Markowitz portfolio theory

Transcription

1 Markowitz portfolio theory Farhad Amu, Marcus Millegård February 9, Introduction Optimizing a portfolio is a major area in nance. The objective is to maximize the yield and simultaneously minimize the risk. One way of optimizing a portfolio was suggested by Harry Markowitz (1927-) who published the article Portfolio selection in Journal of Finance 1952 [1]. In 1990 he received the Nobel Memorial Prize in Economic Sciences due to his contributions to portfolio theory. There have been extensions and developments made on Markowitz model and it is still a widely used model. Our purpose is to explain Markowitz portfolio theory in an understandable and correct way. In order to do so we need some background knowledge of the concepts of portfolio theory. We will also discuss an implementation of the Markowitz model called CAPM. 2 Mean variance portfolio theory Before we present a model for portfolio theory we must be able to handle the basics of probability theory, and also be familiar with the notations used in portfolio theory. This section will therefore mostly consist of denitions and explanation of the theory behind Markowitz portfolio theory. We work with a single period model and there are no tax or transaction costs. 2.1 Some important concepts in portfolio theory An investment instrument that can be bought and thereafter sold is called an asset[2]. The underlying asset could be a stock, currency, option, bond or portfolio. Suppose you purchase an asset at time zero and sell it at a xed time (T 0). We are only interested in this single period and will assume that neither tax nor transaction costs present. Let X 0 be the amount of money invested at time 0, and let X T be the amount of money you receive when you sell the asset at time T. Then the total return, R, on your investment is dened as R = X T. X 0 The rate of return,r, (sometimes referred to as return) is dened as r = X T X 0 X 0, 1

2 hence R = 1 + r. Note that to make a prot r has to be greater than zero. Suppose now that there are n dierent assets, and to each of them corresponds a total return R i, i = 1,..., n. If we spread out our invested money in these assets, we say that we form a portfolio (it is called a portfolio even if we don't invest money in all assets). We select an amount of money X 0i in asset i such that X 0i = X 0, and we restrict ourselves X 0i 0, i = 1,.., n. To make the coming notation simpler we introduce weights w i, i = 1,..., n, such that X 0i = w i X 0. Clearly the sum over all weights is equal to one. Then the amount of money generated at time T by the ith asset is R i w i X 0, and hence the total return R of the portfolio is R = R iw i X n 0 = (1 + r i)w i X 0 = X 0 X 0 Since the sum of all w i is one and R = 1 + r, we have w i + r i w i r = w i r i. 2.2 Expected value and variance of the return of a portfolio The return you get when you sell an asset r is uncertain when you buy the asset. In these cases r is a random variable. Suppose we know the mean for each of the n assets, call it r i, i = 1,..., n. Call the variance of asset i σi 2, and the covariance between asset i and j σ ij. Then the expected return of a portfolio r is simply E[r] = r = w i E[r i ] = w i r i. The variance σ 2 of the portfolio return is σ 2 = E[(r r) 2 ] = E [( n ) 2 ] w i r i w i r i = E [( n w i (r i r i ) )( n w j (r j r j ) )] j=1 = E [ w i w j (r i r i )(r j r j ) ] = w i w j σ ij. 2

3 3 The Markowitz model The fundamental problem of a portfolio can be formulated in 2 ways, either the investor wants to minimize the variance with respect to a xed expected return r or maximize the expected return given a xed variance. This is known as the Markowitz model [2]. To express it mathematically, the problem can be formulated as minimize w iw j σ ij subject to w i r i = r, w i = 1 if you are risk averse, or maximize w i r i subject to w iw j σ ij = σ 2, w i = 1. We will mainly concentrate on the rst one, the second one is rather equivalent. The solution to the Markowitz model is an optimization problem. To be able to solve it and fully understand it, we need to use Lagrange multipliers. 3.1 Lagrange multipliers Lagrange multipliers can be used to nd the extremum of a multivariate function f(x 1,..., x n ) subject to the constraint g(x 1,..., x n ) = 0, where f and g are functions with continuous rst partial derivatives on the open set containing the curve g(x 1,..., x n ) = 0, and g 0 at any point on the curve g(x 1,..., x n ) = 0 (where is the gradient) [3]. For an extremum of f to exist on g, the gradient of f must line up with the gradient of g [3]. If the two gradients are in the same direction, then one is a multiple ( λ) of the other, so f = λ g. The two vectors are equal, so all of their components are as well, giving for i = 1,..., n. f x i = λg x i The extremum is then found by solving the n + 1 equations in n + 1 unknowns λg(x 1,..., x n ) = 0, f x i + λg x i = 0, (i = 1,..., n). If there are several constraints for example if there exist 2 functions g 1 (x 1,..., x n ) = 0 and g 2 (x 1,..., x n ) = 0, the equations become f + λ g 1 + µ g 2 = 0, where µ and λ are constants. 3

4 3.2 Solution to the Markowitz model using Lagrange multipliers We can now apply the Lagrange multipliers and get a solution to the Markowitz model. Using the same notation as in section 3.1, set f(w 1,..., w n ) = w i w j σ ij, λg 1 (w 1,..., w n ) = λ ( n w i r i r ), µg 2 (w 1,..., w n ) = µ ( n w i 1 ). Note that if we want our constraints (namely w i r i = r, w i = 1) to be fullled, g 1 = g 2 = 0 as we moved the right side over to the left side. Also note that we already know σ ij, r i and r for every i, j. If we sum the functions above we get the Lagrangian L(w 1,..., w n ) = w i w j σ ij + λ ( n w i r i r ) + µ ( n w i 1 ). If we dierentiate L with respect to w 1,..., w n, set those n equations equal to zero, and also set λg 1 = µg 2 = 0, we have n + 2 equations with n + 2 unknown variables. As all the equations are linear, a solution can be found easily by using linear algebra. Figure 1: An example of a mean-variance plot. A red dot marks a possible portfolio while the blue line, called the ecient frontier, consists of the set of all portfolios, either a minimal variance with subject to a xed expected return, or a maximal expected return subject to a xed variance. 4

5 4 CAPM The capital asset pricing model (CAPM) is developed mainly by Sharpe, Lintner and Mossin, which follows from the Markowitz mean-variance portfolio theory [2]. This method is applied to investment decision problems. When one chooses to set up a portfolio the expectation of the return has to be studied carefully. The idea is to get better understanding of an asset's return and try to diversify the risk. Later on this result will be input variables in Markowitz model. The CAPM formula is usually written as r i = r f + β i (r M r f ) where β i = σ im σm 2. r i is the expected return on asset i. r f is a risk free rate. β i is a risk measure of asset i. r M is the expected market return. There are many methods to obtain a risk free asset r f. One can lend money with interest of rate or buy bond. The market return, r M, describes the market which contains the asset. For example if we purchase stocks in Ericsson then OMX is our market. We use the value of these two when calculating the β parameter. β a is relation between the asset and market return. 5 Conclusions Markowitz model to minimize risk subject to a given expected return is an optimization problem. One wants to nd the optimal amount of money w i to invest in each asset i. The model is quite simple to understand and can be solved by using Lagrange multipliers. A common way of estimating the expected return, r, is made by CAPM but there are more advanced tools and one is called Arbitrage Pricing Theory (APT). CAPM is a single factor model since one β is generated while APT has many factors. It is important to point out that the Markowitz model is just a tool. The most important part of portfolio theory is the assets and risk factors that one chooses. You have to have knowledge about your asset since there is no guarantee to prot in the model. The risk factors should be chosen carefully because of the correlation between the asset and the risk factor which is not always obvious. A correlation could exist even if the factor and asset are not in the same area of business. A lot of extensions can be made to the Markowitz model. In our example we assumed that there were no transaction costs, this is of course not the case in reality. Braun and Mitchell show a solution to the portfolio problem when the transaction costs are linear [4]. Moreover, we concentrated on the original Markowitz model on just a single period 5

6 where a portfolio is created once and then sold at a xed time T 0. Multiple periods is discussed further by Luenberger [2]. For further reading, David G Luenberger's Investment Science [2] is generally a good source. Sources on the internet should be read with care since the quality of them varies. References [1] Harry Markowitz. Portfolio selection. Journal of nance ; Vol 7; [2] David G Luenberger. Investment science. Oxford university press [3] [downloaded ]. Available: [4] [downloaded ]. Available: mitchj/papers/exact.pdf. 6