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VaR Models: Summary, Applications and Limitations

261

In the second place, the conﬁdence level used in calculating VaR may be increased

at will, if needed, so as to capture a higher percentage of events, if this is considered

to be appropriate in order to have a more comprehensive picture of the possible future

scenarios. If a bank increases its VaR model’s conﬁdence level, and adjusts its capital

endowment accordingly, its creditors will feel more protected, and the cost of debt capital

will be lower; however, it will be necessary to generate a higher income, so as to leave

the return on capital provided by shareholders unchanged.

Finally, it should be noted that a total protection level, i.e., a 100 % conﬁdence level

VaR, would be neither theoretically desirable, nor practically achievable: a ﬁnancial institution wishing such a level of protection (and therefore wishing to eliminate any form

of risk for third parties) should be entirely ﬁnanced with equity capital, and would thus

betray its primary role, i.e., measuring, pricing and managing risks.

9.4.2 VaR models disregard customer relations

A second criticism of VaR is that its mechanical application might lead a bank to abruptly

terminate all positions whose risk-adjusted return is inadequate. Thus, the value of long

term customer relations will not be properly taken into account, and the bank will act

with a short term view (“short-termism”).

In reality VaR models, just like any other management technique, represent a simple

tool which is naturally and necessarily to be complemented by subjective evaluations by

the bank’s management. As a consequence, if a given transaction’s risk-adjusted return

calculated based upon the relevant VaR were insufﬁcient to adequately remunerate the

economic capital, and therefore pointed to value destruction, the bank’s management could

in any case keep the investment in the portfolio based upon a subjective evaluation of the

long-term relation with the counterparty. In any case, the indication of the risk-adjusted

return estimated based upon VaR would be a useful tool, and would force the management

to express the rationale which suggest making the investment, or keeping the asset in the

bank’s portfolio.

9.4.3 VaR models are based upon unrealistic assumptions

A third criticism concerns the assumptions underlying the different criteria to calculate

value at risk. Some institutions may ﬁnd these assumptions as unrealistic and therefore

refuse to adopt a VaR model as a tool for risk measurement, risk control, risk-adjusted

return measurement, and capital allocation among the different business units.

In reply to this criticism, we would argue that, whether one likes it or not, each ﬁnancial

institution chooses to hold a certain level of capital, and to implicitly allocate it to the

different risk taking business units (often based upon a non-transparent perception of

their risk levels). The difference lies in the fact that those who adopt a VaR model are

aware of their capital sizing and allocation logics, as well as the (sometimes questionable)

hypotheses upon which these measures are based; those who do not have a model, proceed

in a considerably more opaque manner.

However, since the assumption underlying a VaR model have been fully expressed

and understood, they can be appropriately changed in the presence of particular market

conditions, or speciﬁc subjective evaluations by the management. In conclusion, awareness

of the models and their limitations is certainly preferable to ignorance, even when the

latter uses precisely the existence of limitations in available models as a shield.

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Risk Management and Shareholders’ Value in Banking

9.4.4 VaR models generate diverging results

VaR models have been criticized implicitly by some researchers who found considerable discrepancies9 when empirically comparing the results generated by the different

approaches presented in the previous chapters. Although these discrepancies are not

an explicit and direct demonstration of the unreliability of the VaR methodology, they

allegedly suggest that the results of these models should be used with caution.

This criticism also disguises poor understanding of the purposes of VaR models. Indeed,

it is true that the results of these models strongly depend on the approach used (as well as

on the speciﬁc hypotheses adopted, the historical sample size, the reference time horizon,

the number of identiﬁed risk factors, how positions are mapped to these factors, and much

more). So, they should not be considered as a univocal (and almost “magical”) measure

of the capital required to support risks.

However, if results are used to consistently and homogeneously introduce risk limits

and risk-adjusted performance measures into all of a ﬁnancial institution’s business units,

what is key is not so much that VaR is univocally determined, but rather that homogeneous criteria are used for all of the bank’s risk taking units. In this way, even if the

risk – meaning the maximum potential loss – should, ex post, turn out to be over- or

underestimated, this over- or under-estimation will have uniformly concerned all business units, and will therefore not have generated any bias in strategic capital allocation

decisions.

9.4.5 VaR models amplify market instability

If all ﬁnancial institutions trading in the ﬁnancial markets adopt a VaR model, in conjunction with any market shock and consequent increase in volatility, the traders from these

institutions will be likely to all receive the same operational signal. Take the example

illustrated in section 9.3, relating to the VaR limit on a government securities portfolio:

when σ increases, the maximum exposure (expressed in terms of market value) which

is compatible with the limit will shrink, requiring the unwinding, or at least a reduction,

of the position. If all banks behave in the same way, generalized sales will heighten the

market downtrend, turning a slump into an actual crash.

In reality, this criticism, which is often raised by the specialized ﬁnancial press, suffers

from two main limitations. First of all, this argument is based on the assumption that

VaR models are all the same, while we know that they differ in many respects (market

factor identiﬁcation, volatility and correlation estimation criteria, hypotheses concerning

the distribution of market return factors, conﬁdence level, etc), and therefore generate

different risk measures.

In any case, it is true that, in the presence of market crises, traders from the different

ﬁnancial institutions tend to adopt uniform behaviors (which, in turn, amplify crises). This

trend, however, does not stem from the development and use of VaR models, but rather

from human nature and from how ﬁnancial markets work.

9

See Beder (1995), Drudi, Generale and Majnoni (1996), Marshall and Siegel (1996), and Jordan and Mackay

(1996). As was noted by Beder, “. . .VAR calculations differ signiﬁcantly for the same portfolio. VARs are

extremely dependent on parameters, data, assumptions and methodology. . .Variances in the VAR statistic ranged

as much as 14 times for the same portfolio, depending on the type of VAR calculation and the time horizon”.

On the other hand, the same discrepancy of results was noticed by the Basel Committee during an experiment

in which some of the major international banks provided their results – in terms of capital at risk – for the

same virtual portfolio.

VaR Models: Summary, Applications and Limitations

263

9.4.6 VaR measures “come too late, when damage has already been done”

One last criticism towards VaR measures concerns the delay with which they reﬂect any

market shocks, and their consequent ineffectiveness in preventing losses. This criticism,

however technically correct, is also affected by a misunderstanding of the real purposes

of a VaR measure, and can be rejected based upon multiple arguments.

First of all, it should be understood that the “delay” mainly comes from the fact that

VaR models are based upon estimated historical volatility to predict future volatility:

from this point of view, if VaR models were fed with more sophisticated predictions (for

instance, based upon the volatility implied in option prices), risk measures capable of

anticipating any market crisis phenomena could be obtained. However, this result would

depend on the quality of the prediction used, and would not be guaranteed in any way.

Also, there are estimation techniques based upon historical data which are nevertheless

sufﬁciently responsive to more recent market conditions (take the example of exponentially weighted moving averages, when lambda is set at a sufﬁciently low value), and

therefore capable of promptly reﬂecting market shocks.

Moreover, regardless of the ability of VaR models to reﬂect and rapidly “incorporate”

any crisis episodes, we should consider two aspects. First, the inability to anticipate

extreme market changes is a limitation which is inherent in any prediction technique.

Second, as has already been highlighted above, the ultimate purpose of a VaR model is

not to anticipate possible crashes by incorporating extreme events, but rather to uniformly

and consistently generate risk measures based upon “normal” conditions of the different

security markets for a bank’s different business units.

9.5 TWO REAL PROBLEMS OF VAR MODELS

If the criticisms illustrated above are mostly the consequence of a misunderstanding of

the real objectives of VaR models, it is also true that these models suffer from some

considerable limitations, which have encouraged the development of alternative risk measures. Below, we are presenting the two main ones (which are closely connected with

each other, as will be explained below).

9.5.1 The Size of Losses

As was illustrated in the previous chapters, VaR is a risk measure whose purpose is to

answer this question:

What is the maximum loss which could be incurred within a given time horizon, except

for a small percentage, for instance 1 %, of worst cases?

A position’s or portfolio’s VaR is therefore a probabilistic measure which takes different

values at different conﬁdence levels. If the conﬁdence level is deﬁned as c and the loss

is deﬁned as L, we will have:

P r(L > V aR) = 1 − c

(9.6)

What really matters is therefore the probability that the actual loss will be in excess of

VaR. If this happens, the model will provide no information about the size of this excess

loss.

264

Risk Management and Shareholders’ Value in Banking

This deﬁciency may be very signiﬁcant. Take the example of two stock portfolios, H

and K, characterized by the same market value (1 million euros) and the same daily VaR

at a 99 % conﬁdence level (50,000 euros). Assume that VaR was estimated by means of a

(historical or Monte Carlo) simulation approach using a (historical or simulated) sample

of 500 observations.

Table 9.5 reports the ten most signiﬁcant losses recorded by the two portfolios – ranked

from the highest to the lowest one. It can be noted that – VaR being equal – the two

portfolios actually show signiﬁcant differences in terms of risk. Indeed, the largest loss

for portfolio H is 150,000 euros, exceeding VaR by 200 %. Conversely, the maximum

loss for portfolio K is 60,000 euros, exceeding VaR only by 20 %. Likewise, the expected

value of the excess losses is 100,000 euros for portfolio H: this mean that the expected

loss in excess of VaR (“expected excess loss”) is 50,000, i.e., equal to VaR itself; vice

versa, the expected excess loss is 5,000 euros (55, 000−50, 000), i.e., 10 % of VaR, for

portfolio K.

Table 9.5 Example of losses relating to two stock portfolios

Losses

Portfolio H

(ranked starting

from the

Portfolio K

worst one)

1

150,000

60,000

2

120,000

56,000

3

100,000

55,000

4

70,000

53,000

5

60,000

51,000

6

50,000

50,000

7

48,000

45,000

8

45,000

40,000

9

42,000

35,000

10

40,000

30,000

VaR 99 %

50,000

50,000

Lmax

150,000

60,000

Lmax - VaR 99 %

100,000

10,000

200 %

20 %

50,000

5,000

100 %

10 %

VaR(99 %)

Maximum Loss

Maximum Excess

Loss

Maximum Excess

Loss/VaR

Expected Excess

E (L-VaR 99 % |L > VaR 99 % )

Loss

Expected Excess

Loss/VaR

VaR Models: Summary, Applications and Limitations

265

9.5.2 Non-subadditivity

A second major limitation of VaR measures is non-compliance with one of the essential

properties of a consistent risk measure,10 i.e., subadditivity.

This term refers to the fact that the risk of a portfolio consisting of multiple positions

must not be higher than the sum of the risks of the individual positions.

For instance, if a portfolio consisting of two positions, X and Y, is considered, a risk

measure F (.) is subadditive if the following condition is always met:

F (X + Y ) ≤ F (X) + F (Y )

(9.7)

This property depends on the fact that any portfolio beneﬁts, to a greater or lesser degree,

from a diversiﬁcation effect related to the fact that there is an imperfect correlation among

the different market factors.

This property must be true not only when multiple positions are “assembled”, but also

when the joint effect of multiple market factors upon a certain asset in the portfolio is

considered. So, for instance, the risk relating to a foreign currency denominated bond

can be “mapped” into an equivalent portfolio consisting of two components: a foreign

currency deposit and an exchange rate risk-free foreign bond; if the two factors (exchange

rate and foreign interest rate curve) are not perfectly correlated, the total risk will be lower

than the sum of the risks of the individual components.

In the previous chapters, we encountered several examples in which application of

VaR gave rise to results in compliance with the subadditivity principle. However, it is

possible to construct examples showing that this property, however intuitive, may not be

guaranteed by VaR. In other words, it may be that:

V aR(X + Y ) > V aR(X) + V aR(Y )

(9.8)

This happens typically when the joint distribution of market factors is characterized by fat

tails, and is therefore different from the multivariate normal.11 Indeed, this phenomenon

occurs more frequently in the case of credit risk than in the case of market risks.

Take the example of two securities, A and B, whose future value probability distribution

is represented in Table 9.6, panel (a). For both, the value in one year’s time could be

100 euros in 95 % of cases, 90 euros in 4 % of cases, 70 euros in 1 % of cases. There

follows an expected value (probability weighted average) of 99.3 euros. For the sake of

simplicity, let us assume that, today, the current value of these securities (panel b) is

exactly 99.3 euros (the example would remain valid anyway, even if the current value

were different). Based upon the difference between the possible future values and the

current value, panel (c) shows the distribution of possible future value changes. In panel

10

Here, consistent risk measure means a risk measure complying with some essential axioms. These axioms

are translation invariance (adding an amount of cash to the portfolio will reduce the risk by the same amount),

positive homogeneity of degree one (if the size of each position is doubled, the portfolio risk will also be

doubled), monotonicity (if the losses on portfolio A are greater than those on portfolio B in every possible

future scenario, then the risk of portfolio A must be greater than that of portfolio B), and subadditivity. For

more details, see Artzner et al. (1993).

11

See Artzner, Delbaen, Eber and Heath (1999).

266

Risk Management and Shareholders’ Value in Banking

Table 9.6 Distribution of the future values of

two securities

Security A Security B

(a) Distribution of future values

Probability

MV

MV

1%

70

70

4%

90

90

95 %

100

100

WWW.

E(MV)

99.3

99.3

V

V

99.3

99.3

V

V

(b) Current value

Value

(c) Future value changes

Probability

1%

−29.3

−29.3

4%

−9.3

−9.3

0.7

0.7

95 %

(d) Synthetic measures

E( V)

0.0

0.0

VaR at 99 %

9.3

9.3

(d), the mean and VaR at 99 % (the value “isolating” one per cent of worst cases) of this

distribution are displayed. The latter value is 9.3 euros for each of the two securities.

Now, let us assume that the two securities are held by the same investor within a

single portfolio. Let us also assume that they are independent, i.e., that the possible future

values of A are uncorrelated to those of B. This lack of correlation must result in a risk

diversiﬁcation effect: in other words, a consistent (and, in particular, subadditive) risk

measure must lead to a lower value for the portfolio than the sum of the risks of the two

securities taken separately. We would therefore expect the VaR of the portfolio comprising

A and B to be lower than 18.6 (9.3 plus 9.3) euros.

Table 9.7 provides the joint probability distribution of the value changes in A and B

(ﬁrst row and ﬁrst column) and in the sum portfolio containing them both – under an

independence hypothesis. For instance, as both securities could experience a decrease

in value of 29.3 euros in 1 % of cases, the Table shows that the sum portfolio would

experience a decrease in value of 58.6 (29.3 plus 29.3) in 0.01 % (1 % by 1 %) of cases.

VaR Models: Summary, Applications and Limitations

267

Table 9.7 Distribution of the future values of the portfolio comprising the two

securities: double-entry table

Security B

−29.3 −9.3

0.7

1%

4%

95 %

−29.3 −58.6 −38.6 −28.6

Security A

1%

0.01 % 0.04 %

−9.3 −38.6 −18.6

0.95 %

−8.6

4%

0.04 % 0.16 %

3.80 %

0.7

−28.6 −8.6

1.4

95 %

0.95 % 3.80 % 90.25 %

Table 9.8 Distribution of the future values of

the portfolio comprising the two securities:

re-ranked table

Probability

Cumulative

probability

Value

0.01 %

0.01 %

−58.6

0.08 %

0.09 %

−38.6

1.90 %

1.99 %

−28.6

0.16 %

2.15 %

−18.6

7.60 %

9.75 %

−8.6

90.25 %

100.00 %

1.4

VaR at 99 %

28.6

Table 9.8 re-ranks the nine values of the possible changes in the sum portfolio, starting

from the worst one (note that events characterized by the same portfolio market value

were aggregated on a single row).

The row in grey highlights the ﬁrst percentile, i.e., the value leaving (at least) 1 % of

worst cases behind it. This value is VaR at 99 %, and is equal to a loss of 28.6 euros,

which is considerably higher than the sum of the individual positions’ VaRs for an equal

conﬁdence level. It was therefore shown that:

V aR(A + B) > V aR(A) + V aR(B)

This result is mainly due to the fact that the individual positions’ VaRs underestimate

their risk by totally disregarding the size of excess losses.

So, it is possible that VaR may not meet the subadditivity requirement. However, this

problem never occurs if VaR is calculated according to the parametric approach, assuming

that portfolio value changes are described by a normal distribution. As a matter of fact,

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268

Risk Management and Shareholders’ Value in Banking

in this case, VaR is simply a multiple α (a function of the desired conﬁdence level) of the

standard deviation. From the formula to calculate the variance of a two-security portfolio

2

2

2

σA+B = σA + 2ρσA σB + σB

(with ρ≤ 1), it therefore follows that:

σA+B ≤ σA + σB

ασA+B ≤ ασA + ασB

V aRA+B ≤ V aRA + V aRB

which ensures subadditivity.

However, when the empirical distribution of value changes takes a form which is clearly

dissimilar to the normal (as in the example in Table 9.6), it is not possible to adopt the

parametric approach just because it would ensure subadditivity. On the contrary, different

distributions (for instance, historical distribution) need to be used, which results in VaR

being exposed to a risk of non-subadditivity.

9.6 AN ALTERNATIVE RISK MEASURE: EXPECTED

SHORTFALL (ES)

The two problems of VaR as described above (non-consideration of the size of excess

losses and violation of subadditivity) can be overcome by having recourse to an alternative

risk measure, referred to as expected shortfall (ES).12 ES can be deﬁned as the expected

value of all losses in excess of VaR.

Hence:

ES = E[−( MV − E( MV ))|−( MV − E( MV ))>V aR]

(9.9)

where “|”, as usual, means “such that”, and market value changes ( MV ) were changed

in sign because they are losses (negative values), and we are interested in their absolute

value.

This is a conditional average which does not consider all possible values of MV, but

only those whose distance from the mean exceeds VaR.

ES can also be written as a function of future values (MV), rather than of their changes

MV:13

ES = E[−(MV − E(MV ))|−(MV − E(MV ))>V aR]

(9.10)

Alternatively, if we assume (as we generally did in the previous chapters) that the expected

value of market value changes (E( MV )) is zero, then ES will be simpliﬁed, and can be

written as

ES = E[− MV |− MV >V aR]

(9.11)

12

Other names of this same measures are average shortfall (AS), conditional VaR (CVaR), or extreme value at

risk (EVaR).

13

In practice, it is sufﬁcient to add and subtract the current portfolio value within the two round brackets.

VaR Models: Summary, Applications and Limitations

269

Note that this measure, like VaR, is characterized by a given conﬁdence level and a given

time horizon.

To understand the meaning and construction logic of Expected Shortfall, it is useful

to go back to Table 9.5. As the reader may remember, for each of the two portfolios

represented therein, there were ﬁve possible scenarios (the ﬁrst ﬁve rows in the Table)

in which the loss was in excess of VaR. Expected Shortfall is but the mean of these

ﬁve losses (value changes VM changed in sign). It will therefore be 100,000 euros

for portfolio H, and 55,000 euros for portfolio K. This is a one-year ES, with a 99 %

conﬁdence level, as the probability distribution of value changes in Table 9.5 refers to

a one-year time horizon, and the VaR used as the “threshold” to select the scenarios on

which the conditional average was to be calculated adopted exactly a 99 % conﬁdence

level.

If, conversely, we take the two securities A and B in Table 9.6, we can immediately see

that, for each of them, one-year ES at a 99 % conﬁdence level is 29.3 euros. Indeed, there

is only one excess loss value, which therefore coincides with the conditional average.

Note that, for the sum portfolio comprising A and B, the value of ES is 40.8 euros: this is

the (probability weighted) average of the loss values (58.6 and 38.6) as shown in the ﬁrst

two rows in Table 9.8 i.e., of all and only the portfolio’s excess losses (28.6). In detail:

ES99 % = 58.6 ·

0.08 % ∼

0.01 %

+ 38.6 ·

= 40.8

0.09 %

0.09 %

Unlike the portfolio VaR (which is higher than the sum of the VaRs for the two individual

securities), the portfolio ES meets the subadditivity condition. It can be demonstrated that

this is not by accident, but that Expected Shortfall ensures compliance with this property,

and is a consistent risk measure.

For this reason, in more recent years, numerous researchers and several ﬁnancial institutions have paid more attention to ES as a risk indicator.

From an economic point of view, while VaR represents the economic capital which

needs to be paid into a bank to limit its probability of bankruptcy to 1-c (where c is the

VaR’s conﬁdence level), some interesting economic meanings can be attached also to ES.

More speciﬁcally, the difference between ES and VaR can be seen as:

• the expected cost that the regulatory authorities should incur to bail out a bank if its

capital (set equal to VaR) were not enough;

• the expected payment that a risk neutral insurer would have to face, if the bank had

insured itself against the risk of excess losses.

SELECTED QUESTIONS AND EXERCISES

1. A bank holds two positions, in stocks and T-bonds respectively. The stock portfolio

is currently worth 80,000 euros and has an average beta of 95 %; the T-bonds have

a market value of 100,000 euros and a modiﬁed duration of 7 years. The volatility

of percent changes in the stock-market index is estimated at 15 %; the volatility of

absolute changes in the yield to maturity of 7-year Treasury bonds is estimated at 2 %.

Based on a 99 % VaR, which of the two portfolios is currently riskier? By how much?

How would the result change if the bank were to adopt a 95 % VaR? And what would

happen if, given a 99 % VaR, the equity portfolio volatility were to change from 15 %

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270

Risk Management and Shareholders’ Value in Banking

to 25 %? [To solve this exercise, recall that N −1 (99 %)∼

=2.33 and N −1 (95 %)∼

=1.64,

−1

where N (.) is the inverse of the normal cumulative density function]

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2. The ﬁxed-income desk of Bank Alpha has been assigned a VaR limit (computed at a

99 % conﬁdence level) of 10 million euros. Currently, the market value and modiﬁed

duration of the portfolio are 100 million euros and 3 years. The yield curve is assumed

to be ﬂat and the volatility of absolute shifts in rates is 1.1 %. First, check if the desk

is exceeding the VaR limit, and if so by how much. Then, outline two alternative

strategies to align the actual VaR with the VaR limit (either increasing or decreasing

risk), involving: (i) a sale/purchase of bonds; (ii) a bond trade to change duration, while

leaving the portfolio market value unchanged. Finally, suppose markets experience a

shock, and volatility increases to 3 %: how could the bank’s portfolio be recalibrated

to comply with the VaR limit?

3. The VaR of a portfolio made of two positions is 50 million euros. In which of the

following cases would parametric VaR increase because one of the two positions has

been sold?

(A) If the two positions have opposite signs (short and long) and are exposed to

negatively-correlated market factors.

(B) If the two market factors, to which the two positions are exposed, are totally independent.

(C) If the two positions have the same sign (both long or short) and are exposed to

negatively-correlated market factors.

(D) Parametric VaR never increases, because it guarantees subadditivity.

4. Which of the following statements are true?

“From an economic perspective, expected shortfall can be interpreted as . . .”

(I.) “..VaR, plus the expected cost that the regulatory authorities would have to sustain

in order to save the bank (paying for its losses) if its capital were not enough to

cover them”.

(II.) “. . .the dividend the bank would have to pay to its shareholders in order to make

it attractive for them to pay in a capital equal to VaR.”;

(III.) “. . .VaR, plus the (risk neutral) cost a bank would face if it wanted to insure itself

against losses larger than VaR.”;

(IV.) “. . .VaR with a 100 % conﬁdence level”.

(A)

(B)

(C)

(D)

I and IV;

II and IV;

I only;

I, III and IV.

5. Based on the following probability distribution of portfolio losses, and using a conﬁdence level of 95 %, compute VaR (deﬁned as the maximum loss L, such that the

probability of experiencing losses greater than L is 5 %) and expected shortfall.

VaR Models: Summary, Applications and Limitations

Probability

Losses, ordered

from the worst

(euro mln)

0.50 %

1000

0.30 %

271

100

1%

80

1.60 %

70

0.80 %

65

0.80 %

60

1%

50

1.30 %

30

0.40 %

20

92.30 %

0

Also, highlight how the VaR and expected shortfall (still with a 95 % conﬁdence) would

change if the maximum loss, rather than 1 billion euros (as stated in the Table) were

500 million euros.

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