Charta is on Google Play: practise on the go. Get the app
Charta

State what beta is and use the single factor model to work out the asset return correlation for any two firms.

FRM® Part II · Credit Risk Measurement and Management · Portfolio Credit Risk · FRM2.CR.11.e

The idea

Beta in the single factor model is the loading of a firm's standardised asset return on the common factor. It equals the correlation between that return and the factor. Beta squared is the share of the return variance explained by the factor. One minus beta² is the idiosyncratic share. Because the idiosyncratic shocks are independent, the only link between two firms is the factor. The asset return correlation therefore follows directly from the two firms' betas. Rho = βi × βj. With equal betas, rho is beta². A beta can be negative for a counter-cyclical firm, which gives a negative correlation. A regression slope in raw units is not the standardised beta. Divide the covariance with the factor by the product of both standard deviations. Typical errors are averaging the betas or multiplying their squares.

Formulas

Asset correlation of two firms in the single factor model

ρ = βi × βj

βi
firm i's standardised loading on the common factor
βj
firm j's standardised loading on the common factor

Holds because idiosyncratic shocks are independent. Averaging the betas or multiplying their squares are the usual errors.

Standardised beta from a covariance

β = Cov(R, M) ÷ (σR × σM)

Cov(R, M)
the covariance of the firm's asset return with the factor
σR
the volatility of the firm's asset return
σM
the volatility of the factor

This is the correlation with the factor. Dividing by the factor variance alone gives the raw regression slope, which is not the model's beta.

A practice question

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

In the single factor model, what is beta for a given firm?

  1. The loading of the firm's standardised asset return on the common factor, equal to the correlation between the two. - correct
  2. The share of the firm's asset return variance explained by the common factor.Why not: That is beta², not beta itself.
  3. The correlation between the asset returns of two different firms.Why not: That is the product of the two firms' betas.
  4. The firm's default probability expressed relative to the average default probability.Why not: Beta is a sensitivity to the factor and has no direct link to the PD level.

Beta measures how strongly the firm's asset return moves with the factor. Its square is the variance share explained, and the product of two betas is the asset correlation of the two firms.

A harder one

Two firms in the same sector have equal betas to a single factor, and the correlation between their asset returns is 0.36. Each firm's beta is:

  1. 0.36Why not: Treats the asset correlation as the beta.
  2. 0.60 - correct
  3. 0.13Why not: Squares the correlation instead of taking its square root.
  4. 0.18Why not: Divides the correlation by two, as if it were a sum of two betas.

With equal betas, rho = beta², so beta = sqrt(0.36) = 0.60.

The hardest tier

Three pairs of firms have the following betas to a single factor: pair X (0.74, 0.74), pair Y (0.97, 0.54) and pair Z (0.99, 0.30). Which statement is correct?

  1. Pair Y has the highest asset return correlation, because its betas have the largest sum.Why not: Ranks by the sum of betas; correlation is the product, which is 0.52 for pair Y.
  2. Pair Z has the highest asset return correlation, because it contains the largest single beta.Why not: Ranks by the largest beta; the product for pair Z is only 0.30.
  3. All three pairs have the same asset return correlation, because they share one factor.Why not: A shared factor does not equalise correlations; each pair's correlation is the product of its own betas.
  4. Pair X has the highest asset return correlation, at about 0.55. - correct

Correlations are 0.74 x 0.74 = 0.548 for X, 0.97 x 0.54 = 0.524 for Y and 0.99 x 0.30 = 0.297 for Z. Balanced loadings give a higher product than one large and one small.

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

Start free →

More in Portfolio Credit Risk