ong (1997) - explaining the assumptions used in the measurement of value at risk

M.K.Ong/Explaining VaR/October 1996 1

Explaining the Rationale Behind the AssumptionsUsed in the Measurement of VaR

Michael K. Ong

This appeared as Chapter 1 of the book,Risk Management for Financial Institutions,Published by RISK Books (London) 1997


Explaining the Rationale Behind the AssumptionsUsed in the Measurement of VaR

Michael K. Ong

Prologue

It is coming.

In the U.S., January 1, 1998 is the date in which internationally active banks must comply withthe risk-based capital standards issued by the Basle Committee on Banking Supervision. Thesupervisory framework for market risks, defined as the risk of losses in on- and off-balance sheetpositions arising from movements in market prices, requires that banks engaging in theseactivities use internal models to measure these potential losses.

The objective in introducing this significant amendment to the Capital Accord of July 1988 is "toprovide an explicit capital cushion for the price risks to which banks are exposed, particularlythose arising from their trading activities. The discipline of the capital requirement impose isseen as an important further step in strengthening the soundness and stability of the internationalbanking system and of financially markets generally."

What have banks really been preparing for all this time?

In my role as a risk manager, I have been asked countless times from regulators, fellowacademics, senior management, and colleagues on the Street to explain what it is that theinternal models -- collectively known by the generic term VaR -- are supposed to measure andwhether or not it is feasible. Can VaR really deliver what it is touted to be? Having deliveredseveral conference talks on the subject, I have finally decided to put some of my thoughts downin writing. Here is the story of VaR -- its assumptions, rationale, foibles, and my own personalreflections in search of truth.


Implementing Variance-Covariance Models

The simplest Value-at-Risk framework, also known as VaR, is based on a so-called analytic orportfolio variance approach. This approach closely resembles the Markowitz framework on theportfolio theory of risk and return. In the original Markowitz formulation, the concept ofportfolio risk is associated with the observed dispersion of the portfolio's return around its meanor average value. Risk, therefore, is a quantifiable entity assumed to be completelyencapsulated by the calculated portfolio variance, a measure of the portfolio return's dispersionor deviation around the mean. Consequently, the square root of the portfolio variance is theportfolio standard deviation -- the dispersion number itself.

Consider a portfolio containing only two assets with prices labelled by S A and S B . Let theportfolio value be denoted by )S ,SU( BA . Suppose due to some market movement, the valuesof these two assets change by the amounts, S A∆ and S B∆ , respectively. Then, if the portfoliovalue U depends only on the asset prices in a linear fashion, the change in portfolio value due tochanges in asset prices is

The stipulation of linear dependence on asset prices necessarily dictates that higher orderedderivatives are all identically zero so that higher order changes in asset prices do not make acontribution to change in portfolio value.

For simplicity, consider only overnight market movements. What is the dispersion or deviationof the change in portfolio value away from the previous day's value, given these overnightchanges in asset prices? The answer lies in the standard deviation of the variance of the changein portfolio value, viz., [ ]U)var( 1/2∆ . With this, one can now begin to ask questions related tothe potential portfolio losses due to market movements. Because of the overnight nature of theassumed market movements, we can define a concept called Daily-Earnings-at-Risk, denotedby DeaR, as defined by

. SsubBU + SSU = U

SA

A B

∆∂∂∆

∂∂∆


U)var( = DeaR B+A ∆

∆

∂∂∆

∂∂

SU + S

SUvar = B

SA

A B

W W = T•Ω• ,

where the weight vector W and the covariance matrix Ω are, respectively, given by

∂∂

∂∂

SU ,

SU = W

BA

∆∆∆

∆∆∆Ω

)Svar()S ,Scov(

)S ,Scov()Svar( =

BAB

BAA .

How should this overnight concept be extended to a holding period of several days?

This question resulted in the invention of Value-at-Risk, or simply VaR. The VaR for ahorizon (or holding period) of T days is then defined as

T* DEaR = VaR B+A ,

which is nothing but a simple scale-up of the daily risk number by a multiplicative factor givenby the square root of the holding period.

Because VaR looks like, but not necessarily is, a standard deviation number, in order tofacilitate a statistical interpretation, one is necessarily forced to make assumptions regarding thestatistical distribution of the changes in the underlying asset prices in the portfolio. In essence,


VaR is a statistical measure of potential portfolio losses, summarized in a single number, givenan assumed distribution. Common wisdom in the market then decided that a multivariate normaldistribution is easiest to deal with. Consequently, the derived VaR number can also be used toattach a confidence level to any estimate of maximum possible loss.

But do all of these make good sense, let alone what it all means?

In our quest for "truth", we need to ask some tough questions along the way. They are:

(1) Correlation

What really is the correlation ρ AB in the expressions for the covariance terms above, given that σσρ BAAB = B) cov(A, ?

(2) Term Structure There is no consideration of the assets' term structure or time to maturity in the formulation above. What if A is a 5-year instrument and B is a 10-year instrument?

(3) Non-linearity

What if assets A and/or B are not linear instruments, e.g., stocks and futures, and what if they are nonlinear instruments, such as options or callable bonds?

(4) Discontinuity

What if assets A and/or B have discontinuous payoffs at some specific market

levels and both derivatives, given by SU

A∂∂ and

SU

B∂∂ , can potentially go to

infinity?

(5) Square Root of Time

What exactly does _T mean and where does it come from?


Questions (2) and (3) can be answered easily by incorporating the following remedies to our VaRformulation earlier:

a) Break positions down into "primitive" components, the so-called risk factors.

b) Incorporate term structure, i.e., maturity buckets, into the underlying assets.

c) Retain higher-ordered derivatives in the change in portfolio value.

With these simple remedies, we are still left with 3 unresolved dilemmas, namely, correlation, discontinuities, and _T .

Before continuing on with the 3 still unresolved issues, let's talk about each of the proposedremedies.

Identifying the Correct Risk Factors

Risk factors that influence the behavior of asset prices can be thought of as primordial "atoms"which make up a material substance. What are these primitive and atomistic components thatcontribute to the observed movements in asset prices? There aren't that many. For example, abond, at first glance, depends on the term structure of interest rates -- zero rates, to be morespecific. The price of a bond is a linear combination of the present value of some periodicstream of cash flows. The zero rates at those specific cash flow dates determine the discountfactors needed to calculate the present value of these cash flows.

What else affects the price of a bond? Volatilities of each of these zero rates is anotherimportant determinant of bond price. Since interest rates are not deterministic but arefundamentally stochastic in nature, they incorporate a random component or "noise". Volatilityis a manifestation of that noise, uncertainty or randomness. In essence, the most primitivefactors one can think of are: zero rates and volatilities, incorporating term structure or timebuckets in both.

Suppose there are n risk factors, denoted by n1,2,3,..., = i ,RF i . We assume that it is possibleto decompose the price of a trading instrument as a linear combination of these primitive risk


factors, viz.,

RFc = Instrument Trading ii

n

=1i∑ ,

for some constants ci .

With the decomposition as illustrated above, it is mandatory that the following assumptions bemade:

a) the linear combination is possible.

b) the decomposition makes sense mathematically.

c) the expression has a meaningful market interpretation.

We give two examples below to illustrate the concept of decomposing an instrument into itsassociated risk factors and then demonstrate, using another example, the calculation of theportfolio variance. The second example, in particular, forces us to ask even more questions.

Example: Decompostion into Primitive Risk Factors

For the bond example earlier, the precise decomposition into its primitive risk factors is

e CF = Price Bond tz -i

n

=1i

ii∑ ,

where CFi and zi are the periodic cash flows and zero rates, respectively, at times ti . Although the volatilities of each of these zero rates and their adjoining intertemporal correlations are not explicitly shown in the expansion above, these relationships are implicitly embedded in the formulation above. Consequently, the price of this bond is ultimately determined by the movements of the zero rates and how each rate interacts with all others across time.

The implication of the decomposition above is that there is a need to perform some type of


Taylor series expansion, to which we immediately need to make a fourth assumption:

d) the series expansion converges about some given initial market condition.

Let's illustrate this formulation using a simple portfolio containing only two assets and only twotime buckets.

Example: Two-Asset Portfolio with Two Time Buckets

Let the discrete time periods be denoted by t1 and t2 . Suppose the portfolio value, ) , , , ,S ,S ,L ,LU( 21212121 ΣΣσσ , depends only on a small set of market variables given below:

US LIBOR rates: resp. ,t and t times at L and L 2121

Spot FX rates: resp. ,t and t times at S and S 2121

$LIBOR volatilities: resp. ,t and t times at and 2121 σσSpot FX volatilities: resp. ,t and t times at and 2121 ΣΣ

Suppose also that the portfolio value's dependency on these market variables need not necessarily be linear in nature, then the change in portfolio value U is

σσ

11

212

1

2

11

U + )L(LU

21 + L

LU = U ∆

∂∂∆

∂∂∆

∂∂∆

Σ∆Σ∂

∂∆∂∂∆

∂∂

11

212

1

2

11

U + )L(SU

21 + S

SU +

terms higher + )t index for (same + 2 .

Notice that we have explicitly kept terms related only to delta and gamma (first and second derivatives, respectively, with respect to either Li or Si , for 1,2 = i ), and vega


(first derivatives with respect to either σ i or Σi , for 1,2 = i ), and lumped the rest of the derivatives as "higher terms". These are the only observable sensitivities in the market.

The variance of the portfolio change can be quite complicated. Some of the expressions in the variance are,

+ ... +] ,Lcov[ULU2 +] var[U +] Lvar[

LU = U)var( 11

111

1

2

11

2

σσ

σσ

∆∆∂∂

∂∂∆

∂∂∆

∂∂∆

... +] ,Scov[USU2 21

21σ

σ∆∆

∂∂

∂∂

which contain an assortment of variances and covariances of the primitive risk factors.

It can be quite exasperating to compute more terms in the Taylor series expansion and thencalculate their contributions to the variance. But let's not go on. Instead, let's ponder on thefollowing immediate questions which are begging to be asked:

1) What the heck are these higher-ordered derivatives, e.g.,

etc.? ,L

U ,LU

21

2

31

3

Σ∂∂∂

∂∂

Which of these sensitivities have market interpretations and should therefore be

incorporated in the calculation of VaR?

2) Does the Taylor series expansion converge about the current market conditions? What is the range of validity of the expansion when the market experiences huge swings about the current conditions?


3) The variance U)var(∆ contains covariance terms like

etc.. ] , ,Scov[ ] ,)L( ,Lcov[ 212

21 Σ∆∆∆∆

What are these things?

From practical experience, it is very clear that the only observable terms in the market are firstderivatives (corresponding to delta and vega) and second derivatives (corresponding to gamma). The cross derivatives and orders higher than second derivatives have no practical interpretationas far as daily risk management of the trading book is concerned. The only possible exception iscross-currency contingent claims. It is quite difficult to assign any meaningful interpretation tomost of the other covariance terms, particularly those involving higher orders as in question (3).

Equivalence

The formulation above shows that the following equivalence holds:

VaR ⇔⇔⇔⇔ Variance-Covariance

In a general sense, the statement above embodies two distinct conditions: necessity andsufficiency.

Necessity: Under the framework in which it is meaningful to decompose a security into a linearcombination of its constituent risk factors, the quantification of value-at-risk by the equationdefined earlier as

T* U)var( = T* DEaR = VaR ∆

is tantamount to expressing the risk associated with the change in portfolio value, given changein market conditions, in terms of the associated variances and covariances of the different riskfactors in the variance of ∆U. This is a necessary condition of the equivalence statement abovesince it is necessary that a meaningful decomposition of the trading instruments in the portfoliointo their constituent primitive risk factors be possible.

Sufficiency: Conversely, if enough of the meaningful and interpretable variances andcovariances can be retained in the calculation of var(∆U), then they are sufficient to encapsulate


the risk due to changes in portfolio value brought about by changes in market conditions.

Finally, for the equivalence statement to be truly useful, the following assumptions need to beimposed on the VaR framework formulated as variance-covariance:

1) linear decomposition of trading instruments into risk factors is possible and the linear combination makes sense.

2) all partial derivatives used in the series expansion exist, are bounded, and have market relevance.

3) correlations in the covariance expressions are estimatable, stable through time, and therefore make sense.

4) for longer horizon risk analysis, the √T- Rule as a scaling factor for the instantaneous change in portfolio value is applicable.

Assumptions on the Potential Loss Distribution

Because VaR is cast in a variance-covariance framework, to interpret it as a measure of risk dueto adverse movements in market conditions requires the estimates of adverse future asset pricemoves using historical information of previous price moves. We immediately have to ask:

1) Can historical data really predict the future ?

2) What statistical assumptions are necessary to ensure that VaR has a probabilistic interpretation ?

Question (1) will be answered in a later section. We'll address question (2) in this section.

The probability distribution of the portfolio's future instantaneous value depends primarily on:

· the linear representation of the primitive risk factors, and· the joint distribution of instantaneous changes in these risk factors.


It is instantaneous because all the derivatives involved in the calculation of var(∆U) aremathematically meaningful only for an infinitesimally small length of time. Although we havestretched this time frame to account for the change in portfolio value due to an overnight changein market conditions, effectively a one day move, the instantaneous nature of the partialderivatives in ∆U still remains irrefutable. The dependence on primitive risk factors is a givenbecause of the equivalence statement between VaR and the variance-covariance formulation. Inaddition, the dependence on some kind of joint distribution is a requirement since these riskfactors, e.g., zero rates and volatilities, do not evolve in time independently of each other. To alarge extent and with very few exceptions, they are inter-related.

What important assumption is required concerning the joint distribution of primitive risk factors?

Practitioners in the market often make one big leap of faith and assume a normal jointdistribution with determinable (?!) return variances and covariances. The argument for a normaldistribution is rather ad hoc although it easily facilitates a statistical interpretation of VaR viaconfidence levels. However, many of these variances and covariances are problematic and aredifficult to infer from the market. In most cases, they cannot even be determined purely fromstatistical analyses of historical data without any kind of subjectivity.

Without an assumption regarding the distribution, the calculated VaR number is but just anumber -- that's all. With the imposition of a normal distribution on the number given by VaR, however, it is possible to interpret this VaR number as a standard deviation from the mean overa small interval of time, whether or not the act of interpretation actually makes sense. Also, with the assumption of normality, the task of estimating the percentile of a probabilitydistribution becomes easy. If a distribution is normal, then all percentiles are simply multiplesof the distribution's standard deviation. In this case, one standard deviation implies aconfidence level of 84.1%, while 1.65 and 2.57 standard deviations can be translated to 95% and99.5% confidence levels, respectively.

Example:

At a confidence level of, say 95%, there is a 95% chance that the change in portfolio value, on an overnight basis, will not exceed the calculated value of U)var(∆ . Conversely, one can also say that there is a 5% chance that the change in portfolio value will exceed this number. Because a change in portfolio value can either be positive or negative, this amount is usually interpreted as a potential gain or loss in portfolio value.


In the next example below, we illustrate further using a little bit of probability.

Example:

The impetus in developing value-at-risk methods is so that trading institutions could ask the following question:

How much money might we lose over the next period of time?

VaR answers this question by rephrasing it mathematically: If X T is a random amountof gain or loss at some future time T, and z is the percentage probability, what quantity v

results in?z = v - < XProb T

The quantity v is the VaR number we seek to find.

Since v(T,z) is both a function of T and z, clearly, in order for the VaR number to have some meaningful interpretation, we need to attach both a time horizon T and a probability distribution.

Alas, with confidence levels circumscribed on it, VaR even has probabilistic meanings attachedto it! With all the statistical accoutrements, it is now a very credible number. Believe it or not.

Figure 1 is a graphical interpretation of VaR using an assumed normal distribution.


Figure 1: Probability of Loss Distribution

Just exactly what is being assumed to have a normal distribution? We had better have an answerto this question if the calculated VaR value were to have a meaning.

To boldly assume that VaR, as a random number, is drawn from a normal distribution can bequite unpalatable. In fact, no where in the procedure discussed earlier, leading to thedetermination of the final VaR result, was there any statistical assumptions made regarding theprimitive risk factors, their changes, or the VaR number itself. So then, how can this VaR

C e n t e rV a R

P r o b a b i l i t y o f g a i n o r l o s s

VZ

- L o s s + G a i n

P r o b X T < - v = z

P r o b a b i l i t y o f L o s s D i s t r i b u t i o n


value suddenly become a random number imbued with statistics? In the ensuing discussions, we present some arguments one way or the other.

Are the Statistical Assumptions about VaR Correct?

Instead of insisting that VaR is drawn from a normal sample, we need to look at the constituentrisk factors which made up the VaR number. In particular, since value-at-risk is defined by

T* U)var( VaR ∆≡ ,

we need to look at ∆U, the change in portfolio value, given changes in market conditions.

Referring back to our earlier example on a portfolio containing only two assets with two timebuckets, the instantaneous change in portfolio value is given by the linear combination,

σσ

11

212

1

2

11

U + )L(LU

21 + L

LU = U ∆

∂∂∆

∂∂∆

∂∂∆

Σ∆Σ∂

∂∆∂∂∆

∂∂

11

212

1

2

11

U + )L(SU

21 + S

SU +

terms higher + )t index for (same + 2 .

The items ,)( ,... ,S ,)L( ,L 22

112

21 Σ∆∆∆∆∆ σ are all quantities related to either first order orsecond order changes in market conditions, e.g., changes in LIBOR rates, changes in spot rates, changes in volatility, etc.. If we first assume that all these market rates are normally distributed, then their associated linear changes are also normally distributed. Unfortunately, the secondorder changes obey a Chi-Square distribution, instead of a normal distribution. Hence, wehave the following observations:

a) if ∆U consists strictly of linear changes alone, then ∆U is also normally distributed.


b) if ∆U also contains second order changes, then ∆U is no longer normally distributed but is combination of both chi-square and normal distributions.

It is, therefore, evident that a VaR framework which attempts to incorporate gamma and otherhigher-order sensitivities cannot establish itself on the familiar foundation of normal distribution. Instead, one has to go out on a limb and make the bold assumption that VaR itself, as anumber, is drawn from a normal distribution without any sound theoretical justification. Thiscan't be right. More importantly, in most cases, the market variables underlying the portfoliovalue are often not normally distributed. Hence, observation (a) is also not quite true unless theportfolio is sufficiently large so that one could, in principle, invoke the central limit theorem.

The central limit theorem asserts that "as the sample size n increases, the distribution of themean of a random sample taken from practically any population approaches a normaldistribution". Interestingly enough, the only qualification for the theorem to hold is that thepopulation have finite variance. Therefore, one can indeed facetiously argue that if var(∆U)were finite (translated loosely as, incurring neither infinite gains nor infinite losses), then thecentral limit theorem holds.

How much can we lose over the next period of time, we ask? Less than infinity -- would be theappropriately terse, but silly, response.

So be it then -- deus ex machina -- large portfolio, ergo, normal distribution.

Non-Normality of Returns

Although the assumption of normality (on either the changes in primitive risk factors or on thedistribution from which the calculated VaR number is drawn) was actually never made during thecalculation of risk exposures, there are distinct consequences for not having a normal or even"near" normal distribution. Most important among these are:

· Predictability of tail probabilities and values.

· Dynamic stability of normal parameters, e.g., mean and standard deviation.· Persistence of autocorrelation.


VaR is intended to be used as a predictor of low probability events (technically known as "tail"probabilities) such as a 5% or less chance of potential change in portfolio value. Becausemarket variables are inherently non-normal, it would be virtually impossible to verify, withcertainty, the accuracy of the probabilities of extremely rare events. The calculated VaRnumber, having not been truly drawn from a normal distribution, is in itself a dubious number, let alone one which entails predictive capability about potential overnight change in portfoliovalue.

In addition, the large literature on the "fat tails" of financial asset returns adds to the problem ofinterpretation. Fat tails exist because there are many more occurrences far away from the meanthan that predicted by a normal distribution. This phenomenon of leptokurtosis (having largermass on the tails and a sharper hump consistent with the normality assumption) tends tounderestimate low probability losses. Furthermore, the inherent bias introduced by skewness(more observations on the left tail or the right tail) can be exacerbated by whether the truedistribution is left or right-skewed and whether the positions in the portfolio are predominantlylong or short.

Secondly, given that returns on assets (as quantified by the changes in the primitive risk factors)are in reality not normally distributed, the dynamic stability of the associated normal parameters, namely, mean and standard deviation, can be called into question. Statistically speaking, ifnormal parameters are stable over time, then past movements can be used to describe futuremovements. In other words, since standard deviation is a measure of dispersion (or uncertainty)of the distribution about the mean, the more stable the standard deviation is over time, the morereliable the prediction of the extent of future uncertainty. This is the fundamental principleunderlying the VaR vis-a-vis the variance-covariance framework. Because the predictivecapability of the calculated VaR number depends on the calculation of the portfolio covariancematrix using historical data, the predictive power of VaR is doomed from the very start if thenormality assumption is not made. The argument has now become rather nonsensical andcircuitous. There is no way out.

Paradoxically, the important thing to bear in mind is that in our previous description of the VaRframework using historical data to calculate the associated covariance matrix, an assumption ofnormality was never made as a basis for estimation -- it just wasn't necessary. So, why is anassumption of normality now so urgent and important?Thirdly, are today's changes in the risk factors related to yesterday's changes? For most financialproducts, the answer is more likely to be in the affirmative. This implies that asset returns arenot serially independent. Consequently, the time series of price changes are correlated from one


day to the next. This persistence of autocorrelation comes about because the parameters of thedistribution are not constant but are time-dependent or time-varying. The next section addressesthis issue.

Time-Varying Parameters

Asset returns are temporally dependent. More specifically, volatilities and correlations of assetprices are not constant over time and they tend to bifurcate or make rapid jumps to differentplateaus as dictated by the different regimes in market conditions. They can vary through timein a way that is not easily captured through simple statistical calculations. This means thatsystemic time-dependent biases will vary with the holding period and will be conditioned on thetime of market risk prediction of losses as calculated by VaR. This insight forced us earlier toincorporate term structure or time buckets into our variance-covariance formulation, with thehope that by incorporating time dependencies of both volatilities and correlations, shifts inmarket regimes would also be captured as a consequence.

Unfortunately, the degree of time-dependent effects varies across instruments and there exists nosingle cure-all panacea. Timely and accurately determined estimates of both volatilities andcorrelations are especially important after a change in market regime, but this requires vigilanceand some level of subjectivity on exactly how they are to be calculated. This brings us full circleto the unanswered question we asked earlier: can historical data really predict the future?

Historical Data: Does History Determine the Future?

If it is true, as expressed by George Santayana, that "those who cannot remember the past arecondemned to repeat it", can we twist this question around and ask, "does knowing the pastinfluence the future?". The crux of the philosophy lies in the √T- Rule.

The √T- Rule, given by

T* DEaR = VaR ,

implies that the risk exposure in the portfolio over a period of T successive days into the futurecan be determined historically from today's risk position given by the daily-earnings-at-riskvalue, DEaR. As usual, the following tough questions need to be asked:


1) Under what conditions can this claim be true?

2) Where did the √T- Rule come from?

3) What is the range of validity for the claim?

4) What are the implications on how historical data should be handled?

We will answer Questions (1) and (2) first.

Serial Independence and Origins of √T- Rule

Two sections ago, we talked about serial dependence -- that the time series of price changes arecorrelated from one day to the next. In a Black-Scholes environment for pricing contingentclaims, the assumption of a true random walk (i.e., Brownian motion) is always invoked. Whilethe market, to a large extent, is not truly random, the assumption of Brownian motion is veryconvenient. It certainly facilitates nice-looking and easy formulas for use in valuation.

For risk management purposes, a fundamental component of the random walk assumption relieson treating each incremental change ∆S in asset price over a small time interval ∆t to beindependent and identically distributed in a normal fashion. In simple words, each incrementalchange in asset price is assumed to be normally distributed and unrelated to preceding changes atearlier times. A time series of these changes is, therefore, serially uncorrelated. Let'sdemonstrate this point.

Suppose the change in asset price at time T is denoted by ST∆ . Suppose also that the changefollows a random walk so that the change at time T is a result of a change in previous time T-1, triggered by some "noise" εT , viz.,

εT1-TT + S = S ∆∆ .

Each of the noise terms, being Brownian motions, is uncorrelated with zero mean and has thesame constant variance, say σ 2 .

We can now iterate successively on the equation above to yield


εεεε 12-T1-TT0T + + ... + + + + S = S ∆∆

ε i

T

=1i0 + S = ∑∆ .

This equation is quite insightful -- the change at time T is due to the change at an initial time 0plus the incremental sum of past "noises" generated between time 0 and time T.

Taking the expectation of both sides of the equation results in

SE[ +] SE[ =] SE[ 0i

T

=1i0T =] ∆∑∆∆ ε ,

and hence, the dispersion of the change from the mean is

σε 2i

T

=1i

22

0T * T = E =] )S - SE[(

∆∆ ∑ .

This is again very insightful -- the dispersion or uncertainty of the change in asset price at time Tis nothing but the uncertainty of the individual "noises" σ 2 multiplied by the length of theobservation time T. In other words, the uncertainty is an accumulation of each successive noisecontribution leading from the initial time to the final observation time T. The past does, indeed,foretell the future, albeit relying only on the ramblings of the past. Also, the standard deviation, being the square root of the variance, is simply T* σ . Voila! Herein lies the square root ofT!

What have we learned?

Only through the asssumption of a true random walk (via Brownian motion) is the time series ofchanges in asset prices serially uncorrelated, resulting in the √T- Rule. Serial independence orthe absence of autocorrelation is, therefore, a required assumption for using the √T- Rule.


Later on, we will examine the √T- Rule in greater mathematical details and reveal one moreintriguing requirement for it to be applicable.

Historical Data Usage

Given the discussion above, does it matter how historical data are used in the calculation ofVaR? It does indeed. The length of the historical time series used is also relevant.

Traditional methods of estimating volatility and correlation using time series analysis have reliedon the concept of moving average. Moving averages come in two favorite flavors: fixedweights and exponential weights.

Consider a time series of observations, labelled by x t . Define the exponential movingaverage, as observed at time t, by a weighted average of past observations, viz.,

x x i-ti

n

=1it ω∑≡ˆ

where the weights are 1 < < 0 ,) - (1 = ii λλλω , and which must sum to 1. The parameter λ is

chosen subjectively to assign greater or less emphasis on more recent or past data points. Inpractice, the choice of λ is dictated by how fast the mean level of the time series x t changesover time.

A simple arithmetic average results if all the weights are set equal to 1/n, where n is the numberof observations. The choice of the number of data points n can also be subjective. The decisiondepends on both the choice of the parameter λ and the tolerance level on how quickly decliningthe exponential parameter should be.

Common practice, being more art than scientific statistics, is to choose n=20 and λ between[0.8, 0.99], regardless of what kind of assets are being analyzed. Since this paper is not aboutthe statistics of VaR, we'll have to content ourselves with keeping these comments on a cursorylevel without asking any questions.

One comment we must make, however, is that since VaR is scaled up from an overnight riskexposure number DEaR, it is preferable to place relatively heavier emphasis on more recent datapoints. After all, using an excessively large number of past observations tends to smooth out


more recent events and is, therefore, seemingly contradictory to the overnight intent of DEaR.

The exponential moving average places relatively more weight on recent observations. Notsurprisingly, the goal of an exponentially moving average scheme is to capture more recent shortterm movements in volatility along the lines of a popular and more sophisticated estimationmodel called GARCH (1,1). The moral lesson seems to be pointing to practicalities in riskmanagement using simple but sensible tools, instead of reliance on more sophisticated but ratherintrepretative estimation models.

To round up our discussion on the alleged predictive power of historical data and serialdependency, we need to examine the √T- Rule more closely.

√√√√T- Rule Further Examined

Where does the √T- Rule come from? We have partially addressed this question in an earliersection on serial independence, albeit heuristically. We need to bring in more mathematics thistime.

Consider the price of an underlying, denoted by F, which diffuses continuously as,

dz + dt = dF σµ , (continuous version)

where the Brownian motion of F is governed by the so-called Wiener measure dz. The Wienerprocess is a limiting process of infinitely divisible and compact normal stochastic process withincrements modeled as

t)O( + t + t = F ∆∆∆∆ εσµ ~ . (discrete version)

By "infinitely divisible" we mean each small increment of time ∆t can be chopped into evensmaller pieces, ad infinitum. The symbol O(∆t), read as "order of ∆t", is a short-hand notationfor ignoring higher orders beginning with ∆t. This is, in fact, a unique feature of Brownianmotions in which disturbances or noises larger than a small increment proportional to t∆ donot contribute to the path taken by the price F of the asset. Mathematically, the discretizedversion of the continuous Wiener measure dz is


t =z ∆∆ ε~ , where N(0,1)~ ε~ .

Thus, the variance of ∆F ist)O( + t =F] var[ 2 ∆∆∆ σ .

The volatility or standard deviation is the square root of the variance, viz.,

0 >- t smallfor ,t~ F)vol( ∆∆∆ σ .

Observe that we have recovered the √T - Rule, but this time with a very important caveat -- theinterval of time ∆t considered is required to be small. The scale factor of √T in VaR, however, is not intended to be small. Regulatory pressures insist that the holding period T should be 10days!

There are some serious implications regarding the applicability of the √T - Rule. We can nowsummarize them:

a) Serial independence of the time series of changes in primitive risk factors needs to be assumed although, for the most part, changes are primarily autocorrelated.

b) Each change in the primitive risk factors must be assumed to have a normal distribution so that the principle of random Brownian motion is applicable.

c) The rule can only be used if the time period of observation is sufficiently small although, in contradiction, the practical intent is to use a 10-day holding period.


Incorporating Options Positions in VaR

In contrast to "linear" instruments such as stocks, futures, bonds, and forward contracts, optionsare derived instruments with payoffs contingent on the future paths taken by the linearinstruments mentioned. For instance, a call option, on a stock S and struck at price K, has aterminal payoff at maturity time T of K] - S [0, Tmax . This contingency of non-negative payoffat maturity forces the value of the call option to be nonlinear for all times prior to maturity -- thesharpest nonlinearity or convexity occurring around the strike price. This is the reason optionsare considered "nonlinear" instruments. In the context of the value-at-risk framework, theimmediate questions to ask are:

1) Can an option be decomposed into a linear combination of primitive risk factors?

2) Does the decomposition make sense?

3) Do the higher-order terms in the decomposition have any market interpretation?

Although these questions were asked earlier in regard to linear instruments, they need to beasked again even more so in the context of nonlinear instruments. To answer these questions, one needs to be aware of the following points:

· Naively incorporating higher-ordered partial derivatives can have misleading and disastrous effects.

· Since not all contingent claims are created equal, the degree of nonlinearity has to be taken into account properly.

· Interpretation of non-market observable partial derivatives needs to be carefully thought out.

· Discontinuities in option payoffs need to be considered in light of unbounded partial derivatives.

· Incorporating different kinds of "greeks" is not a trivial matter.

The simple example below illustrates the significance of these questions and remarks.


Example:

Suppose the only parameter of importance is the underlying price. Then, the change in the value V of an option, with respect to a change in the underlying price u, is given by

... + )u(uV

21 +u

uV =

V(u) - u) +V(u = V

22

2

∆∂∂∆

∂∂

∆∆

The first and second partial derivatives have market interpretations of "delta" and "gamma", respectively, so we rewrite it as

... + )u(* gamma* 21 + u)(* delta >- V 2∆∆∆

Other higher derivatives with respect to the underlying price have no market interpretations. For a hedged portfolio, if the magnitude of the changes in the underlying price and price volatility of the underlying asset is sufficiently small, the approximation for ∆V up till the second order term is sufficient to capture the change in value of the portfolio.

Incorporating Other Risks

To incorporate volatility risk, one normally includes a correction of the following form,

σ

σσ

σσσ

∆

∆∂∂∆∆

(vega) >-

V =

)V( - ) + V( = V


In practice, a second order correction to vega is not necessary since, for most options, therelationship tends to be almost linear in nature and the order of magnitude is insignificant.

Option payoffs are curvilinear in nature and are therefore not symmetric about the currentunderlying price u. To take into account the difference in either a price increase or a pricedecrease, the derivatives -- as approximated by finite difference -- need to be taken along bothsides of the current underlying price.

In general, to make the approximation of the change in portfolio value more robust, one needsto consider a multivariate Taylor series expansion of the option value V(u,r,σ,t) as a function ofthe underlying price u, interest rate r, volatility σ, and time t, among other things, viz.,

... + t(theta) + (vega) + r(rho) +u (delta) =

terms order higher + ttV + V + r

rV +u

uV =

t),r,V(u, - t) + t , + r, + r u, +V(u = t),r,V(u,

∆∆∆∆

∆∂∂∆

∂∂∆

∂∂∆

∂∂

∆∆∆∆∆

σ

σσ

σσσσ

Term Structure Effects: Greek Ladders

Term structure effects are very important and must not be neglected. An expansion similar toour earlier example on two-asset portfolio with two time buckets needs to be performed when theportfolio contains options. In effect, each "greek" expression above needs to be replaced by itscoresponding "ladder" -- the rungs of the ladders increasing with time to maturity.

For instance, consider an option position which matures at time t5 and uses only 5 distinct timebuckets, namely, t ,t ,t ,t ,t 54321 . Since these 5 time buckets are used as distinct key points forrisk managing the position, the delta associated with the option position requires 5 rungs in itsdelta ladder, viz.,

]delta ,delta ,delta ,delta ,delta[ ttttt 54321 .

The same term structure effects should be applied to the rest of the "greeks".


Assumptions Required for Incorporating Options Positions

The discussion above forces one to reckon with the assumptions required for the VaR frameworkbefore it can properly capture risk exposures introduced by the presence of options in theportfolio. The assumptions (in principle, more like common sense rules than assumptions) arenow self-evident:

1) Incorporate only those "greeks" which are observable in the market and which are actually used for risk managing the option positions. The common "greeks" are delta ladders, gamma ladders, vega ladders, rho ladders, and to a lesser extent,

theta.

2) Assume that the observable greeks are sufficient to capture the various degrees of curvilinearity in the option positions. This, in turn, requires the assumption that the Taylor series expansion of the option value in terms of the observed greeks converges for small perturbations around the set of current market parameters.

3) Assume that discontinuous payoffs in options do not lead to unbounded partial derivatives so that the series expansion is meaningful.

Assumption (3) is very interesting and deserves additional analysis. Plain vanilla optionswithout discontinuous payoffs have very smooth and bounded partial derivatives. That is not thecase for many exotic options and their various embeddings in complex structures. We need toaddress this next.

Incorporating Non-Standard Structures

Overall, the VaR vis-a-vis variance-covariance framework is generally not suitable for non-standard structures with discontinuous payoffs. For instance, consider a so-called digital calloption, defined by the payoff function at maturity time T,

K < S if ,0 T

K S if ,1 T ≥ .


There is a sudden jump in payoff from zero to one if, at expiry, the stock price ST is greater thanor equal to the strike price K. Figure 2 illustrates the terminal payoff function for the digital calloption.

Figure 2: Digital Call Option

As illustrated in Figure 2, the value of the option at any time prior to expiration is smooth andcontinuous. Prior to expiry, all partial derivatives with respect to the underlying price S arebounded and well-defined. As the time to expiry diminishes, these derivatives increase

K

D e r i v a t i v e s “ b l o w u p ”

s

D i g i t a l C a l l O p t i o n


dramatically without bound until they finally "blow up" at expiration, in accordance to thepayoff function given above. Therefore, for a digital call, we ask what is the meaning of

T >- t as , + >- SC

K=S

∞

∂∂≡∆ .

Does it make sense to incorporate this quantity with other well-behaved and bounded greeks?

Carelessly mixing and matching greeks from non-standard structures (which have a naturalpropensity to "blow up") with well-behaved bounded ones from vanilla options has some seriousconsequences. Because same-letter greeks are normally treated as additive when risk managinga portfolio, increasingly larger and larger greeks due only to a single option position cloud thetrue risk profile of the portfolio. Close to the discontinuity, the contribution of, say, the delta, to the portfolio variance can either dominate or completely overwhelm the total risk profile. Secondly, because the Taylor series associated with such non-standard structures are potentiallymeaningless, the calculated VaR number (being the portfolio variance) is also likely to make nosense.

Incompatibility of "Greeks"

There are fundamental differences between a Black-Scholes delta and a "delta" calculated from aone basis point parallel shift (or some other kinds of non-parallel shifts) in the yield curve. Vegasensitivities implied from the Black-Scholes world and those resulting from a calibrated lattice ofinterest rate model for valuing interest rate contingent claims are also fundamentallyincompatible. The same fundamental differences also hold for other risk sensitivities. If thesedifferences are not fully recognized and then resolved, the variance-covariance framework willnot be able to successfully incorporate these disparate kinds of risk sensitivities into a meaningfulmeasurement of risk.

The variance-covariance framework, by its very construction, forces one to aggregate thevarious risk sensitivities of all instruments within the portfolio -- regardless of the inherentnature of these instruments. The VaR formulation does not even distinguish among the distinctshades of quirkiness of non-standard structures since the basic tenet of the formulation has


always been that "it is possible to decompose any instrument in the portfolio into a linearcombination of its underlying primitive risk factors". Of course, the primitive risk factors arethe atoms that the structures -- vanilla, non-standard or otherwise -- are made up of. However, from a risk management perspective, risk sensitivities (e.g., delta, vega or gamma) are neithercalculated nor used by the various trading desks (even within the same institution) in a consistentand uniform manner.

As an example, consider the case of a swaption, requiring an in-house calibrated stochasticinterest rate model to determine its value and its day-to-day risk sensitivities. The model isnormally calibrated to observed market traded caps/floors and European swaptions, after whichthe swaption in question is then valued using the model. Also, market prices are quoted in termsof one single volatility as implied from Black's 76 formula.

The implied Black volatilities of the calibrating caps and floors constitute a linear array, arranged accordingly by tenor. For swaptions, it is not as simple. A swaption is an option onsome underlying swap. Thus, swaption volatilities do not form linear arrays but rectangularmatrices instead. The two-dimensionality is necessary to take into account both the option tenorand the term to maturity of the underlying swap.

From a variance-covariance perspective, there are immediately two issues to confront:

a) the meaning and construction of the vega ladder.

b) compression of two-dimensional swaption volatility into a one dimensional risk exposure ladder.

On one hand, vega ladders for swaptions are clearly not acceptable, as swaption volatilitiesgeometrically constitute a two dimensional surface and not a linear array. On the other, thevariance-covariance framework, as constructed in earlier sections, is a quadratic multiplicationof the weight vector W and the covariance matrix Ω, viz.,

W W T•Ω•

where the weight vector W is composed of the various greek ladder sensitivities, arranged in alinear array. Clearly, the linear framework of VaR does not allow for a matrix structure.


A compromise, albeit not an ideal one, is to "collapse" or compress the matrix array into a lineararray along the tenors of the swaptions, ignoring the terms to maturity of the underlying swaps. Doing the compression, however, introduces mixing and matching of vegas into the variance-covariance framework, thereby rendering the calculated VaR difficult to interpret. There is nosimple way out and vega, unfortunately, is not the only greek sensitivity afflicted with thisproblem.

Using Proxies and Short Cuts

Earlier we discussed the handling of historical data and their predictivity in loss estimates usingthe VaR framework. In this section, we discuss issues concerning data quality and what to do inthe absence of sufficient amounts of historical data. The proper usage of statistics aside, theconstruction of the variance-covariance framework, as articulated so many times before, requires careful usage of historical data and myriads of assumptions about the data's behavior. What happens then when either market and/or position data are hard to come by?

For highly illiquid instruments, such as many of emerging markets bonds and currencies data, "clean" historical data are simply unavailable. In general, data streams from different sourcescould have gaps or missing data which then require some subjective patch work. Data which areparticulary "noisy" also require some kind of subjective "scrubbing" and cleaning. Arguments always ensue when deciding how much "bad" data to remove or how much patch work need tobe quilted onto the database.

Many of the data streams warehoused in a database may be asynchronous -- i.e., not collected atthe same time during the course of a trading day. They are, therefore, contaminated by intra-daytrading activities. In addition, an institution's daily collection and aggregation of trading positionsummaries from the various in-house trading systems may not be performed at the same timeacross all trading desks and installations. This means that both market data and an institution'sown trading position data could potentially suffer from asynchronicity. The impact of timingmisalignment certainly could be significant in the estimate of the portfolio covariance matrix.

Some proxies and short cuts commonly used in practice are:

a) use in-house default factors for minor foreign exchange exposures instead of using undesirable and incomplete market data.


b) when long term rates, say beyond 10 years or so, are not available, perform linear extrapolation to obtain longer periods on a flat basis.

c) ignore basis risk by using LIBOR-based curve as the underlying curve and adding a constant spread to it in lieu of appropriate curves for other fixed income securities.

d) since short rates tend to be noisy, retain short rate information without change for several days worth of VaR calculation.

e) since some implied volatilities cannot be inferred from the market, use calculated historical volatilities as proxies.f) aggregate certain gamma or vega buckets into one bigger bucket when the entire ladders are not available from trading systems.

For a myriad of technical reasons, proxies and short cuts are often used to represent actualmarket data, whenever possible. It is, therefore, imperative to ask: is it prudent to use proxieswhen market data are sparse or unclean? There is, unfortunately, no good answer. Becauseproxies and short cuts are a necessary evil and may have to be used despite the lack of fulljustification, one is usually left with no satisfactory nor defensive explanations.

Some Evident Proxy Dangers

In the context of the variance-covariance framework, we point out one particular undersirableeffect of using proxies for interest rates. Similar arguments hold for other proxies. Let the realor true (but unobtainable) rates be labelled ri at time ti be proxied by some rates li , viz.,

s+lr iii ≈ ,

where si represents the spreads (or "errors") over the unobtainable true rates. Using this proxy, the change in portfolio value U is

ssU + l

lU U i

iii

iiri

∆∂∂∆

∂∂≈∆ ∑∑ .

The portfolio variance, ]Uvar[ ri∆ , naturally contains terms like


)l ,lcov( ,)lvar( jii ∆∆∆)s ,scov( ,)svar( jii ∆∆∆

)l ,scov( ji ∆∆ .

The presence of these covariance terms points to a potential problem with

a) the persistence of autocorrelation, andb) correlation between the time series of proxies and the error terms.

Depending on the instruments being proxied, the error terms may not be small. One isnecessarily forced to make very strong assumptions regarding the behavior in items (a) and (b)before a variance-covariance framework can be used to determine VaR. There is no further needto elaborate on these points since we have already made several arguments earlier along this linewhen a discussion of serial independence was made.

Validating Risk Measurement Model Results

Model validation, from a scientific perspective, is difficult and problematic. Given all thereservations, ranging from data usage to model assumptions to the probabilistic interpretation ofrisk measurement model outputs, it is extremely difficult to keep a scientific straight face andsay "all is good". In fact, all is not good if the intent of using the risk measurement model is notclear from the outset.

The recent regulatory impetus to use the VaR number for market risk capital purposes is oneproblematic and unjustifiable usage of risk measurement models. For a variety of very goodreasons, trading desks in reality do not use this highly condensed VaR number for risk managingtheir trading positions. Why, then, are we so enamored of this number?

We definitely need to ask the BIG question: What is this calculated VaR number used for?

Following up on this query, any sane modeler or user of the model needs to ask: Is thissupposed to be rocket science or is it an exercise in prudent risk management?

It is reckless to rely too heavily upon risk measurement models as crystal balls to predict


potential losses or to assess capital charges for trading activities. It is equally reckless not tobecome fully aware of the myriad limitations imposed on such a risk measurement model. Thehuman elements of prudence, sound judgment, and trading wisdom -- all non-quantifiablesubjectivity -- cannot be more emphatically stressed than what we already have throughout thisarticle.

Back-Testing

In practice, many institutions with internal risk measurement models routinely compare theirdaily profits and losses with model-generated risk measures to gauge the quality and accuracy oftheir risk measurement systems. Rhetoric aside, given the known limitations of riskmeasurement models, historical back-testing may be the most straightforward and viable meansof model validation.

Back-testing has different meanings to different people. Regulatory guidelines suggests back-testing as "an ex post comparison of the risk measure generated by the model against actual dailychanges in portfolio value over longer periods of time, as well as hypothetical changes based onstatic positions". The essence of all back-testing efforts is, therefore, to ensure a soundcomparison between actual trading results and model-generated risk measures. The logic is thata sound VaR model, based on its past performance and under duress of the back-testingprocedure, will accurately portray an institution's estimate of the amount that could be lost on itstrading positions due to general market movements over a given holding period, measured usinga pre-determined confidence level. Events outside of the confidence intervals are deemedcatastrophic market phenomena unforeseen by the model. But how many institutions really havethe ability to combine their entire trading portfolios into one single risk measurement system?

Figure 3 illustrates the result of one instance of back-testing for a sample portfolio. The graphicpresentation is intuitively appealing. The "envelope" is an indication of the theoretical bound ofpotential change in portfolio value vis-a-vis actual observed P/L. There is really no good reasonto develop further statistical tests -- disguised in the name of science -- as currently beingadvocated in the market by some people to validate a relatively unscientific risk measurementmodel. Of course, one could facetiously push the borders of the envelope further away fromactual P/L observations by "tweaking" some key data inputs into the calculation of the VaRnumber or by simply multiplying some unjustifiable scaling factors to the VaR number. Common sense appears to be the key judgment factor that determines what is proper and what isnot.


Figure 3: Back-Test Results

R e s u l t s o f B a c k T e s t i n g f o r a S a m p l e P o r t f o l i o

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- 1 5 0 0

- 1 0 0 0

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Some "Final" Thoughts

It is true that senior management tends to prefer a single condensed measure of the institution'strading risk. Since VaR has all this intuitive appeal as a condensed big picture of risk exposure, VaR has gained significant support primarily from the regulatory sectors and has been touted tosenior management as a risk control device. But what is missing is that trading risk cannot bequantified strictly in terms of a single measure. Trading decisions involve a dynamic multi-faceted interaction of several external factors. The shift of a specific point on the yield curve, the jump in the payoff characteristic of a structured note, or the specific tenor of an embeddedoption and its path-dependencies, to name just a few, are the contributory factors to the riskprofile of a trading portfolio. VaR is none of these.

This value-at-risk number attempts to express the magnitude of multi-dimensional market risk asa scalar. Reliance on a one-dimensional VaR number -- by sheer faith -- is both perplexingand mind-numbing, even to myself who is instrumental in constructing such an internaltheoretical framework for risk measurement. As defined through its equivalence with thevariance-covariance framework, value-at-risk, regardless of confidence levels, changessignificantly depending on

· the time horizon considered· correlation assumptions· integrity of database and statistical methods employed,· the choice of mathematical models.

Consequently, VaR does not and cannot provide certainty or confidence outcomes, but rather, an expectation of outcomes based on a specific set of assumptions, some of which are verydifficult to justify. Other equally important risks, e.g., liquidity risk, model risk, opertationalrisk, etc., are very difficult to quantify and are, therefore, not properly taken into account by acondensed VaR number.

So, have we learned something about VaR or from VaR?

Indeed we have -- very important ones. The lessons are not in the mathematics nor probabilitynor statistics nor in the soothsaying power of VaR nor the numerous highfalutin ways ofextracting, smoothing, and interpolating data. All of these tools are there, but they are notimportant.


Our quest for truth has ended right here.

The recent attention focused on VaR, together with its implication on trading losses andregulatory capital requirement, has triggered some serious queries and attracted keen attentionfrom senior management. For a scientist, an academic, a risk manager, and a seeker of truth, such as myself, I am utterly delighted when ranking executives of the Bank become moreinvolved and begin to ask serious questions about mathematical models, multi-factor mean-reverting diffusion processes, implied volatilities, and the like -- none of which, of course, arewithin their technical comprehension.

How refreshing it is for a member of the Board to query, with genuine awe, about the impact ofCMT caps on the trading portfolio and in so doing, asks "What is a CMT cap?". Howenlightening it is for the CEO to inquire whether the current condition of systems infrastructure issufficient to support more American style swaption trades. The heightened awareness of seniormanagement brought about by VaR, and their participation and concerns in the day-to-day riskmanagement process, stands to benefit all of us who are in the financial industry. For it is in theawareness and support of senior management -- and through the prudent day-to-day riskmanagement functionalities -- that an institution engaged in market risk activities can trulyprotect itself from losses and from its own trading follies. It is not through a single value-at-risknumber generated by some internal model, however rational its assumptions may be.

Indeed, a dialog has begun -- thanks to VaR.


References

Basle Committee on Banking Supervision, Amendment to the Capital Accord to IncorporateMarket Risks, January 1996.

International Institute of Finance, Specific Risk Capital Adequacy, Sept. 20, 1996.

RiskMetrics -- Technical Document, 2nd Edition, J. P. Morgan, November 1994.

Kupiec, P. H. and J. M. O'Brien, The Use of Bank Trading Risk Models for Regulatory CapitalPurposes, Board of Governors of the Federal Reserve System, Finance and EconomicsDiscussion Series, March 1995.

Acknowledgement: The author wishes to thank Art Porton for the outstanding graphic work and his critique of the manuscript.

ong (1997) - explaining the assumptions used in the measurement of value at risk

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