Portfolio Theory Part 1 - Setting up the Problem

INVESTMENTS A Little History In March 1952, Harry Markowitz, a 25 year old graduate student from the University of Chicago, published ”Portfolio Selection” in the Journal of Finance. The paper opens with: ”The process of selecting a portfolio may be divided into two stages. The first stage starts with observation and experience and ends with beliefs about the future performances of available securities. The second stage starts with the relevant beliefs about future performances and ends with the choice of portfolio”. Thirty eight years later, this paper would earn him a Nobel Prize in economic sci­ ences. Introduction Two basic elements of investments: • the investment opportunity; • the investor. Our task for this class: • a model for financial assets; • a model for investors; • optimal portfolio selection. Modelling Financial Returns Virtually all real assets are risky. Financial assets, claims on real assets, bear such risk: • some are designed to minimize risk • some are designed to capture risk 25% 20% 15% Daily returns (ln) 10% 5% 0% -5% -10% -15% 6 10 /4 /1 99 96 96 19 19 4/ 7/ 4/ 4/ 5 96 99 4/ 1/ /1 /4 10 19 95 95 19 4/ 19 4/ 4/ 7/ 4 1/ 4/ 19 95 94 10 /4 /1 99 94 19 4/ 7/ 4/ 4/ 19 94 3 19 4/ 1/ 10 /4 /1 99 93 93 19 4/ 7/ 19 4/ 4/ 1/ 4/ 19 93 -20% 30% 30% 25% 25% 20% 20% Probability Probability Figure 1: Return of Mexican Peso, Source: Bloomberg Professional. 15% 10% 15% 10% 5% 5% 0% 0% -5% -0.080 -0.060 -0.040 -0.020 - 0.020 0.040 0.060 Daily Returns Current Distribution Normal-Distribution 0.080 -5% -0.080 -0.060 -0.040 -0.020 - 0.020 0.040 0.060 Daily Returns Current Distribution Normal-Distribution Figure 2: Return of S & P 500 Index, Figure 3: Return of 10 Year Treasury Bills, Source: Bloomberg Professional. Source: Bloomberg Professional. 0.080 Modelling Investors Overall, investors are risk averse, although some are more so than the others. ”We next consider the rule that the investor does (or should) consider expected return a desirable thing and variance of return an undesirable thing”. - Markowitz (1952). Heterogeneity of investors: • individual investors vs. corporations • investors with different marginal tax rates • informed vs. uninformed • young vs. old • Behavior issues: loss aversion, mental accounting, over confidence, over reaction, under reaction, etc. Choose A or B A: B: � $240� 000 with probability 100% � $1� 000� 000 with probability 25% $0 with probability 75% Choose C or D C: D: � � −$750� 000 with probability 100% $0 with probability 25% −$1 000 000 with probability 75% � � Equivalent Choices: A+D : � B+C : � $240� 000 with probability 25% −$760� 000 with probability 75% $250� 000 with probability 25% −$750� 000 with probability 75% Setting up the Problem What do we need . . . a recipe and some ingredients. The investment opportunity: • riskfree rf = 7% • risky rp : E (rp ) = 15%, std (rp ) = 22%. BKM, A mean-variance investor: 157 ff. 1 U (r) = E(r) − 0.005 · A · var (r) (1) The optimal portfolio selection: • invest a portion y of the total wealth in the risky asset, leaving the rest in the riskfree account • possible portfolios: ry = (1 − y) · rf + y · rp • the optimal portfolio? max y∈R U (ry ) where R stands for the space of real numbers. The coefficient 0.005 is in the literature as well written as calibrate the subjective risk aversion coefficient A. 1 1 2. It is a calibration coefficient to p. Portfolio Construction The opportunity set is fixed: rf and rp Our only choice variable: y [how much to invest in risk portfolio] The end product: ry = (1 − y) · rf + y · rp E (ry ) = E ((1 − y) · rf ) + E (y · rp ) (2) (3) = (1 − y) · 0.07 + y · 0.15 (4) = 0.07 + 0.08 · y (5) var (ry ) = var ((1 − y) · rf ) + var (y · rp ) +2 · cov ((1 − y) · rf , y · rp ) (6) = 0 + y 2 · 0.222 + 0 (7) = 0.222 · y 2 (8) std (ry ) = � var (ry ) = 0.22|y| (9) The Risk Return Combinations Every choice of y gives rise to one pair of return E and risk std. • For y � 0, we have: y= std (ry ) E (ry ) − 0.07 = 0.08 0.22 (10) • More generally, we have, for any y ¿ 0: y= std (ry ) E (ry ) − rf = E (rp ) − rf std (rp ) (11) y may vary over the entire positive real line, but this relation holds regardless. A linear relation between E and std: E (ry ) − rf = E (ry ) − rf std (ry ) std (rp ) (12) The Capital Allocation Line Collecting all y ∈ R,we get all of the risk-return (µ, σ) combinations available to in­ vestors. E(r) E(rP ) rf CAL P Line) n o i t a Alloc l a t i (Cap E(rp) ­ rf S σ σP Figure 4: Capital Allocation Line e) weight y CA P L( Ca pit al A llo cat ion Lin wy E(rP ) σP 0 0 rf Figure 5: Capital Allocation Line, a different view. σ The Sharpe Ratio One measure of the attractiveness of a portfolio r is its Sharpe Ratio (S): Intuitively, S measures extra return per extra risk. S= E (r) − rf std (r) (13) Recall that the CAL can be re-written as: E (ry ) − rf = E (ry ) − rf std (ry ) std (rp ) (14) For the extra risk std (ry ) chosen (through y), the extra reward is: Sp · std(ry ). Moreover, the Sharp Ratio Sy of any portfolio thus constructed from rf and rp is the same: Sy = Sp = E (ry ) − rf std (rp ) for any y � 0. Does that make sense to you? (15) Forming the Optimization Problem We are now ready to ”feed” our portfolio to the optimization machine: max y∈R U (ry ) (16) where U (r) = E(r) − 0.005 · A · var (r) (17) From our earlier derivation, we know that: E(ry ) = 0.07 + 0.08 · y; var(ry ) = 0.222 · y 2 (18) Our optimization problem therefore becomes max f (y): max y∈R U (ry ) f (y) = 0.07 + 0.08 · y − 0.0005 · A · y 2 (19) The Optimization Machine Three components of an optimization problem: • the objective function f (y); • the variable y; and • the search space R Three ways to solve an optimization problem: • analytical; • numerical; and • graphical. The mathematical foundation: • let y* be the solution of f � (y) = 0; • if f �� (y∗) < 0, then y* is truly the optimal solution. A Pictorial Solution f (y) 0 0 f ' (y) y optimal portfolio weight y* 0 0 0 optimal portfolio weight y* Figure 6: Optimal portfolio weight y An Analytical Solution The risk aversion coefficient is set at A = 4 and the optimal weight is y*=0.41. Let’s take some derivatives: f � (y) = ∂f (y) = 0.08 − 0.222 · A · y ∂y (20) ∂f � (y) = −0.222 · A ∂y (21) f �� (y) = 1. look for y∗ that satisfies f � (y∗) = 0: y∗ = 0.08 0.222 · A 2. check the optimality of y∗: f ”(y∗) < 0? (22) Determinants of Portfolio Weights More generally, the optimal solution can be expressed as E (rp − rf ) y∗ = var (rp ) · A (23) A more risk-averse investor (with a larger risk-aversion coefficient A) will invest less in the risky asset. least risk aversion Utility medium risk aversion most risk aversion 0 portfolio weight y y Figure 7: Utility-function. If the risk premium, E (rp ) - rf , of the original risky asset decreases, a risk-averse investor will reduce his holdings in the risky asset accordingly. If the original asset is risky (with var(rp ) > 0), but pays zero risk premium, then no risk-averse investor will hold the risky asset. If the risk premium is negative, a risk-averse investor will start shorting the asset. What is the optimal portfolio weight y* of an investor with the following utility func­ tion? HINT: This investor cares only about the Sharpe Ratio. Going Beyond Our setup assumes the following: 1. A mean variance investor; 2. Investment horizon is fixed to one year; 3. No dynamic rebalancing in between. Of course, this setup is a very rough characterization of the real investment problem. Nevertheless, this example is valuable: • First, it provides a framework for us to think about the portfolio optimization problem. • Second, although simple, it provides rich intuition. Now we can go beyond to the next step. Three Extensions 1. The Skewness Extension: allow skewed asset returns and add preference for posi­ tive skewness and aversion to negative skewness. 2. The Horizon Extension: allow investment horizon to vary. 3. The Dynamic Extension: allow for dynamic rebalancing.