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Describe what goes into a multifactor model, factor betas included, and explain why using such models for hedging is difficult.

FRM® Part I · Foundations of Risk Management · The Arbitrage Pricing Theory and Multifactor Models of Risk and Return · FRM1.FR.6.b

The idea

A multifactor model writes an asset's return as a constant, plus a beta times a factor for each factor, plus a residual. R = a + b1 × F1 + b2 × F2 + … + bk × Fk + e. The factors are systematic drivers such as market return, GDP growth, inflation or interest rates, usually entered as surprises. Each factor beta is the expected change in the asset's return for a one-unit change in that factor, holding the others constant. It is estimated by regression. The residual e is idiosyncratic, uncorrelated with the factors and diversifiable. The model should explain risk, but hedging with it is hard. Betas are estimated with error and drift over time. Factors may be correlated. Macroeconomic factors are not directly tradable, so proxies leave basis risk. A hedge instrument usually loads on several factors at once. Exam point: a hedge is only as good as the beta estimates and the instruments behind it.

A practice question

From the question bank, with the reason each wrong option is wrong.

In a multifactor model of asset returns, an asset's beta with respect to a given factor measures:

  1. the asset's total volatility divided by the volatility of the factor.Why not: That ratio ignores correlation and the other factors; a beta is a regression slope, not a volatility ratio.
  2. the correlation between the asset's return and the factor.Why not: Correlation is bounded by -1 and +1 and is unit-free; a beta carries the units of return per unit of factor.
  3. the expected change in the asset's return for a one-unit change in that factor, holding the other factors constant. - correct
  4. the risk premium that investors earn for bearing exposure to the factor.Why not: Confuses beta (the sensitivity) with the factor risk premium (the price per unit of exposure).

Return = alpha + b1 x F1 + ... + bk x Fk + e. Each beta is the partial sensitivity of return to one factor with the others held fixed; expected return is then the risk-free rate plus sum of beta times premium.

A harder one

A stock's annual return volatility is 25%. A multifactor regression of its returns on the systematic factors has an R² of 0.64. The volatility of the stock's residual (idiosyncratic) return is closest to:

  1. 15.0% - correct
  2. 9.0%Why not: Multiplies volatility by (1 - R²); variances, not volatilities, add up in proportion to R².
  3. 20.0%Why not: This is the systematic volatility (sqrt(R²) x 25%), not the residual.
  4. 16.0%Why not: Multiplies total volatility by R²; the proportions apply to variance, so a square root is needed.

Residual variance = (1 - R²) x total variance, so residual volatility = 25% x sqrt(0.36) = 15.0%. Systematic volatility is 25% x sqrt(0.64) = 20%; the two combine as 15² + 20² = 25².

The hardest tier

A manager hedges a portfolio's market-factor exposure by selling equity index futures sized to the portfolio's market beta, estimated in a regression that also includes an interest-rate factor. The futures contract itself has a significant loading on the interest-rate factor. Which is the main hazard of this approach?

  1. The futures eliminate idiosyncratic risk as well, so the hedged portfolio becomes over-hedged.Why not: Futures do not remove or add idiosyncratic risk in the way described; the problem is cross-factor loading.
  2. The market factor is not tradable, so index futures cannot be used to hedge it.Why not: Equity index futures are the standard tradable proxy for the market factor.
  3. Under the APT, hedging a factor automatically sets the portfolio's alpha to zero.Why not: Hedging changes factor exposure, not alpha; alpha is the intercept left after the factor terms.
  4. The futures add an unintended interest-rate exposure, so hedging one factor shifts exposure to another; all factor exposures must be hedged jointly. - correct

Real hedging instruments rarely load on only one factor. Selling index futures to neutralise the market beta also changes the interest-rate beta, so the hedge ratios for all factors must be solved together, using reliable estimates of each instrument's factor loadings.

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