Mean-Variance Theory at Work: Single and Multi-Index (Factor) Models

Mean-Variance Theory at Work: Single and Multi-Index (Factor) Models Prof. Massimo Guidolin Portfolio Management Spring 2017

Outline and objectives The number of parameters in MV problems and the curse of dimensionality o Guidolin-Pedio, chapter 5, sec. 5.1 Using asset pricing, factor models to win the dimensionality curse o Guidolin-Pedio, chapter 5, sec. 5.1 Single-factor models o Guidolin-Pedio, chapter 5, sec. 5.2 Multi-factor models and factor-mimicking portfolios o Guidolin-Pedio, chapter 5, sec. 5.3 2

Key Concepts/1 Mean-variance analysis is subject to a dimensionality curse because it normally requires the estimation of 0.5N 2 + 1.5N inputs/parameters These are N means, N variances, and 0.5N(N-1) correlations The curse means that occasionally it may occur that the implementation of MV analysis requires one to estimate more parameters than available observations, which is impossible The use of high-frequency data tends to create more problems (related to the econometrics of the estimation of parameters) than it solves When inputs are produced by financial analysts, they tend to specialize by firms or at least sectors, so that cross-sector, or even cross-country correlations are hard to get Because of such difficulties, financial economists look for benefits in deriving parameters from asset pricing models, factor models 3

Key Concepts/2 The general structure of a factor model is: If a factor model contains M factors, with N assets or portfolios, it is convenient to resort to asset pricing models iff: MN + N << 0.5N 2 + 1.5N The most commonly used asset pricing framework is the market (single-index) model, where the only factor is the market ptf.: The economic content is that different assets co-move because they relate in the same way with general market movements Under additional assumptions, the market model becomes the famous CAPM: Note that deriving inputs from the CAPM does not imply the passive, market-based investment policy that the CAPM advises 4

Key Concepts/3 It is then easy to go from the estimates of the parameters of the market model to the inputs required by MV applications: These expressions require the estimation of 3N + 2 parameters (the +2 comes from E[R m,t+1 ] and Var[R m,t+1 ]) 3N + 2 << 0.5N 2 + 1.5N very easily, one can show for N > 4 When N = 100, the one-factor model implies a need to estimate 302 parameters, while the standard approach that separately calibrates means, variances, and covariances, implies that 5,150 different inputs need to be provided, i.e., 17 times as many! 5

Key Concepts/4 An obvious alternative is to use multi-index, multi-factor models The idea is to expand the factors to include systemic, aggregate macroeconomic or industry differences Typically the factors on the RHS consists of shocks (non anticipated news) to either macroeconomic or sectorial variables Because macro indices are rarely tradable (apart from inflation, and yield curve-related factors), often portfolio of assets (stocks) that mimic factors and highly correlated with them are set up and used Such portfolios often short assets with max negative exposure to the factor and go long in high, positive exposures and may be selffinancing, zero-cost ones 6

Key Concepts/5 The way in which a multi-index model helps with the curse is similar to single-index models under factor orthogonality: These estimation of (K+2)N + 2K = (2 + N)K + 2N parameters To save parameters and efforts is easy because (2 + N)K + 2N grows linearly in N, differently from 0.5N 2 + 1.5N which grows at a quadratic rate 7

Key Concepts/6 The general multi-index model bears close links to the APT (Arbitrage Pricing Theory) developed by Ross (1976) The APT is a no-arbitrage model that restricts asset risk premia to prevent the possibility of arbitrage strategies Modern implementation of the APT use (among many) also investment style factors a la Fama and French: size, value, momentum, and recently volatility-sorted portfolios Recent literature and practice have emphasized the usefulness of controlling not the optimal weights to different assets Instead need to go from optimizing factor exposures to optimal portfolio selection While financial economics has given us a very sensible and mathematically rigorous definition of what risk aversion is, the notion of risk aversion with respect to any special risk factor is hard to pin down 8

Inputs to MV analysis: the curse of dimensionality The efficient frontier mapping expected returns, variances, and covariances of (among) all the N 2 securities/assets into expected returns and variance of all possible portfolios, for some weights {w j } Simple accounting reveals that formulas N+N+N(N-1)/2 parameters o N means, E[R i ], i = 1, 2,, N o N variances, σ 2 i, i = 1, 2,, N o N(N 1) correlations, ρ ij, i < j = 1, 2,, N Estimating N[2 + 0.5(N -1)] = 0.5N 2 + 1.5N parameters poses 2 issues: 1 The first is purely quantitative, they are a lot of parameters! Often we may even lack a sufficient number of observations to make estimation possible, aka curse of dimensionality o E.g., with N = 25 assets/ptfs./indices, 0.5(25 2 ) + 1.5x25 = 350 inputs o How many will be needed for such a task? Assume a minimal required saturation ratio of 20 9

Inputs to MV analysis: the curse of dimensionality o The saturation ratio is the ratio between the total number of observations across all assets/securities and the number of parameters/inputs o With N = 20, then 350 x 20 = 7,000 observations, i.e., 280 obs. per asset o At a monthly frequency, this implies access to time series that exceed an overall length of 23 years, not a negligible data requirement The plot visualizes the trade-off between the realism of the size of the asset menu (N) and the data requirements to achieve a reliable estimation of MV inputs The number of inputs required increases in a convex fashion For ptf. Mgmt purposes, most institutions follow at least between 150 and 250 stocks When N = 200, the number of parameters >> 20,000 10

Inputs to MV analysis: the curse of dimensionality It is occasionally possible to resort to higher frequency time series o E.g., imposing again a saturation ratio of 20, because there are 52 weeks in a year, the number of years of weekly observations required to provide adequate inputs to a MV problem is o Even using 20-year long weekly time series, the largest admissible asset menu includes only 87-88 different assets The use of daily data appears to be uncommon in asset management High-frequency data (especially daily) imply complex technicalities 2 The structure of security research industry implies that analysts specialize by security, so that while they provide expected return and risk forecasts, unclear where correlations may be coming from o E.g., one analyst specializes in automotive and another one in bank stocks o While they may be both skilled in yielding precise forecasts of mean and variance of the returns on the stocks they cover, to them producing forecasts of correlations between pairs of such stocks represents a harder challenge because it requires an understanding of how stocks interact 11

Inputs to MV analysis: the role of asset pricing models Because of these difficulties, financial economists look for support in asset pricing models to estimate means, variances, and correlations A model is defined is of a no-arbitrage type, when the relationships pinning down the dynamics of asset returns can be simply derived by imposing that no prices can cause the existence of arbitrage o Arbitrage: when a riskless, zero net investment strategy may be set up to yield sure profits A model is as of an equilibrium type, when asset prices are directly or indirectly derived from equating the supply of assets to their demand Fortunately, the randomness displayed by the returns of risky assets often can be traced back to a smaller number of underlying sources of risk (often called factors) that influence individual returns A factor model that represents this connection between factors and returns leads to a simplified structure for the covariance matrix: f( ) is a generic return generating function that maps K factors (also called indices) into estimates/forecasts of asset returns (i = 1, 2,, N) 12

Inputs to MV analysis: the role of asset pricing models The εε ii,tt+1 are additive, idiosyncratic shocks that are assumed to be orthogonal to the component captured the function of indices, so that The coefficients θθ ii,1, θθ ii,2,, θθ ii,mm are M asset-specific parameters typically inferred (estimated) from the data What is the benefit of switching from the initial 0.5N 2 + 1.5N input set up to the asset pricing model? Assuming that, the answer is a purely quantitative one: the shift will be advantageous when MN + N << 0.5N 2 + 1.5N, i.e., when the models imply the estimation of many less parameters than the inputs that would otherwise be required by a straight implementation of the MV framework o M is the number of factors o N is the number of assets/ptfs. for which estimation has to be performed o It is not only MN, but +N because of the N variances: 13

Inputs to MV analysis: single-index model Given N, it is then sufficient to set out models with a modest number of factors M << 0.5N + 0.5, for the advantage of using factor models to be considerable o E.g., when N = 25, the factors need to be below 13 and simple models with at most 3-4 factors will imply considerable reduction of parameters o To be precise: a 4-factor model implies estimating 4x25 + 25 = 125 parameters vs. 0.5(25 2 ) + 1.5x25 = 350 inputs without the factor model The first class of models are single-index (aka market) models: o R m,t is the return on the market portfolio o β i is the beta that measures the loading of returns on the specific asset i on market ptf. returns, i.e., the reaction of R i,t to a unit change in R m,t o The function f( ) in this case is linear affine o We also assume idiosyncratic risk conveys no information on market risk o Such an (orthogonality) assumption on the residuals makes OLS estimation natural (e.g., using the Excel regression tool) 14

Inputs to MV analysis: single-index model Using the linear properties of expectation operators: These expressions require the estimation of 3N + 2 parameters (the +2 comes from E[R m,t+1 ] and Var[R m,t+1 ]): 3N + 2 << 0.5N 2 + 1.5N While predicted expectations and variances not only depend on market risk, but also display an asset-specific component (α ii and σ ii ), the covariance between a pair of assets only depends on market risk Assets co-move as a result of a common response to the market o The (absolute value of this) expression degenerates to 1 when σ ii = σ j = 0 15

Inputs to MV analysis: single-index model These expressions can be used to gain insight into how far one can push diversification: This implies that risk cannot be simply eliminated by holding larger and larger ptfs. based on equal weights Note that when idiosyncratic risk is bounded, even when N, portfolio risk does not disappear and equals exposure to market risk With increasing frequency, analysts that specialize in specific stocks or asset classes tend to supply not only estimates of expected return and predicted risk, but also betas vs. general market movements Betas can be purchased through subscriptions to data services 16

Inputs to MV analysis: single-index model When there is a riskless asset and a few more perfect, frictionless markets assumptions, the single-index model turns into the celebrated CAPM: o In essence, the intercepts should be equal to the riskless rate for all assets o Or, the (unconditional) risk premium on all assets is proportional to the market risk premium Although the CAPM is used to derive the inputs to MV analysis, CAPMstyle implications for portfolio management are not taken seriously o Under the CAPM strong assumptions, all investors are better off holding the market ptf. Vs. individual stocks according to optimal weights o Investors can diversify away the idiosyncratic part and increase their returns by simply holding the market factor portfolio o Therefore under CAPM, no MV analysis can be useful the optimal risky ptf. is just the market, normally through index mutual funds and ETFs that passively replicate wide market indices Under a single-index model, whether and by how much an asset ought to be included in a MV ptf. depends on its Treynor ratio 17

The single-index model and the Treynor ratio s rule The Treynor ratio is (for β i 0): o If an asset is included in an optimal portfolio, all stocks with a higher ratio will also be included o Conversely, all stocks with lower ratios than a stock excluded from an optimal ptf. will be excluded (or if short selling is allowed, sold short) o How many stocks are selected depends on a unique cut-off rate: o CC * jj is the only cut-off that, when used as a cut-off rate, selects only the assets used to construct it, for example: 18

The single-index model and the Treynor ratio s rule o The optimal MV portfolio ought to include only the securities, A and C o When short sales are instead admitted, all the securities will be traded: while securities A and C are held long, assets B, C, and D are to be shorted It is well-known that CAPM is rejected by most data on a variety of asset classes (not only the cross-section of US stock returns, in fact) The natural alternative is multi-index models, an attempt to capture some of the non market influences (macro)economic factors or structural groups (industries) that move securities together o No real opposition btw. them and single-index ones, as also multi-factor models are expected to include the market portfolio returns As the number of factors grows the parsimony advantage offered by index models over 0.5N 2 + 1.5N, declines 19

Inputs to MV analysis: multi-index models The general structure of a K-factor model is: o The first index II 1,tt+1 often corresponds (although it does not have to) to the return on the market portfolio o The betas measures asset exposures to each of the K factors o E.g., stocks of cyclical firms (say, producers of TV plasma screens) will have larger betas to the growth in GDP factor than noncyclical firms will, such as grocery store chains o Likewise, you will hear about interest-sensitive stocks, like real estate o Typical examples of indices are macroeconomic variables such as the (surprises in, i.e., the unanticipated shocks to the) rate of growth of industrial production, the change of the unemployment rate, the change in consumer or production price inflation, the riskless term spread implied by government bond yields, default spreads implied by corporate yields (i.e., the spread between junk and investment grade yields), etc. Optimal Portfolio Selection in a MV Framework 20

Inputs to MV analysis: multi-index models There are appropriate statistical techniques (factor and principal component analyses) to infer directly from the data a set K of indices o E.g., a number K such that most of the variability of a given set of macroeconomic indicators can be explained o Most literature has found that between 3 and 5 principal components can explain a significant portion of the variance of stock returns o PCA has been criticized because it is common to find that in many cases, the principal components lack a clear economic interpretation Unfortunately, macroeconomic variables are not tradable Hence common to use returns on (factor-mimicking) ptfs. of securities (stocks) that are first sorted on the basis of their correlation with each given factor, and then turned into (long-short) spreads o Spreads are typically based on going long in the top decile(s) and short in the bottom ones, with regular updating of what the deciles include o Why spreads? Long-short dynamic strategies are market-neutral o However, this requires shorting a large number of assets, which may difficult or costly Optimal Portfolio Selection in a MV Framework 21

Inputs to MV analysis: multi-index models The model has convenient properties when indices are uncorrelated o It is always possible to take any set of correlated indices and convert them into a set of uncorrelated indices o Once more, the covariance between a pair of assets only depends on their betas, although correlations do not: Optimal Portfolio Selection in a MV Framework 22

Inputs to MV analysis: multi-index models These equations estimation of (K+2)N + 2K = (2 + N)K + 2N parameters o N intercept coefficients, β 0 i, i = 1, 2,, N o NK asset betas measuring the exposure to each of the K factors in the model, β k i, k = 1, 2,, K i = 1, 2,, N o N measures of idiosyncratic variance/risk, σ 2 i, i = 1, 2,, N o K factor-specific means E[I k,t+1 ] and K factor-specific variances Var[I k,t+1 ] Also in this case, (2 + N)K + 2N grows linearly in N, differently from 0.5N 2 + 1.5N which grows at a quadratic rate Given a selection of K, compute for which minimal value of N it becomes advantageous to use a K- factor model over no model: Optimal Portfolio Selection in a MV Framework 23

Inputs to MV analysis: multi-index models o E.g., in a 4-factor model, it takes a minimum N = 11 assets for the number of inputs implied, 2 x 4 + 11 x 4 + 2 x 11 = 74 to be inferior to the inputs in a MV framework, 0.5N 2 + 1.5N = 0.5(112) + 1.5 x 11 = 77 o Obviously, in the same model, as the number of assets grows, the advantage increases: with N = 100, 2 x 4 + 100 x 4 + 2 x 100 = 608 is much less than 0.5(1002) + 1.5 x 100 = 5,150 parameters The general multi-index model bears close links to the APT (Arbitrage Pricing Theory) developed by Ross (1976) The APT is a no-arbitrage model that restricts asset risk premia to prevent the possibility of arbitrage strategies De-meaned, orthogonalized factors Return on factormimicking ptf. o The APT obtains under weaker assumptions than the CAPM Optimal Portfolio Selection in a MV Framework 24

Inputs to MV analysis: multi-index models Some famous factors that have received a clear interpretation either behavioral, i.e., not necessarily based on the theory of efficient and maximizing investors, or rational in an APT perspective are: o Inflation rate o Rate of growth of the economy (e.g., industrial production, aggregate sales, employment numbers or the unemployment rate change) o Volatility (e.g., as captured by the option-implied VIX index, characterized by a negative price of risk) o Real productivity risk (e.g., as measured by total factor productivity) o Demographic risk (i.e., shocks to a birth rate index) o Market portfolio o Size, value, and momentum factors, the investment style factors by Fama and French (1993)-Carhart (1997), aka smart beta factors Recent literature and practice have emphasized the usefulness of controlling not the optimal weights to different assets Instead need to go from optimizing factor exposures to optimal portfolio selection Optimal Portfolio Selection in a MV Framework 25

Inputs to MV analysis: multi-index models While financial economics has given us a very sensible and mathematically rigorous definition of what risk aversion is, the notion of risk aversion with respect to any special risk factor is hard to pin down o For instance, factor allocation usually produces returns that are highly left-skewed, i.e., they can occasionally produce some very large losses o However, how such asymmetries in the distribution of portfolio returns might be traded-off vs. the potential increase in Sharpe ratios that factor models make possible, remains complex o The tension between asset vs. risk factor-based allocations derives from the existence of some informational spread between the two problems o Idrozek and Kowara (2013) prove that when there are as many risk factors as assets and hence the risk factors perfectly explain the returns of asset classes, there is no inherent advantage from either approach: we can start with either risk factors or asset classes, derive an optimal portfolio, and move from one space to the other with no gain or loss o However, when there are more assets than risk factors and hence the latter cannot provide a perfect fit to the former, a difference may exist Optimal Portfolio Selection in a MV Framework 26