Introduction to Computational Finance and Financial Econometrics Introduction to Portfolio Theory

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1 You can t see this text! Introduction to Computational Finance and Financial Econometrics Introduction to Portfolio Theory Eric Zivot Spring 2015 Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 1 / 46

2 Outline 1 Portfolios of Two Risky Assets 2 Efficient Portfolios with Two Risky Asssets 3 Efficient Portfolios with a Risk-free Asset 4 Efficient Portfolios with Two Risky Assets and a Risk-free Asset Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 2 / 46

3 Investment in Two Risky Assets R A = simple return on asset A R B = simple return on asset B W 0 = initial wealth Assumptions: R A and R B are described by the CER model: R i iid N(µ i, σ 2 i ), i = A, B cov(r A, R B ) = σ AB, cor(r A, R B ) = ρ AB Investors like high E[R i ] = µ i Investors dislike high var(r i ) = σ 2 i Investment horizon is one period (e.g., one month or one year) Note: Traditionally in portfolio theory, returns are simple and not continuously compounded Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 3 / 46

4 Portfolios x A = share of wealth in asset A = $ in A W 0 x B = share of wealth in asset B = $ in B W 0 Long position: x A, x B > 0 Short position: x A < 0 or x B < 0 Assumption: Allocate all wealth between assets A and B: x A + x B = 1 Portfolio return: R p = x A R A + x B R B Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 4 / 46

5 Portfolios cont. Portfolio Distribution: µ p = E[R p ] = x A µ A + x B µ B σ 2 p = var(r p ) = x 2 Aσ 2 A + x 2 Bσ 2 B + 2x A x B σ AB = x 2 AσA 2 + x 2 BσB 2 + 2x A x B ρ AB σ A σ B R p iid N(µ p, σp) 2 End of Period Wealth: W 1 = W 0 (1 + R p ) = W 0 (1 + x A R A + x B R B ) W 1 N(W 0 (1 + µ p ), σpw ) Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 5 / 46

6 Portfolios cont. Result: Portfolio SD is not a weighted average of asset SD unless ρ AB = 1: ) 1/2 σ p = (x 2 AσA 2 + x 2 BσB 2 + 2x A x B ρ AB σ A σ B x A σ A + x B σ B for ρ AB 1 If ρ AB = 1 then: σ AB = ρ AB σ A σ B = σ A σ B and, σp 2 = x 2 AσA 2 + x 2 BσB 2 + 2x A x B σ A σ B = (x A σ A + x B σ B ) 2 σ p = x A σ A + x B σ B Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 6 / 46

7 Example data µ A = 0.175, µ B = σ 2 A = 0.067, σ 2 B = σ A = 0.258, σ B = σ AB = , ρ AB = σ AB σ A σ B = Note: Asset A has higher expected return and risk than asset B. Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 7 / 46

8 Example Example: Long only two asset portfolio Consider an equally weighted portfolio with x A = x B = 0.5. The expected return, variance and volatility are: µ p = (0.5) (0.175) + (0.5) (0.055) = σ 2 p = (0.5) 2 (0.067) + (0.5) 2 (0.013) + 2 (0.5)(0.5)( ) = σ p = = 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. This reflects risk reduction via diversification. Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 8 / 46

9 Example Example: Long-Short two asset portfolio Next, consider a long-short portfolio with x A = 1.5 and x B = 0.5. In this portfolio, asset B is sold short and the proceeds of the short sale are used to leverage the investment in asset A. The portfolio characteristics are µ p = (1.5) (0.175) + ( 0.5) (0.055) = σ 2 p = (1.5) 2 (0.067) + ( 0.5) 2 (0.013) + 2 (1.5)( 0.5)( ) = σ p = = This portfolio has both a higher expected return and standard deviation than asset A. Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 9 / 46

10 Portfolio Value-at-Risk Assume an initial investment of $W 0 in the portfolio of assets A and B. Given that the simple return R p N(µ p, σ 2 p). For α (0, 1), the α 100% portfolio value-at-risk is VaR p,α = q R p,αw 0 = (µ p + σ p q z α) W 0 where q R p,α is the α quantile of the distribution of R p and q z α = α quantile of Z N(0, 1). Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 10 / 46

11 Relationship between Portfolio VaR and Individual Asset VaR Result: Portfolio VaR is not a weighted average of asset VaR: VaR p,α x A VaR A,α + x B VaR B,α unless ρ AB = 1. Asset VaRs for A and B are: VaR A,α = q R A 0.05 W 0 = (µ A + σ A q z α)w 0 VaR B,α = q R B 0.05 W 0 = (µ B + σ B q z α)w 0 Portfolio VaR is: VaR p,α = (µ p + σ p q z α) W 0 = [ (x A µ A +x B µ B )+(x 2 A σ2 A +x2 B σ2 B +2x Ax B σ AB) 1/2 q z α ] W 0 Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 11 / 46

12 Relationship between Portfolio VaR and Individual Asset VaR cont. Portfolio weighted asset VaR is: x A VaR A,α + x B VaR B,α = x A (µ A + σ A q z α)w 0 + x B (µ B + σ B q z α)w 0 = [(x A µ A + x B µ B ) + (x A σ A + x B σ B ) q z α] W 0 provided ρ AB 1. (µ p + σ p q z α) W 0 = VaR p,α If ρ AB = 1 then σ AB = ρ AB σ A σ B = σ A σ B and: σp 2 = x 2 AσA 2 + x 2 BσB 2 + 2x A x B σ A σ B = (x A σ A + x B σ B ) 2 σ p = x A σ A + x B σ B and so, x A VaR A,α + x B VaR B,α = VaR p,α Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 12 / 46

13 Example Example: Portfolio VaR and Individual Asset VaR Consider an initial investment of W 0 =$100,000. The 5% VaRs on assets A and B are: VaR A,0.05 = q R A 0.05 W 0 = ( ( 1.645)) 100, 000 = 24, 937, VaR B,0.05 = q R B 0.05 W 0 = ( ( 1.645)) 100, 000 = 13, 416. The 5% VaR on the equal weighted portfolio with x A = x B = 0.5 is: VaR p,0.05 = q Rp 0.05 W 0 = ( ( 1.645)) 100, 000 = 10, 268, and the weighted average of the individual asset VaRs is, x A VaR A,0.05 +x B VaR B,0.05 =0.5( 24,937)+0.5( 13,416)= 19,177. Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 13 / 46

14 Outline 1 Portfolios of Two Risky Assets 2 Efficient Portfolios with Two Risky Asssets 3 Efficient Portfolios with a Risk-free Asset 4 Efficient Portfolios with Two Risky Assets and a Risk-free Asset Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 14 / 46

15 Portfolio Frontier Vary investment shares x A and x B and compute resulting values of µ p and σ 2 p. Plot µ p against σ p as functions of x A and x B. Shape of portfolio frontier depends on correlation between assets A and B If ρ AB = 1 then there exists portfolio shares x A and x B such that σ 2 p = 0 If ρ AB = 1 then there is no benefit from diversification Diversification is beneficial even if 0 < ρ AB < 1 Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 15 / 46

16 Efficient Portfolios Definition: Portfolios with the highest expected return for a given level of risk, as measured by portfolio standard deviation, are efficient portfolios. If investors like portfolios with high expected returns and dislike portfolios with high return standard deviations then they will want to hold efficient portfolios Which efficient portfolio an investor will hold depends on their risk preferences Very risk averse investors dislike volatility and will hold portfolios near the global minimum variance portfolio. They sacrifice expected return for the safety of low volatility. Risk tolerant investors don t mind volatility and will hold portfolios that have high expected returns. They gain expected return by taking on more volatility. Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 16 / 46

17 Globabl Minimum Variance Portfolio The portfolio with the smallest possible variance is called the global minimum variance portfolio. This portfolio is chosen by the most risk averse individuals To find this portfolio, one has to solve the following constrained minimization problem min x A,x B σ 2 p = x 2 Aσ 2 A + x 2 Bσ 2 B + 2x A x B σ AB s.t. x A + x B = 1 Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 17 / 46

18 Review of Optimization Techniques: Constrained Optimization Example: Finding the minimum of a bivariate function subject to a linear constraint y = f(x, z) = x 2 + z 2 min x,z y = f(x, z) s.t. x + z = 1 Solution methods: Substitution Lagrange multipliers Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 18 / 46

19 Method of Substitution Substitute z = x 1 in f(x, z) and solve univariate minimization: y = f(x, x 1) = x 2 + (1 x) 2 min x f(x, x 1) First order conditions: 0 = d dx (x2 + (1 x)) = 2x + 2(1 x)( 1) = 4x 2 x = 0.5 Solving for z: z = = 0.5 Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 19 / 46

20 Method of Lagrange Multipliers Idea: Augment function to be minimized with extra terms to impose constraints. 1 Put constraints in homogeneous form: x + z = 1 x + z 1 = 0 2 Form Lagrangian function: L(x, z, λ) = x 2 + z 2 + λ(x + z 1) λ = Lagrange multiplier 3 Minimize Lagrangian function: min x,z,λ L(x, z, λ) Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 20 / 46

21 Method of Lagrange Multipliers cont. First order conditions: 0 = 0 = 0 = L(x, z, λ) x L(x, z, λ) z L(x, z, λ) λ = 2 x + λ = 2 z + λ = x + z 1 We have three linear equations in three unknowns. Solving gives: 2x = 2z = λ x = z 2z 1 = 0 z = 0.5, x = 0.5 Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 21 / 46

22 Example Example: Finding the Global Minimum Variance Portfolio Two methods for solution: Analytic solution using Calculus Numerical solution use the Solver in Excel use R function solve.qp() in package quadprog for quadratic optimization problems with equality and inequality constraints Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 22 / 46

23 Calculus Solution Minimization problem: min x A,x B σ 2 p = x 2 Aσ 2 A + x 2 Bσ 2 B + 2x A x B σ AB s.t. x A + x B = 1 Use substitution method with: x B = 1 x A to give the univariate minimization, min x A σ 2 p = x 2 Aσ 2 A + (1 x A ) 2 σ 2 B + 2x A (1 x A )σ AB Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 23 / 46

24 Calculus Solution cont. First order conditions: 0 = d dx A σ 2 p = d ( ) x 2 dx AσA 2 + (1 x A ) 2 σb 2 + 2x A (1 x A )σ AB A = 2x A σ 2 A 2(1 x A )σ 2 B + 2σ AB (1 2x A ) x min A = σ 2 B σ AB σ 2 A + σ2 B 2σ AB, x min B = 1 x min A Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 24 / 46

25 Excel Solver Solution The Solver is an Excel add-in, that can be used to numerically solve general linear and nonlinear optimization problems subject to equality or inequality constraints. The solver is made by FrontLine Systems and is provided with Excel The solver add-in may not be installed in a default installation of Excel Tools/Add-Ins and check the Solver Add-In box If Solver Add-In box is not available, the Solver Add-In must be installed from original Excel installation CD Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 25 / 46

26 Outline 1 Portfolios of Two Risky Assets 2 Efficient Portfolios with Two Risky Asssets 3 Efficient Portfolios with a Risk-free Asset 4 Efficient Portfolios with Two Risky Assets and a Risk-free Asset Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 26 / 46

27 Portfolios with a Risk Free Asset Risk Free Asset: Asset with fixed and known rate of return over investment horizon Usually use U.S. government T-Bill rate (horizons < 1 year) or T-Note rate (horizon > 1 year) T-Bill or T-Note rate is only nominally risk free Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 27 / 46

28 Properties of Risk-Free Asset R f = return on risk-free asset E[R f ] = r f = constant var(r f ) = 0 cov(r f, R i ) = 0, R i = return on any asset Portfolios of Risky Asset and Risk Free Asset: x f = share of wealth in T-Bills x B = share of wealth in asset B x f + x B = 1 x f = 1 x B Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 28 / 46

29 Properties of Risk-Free Asset cont. Portfolio return: R p = x f r f + x B R B = (1 x B )r f + x B R B = r f + x B (R B r f ) Portfolio excess return: R p r f = x B (R B r f ) Portfolio Distribution: µ p = E[R p ] = r f + x B (µ B r f ) σ 2 p = var(r p ) = x 2 Bσ 2 B σ p = x B σ B R p N(µ p, σ 2 p) Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 29 / 46

30 Risk Premium µ B r f = excess expected return on asset B = expected return on risky asset over return on safe asset For the portfolio of T-Bills and asset B: µ p r f = x B (µ B r f ) = expected portfolio return over T-Bill The risk premia is an increasing function of the amount invested in asset B. Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 30 / 46

31 Leveraged Investment x f < 0, x B > 1 Borrow at T-Bill rate to buy more of asset B. Result: Leverage increases portfolio expected return and risk. µ p = r f + x B (µ B r f ) σ p = x B σ B x B µ p & σ p Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 31 / 46

32 Determining Portfolio Frontier Goal: Plot µ p vs. σ p. σ p = x B σ B x B = σ p σ B µ p = r f + x B (µ B r f ) where, ( µb r f = r f + σ p σ B (µ B r f ) ( µb r f = r f + σ B σ B ) σ p ) = SR B = Asset B Sharpe Ratio = excess expected return per unit risk Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 32 / 46

33 Determining Portfolio Frontier cont. Remarks: The Sharpe Ratio (SR) is commonly used to rank assets. Assets with high Sharpe Ratios are preferred to assets with low Sharpe Ratios Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 33 / 46

34 Outline 1 Portfolios of Two Risky Assets 2 Efficient Portfolios with Two Risky Asssets 3 Efficient Portfolios with a Risk-free Asset 4 Efficient Portfolios with Two Risky Assets and a Risk-free Asset Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 34 / 46

35 Efficient Portfolios with 2 Risky Assets and a Risk Free Asset Investment in 2 Risky Assets and T-Bill: R A = simple return on asset A R B = simple return on asset B R f = r f = return on T-Bill Assumptions: R A and R B are described by the CER model: R i iid N(µ i, σi 2 ), i = A, B cov(r A, R B ) = σ AB, corr(r A, R B ) = ρ AB Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 35 / 46

36 Efficient Portfolios with 2 Risky Assets and a Risk Free Asset cont. Results: The best portfolio of two risky assets and T-Bills is the one with the highest Sharpe Ratio Graphically, this portfolio occurs at the tangency point of a line drawn from R f to the risky asset only frontier The maximum Sharpe Ratio portfolio is called the tangency portfolio Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 36 / 46

37 Mutual Fund Separation Theorem Efficient portfolios are combinations of two portfolios (mutual funds): T-Bill portfolio Tangency portfolio - portfolio of assets A and B that has the maximum Shape ratio Implication: All investors hold assets A and B according to their proportions in the tangency portfolio regardless of their risk preferences. Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 37 / 46

38 Finding the tangency portfolio max SR p = µ p r f x A, x B σ p subject to µ p = x A µ A + x B µ B σ 2 p = x 2 Aσ 2 A + x 2 Bσ 2 B + 2x A x B σ AB 1 = x A + x B Solution can be found analytically or numerically (e.g., using solver in Excel). Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 38 / 46

39 Finding the tangency portfolio cont. Using the substitution method it can be shown that: x tan A = (µ A r f )σ 2 B (µ B r f )σ AB (µ A r f )σ 2 B + (µ B r f )σ 2 A (µ A r f + µ B r f )σ AB x tan B = 1 x tan A Portfolio characteristics: ( σ tan p µ tan p = x tan A ) 2 = ( x tan A µ A + x tan B µ B ) 2 σ 2 A + ( x tan B ) 2 σ 2 B + 2x tan A x tan B σ AB Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 39 / 46

40 Efficient Portfolios: tangency portfolio plus T-Bills x tan = share of wealth in tangency portfolio x tan + x f = 1 x f = share of wealth in T-bills µ e p = r f + x tan (µ tan p r f ) σ e p = x tan σ tan p Result: The weights x tan and x f are determined by an investor s risk preferences Risk averse investors hold mostly T-Bills Risk tolerant investors hold mostly tangency portfolio Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 40 / 46

41 Example For the two asset example, the tangency portfolio is: ( σ tan p x tan A =.46, x tan B = 0.54 µ tan p = (.46)(.175) + (.54)(.055) = 0.11 ) 2 = (.46) 2 (.067) + (.54) 2 (.013) + 2(.46)(.54)(.005) = σ tan p =.015 = Efficient portfolios have the following characteristics: µ e p = r f + x tan (µ tan p r f ) = x tan ( ) Eric Zivot e (Copyright tan 2015) Introduction to Portfolio Theory 41 / 46

42 Problem Find the efficient portfolio that has the same risk (SD) as asset B? That is, determine x tan and x f such that σ e p = σ B = = target risk. Note: The efficient portfolio will have a higher expected return than asset B. Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 42 / 46

43 Solution.114 = σ e p = x tan σ tan p = x tan (.124) x tan = =.92 x f = 1 x tan =.08 Efficient portfolio with same risk as asset B has: (.92)(.46) =.42 in asset A (.92)(.54) =.50 in asset B.08 in T-Bills If r f = 0.03, then expected Return on efficient portfolio is: µ e p =.03 + (.92)( ) =.104. Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 43 / 46

44 Problem Assume that r f = Find the efficient portfolio that has the same expected return as asset B. That is, determine x tan and x f such that: µ e p = µ B = = target expected return. Note: The efficient portfolio will have a lower SD than asset B. Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 44 / 46

45 Solution = µ e p = x tan (.11.03) x tan = =.31 x f = 1 x tan =.69 Efficient portfolio with same expected return as asset B has: (.31)(.46) =.14 in asset A (.31)(.54) =.17 in asset B.69 in T-Bills The SD of the efficient portfolio is: σp e =.31(.124) =.038. Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 45 / 46

46 You can t see this text! faculty.washington.edu/ezivot/ Eric Zivot (Copyright 2015) Introduction to Portfolio Theory 46 / 46