Shockwaves

The value of stress test analysis

In the classic Black-Litterman approach it is possible, using reverse engineering, to obtain the expected asset equilibrium returns implied by the weights of the market portfolio — that is, the benchmark. However, given that analysts may have different views on some of the expected returns implied by the benchmark weights, it is possible to obtain the posterior distribution by combining analysts’ views and prior market information. In this article we propose a methodology for stress test analysis in asset management companies: more precisely, we shock a set of factors which affect the asset returns (such as oil price, interest rates, inflation, volatilities and so on) imposing the analysts’ view on their variation from the expected level.

Introduction
Stress testing is becoming a very important analysis tool for the financial institutions (banks, asset management companies, etc). First of all, stress testing is required from regulators (eg, Basel Committee on Banking Supervision Supervisory guidance for assessing banks’ financial instrument fair value practices (April 2009); Icaap Stress Testing Framework; Banca d’Italia Eligible Assets (December 2008)). But stress test analysis could also be a very useful tool for risk managers, fund managers and even for client reporting.

Stress test approaches differ among institutions, owing to the nature of the tested problem and the way the stress scenarios have been selected. Scenario tests can be constructed on the basis of historical events (a crisis observed in the past), historical stress scenarios or on scenarios that may be judged as possible in a future owing to changes in macroeconomic, socioeconomic or political factors — prospective or hypothetical stress scenarios.

Events commonly used to build historical scenarios include the large US stock market declines in October 1987, the Asian financial crisis of 1997, the financial market fluctuations surrounding the Russian default of 1998 and the financial market developments following the 11 September 2001 terrorist attacks in the US.

Prospective scenarios can be constructed according to an event-driven approach that identify risk sources, or factors, causing changes in asset returns. By assuming large factor movements, it is possible to investigate the extent risk parameters change if such an event occurs. Therefore, stress scenarios are based on plausible, even if unlikely, events and are suitable for a portfolio sensitivity analysis. Risk managers identify a portfolio’s key financial drivers and then formulate scenarios where these drivers are stressed.

One of the main problems in stress test analysis is the simulation of consistent scenarios that are able to integrate historical and private information, and preserve the data correlation structure as well as capture the direct effects of movements in drivers and indirect effects owing to correlation among portfolio assets. An appropriate framework is provided by the Black-Litterman approach, which can be adapted to stress testing. The general idea behind our approach is that asset returns depend on a number of financial or macroeconomic ‘core’ factors which act as drivers. Therefore, it is possible to stress asset returns by imposing shock on the drivers.

First of all, let us briefly describe the classic Black-Litterman model (BL model), then describe how we adapted the BL model for a ‘what if?’ analysis in order to monitor the portfolio’s reaction to shocks. Finally we will illustrate some empirical examples on an effective portfolio.

Review of the classic Black-Litterman model
The BL model was mainly introduced as a response to two problems in asset allocation. The first is the need to overcome the critical step of expected return estimation, critical mainly for the presence of estimation errors. The second is the need to integrate subjective information (the experts’ views) with market information. The main idea behind this approach is to extract the equilibrium results, given the Sharpe ratio, as the returns implicit in the benchmark.
The BL model argues the only sensible definition of neutral returns is the set of expected returns that would clear the market if all investors had identical views. If the Capital Asset Pricing Model holds and if the market is in equilibrium, the weights based on market capitalisations are also the weight of the optimal portfolio. Afterwards, via reverse optimisation, one can recover the equilibrium returns (prior returns). The theoretical reason for this is that if the benchmark is a good proxy for the market portfolio, its composition is the solution for an optimisation problem for a vector of unknown equilibrium returns. The equilibrium returns Π of the stocks making up the benchmark are obtained by solving the unconstrained maximisation problem faced by an investor with a quadratic utility function or by assuming normally distributed returns1.

We consider a market of N assets whose returns are normally distributed. The expected returns E(R) are assumed to be normally distribute  with the covariance matrix proportional to the historical one, rescaled by a shrinkage factor; as the uncertainty on the mean is lower than the uncertainty of the returns themselves, the value of  should be close to zero2.
The equilibrium returns provide a neutral reference point for asset allocation. In the event of no views on the market, there is no reason to deviate from the benchmark (the benchmark is a proxy of the equilibrium portfolio). However, an active asset manager can deviate from the benchmark tracking strategy, according to his/her economic reasoning in tactical asset allocation. The BL model combines equilibrium returns with uncertain views about expected returns.
Assume we have k views expressed by a set of linear constraints (1):

 with  and  
 (1)
where  is a matrix k x , where each row correspond to one view;  is a random vector k x 1 of errors of the views; and  is the matrix k x k containing the covariance, or uncertainty, of the views. The views are expressed on the expected returns and are normally distributed, so the joint distribution is:


(2)
Using the Bayes’s Theorem, it is possible to generate a vector of ‘posterior’ returns for all asset returns. The conditional distribution of the returns is:

(3)
where:


The investor’s views have the effect of modifying equilibrium returns according to the degree of uncertainty. The greater the uncertainty, the less the deviation from natural views.

The main results are that views must not be expressed for each asset and the conditional expected returns do not suffer from the typical problems of corner solutions. An assumption of the BL model is the normal distribution of uncertainty with regard to views. If all views are independent, the covariance matrix is diagonal. However, in practice, it can be difficult to specify the degree of confidence of each view. A more convenient approach is to consider  is proportional to assets’ volatility. The more volatile an asset, the more uncertain is the view as to its expected return. This can be easily implemented assuming:
    
(4)

The conditional expected return — that is, the equilibrium return adjusted according to views — can be easily expressed as:

(5)
We can observe that for , the views are certainty views; for  the views are unreliable and .  If S is invertible — that is, if we have a number of linearly independent views equal to the number of assets — (5) becomes:

(6)

Black-Litterman model adapted to stress test analysis
Following the event-driven approach, risk managers identify a portfolio’s key financial drivers and then formulate scenarios where these drivers are stressed. In this section, we explain how it is possible to include factors in the original model without affecting asset returns in the absence of shocks, and, at the same time, how we can shock the factors and observe the effects on asset returns. This framework solves one of the main problems in stress test analysis — the simulation of consistent scenarios able to integrate historical and private information and to preserve the data correlation structure, while being able to capture the direct effects of movement in the driver and the indirect effects owing to correlation among portfolio assets.

Starting from the classic BL model, we introduce K factors that can influence portfolio performance, and we can assume the percentage variations in the factors are jointly normally distributed. Therefore, the factors’ prior distribution is . In order not to influence asset returns directly, we model centred factors, , so the expected variation of the factors is null.
We can express the multivariate distribution of N assets and the K factor as in (7), where the covariance matrix is defined in term of blocks. This way we explicitly isolate the correlations among assets from correlations among exogenous factors and cross-correlations:

(7)
Now, we assume experts express views only on factors, in terms of deviation from the equilibrium level.
In equilibrium all views are null. In
other words, the views are a shock on the factors.
The shocks are expressed as a set of linear constraints:


(8)
Where S is a matrix k x F, where each row corresponds to one shock; u is a random vector k x 1 of errors on the shock; and  is the matrix k x k containing the covariance, or uncertainty, of the shock.
If we do not shock the factors, we get the observed distribution for asset returns. When we expect variations in the driver, we investigate the effects of those variations in asset returns. The expected value of the posterior distribution is:

(9)

As we are mainly interested in the reaction of the asset returns to factor shocks, we focus our attention on the expected return (10):


(10)

We observe that assuming an absence of shocks, a view given by a vectors of zeros, we get the unconditional equilibrium returns. The extra return resulting from a shock can be easily obtained as:

 
where:

(11)

H contains all information regarding the delta performance of each single asset.
Moreover, if we assume

, with  , (11) becomes:

(12)

we may point out that:
given any differential composition of an active managed portfolio with respect to the benchmark, we can compute the differential (portfolio vs benchmark) impact, security by security, caused by factors’ shocks; the covariance matrix of asset returns is not directly involved in the computation of the delta performance. This result is extremely important because it allows the asset correlation stress test to be disentangled from the factor correlation stress test. We can observe the effects of large movements in the factors by assuming correlations among factors are unchanged; at a second stage we can stress the correlations of factors; and finally we can stress the correlations of returns as well.

Empirical results
We describe how to apply the stress test analysis to a managed portfolio and evaluate his conditional return with respect to his benchmark. The results of the analysis would be very useful for the risk manager and fund manager, but could also be a very useful tool for institutional client reporting.

First we analyse the portfolio and his benchmark in terms of asset class composition (Table 1, opposite) and risk profile (Table 2). The portfolio has a slight overweight in equity and is short duration with respect to the benchmark. There are several available factors for the scenario construction:
Equity Indexes, Zero Coupon Rates, OAS Indexes, Exchange Rates, European Dated Brent Forties Oseberg (BFO) Spot Price, Commodity (CRB) Future, CBOE SPX and CBOE NDX Volatility Index (VIX and VXN), for example. Using the available factors, it is possible to construct theoretical or historical scenarios and analyse the portfolio with respect to those scenarios. We construct three scenarios (Equity Shock, Commodity Shock, Bond Shock) and a fourth scenario which combines the previous three:

Equity Shock: we suppose a large decline of equity markets;
Commodity Shock: we suppose an increase of commodity prices;
Bond Shock: we suppose a parallel shift in the Euro curve of interest rates;
Equity Shock + Commodity Shock + Bond Shock.

These scenarios are translated into variations, percentage or absolute, of the available factors (Table 3, above). For the four scenarios we measure the expected excess return of the portfolio with respect to its benchmark due only to the shock we have applied to the factors (Table 4, page 23). The results can be interpreted in relation to the portfolio and benchmark compositions, and the relative risk profile. For example, in the first scenario the excess return is negative, and this is coherent with the portfolio’s equity overweight and the negative view on equity markets of the scenario.
As we have pointed out above, we can decompose the excess return, caused by factors shock, security by security (Table 5, above). We could also run the same analysis using historical scenarios (Table 6, below).

Conclusion
This article proposes a stress testing model: we extend the BL model in order to translate shocks on factors into variations in asset performance with respect to their benchmark. The general idea behind our approach is that asset returns depend on a number of exogenous financial or macroeconomic ‘core’ factors acting as drivers (such as oil price, volatilities, interest rates, inflation, equity indexes and so on). The stress testing is undertaken by shocking a subset of factors. The stress test analysis becomes a very important tool for risk management purposes particularly in periods of crisis such as the second half of 2008. I PMCRR


References
Black F, Litterman R. (1991) Asset allocation: combining investors views with market equilibrium. Fixed Income Research. Goldman Sachs & Company, September

Black F, Litterman R. (1991) Global asset allocation with equities, bonds and currencies. Fixed Income Research. Goldman Sachs & Company, October

Meucci A. (2005) Risk and Asset Allocation. Springer

Meucci A. (2006) Beyond Black-Litterman in practice. Risk Magazine; 19(9): 114-119

Meucci A. (Forthcoming), Enhancing the Black-Litterman and related approaches: Views and stress-test on risk factors. Journal of Asset Management, in press.

Footnotes:
1More precisely  , where  is the Sharpe ratio.
2We assume that  is the expected value of the distribution of the expected returns, following the original Black-Litterman formulation. In other approaches (see Meucci) the expected returns are assumed to be constant. In both cases the analysis is similar and it is possible to derive the conditional distribution of the returns, given the views, and to compute the moments of the conditional distribution.