fin5123 - Outlines of Lecture 3 Objectives 1. Introduce...

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Unformatted text preview: Outlines of Lecture 3 Objectives 1. Introduce matrix calculation in portfolio expected return and variance 2. Discuss the best feasible investment opportunity set 3. Construct an efficient frontier portfolio and the optimal risky portfolio by using Excel Solver References Chapter 7 & 8, BKM; Chapter 13, EGBG; Handouts Hsiu-lang Chen Portfolio Analysis 1 What is a portfolio? ω1 1st security ω2 2nd security ω3 3rd security · · · ωN Nth security Hsiu-lang Chen A portfolio is a group of assets. A portfolio is described by the weights ωi allocated to securities. Portfolio Analysis 2 Portfolio Expected Returns E ( ~ ) = ∑ϖ E ( ~) r r N i =1 P i i Portfolio Variance (Portfolio Risk) Var ( ~ ) = Cov ( ~ , ~ ) rP rP rP N N i =1 j =1 N N = Cov (∑ϖ i ~ , ∑ϖ j ~j ) = ∑ ∑ϖ iϖ j Cov ( ~ , ~j ) ri r ri r N i =1 j =1 N N = ∑ϖ i2Var ( ~ ) + ∑ ∑ϖ iϖ j Cov ( ~ , ~j ) ∀ i ≠ j ri ri r i =1 N i =1 j =1 N N = ∑ϖ i2σ i2 + ∑ ∑ϖ iϖ jσ i j ∀ i ≠ j i =1 Hsiu-lang Chen i =1 j =1 Portfolio Analysis 3 In Terms of Matrix Expression E (~ ) = [ϖ 1 ϖ 2 rP Var (~ ) = [ϖ 1 ϖ 2 rP r1 E (~) E (~ ) r 2 • • ϖ N ] • • E (~ ) rN σ 11 σ 12 σ σ 22 21 • • ϖ N ] • • • • σ N 1 σ N 2 • σ 1 N ϖ 1 • σ 2 N ϖ 2 • • • • • • • • σ NN ϖ N • • • • How to do matrix multiplication in Excel? Hsiu-lang Chen Portfolio Analysis 4 Portfolio Theory (The Optimal Asset Allocation) What is the best feasible investment opportunity set given risky assets? −A minimum variance frontier (MVF)− ~) Var ( r ϖ s.t. E (~ ) = r r Min P P N ∑ ϖ =1 i =1 Hsiu-lang Chen i Portfolio Analysis 5 Construction of MVF What are all feasible investment combinations given two risky assets? E(r1)=25%,E(r2)=10%,σ1=75%,σ2=25%, ρ12 In a plane of {σP, E(rp)}, a portfolio of these two assets can be described by E(rp)=wE(r1)+(1-w)E(r2) σP = [w2σ12 +2w(1-w) ρ12σ1 σ2+ (1-w)2σ22]0.5 Hsiu-lang Chen Portfolio Analysis 6 The minimum variance frontier (MVF) can be traced by varying w. Case 1: ρ= 1 E(rp)=wE(r1)+(1-w)E(r2) σp=wσ1+(1-w) σ2 Case 2: ρ= -1 E(rp)=wE(r1)+(1-w)E(r2) σp=|wσ1-(1-w) σ2| Hsiu-lang Chen Portfolio Analysis 7 MVF of two risky assets E(r1)=25%,E(r2)=10%,σ1=75%,σ2=25%,ρ12 -1.0 What does MVF tell us? How to construct it? Hsiu-lang Chen Portfolio Analysis 8 In Excel 1. Randomly choosing w 2. Trace all feasible investments in a plane of E(rP) and σP for the four cases: (1) ρ12 =1, (2)ρ12 = 0.2, (3) ρ12 = - 0.2, (4) ρ12 = -1 3. What do you see the MVF? 4. Are all feasible investment combinations on the MVF? Hsiu-lang Chen Portfolio Analysis 9 Extend to a portfolio of three assets Expected Returns Return Covariance Matrix Asset A Asset B Asset C Asset A 0.05 Asset A 0.04 0.02 0 Asset B 0.1 Asset B 0.02 0.1 0.06 Asset C 0.15 Asset C 0 0.06 0.16 1. Trace all feasible investments in a plane of {σP, E(rp)} in Excel for two cases: (1) Short-selling is allowed, (2) Short selling is prohibited. 2. What do you see the MVF ? Hsiu-lang Chen Portfolio Analysis 10 Three risky assets E(r) Add 0.1 to the σ coordinates Hsiu-lang Chen Portfolio Analysis 11 MVF of Three Risky Assets 0.4 E(r) How to construct a portfolio on the efficient frontier in Excel? 0.35 0.3 0.25 No Short-selling Var (~ ) r s.t. E (~ ) = r r 0.2 Min ϖ 0.15 P P 0.1 N ∑ 0.05 ϖ =1 i =1 i σ 0 -0.1 Hsiu-lang Chen 0.1 0.3 0.5 0.7 0.9 Portfolio Analysis 1.1 1.3 1.5 12 MVF of Risky Assets E(r) Efficient The best feasible investment opportunity set given risky assets frontier Individual assets Global minimum variance portfolio Hsiu-lang Chen Minimum variance frontier Portfolio Analysis St. Dev. 13 Concepts What are the required input estimates to generate the efficient frontier? What would be the matrix expression of E(r̃p) & σp2 ? N expected returns; N Variances; N(N-1)/2 Correlations How to estimate them in the real world? Hsiu-lang Chen Portfolio Analysis 14 Without a Prior Belief Construct return sample distribution Sample Mean 1 T r = ∑ rt T t =1 Sample Variance 2 = 1 T ( r − r )2 σ T − 1 t∑1 t = Sample Covariance 1 T (r − r )(r − r ) σ XY = T − 1 t∑1 xt x yt y = Hsiu-lang Chen Portfolio Analysis 15 With a Prior Belief Construct return distribution based on some models The market index model Fama and French 3-factor model plus the Momentum factor The macroeconomic factor model Hsiu-lang Chen Portfolio Analysis 16 Another Example of Construction of an Efficient Frontier in Excel Using the stock market data from seven countries in the file “ch7a_BKM7.xls” to construct the global efficient frontier. Examining the impact of short-selling prohibition on the efficient frontier. Constructing the optimal risky portfolio given the risk free rate of 5% when (1) the short-selling is prohibited; (2) the short-selling is allowed. (To be discussed next!) Hsiu-lang Chen Portfolio Analysis 17 What is the best feasible investment opportunity set given risky assets and a risk-free asset? Case 1: Lending rates and borrowing rates are identical. Case 2: Lending rates and borrowing rates are different. Hsiu-lang Chen Portfolio Analysis 18 Rf The CAL, not the efficient frontier, describes the best feasible investment opportunity now! The optimal risky portfolio is the tangency portfolio. Hsiu-lang Chen Portfolio Analysis 19 Construction of the Optimal Risky Portfolio Max ϖ SP = E ( ~ ) − rf rP σP i N s.t. ∑ϖ i =1 i =1 Equation (7.13) at BKM (8th & 9th Edition) on the next slide gives the formula to construct the optimal risky portfolio in a case of two risky assets. Hsiu-lang Chen Portfolio Analysis 20 The optimal risky portfolio in the case of two risky assets B & C Solutions for the general case? Hsiu-lang Chen Portfolio Analysis 21 Construction of the Optimal Risky Portfolio (Short-Selling Allowed) The solution involves solving Zi for the following system of N simultaneous equations. E ( ~) − rf = Z1σ i1 + Z 2σ i 2 + Z 3σ i 3 + ⋅ ⋅ ⋅ + Z N σ iN , i = 1,2,3,..., N ri Where ϖ k = Zk N ∑Z i =1 Hsiu-lang Chen i Portfolio Analysis 22 In Terms of Matrix Expression r1 E (~ ) − rf E (~ ) − r r2 f • • E (~ ) − rf rN σ 11 σ 21 = • • σ N 1 Z1 σ 11 Z σ 2 21 ⇒ • = • • • Z N σ N 1 Hsiu-lang Chen σ 12 σ 22 σ N2 • • • • • • • • • • • • • • • • • σ 12 σ 22 • • σ N2 Portfolio Analysis • σ 1 N Z1 • σ 2N Z2 • • • • • • • σ NN Z N −1 r1 σ 1N E (~ ) − rf σ 2 N E (~ ) − rf r2 • • • • σ NN E (~ ) − rf rN 23 The mutual fund theorem (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 T- bills versus the optimal risky portfolio depending on personal preference. Hsiu-lang Chen Portfolio Analysis 24 In practice, however, different managers will estimate different input lists, thus deriving different efficient frontiers, and offer different “optimal” portfolios to their clients. The source of the disparity mainly lies in the security analysis. Hsiu-lang Chen Portfolio Analysis 25 The Derivation of CAPM • If there are homogeneous expectations, then all investors must select the same optimal risky portfolio, the market portfolio. [EGBG, p. 291][BKM, p. 293] E ( ~) − r = λCov ( ~, ~ ), i = 1,2,3,..., N r r r E ( ~ ) − r = λCov ( ~ , ~ ) = λσ r r r i f i M 2 M f M ⇒ E ( ~) − r = r i M M E (~ ) − r r M σ f f Cov ( ~, ~ ) r r i 2 M ⇒ E ( ~) = r + β (E ( ~ ) − r r r i Hsiu-lang Chen f i M Portfolio Analysis f M ) 26 Construction of the Optimal Risky Portfolio (Short-Selling Prohibited) Max ϖ θ= E (~P ) − r f r σP N s.t. ∑ϖ i =1 i =1 ϖ i ≥ 0, ∀i This is a quadratic programming problem! You can accomplish this in the Excel or other statistical software. Hsiu-lang Chen Portfolio Analysis 27 On Diversification N 2σ 2 + N N ϖ ϖ σ ∀ i ≠ j Var (~ ) == ∑ ϖ r ∑ ∑ P i i i j ij i =1 i =1 j =1 For an equally-weighted portfolio (ωi =1/N), Var ( ~ ) = rP 1 N σ + 2 N −1 N C O V as N → C O V →∞ N Where σ 2 ≡ σ i2 ∑ i =1 N COV ≡ N 1 N ( N −1) N ∑ ∑σ i =1 j =1 ij ∀i ≠ j Thus only the average covariance matter for large portfolios. If the average covariance is zero, then the portfolio variance is close to zero for large portfolios! Hsiu-lang Chen Portfolio Analysis 28 On Diversification The component of risk that can be diversified away we call the diversifiable or non-systematic risk. Empirical facts: – The average (annual) return standard deviation is 49%. – The average (annual) covariance between stocks is 0.037, and the average correlation is about 39%. Since the average covariance is positive, even a very large portfolio of stocks will be risky. We call the risk that cannot be diversified away the systematic risk. Hsiu-lang Chen Portfolio Analysis 29 Multiple Risky Securities Basic Message: Your risk/return tradeoff is improved by holding many assets with less than perfect correlation. Far from everybody agrees: Hsiu-lang Chen BERKSHIRE HATHAWAY INC. Portfolio Analysis BRK/A: $104k; BRK/B: $69.93 09/07/201130 All eyes were on Warren Buffett at Berkshire Hathaway's 2008 annual meeting at the Qwest Center in Omaha, Neb. Hsiu-lang Chen Portfolio Analysis 31 Less Hedging, Bigger Fees Single-Stock Funds Defy Diversification Doctrine, Make Ackman a Winner June 7, 2007; WSJ; Page C12 Hsiu-lang Chen Portfolio Analysis 32 Illustration of Ackman Hedge Fund Hedge funds typically collect 20% of profits they generate. Suppose an investor want to put the entire $1 billion into two different fund investments, a typical hedge fund or Ackman hedge fund. A Typical Hedge Fund Ackman Hedge Fund Invest $100m in each of 10 stocks (5: +50%; 2: +0%; 3: -50%) Invest $100m in each of 10 funds held by Ackman, where each fund holds one stock only (5: +50%; 2: +0%; 3: -50%) Overall Profit: $100m*2*50%=$100m 20% Profits: 0.2*$100m=$20m Profits on 5 Funds: $100m*5*50%=$250m 20% Profits: 0.2*$250m=$50m Hsiu-lang Chen Portfolio Analysis 33 Focused Funds Find Less Can Be More Ups, and Downs, of Concentrated Investing By JONATHAN BURTON; WSJ; AUGUST 18, 2009; C9 Focused funds -- portfolios with only a couple of dozen holdings -- are getting attention in a market where stock selection is more important than ever. Diversification's downside is that it limits a fund's chance to meaningfully outperform its index. Moreover, an overly diversified portfolio can mimic an index fund, but at a much higher cost. Hsiu-lang Chen Portfolio Analysis 34 Appendix - STATS FACTS Hsiu-lang Chen Portfolio Analysis 35 ...
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