Advances in Risk Management

128 IDIOSYNCRATIC RISK, SYSTEMATIC RISK AND STOCHASTIC VOLATILITY

and ratios. In the same way, Phoa (2003) underlines the coherency of Merton-
type structural models with observed risky debt market data. This author
points out the usefulness of such models as risk management tools. In con-
trast, Eom, Helwege and Huang (2004) find that new simplified structural
models misestimate credit risk. Therefore, our stochastic volatility frame-
work can solve this problem by better fitting to empirical behavior of credit
spreads, and then reconciling all points of view. To investigate this issue,
future research should estimate the result of our model using risky debt
data, and then test its performance.
Our work’s significance and future implementation are of major impor-
tance for a sound assessment of credit risk. First, credit spreads and default
ratesarekeydeterminantsforbothpricingandhedgingofcreditinstruments
along with dynamic credit portfolio management. Second, as idiosyncratic
risk is diversifiable, systematic risk is more important at a portfolio level
(Jarrow, Lando and Yu, 2005; Frey and McNeil, 2001; Lucas, Klaassen, Spreij
and Straetmans, 2001; Giesecke and Weber, 2004). However, Goetzmann
and Kumar (2001) show the existence of many under-diversified portfo-
lios. Such portfolios are usually naively diversified and bear an important
idiosyncratic risk. Consequently, credit portfolio management has to inte-
grate idiosyncratic and systematic risk trade-off. Such a consideration is all
the more important at an individual firm viewpoint.

NOTES

1. Volatility is represented by a combination of both an autoregressive process and an
   additional ARCH process.


2. Such rules require computing the frequency of occurrence of excessive losses (for
   example, observedlossesthatlieabovethelossforecastscomputedfromValue-at-Risk
   models). The lower this frequency, the better the model performs.


3. Sharpe (1963) proposes a two-factor model where only one factor plays a role on
   an average basis. Analogously to the asset pricing theory, we propose a two-factor
   model where idiosyncratic risk is explicitly taken into account. Moreover, our general
   framework reduces to Sharpe’s (1963) setting whenE[dlnIt]=(1−β)r. Notice also
   thatE[dlnIt|Ft]=μI(t,V)−σ
   I^2 (t,V)
   2 dt.


4. Such a characterization is only valid whenβis non-zero (for example, 1/βis defined).
   Whenβis zero, driftμV(t,Vt,It) reduces toμV(t,It)=μI(t,It), and global volatility
   σV(t,Vt,It) reduces toσV(t,It)=σI(t,It) sinceVt=It.


5. In general, stochastic variance and firm value are non-perfectly correlated. First,
   assume that the volatility of the systematic risk factorXis at best a determinis-
   tic function of time, the firm value’s global variance is then independent ofX.In
   this case, we have a non-perfect correlation between firm value and its variance
   Corr(dσ^2 V(t,Vt,It),dVt)=ρ(t,Vt,It) sinceRV(t,Vt,It)=0. Second, assume that the
   volatility of idiosyncratic risk factorIis at best a deterministic function of time, then
   the firm value’s global variance is independent ofI. In this case, we have a perfect
   correlation between firm value and its varianceCorr(dσ^2 V(t,Vt,It),dVt)=1 since
   RI(t,Vt,It)=0.