases / decreases the likelihood of extreme events* * Note: 1) Extreme events can be both positive or negative 2) Correlation does not affect the mean or Expected Loss of the underlying pool of assets. Correlation (along with Recovery / Loss Given Default / Loss Severity) determines the risk of different tranches Structured Credit Instruments are thus sometimes called “correlation products”. CODING ASSIGNMENT Example Portfolio of 200 assets 5 year horizon Asset default probability of 2.3% (BBB level) Generate default distribution using different levels of correlations and a one factor Gaussian copula default time model CODING ASSIGNMENT Implement a more generic version of the above example where information on N assets can be read from an Excel File Horizon for the analysis should be a user input Asset default probabilities could come from the same input Excel file Generate and plot a default distribution using correlation information from the input Excel file (i.e using factor loadings for each asset) and the option to select up to 5 factors (user input). IMPACT OF CORRELATION ON THE DEFAULT DISTRIBUTION Increase / decrease in correlation increases / decreases the likelihood of extreme events Again, extreme events can be both positive (no or low defaults) or negative (high defaults) Correlation impacts different tranches differently Beneficial for Equity/First Loss tranche Detrimental for Senior tranches Correlation Effect on Default Distribution 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Number of Defaults Pr ob ab ilit y 0% Correlation 10% Correlation 20% Correlation THE DEFAULT TIME MODEL With a Default Time Model, the objective is to model the timing of losses on a collateral portfolio taking into account the primary CDO risk drivers Default probability term structure Correlations Loss Given Default or Recovery THE GAUSSIAN COPULA DEFAULT TIME MODEL Define: where Si and Zi ~ N(0,1) By construction, correlation between Xi and Xj = ai1.aj1 + ai2.aj2 +…+ain.ajn and Xi ~ N(0,1) Transform the correlated Xi’s to correlated uniform variates Ui’s Ui = Φ(Xi), where Φ is the standard cumulative normal distribution function If Ui > Pi(t) , then firm i does not default by time t If Pi(t1) < Ui < Pi(t2) , then firm i defaults between time t1 and t2 Note: It is more efficient to implement the model in terms of returns rather than default probabilities If Xi > Φ 1(Pi(t)) , then firm i does not default by time t If Φ 1(Pi(t1)) < Xi < Φ 1(Pi(t2)), then firm i defaults between time t1 and t2 2 2 2 1 1 2 2 1 2. . ... . . 1 ...i i i in n i i i inX a S a S a S Z a a a= + + + + FULL IMPLEMENTATION OF THE MODEL Aggregate Simulation Trials to: Calculate Default Likelihood / Expected Losses for each tranche One can also compute a ‘fair’ tranche price i.e. the spread level that equates discounted expected spread earnings to discounted expected losses (if risk-neutral PDs are used)