Portfolio Analysis. M. Kateregga

Transcription

1 Portfolio Analysis M. Kateregga November 21, AIMS South Africa

2 Outline Introduction The Optimization Problem Return of a Portfolio Variance of a Portfolio Portfolio Optimization in n-stocks case The Lagrangian Method R Implementation of portfolio Theory

3 Introduction A portfolio is an investment made in n assets A k with the returns R k, k = 1,, n using some amount of wealth W. Suppose W k is the amount of wealth invested in asset A k, then n W k = W. k=1 It is better to represent the investments in terms of relative values i.e. w k = W n k W, such that w k = 1 k=1 and n w k R k = R p. We shall refer to w k as the weight of asset A k with respect to the total portfolio. k=1 M. Kateregga AIMS South Africa Explore Financial Markets with Maths 3/21

4 Preliminary The return R k of a financial asset A k is described as a random variable. The mean and variance of R k can be estimated from historical data. Individual stocks in a portfolio may not be normally distributed but a portfolio of many of them tends to normal distribution. Performance is measured in terms of realized return and variance of a portfolio. The expected return is interpreted as the reward of the investment and variance as its risk. M. Kateregga AIMS South Africa Explore Financial Markets with Maths 4/21

5 The Optimization Problem The problem can be formulated as follows: Consider a set of financial assets, characterized by their expected mean and their covariances. We need to find the optimal weight of each asset, such that the overall portfolio provides the smallest risk for a given overall return. Therefore, the problem is to find the efficient frontier or the set of all achievable portfolios that offer the highest rate of return for a given level of risk. M. Kateregga AIMS South Africa Explore Financial Markets with Maths 5/21

6 Return of a Portfolio The expected return of the portfolio, E[R p ] is the weighted average of the expected returns of the individual components i.e. E[R p ] = n w k E[R k ], k=1 w k is the proportion of the portfolio invested in security k and we require that n w k = 1. k=1 The realized return is a linear combination of the returns of the individual investments. M. Kateregga AIMS South Africa Explore Financial Markets with Maths 6/21

7 Variance of a Portfolio Let σ ij = E[(R i E[R i ])(R j E[R j ])]. The portfolio variance is given by n n σp 2 = E[ R p E[R p ] 2 ] = w i w j σ ij. i=1 j=1 w i is the proportion of total investment in security i. σ ij = σ ji = ρ ij σ i σ j is the covariance between assets A i and A j ρ ij [ 1, 1] is the correlation coefficient between securities i and j. σ i is the standard deviation of security i. M. Kateregga AIMS South Africa Explore Financial Markets with Maths 7/21

8 Example: The 3-Assets portfolio We consider a portfolio of stocks 1, 2 and 3 respectively. The portfolio variance is then given by 3 3 σp 2 = w i w j ρ ij σ i σ j. i=1 j=1 = w 1 w 1 ρ 11 σ 1 σ 1 + w 1 w 2 ρ 12 σ 1 σ 2 + w 1 w 3 ρ 13 σ 1 σ 3 + w 2 w 1 ρ 21 σ 2 σ 1 + w 2 w 2 ρ 22 σ 2 σ 2 + w 2 w 3 ρ 23 σ 2 σ 3 + w 3 w 1 ρ 31 σ 3 σ 1 + w 3 w 2 ρ 32 σ 3 σ 2 + w 3 w 3 ρ 33 σ 3 σ 3. Observe that ρ ii = 1 for all i. M. Kateregga AIMS South Africa Explore Financial Markets with Maths 8/21

9 The 3-Asset portfolio Table 1: 3-Asset portfolio Stock 1 Stock 2 Stock 3 Expected Return Variance Standard deviation Weight 35% 45% 20% Correlation coefficients ρ 12 = 0.4 ρ 23 = 0.5 ρ 13 = 0.6 Exercise 1: Compute the expected return and variance of this portfolio. M. Kateregga AIMS South Africa Explore Financial Markets with Maths 9/21

10 Minimum Variance Portfolio The task is to determine a particular combination of stocks that will result in a least possible portfolio variance. This is a minimization problem. We apply basic calculus principles i.e. We make the following restrictions: dσ 2 p dw 1 = 0. w 1 = 0.4w 2 and w 3 = 1 w 1 w 2. Exercise 2: Determine the weights w 1, w 2 and w 3 that constitute the optimal portfolio. M. Kateregga AIMS South Africa Explore Financial Markets with Maths 10/21

11 Matrix notation: 3-Asset portfolio The weight and covariance matrices are given by w 1 σ 11 σ 12 σ 13 w = w 2 and V = σ 21 σ 22 σ 23. w 3 σ 31 σ 32 σ 33 where σ ij = ρ ij σ i σ j thus σ ii = σi 2. Therefore the portfolio variance is given by ) σ 11 σ 12 σ 13 σp 2 = (w w 1 1 w 2 w 3 σ 21 σ 22 σ 23 w 2. σ 31 σ 32 σ 33 w 3 M. Kateregga AIMS South Africa Explore Financial Markets with Maths 11/21

12 General Case: n-asset portfolio We calculate the covariance matrix which is a pairwise combination of all portfolio constituents. A n-stock portfolio requires (n 2 n)/2 covariances of its components to compute its variance. The portfolio variance σ 2 p in this case is given by σ 11 σ 12 σ 1n ) w 1 σ 21 σ 22 σ 2n σp 2 = (w 1 w n w n σ n1 σ n2 σ nn M. Kateregga AIMS South Africa Explore Financial Markets with Maths 12/21

13 In Summary We shall denote: w to be the vertical vector of weights of the different stocks in the portfolio µ as the vertical vector of expected returns. V to be the variance-covariance matrix. the variance of the portfolio is therefore given by σ 2 p = w t V w, where w t denotes the transpose of w. The idea is to minimize this variance subject to the constraints of achieving a given expected return [Portfolio Optimization]. M. Kateregga AIMS South Africa Explore Financial Markets with Maths 13/21

14 Portfolio Optimization in n-stocks case We wish to minimize w t V w over w i.e. subject to: Minimize w w t V w, { 1 t w = 1 µ t w = E[R p ] where E[R p ] is the desired return on the portfolio, 1 is a vector of ones, E[R 1 ] w 1 µ =. and w =.. E[R n ], w n M. Kateregga AIMS South Africa Explore Financial Markets with Maths 14/21

15 The Lagrangian Method The Lagrangian is defined as L(w, λ, γ) = w t V w + (E[R p ] w t µ)λ + (1 w t 1)γ, where λ and γ are Lagrangian multipliers. The critical point of the Lagrangian is obtained by solving a system of equations: w L(w, λ, γ) = 2V w µλ 1γ = 0. (1) L(w, λ, γ) = E[R p ] w t µ = 0. λ (2) L(w, λ, γ) = 1 w t 1 = 0. γ (3) M. Kateregga AIMS South Africa Explore Financial Markets with Maths 15/21

16 The Lagrangian Method From equation (1), we obtain w = 1 2 V 1 [µλ + 1γ]. Then from equations (2) and (3) we deduce µ t V 1 µλ + 1 t V 1 µγ = 2E[R p ]. µ t V 1 1λ + 1 t V 1 1γ = 2. We can rewrite this as ( ) ( ) λ E[R p ] A = 2 γ 1 ( ) µ t V 1 µ 1 t V 1 µ where A = µ t V t V 1. 1 M. Kateregga AIMS South Africa Explore Financial Markets with Maths 16/21

17 The Lagrangian Method For the system to have a solution, the following matrix A must be invertible The matrix V is positive definite and so is its inverse V 1. Thus, x t V 1 x > 0 for any vector x 0. The determinant d of A is greater than zero and is given by where 1 t V 1 µ = µ t V 1 1. d = ([µ t V 1 µ][1 t V 1 1] [1 t V 1 µ] 2 ) > 0. M. Kateregga AIMS South Africa Explore Financial Markets with Maths 17/21

18 The Lagrangian Method Thus the multipliers are given by ( ) ( ) λ = 2 [1 t V 1 1] [1 t V 1 µ]e[r p ] γ d [1 t V 1 µ] + [µ t V 1. µ]e[r p ] Therefore the weights of the optimal portfolio can then be given by w = f + E[R p ]g, where f = 1 ) ([1 d V 1 t V 1 1]1 [1 t V 1 µ]µ. g = 1 ( ) d V 1 [1 t V 1 ]1 + [µ t V 1 µ]µ. M. Kateregga AIMS South Africa Explore Financial Markets with Maths 18/21

19 The Lagrangian Method Let ( ) ( ) a 11 a 12 µ t V 1 µ 1 t V 1 µ = a 21 a 22 µ t V t V 1 1 and q = E[R p ]. The variance σ 2 p is then given by σ 2 p = w t V w. = q 2 g t V g + q(g t V f + f t V g) + f t V f. = a ( 11 q a ) d a 11 a 11 = 1 d (a 11q 2 2a 12 q + a 22 ). M. Kateregga AIMS South Africa Explore Financial Markets with Maths 19/21

20 The Lagrangian Method The equation below of a hyperbola in the space (σ, q)-plane represents the "efficient frontier" σ 2 p = 1 d (a 11q 2 2a 12 q + a 22 ). We can now obtain the weights of the minimum variance portfolio as w(q) = f + a 12 a 11 g, and the corresponding risk and return values: σ(q) = 1/a 11 and q = a 12 /a 11. The sharpe ratio η of a portfolio is defined by η = q/σ(q). M. Kateregga AIMS South Africa Explore Financial Markets with Maths 20/21

21 R Implementation of portfolio Theory This session will be conducted in the computer lab. Learners will be provided with a set of step by step instructions for constructing an optimal portfolio in R and plotting the efficient frontier. The objective of the session is to help learners develop a fully working routine from an algorithm and, learn how to construct optimal portfolios using R. M. Kateregga AIMS South Africa Explore Financial Markets with Maths 21/21