The mean-variance portfolio choice framework and its generalizations Prof. Massimo Guidolin 20135 Theory of Finance, Part I (Sept. October) Fall 2014
Outline and objectives The backward, three-step solution to modern portfolio theory problems Exact and approximate foundations of mean-variance utility functionals The normality assumption applied to asset returns Building the mean-variance efficient frontier in the two-asset case Generalizing the MVF to the N-asset case and to the presence of a riskless asset Quadratic Programming construction and the role of constraints The separation theorem : univariate vs. multivariate Extensions and the power/log-normal case 2
Modern portfolio theory Consider the general problem of portfolio selection among N risky assets: This is equivalent to: Portfolio rate of return Modern Portfolio Theory (MPT) explores the details of a portfolio problem such as this under the mean-variance utility hypothesis o This is normally done for an arbitrary number of risky investments, with or without a risk-free asset Assume that utility is provided by bundles of consumption goods, where the indexing is cross dates and states 3
Modern portfolio theory: three steps o o States of nature are mutually exclusive For each date and state of nature ( ) there is a traditional budget constraint: where the indexing runs across goods for a given state ; the m quantities c i ; and the m prices p i ; (i = 1, 2,, m) correspond to the m goods available in state of nature o Y is the end of period wealth level available in that same state MPT summarizes an individual's decision problem as being undertaken sequentially, in three steps. 1 The Consumption-Savings Decision: how to split period zero income/wealth Y 0 between current consumption now C 0 and saving S 0 for consumption in the future where C 0 + S 0 = Y 0 : 2 The Portfolio Problem: choose assets in which to invest one's savings so as to obtain a desired pattern of end-of-period wealth across the various states of nature o This means allocating (Y 0, - C 0 ) btw. a risk-free and N risky assets 4
Modern portfolio theory: three steps 3 The Consumption Choice: Given the realized state of nature and the wealth level obtained, the choice of consumption bundles to maximize the utility function It is fruitful to work by backward induction, starting from step 3 Step 3 is a standard microeconomic problem and its solution can be summarized by a utility-of-money function U(Y ) representing the (maximum) level of utility that results from optimizing in step 3 given that the wealth available in state is Y : Maximizing EU(Y ) across all states of nature becomes the objective of step 2: 5
Modern portfolio theory: backward solution o The end-of-period wealth can be written as Clearly an appropriate redefinition of the utility function leads to The level of investable wealth, (Y 0 - C 0 ), becomes a parameter of the U-hat representation Finally, given the characteristics (e.g., expected return, standard deviation) of the optimally chosen portfolio, the optimal consumption and savings levels can be selected, step 1 From now on we work with utility functions defined on r P This utility index can be further constrained to be a function of the mean and variance of the probability distribution of r P This simplification can be accepted either as a working approximation or it may result from two further (alternative) hypotheses made within the expected utility framework 6
Mean-variance analysis: foundations In the exact case, we have two avenues: 1 A decision maker's utility function is quadratic, see Appendix A 2 Asset returns are (jointly) normally distributed, see Appendix B The main justification for using a mean-variance approximation is its tractability o Probability distributions are cumbersome to manipulate and difficult to estimate empirically o Summarizing them by their first two moments is appealing In the approximate case, using a simple Taylor series approximation, one can also see that the mean and variance of an agent's wealth distribution are critical to the determination of his expected utility for any distribution: 7
Mean-variance analysis: foundations Let us compute expected utility using this approximation: If U(Y) is quadratic, U ( ) is a constant and, as a result, EH 3 = 0, so E(Y) and σ 2 (Y) are all that matter. If Y is normally distributed, EH 3 can be expressed in terms of E(Y) and σ 2 (Y), so the approximation is exact in this case as well o See Appendices A and B for these claims in greater detail Under either of the above hypotheses, indifference curves in the mean-variance space are increasing and convex to the origin 8
Mean-variance analysis: does normality hold? Assuming U( ) is quadratic, however, is not fully satisfactory because it implies increasing absolute risk aversion (IARA) The normality hypothesis on returns is easy to test, but we know it cannot be satisfied exactly o E.g., limited liability instruments such as stocks can pay at worst a negative return of -100% (complete loss of the investment) o Option-based instruments, which are increasingly prevalent, are also characterized by asymmetric probability distributions o While the normal is perfectly symmetric about its mean, highfrequency returns are frequently skewed to the right and index returns appear skewed to the left o Sample high-frequency return distributions for many assets exhibit excess kurtosis or fat tails We now turn to use the basic insight of mean-variance analysis to illustrate the principle of diversification 9
Mean-variance dominance In a mean-variance (M-V) framework, an investor's wants to maximize a function U(µ R, P ), where U 1 U/ µ R > 0 and U 2 U/ P < 0 She likes expected return (µ R ) and dislikes standard deviation ( P ) From lecture 1, an asset (or portfolio) A is said to mean-variance dominate an asset (or portfolio) B if µ A µ B and A < B, or if µ A > µ B while A B 10
Mean-variance efficient frontier: two-asset case An asset (or portfolio) A is said to mean-variance dominate an asset (or portfolio) B if µ A µ B and A < B, or if µ A > µ B while A B We can then define the efficient frontier as the locus of all nondominated portfolios in the mean- standard deviation space By definition, no ( rational ) mean-variance investor would choose to hold a portfolio not located on the efficient frontier The shape of the efficient frontier is of primary interest; let us examine the efficient frontier in the two-asset case for a variety of possible asset return correlations Recall that the variance of a portfolio of two assets, 1 and 2, is: where w i is the weight allocated to asset I Theorem. In the case of two risky assets with perfectly positively correlated returns (ρ 1,2 = 1), the efficient frontier is linear; in that extreme case the two assets are essentially identical, there is no gain from diversification 11
Mean-variance efficient frontier: two-asset case The portfolio's standard deviation is nothing other than the average of the standard deviations of the component assets: The equation of the efficient frontier is The algebra is as follows: 12
Mean-variance efficient frontier: two-asset case Theorem. In the case of two risky assets with imperfectly correlated returns (-1 < ρ 1,2 < 1), the standard deviation of the ptf. is necessarily smaller than it would be if the two component assets were perfectly correlated: (provided the weights are not 0 or 1) Thus the efficient frontier must stand left of the straight line for the case ρ 1,2 = 1 The smaller the correlation (further away from +1), the more to the left is the efficient frontier It is useful to distinguish the minimum variance frontier from the efficient frontier. 13
Mean-variance efficient frontier: two-asset case In the picture, all portfolios between P 1 and P 2 belong to the minimum variance frontier However, certain levels of expected returns are not efficient targets as higher levels of returns can be obtained for identical levels of risk Thus portfolio P 1 is minimum variance, but it is not efficient, being dominated by portfolio P 3 Theorem. If the two risky assets have returns that are perfectly P negatively correlated (ρ 1,2 = -1), 3 the minimum variance ptf. is risk free while the frontier is linear If one of the two assets is risk free, then the efficient frontier is a straight line originating on the vertical axis at the level of the risk-free return 14
Mean-variance efficient frontier: multi-asset case o In the absence of a short sales restriction, the overall ptf. can be made riskier than the riskiest among the existing assets o In other words, it can be made riskier than the one risky asset and it must be that the efficient frontier is projected to the right of the (µ 2, 2 2) point o See Appendix C for details Because a portfolio is also an asset fully defined by its expected return, its standard deviation, and its correlation with other existing assets or portfolios o The previous analysis with 2 assets is more general than it appears as it can easily be repeated with one of the two assets being a ptf. Theorem. In the general, N-asset case, as long as these are imperfectly correlated, the efficient frontier will have a curved shape 15
Mean-variance efficient frontier: the risk-free rate o Adding an extra asset to the 2-asset framework implies that the diversification possibilities are improved o The efficient frontier may move up, to the left Theorem. With N risky assets and a risk-free one, the efficient frontier is a straight line To arrive to this conclusion, pick one portfolio on the efficient frontier, say P T in the picture The new efficient frontier collects all possible ptfs combining P T and the risk-free asset, on the line joining the point (0, r f ) to P T Notice that if we had picked any other efficient portfolio different from P T, the resulting line would be dominated 16
Mean-variance efficient frontier : the risk-free rate 17
Mean-variance efficient frontier : the risk-free rate We call P T the tangency portfolio o As before, if we allow short position in the risk-free asset, the efficient frontier extends beyond P T With N assets (possibly one of them risk free), the EF is obtained as the relevant (non-dominated) portion of the MVF, the latter being the solution, for all possible expected returns, to the following quadratic program (QP): o In (QP) we search for the vector of weights that minimizes the variance of the portfolio under the constraint that the expected return on the portfolio must be µ; this defines one point on the MVF o One can then change the value of µ equating it successively to all plausible levels of portfolio expected return 18
Quadratic programming construction o In this way one effectively draws the MVF o Program (QP) is the simplest version of a family of similar quadratic programs used in practice because (QP) includes only equality, natural constrains Many other constraints can be added to customize the ptf. selection process without altering the basic structure of problem (QP) o One might maximize a ptf. expected return for given levels of standard deviation, but it is more efficient to do the reverse The most common implicit or explicit constraint for an investor involves limiting her investment universe o Example 1: home bias, local investment constrains o Example 2: ethical mutual funds o Example 3: non-negativity constraints (w i 0), indicating the impossibility of short selling some or all assets We now ask whether and how the MVF construction can be put to service to inform actual portfolio practice: one result is surprising 19
The separation theorem The optimal portfolio is naturally defined as that portfolio maximizing the investor's (mean-variance) utility That portfolio for which he is able to reach the highest indifference curve in MV space Such curves will be increasing and convex from the origin They are increasing because additional risk needs to be compensated by higher means They are convex iff the investor is characterized by increasing absolute risk aversion (IARA), which is the case under MV preferences, as we have claimed 20
The separation theorem Any risk averse investor, independently of her risk aversion, will diversify between a risky (tangency ptf.) fund and the riskless asset However, it is natural to realize that if there is a risk-free asset, then all tangency points must lie on the same EF, irrespective of the coefficient of risk aversion of each specific investor o Let there be two investors sharing the same perceptions as to expected returns, variances, and return correlations but differing in their willingness to take risks o The relevant efficient frontier will be identical for these two investors, although their optimal portfolios will be represented by different points on the same line o With differently shaped indifference curves the tangency points must differ. However, it is a fact that our two investors will invest in the same two funds, the risk-free asset on the one hand, and the risky portfolio (P T ) identified by the tangency point between the straight line originating from the vertical axis and the EF 21
The separation theorem This result is also called the two-fund theorem, or separation theorem It implies that the optimal portfolio of risky assets can be identified separately from the knowledge of the risk preference of an investor This result will play a significant role when constructing the CAPM Notice that this important result applies regardless of the (possibly non normal) probability distributions of returns representing the subjective expectations of the particular investor Can view financial analysis as providing plausible figures for the relevant inputs to MV calculations 22
Myopic portfolio choice: canonical case The risky share should equal the risk premium, divided by conditional variance times the coefficient capturing aversion to risk Suppose the conditional mean of a single risky portfolio is E t [r t+1 ] and the conditional variance is 2 t The investor only cares for conditional mean and conditional variance, The problem has the classical solution: The portfolio share in the risky asset should equal the expected excess return, or risk premium, divided by conditional variance times the coefficient that represents aversion to variance 23
Myopic portfolio choice: canonical case If we define the Sharpe ratio of the risk asset as: then the MV solution to the problem can be written as: The corresponding risk premium and Sharpe ratio for the optimal portfolio are as follows: Hence all portfolios have the same Sharpe ratio because they all contain the same risky asset in greater or smaller amount 24
Myopic portfolio choice: multivariate case These results extend straightforwardly to the case where there are many risky assets o We define the portfolio return in the same manner except that we use boldfaced letters to denote vectors and matrices o R t+1 is now a vector of risky returns with N elements. o It has conditional mean vector E t [R t+1 ] and conditional covariance matrix t Var t [R t+1 ] o We want to find the optimal allocation w t The maximization problem now becomes Vector of 1s repeated N times with solution 25
Myopic portfolio choice: multivariate case A straightforward generalization of the single risky asset case o The single risk premium return is replaced by the vector of risk premia and the reciprocal of variance is replaced by -1 t, the inverse of the covariance matrix of returns o Investors preferences enter the solution only through the scalar Thus investors differ only in the overall scale of their risky asset portfolio, not in the composition of that portfolio o This is the two-fund, separation theorem of Tobin (1958) again The results extend to the case where there is no completely riskless asset: we call still define a benchmark asset with return R 0,t We now develop portfolio choice results under the assumption that investors have power utility and that asset returns are lognormal We apply a result about the expectation of a log-normal random variable X: 26
The power utility log-normal case o The log is a concave function and therefore the mean of the log of a random variable X is smaller than the log of the mean, and the difference is increasing in the variability of X Assume that the return on an investor s portfolio is lognormal, so that next-period wealth is lognormal Under of power utility, the objective is Maximizing this expectation is equivalent to maximizing the log of the expectation, and the scale factor 1/(1 - ) can be omitted Because next-period wealth is lognormal, we can apply the earlier result to rewrite the objective as The standard budget constraint can be rewritten in log form: (*) 27
The power utility log-normal case Under power utility and log-normal portfolio returns (hence, terminal wealth), a mean-variance result applies under appropriate definition of the mean of portfolio returns relevant to the portfolio choice Dividing (*) by (1 - ) and using the new constraint, we have: Just as in mean-variance analysis, the investor trades-off mean against variance in portfolio returns Notice that this can be further transformed as: so that 28
The power utility log-normal case o The appropriate mean is the simple return, or arithmetic mean return, and the investor trades off the log of this mean linearly against the variance of the log return o When = 1, under log utility, the investor selects the portfolio with the highest available log return (the "growth optimal portfolio) o When > 1, the investor seeks a safe portfolio by penalizing the variance of ln(1 + R p t+1) o When < 1, the investor actually seeks a riskier portfolio because a higher variance, with the same mean log return, corresponds to a higher mean simple return o The case = 1 is the boundary where these two opposing considerations balance out exactly Problem: To proceed further, we would need to relate the log ptf. return to the log returns on the underlying assets But while the simple return on a ptf. is a linear combination of the simple returns on the risky and riskless assets, the log ptf. return is not the same as a linear combination of logs 29
Summary In this lecture we have learned a number of things 1. There is no contradiction between the way in which an economist looks at portfolio problems and what is typically done in practice in finance 2. We have defined mean-variance preferences and analyzed their microeconomic foundations, which may be exact (quadratic utility, jointly normally distributed returns) or approximated (Taylor) 3. We have built the minimum variance and mean-variance efficient frontiers for a variety of cases, with and without constrains 4. We have examine how a risk-averse, IARA investor should be optimizing her portfolio with and without a riskless asset 5. The separation, aka two-fund theorem emerged rather naturally from our work; we have discussed its implications for the asset management industry 6. We developed mean-variance closed-form asset allocation formulas 7. We have seen that under special definitions of mean returns, also power utility leads to mean-variance reults 30
Appendix A: Quadratic utility functions If the utility function is quadratic, it can be written as: where r P is the portfolio's rate of return Let the constant a = 0 because it does not play any role For this functioto make sense we must have b > 0 and c < 0 The first and second derivatives are, respectively, Expected utility is then of the following form: that is, of the form g(µ P, P ) This function is strictly concave. But it must be restricted to ensure positive marginal utility: Moreover, the coefficient of absolute risk aversion is increasing These two characteristics are unpleasant 31
Appendix B: Normally distributed returns If individual asset returns r i are normally distributed, is normally distributed as well Let r P have density f(r P ), where The standard normal variate Z is: Thus, r P = P Z + µ P, The quantity E(U(r P )) is again a function of µ P and P An indifference curve in the mean-variance space is defined as the set: for some utility level k This can be rewritten as This defines the set of points (µ P, P ) located in the circle of radius 32
Appendix C: The case of perfect negative correlation The case is defined by and it implies: As a result, we have: 33
Appendix C: The case of perfect negative correlation The presence of coefficients preceded by ± indicates the existence of two (linear) branches characterizing the MVF, as shown in the now familiar picture 34