Example 8.4 (Default Probability and Default Threshold) Suppose a firm has β(i) = 0.4 and k(i) = −2.33, it is a middling credit, but cyclical (relatively high βi). Its unconditional probability of default is Φ(−2.33) = 0.01. If we enter a modest economic downturn, with m = −1.0, the conditional asset return distribution is N(−0.4, sqrt[1 − 0.402]) or N(−0.4, 0.9165), and
the conditional default probability is found by computing the probability that this distribution takes on the value −2.33. That probability is 1.78 percent.
If we were in a stable economy with m = 0, we would need a shock of −2.33 standard deviations for the firm to die. But with the firm’s return already 0.4 in the hole because of an economy-wide recession, it takes only a 1.93 standard deviation additional shock to kill it.
Now suppose we have a more severe economic downturn, with m =−2.33. The firm’s conditional asset return distribution is N(−0.932, 0.9165) and the conditional default probability is 6.4 percent. A 0.93 standard deviation shock (ϵ(i) ≤ −0.93) will now trigger default [dharper: I disagree with this value]