Key investment insights

Transcription

1 Basic Portfolio Theory B. Espen Eckbo 2011 Key investment insights Diversification: Always think in terms of stock portfolios rather than individual stocks But which portfolio? One that is highly diversified But how much portfolio risk? Allocate your in vestment between the riskfree asset and your diversified portfolio depending on your tolerance for risk Eckbo (43) 2 1

2 Optimal portfolios Step I: Find the portfolio opportunity set consisting of risky assets only Cases with two risky assets Arbitrary number of risky assets Effect of diversification Computation of optimal risky portfolio weights Separation theorem Step II: Find the allocation between risky portfolio and risk-free assets Requires specifying investor preferences Eckbo (43) 3 2-asset portfolio opportunities with no risk free asset Will show that there is a single optimal risky portfolio Will derive the Minimum Variance Frontier (MVF) The MVF is the set of portfolios with the lowest variance for a given expected return Will show how the shape of MVF depends on the correlation between the risky securities Eckbo (43) 4 2

3 Probability h(r t ) Probability density Area under the curve is the cumulative probability p t = stock price E(R t )=[E(p t )-p t-1 ]/p t-1 E(R)= t R t [h(r t )] Mean=(1/T) t R t -100% Variance=σ 2 (R)= (1/T) t [R t -E(R)] E(R ) Eckbo (43) 5 R t Notation Subscript i denotes stock i (i=1,2) E i = E(r i ) (expected return) 2 i = 2 (r i ) (variance) i = 2 i ij = cov(r i,r ij = cov(r i,r j )/ i j -1 ij 1 (standard deviation),r j ) (covariance) x i = portfolio weight of stock i (correlation coefficient) i x i =1 (where is the summation function) With two stocks only: x 2 =1-x 1 Eckbo (43) 6 3

4 Mean and variance of portfolio p s return: E p = x 1 E 1 + x 2 E 2 2 p = x x x 1 x 2 12 Using the definition iti of the correlation coeff.: 2 p = x x x 1 x Will derive the minimum variance frontier (MVF) for three different values of 12: 12 = 1 (perfect positive correlation) 12 = -1 (perfect negative correlation) 12 = 0 (uncorrelated assets) Eckbo (43) 7 Case 1: 12 = 1 2 p = x x x 1 x p = (x x 2 2 ) 2 p = x x 2 2 E p = x 1 E 1 + x 2 E 2 Let E 1 >E 2 and 1 > 2 (1 most risky asset) Since x 2 =1-x 1, and substituting into E p : E p = E 2 + [(E 1 -E 2 )/( 1-2 )]( p - 2 ) MVF is a straight line w/positive slope p Eckbo (43) 8 4

5 E(r) 2-asset MVF for 12 = x 1 = Asset 2 x 1 =0 x 1 =1 Asset Risk-free return for x 1 = -0.5 (r) Eckbo (43) Case 2: 12 = -1 2 p = x x x 1 x p = (x 1 1 -x 2 2 ) 2 Since p is nonnegative, take absolute value: p = x 1 1 -x 2 2 E p = x 1 E 1 + x 2 E 2 Note: p = x 1 1 -x 2 2 = 0 for x 1 = 2 /( ) We just created a risk free asset with a long position in both risky assets: Eckbo (43) 10 5

6 E(r) 2-asset MVF for 12 = -1 Asset Asset Risk-free return for x 1 = 0.25 (r) Eckbo (43) 11 Case 3: 12 = 0 2 p = x x p =(x x ) 1/2 E p = x 1 E 1 + x 2 E 2 MVF is no longer a straight line. It s a parabola when plotting variance and a hyperbola when plotting standard deviation. There are no risk free opportunities as long as Eckbo (43) 12 6

7 E(r) 2-asset MVF for 12 = 0 Asset Important property of MVF: Combinations of MV portfolios are themselves MV portfolios Asset (r) Eckbo (43) 13 E(r) 2-asset MVF summary Asset 1 12 = 1 12 = Asset 2 12 = 1 12 = = = -1 (r) Eckbo (43) 14 7

8 With a risk-free asset, the weights (x) in the tangency portfolio maximizes the slope of the straight line, also called the Sharpe Ratio How to find these weights (x * 1): max(x) (E p -r f )/ p E p = x 1 E 1 + x 2 E 2 subject to 2 p = x x x 1 x Solution given two risky assets only (e denotes excess return r-r f ): x * 1= (E e E e 2 12 )/[E e E e (E e 1 +E e 2)) ] Eckbo (43) 15 Example: Asset E i i 1 10% 20% 2 15% 30% Also: r f =3% and 12 =05 =0.5. The Sharpe Ratio of the MVE-portfolio is SR MVE =E e MVE / MVE =0.1250/0.2179= Eckbo (43) 16 8

9 Portfolio with N risky assets, i=1,..,n: i x i =1 E p = i x i E i (sometimes we also use μ) 2 = 2 2 p i x i i + i x i j x j ij (where i j) i x 2 i 2 i + i j x i x j ij (where i j) x 1 x 2 x 3 Variance- covariance matrix V x x x Eckbo (43) 17 Rule: for each in the matrix, premultiply by the x i (same row) and x j (same column) and then sum over all such products Thus (verify!): 2 p = i j x i x j ij Note also: 2 p = i x i cov(r i, j x i r j )= i x i ip where p is the portfolio of all N assets. x i ip is asset i s contribution to p s total risk ip is therefore a marginal risk concept Later: β i ip / 2 p (standardized marginal risk) Eckbo (43) 18 9

10 Optimal portfolio weights ( excess return over variance rule ) E e = the expected excess return vector V = the full variance-covariance covariance matrix x = optimal portfolio weights Step 1: Compute the raw weights: w= E e /V Step 2: The weights w do not sum to 1. Thus, normalize: x=w/w I, where I is the unit vector [1,1,1,1,1,,,1] Sharpe Ratio: SR x =x E e /(x Vx) 1/2 Eckbo (43) 19 Examples of excess return over variance rule Ex 1: E A =10%, E B =20%. 2 A= 0.04, 2 B=0.09. A and B are uncorrelated r F =5% Compute (E -r 2 i F )/ i (i=a,b) and standardize Optimal portfolio: MVE,A =42.86%, x MVE,B =57.14% x MVE,A Eckbo (43) 20 10

11 Ex 2: Asset E i i 1 5% 10% 2 10% 20% 3 15% 30% r f =3.5%, 12 =0, 13 =05 =0.5, 23 =0.5 x MVE = [ ] SR MVE =E e MVE / MVE = /0.2163= Eckbo (43) 21 Ex 3: Add security 4 E 4 =15%, 4 = 45% 41 = 42 = 43 =0 x MVE = [ ] SR MVE =E e MVE / MVE =0.1302/0.1961= Why would anyone would hold security 4 (i.e., why is it not dominated by security 3)? Eckbo (43) 22 11

12 Ex 4: Another security 4 E 4 =5%, 4 = 45% 41 = 42 = 43 = -0.2 x MVE = [ ] SR MVE =E e MVE / MVE =0.1065/0.1646= Again, why would anyone would hold security 4 (this one seems even more dominated by security 3)? Eckbo (43) 23 Effect of Diversification What happens to 2 p when N? 2 p = i x 2 i 2 i + i j x i x j ij (where i j) Let x i =x j =1/N (equal-weighted portfolio) 2 p = (1/N 2 ) i 2 i + i (1/N 2 ) j ij (where i j) Substitute in the average 2 p og ij AV( 2 i) = (1/N) i 2 i AV( ij ) = [1/N(N-1)] 1)] i ij (where i j) 2 p = (1/N)AV( 2 i) + [(N-1)/N]AV( 1)/N]AV( ij ) so, as N, 2 p AV( ij ) Eckbo (43) 24 12

13 N, 2 p AV( ij ): In large portfolios, stocks own-variances cancel out (is diversified away) and total portfolio risk reduces towards the average covariance The remaining covariance is called the portfolios systematic (nondiversifiable) risk We will see later that, in asset pricing models, systematic risk is the only priced risk, i.e., the only risk that generates a compensation in terms of expected return Eckbo (43) 25 2 p Fig. 8: Diversification and the Number of stocks N in the portfolio N Eckbo (43) 26 13

14 II: Allocation between the risk-free asset and the optimal risky portfolio So far, we did not introduce investor preferences (tolerance for risk) Now we need to model investor demand Will assume preferences over mean and variance of wealth W (MV-preferences) Holds if returns are jointly normally distributed ib t d (only two parameters) Maximize expected utility: E[U(W)] Eckbo (43) 27 Investor s general objective: c t is consumption at time t max E u( c0,..., ct x t Returns and consumption related by wealth dynamics: In last period T, consume c T = W T-1 (1+r P ) Work backwards to time 0 For simplicity, we will use: 1-period time horizon Mean-variance preferences over returns ) Eckbo (43) 28 14

15 Eckbo (43) 29 Eckbo (43) 30 15

16 Eckbo (43) 31 Eckbo (43) 32 16

17 MV preference function over returns E[U(r)] = E(r) - 0.5A 2 (r) A = risk aversion coefficient: E[U]/ E[U]/ =A =A Risk averse investor: A>0 Risk neutral investor: A=0 Risk prone investor: A<0 The 0.5 scales the marginal utility (first derivative) and here reflects use of fractional returns, i.e., r=0.10 for 10%. If you use r=10 for 10%, then change to 0.005A 2 (r). Eckbo (43) 33 U(r) Risk-averse (concave) utility function U[E(r)] E[U(r)] Certainty equivalent return Risk premium -50% 25% 100% Project with two equally likely outcomes E(r) = 25% Var(r) = 56% (r) = 75% r Eckbo (43) 34 17

18 Certainty equivalent return: r CE =E[U(r)] The investor is indifferent between receiving r CE with certainty or investing in the risky asset If A=0.50, will you hold a risk free asset yielding 3%? A r CE 24% 11% 3% -3% What A-value makes you indifferent between holding the risky and risk free assets? Eckbo (43) E(r) Risk free asset A>0.78 A=0.78 A=0 Risky asset Fig 10: Indifference curves with risk aversion coefficient A, E(r)= 25 (r)= E(r).25, (r) r f =r CE at A=0.78 (r) Eckbo (43) 36 18

19 Capital Allocation Line (CAL) What combinations of E and result from combining the risk-free and risky assets in a portfolio? y= portfolio weight in risky asset r p = yr+(1-y)ry)r f E p =ye+(1-y)ry)r f = r f +y[e-r f ] 2 p=y 2 2 p or y= p / E p = r f +[(E-r f )/ ] p Eckbo (43) 37 E(r) Portfolio opportunities w/risk-free asset Mean-Variance Efficient (MVE) Portfolio Asset 1 R f Asset 2 Suboptimal CALs Thus, although 12 = 0 as in Case 3, MVE is a straight line 0 (r) Eckbo (43) 38 19

20 E(r) 2-asset CAL Indifference curve at A= Risk free asset CAL (slope = Sharpe Ratio) Risky asset Optimal allocation r f =r CE at A=0.78 (r) Eckbo (43) 39 Find the optimal portfolio weight y*: max(y){e[u(r p )]}=max(y){e p -0.5A 2 p} Substituting for E p and 2 p we get =max(y){[r f +ye(r-r f )-0.5Ay 2 2 ]} Solution: y * =E(r E(r-r f )/A 2 Stock share = (1/risk aversion)(excess return/variance) Eckbo (43) 40 20

21 Two-fund separation theorem Two funds: (1) risk-free asset and (2) MVE portfolio x = V - 1 μ e Place the fraction y of your total investment amount in the MVE portfolio Place the rest (1-y) in the risk-free asset Need to specify investor s risk aversion to determine y, while x = V - 1 μ e is independent d of risk preferences (double- check) Thus, you can separate the computation of x and y. Find x first and then y Eckbo (43) 41 In our example: A y * E p p What is the meaning of y * =1.56? Can you ever get y * <0? Eckbo (43) 42 21

22 Summary Marginal vs. total risk: The risk of an individual asset in a portfolio is its marginal (covariance) contribution to total portfolio risk 2 p= i j x i x j ij = i x i cov(r i, j x i r j )= i x i im MVE portfolio: You should hold the same portfolio x of risky assets no matter what your risk tolerance A x = V - 1 μ e Two-fund separation: ation Use your risk tolerance to allocate your investment between the risky portfolio and the risk-free asset y * =E(r-r f )/A 2 Eckbo (43) 43 22