310.1. The single-factor model measures portfolio credit risk by assuming each firm (i = 1, 2, ...) has its own sensitivity to the common market factor. The sensitivity is denoted by beta (i), β(i), and the market factor is denoted by (m):³

Further, as qualified above, the market and idiosyncratic shock, e(i) are random standard normal variates that are uncorrelated with one another. Assume our single-factor portfolio contains only three credits with the following betas: β(1) = 0.35, β(2) = 0.40, β(3) = 0.56. What is the implied correlation directly between credits (1) and (2), ρ(1,2)?

$$a_i = \beta_i m + \sqrt{1 - \beta_i^2} \, \varepsilon_i \quad i = 1, 2, \dots$$

where

• $m, \varepsilon_i \sim N(0, 1)$
• $\text{cov}[m, \varepsilon_i] = 0$
• $\text{cov}[\varepsilon_i, \varepsilon_j] = 0$
• $i, j = 1, 2, \dots$