Investment managers are typically interested in holding periods longer than those used by the derivatives dealers who originally developed value-at-risk. The longer holding period makes the model of expected returns more important and gives a strong push in the direction of using Monte Carlo simulation to compute VaR measures. One role of the expected-returns model is obvious through the appearance of E[r] in for- mulas such as
VaR −(E[r] −k).
Less obviously, the expected returns model also impacts the estimate of . To see this, consider a simple example with an equity portfolio, a portfolio manager, and a risk manager, perhaps employed by a plan sponsor. The portfolio manager is skilled at predicting expected returns and selecting stocks. Let denote the component of the portfolio return that he or she is able to predict (i.e., he or she knows ), and let denote the unpredictable component. The portfolio return r is the sum of the predictable and unpredictable components. For concrete- ness, one can think of either or as the return on an industry factor and the other term as the idiosyncratic component of return. Assume is normally distributed, and the conditional expectation of , given , is E[ ] 0.
From the perspective of the portfolio manager, the unpredictable com- ponent is the only source of uncertainty, because he or she knows . Thus, from his or her perspective, the expected value and standard deviation of portfolio return are E[ ] E[ ] and , and the 95%-confidence value-at-risk is VaR1 −(−1.645) 1.645−.
∂VaR( )w
∂wi
---wi = ri1−wi.
ri1−
=
= +
=
r = + = = var( )
= =
The risk manager does not have the information to predict expected returns and therefore does not know . Further, the risk manager either does not believe or is unwilling to assume that the portfolio manager knows , and therefore calculates value-at-risk as if is unknown. Again for sim- plicity, assume that, from the portfolio manager’s perspective, is normally distributed with expected value E[] , where is the expected return on a passive benchmark. Thus, to the risk manager the expected value and standard deviation of portfolio returns are E[r] E[ ] and >, and the 95%-confidence value-at-risk is VaR2 −( −1.645) 1.645− .
In the portfolio manager’s computation VaR1 −(−1.645) 1.645−, the term will always be positive because, if the portfolio manager predicted returns less than on some stocks, he or she could either select different stocks, establish short positions in the stocks for which he or she predicted negative returns, or simply invest in the pas- sive benchmark. Comparing this to the risk manager’s computation VaR2 −( −1.645) 1.645− , one can see that the risk manager’s value-at-risk estimate will always exceed that of the portfolio manager, because > and <. The result that the risk manager’s value-at-risk estimate exceeds that of the portfolio manager will remain true if one uses the expected returns as the benchmarks for defining loss, because then the VaRs are proportional to the standard deviations VaR1 1.645 and VaR2 1.645.
Though this is only a simple example, the point holds more generally.
To the extent that a portfolio manager is able to predict expected returns, the predictive model will explain some of the variability of returns. Thus, the variability of the unexplained component of returns, namely, the resid- ual, will be less than the variability of returns computed when assuming that the expected return is constant. As a result, VaR estimates computed with the use of forecast errors from the predictive model of returns will indicate less risk than VaR estimates computed when assuming that the expected return is constant.
Given this, how should VaR be computed? For a bank that is ware- housing a portfolio of interest-rate and other derivatives, the answer is clear. In this setting, a key use of VaR is to monitor and control the indi- vidual risk-takers and risk-taking units, for example, traders, trading
“desks,” and larger business units. To the extent that the VaR estimates used for internal risk-management rely on the traders’ models, forecasts, and beliefs, this monitoring and control function is defeated. More gen- erally, “best practice” is for the internal risk-management unit to be independent of the risk-taking units. Thus, it is standard for a bank’s
=
= + =
var( ) +var( )
=
= =
= =
= =
=
=
internal VaR models to use naïve estimates of expected returns, typically setting them to zero. This argument in favor of the use of naïve esti- mates of expected price changes is reinforced by the fact that assump- tions about expected returns have a limited impact on the VaR model when a one-day holding period is used, because volatilities are roughly proportional to the square root of time and expected price changes are proportional to time.
In portfolio-management uses of VaR, the answer is less clear. Investors or plan sponsors acting on their behalf entrust portfolio managers with their funds, expecting returns; the portfolio manager, acting on behalf of the investors or plan sponsors, must construct portfolios that balance the anticipated returns against the risks accepted to earn those returns. This portfolio construction is based, explicitly or implicitly, on the manager’s beliefs about expected returns. In this situation, the risk is due to the fact that the realized asset-returns may deviate from the manager’s forecasts, and VaR measures should measure the risk of such deviations. These are determined by the extent to which the realized values of the market factors deviate from the expected values, given the manager’s information, unlike traditional VaR measures, which are based on the magnitudes of deviations from naïve expectations.
This approach to computing VaR numbers is used at two different lev- els, for two different purposes. First, the discussion above implies that it is the correct approach at the level of an individual fund, when VaR estimates are being computed for use in portfolio construction and managing the risk-taking process. In this case, the VaR would be computed based on deviations from the model used by the portfolio manager in forecasting returns. The next chapter provides several examples of this use of VaR. Sec- ond, this is also a reasonable approach at the level of the plan sponsor, both when it makes asset-allocation decisions among asset classes and when it allocates funds to portfolio managers within each asset class. When it does this, the plan sponsor is playing a role analogous to a portfolio manager, but selecting among asset classes and portfolio managers rather than among individual securities. In this case, the VaR would be computed based on deviations from the expected returns in the plan sponsor’s strategic asset-allocation model. Chapter 13 provides an extended example of this use of VaR.
Plan sponsors also play a second role, that of monitoring their portfo- lio managers. This is akin to that of the internal risk-management function at a bank or other derivatives dealer. For this purpose, it is reasonable to use naïve estimates of expected returns. This provides a check on the man- agers in case the portfolio manager incorrectly thinks that he or she has the
ability to predict expected returns when he or she does not, and so that the portfolio manager’s model understates the risk of deviations from his or her forecasts.
NOTES
The approach to risk decomposition in this book is that of Litterman (1996), which has become standard. Adding confusion, some (e.g., Mina and Xiao 2001) defy standard usage of the word marginal and switch the definitions of marginal and incremental risk. That is, in RiskMetrics, mar- ginal VaR is the change in risk resulting from selling the entire position and incremental VaR measures the effect of a small change in a position (Mina and Xiao 2001: Sections 6.2–6.3).
Euler’s law is obvious in the one-variable case, because only functions of the form f(w) bw are homogenous of degree 1, implying that, for such functions, f(kw) kbw and ∂f(kw) ∂k bw f(w).
To obtain Euler’s law in the general case, one starts with the statement that the value-at-risk of a portfolio kw is VaR(kw) kVaR(w) and differ- entiates both sides of this equation with respect to k. This differentiation amounts to asking what the effect is of a proportionate increase in all positions.
The right-hand side of the equation means that the portfolio value-at- risk is proportional to k and thus increases at the rate VaR(w) as k changes.
Thus, the derivative of the right-hand side is
(10.6) To compute the derivative of the left-hand side, write it as VaR(kw) VaR(y1,y2, . . . , yN,) VaR(y), where y1 kw1, y2 kw2, . . . , yN kwN, and y is the vector y (y1,y2, . . . , yN)′. Then,
.
This says that the effect on the value-at-risk of increasing k is determined by the effects of yi on VaR and the effects of k on yi. Then, using the facts that ∂y1 ∂k w1, ∂VaR(y) ∂y1 ∂VaR(w) ∂w1, and similarly for the other yis, one obtains
=
= ⁄ = =
=
∂(kVaR( )w )
∂k
--- = VaR( ).w
=
= = = =
=
∂VaR( )y
---∂k ∂VaR( )y
∂y1
--- ∂y1
∂k ---
× ∂VaR( )y
∂y2
--- ∂y2
∂k---
× . . . ∂VaR( )y
∂yN
--- ∂yN ---∂k
×
+ + +
=
⁄ = ⁄ = ⁄
(10.7)
Combining (10.6) and (10.7) yields equation (10.2) in the body of the chapter. A similar analysis will yield the decomposition of the standard deviation in equation (10.1)
∂VaR( )y
∂k
--- ∂VaR( )y
∂y1
---w1 ∂VaR( )y
∂y2
---w2 . . . ∂VaR( )y
∂yN ---wN
+ + +
=
∂VaR( )X
∂w1
---w1 ∂VaR( )w
∂w2
---w2 ∂VaR( )X
∂wN ---wN.
+ +
=
11
163
A Long-Short Hedge Fund Manager
We start our series of examples illustrating the use of value-at-risk in risk decomposition and risk budgeting by looking at the simplest possible case, a quantitative portfolio manager who uses value-at-risk internally in order to measure, manage, and optimize the risks of its portfolios. This manager, MPT Asset Management (MPT), specializes in predicting the relative returns in the stock, bond, and currency markets of some of the developed countries over short horizons and uses these predictions in managing long- short hedge funds. Both MPT’s predictions of expected returns and its esti- mates of market risks change frequently, requiring that it rapidly alter its portfolios to reflect the changes in its beliefs and optimize the risk-return tradeoff. Due to their high liquidity, low transaction costs, and (for the futures contracts) lack of credit risk, futures and currency forward contracts are MPT’s preferred investment vehicles. In particular, for each country it follows it uses the leading stock index futures contract, a futures contract on a benchmark government bond, and currency forward contracts.
The futures and forward contracts involve no cash outlay (other than the required margin or collateral), so in addition to the futures contracts each of MPT’s funds includes a portfolio of securities. These asset portfo- lios are not actively managed but rather are chosen to match the returns on the benchmarks used by the various funds. For example, some of MPT’s funds are absolute return funds, benchmarked to the riskless return; for these the asset portfolio consists of high-grade money market instruments.
Other funds are benchmarked to large-capitalization U.S. equity indexes, and for these the asset portfolios consist of passively managed portfolios of U.S. equities. Thus, each of MPT’s portfolios consists of an active portfolio of liquid futures (and sometimes forward) contracts, along with a passively managed benchmark asset portfolio. Such simple portfolios provide a good setting in which to begin illustrating the use of value-at-risk in portfolio
management because they allow us to illustrate some of the main ideas without requiring factor models to aggregate the risks across different instruments and portfolios.
As suggested by its name, MPT optimizes its portfolios using mean- variance optimization techniques. MPT is well aware that the optimal portfolios produced by mean-variance optimizers are sensitive to estima- tion errors in both the expected returns and covariance matrices and that these problems can be especially severe in portfolios that mix long and short positions. For this reason, it uses proprietary Bayesian statistical approaches to estimate the parameters. These combine the historical data with prior information about the parameter values and thereby reduce the sampling variation from the historical sample that is the source of the esti- mation error. For this example, we do not worry about the source or qual- ity of the estimates of the mean returns and covariance matrix but simply use them to illustrate the use of value-at-risk and risk decomposition.