Lecture 3: Factor models in modern portfolio choice

Transcription

1 Lecture 3: Factor models in modern portfolio choice Prof. Massimo Guidolin Portfolio Management Spring 2016

2 Overview The inputs of portfolio problems Using the single index model Multi-index models Portfolio allocation in practice 2

3 The inputs of portfolio problems We now turn to issues of implementation and often, simplification, of mean-variance portfolio theory The inputs required by modern portfolio theory are return means, variances, and correlations o As discussed by Elton et al., the set up of the financial analysis industry makes estimating and forecasting correlations problematic o Whether the analyst's estimates contain information or whether one is better off estimating returns from an equilibrium model (from your asset pricing courses) is an open question The problem is made irksome by the number of estimates required o Most financial institutions follow between 150 and 250 stocks: to employ portfolio analysis, they would need forecasts of between 150 and 250 expected returns and 150 and 250 variances o Worse, they will need as many as N(N -1)/2 correlations, btw. 11,175 and 31,125 inputs 3

4 The single-index model The sheer number of inputs is staggering and it seems unlikely that analysts will be able to directly estimate correlation structures This has trigged a search for tricks and shortcuts able to simplify the problem The most widely used technique assumes that the co-movement between stocks is due to a single common influence or index: o Here I is the component of security i's return that is independent of the market's performance o R m is the rate of return on the market index o i measures the expected change in R i given a change in R m It is convenient to have e i uncorrelated with R m, formally, o e i uncorrelated with R m, implies that how well the model describes the return of any security is independent of what the return on the market happens to be 4

5 The single-index model Key properties: The key assumption of the single-index model is that e i is independent of e j for all values of, E(e i e j ) = 0, for all pairs i j The only reason assets vary together, systematically, is because of a common co-movement with the market There are no effects beyond the market (e.g., industry effects) that account for co-movement between securities This is an empirically refutable restriction: nothing in the structure of the single-index model forces asset returns to conform If we define then we obtain the properties listed at the top of this slide 5

6 The single-index model As stated already, while a security's mean and variance are the same two parts, an asset-specific and a market related ones, in contrast, the covariance depends only on market risk o As an exercise, let s derive the result concerning the covariance o Carrying out the multiplication, simplifying by cancelling the 's and combining the terms involving s yields: o However, the last three terms are zero, and hence See a simple example We assume = 1.5 6

7 The single-index model A single index model applied to N assets implied a need to estimate 3N + 2 parameters vs. 2N + N(N-1)/2 in the original case o When is set equal to 1.5, the market return is independent of the residual return e i o A lower value of e i leaves some market return in e and the covariance of e with the market is positive; hence a in excess of 1.5 removes too much market and yields a negative covariance btw. e and the market o The value of is unique and is the value that exactly separates market from unique return, making the covariance between R m and e i zero Plugging now single-index results into the general input, we have: All these inputs will now be derived provided that we obtain an estimate of I, I, and σ 2 i for each asset and, finally, an estimate of both the expected return (R m ) and variance (σ 2 m) for the market This is a total of 3N + 2 estimates << 2N + N(N-1)/2 found before 7

8 The single-index model o For an institution following between 150 and 250 stocks, the singleindex model required between 452 and 752 parameters o Compare this to the 11,175-31,125 correlation estimates or 11,475-31,625 estimates required when no simplifying structure was assumed There is one informative way to re-formulate the results we have just obtained, by defining the Beta on a portfolio P as a weighted average of the individual s on each stock: Similarly, define ptf Alpha as: Notice now that the variance of ptf. can be written as: and Yet this shows: 8

9 The single-index model Assume for a moment that an investor forms a portfolio by placing equal amounts of money into each of N stocks, then: The last term can be expressed as 1/N times avg. residual risk As the number of stocks increases, the importance of avg. residual risk diminishes drastically The table shows that the residual risk fails so rapidly that most of it is eliminated on even moderately sized portfolios The risk that is not eliminated is risk associated with the term P If we assume that residual risk approaches zero, the risk of the portfolio approaches: 9

10 Multi-index models Multi-index models are an attempt to capture some of the nonmarket influences that cause securities to move together The search for nonmarket influences is a search for a set of economic factors or structural groups (industries) that account for common movement in stock prices beyond that accounted for by the market index itself The cost of introducing additional factors is the chance that they are picking up random noise rather than real influences Any additional sources of covariance between securities can be introduced into the equations for risk and return, simply by adding these additional influences to the general return equation: where b * ij is a measure of the responsiveness of the return on stock i to changes in the index j c i is the random component of the asset i s return with a mean of zero and a variance we will designate as σ 2 ci 10

11 Multi-index models Multi-index models are best defined in terms of orthogonal factors While a multi-index model of this type can be employed directly, the model would have convenient mathematical properties if the indexes were uncorrelated (orthogonal) It is always possible to take any set of correlated indexes and convert them into a set of uncorrelated indexes (see Appendix A): where Cov[I k, I j ] = 0 j k The new indexes still have an interesting economic interpretation o Assume I * 1 is a stock market index and I * 2 an index of interest rates o The two are orthogonal if I * 2 is turned into an index of the difference between actual interest rates and the level of interest rates that would be expected given the rate of return on the stock market (I * 1) o b i2 is a measure of the sensitivity of return on stock i to this difference, the sensitivity of stock i's return to a change in interest rates when the rate of return on the market is fixed 11

12 Multi-index models Not only is it convenient to make the indexes uncorrelated, but it is also convenient to have the residual uncorrelated with each index, E[c i (I * j E[I*j])] = 0 for all i, j Similarly to single-index models, E(c i c j ) = 0 stocks vary together because of common co-movement with the set of factors This is equivalent to say that there are no factors beyond these indexes that account for co-movement between any two securities Nothing in the estimation of the model forces this to be true and the performance of the model is determined by this approximation Under these conditions, the key properties above obtain 12

13 Multi-index models Expected return and risk can be estimated for any portfolio if we have estimates of: o a i for each stock o estimates of b ik for each stock with each index o an estimate of 2 ci for each stock o an estimate of the mean I j and variance 2 Ij of each index This is a total of 2N + 2L + LN estimates o For an institution following between 150 and 250 stocks and employing 10 factors, this calls for between 1820 and 3020 inputs o Larger than the inputs required for the single-index model but less than the inputs needed when no simplifying structure was assumed For instance, very popular multi-index models set I * 1 = R m and identify the remaining factors with industry factors: 13

14 Multi-index models How many factors L ought to be selected? Literature started by Elton and Gruber (1971, 1973) found (that on both statistical grounds and economic grounds) adding additional indexes to the single-index model led to a decrease in performance Although adding more factors led to a better explanation of the historical correlation matrix, it led both to a poorer prediction of future correlations matrix and to worse portfolios o In short, these additional factors introduced more random noise than real information into the forecasting process The evidence that multi-index models, where the indexes are extracted according to explanatory power from past data, does not perform as well as a single-index model is very strong Chen, Roll, and Ross (1986) and others have produced a set of multi-factor models based on an a priori-hypothesized set of macroeconomic variables o See my recent work with Bianchi and Ravazzolo, but no ptf. choice 14

15 Portfolio allocation in practice Suppose you have some model to forecast means, variances, and covariances The model does not have to be of the single- or multi-index types seen above o E.g., you shall see that when x t is a predictor that may take positive or negative values (e.g., a rate of growth), then it may be viable to use E t-1 [r i,t ] = i + i x t-1 + i,t Var t-1 [r i,t ] = i + i x 2 t-1 + v i,t cov t-1 [r i,t, r j,t ] = jk + jk x t-1 + jk,t Given a set of risky assets and of weights that describe how the portfolio investment is split, the general formulas for N assets are: t-1 t-1 t-1,t-1 t-1 t-1 Working in a spreadsheet, it helps to have expressions for portfolio return and risk that are easy to enter Summed formulas above are most unsuitable for cell entry in Excel 15,t-1

16 Portfolio allocation in practice Therefore we use cell formulas based on Excel s vector and matrix multiplication (see Appendix B) If the expected returns and the portfolio weights are represented by column vectors (e t-1 and w t-1 respectively, with row vector transposes e T t-1 and w T t-1), and the covariance terms by matrix V t-1, then the expressions can be written as simple matrix formulas: t-1 t-1 At this point, suppose we want to construct an efficient (minimum variance) portfolio producing a target return of 7% The problem is to find the split across the assets that achieves the target return whilst minimising the variance of return This is a standard optimisation problem amenable to Excel s Solver, which contains a range of iterative search methods for optimization o An alternative approach is to determine portfolio weights to maximiwe expected return for a specified level of risk 16

17 Portfolio allocation in practice If there are no constraints on individual asset weights, the efficient frontier can also be produced elegantly from algebra Huang and Litzenberger (1998, HL) have described how to find two points on the frontier, and then to generate the whole of the frontier from these points The HL method for finding efficient portfolios requires the inverse of the variance covariance matrix, which is written as V t-1 1 In order to find two frontier portfolios (labelled g t-1 and g t-1 +h t-1), HL start by calculating four scalar quantities (A, B, C and D ): where u is a vector of ones Define 2 intermediate column vectors The expressions simplify to: t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 t-1 The two points on the envelope, namely portfolio g and g + h are special: g has an expected return of 0% and g + h of 100% 17

18 Portfolio allocation in practice Their formulas are: (the time index of the conditioning has been dropped) HL show that using the vectors g t-1 and h t-1, a linear combination of them in the form g t-1 + H t-1 E[R*], where E[R*] is the target mean, can be constructed to give the weights for a portfolio on the frontier producing any specified expected return, E[R*] If one changes E[R*] starting with the expected return of the GMPV and increasing the target in steps, the entire efficient set results If there are constraints on the weights (e.g., weights 0), the analytical solution does not apply However, Solver will produce the optimum set of weights By repeating the use of Solver for a number of different target expected returns we can points on the constrained frontier 18

19 Summary and conclusions If one tries and base optimal portfolio choice on historical estimates of means, variance and covariances, too many parameters need to be estimated Often these may even exceed the number of available data points! Therefore simplifications need to be introduced that consist of imposing structure on the correlation matrix One option consists of single-index models, often based on projecting individidual asset returns on the returns on a market portfolio, often a large stock index However, in the industry, multi-factor models that supplement the market index with additional economic indices have become popular In multi-factor models, the factors are often either orthogonalized or consist of principal components Two widespread possibilities consist of industry-based and macro economic factor based models 19

20 Appendix A: Building orthogonal factors We illustrate the procedure with a two-index model: For example, I * 1 might be a market index and I * 2 a sector index (e.g., aggregate index for all firms producing capital goods) If two indexes are correlated, the correlation may be removed from either index Define I 1 as equal to I * 1; to remove the impact of the market from the second index, estimate by a regression the parameters 0 and 1 of: By standard regression analysis results, d t is uncorrelated with I 1 Hence d t = I * I 1 is an index of the performance of the sector index with the effect of I 1 (the market) removed Solving for I * 2 and substituting into the return equation yields Re-arranging terms gives 20

21 Appendix A: Building orthogonal factors The first term is a constant we define as a i ; the coefficient on the second term is a constant we define as b i1 Now let b i2 = b * i1; then this equation becomes where I 1 and I 2 have been defined so that they are uncorrelated, and we have accomplished the task If the model contained a third index, for example, an industry index, then this index will be made orthogonal to the other two indexes by running the following regression: The index I 3 could be defined as This leads to a three-index model with uncorrelated indexes of the form 21

22 Appendix B: A quick review of matrix algebra In algebra, rectangular arrays of numbers are referred to as matrices A single column matrix is usually called a column vector; similarly a single row matrix is called a row vector x y x is a column vector and y a row vector; matrix A has three rows and three columns and hence is a square matrix; B is not square since it has four rows and three columns, i.e. B is a 4 by 3 matrix Transposition of a matrix converts rows into columns (and vice versa) The transpose of column vector x will be a row vector, denoted as x T Adding two matrices involves adding their corresponding entries; for this to make sense, the arrays being added must have the same dimensions (e.g., x and y cannot be added): A B 22

23 Appendix B: A quick review of matrix algebra To multiply vector y by 10 say, every entry of y is multiplied by 10: For two matrices to be multiplied they have to have a common dimension, that is, the number of columns for one must equal the number of rows for the other, e.g., In contrast, the product yx has dimensions (1x2) times (2x1), that is (1x1), i.e. it consists of a single element: These results demonstrate that for matrices, xy is not the same as yx, the order of multiplication is critical Considering two more matrices, 23

24 Appendix B: A quick review of matrix algebra However, the product DC cannot be formed because of incompatible dimensions (the number of columns in D does not equal the number of rows in C) In general, the multiplication of matrices is not commutative, so that usually CD DC A square matrix I with ones for all its diagonal entries and zeros for all its off-diagonal elements is called an identity matrix Suppose D is the (2x3) matrix used above, and I is the (2x2) identity matrix, then: Multiplying any matrix by an identity matrix of appropriate dimension has no effect o the original matrix (similar to multiplying by one) Suppose A is a square matrix of dimension n, an nxn matrix, then, the square matrix A -1 is called the inverse of A if: E.g., if: 24

25 Appendix B: A quick review of matrix algebra One use for the inverse of a matrix is in solving a set of equations such as the following: These can be written in matrix notation as Ax = b where: The solution is given by premultiplying both sides by the inverse of A: Not every system of linear equations has a solution, and in special cases there may be many solutions The set Ax = b has a unique solution only if the matrix A is square and has an inverse A 1 In general, the solution is given by x = A 1 b 25