Stock return risk structure

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Stock return risk structure

Liuren Wu

2021

Liuren Wuc (Baruch) Stock return risk structure 2021 1 / 8

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The mean-variance framework

Markowitz’s mean-variance framework forms the foundation of modern finance.

A veryintuitive framework for analyzing atypicalportfolio Number of independent bets

Effects of correlation Long-short portfolio

Forms the building block of asset pricing theory

Exceptions (tails/jumps/black swan) need special treatment (e.g., scenario/stress analysis)

For a long time, mean-variance portfolio optimization was rarely used in practice.

Academic critique: A simple average portfolio performs better out of sample than a mean-variance optimized portfolio.

Academic practice goes back to stone age: deciles, quintiles, HmL The issue: Mean/variance estimates are not trustworthy

Small estimation errors are amplified in matrix inversion, wt ∝ Σ⁻¹α

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The factor structure

One way to reduce estimation error in the covariance matrix is by imposing some factor structure

r = α + βf +p Σ_εε

so that stock return correlations are captured by the factor structure with the residual risk Σεshrunk to a diagonal matrix.

1 Principal component analysis, cluster analysis — Identify statistical structures (β, f)

2 Identify influential financial/macroeconomic indices as factors f (SPX, oil, rates,...) , usetime-series regressionto estimate loading β

3 Identify similarities in “observable” firm characteristics (Xt, e.g., industry, size, vol, beta, styles), and assume that firms with similar characteristics have similar risk loadings (β).

Cross-sectionally standardize Xt and regard them as βt

Performcross-sectional regressionof stock returns rt+1 against standardized characteristics βt. The slope estimate represents factor portfolio return ft+1. The residual is idiosyncratic risk√

Σεε.

The 3rd approach has been widely adopted in the industry nowadays for portfolio construction and P&L attribution. (BARRA, Bloomberg,...)

Commercial packages: BARRA, Bloomberg, ...

Many factors found by academics (“styles”) are used as

“characteristics” to build robust covariance matrix

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The cross-sectional stock return risk factor structure

r_t+h= β_tf_t+h+p Σ_εε_t+h

Barra characteristic choices (βt) — this is constantly evolving

1 Market/country, intercept

2 Industries(∼ 62): 0, 1, or fraction

3 Style (∼ 21): each is cross-sectionally standardized (winsorized).

Barra style exposures are mostly from academic findings on characteristics that predict future returns on average, e.g., market, size, value, growth, momentum, sentiment...

One should also identify exposures that have high explanatory powers at each period, but the factor returns ft+h have small average values.

Industry classification serves this purpose, maybe there are also style exposures that can serve similar purpose.

Identification: Time-series averages of R²(or absolute t-values) are high, but average factor returns may not be significant.

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Barra factors on USMEDS

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The cross-sectional stock return risk factor structure

rt+h= βtft+h+p Σεεt+h

Econometrics

r_t+h can be gross or excess return over different horizons

In principle, one can use the same factor structure (βt), but estimate the factor return over different horizons

But it is also possible that the best factor structure can change for different investment horizons

Weighting (W ): Square root of market capitalization

Constraints: P Wgf_t+1^g = 0 so that the intercept (market) captures the average market return whereas the industry dummy captures the excess return of each industry net of market

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Asset pricing from two perspectives

The pricing of a security can be analyzed from two perspectives:

1 How much should an asset/security be priced/valued at, given its projected future cash flows and risks?

Generate fair valuation and compare with themarket price.

2 How much return should an investor ask for investing in a security?

The higher the risk, the higher should be therequired return.

The task: Define/classify risk types and determine the pricing per unit of each type of risk, i.e.,market pricing of risk.

Theory: CAPM, APT

Empirical work: Identify (test?) factors (or firm characteristics as proxies of risk exposures) that is highly priced by the market Invest decisions from two perspectives

1 Buy cheap and sell expensive based on valuation

Alternatively: Compute implied cost of capital (ICC) from market price, and compare with required return (ECC) from risk estimates

2 Target exposures to highly compensated risks

Long equity market; sell stock index options; sell out-of-the-money puts on index or highly levered stock; sell (credit) insurance

Combining stock valuation with Barra-type return risk structure allows one to target either or both.

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Comparing the factor structures for value and return

Value : qt = ϕtct−√ Σeit

Return : rt+h = βtft+h+√ Σεεt+h

The valuation model is a contemporaneous relation, whereas the return model is a forecasting regression.

The valuation relation has much stronger explanatory power (R²).

Identification should be easier.

The value pricing ct should be more persistent over time, whereas the factor returns contain a lot more transient noise.

Tradingenhancesthe valuation relation, but candestroypredictions of

“return abnormality.”

The construction process is similar. The mechanisms can be related.

CAPM/APT: If a risk exposure β_t induces a large positive risk premium, we should embed this into the discount rate for valuation.

Risks and risk premiums can also interact with cash flows:

Firms, just like investors, should take more risk for better opportunities.

Risk premium ∝ growth rate... in the long run (on average)?