Advances in Risk Management

124 IDIOSYNCRATIC RISK, SYSTEMATIC RISK AND STOCHASTIC VOLATILITY

9000

8000

7000

6000

5000

4000

3000

2000

1000

0

0.51.5




0.5

2.251.5

3.753.00

4.50

6.005.25

7.006.58

8.507.75

10.009.25

Time to maturity (years)

Credit spreads (basis points)

Beta

Figure 6.4 Credit spread whenλ= 5

Tincreases from current timetto (t+τ 0 ). Moreover, current timetis equal to
(T−τ 0 ) andTwhenτis respectivelyτ 0 and 0. Such a behavior is consistent
with the work of Collin-Dufresne and Goldstein (2001) and Gemmill (2002).
Indeed, Gemmill (2002) exhibits an upward-sloping credit spread term
structure, which is consistent with Merton-type profiles provided that firms’
leverages exhibit a drift over time. Collin-Dufresne and Goldstein (2001) find
a convex decreasing shape (relative to time to maturity) for credit spreads of
firms with stationary leverages. Moreover, the convexity pattern describes
investment grade bonds whereas a concavity pattern describes speculative
grade bonds (junk bonds). Analogously to Collin-Dufresne and Goldstein

(2001), we consider the quasi-debt leverage ratiodt=Be

−rτ
Vt of (Merton, 1974).
Our log-leverage ratiodt=ln(dt) follows a stationary normal process such

thatdt∼N

(

σ^2 V(t,Vt,It)

2 dt,σ

2

V(t,Vt,It)dt

)

conditional onFt. The starting value

of our quasi-debt leverage is 11.23% under our assumptions. Thus, we con-
siderinvestmentgrade-typedebtinahighvolatilityframeworkwherefirm’s
global volatility stabilizes after the first five years following debt’s issue.
Notice that the first two moments describingdt’s lognormal law depend on
firm value’s global varianceσV^2 (t,Vt,It), which follows an asymptotically
mean reverting process. However, unlike Collin-Dufresne and Goldstein
(2001), our quasi-debt leverage is not mean-reverting, which allows to take