Portfolio Theory Outline - Investment Analysis and Portfolio Management | FINC 852, Study notes of Finance

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Chapters 5-8 Portfolio Theory Section 2 - Outline Chapter 5—Interest Rates and Risk Premiums – See notes from first lecture Chapter 6--Risk and Risk Aversion » Utility theory » Indifference Curves Chapter 7--Capital Allocation » The opportunity set (Capital Allocation Line) » The Optimal choice Chapter 8--Optimal Risky Portfolios » Diversification » The Opportunity Set (Efficient Frontier) Two risky assets Many risky assets » The Optimal choice More multiple period returns Geometric average does not ignore compounding: In the case of EXDS: rg = 3.7987911/11-1 = 12.90 % the future value of $1 invested at 12.9% for 11 months is » = (1+.129)11 = 3.798791 1)]1)...(1)(1[( /121 −+++= n ng rrrr Date Close Return Cumulative (1+r) Sep-99 18.0156 1 Oct-99 21.5 0.19341 1.19341 1.19341 Nov-99 26.9531 0.253633 1.496098 1.253633 Dec-99 44.4062 0.647536 2.464875 1.647536 Jan-00 57.4375 0.293457 3.188209 1.293457 Feb-00 71.1875 0.239391 3.951437 1.239391 Mar-00 70.25 -0.01317 3.899398 0.986831 Apr-00 44.2188 -0.37055 2.454473 0.629449 May-00 35.2812 -0.20212 1.958369 0.797878 Jun-00 46.0625 0.305582 2.556812 1.305582 Jul-00 44.4375 -0.03528 2.466612 0.964722 Aug-00 68.4375 0.540084 3.798791 1.540084 0.168361 3.798791 Chapter 6 Risk and Risk Aversion Suppose an Investor has $100,000 to invest and she has the following investment choices: (A) A risk-free asset that pays a guaranteed 3% (B) A risky asset which pays either double or half with equal probability – What is the expected end-of-period wealth from (A) – What is the expected end-of-period wealth from (B)? – What is the expected return from (B)? Simple Example How should the investor decide which investment to take? – First, calculate the risk-premium (or excess return): – Next, calculate a measure of risk. One possible measure is standard deviation (or variance). Is the risk-premium enough to compensate the investor for the risk? 75.5625.)25.5.( 2 1)25.1( 2 1 22 ==−−+−=σ E R Rf( ) . . .− = − =25 03 22 Portfolio returns In general: The portfolio return is a weighted average of the individual asset returns 1 1 1 = = ∑ ∑ = = n i i i n i ip w where rwr Scenario analysis Three basic steps: – Consider the scenarios that may occur – Attach probabilities and returns to each scenario – calculate the E(r) – “Rule 1” scenario p(s) HPR 1 0.2 -0.05 2 0.3 0.25 3 0.4 0.05 4 0.1 -0.1 s n s s rprE ∑ = = 1 )( scenario p(s) HPR p(s)*HPR 1 0.2 -0.05 -0.01 2 0.3 0.25 0.075 3 0.4 0.05 0.02 4 0.1 -0.1 -0.01 sum 0.075 Variance and standard deviation Historical data for EXDS and ENE: Use a spreadsheet! Our ex-ante measure is “Rule 2” Date EXDS ENE Sep-99 Oct-99 0.19341 -0.0274 Nov-99 0.253633 -0.04378 Dec-99 0.647536 0.165847 Jan-00 0.293457 0.529578 Feb-00 0.239391 0.014824 Mar-00 -0.01317 0.089091 Apr-00 -0.37055 -0.06928 May-00 -0.20212 0.047613 Jun-00 0.305582 -0.11492 Jul-00 -0.03528 0.143411 Aug-00 0.540084 0.150848 mean 0.168361 0.08053 var 0.091732 0.031104 std 0.302872 0.176364 2 1 22 ))(( σσσ =−=∑ = andrErp n s ss scenario p(s) HPR-A p(s)*HPR p(r-E(r )) 2̂ 1 0.2 -0.05 -0.01 0.003125 2 0.3 0.25 0.075 0.009188 3 0.4 0.05 0.02 0.00025 4 0.1 -0.1 -0.01 0.003063 sum 0.075 0.015625 std 0.125 Portfolio variance “Rule 5” For 2 assets, portfolio variance is: In a 3-asset portfolio, we must consider the covariance between each asset pair (1,2), (1,3), and (2,3). In a n-asset portfolio, or, re-stated as 2,121 2 2 2 2 2 1 2 1 2 σσσσ wwwwp ++= 3,2323,1312,121 2 3 2 3 2 2 2 2 2 1 2 1 222 σσσσσσσ wwwwwwwwwp +++++= ∑ ∑∑ = = ≠ = += N i N i N ij j jijiiip www 1 1 1 , 222 σσσ jij N j i N i P ww , 11 2 σσ ∑∑ == = More portfolio variance In general, the portfolio variance has N2 terms – N variance terms – N2 - N covariance terms For an equally weighted portfolio: In a well-diversified portfolio, we only bear “covariance risk”. This is risk is most often referred to as systematic risk or market risk. ( )= ∞→ 2lim PN σ average covariance Conceptual Portfolio Variance NxN terms mostly covariances! Var Cov Cov Cov Cov Cov Cov Var Cov Cov Cov Cov Cov Cov Var Cov Cov Cov Cov Cov Cov Var Cov Cov Cov Cov Cov Cov Var Cov Cov Cov Cov Cov Cov Cov Cov Var Indifference Curves (remember ECON?) We can use utility to define the combinations of risk and return (investment choices) that would provide the investor with the same utility level. Example: What combinations of risk and return would provide a utility level of .05 for an investor with A=2? E(R) σ Utility: U(R)=E(R)-.5Aσ2 0.05 0.00 U = .05 - 0 = .05 0.10 ? .05 0.15 ? .05 0.20 ? .05 Indifference Curves A graphical representation For our example with U=.05 and A=2: What about A=4? What about U=.10? E(R) σ 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.1 0.2 0.3 0.4 0.5 Capital Allocation Assume there are two securities (i) One risky portfolio, P: E(RP)=15%, (ii) One risk-free asset, F: RF=7%, What proportion of our investment do we allocate to P (weight w) and what proportion to F (1-w)? Rule 1 : E(RC) = w·E(RP) + (1-w)·RF = RF + w{E(RP)-RF} Rule 2 : (only true in the case where one asset σ=0) %22=Pσ 222 PC w σσ ⋅= Specific Investment Objectives What portfolio weights would we choose to create a portfolio with a standard deviation of 10%? – What is the expected return on this portfolio? What portfolio weights would we choose to create a portfolio with an expected return of 20%? – What is the risk of this portfolio? .1 = w(.22) ⇒ w = .4545 E(R) = .07 + .4545(.15-.07) = .1063 .2= .07 + w(.15-.07) ⇒ w = 1.625 σC = 1.625(.22) = 0.3575 Optimal Capital Allocation An Investor confronted with the Capital Allocation Line of investment opportunities must choose an optimal w. How?? Find the weight (w*) that maximizes utility given all possible portfolios on the CAL (i.e., maximize utility subject to rules 1 and 2). Example: If Then 2 2 12 )(),( σσ ARERU −= 2 * )( P fP A RRE w σ − = Optimal Capital Allocation How do we graphically represent the optimal capital allocation decision? – Utility is maximized at the point of tangency between the CAL and the investor’s indifference curves. E(R) Std. Dev. CAL Indifference Curves P F w* = Optimal Portfolio Choice Two Risky Assets Consider the following two assets: A - E(RA) = 8% B - E(RB) = 16% How do we describe the expected return and risk of a portfolio in terms of the returns on the individual securities that make up the portfolio? %7=Aσ %20=Bσ ),()1(2)1( :*2 Rule )()1()()( :*1 Rule 22222 BABAP BAP RRCovwwww REwRwERE −+−+= −+= σσσ With perfectly negative correlation ρ=-1 Note: If ρ=-1, we can create a perfect hedge. What must be the risk-free rate in this economy? 0.00% 4.00% 8.00% 12.00% 16.00% 20.00% 0.00% 10.00% 20.00% 30.00% 40.00% Std. Dev. E( R ) w E(RP) σP -0.5 20.0% 33.5% 0.0 16.0 20.0 0.5 12.0 6.5 0.741 10.1 0.0 1.0 8.0 7.0 1.5 4.0 20.5 With zero correlation ρ =0 Note: In general, 0<ρ<1. How would this case look? 0.00% 4.00% 8.00% 12.00% 16.00% 20.00% 0.00% 10.00% 20.00% 30.00% 40.00% Std. Dev. E( R ) w E(RP) σP -0.5 20.0% 30.2% 0.0 16.0 20.0 0.5 12.0 10.6 1.0 8.0 7.0 1.5 4.0 14.5 Optimal Portfolio Selection (A) Case A: With a Risk-Free Asset σ E(R) RF Optimal Tangency Portfolio M CAL Tthe optimal risky portfolio is the one with the steepest CAL. Investors will choose to hold a combination of portfolio M and the risk-free security. Optimal Portfolio Selection (B) Case B: No Risk-Free Asset σ E(R) Indifference CurvesOptimal Portfolio Example: Portfolio Choice with 2 risky assets and a risk-free asset State Probability RA RB RC 1 .3 -.05 0.00 .04 2 .5 .18 .12 .04 3 .2 .275 .10 .04 How should Jana allocate her capital between assets A, B, and C in order to maximize U=E(R)-.005(10) σ2 ? Example Continued Step 3: Allocate between Rf and P* to max U. 20.1 )36.6)(10(01. 486.8 2 * * = − =Pw WAmaxU =1.2(.172) = .2064 WBmaxU =1.2(.828) = .9936 WCmaxU = = -.2000 1.0000 Example Continued E(RmaxU) = .2064(.13) + .9936(.08) + -.2(.04) = .0983 σ2maxU = (.2064)2(.0152)+(.9936)2(.0028)+(-.2)2(0) +2(.2064)(.9936)(.0059) + 0 + 0 = .00583 σmaxU = .0763 Step 3 Continued Conclusions Investment decisions includes two tasks: 1. Determine the optimal risky portfolio - M This is independent of investor preferences 2. Allocate Capital between M and the risk-free asset Depends on individual investor’s preferences Two-Fund Separation: Every investor allocates wealth between two mutual funds: M and RF Are there benefits from access to risk-free borrowing/lending? » When we introduce a risk-free asset, each investor is able to reach a higher level of utility.