Lecture IV Portfolio management: Efficient portfolios. Introduction to Finance Mathematics Fall 2014
Reduce the risk, one asset Let us warm up by doing an exercise. We consider an investment with σ 1 = 0.27. What reduce the risk more: Moving half the capital into 1. cash (volatility 0, covariance 0), 2. an second asset with σ 2 = 0.16 and ρ 1,2 = 0.5. 2/30
Reduce the risk, one asset For this we use: 3/30 σ 2 V = w 2 1σ 2 1 + w 2 2σ 2 2 + 2w 1 w 2 Cov(K 1, K 2 ) Strategy 1: w 1 = w 2 = 0.5, σ 1 = 0.27, σ 2 = 0 and Cov(K 1, K 2 ) = 0 and compute σ 2 V = 1 4 0.272 + 1 4 0 + 2 4 0 = 0.002916 σ V = 13.5% Strategy 2: w 1 = w 2 = 0.5, σ 1 = 0.27, σ 2 = 0.16 and ρ = 0.5 and compute σ 2 V = 1 4 0.272 + 1 4 0.162 + 2 4 σ V = 11.78% ( 0.5) 0.27 0.16 = 0.013825
Reduce the risk, two assets+bond 4/30 Two stocks have correlation ρ < 0, and the variances σ 1 and σ 2, respectively. Assume w 1 > 0, determine w 2 such that σ v is minimised. Further, we put the weight (1 w 1 w 2 ) into the risk-less bond. σ 2 V = w 2 1σ 2 1 + w 2 2σ 2 2 + 2w 1 w 2 σ 1 σ 2 ρ We want to find the minimum of this function, so we compute dσ 2 V dw 2 = 2w 2 σ 2 2 + 2w 1 σ 1 σ 2 ρ for completeness dσ 2 V d 2 w 2 = 2σ 2 2 > 0 This is zero for w 2 = w 1σ 1 σ 2 ρ, with a value of σ2 2 σ 2 V (w 2) = w 2 1σ 2 1 + w2 1σ 2 1σ 4 2ρ σ 4 2 2 w2 1σ 1σ 2 2ρ 2 ρ = w σ 1σ 2 2 2 4 1 ) (1 + ρ 2 ρ2. σ 2 2
Portfolio, weights We consider the portfolio with weights w i. We write these weights into a vector w = w 1 w 2 w 3. w N We analyse now all portfolios on the level of these vectors. We have to add the constrain N i=1 w i = 1, as only these represent portfolios. We call all w R N with N i=1 w i = 1, attainable portfolios. 5/30
Portfolio, Notation The weights of a portfolio 6/30 w T = ( w 1 w 2 w 3 w N ). The expected returns of the assets, µ i = E(S i ), are noted in µ T = ( µ 1 µ 2 µ 3 µ N ). The variation σi 2 = σ i,i = Cov(S i, S i ) = Var(S i ) and covariation σ i,j = Cov(S i, S j ) are put into a matrix σ 1,1 σ 1,2... σ 1,N C = σ 2,1 σ 2,2... σ 2,N......, σ N,1 σ N,2... σ N,N called covariance matrix.
Portfolio, expected return For the expected return we see that 7/30 ( N ) E(K V ) = E w i K i = i=1 N E (w i K i ) = i=1 For the variation we see ( N ) σ 2 V = Var(K V ) = Var w i K i = N i=1 i=1 N Cov (w i K i, w j K j ) = j=1 N i=1 N w i E (K i ) = µ T w = w T µ. i=1 N w i w j σ i,j = w T C w Remark to the notation in the book 1 = u and we work with column vectors. j=1
Minimal Variance Portfolio The portfolio with the smallest variance in the attainable set has weights w T min = 1 T C 1 1 T C 1 1, provided that the denominator is non-zero. Its variance is given by σ 2 min = 1 1 T C 1 1. 8/30
Minimal Variance Portfolio, proof* The proof uses the method of Lagrange multipliers, named after Lagrange (1736-1813). It is a strategy to find minima or maxima of function subject to constraints. We want to minimize: f( w) = w T C w und the constraint g( w) = 1 T w = 1. Using the Lagrange multipliers λ, we want to find the minimum of F ( w, λ) = w T C w + λ( 1 T w 1) For general interest let me explain the method first at the black board. 9/30 Picture taken from Wikipedia.
Minimal Variance Portfolio, proof* We compute d dw i F ( w, λ) = d dw i ( w i w j σ i,j + λ i,j i w i ) 10/30 = 2 j w j σ i,j + λ = (2 w T C λ 1 T ) i d dλ F ( w, λ) = ( 1 T w 1) where ( v) i is the i-th entry of the vector. The first equations implies 0 T = 2 w T C λ 1 T = λ 2 1 T C 1 = w T Further, 0 = ( w T 1 1) 1 = λ 2 1 T C 1 1 1 λ = 2 1 T C 1 1
Portfolio Variance, Example I We have the covariance matrix C = σ2 1 0 0 0 σ2 2 0 with inverse C 1 = 0 0 σ3 2 1 σ 2 1 0 1 σ 2 2 So we consider three independent assets, so that 1 0 0 0 0 0 1 σ 2 3. 11/30 σ 2 min = 1C 1 1 = 1 σ 2 1 + 1 σ 2 2 + 1 σ 2 3 The portfolio, with smallest variance has weights: w T min = σ 2 min 1 T C 1 = 1 1 σ1 2 + 1 σ2 2 + 1 σ3 2 ( 1 σ 2 1 1 σ 2 2 ) 1 σ. 3 2
Portfolio Variance, Example II We have the covariance matrix 0.2 0.1 0.1 C = 0.1 0.2 0.1 with inverse C 1 = 5 3 1 1 1 3 1. 0.1 0.1 0.2 2 1 1 3 12/30 The minimal variance is 1 = 1 T C 1 1 = 5 ( ) 1 (3 1 1) 1 1 1 = 7.5 σmin 2 } 2 {{} 1 1 T C 1 and weights to the minimal portfolios are w T min = σ 2 min 1 T C 1 = 2.5 7.5 1 T = 1 3 1 T.
Portfolio Variance, Example III We have the covariance matrix 0.2 0 0 C = 0 0.2 0.1 with inverse C 1 = 1 15 0 0 0 20 10. 0 0.1 0.2 3 0 10 20 13/30 The minimal variance is 1 = 1 T C 1 1 = 1 ( ) 1 15 10 10 1 = 35 σmin 2 3 1 3 and weights to the minimal portfolios are w T min = σ 2 min 1 T C 1 = 1 35 ( 15 10 10 ).
Minimal Variance Line* Let 14/30 c 1,m = 1 T C 1 µ = µ T C 1 1 c m,m = µ T C 1 µ c 1,1 = 1 T C 1 1. The portfolio with the smallest variance among attainable portfolios with expected return µ V has weights w T = c m,m µ V c 1,m 1 T C 1 + µ V c 1,1 c 1,m µ T C 1. c 1,1 c m,m c 2 1,m c 1,1 c m,m c 2 1,m
Minimal Variance Line, to the proof* We use again the method of Lagrange multipliers, We want to minimise w T C w, with the constraints w T 1 = 1 and w T µ = µ V. So we want to minimise G( w, λ, ϑ) = w T C w λ( w T 1 1) ϑ( w T µ µ V ) 15/30 Similar to the earlier computations: From d dw i = 0 we obtain w T = λ 2 1 T C 1 + ϑ 2 µt C 1 Using also d dλ = 0 and d dϑ 1 = w T 1 = λ 2 1 T C 1 1 + ϑ 2 µt C 1 1, µ V = w T µ = λ 2 1 T C 1 µ + ϑ 2 µt C 1 µ. = 0 (extra condition) we obtain Solving for λ and ϑ gives the stated result.
Minimal variance Give is a market with µ T = ( 0.2 0.14 0.17 ) σ 1 = 0.25 σ 2 = 0.28 σ 3 = 0.2 ρ 1,2 = 0.3 ρ 1,3 = 0.15 ρ 2,3 = 0.2 16/30 Compute the minimal variance portfolio and its return. First we use this data to create the covariance matrix C = σ2 1 ρ 1,2 σ 1 σ 2 ρ 1,3 σ 1 σ 3 0.0625 0.021 0.0075 ρ 1,2 σ 1 σ 2 σ2 2 ρ 2,3 σ 2 σ 3 = 0.021 0.0784 0.0112 ρ 1,3 σ 1 σ 3 ρ 2,3 σ 2 σ 3 σ3 2 0.0075 0.0112 0.04 We compute σ 2 min = 1 T C 1 1 = 38.9695, so σ min = 0.1602. w T min = 1 σ 2 min w T min µ = 0.173433 1 T C 1 = ( 0.2876 0.1743 0.539 )
Portfolio to a given return* Compute the portfolio with µ V we compute = 0.2 and the smallest variance. For this c 1,m = 1 T C 1 µ = 6.7586 c m,m = µ T C 1 µ = 1.20867 c 1,1 = 1 T C 1 1 = 38.9695 Then, we compute c 1,1 c m,m c 2 1,m = 1.42238 c m,m µ V c 1,m = 0.143055 µ V c 1,1 c 1,m = 1.0353 17/30 w T = (c m,m µ V c 1,m ) 1 T C 1 + (µ V c 1,1 c 1,m ) µ T C 1 c 1,1 c m,m c 2 1,m = ( 0.7445 0.2555 0.511 ) w T µ = 0.2
Growth portfolios One does not buy stocks not in order to minimise risk, but perhaps to maximise profit/return. For mathematical reason we will not maximise return r, but the gain g: r = S t+1 S t S t g = log (S t+1 /S t ) = log (S t+1 ) log (S t ). The expected gain is also called drift ν V. The drift and expected return are connected by: ν V = µ V 1 2 σ2 V = µ T w 1 2 wt C w. Using the techniques demonstrated earlier compute first the derivative with respect to w i and set them to zero to obtain. µ T w T C = 0 w T max = µt C 1 µ T C 1 1. 18/30
Maximal drift, independent assets Let us assume that we have three independent assets, who does w max look like? If the assets are independent, then C are of the form: 1 C = σ2 1 0 0 0 0 σ 0 σ2 2 0 with inverse C 1 1 2 1 = 0 0 0 0 σ3 2 σ2 2. 1 0 0 σ3 2 So that w T max = 1 µ T C 1 1 ( µ1 σ 2 1 µ 2 σ 2 2 ) µ 3 σ. 3 2 At this example we also see that we can not consider a bond, as in this formula all σ should be non-zero. Further, not that ( w max ) i = µ i /σ 2 i, what is a ration we also optimised in the last lecture (Risk.vs.variance). 19/30
Growth portfolios, with bonds Let us consider the same also with a weight (1 w i ) in a bond with interest r f, then 20/30 µ P = (1 i w i )r f + µ T w = r f + ( µ r f 1) T w, ν P = r f + ( µ r f 1) T w 1 2 wt C w. Taking the derivates and solving for the minimum we obtain: w T = ( µ r f 1) T C 1 ( µ r f 1) T C 1 1 = µ T C 1 1 T C 1 ( µ r f 1) T C 1 1 r f ( µ r f 1) T C 1 1 = a w T max b w T min for some numbers a, b.
Growth portfolios, with bonds 21/30 For the following computation we define the notation ω = ( µ r f 1) T C 1 1: w T max = ( µ r f 1) T C 1 ω σ 2 max = ( µ r f 1) T C 1 (µ r f 1) ω 2 µ max = w max µ T = ( µ r f1) T C 1 µ ω = ( µ r f1) T C 1 ( µ r f 1) ( µ r f 1) T C 1 1 + r f ω ω = ωσ 2 max + r f This µ min > r f this function has a special function and is called Markowitz portfolio.
Maximal drift, constrained volatility Analog to the minimal variance line we can also compute the portfolio that has the maximal drift in the set of all portfolios that have a given variance 0 < σ 2 V < σ max. Using the Lagrange multipliers we want to optimise F ( w, λ, ϑ) = ν( w) + λ( 1 T w 1) + ϑ(σ( w) σ V ) We find the stock weights are given by w = σ V σ max w max. This means we just scale the maximal variance and put the proportion into the bond. σ max σ V σ max 22/30
Maximal drift, other constraints Using can use similar linear optimization techniques and other well-known techniques to compute all kinds of portfolios. Further, we can add multiple the constraint and optimize under these constraints. Some key words: optimize drift, optimize variance, optimize return, no-short selling, limited weight in one/multiple asset. 23/30
Attainable portfolios In the plot below all combination of return and deviation are displayed that can be reached by a combination of assets. The thick line corresponds the the return/deviation of the Minimal Variance Line. 24/30 The three dots correspond to the portfolios that consist of just one asset.
Attainable portfolios, No short-selling If we forbid short-selling (so condition on w i 0), then the set of return and deviation combination are as displayed below. 25/30 The three dots correspond to the portfolios that consist of just one asset.
Definition of dominant portfolio 26/30 Definition 5.1: We say that a security with expected return µ 1 and standard deviation σ 1 dominates another security with expected return µ 2 and standard deviation σ 2, whenever µ 1 µ 2 and σ 1 σ 2. Definition 5.2: A portfolio is called efficient if there is no other portfolio, expect itself, that dominates it. The set of efficient portfolios among all attainable portfolios is called the efficient frontier. Ever rational investor will choose an efficient portfolio. However, the different investors may select different portfolios, depending on their individual preference. In the following we show that all efficient portfolio are on the minimal variance line and that any combination of w min and w max is an efficient portfolio.
Minimal variance line is convex Take any two different portfolios on the minimum variance line, with weights w and w. Then, the minimum variance line consists of portfolios with weights c w + (1 c) w for any c R and only of such portfolios. Proof: We know that the points are of the form: w T = a 1 T C 1 + b µ T C 1 This means that C w = a 1 + b µ, C w = a 1 + b µ. Further, we know for the combined portfolio w = cw + (1 c)w. C w = c(a + a ) 1 + (1 c)(b + b ) µ µ = cµ + (1 c)µ, so that also w is on the mean variance line. Remark: The portfolio w might have the minimal variance under the class of all portfolios with µ v = µ, but this statement makes no statement about the size of the variance. 27/30
Characterisation of efficient portfolios We can conclude that all efficient points=portfolios in the efficient frontier w w min, satisfy γ w T = µ T C 1 µ 1 T C 1 For some real numbers µ, γ. 28/30
Computation of the factors* Consider any given marker and a portfolio on the efficient frontier with given expected return µ V. Compute the values of γ and µ such that the weights w in this portfolio satisfy γ w T C = µ µ 1. Solution: γ w T C = µ T µ 1 T C 1 1 γ = µ T C 1 1 µ 1 T C 1 1 γ w T C = µ T µ 1 T C 1 µ γµ V = µ T C 1 µ µ 1 T C 1 µ We combine these and solve for µ, which we could then compute with given data µ V ( µ µ 1) T C 1 1 = Having computed µ we can compute γ using ( ) γ = µ T µ 1 T C 1 1 ( ) T µ T C 1 ( µ µ V 1) µ µ 1 µ µ = 1 T C 1 ( µ µ V 1) 29/30
Summary to portfolio selection We define the set efficient frontier as the set of all portfolio, such that there exist no other attainable portfolio ( 1 T w = 1), that has better return, but lower risk. By common sense these are the only portfolios that we would invest into. The name frontier comes from the characterisation of portfolios on the (µ, σ) plain, see Slide 24/25. We have shown that all these portfolio are of the form w T = a µ T C 1 + b 1 T C 1 = a w max + (1 a) w min, for some numbers a, b, a, b, where w min is the market portfolio with the minimal variance and b w max is the portfolio with the biggest drift. 30/30