Portfolio Theory and Risk Investment Analysis and Portfolio Management Instructor: Attila Odabasi Portfolio Theory and Risk Main Points: Portfolio expected return and SD Investment opportunity set Optimal Risky Portfolio with two risky assets Power of diversification

What is a Portfolio and Why is it Useful? A portfolio is simply a specific combination of securities, usually defined by portfolio weights that sum to 1: w1 + ...+ wN = 1 (feasibility constraint) Where asset weights {w1,..,wi,...wN} defined as: Weights can be positive (long positions) and negative (short positions).

Example Your investment account of $100,000 consists of three stocks A, B and C as follows: Your portfolio is summarized by the following weights: Asset Shares Price/Share Dollar Investment Portfolio Weight A 200 $50 $10,000 Wa= 0.1 B 1000 $60 $60,000 Wb= 0.6 C 750 $40 $30,000 Wc= 0.30 Total $100,000 Sum= 1

Measuring risk and reward - Motivation Assumptions: Investors like high expected returns but dislike high volatility Investors care only about the expected return and volatility of their overall portfolio Why not pick the best stock instead of forming a portfolio? Portfolios provide diversification, reducing unnecessary risks. Portfolios can customize and manage risk/reward trade-offs.

Basic Properties of Mean and Variance for Portfolio Returns The expected return of the risky portfolio: Assuming two assets: A & B

Portfolio Variance and SD Variances don’t add – Covariances matter! The portfolio variance: The standard deviation is:

Covariance and Correlation The problem with covariance Its magnitude depends on the units of measurement, It is not possible to simply deduce from the size of the number (covariance) how much the stock returns move together, it only gives the direction – positive, negative etc. However, we can standardize the covariance and calculate the correlation coefficient. Corr Coef tells both the direction and the degree that the stocks move together. I can’t look at a Covariance and tell you whether it is ‘big’ or not, because its ‘bigness’ is a function of the standard deviations of the two stocks.

Measuring the Correlation Coefficient For example for the returns of Stock A and Stock B: Range of values for A,B are: [1.0 > r > -1.0] If r = 1.0, the securities are perfectly positively correlated and move in the same direction in proportion If r = - 1.0, the securities are perfectly negatively correlated and move in opposite directions in proportion If r = 0, on average they don’t move together or in opposite directions.

Diversification with 2 assets: Example Ex: Suppose we have two assets, A and B, with: Asset E(r) Std Var A 0.08 0.2 0.04 B 0.13 0.2 0.04 r(rA, rB) = 0.3 Suppose we have $100 to invest. How would a portfolio of the two stocks perform? That means deciding on wA and wB.

Diversification with 2 assets: Example Asset E(r) Std Var A 0.08 0.2 0.04 B 0.13 0.2 0.04 r(rA, rB) = 0.3

Portfolio: 50:50 Consider an equally weighted portfolio with wA=wB= 0.5. The expected return, variance and volatility are: E(r) = (0.5)*(0.08) + (0.5)*(0.13) = 0.105 Var(r) = (0.5)2 * (0.04) + (0.5)2*(0.04) + 2*(0.5)(0.5)(0.3*0.2*0.2) = 0.0259 Std = (0.0259)1/2 = 0.161 This portfolio has expected return half-way between the expected returns on Assets A and B, but the portfolio standard deviation is less than half-way between the asset standard deviations.

Portfolio 50:50 The 50:50 portfolio yields the average of the individual returns ….. but bears less risk than the average of the individual securities! Asset E(r) Std Var A 0.08 0.2 0.04 B 0.13 0.2 0.04 Port 0.105 0.161 Diversification effect!

Portfolio Frontier – 2 risky assets Vary investment shares wA and wB and compute resulting values of E(r) and Std. Plot E(r) against Std as functions of wA and wB. The portfolio parameters are: E(Rp) = wA*0,08 + (1-wA)*0,13 p2 = wA20,22 + (1-wA)20,22 + 2wA(1-wA) 0,3(0,2)(0,2) What happens when we vary wA , the choice variable?

Portfolio Frontier What happens when we vary the investment shares, wA and wB (for a given correlation)?

Portfolio Frontier, Terms What happens when we vary the investment shares, wA and wB (for a given correlation)? The Investment Opportunity set consists of all available risk-return combinations. The envelope contains all the portfolios that minimize the level of risk for each level of expected return. MVP: The minimum variance portfolio is the portfolio that provides the lowest variance (or SD) among all possible portfolios of risky assets. The efficient frontier is the upper part of the envelope starting by the minimum variance portfolio. An efficient portfolio is a portfolio that has the smallest risk for a given level of expected return for a given SD.

Given two risky assets, the Optimal Risky Portfolio? Reminder: The Sharpe Ratio (SR) gives excess expected return per unit of risk. Assets with high SRs are preferred to assets with low SRs. The optimal risky portfolio is the one with the highest SR Graphically, this portfolio occurs at the tangency point of a line drawn from rf to the portfolio frontier of risky assets, hence tangency portfolio.

Given two risky assets, the Optimal Risky Portfolio? Example: Optimal Risky Portfolio Two-risky asset portfolio Assets A B Rf wA wB E(R) Std exp ret SR 0.08 0.13 0.05 -0.5 1.5 0.155 0.286 0.3667 0.2 -0.2 1.2 0.140 0.231 0.14 0.3893 Cov(1,2) 0.01 -0.1 1.1 0.135 0.215 0.3956 Corr 0.3 all in B 1 0.130 0.200 0.4000 0.1 0.9 0.125 0.187 0.4011 0.7 0.115 0.168 0.3868 0.4 0.6 0.110 0.163 0.11 0.3682 MVP 0.5 0.105 0.161 0.3411 0.100 0.3068 0.8 0.090 0.176 0.09 0.2270 0.085 0.1872 all in A 0.080 0.1500 0.075 0.1164 0.070 0.07 0.0865

The case of two risky and one risk-free asset: New Efficient Frontier, also called CAL Optimal Risky Portfolio: E(R)= 12.5%, SD= 18.7%

Efficient frontier with a risk-free asset The least risky portfolio is now constituted only of the risk-free asset. The efficient frontier is now a straight line because the risk-free asset has a risk of 0. It includes the risk-free asset and the risky securities. There is one portfolio that belongs to both efficient frontiers (with and without the risk-free asset) It is called the tangency portfolio and contains only risky assets. In fact, it is the only portfolio on the blue line to contain only risky assets.

Efficient Frontier Line Parameters of an efficient portfolio

P or combinations of P & Rf offer a return per unit of risk of 7. 5/18 CAL (Capital Allocation Line) P E(rp) = 12.5% E(rp) - rf = 7.5% This is the complete Capital Allocation Line. The slope says that for an increase in return of 8% we will have an increase in the SD of our portfolio by 22%. The risk premium is also shown. ) Slope = 7.5/18.7 rf = 5% F s rp = 18.7%

Tangency portfolio: Analytic Solution The tangency portfolio can be found analytically as well (e.g.solver): problem:

How to allocate funds between Rf and Optimal Risky Portfolio? Every portfolio on the straight line is actually composed of only two portfolios. risk-free asset and the tangency portf. They all have the same SR E(r) Utility Thus, the investor’s decision boils down to choosing a relative allocation between these portfolios. C P F 

How to allocate funds between Rf and Optimal Risky Portfolio? Assuming the risky portfolio ‘P’ has been optimized, we have to combine the risk-free asset and the tangency portfolio to obtain portfolio ‘C’. Risk and return tradeoff of portfolio ‘C’. E(rc) = yE(rp) + (1 - y)rf 2c = y22rp + (1-y)22rf + 2 y(1-y) Cov(rp, rf) 2c = y22rp c = yrp E(rc) = Return for complete (combined) portfolio ‘C’, a linear combination of component returns c = Standard deviation of the complete portfolio;

Example, contn’d Given E(rp), rf, rp, and rf , let us define a (combined) portfolio with various y’s and ask: What are the return for the complete portfolio, E(rC), and the standard deviation, C, for each y? Assets E(r) Std % invested Risk-free 5% 0% (1-y) Tangent, P 12.5% 18.7% (y) Combined ?

Possible Combinations… E(r) Capital Allocation Line (CAL) E(rp) = y=1.2 P E(rp) = y = 1 Expected return on the vertical axis, and standard deviation of total portfolio on the horizontal axis. With all of your money in the risk free you will have 7% return and no standard deviation. With all in the risky asset will have 15 expected return and 22% standard deviation. Can move along this line. y =.7 rf = F y = 0 s

Risk Aversion and Allocation: Choosing ‘C’ Greater levels of risk aversion => larger proportions of the risk-free asset Lower levels of risk aversion => larger proportions of the portfolio of risky assets Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations Q: How to choose the optimal complete portfolio for? We can assign a welfare or utility score to competing portfolios on the basis of expected return and risk of those portfolios, Many particular scoring systems are legitimate. A particular one used by CFA is as follows:

Utility Function U = utility E (r) = expected return on the asset or portfolio A = coefficient of risk aversion s2 = variance of returns ½ = a scaling factor

Risk-Return Trade-off, The Indifference Curve Portfolios that are equally preferred by an investor will lie on a curve called the indifference curve, which connects all portfolios with the same utility value. Northwest, preferred direction Indifference Curve E(r) Q P II I C B A III IV std 29

s E(r) CAL (Capital Allocation Line) rf = 5% F A=3 A=3 Risky portf. P This is the complete Capital Allocation Line. The slope says that for an increase in return of 8% we will have an increase in the SD of our portfolio by 22%. The risk premium is also shown. C rf = 5% F s 30

s E(r) C2 rf = 5% F A=3 CAL A=2 (Capital Allocation Line) Risky portf. P C2 This is the complete Capital Allocation Line. The slope says that for an increase in return of 8% we will have an increase in the SD of our portfolio by 22%. The risk premium is also shown. C1 rf = 5% F s

Summary: “A” and Indifference Curves Indifference curves describe different combinations of return and risk that provide equal utility (U) or satisfaction. U = E[r] - 1/2Ap2 «A» and indifference curves: For a given ‘A’, investor wants the most return for the least risk. Hence indifference curves higher and to the left are preferred. The greater the A the steeper the indifference curve and all else equal, such investors will invest less in risky assets. The smaller the A the flatter the indifference curve and all else equal, such investors will invest more in risky assets. This is not in the chapter but it should be. Note this is just one version of U, many forms are possible.

Setting up the Problem for ‘C’ The investment opportunity: Risk-free rate = 5% Risky portf.: E(rp) = 12.5%, std(rp) =18.7% A mean-variance investor: U = E[r] - 1/2Ap2 The optimal portfolio selection: Invest a portion ‘y’ of the total wealth in the risky asset, leaving the rest in the risk-free asset. Optimal portfolio? Max U(rc)

Portfolio Construction Expected return of the complete portfolio: rC = (1-y)rf + yrp rC = rf + y(rp – rf) E[rC] = rf + y(E[rp] – rf) The risk of the complete portfolio: sC = ysp

Numerical Solution: Various Positions in Risky Assets (y) for an Investor with Risk Aversion A = 4 E(rC) = y E(rp) + (1 - y) rf c = y rp y (1-y) E(rc) Std(rc) U 1 0.05 0.0500 0.1 0.9 0.0575 0.0187 0.0568 0.2 0.8 0.065 0.0374 0.0622 0.3 0.7 0.0725 0.0561 0.0662 0.4 0.6 0.08 0.0748 0.0688 0.4650 0.5350 0.0849 0.0870 0.0698 0.535 0.465 0.09013 0.10005 0.070107 0.095 0.1122 0.1025 0.1309 0.0682 0.11 0.1496 0.0652 0.1175 0.1683 0.0609 0.125 0.187 0.0551 1.1 -0.1 0.1325 0.2057 0.0479 1.2 -0.2 0.14 0.2244 0.0393

Analytic Solution: Forming the Optimization Problem Which combination is optimal if our objective is to maximize our utility function? max U(rC) = E(rC) – 0,5 A C2 Where E(rC) = rf + y [E(rp) – rf ] C2 = y 2p2 Substitute these two equations in U(rC) and get: max U(rC) = rf + y [E(rp) – rf ] – 0,5 A y 2p2

Ex:Forming the Optimization Problem max U(rC) = rf + y [E(rp) – rf ] – 0,5 A y 2p2 Choice variable: y For the previous example the function to maximize is: Max U(rC) = 0.05 + y [0.125 – 0.05] – 0,5 A y 2(0.187)2 Max U(rC) = 0.05 + 0.075y – 0,5A(0.1872)(y2)

Analytical Solution Set f’(y) = 0 and calculate the value of choice variable. Verify if f’’(y*) < 0, then y* is truly the optimal solution. Taking derivatives: 1. Look for y* that satisfies f’(y*)= 0: 2. Check for optimality of y*: f’’(y*)< 0 ?

Second phase:Optimal Complete Portfolio 04.12.2018 Second phase:Optimal Complete Portfolio We know that: y* = 0.535 The funds would be allocated among the assets as follows: 46.5% in risk-free asset, 53.5% in risky portfolio in fixed proportions. Assume that the risky portfolio composition is known as 10% in A and 90% in B. Then overall allocation: Risk-free asset: 46.5% A= 0.1 * 0.535 = 0.054 or 5.4% B= 0.9 * 0.535 = 0.482 or 48.2% TOTAL 100.0%

Optional Slides

Variance of a portfolio (n=2) derived

Covariance A measure of how random variables move together is covariance. Given two random variables, A and B, their covariance is defined as:

Expost Covariance Calculations If when r1 > E[r1], r2 > E[r2], and when r1 < E[r1], r2 < E[r2], then COV will be positive. If when r1 > E[r1], r2 < E[r2], and when r1 < E[r1], r2 > E[r2], then COV will be negative. Sample covariance, adjusted for loss of df Which will “average away” more risk?

r and Diversification in a 2 Stock Portfolio The lower the correlation the greater the reduction in risk from putting stocks together in a portfolio. Typically r is greater than zero and less than 1.0. r(A,B) = r(B,A) and the same is true for the COV The covariance between any stock such as Stock A and itself is simply the variance of Stock A, r(A,A) = +1.0 by definition We have no measure for how three or more stocks move together. Note WB = 1 – WA; can use this to solve for min var. weights when  = -1.

The Effects of Correlation & Covariance on Diversification Asset A Asset B Portfolio AB Assets A and B have positive standard deviations and the correlation between A and B is +1. Thus, the standard deviation of Portfolio AB is a simple weighted average of the standard deviations of A and B and no risk is reduced by combining the two.

The Effects of Correlation & Covariance on Diversification Asset C Asset D Portfolio CD Assets C and D have positive standard deviations and the correlation between C and D is -1. In this case the standard deviation of Portfolio CD is much less than a simple weighted average of the standard deviations of C and D and in this specific case CD has no risk. All of the risk has been averaged or diversified away.