Risk Factor Investing ASSET PRICING AND PORTFOLIO CHOICE Dipartimento di Scienze Economico-Sociali e Matematico-Statistiche Anno Accademico 2014/2015.

Risk Factor Investing

ASSET PRICING AND PORTFOLIO CHOICE

Dipartimento di Scienze Economico-Sociali e Matematico-Statistiche

Anno Accademico 2014/2015

2Risk Factor Investing - A.A.2014/2015

Introduction

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Initial considerations (1)

Risk factors have long been used for risk and performance evaluation of actively managed portfolio (Sharpe 1983).

Recently a new approach know as factor investing has emerged, which recommends that allocation decisions be expressed in terms of risk factors, as opposed to standard asset class decomposition.

The main idea is that the most meaningful way for grouping individual securities is not by forming arbitrary asset class indices, but instead by forming replicating portfolios for a set of suitable designed asset pricing factor indices.

Factor indices explicitly seek an efficient exposure (diversification) to rewarded risk factors while diversifying away unrewarded risks.

Risk Factor Investing - A.A.2014/2015

Introduction

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Initial considerations (2)

In Sharpe’s CAPM there is a simple rewarded risk factor (market risk) so only the latter objective is relevant and the focus should be holding a well-diversified proxy of the market portfolio.

In a multi-factor world, where the risk premium is multi-dimensional (including not only market risk but also size, B/E, momentum, volatility, and so on) an important component of an investor’s investment process is the determination of the appropriate allocation to these rewarded risk exposures.


Introduction


Introduction

FACTORS from risk and performance evaluation tools to asset allocation tools

FACTOR INVESTING


Theoretical Background

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Fundamental Asset Pricing Equation

Any asset pricing model, including pricing models as diverse as Sharpe’s CAPM (1964) or Black-Scholes-Merton’s option pricing model (1973), boil down to a single equation, stating that the price p of any payoff x is the conditional expected value of the payoff with a stochastic discount factor (SDF) m:

The existence of a SDF is guaranteed by the absence of arbitrage, and the uniqueness by complete markets.



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Marginal Utility of Consumption

In an equilibrium model, we interpret the SDF as marginal utility of consumption:

• an asset that has a high expected return is an asset that tends to perform poorly in bad times, that is when consumption is low (marginal utility of consumption is high)

• for example, value stocks and small cap stocks outperform over multiple market conditions but/because they take a bigger dip down in the worst market conditions

• conversely, an investor is ready to pay a higher price and get a lower return to hold an asset that pays off well in bad times.



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Asset Pricing Factors

Formally, a factor model postulates that the SDF can be written as a linear combination of factors, called asset pricing factors.

One key implication is that expected excess returns are linear in the betas (APT line that generalized the security market line):



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Factor Fishing

So properly select a set of rewarded asset pricing factors is crucial in factor investing: FACTOR FISHING.

Two principles should always kept into consideration:

• EQUILIBRIUM: factors should proxy for growth in marginal utility

• ARBITRAGE: factors should explain cross-section of expected returns.

Although asset-pricing theory provides a sound rationale for the existence of multiple factors, it provides little guidance on which factors investors should expected to be rewarded.




Factor Selection and Exposure

Methodology

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The Factor Investing Process

A sound approach to factor investing requires the proper execution of three different steps:


Factor Selection and Exposure Methodology

1. Choice of factors that are rewarded in the long term

2. Designed factor-tilted portfolios capturing the fair risk-adjusted exposure to the factor

3. Choice of a methodology for deriving the optimal multi-factor exposures

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Factor Selection Criteria

The selection of rewarded factors should rely on both empirical and economic rational:

• first order condition for a factor to be deemed important is the existence of empirical evidence of its relevant effect on the cross-section of returns (Fama – MacBeth two-pass regression: time-series regression of asset returns on factors to get beta estimates, and then cross-sectional regression of returns on betas to get premia estimates)

• the economic intuition for the existence of a reward for a given risk factor is that the exposure to such factor is undesirable for the average investor, because it leads to losses in bad times when marginal utility is high (Cochrane 2001)



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Equity Factors: empirical evidence

The most common and well-know rewarded equity factors are four: size, value, momentum and low volatility.

• Fama and French (1993) showed that value (book to market) and size ( market cap) explain average asset returns, as a complement to the market beta.

• Carhart (1997) empirically proved the existence of another priced factor: the momentum factor.

• The low-volatility factor, which qualifies as an anomaly rather than as a risk factor, is the so-called “volatility puzzle”, which states that low-volatility stocks tend to outperform high-volatility stocks in the long-run (Ang et al. 2006).



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Equity Factors: historical performance



Amenc, Goltz, Lodh and Martellini (2014)

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Equity Factors: Economic Rational (1)

The debate about the existence of positive premia for these factors is far from closed and although they are documented in a extensive literature, some authors question the robustness or the persistence of the reward associated with these factors.

So the choice of relevant factors should always consider the economic rational behind the expected reward.

• Choi (2013) shows that value firm suffer from losses in bad times because they have increasing betas in down market, due to rising asset betas and leverage.

• Zhang (2005) argues that value firms are much more affected by the bad times because their stock prices are mainly made up tangible asset that are hard to reduce.



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Equity Factors: Economic Rational (2)

• Lakonishok et al. (1994) conclude that “value strategies exploit the suboptimal behavior of the typical investor”, that is, their psychological tendency to extrapolate recent developments into the future and to ignore evidence that is contrary to the extrapolation.

• Fama and French (1995) show that small stock, even after adjusting for book-to-market effects, tend to have lower return on equity and greater uncertainty of earnings and are therefore more sensitive to economic shocks.

• Frazzini and Pedersen (2014) provide a model in which liquidity-constrained investors are able to invest in leveraged position of low-beta assets but are forced to liquidate these asset in bad times, when their liquidity constraints can no longer sustain the leverage: so low-risk assets are exposed to liquidity shocks.



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Commonly Used Risk Factors

Even if by far more extensive empirical studies and literature have been produced about equity risk factors, few quite simple and straightforward rewarded factors can be identified for bond and commodities too.



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Factors Exposure MethodologyOnce that relevant rewarded factors have been indentified, the next step consist in designing factor tilted portfolios that capture the fair risk-adjusted reward, so called factor indices.

Factor indices may fall into three main categories:

1. cap-weighted indices, selecting stocks that are most exposed to the desired factor and applying a cap-weighting scheme to this selection

2. “factor-weighted”, maximizing the exposure to a factor, either by weighting the whole of the universe on the basis of the exposure to this factor (score/rank weighting) or by selecting and weight according to the exposure score of the stock to that factor.

3. Alternative (“smart”) methodology, like equally weighting, equally risk contribution, minimum volatility and so on.



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“Smart” Beta: an example

Amenc, Goltz, Lodh and Martellini (2014) proposed a diversified multi-strategy approach that combines five popular smart beta weighting schemes in equal proportions , letting investors diversifying away the non rewarded risk associated with each of the weighting scheme.

The five “smart” beta weighting methodologies are:

1. Maximum Deconcentration

2. Diversified Risk Weighted

3. Maximum Decorrelation

4. Efficient Minimum Volatility

5. Efficient Maximum Sharpe Ratio



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Diversified Multi-Strategy approach




Risk Factors in Allocation Decisions

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Asset and Risk Allocation to Risk Factors (1)

When a set of rewarded risk factors with their relative efficient exposure methodologies has been properly defined, a methodology to derive the optimal risk-factor exposure may be indentified.

Both a traditional mean-variance approach or a pure risk allocation model (like GMV or Risk Parity) but different dimensions should be always taken into consideration:

• factor allocation may be intra-asset class (for example combining value, P/E, momentum and volatility in an equity portfolio) or across different asset class (equity + bond + commodities)

• long-only or long-short factors can be used, depending on the desiderate factor exposure



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Asset and Risk Allocation to Risk Factors (2)

• long-only or long-short allocation to the factors can be achieved, depending on model’s constraints, operational constraints and availability of investable instruments

• factors can be added to the relevant investment universe, together with more traditional asset classes (beware of overlapping) or

• traditional asset classes can be substituted by a set of corresponding risk factors indices, better representing actually rewarded risk premia.

Whatever the mean goal, the end goal is always the same: construct a well-diversified portfolios.




References


Amenec, N., Goltz, F., Lodh, A. and Martellini, L. “Towards Smart Equity Factor Indices: Harvesting Risk Premia without Taking Unrewarded Risks” The Journal of Portfolio Management, Volume 4 Number 4, Summer 2014
