Work out beta, the adjustment factor in a regression hedge.

FRM® Part II · Market Risk Measurement and Management · Regression Hedging and Principal Component Analysis · FRM2.MR.11.c

The idea

Beta is the slope of the regression of target yield changes on hedge yield changes. Write the covariance of target and hedge yield changes as Cov, and the variance of hedge yield changes as Var. In terms of those two quantities, the slope is the covariance divided by the variance. Beta = Cov ÷ Var. Using the correlation ρ and volatilities σT and σH, the slope is also as follows. Beta = ρ × (σT ÷ σH). Volatilities are of yield changes, in basis points, over the same horizon. Beta says how many basis points the target yield tends to move for a one basis point move in the hedging yield. Take correlation 0.8, target volatility 5 bp and hedge volatility 8 bp. Beta = 0.8 × (5 ÷ 8) = 0.50. If you are given R², take its square root to recover the correlation, keeping the sign of the slope. The usual errors are inverting the volatility ratio, leaving out the correlation, and using R² where the correlation belongs.

Formulas

Regression hedge beta from correlation and volatilities

β = ρ × (σT ÷ σH)

ρ: the correlation between target and hedge yield changes
σT: the volatility of the target's yield changes
σH: the volatility of the hedging instrument's yield changes

Volatilities are of yield changes over the same horizon. Inverting the ratio, or using R² in place of ρ, are the usual slips.

Regression hedge beta from covariance and variance

β = Cov ÷ Var

Cov: the covariance of target and hedge yield changes
Var: the variance of the hedge yield changes

Divide by the variance of the hedging instrument's changes, not the target's. Stopping at the covariance is the other slip.

A practice question

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

In a regression hedge, beta is the slope from regressing target yield changes on hedge yield changes. Which expression gives it?

Correlation x (volatility of hedge yield changes / volatility of target yield changes).Why not: Inverts the volatility ratio, which is the slope for the reverse regression.
Volatility of target yield changes / volatility of hedge yield changes.Why not: Leaves out the correlation, which equals the slope only when the yields are perfectly correlated.
Correlation x (volatility of target yield changes / volatility of hedge yield changes). - correct
Correlation x volatility of target changes x volatility of hedge changes.Why not: This is the covariance; it has not been divided by the variance of the hedge yield changes.

Beta = cov(target, hedge) / var(hedge) = rho x sigma_target x sigma_hedge / sigma_hedge² = rho x sigma_target / sigma_hedge.

A harder one

Daily yield changes have a standard deviation of 7.5 bp for the target and 9.0 bp for the hedging instrument, and the correlation is 0.85. Beta for the regression hedge is closest to:

0.71. - correct
1.02.Why not: Uses correlation x (hedge volatility / target volatility), the inverted ratio.
0.83.Why not: Uses the volatility ratio 7.5/9.0 and omits the correlation.
57.4.Why not: Multiplies correlation by both volatilities, the covariance, and does not divide by the hedge variance.

Beta = correlation x target volatility / hedge volatility = 0.85 x 7.5 / 9.0 = 0.7083.

The hardest tier

A regression of daily changes in a target yield on daily changes in a hedging yield, estimated over two years, reports R² of 0.64, a positive slope, and daily standard deviations of 5.2 bp (target) and 6.8 bp (hedge). The beta used in the regression hedge is closest to:

0.49.Why not: Multiplies R² (0.64) by the volatility ratio instead of the correlation, which is the square root of R².
1.05.Why not: Uses the correlation 0.8 times the inverted ratio 6.8/5.2.
0.76.Why not: Uses the volatility ratio alone, forgetting to bring in the correlation (the square root of 0.64).
0.61. - correct

Correlation = sqrt(0.64) = 0.80 (positive slope). Beta = 0.80 x 5.2/6.8 = 0.612.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →