Quantitative Portfolio Theory & Performance Analysis

Transcription

1 Quantitative ortfolio Theory & erformance Analysis Week February 18, 2013 Basic Elements of Modern ortfolio Theory Assignment For Week of February 18 th (This Week) Read: A&L, Chapter 3 (Basic Elements of Modern F Theory) Read: E&G, Chapters 4, 5, 6 and 11 roblems: E&G11: 1, 7 (Due Feb 28 th ) Assignment For Week of February 25 th (Next Week) Read: Chapters 5, 6 & 9 from Hull (on Estimation & EVT) See Supplemental on Website roblems: E&G11: 1, 7 (Due Feb 25 th ) roblems: H5: 2, 5, 9, 11; 19 (Due Mar 4 th ) roblems: H6: 6 (Due Mar 4 th ) roblems: H9: 3, 6 (Due Mar 4 th ) 1.3 Assignment For Week of February 25 th (Next Week) Read: A&L, Chapter 3 (Basic Elements of Modern F Theory) Read: E&G, Chapters 9 & 7 (Efficient Frontier and Correlation Structure for Single-Index Model ) See Supplemental on Website roblems: EG11: 1, 7 (Due Feb 25 th ) roblems: H5: 2, 5, 9, 11; 19 (Due Mar 4 th ) roblems: H6: 6 (Due Mar 4 th ) roblems: H9: 3, 6 (Due Mar 4 th ) roblems: EG7: 1, 2, 4 (Due Mar 4 th ) roblems: EG9: 1, 2 (Due Mar 4 th ) 1.4 1

2 Assignment For Week of March 4 th (In Two Week) Read: A&L, Chapter 4 (Capital Asset ricing Model and its Application to erformance Measurement) Read: Dowd material on the website roblems: H5: 2, 5, 9, 11; 19 (Due Mar 4 th ) roblems: H6: 6 (Due Mar 4 th ) roblems: H9: 3, 6 (Due Mar 4 th ) roblems: EG7: 1, 2, 4 (Due Mar 4 th ) roblems: EG9: 1, 2 (Due Mar 4 th ) Basic Elements of Modern ortfolio Theory (MT) Basic rinciples and Tradeoffs of Risk & Return Markowitz Model Efficient Frontier Optimal ortfolio Selecting from the Efficient/Opportunity Set Utility and Alternatives Sharpe s 1-factor model to construct efficient frontier We have looked at elements of F return and risk These allow us to evaluate the results of F management ex post Now we turn our attention to rational F choice criteria In particular, the arbitrage between risk and return The first to quantify this link was Markowitz From his model, numerous developments in finance evolved In particular, from it evolved the Capital Asset ricing Model (CAM), the basis for the first performance analysis models ublication by Markowitz in 1952: the source of MT With Sharpe & Miller: awarded the 1990 Nobel rize 1.7 rinciples Developed a theory of portfolio choice in an uncertain future Quantified the difference between the risk of F assets taken individually and the overall F risk That F risk comes from the covariance of the assets Marginal contribution of a security to the F return variance is therefore measured by the covariance between the securities return and the F s return Not by the variance of the security itself F risk is less than the average of the risks of each asset1.8 2

3 The case of 2 risky assets, A & B (fully invested) R xara xbrb xara 1xARB x x 2x x x 1x 2x 1x A A B B A B AB A A A B A A AB x x x x A A A B A A AB A B Let the assets be The case of 2 risky assets, C & S (fully invested) ρ = +1 : With corr = +1, -1, 0, & otherwise The case of 2 risky assets, C & S (fully invested) ρ = -1 : The case of 2 risky assets, C & S (fully invested) ρ = 0 :

4 The case of 2 risky assets, C & S (fully invested) In Summary: The Efficient Frontier Rather than just 2 risky assets, we could have many The Efficient Frontier Furthermore, assuming investors prefer more return to less and less risk than more We could reduce the set of all portfolios to a subset of portfolios that offered The biggest return for a given level of risk The lowest risk for a given level of return These would be the portfolios that an investor could consider holding From the cloud, C, E, and B would be such portfolios The Efficient Frontier Thus the efficient set consists of the envelope curve of all portfolios that lie between the global minimum variance portfolio and the maximum return portfolio This set is called the Efficient Frontier The Efficient Frontier with Short Sales Consider again S & C, but ρ = 0.5 D, A, and F would not

5 The Efficient Frontier with Short Sales Which leads to an efficient set with no finite upper limit 1.17 The Efficient Frontier with a riskless asset Up to now we have only considered risky assets Suppose we consider lending or borrowing at a risk free rate, R F If we consider this investment in conjunction with a risky asset, our return and risk equations become R xara 1xA RF xa A 1xA RF 2xA1xAARF A RF x since A A RF As the risk free rate has a certain rate of return 1.18 The Efficient Frontier with a riskless asset C Solving xa A So R R 1 R A A RA R F RF A A F 1.19 The Efficient Frontier with a riskless asset Rather than an arbitrary portfolio, A, let us only choose ones from the efficient set of risky assets We find that F G is superior to all others The ability to determine the optimal F of risky assets without knowing anything about the investor is the Separation Theorem

6 The Efficient Frontier with a riskless asset Investors who believed they faced the efficient frontier and riskless lending and borrowing would hold the same F of risky assets Those wanting less risk than that would hold the F on R F G with proportions in G & the riskless deposit Those wanting a higher return with added risk would borrow at the riskless rate and buy more of G and be on G H The Efficient Frontier with a riskless asset Those investors that could not borrow (or use leverage) would find the efficient frontier as The Efficient Frontier with a riskless asset Alternatively, if borrowing is at a higher rate than lending, the efficient frontier appears as 1.23 The Efficient Frontier We next look at a simplified approach to finding the efficient frontier finding the F G in the case of lending, borrowing and when short sales are permitted where many asset choices are possible Afterwards, we consider, in turn The case of short sales, but no lending or borrowing The case where short sales are forbidden, but riskless lending and borrowing exist Neither short sales, lending or borrowing are permitted Curiously, the situation progresses from the simplest, to the more complex

7 The Efficient Frontier Returning to the idealization from before We know there is a single optimal F of risky assets, B This F is the one with the steepest slope maximize R RF Subject to the constraint N i i1 x 1 Solution for weightings is straightforward, tedious and 1.25 sensitive to data covariances & expected returns The Efficient Frontier When short sales are allowed, but no riskless lending and borrowing Can use previous method Vary riskless rate to generate the frontier Even more tedious 1.26 The Efficient Frontier When short sales are not allowed, but riskless lending and borrowing are permitted We use the same procedure, over F weightings R RF Maximize With the constraints N x 1 i i1 xi 0 Even more tedious as the objective function is nonlinear and weightings are non-negative 1.27 The Efficient Frontier When short sales are not allowed and riskless lending/borrowing are not permitted As the efficient set is determined by minimizing the risk for any level of expected return If we specify the return at some level and minimize risk, we have on point on the frontier If, in addition, we insure that all the constraints are imposed N N N 2 2 Maximize x x x Subject to N i i i j ij i1 i1 j1 ji N x 1 and x R R and x 0 i i i i i1 i

8 Theory is based on maximizing the utility of the investor s terminal wealth Utility function can be defined according to the expected return and the standard deviation of wealth A solution to the problem of F choice for the riskaverse investor The optimal F for a rational investor are defined as those that have the lowest risk for a given return These are said to be mean-variance efficient For simplicity, the model assumes that there is only one investment period 1.30 Utility Function: The expected utility of an investor s wealth Where EUW puw i i i p i : probability of obtaining wealth equal to w i at the end of the period and i is the summation index over all outcomes Investors always seek to maximize their expected utility Utility functions are specific to each individual Can adapt each investors preferences Markowitz s mean-variance approach assumes Investors have quadratic utility function, or The distribution of F returns is normal 1.31 Utility Function (some intuition) The Utility Function facilitates solving the decision problem of selecting the optimum from the opportunity set of efficient portfolios Can t we just choose directly? erhaps Consider the 3 portfolios selected from an efficient frontier where there is riskless lending and borrowing Optimum F of risky assets (tangency F) Expected return of 10% Risk (standard deviation) of 20 Utility Function (Directly Instead?) The 3 portfolios Assuming normally distributed returns yields distributions Risk free rate of 4%

9 Utility Function (Directly Instead?) The 3 portfolios If chief among investor concerns is possibility of loss, then one way to deal with choice is to choose the F with the lowest probability of negative outcome Of the 3, the F w/7% return is best 7: mean is 7/10 SD from 0 r (R<0) = 24% 8.5: mean is 8.5/15 from 0 Utility Function (Directly Instead?) The 3 portfolios If we carry this to the extreme, though, the best investment of all is the risk free rate Shouldn t there be some utility to more return then less? This where the idea of utility becomes important when there are tradeoffs; perhaps complicated ones We want to be able to assign a score to the various outcomes of an investment to determine its expected r (R<0) = 28% 10: r (R<0) = 31% 1.34 utility then choose the one with the highest 1.35 Utility Function Using the Markowitz formulation U r EU Where the expected utility is taken over all events Consider the first case of a weighting function Let the investment choices be Utility Function And the weighting function be Then the utility of investment A is highest U(15) / 3 U(10) / 3 U(5) / 3 34 / 3 Vs B: U(20) / 3 U(12) / 3 U(4) / /

10 Utility Function As a second example consider the case of a quadratic utility function of wealth, W 1 2 UW ( ) 4W W 10 Also consider the investment options with wealth outcomes Utility Function So the outcomes, including utility are 1 UW ( ) 4W W 10 Where we would find Expected Utility A = 36.3 ; B = ; C = 34.4 And the one with maximum utility is A Utility Function We have noted that Utility functions are specific to each individual and can be adapted to each investor s preferences What are the fundamental elements of a rational Utility Function for F selection More wealth to less Money with certainty rather than to engage in a gamble with the same expected value It doesn t matter how much I have, it doesn t change Economic roperties of Utility Functions More Wealth to Less Utility of more dollars is always higher than less First derivative of utility w/r to wealth is positive Appetite for Risk Investors are either Risk Averse Risk Neutral Risk Seeking More formally

11 Economic roperties of Utility Functions Appetite for Risk The Risk Averse (RA) Consider the Gamble The RA investor will reject the fair gamble Will always take 1 for certain Do Not Invest over an equal chance to double or loose everything Invest Thus, the second derivative of utility w/r to wealth is negative Indeed U(1) >.5 U(2) +.5 U(0) or U(1) U(0) > U(2) U(1) A 1 unit of change from 0 to 1 is worth more than a 1 unit of change from 1 to 2 A function where each additional unit has less value than the last has negative 2 nd derivative 1.42 Economic roperties of Utility Functions Appetite for Risk The Risk Neutral: 2 nd derivative w/r to wealth is zero The Risk Seeker: 2 nd derivative w/r to wealth is positive 1: Seeker ; 2: Neutral ; 3: Risk Averse 1.43 Economic roperties of Utility Functions Investors reference to a Change in Wealth as Wealth Changes It remains the same as wealth changes (grows) 50% of wealth in risky assets at W = 10,000, and 50% of wealth in risky assets at W = 20,000 This is constant relative risk aversion Economic roperties of Utility Functions Measures of Risk Aversion Absolute Risk Aversion (ARA) U ( W) ARA U ( W) U( W) 0 and U ( W) 0 for risk averse Risk Aversion for a given level of wealth U ( W) Relative Risk Aversion (RRA) RRA W U ( W ) Constant RRA means that the loss amount tolerated increases with proportion to individual wealth

12 Economic roperties of Utility Functions Consider the Logarithmic Utility Function UW ( ) ln( W) U( W) 1 U( W) 1 2 W W U ( W) 1 ARA a decreasing function of Wealth U( W) W U ( W) RRA W 1 constant U ( W ) Consistent with a risk averse investor Not so with quadratic utility functions 1.46 Utility Function Attempts have been made to extract investor utility functions (by banks and brokerages) Subject chooses between a number of alternatives to extrapolate utility, but Subjects may not be consistent in a series of choices And as situations become more complex, subject s selection rules can become discontinuous 1.47 Utility Function As a consequence alternatives to utility functions have been studied as a means for F selection These include Risk Tolerance Functions So called, Safety First Maximizing the Geometric Mean Risk Tolerance Function 2 2 Let there be a utility function UR, R T To capture the tradeoff between return and risk Where T is risk tolerance for an investor; more T, more tolerant of risk The optimization is to maximize U Suppose investor A has T = 100, B has T = 150 For the example 1.48 A selects 2 and B selects

13 Risk Tolerance Function More generally than the 3 outcomes for the last problem we have Optimum F of risky assets (tangency F) Expected return of 10% Risk (standard deviation) of 10 Risk free rate of 4% 2 2 For the utility with risk tolerance UR, R & T Where R xtrt 1 xtrf RF xt RT RF and xt T 2 2 xt T So U RF xt RT RF T 1.50 Risk Tolerance Function For the utility with risk tolerance 2 2 U R, R T And seeking the amount of the risky F to maximize the utility xt T U 2xTT U RF xt RT R F RT RF 0 T xt T And T RT R F xt 2 2 T For Investor A (T = 100): x 3 1 T and 1 2 xt For Investor B (T = 150): xt 1 and 1xT To implement, we need the investor s risk tolerance, T - possible, but not rigorous Safety First These alternatives are founded on the principle that investors are unable/unwilling to go through the mathematics of utility theory, but would use a simpler theory focused on bad outcomes Three Safety First Criteria Safety First 1 (Roy) The best F is the one with the smallest probability of producing a return below some specified level, R L min r( R R ) If returns are normally distributed, optimum is the F where R L is the maximum number of std dev away from the mean For the example if R L = 5% then F B is preferred L

14 Safety First 1 (Roy) The criteria could be restated as: find the F that R R minimize L Which is equivalent to R R maximize L If we had the riskless return R F vs R L we would have the efficient set problem from before (& techniques) All F that have the same value are equally desirable so they could be described by R RL K 1.54 Safety First 1 (Roy) Rearranging gives the straight line R RL K Where we are seeking the highest slope 1.55 Safety First 2 (Kataoka) Maximize the lower limit subject to the constraint that the probability of a return less than or equal to the lower limit is not greater than some predetermined value α Maximize R L subject to the constraint that the chance of a return less than R L is lower than α or maximize RL subject to rob ( R R ) L Safety First 2 (Kataoka) If returns are normally distributed we can proceed Suppose α = 5% If the lower limit R L is at least 1.65 std dev below the mean the condition is met With α = 5%, the constraint becomes RL R 1.65 Since we want to make R L as large as possible we rewrite the constraint as R RL 1.65 So we want the highest line that intercepts the risky frontier

15 Safety First 2 (Kataoka) So we want the highest line that intercepts the risky frontier Safety First 3 (Telser) Maximize expected return subject to the constraint that the probability of a return less than or equal to some predetermined limit was not greater than some predetermined number maximize R subject to rob ( R R ) L With normally distributed returns, and as before Safety First 3 (Telser) The equation is R RL constant For Telser, R L is fixed and we seek the highest return portfolio; so the straight line interpretation is finding the F A on the risky frontier The F with the highest return Maximize the Geometric Mean Return Select the F with the highest expected geometric mean return Many put this forth as a universal criterion For a retirement F the highest terminal wealth is the one with the highest geometric mean return Has the highest prob of reaching or exceeding any given wealth in the shortest possible time Has the highest prob of exceeding any given wealth level over any given period

16 Maximize the Geometric Mean Return However, maximizing the geometric mean implicitly assumes a particular tradeoff between the expected value of wealth and the occurrence of really bad outcomes Maybe this isn t appropriate in all cases Critics observe that maximizing terminal wealth is not the same as maximizing the utility of terminal wealth Never-the-less 1.62 Maximize the Geometric Mean Return Geometric Mean ortfolio Return N 1 2 N i RG 1R1 1R2 1RN 11Ri 1 And i = 1/N if all observations are equally likely roperties of maximum geometric mean F Usually well diversified it penalizes extreme outcomes Indeed, the investment with failure possibility (-1 return) is a minimal geometric mean F with 0 return While the maximum geometric mean return F may be highly diversified, it is not mean variance efficient Except in 2 special cases But first i Maximize the Geometric Mean Return Maximizing geometric mean return is equivalent to maximizing the expected value of a log utility function max EUW ( T) max Eln( WT) reserved under linear transformation vs initial wealth max Eln( WT) ln( W0) max Eln WT W0 max Eln 1R N N N i max ln 1R max ln 1R max ln 1R i max ln 1R i i i i G i1 i1 i1 Since taking the log of a set of numbers maintains rank order, F with highest geometric mean will also be the preferred F if the investor has a log utility function 1.64 Maximize the Geometric Mean Return Special Case 1: normally distributed returns If returns are normally distributed, then mean variance F analysis is appropriate to maximize expected utility Investors with log utility functions are such investors Thus, investors who wish to maximize the geometric mean return could use mean-variance analysis if returns were normally distributed Special Case 2: log-normally distributed returns

17 rinciples The Markowitz Model Addresses the portfolio vs. the market The efficient portfolio A portfolio with minimum risk for a given return Equivalently, a portfolio with the highest return for a given level of risk The complete set of these portfolios forms the efficient frontier The convex set of all portfolios that can be produced As investors use different forecasts for returns, their rinciples The Markowitz Model Assumptions Investors choose portfolios to maximize the expected utility of their terminal wealth Their utility function is an increasing function of their wealth They are risk averse Efficiency is based only on the 1 st two moments of the random distribution of their wealth expectation and variance Expected F return and variance of F return Valid if returns are normally distributed As investors have an asymmetric view of risk, we see why efficient frontiers will be different assume not 1.66 Markowitz preferred semi-variance analytically simple 1.67 rinciples The Markowitz Model Formulation of the model E R n xer rinciples The Markowitz Model With this definition of F return and risk we have that, for the efficient frontier, we Minimize the risk for a given return, or Maximize return for a given risk i i i1 n n var R xx i j cov Ri, Rj For the former, in its simplest version, seek the weightings x i i1 j1 Where min var R with x i the proportion of asset i in the F ER E and E R i the expected return of asset i, and n cov Ri, R j the covariance between assets i and j xi 1 i1 Also, we know We can define more elaborate linear constraints, i.e., nonnegativity cov Ri, Rj ij var Rivar Rj

18 rinciples The Markowitz Model The choice, by an investor, of a particular F on the frontier is a function of utility and risk aversion The F is found by maximizing the expectation of wealth utility Quadratic utility functions have their expected utility as a function of the F mean and variance For example, we maximize E(R ) λ var(r ), where λ denotes the investor s aversion to risk higher λ means more averse Once λ is set, the optimal F is uniquely characterized Aversion to risk is difficult to define and it is proposed to rinciples The Markowitz Model Aversion to risk is difficult to define and it is proposed to characterize it from shortfall constraints some criteria Best F is one which has the lowest probability of falling below a target level The one that maximizes the floor value of return, with the constraint that the probability of falling below that floor has probability lower than a target value One that maximizes return, with the constraint that the probability of falling below a floor is lower than a target Whew! This Markowitz model is complicated! We re characterize it from shortfall constraints 1.70 engineers, can we come up with something? 1.71 rinciples Simplified F Modeling Methods Sharpe s single-index (factor) model One issue with Markowitz model is complexity of working with the complete correlation matrix Sharpe s idea is that asset returns are made up of a factor common to all and a component unique to each No real theoretical basis, but works well if one sees the unknown factor from the perspective of the market ; (AT) 1.72 rinciples Sharpe s Model An empirical model where Rit i irmt it R it return on asset i R Mt return on market index it specific return on asset i i, i coefficients to be determined The coefficients are obtained by linear regression of market returns on the asset returns for the same period using least squares, the beta coefficient is cov Rit, RMt i var RMt And alpha R R i it i Mt

19 rinciples Sharpe s Model From the model definition, the residual term is non-correlated with the market return; so the total risk of an asset has a The F var and return n n systematic component (market) and idiosyncratic term E R xe i RixE i i irmt ER 2 var Rit i var RMt var i1 i1 it n n n The Markowitz correlation matrix problem now becomes var 2 R cov Rit, Rjt xixjcov Ri, Rj M xi i M ij i jm i j i1 j1 i1 As cov i, j 0 and the residuals are uncorrelated with Which keeps a linear formulation the market And on the diagonal var Rit i i M var it And instead of having n(n+1)/2 terms, we have only (2n+1) 2 2 the n ; the n var ; and var R i it i Mt M rinciples Sharpe s Model 19