The discussion above focused on designing sensible stress scenarios that are relevant to both the current portfolio and the current market environment.
This point is worth emphasizing: the success of stress testing depends cru- cially upon the selection of sensible scenarios. The only meaningful stress scenarios are those that could occur in the current market environment.
Useful stress tests also reveal the details of the exposures, for example, which positions are responsible for the profits and losses. Absent this infor- mation, it is difficult to identify which positions to eliminate or hedge in order to reduce any undesirable exposures. In fact, a great part of the value of stress testing stems precisely from the process of systematically thinking through the effects of shocks on different positions and markets and what can be done to ameliorate them.
In addition, organizations whose risk-management strategies depend on the ability to frequently adjust or rebalance their portfolios need to con- sider the impact of crises on market liquidity, because it may be difficult or impossible to execute transactions at reasonable bid/ask spreads during cri- ses. Finally, organizations that use futures contracts to hedge relatively illiq- uid securities or other financial instruments should consider the cash flow requirements of the futures contracts. Gains or losses on futures contracts are received or paid immediately, while gains or losses on other instruments are often not received or paid until the positions are closed out. Thus, using
futures contracts to hedge the changes in the values of other instruments can lead to timing mismatches between when funds are required and when they are received.
A common criticism of stress tests is that stress tests as usually con- ducted are not probabilistic and thus lie outside the value-at-risk frame- work. The typically subjective choice of stress scenarios complicates external review of a stress-testing program, and the failure to assign proba- bilities renders back testing impossible. In response to these concerns, vari- ous methods of attaching probabilities to stress scenarios have been proposed. While these approaches provide either objectivity or its appear- ance, they are a step away from stress testing, back toward the computa- tion of value-at-risk using a low probability or high confidence level. While extreme tail value-at-risk estimates are useful, stress testing is a distinct activity.
NOTES
The zero-out and historical stress test approaches have been standard (see e.g., Kupiec 1998) for some time. Kupiec (1998) proposes the use of predic- tive stress scenarios and also discusses the possibility of combining them with stressed volatilities and correlations. All of these approaches are implemented in at least some risk measurement software. The more recent broken arrow stress test is due to Kim and Finger (2000).
Finger (1997) proposes an approach for modifying parts of a correla- tion matrix while ensuring that the matrix remains positive definite. The drawback, pointed out by Brooks, Scott-Quinn, and Whalmsey (1998), is that the property of positive definiteness is maintained by changing the other parts of the correlation matrix in an unintuitive and uncontrolled fashion. Rebonato and Jäckel (1999/2000) present an efficient method for constructing a valid correlation matrix that is as close as possible to a desired target correlation matrix, where closeness can be defined in a num- ber of different ways. They also explain how principal components analysis can be used to obtain easily approximately the same correlation matrix as their proposed approach.
The conventional wisdom is that correlations increase during periods of market stress. For example, Alan Greenspan’s oft-quoted speech on measuring financial risk includes the phrase, “. . . joint distributions esti- mated over periods without panics will misestimate the degree of asset correlation between asset returns during panics,” while Bookstaber (1997) writes, “During periods of market stress, correlations increase
dramatically.” Brooks and Persand (2000) document the existence of such changes in measured correlations. However, the correct interpretation of such statements and evidence is delicate. They are certainly correct if understood to mean that the measured correlations are higher when returns are large; however, it is not clear that the distributions generating the data actually change.
Boyer, Gibson, and Loretan (1997) show that apparent correlation breakdowns can be found in data for which the true underlying distribu- tion has a constant correlation coefficient. Specifically, they derive an expression for the conditional correlation coefficient of data generated from a bivariate normal distribution and find that the conditional correla- tion is larger when one conditions on large rather than small returns. Thus, one expects measured correlations to be larger during periods of high returns or market volatility, even if the distribution generating the data has not changed. A similar result is cited by Ronn, Sayrak, and Tompaidis (2000), who credit Stambaugh (1995) for it. Boyer, Gibson, and Loretan (1999) find no evidence for nonconstant correlations in a limited empirical analysis of exchange rate changes, while Cizeau, Potters, and Bouchaud (2001) argue that a simple non-Gaussian one-factor model with time-inde- pendent correlations can capture the high measured correlations among equity returns observed during extreme market movements. In contrast, Kim and Finger (2000) estimate a specific (mixture of distributions) model that allows for correlation changes and find evidence of such changes for four of the 18 market factors they consider. Thus, there is evidence of cor- relation changes, though overall the evidence argues against their ubiquity.
Berkowitz (1999/2000) argues that the typically subjective choice of stress scenarios makes external review of a stress-testing program difficult, and the failure to assign probabilities renders back testing impossible. In response to these concerns, Berkowitz proposes that risk managers explic- itly assign (perhaps subjective) probabilities to stress scenarios and then combine the resulting stress distribution with the factor distribution for normal market conditions to generate a single forecast distribution. He argues that this will impose needed discipline on the risk manager and enable back testing. This approach is also advocated by Aragonés, Blanco, and Dowd (2001). Similar approaches are suggested by Cherubini and Della Lunga (1999) and Zangari (2000), who use Bayesian tools to com- bine subjective stress scenarios with historical data.
A different approach is taken by Longin (2000), who proposes the use of extreme value theory (see chapter 16) to model the extreme tails of the distribution of returns. As discussed there, multivariate extreme value theory is not well developed, and current proposals for using EVT involve
fitting the tails of the distribution of portfolio returns, without consider- ation of the detail that causes these returns. A limitation is that this amounts to an extreme tail VaR and thereby sacrifices some of the bene- fits of stress testing.
three
Risk Decomposition and
Risk Budgeting
1O
153
Decomposing Risk
Risk decomposition was introduced in Chapter 2 in the context of a simple equity portfolio exposed to two market factors. The next few chapters turn to more realistic examples of the use of risk decomposition, without which risk budgeting cannot exist. As a preliminary, this chapter summarizes the mathematics of risk decomposition. Risk decomposition is crucial to risk budgeting, because the aggregate value-at-risk of the pension plan, or other organization, is far removed from the portfolio managers. At the risk of stating the obvious, the portfolio managers have control over only their own portfolios. For them, meaningful risk budgets are expressed in terms of their contributions to portfolio risk.
In fact, meaningful use of value-at-risk in portfolio management almost requires risk decomposition. Value-at-risk, or any other risk measure, is useful only to the extent that one understands the sources of risk. For exam- ple, how much of the aggregate risk is due to each of the asset classes? If we change the allocations to asset classes, what will be the effect on risk? Alter- natively, how much of the risk is due to each portfolio manager? How much is due to tracking error? Is it true that the hedge fund managers do not add to the overall risk of the portfolio? All of these questions can be answered by considering risk decomposition. This chapter first describes risk decom- position and then turns to another issue, the model for expected returns.
RISK DECOMPOSITION
Let w= (w1, w2, . . . , wN)′ denote the vector of portfolio weights on N assets, instruments, asset classes, or managers, and let (w) and VaR(w) denote the portfolio standard deviation and value-at-risk, which depend on the positions or weights wi. Imagine multiplying all of the portfolio weights by the same con- stant, k, that is, consider the vector of portfolio weights kw= (kw1, kw2, . . . , kwN)′ and the associated portfolio standard deviation (kw) and value-at-risk VaR(kw). A key property of the portfolio standard deviation is that scaling
all positions by the common factor k scales the standard deviation by the same factor, implying (kw) k(w). This is also true of the value-at-risk, because scaling every position by k clearly scales every profit or loss by k, and thus scales the value-at-risk by k.
In mathematical terminology, the result that VaR(kw) kVaR(w) for k>0 means that function giving the value-at-risk is homogenous of degree 1, or linear homogenous. From a financial perspective, this property of value-at-risk is almost obvious: if one makes the same proportional change in all positions, the value-at-risk also changes proportionally.
Though very nearly obvious, this property has an important implica- tion. If value-at-risk is linear homogenous, then Euler’s law (see the notes to this chapter) implies that both the portfolio standard deviation and VaR can be decomposed as
(10.1)
and
(10.2)
respectively. The ith partial derivative, ∂(w) ∂wi or ∂VaR(w) ∂wi, is interpreted as the effect on risk of increasing wi by one unit; in particular, changing the ith weight by a small amount, from wi to wi*, changes the risk by approximately (∂(w) ∂wi)(wi*−wi), or (∂VaR(w) ∂wi)(wi*−wi). The ith term, (∂(w) ∂wi)wi or (∂VaR(w) ∂wi)wi, is called the risk contri- bution of the ith position and can be interpreted as measuring the effect of percentage changes in the portfolio weight wi. For example, the change from wi to wi* is a percentage change of (wi* −wi) wi, and the change in portfolio standard deviation resulting from this change in the portfolio weight is
the product of the risk contribution and the percentage change in the weight.
=
=
( )w ∂( )w
∂w1
---w1 ∂( )w
∂w2
---w2 . . . ∂( )w
∂wN ---wN
+ + +
=
VaR( )w ∂VaR( )w
∂w1
---w1 ∂VaR( )w
∂w2
---w2 . . . ∂VaR( )w
∂wN ---wN,
+ + +
=
⁄ ⁄
⁄ ⁄
⁄ ⁄
⁄
∂( )w
∂wi
---(wi*–wi) ∂( )w
∂wi
---wi (wi*–wi)
wi ---
× ,
=
A key feature of the risk contributions is that they sum to the portfolio risk, permitting the portfolio risk to be decomposed into the risk contribu- tions of the N positions wi. Similarly, we can define ((∂(w) ∂wi)wi) (w) or ((∂VaR(w) ∂wi)wi) VaR(w) to be the percentage contribution to portfo- lio risk of the ith position. It is straightforward to compute these risk con- tributions when risk is measured by standard deviation. Computing the derivative with respect to the ith portfolio weight,
(10.3)
where the numerator wicov(ri,ri), is the covariance between the return ri and the portfolio return wjrj. Thus, the risk contribution of the ith position or asset
is proportional to the covariance between the return on the ith position and the portfolio, and is zero either when that position is uncorrelated with the portfolio or when the weight wi=0. The percentage contribution to portfo- lio risk is
where i is the regression coefficient, or beta, of the ith market return on the return of the portfolio. By construction, the weighted sum of the betas is 100% or one, that is, iwi 1. This is exactly analogous to the stan- dard result in portfolio theory that the market beta of one is the weighted average of the betas of the stocks constituting the market.
Decomposing delta-normal value-at-risk is almost equally easy. Recog- nizing that delta-normal VaR is of the form
⁄ ⁄
⁄ ⁄
∂( )w
∂wi ---
wi cov(ri,rj)
j=1
∑
N( )w ---,
=
j=1N i=1N∂( )w
∂wi ---wi
wicov(ri,rj)
j=1
∑
N( )w ---wi
=
∂( )w
∂wi --- wi
( )w ---
wicov(ri,rj)
j=1
∑
N2( )w ---wi
= ii,
=
i=112 =the ith partial derivative and risk contribution of the ith asset are
and
respectively. The only difference (besides the constant k) between this and equation (10.3) is the expected return component −E[ri]wi, which appears because larger expected returns shift the distribution upward and reduce the probability of loss.
The key insight from these results is that the risk contribution of a position crucially depends on the covariance of that position with the existing portfolio. This covariance is zero when the position is uncorre- lated with the existing portfolio, in which case the risk contribution is zero. When the correlation is positive, the risk contribution is positive;
when it is negative, the position serves as a hedge, and the risk contribu- tion is negative.
In interpreting risk decomposition, it is crucial to keep in mind that it is a marginal analysis; a small change in the portfolio weight from wi to wi* changes the risk by approximately (∂(w) ∂wi)(wi*−wi), or (∂VaR(w) ∂wi)(wi*−wi). Alternatively, if the risk decomposition indi- cates that the ith position accounts for one-half of the risk, increasing that position by a small percentage will increase risk as much as increasing all other positions by the same percentage. The marginal effects cannot be extrapolated to large changes, because the partial derivatives ∂(w) ∂wi
and ∂VaR(w) ∂wi change as the position sizes change. In terms of correla- tions, a large change in market position changes the correlation between the portfolio and that market. For example, if the ith market is uncorre- lated with the current portfolio, the risk contribution of a small change in the allocation to the ith market is zero. However, as the allocation to the ith market increases, that market constitutes a larger fraction of the port- folio and the portfolio is no longer uncorrelated with the ith market. Thus,
VaR( )w wiE r[ ]i
i=1
∑
N –k( )w ,–
=
∂VaR( )w
∂wi
--- –E r[ ]i k∂( )w
∂wi --- +
=
∂VaR( )w
∂wi
---wi –E r[ ]wi i k∂( )w
∂wi ---wi, +
=
⁄ ⁄
⁄ ⁄
the risk contribution of the ith market increases as the position in that market is increased.