In a one-factor model, characterise the correlations among variables that are normally distributed.

FRM® Part I · Quantitative Analysis · Measuring Returns, Volatility, and Correlation · FRM1.QA.12.g

The idea

In a one-factor model each variable is built from a common factor F and its own idiosyncratic term Zi. Xi = ai × F + √(1 − ai²) × Zi. The factor and every Zi are standard normal and independent of one another. Each Xi is then standard normal, ai is its loading on the factor, and ai² is the share of its variance explained by F. The correlation between two variables is the product of their loadings. Correlation = ai × aj. Conditional on F, the variables are independent, so all the correlation comes through the common factor. The model needs only n loadings instead of n(n − 1) ÷ 2 pairwise correlations, which makes large correlation matrices manageable and always consistent. It underlies Gaussian copula models of default correlation. Do not add the loadings or square their product.

Formulas

Variable in a one-factor model

Xᵢ = aᵢ × F + √(1 − aᵢ²) × Zᵢ

aᵢ: the loading of variable i on the common factor
F: the common factor, standard normal
Zᵢ: the idiosyncratic term of variable i, standard normal and independent of F and of every other Z

The coefficient on Zᵢ is chosen so that Xᵢ keeps unit variance; aᵢ² is the share of variance the factor explains.

Correlation between two variables in a one-factor model

ρᵢⱼ = aᵢ × aⱼ

aᵢ: the factor loading of variable i
aⱼ: the factor loading of variable j

All correlation comes through the common factor, so conditional on F the variables are independent. Do not add the loadings or square their product.

A practice question

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

In a one-factor model, X_i = a_i F + sqrt(1 - a_i²) Z_i, with F and the Z_i independent standard normal variables. The correlation between X_i and X_j is:

a_i x a_j - correct
a_i + a_jWhy not: Adds the loadings, which could exceed 1.
(a_i x a_j)²Why not: Squares the product of the loadings; the correlation is the product itself.
a_i / a_jWhy not: Takes the ratio of loadings, which is not bounded by 1.

Cov(X_i, X_j) = a_i a_j Var(F) = a_i a_j because the Z terms are independent of F and of each other. Both variances equal 1, so the correlation is a_i a_j.

A harder one

A standard normal variable has a loading of 0.7 on the common factor in a one-factor model. The coefficient on its idiosyncratic standard normal term is closest to:

0.510Why not: Gives the idiosyncratic variance share, 1 - a², instead of the coefficient (its square root).
0.300Why not: Takes 1 - a as the coefficient.
0.714 - correct
0.490Why not: Gives the share of variance explained by the factor, a², rather than the idiosyncratic coefficient.

To keep unit variance, a² + b² = 1, so b = sqrt(1 - 0.49) = sqrt(0.51) = 0.714.

The hardest tier

In a one-factor model, X_i and X_j are standard normal, each with a loading of 0.6 on the factor F. Given that F takes the value -2, which description of X_i and the relationship between X_i and X_j is correct?

Conditional mean -1.2 and variance 0.64, with the conditional correlation still 0.36.Why not: Treats the unconditional correlation as persisting; once F is fixed the variables are independent.
Conditional mean -1.2 and variance 1.00, with X_i and X_j conditionally uncorrelated.Why not: Forgets that the factor explains 36% of the variance, which is removed once F is known.
Conditional mean -2.0 and variance 0.64, with X_i and X_j conditionally uncorrelated.Why not: Ignores the loading when computing the conditional mean (should be 0.6 x -2).
Conditional mean -1.2 and variance 0.64, with X_i and X_j conditionally uncorrelated. - correct

E[X_i | F = -2] = 0.6 x -2 = -1.2 and Var = 1 - 0.36 = 0.64. Given F, X_i and X_j depend only on independent Z terms, so their conditional correlation is 0.

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