Portfolio optimization

A complementary approach to improve portfolio quality is algorithmic portfolio optimization under certain constraints such as non-negativity of exposures, the size of expected returns or others. This was long an unsolved problem because the standard approach of minimizing port-folio variance under side-constraints was difficult to apply to credit portport-folios due to the lack of interpretability of portfolio variance (or standard deviation). However, in their 1999

semi-nal paper Stanislav Uryasev and Tyrell Rockafellar²⁸⁹ proposed a methodology to minimize portfolio shortfall²⁹⁰ under the respective side-constraints.

a) Optimization approach

Indicate clients (or portfolio segments of interest²⁹¹) by i for i=1,...,n. Let

(

X Xn

)

ⁿ

X = ₁,..., ∈Ω⊂R_≥0 be the vector of clients’ exposures in the portfolio chosen out of a certain subset Ω⊂Rⁿ_≥0. In the sense of the optimization problem, X is a decision vector re-flecting the composition of the credit portfolio. Let Y =

(

Y₁,...,Yn

)

∈Rⁿ be a random vector that stands for the uncertainties of the future value of the exposures. For ease of exposition we assume in the following that the distribution of Y is continuous with density ^p

( )

^{Y . In}

prac-tice, the distribution of Y need not be known analytically, but is being simulated in accor-dance with the portfolio model used²⁹². Let f

(

X,Y

)

∈^Rbe the portfolio loss associated with the exposure vector X and the random ‘creditworthiness’ vector Y. For given X, the distribu-tion of ^f

(

^X,Y

)

is the portfolio loss distribution.

289 Rockafellar and Uryasev 1999.

290 Minimization of portfolio shortfall is not equivalent to minimization of portfolio value at risk. The shortfall always domi-nates the respective value at risk.

291 In practice, it may be difficult to do a portfolio optimization at the client level because single clients’ exposures usually cannot be modified easily without the respective client’s agreement. For this reason, it is rather more practicable to consider portfolio segments at an intermediate level of aggregation in the portfolio where exposures can be more freely adapted.

292 Note in particular, that the methodology suggested by Rockafellar and Uryasev is independent of the actual portfolio model that is used by a financial institution to assess portfolio risk. The only place where the portfolio model enters in is the simulation of clients’ future creditworthiness.

The key to the optimization approach of Rockafellar and Uryasev is the characterization of optimal shortfall and the corresponding value at risk in terms of the function F_β defined as

( ) ∫ [ ( ) ] ( )

The relationship between optimal shortfall, corresponding value at risk and the function F_β is captured in the following

Theorem 15 (Rockafellar and Uryasev 1999):

Minimizing the β-shortfall of the loss associated with X over all X∈Ω is equivalent to minimizing Fβ

(

X,α

)

over all

(

X,α

)

^∈Ω×R, in the sense that

According to the theorem, it is sufficient to do the minimization of the β-shortfall with the function F_β that does not contain the β-VaR, which often is mathematically troublesome to handle, instead of using the β-shortfall directly. The theorem also states that the value at risk corresponding to the optimal β-shortfall is automatically given as one of the calculation re-sults, quasi as a byproduct of the optimization.

In practice, the distribution of Y in F_β can be approximated by a sampled empirical distribu-tion due to the theorem of Glivenko-Cantelli. With simuladistribu-tion results Y ,...,₁ Y_m the approxima-tion F~_β

b) A portfolio optimization

As an example, we consider a heterogeneous portfolio of 10,000 clients²⁹³ in n = 20 segments with a total portfolio exposure of € 1,000 in the normal correlation model²⁹⁴. Risk index

293 Individual default probabilities, exposures and segment adherence were assigned randomly.

relations were set to 30%. The portfolio loss distribution was sampled m = 2,000 times²⁹⁵. Exposures and losses were aggregated per segment in each simulation run so that the optimi-zation can be performed at the segment level²⁹⁶. The coefficients of the random vector

(

^{j jn}

)

ⁿ

j Y Y

Y = ¹,..., ∈R signify which fraction of the portfolio exposure in the respective segment was lost in simulation run j for j = 1, …, m. The portfolio loss function f then simply is

(

^{X Y}

)

^{X Y}

f , = ^t

so that the optimization problem can be stated as

( ) ( ) _∑ [ ]

The feasibility set _Ω⊂Rⁿ_≥0 was specified as a linear return-constraint²⁹⁸

∑

₌ for fixed values of θ >0 and the trivial constraint

≥0 Xi

294 The normal correlation model was used as a pure default model so that rating migrations other than to default were ig-nored.

295 The optimization was performed with an own implementation of the simplex algorithm. It turned out that the algorithm could not handle problems with more than about 2,000 constraints, thus, limiting the number of simulation runs (see below). 2,000 runs are, however, too few to precisely forecast the 95%-shortfall. Optimization results are, therefore, rather illustrative in nature. A serious implementation would need a high-end linear solver such as IBM-OSL or CPLEX.

296 We suppose that the composition of the segments remains unchanged by the optimization.

297 Note that the simplex algorithm assumes non-negativity of all variables so that the number of non-trivial constraints is approximately equal to the number of simulation runs.

298 Other possible linear constraints are exposure limits or exposure concentration limits. Risk or risk concentration limits usually are non-linear in nature.

for i = 1, …, n.

Marginal VaR before and after optimisation

0%

Marginal VaR in percent of marginal exposure

Marginal VaR before optimisation Marginal VaR after optimisation

Figure 76: Marginal VaR before and after optimization

Figure 76 shows the segments’ marginal values at risk before and after optimization of the 95%-portfolio shortfall. It is particularly striking that after the optimization the number of small exposures with high marginal risks has decreased substantially. Note moreover that one segment has an exposure of zero after the optimization and has completely dropped out of the portfolio.

VaR and shortfall efficient frontiers

8,0%

Risk in percent of portfolio exposure

Expected return

VaR efficient frontier shortfall efficient frontier VaR before optimisation shortfall before optimisation original portfolio return

Figure 77: VaR and shortfall efficient frontiers

Figure 77 illustrates the efficient frontiers²⁹⁹ of the example portfolio for different values of the minimum expected return θ. It turns out that the optimization had a remarkable effect on the portfolio shortfall and also on the portfolio value at risk by reducing risk by more than half in both cases. Overall, the optimization greatly improved portfolio quality both from the point of view of portfolio structure and total portfolio risk.

Conclusion

The discussion has shown that credit risk modeling and management is a complex task where many different aspects play an important role and where many errors can be committed. The analysis has identified some of these errors, particularly in the estimation of default probabili-ties and the concepts of dependence among clients. Most importantly, the thesis proposes so-lutions for some of the deficiencies of current risk models.

Moreover, the entire risk management process has been assessed and methods for portfolio analysis and management have been described in detail. We are convinced that some of our results can help financial institutions to avoid some of the pitfalls of credit risk modeling and management and to improve the quality of their risk management methods and in turn the credit risk quality of their portfolios.

299 Only the portfolio shortfall was optimized. The values for the value at risk just correspond to the respective shortfalls and may not be optimal for a value at risk criterion. Experiments with various portfolios also showed that the value at risk efficient frontier is not necessarily convex. Moreover, we also found examples where the portfolio value at risk effi-cient frontier did not even fall into a part that was monotonously decreasing and another that was increasing in the ex-pected return. In practice, the VaR-efficient frontier can take a great variety of shapes.

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195

Uwe Wehrspohn

is managing partner at the Center for Risk & Evaluation, a consulting firm specializing in risk management strategies, methodology and technology based in Heidelberg and Eppingen.

He also holds a research and teaching position at the University of Heidelberg.

Uwe Wehrspohn has studied mathematics, economics and

theology in Montpellier, St. Andrews, Munich and Heidelberg. He then worked for several years as a senior consultant at the Competence Center Controlling and Risk Management of Computer Sciences Corporation in Europe.

CRE Center for Risk & Evaluation GmbH & Co. KG

Berwanger Straße 4 D-75031 Eppingen Germany

Tel. +49 7262 20 56 12 Mobile +49 173 66 18 784 Fax + 49 7262 20 69 176

Email [email protected] www.risk-and-evaluation.com

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