C. Quantification of portfolio risk
3. Standard deviation 272
The portfolio standard deviation is another widely used risk measure. It is defined as the stan-dard deviation of the portfolio loss distribution. Other than the expected loss, the size of the portfolio standard deviation changes with the composition of the portfolio even if the portfo-lio’s expected loss is held constant. Hence, by considering clients’ marginal contributions to the portfolio standard deviation, the standard deviation can be used to analyze the portfolio structures and to localize components that are well diversified in the portfolio and others where risk is particularly concentrated.
271 Two simulations were performed sequentially. First, a sample of n points was drawn from the portfolio loss distribution and the expected value was estimated as described above. Then, this was repeated 1,000 times in order to obtain 1,000 independent estimates for the expected loss as an empirical proxy for the distribution of the estimator.
272 On the applicability of the portfolio standard deviation in credit risk management confer to Wehrspohn 2001.
One of the major advantages of the standard deviation that founded its popularity among prac-titioners is that it is analytically computable in elementary portfolio models such as the Va-sicek-Kealhofer model273. In more complicated models such as the CRE model where several influences on the clients’ credit risk superimpose one another such as country risk, sectorial correlations and individual dependencies with other clients, the analytical calculation of the portfolio standard deviation is considerably more complicated and no longer practicable. Be-low we show how the standard deviation and confidence intervals can be estimated based on the simulation results.
A second great advantage of the portfolio standard deviation is that it expresses a feature of the entire loss distribution. This is particularly valuable if it comes to the calculation of the individual clients’ contributions to total portfolio risk because for a client with positive prob-ability of default and positive exposure his marginal standard deviation is also positive. This is not necessarily the case with a client’s marginal value at risk or shortfall which much stronger depend on the specific simulation results and the number of simulation runs per-formed since they only consider a specific percentile or the tail of the loss distribution, respec-tively.
However, in all credit risk models, the standard deviation has the important drawback that it cannot be interpreted easily. This is a decisive difference between the application of portfolio standard deviation in the risk management of, say, equity portfolios as compared to the risk management of credit portfolios. In popular equity portfolio models such as the capital asset pricing model (CAPM), the distribution of portfolio returns is always a normal distribution so that there is a constant ratio between portfolio standard deviation and any value at risk to a constant confidence level. For example, no matter how the portfolio is composed, the 99%-value at risk is always 2.32 standard deviations above the mean in a normally distributed
273 In general, the portfolio variance is the sum of the variances and covariances of the portfolio components and can be written as and ρijdef the default correlation between client i and j. In the Vasicek-Kealhofer model the default correlation ρijdef can be directly computed as
( ) ( )
Here Φ2(⋅,⋅,ρ) is the bivariate normal cumulative distribution function and ρij the risk index correlation between cli-ent i and j.
ting. The portfolio standard deviation and portfolio value at risk are, therefore, exchangeable in their informational content so that the standard deviation inherits the interpretability of the value at risk.
In credit portfolio management things are different. Here the portfolio loss distribution does not have a fixed shape, but permanently substantially changes with the composition of the portfolio and the quantitative and qualitative structure of the risk factors. As a consequence, the relationship between standard deviation and portfolio value at risk is not predetermined. It is not even monotonous, meaning the value at risk may decrease while the standard deviation increases.
Densities of loss distributions of highly diversified portfolios
(Correlations = 0.2, p = default probability, EL = Expected Loss, 99%-VaR)
0
Loss in percent of portfolio value
p=0.01 p=0.02 p=0.03 p=0.04 EL p = 0.01 EL p = 0.02
Densities of loss distributions of highly diversified portfolios
(Correlations = 0.2, p = default probability, EL = Expected Loss, 99%-VaR)
0
Loss in percent of portfolio value
p=0.01 p=0.02 p=0.03 p=0.04 EL p = 0.01 EL p = 0.02
Figure 64: The ratio of value at risk and standard deviation of credit portfolios
Figure 64 shows the ratio of the 99%-value at risk to the portfolio standard deviation in ho-mogenous portfolios in the normal correlation model. In this example, the ratio decreases while the clients’ default probability increases. Hence, even in the highly idealized case of the normal correlation model, the standard deviation cannot be used as a proxy for the value at risk and is a rather blurred measure for the variability and the width of a distribution than a clear-cut measure of portfolio risk.
a) Estimation
Let X ,...,1 Xn be the simulated independent draws from the portfolio loss distribution. Then the portfolio variance σ2 can be unbiasedly and consistently estimated through the variance of the simulated loss distribution. Let
( )
=∑
=be the mean of the simulated loss distribution. Then
( )
= −∑
=n(
−)
where κ is the kurtosis of the portfolio loss distribution defined as
(
µ)
4for all ε >0. Due to the concavity of the square-root-transformation, the portfolio standard deviation can be consistently estimated with a small sample bias E
(
σˆ(
X ,...,1 Xn) )
<σ .b) Confidence intervals
It follows from the central limit theorem, that σˆ2
(
X ,...,1 Xn)
is asymptotically normally dis-tributed with mean σ2. The variance of the estimator was already calculated in the previous section. Note that for large n the variance approximately simplifies to( )
and that the kurtosis κ of the portfolio loss distribution can be consistently estimated as
(
X1,..., Xn)
= n1∑
i=n1(
Xi− ˆ)
4ˆ µ
κ
from the simulation results.
Upper and lower limits of two-sided confidence intervals for the portfolio standard deviation at the confidence level α then are
( )
0 10.000 20.000 30.000 40.000 50.000 60.000 70.000 80.000 90.000 100.000 Simulation runs
Loss in percent of portfolio exposure
Lower 95%-CI Standard deviation Upper 95%-CI Span
Standard Deviation
0 10.000 20.000 30.000 40.000 50.000 60.000 70.000 80.000 90.000 100.000 Simulation runs
In percent of portfolio exposure
Lower 95%-CI Standard deviation Upper 95%-CI Span True value
Figure 65: Accuracy of the estimation of the standard deviation of portfolio losses dependent on the number of simulation runs
Figure 65 shows the analytic and simulated results for the portfolio standard deviation and confidence bands for the estimator at the 95%-confidence level for a homogenous portfolio with default probability 0.5% and risk index correlations of 30% in the normal correlation model. The span is defined as the difference between the upper and the lower bound of the confidence interval. To get the simulated values of the portfolio standard deviation and its confidence intervals in the right chart, the distribution of the estimator was simulated 1,000 times and the mean, the 2.5%-percentile and the 97.5%-percentile were plotted274.
274 Two simulations were performed sequentially. First, a sample of n points was drawn from the portfolio loss distribution and the standard deviation was estimated as described above. Then, this was repeated 1,000 times in order to obtain 1,000 independent estimates for the standard deviation as an empirical proxy for the distribution of the estimator.
Similar to the estimation of expected portfolio losses, it is evident from the exhibits that the simulation results are well in line with the analytic solution and that the estimation of the port-folio standard deviation is consistent. Note that the small sample bias of the estimation al-ready disappears after a few thousand simulation runs. The span of the confidence intervals decreases strictly monotonously in the number of simulation runs as should be expected from the consistency of the estimator, though the increase in precision from an additional run de-clines rapidly.