fin5124 - Outlines of Lecture 4 Objectives 1. Determine the...

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Unformatted text preview: Outlines of Lecture 4 Objectives 1. Determine the optimal asset allocation given a client’s risk preference 2. Estimate a stock’ market beta empirically 3. Apply the market index model in practice References Chapter 6-8, EGBG; Chapter 7-9, BKM; Handouts Hsiu-lang Chen Portfolio Analysis 1 2nd Task in the Separation Property A portfolio choice problem can be separated into two independent tasks. #1: Determining the optimal risky portfolio, which is purely technical. Given the manager’s input list, the best risky portfolio is the same for all clients, regardless of risk aversion. #2: Allocating of the complete portfolio to Tbills versus the optimal risky portfolio depending on personal preference. Hsiu-lang Chen Portfolio Analysis 2 Portfolio Selection & Risk Aversion U’’’ U’’ U’ E(r) S P Q Less risk-averse investor More risk-averse investor Hsiu-lang Chen Efficient frontier of risky assets Portfolio Analysis St. Dev 3 Efficient Frontier with Lending & Borrowing CAL E(r) B Q P 1. What is the expression of CAL? A rf 2. How to construct the tangency portfolio A? Equation 7.14 of BKM F Hsiu-lang Chen Portfolio Analysis 4 Optimal Asset Allocation Given Risk Tolerance Construction of the Tangency Portfolio (Optimal Complete Portfolio) between the indifference curve and the capital allocation line—a combination of a risky asset (P) and a risk-free asset (rf) MaxU (r C ) = E (rC ) − 0.5 Aσ 2 C y = rf + y[ E (rP ) − rf ] − 0.5 Ay σ 2 ⇒ y* = E (rP ) − rf Aσ 2 P 2 P I ntuition? (y*: the proportion allocated to P) Hsiu-lang Chen Portfolio Analysis 5 Example • Suppose that S&P 500 index has an expected return of 13% and the standard deviation of 25%. The risk-free rate (Rf) is 5% but the borrowing rate that your client faces is 9%. What is the range of risk aversion for which a client will (1) always borrow, (2) always lend, and (3) neither borrow nor lend (i.e. for which y=1)? Hsiu-lang Chen Portfolio Analysis 6 CAL with Higher Borrowing Rate E(r) S&P 500 ) S = .16 13% 9% 5% ) S = .32 σ σp = 25% Hsiu-lang Chen Portfolio Analysis 7 Markowitz Portfolio Selection vs. CAPM So far, we have looked at the problem of how to select a portfolio, given that we know what the expected returns and return covariances are. Expected returns are very hard to measure, so it would be useful to have a model of what returns should be (absent any special information). Such a model would provide baseline estimates to be modified based on specific information about particular assets. The CAPM is an equilibrium model of the relation between the expected rate of return and the return covariances for all assets. Hsiu-lang Chen Portfolio Analysis 8 CAPM Basics - Assumptions No transaction costs. Assets are all tradable and are all infinitely divisible. No taxes. No individual can affect security prices (perfect competition). Decisions are made solely in terms of expected returns and variances. Unlimited short sales. Unlimited borrowing and lending at the risk-free rate. Homogeneous expectations. Hsiu-lang Chen Portfolio Analysis 9 CAPM Basics - The market portfolio What do we know from the passive portfolio problem? Everyone holds two portfolios: the risk-free security and the tangency portfolio. If everyone sees the same CAL, then everyone has the same tangency portfolio. So, what must the tangency portfolio be? It must be the market portfolio. What is it? – A portfolio of all risky securities held in proportion to market cap. ( Important: “all” include stocks, bonds, real-estate, human capital, etc.) Hsiu-lang Chen Portfolio Analysis 10 CAPM Basics - The Capital Market Line 2 E (~ ) − rf ≈ A σ M rM • Note that this says that all investors should only hold combinations of the market and the risk-free asset. • How does this relate to the increased popularity of index funds? Hsiu-lang Chen Portfolio Analysis 11 CAPM Basics - The Security Market Line Describe the relationship between expected returns and systematic risk of individual assets. Investors will only want to hold a security in their portfolio if it provides a reasonable amount of extra reward. For all securities, what a security adds to the risk of a portfolio will be just offset by what it adds in terms of expected return. The ratio of marginal return contribution to marginal variance contribution must be the same for all assets. – What it adds in expected return is E(ri)-rf. – What it adds in terms of risk is proportional to its covariance with the market portfolio (i.e. its β). ~) − r = β (E ( ~ ) − r ) E ( ri rM f i f Hsiu-lang Chen Portfolio Analysis 12 CAPM Basics - CML vs. SML Hsiu-lang Chen Difference? Portfolio Analysis 13 Disequilibrium in CAPM E(r) Undervalued 15% SML +α{ Rm=11% -α Overvalued 9% rf=3% β 1.0 Hsiu-lang Chen 1.25 Portfolio Analysis 14 Hsiu-lang Chen Portfolio Analysis 15 β - Estimation Beta is usually estimated using a Security Characteristic Line (SCL) regression: ri,t –rf,t = αi + βi[rm,t –rf,t ] + εi,t The part of ri-rf that is “explained” by the market return is βi[rm-rf], a component related to the systematic or market risk of the asset. The part of ri-rf not explained by the market return is εi, a component related the unsystematic or idiosyncratic risk. What’s the difference between SML and SCL? What are the expected return and variance/covariance of stocks under this specification? Hsiu-lang Chen Portfolio Analysis 16 Example Hsiu-lang Chen Portfolio Analysis 17 ß-Estimation Can we read off αˆi, βˆi, and εˆi,t from this scatter plot? Hsiu-lang Chen Portfolio Analysis 18 Regression Results rGM - rf = α + ß(rm - rf) α Estimated coefficient -2.590 Std error of estimate (1.547) Variance of residuals = 12.601 Std dev of residuals = 3.550 R-SQR = 0.575 Hsiu-lang Chen Portfolio Analysis ß 1.1357 (0.309) 19 ß - Estimation The systematic variance is βi2σm2. The unsystematic/idiosyncratic variance is σε2. The CAPM says: only the systematic component is priced. Note that the total return variance is βi2σm2+σε2. Why is the correlation between the two parts equal to zero? Note that the return covariance is βiβjσm2. Why is the correlation between εi and εj equal to zero? Why don't investors care about unsystematic risk? What must αi be, according to the CAPM? What is the R2 of the regression? Hsiu-lang Chen Portfolio Analysis 20 Hsiu-lang Chen Portfolio Analysis 21 ß-Estimation To estimate ß, typically we could use monthly data. It is standard use 5 years (60 months) of monthly data or sometimes weekly data. Why not use more data? (10 or 20 years). • Parameter stationarity. Can we use daily or intraday data? • Non-synchronous prices (Scholes & Williams, JFE 1977). • Bid-ask bounce. • Weekend/holidays. Sum-betas: to capture the potential delayed reaction of a stock to the market. Analysis Hsiu-lang Chen Portfolio 22 Regression Output in Excel Regression rPNC = α + β rS&P500 +ε estimated from Jan 1998 to Dec 2010. Hsiu-lang Chen Portfolio Analysis 23 The Single Index Model for Asset Allocation R =α +β R +e i i i M i The expected return on a portfolio: N N E(R ) = ∑ϖ α + ∑ϖ β E(R ) M P i =1 i i i =1 i i The variance on the portfolio: N N N 2 2 2 2 σ = ∑ ∑ ϖ ϖ β β σ + ∑ϖ σ P i =1 j =1 i j i j M i =1 i ei How many input estimates are required to generate the efficient frontier now? 3N+2! Hsiu-lang Chen Portfolio Analysis 24 How does an index model reduce # of input estimates for the efficient frontier? An efficient frontier of 500 stocks • Data collection: • Input estimates in expected returns • Input estimates in return covariance matrix Hsiu-lang Chen Portfolio Analysis 25 ß-Estimation There are a number of Institutions that supply β's: – Value-Line uses the past five years (with weekly data) with the Value-Weighted NYSE as the market. Merrill Lynch uses 5 years of monthly data with the S&P 500 as the market. Hsiu-lang Chen Portfolio Analysis 26 ß-Estimation Note: Merrill uses adjusted β's, which are equal to: βiAdj ≈ 1/ 3 + (2/3)βˆi Why do we need an adjustment? What is the Intuition? – Statistical Bias. – diversification of operations. Why 1/3 and 2/3 ? – What should these numbers be based on? – Other more advanced ``shrinkage'' techniques. – Using characteristics (e.g. industry) to estimate/forecast’s. • BARRA and Ibbotson forecast β based on firm size, growth, leverage etc. – How would you estimate the β of a new company? Hsiu-lang Chen Portfolio Analysis 27 Time Varying Beta • Typical time series of beta estimate • Here shown for AT&T using a rolling 60 month window Hsiu-lang Chen Portfolio Analysis 28 Time Varying Beta Suppose CAPM holds year by year Consider only 2 stocks in the economy: A and B Betas and the expected risk premium on the market change form year to year: If we only look at average betas, we’ll mistakenly conclude that the risk premium is independent of beta. Hsiu-lang Chen Portfolio Analysis 29 Accuracy of Historical Betas Hsiu-lang Chen Portfolio Analysis 30 Why might observed betas in one period differ from betas in an adjacent period? [EGBG, p. 139] The risk (beta) of a security or portfolio might change Fundamental Betas. β i = a0 + a1 X 1 + a2 X 2 + a3 X 3 + ⋅ ⋅ ⋅ Where Xi could be market variability, earning variability, firm characteristics, industry dummy variables, and so on. The beta in each period is measured with a random error Portfolio grouping technique, and Vasicek’s technique. Hsiu-lang Chen Portfolio Analysis 31 Vasicek's Technique 2 2 σ β1 σ βi1 βi2 = 2 β1 + 2 β i1 2 2 σ β 1 + σ βi1 σ β 1 + σ βi1 Where βit is the i-th security’s beta at time t. β¯1 is the cross-sectional mean beta. This weighting procedure adjusts observations with large standard errors further toward the mean than it adjusts observations with small standard errors. It is a Bayesian estimation technique. Hsiu-lang Chen Portfolio Analysis 32 A "Bayesian" • Greenspan's Legacy Rests On Results, Not Theories • • Fed Chief's Biggest Idea Was to Avoid Having One; Embracing the Ambiguity By GREG IP Staff Reporter of THE WALL STREET JOURNAL January 31, 2006; Page A1 Thomas Bayes was an 18th-century British Presbyterian minister who had early insights into making decisions when key determinants of the outcome are unknown. A Bayesian makes a decision based not on the most probable outcome but on a range of possible outcomes. Hsiu-lang Chen Portfolio Analysis 33 'He Has Set a Standard' By MILTON FRIEDMAN January 31, 2006; Page A14 Hsiu-lang Chen Portfolio Analysis 34 “I was praised for things I didn’t do.” “I am now being blamed for things that I didn’t do.” His Legacy Tarnished, Greenspan Goes on Defensive Future of U.S. Financial Reform Is at Stake; 'I Am Right' By GREG IP; W SJ, April 8, 2008; P age A1 Hsiu-lang Chen Portfolio Analysis 35 ...
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