<< стр. 11(всего 16)СОДЕРЖАНИЕ >>
(unfavourable) for him.
In the same way as the VBP for rate products, the sensitivity of a spot exchange position
can be valued by calculating the effect of a variation in a вЂ˜pipвЂ™, upwards or downwards
on the result for the position. The calculations are simple:

75 000 000 Г— 1.0267 = 77 002 500
75 000 000 Г— 1.0268 = 77 010 000
77 010 000 в€’ 77 002 500 = 7500
Example of historical returns
On 19 January 2001, the spot EUR/USD was worth 0.9336, while on 26 January 2001
it stood at 0.9238. The historical return (вЂ˜ReturnsвЂ™ sheet, cell AO4) is в€’1.0497 %. This
means that: 0.9336 Г— (1 + (в€’1.0497)) = 0.9238.
By applying Position 3 of the treasury portfolio to the current rate, we have: 1.0267 Г—
(1 + (в€’1.0497)) = 1.01592273.
The investorвЂ™s position is вЂ˜longвЂ™, so a fall in the EUR/USD rate will be unfavourable
for him, and the loss (in USD) will be:

75 000 000 Г— ((1.0267 Г— (1 + (в€’1.0497)) в€’ 1.0267))
= 75 000 000 Г— (1.01592273 в€’ 1.0267) = в€’808 295.31

This result is displayed in cell F3 of the вЂ˜Treasury RevalвЂ™ sheet.

8.2.1.4 Revaluation of the portfolio
The revaluation of the treasury portfolio is shown in the table produced by cells from B2
to G102, on the вЂ˜Treasury RevalвЂ™ sheet.
248 Asset and Risk Management

For each of the positions, from 1вЂ“3, we have applied 100 historical returns (from 26
January 2001 to 20 December 2002) in the currency in question (GBP, USD, EUR). The
total shown is the loss (negative total) or proп¬Ѓt (positive total) as calculated above taking
account of past returns.
Let us take as an example the п¬Ѓrst revaluation (corresponding to the 26 January 2001
return) for Position 1 of the portfolio (cell D3 in the вЂ˜Treasury RevalвЂ™ sheet).
The formula that allows the loss or proп¬Ѓt to be calculated consists of the difference in
interest receivable at the current (initial) price of the position and the interest receivable
in view of the application to the initial price of the corresponding historical return on 26
January 2001. We therefore have the general formula:
ND ND
L = C В· R В· (1 + HR) В· в€’ CВ·RВ·
DIV DIV
ND ND
=CВ· R В· (1 + HR) В· в€’ RВ·
DIV DIV
Here:

L is the loss;
C is the total to which the transaction relates;
L is the current rate (initial price) of the transaction;
HR is the historical return.

It is this last formula that is found in cells D3 to F102. Of course we could have simpliп¬Ѓed3
it here:
ND ND
L=CВ· R В· (1 + HR) В· в€’ RВ·
DIV DIV
ND
=CВ·RВ· В· ((1 + HR) в€’ 1)
DIV
ND
=CВ·RВ· В· HR
DIV
If the investor is вЂ˜longвЂ™, he has borrowed and will wish to cover himself by replacing
his money at a higher rate than the initial price. Therefore, if HR is a negative (positive)
total, and he has realised a loss (proп¬Ѓt). This is the case for Position 1 of the portfolio on
26 January 2001.
On the other hand, if the investor is вЂ˜shortвЂ™, he has lent and will wish to cover himself
by borrowing the money at a lower rate than the initial price. Therefore, if HR is a
negative (positive) total, P must be positive (negative) and the preceding formula (valid
if the investor is вЂ˜longвЂ™) must be multiplied by в€’1.
For Position 2 of the portfolio, the investor is вЂ˜shortвЂ™ and we have (cell E3 of the
вЂ˜Treasury RevalвЂ™ sheet:

ND ND
L = (в€’1) В· C В· R В· (1 + HR) В· в€’ RВ·
DIV DIV
3
We have not simpliп¬Ѓed it, so that the various components of the difference can be seen more clearly.)
Setting Up a VaR Methodology 249

On each past date, we have a loss or proп¬Ѓt expressed in the currency of the operation for
each position. As the investor has the euro for his national or accounting currency, we
have summarised the three losses or gains in EUR equivalents at each date. The chosen
FX rate for the euro against the other currencies is of course the rate prevailing on the
date of calculation of the VaR, that is, 20 December 2002. The overall loss is shown in
column G of the вЂ˜Treasury RevalвЂ™ sheet.

8.2.1.5 Classifying the treasury portfolio values and determining the VaR
When all the revaluations have been carried out, we have (see вЂ˜Treasury RevalвЂ™ sheet) a
series of 100 losses or proп¬Ѓts according to historical return date.
One has to classify them in increasing order, that is, from the greatest loss to the small-
est. The reader will п¬Ѓnd column G of the вЂ˜Treasury RevalвЂ™ sheet classiп¬Ѓed in increasing
order on the вЂ˜Treasury VaRвЂ™ sheet, in column B. To the right of this column, 1 в€’ q appears.

A. Numerical interpretation
We think it important to state once again that when 1 в€’ q corresponds to a plateau of the
loss distribution function, we have chosen to deп¬Ѓne VaR as the left extremity of the said
section (see Figure 6.7).
We therefore say that:

вЂў There are 66 chances out of 100 that the actual loss will be в€’EUR360 822 or less
(1 в€’ q = 0.34), or VaR 0.66 = в€’360 822.
вЂў There are 90 chances out of 100 that the actual loss will be в€’EUR1 213 431 or less
(1 в€’ q = 0.10), or VaR 0.90 = в€’1 213 431.
вЂў There are 99 chances out of 100 that the actual loss will be в€’EUR2 798 022 or less
(1 в€’ q = 0.01), or VaR 0.99 = в€’2 798 022.
B. Representation in graphical form
If the forecast of losses is shown on the x-axis and 1 в€’ q is shown on the y-axis, the
estimated loss distribution will be obtained. Figure 8.1, also appears on the вЂ˜Treasury
VaRвЂ™ sheet.

Treasury VaR
1.00
0.90
0.80
0.70
0.60
Probability

0.50
0.40
0.30
0.20
0.10
0.00
вЂ“3 000 000 вЂ“2 000 000 вЂ“1 000 000 0 1 000 000 2 000 000
Estimated loss

Figure 8.1 Estimated loss distribution of treasury portfolio
250 Asset and Risk Management

8.2.2 Bond portfolio case
The п¬Ѓrst stage once again consists of determining the past returns (in this case, weekly).

8.2.2.1 Past variations to be applied
The main difп¬Ѓculty connected with this type of asset, in terms of determining VaR, is the
question of whether or not the historical prices or rates are available.
When a bond is п¬Ѓrst issued, for example, it has to be acknowledged that we do not
have any historical prices.
As the aim of this chapter is merely to show how VaR can be calculated using the
historical simulation method, using deliberately simpliп¬Ѓed examples, we have used a range
of rates for deposits and swaps on the basis of which we will construct our example. We
did not therefore wish to use bond historical prices as a basis.
A. Yield
The price of a bond is known on at least one date: that for which we propose to determine
the VaR (in our example, 20 December 2002).
Using this price, and by taking into account the calculation date, the maturity date,
coupon date, the price on maturity, the basis of calculation and the frequency of the coupon
payments, the вЂ˜yield to maturityвЂ™ or YTM, can be calculated as shown in Section 4.1.2.
Columns H3 to H9 of the вЂ˜PortfoliosвЂ™ sheet show the relative yields for the bond in
our п¬Ѓctitious portfolio. As not all versions of Excel contain the вЂ˜yieldвЂ™ п¬Ѓnancial function,
we have copied the values into columns I3 to I9.
It is to this yield to maturity that we intend to apply the variations relative to corre-
sponding rates of deposit and/or swaps, in terms of maturity, for the remaining period of
the corresponding bond.
We are of course aware that this method is open to criticism as the price, if we had
used it, not only reп¬‚ects general interest-rate levels, but also carries a dimension of credit
risk and lack of liquidity.
B. Interpolation of rates
We cannot deduce from the вЂ˜RatesвЂ™ sheet the returns to be applied to the yield to maturity;
the remaining periods are in fact broken.
We have determined (in the вЂ˜Bonds InterpвЂ™ sheet), the two maturity dates (columns I
and J in that sheet) that straddle the remaining period, together with the portion of rate
differential to be added to the lower rate (column F divided by column H).
Readers will п¬Ѓnd the value of the rates to be interpolated (taken from the вЂ˜RatesвЂ™ sheet)
in the вЂ˜Variation BondsвЂ™ sheet, the rate differential to which the rule of interpolation
mentioned above is applied. For bond 1 in our portfolio, this calculation is found in
column G. All that remains now is to determine the return relative to the series of
synthetic rates in exactly the same way as shown in Section 8.2.1. The returns applicable
to the yield to maturity for bond 1 in the portfolio are thus shown in column H of the
вЂ˜Variation BondsвЂ™ sheet.

8.2.2.2 Composition of portfolio
The вЂ˜bondвЂ™ portfolio is found on the вЂ˜PortfoliosвЂ™ sheet in CH8.XLS. This sheet, like the
rest of the portfolio, is purely п¬Ѓctitious. The investorвЂ™s national currency is the euro.
Setting Up a VaR Methodology 251

The portfolio is вЂ˜longвЂ™, with six bonds, for which the following are given:
вЂў currency;
вЂў coupon;
вЂў maturity date;
вЂў ISIN code;
вЂў last known price (this is the вЂ˜bidвЂ™, because if the position is closed, one should expect
to deal at the bid price);
вЂў yield on maturity (in formula form in column H, in copied value in column I);
вЂў basis of calculation (current/current or 30/360);
вЂў frequency of payment of the coupon.

8.2.2.3 Portfolio revaluation
In Table B2вЂ“H9, the вЂ˜Losses BondsвЂ™ sheet summarises the portfolio data that we need in
order to revalue it.
Remember that we propose to apply the relative variations in rates (column L for bond
1, column Q for bond 2 etc.), to the yield to maturity (column C) of each bond that
corresponds, in terms of maturity, to the period still outstanding.
A new yield to maturity is therefore deduced (column M for bond 1); it is simply the
current total to which a past variation has been applied.
We explained above that starting from the last known price of a bond, and taking
account of the date of the calculation as well as the expiry date, the coupon date, the
price on maturity, the basis of calculation and the frequency of the coupon, we deduce
the yield to maturity.
It is possible, in terms of correlations, to start from our вЂ˜historicalвЂ™ yields to maturity in
order to reconstruct a synthesised price (column N). The вЂ˜PriceвЂ™ function in Excel returns
a price on the basis of the given yield to maturity (column M) and of course that of the
date of the calculation as well as the expiry date, coupon date, price on maturity, basis
of calculation and frequency of coupon. As not all versions of Excel contain the вЂ˜PriceвЂ™
function, we have copied the values from column N into column O for bond 1, from
column S into column T for bond 2, etc.
All that now remains is to compare the new price to the last known price, and to
multiply this differential by the nominal held in the portfolio in order to deduce the
resulting proп¬Ѓt or loss (column P for bond 1). As indicated in cell B11, we assume that
we are holding a nominal of В¤100 million on each of the six bond lines.
Note
It may initially seem surprising that the nominal used for bond 1 (expressed in PLN) is also
В¤100 million. In fact, rather than expressing the nominal in PLN, calculating the loss or
proп¬Ѓt and dividing the total again by the same EUR/PLN rate (that is, 3.9908 at 20 Decem-
ber 2002), we have immediately expressed the loss for a nominal expressed in euros.
It is then sufп¬Ѓcient (column AP) to summarise the six losses and/or proп¬Ѓts for each of
the 100 dates on each line (with respect for the correlation structure).

8.2.2.4 Classifying bond portfolio values and determining VaR
Once all the new valuations have been made, a series of 100 losses or proп¬Ѓts (вЂ˜Losses
BondsвЂ™ sheet) will be shown according to historical return date. One has to classify them
252 Asset and Risk Management

Bond portfolio VaR
1.00
0.90
0.80
0.70
0.60
Probability
0.50
0.40
0.30
0.20
0.10
0.00
вЂ“1 600 000 вЂ“1 100 000 вЂ“600 000 вЂ“100 000 400 000 900 000 1 400,000 1 900,000
Estimated loss

Figure 8.2 Estimated loss distribution of bond portfolio

in ascending order, that is, from the greatest loss to the smallest. Readers will п¬Ѓnd column
AP in the вЂ˜Losses BondsвЂ™ sheet classiп¬Ѓed in ascending order on the вЂ˜Bonds VaRвЂ™ sheet
in column B. 1 в€’ q is located to the right of that column.

A. Numerical interpretation
We say that:

вЂў There are 66 chances out of 100 that the actual loss will be в€’EUR917 or less (1 в€’ q =
0.34), or VaR 0.66 = в€’917.
вЂў There are 90 chances out of 100 that the actual loss will be в€’EUR426 740 or less
(1 в€’ q = 0.10), or VaR 0.90 = в€’426 740.
вЂў There are 99 chances out of 100 that the actual loss will be в€’EUR1 523 685 or less
(1 в€’ q = 0.01), or VaR 0.99 = в€’1 523 685.

B. Representation in graphical form
If the loss estimates are shown on the x-axis and 1 в€’ q is shown on the y-axis, the esti-
mated loss distribution will be obtained. Figure 8.2 also appears on the вЂ˜Bonds VaRвЂ™ sheet.

8.3 THE NORMALITY HYPOTHESIS
We have stressed the hidden dangers of underestimating the risk where the hypothesis of
normality is adopted. In fact, because of the leptokurtic nature of market observations,
the normal law tails (VaR being interested speciп¬Ѓcally in extreme values) will report the
observed historical frequencies poorly, as they will be too п¬‚at. It is prudent, when using
theoretical forecasts to simplify calculations, to overstate market risks; here, however, the
opposite is the case.
In order to explain the problem better we have compared the observed distribution for
the bond portfolio in CH8.XLS with the normal theoretical distribution. The comparison
is found on the вЂ˜Calc NвЂ™ sheet (N = normal) and teaches us an interesting lesson with
regard to the tails of these distributions.
Setting Up a VaR Methodology 253

We have used the estimated loss distribution of the bond portfolio (copied from the
вЂ˜Bonds VaRвЂ™ sheet). We have produced 26 categories (from в€’1 600 000 to в€’1 465 000,
from в€’1 465 000 to в€’1 330 000 etc., up to 1 775 000 to 1 910 000) in which each of these
100 losses will be placed. For example, the loss of в€’1 523 685.01 (cell D4) will belong
to the п¬Ѓrst class (from в€’1 600 000 to в€’1 465 000, column G).
The table G2вЂ“AF103 on the вЂ˜Calc NвЂ™ sheet contains one class per column (lines
G2вЂ“AF3) and 100 lines, that is, one per loss (column D4вЂ“D103). Where a given loss
intersects with a class, there will be a п¬Ѓgure of 0 (if the loss is not in the category in
question) or 1 (if otherwise).
By п¬Ѓnding the total of 1s in a column, we will obtain the number of losses per class,
or the frequency. Thus, a loss of between в€’1 600 000 and в€’1 465 000 has a frequency of
1 % (cell G104) and a loss of between 425 000 and 560 000 has a frequency of 13 %
(cell V104).
Cells AH2вЂ“AJ29 carry the category centres (в€’1 532 500 for the class в€’1 600 000 to
в€’1 465 000), and the frequencies as a п¬Ѓgure and a percentage.
If we look at AH2 to AI29 in bar chart form, we will obtain the observed distribution
for the bond portfolio (Figure 8.3) located in AL2 to AQ19.
Now the normal distribution should be calculated. We have calculated the mean and
standard deviation for the estimated distribution of the losses in D104 and D105, respec-
tively. We have carried the losses to AS4 to AS103.
Next, we have calculated the value of the normal density function (already set out in
1 xв€’Вµ 2
1
Section 3.4.2 вЂ˜Continuous modelвЂ™), that is, f (x) = в€љ exp в€’ , to each
Пѓ
2
2ПЂПѓ
loss in the bond portfolio (AT4 to AT103). If we plot this data on a graph, we will obtain
(Figure 8.4) the graph located from AV2 to BB19.
In order to compare these distributions (observed and theoretical), we have superim-
posed them; the calculations that allow this superimposition are located in the вЂ˜Graph
NвЂ™ sheet.
As can be seen in Figures 8.3 and 8.4, the coordinates are proportional (factor 135 000
for class intervals). We have summarised the following in a table (B2 to D31 of the
вЂ˜Graph NвЂ™ sheet):

Observed distribution (Bonds Pf.)
14 %
12 %
10 %
8%
6%
4%
2%
0%
вЂ“1 532 500
вЂ“1 397 500
вЂ“1 262 500
вЂ“1 127 500
вЂ“992 500
вЂ“857 500
вЂ“722 500
вЂ“587 500
вЂ“452 500
вЂ“317 500
вЂ“182 500
вЂ“47 500
87 500
222 500
357 500
492 500
627 500
762 500
897 500
1 032 500
1 167 500
1 302 500
1 437 500
1 572 500
1 707 500
1 842 500

Figure 8.3 Observed distribution
254 Asset and Risk Management
Normal distribution (Bonds Pf.)

0.0000007
0.0000006
0.0000005
0.0000004
0.0000003
0.0000002
0.0000001
0
вЂ“1 600 000 вЂ“850 000 вЂ“100 000 650 000 1 400 000

Figure 8.4 Normal distribution

Observed and normal distribution (Bonds Pf.)
2.E вЂ“ 06
Observed dist.
1.E вЂ“ 06 Normal dist.

8.E вЂ“ 07

6.E вЂ“ 07

4.E вЂ“ 07

2.E вЂ“ 07

0.E + 00
вЂ“1 532 500
вЂ“1 397 500
вЂ“1 262 500
вЂ“1 127 500
вЂ“992 500
вЂ“857 500
вЂ“722 500
вЂ“587 500
вЂ“452 500
вЂ“317 500
вЂ“182 500
вЂ“47 500
87 500
222 500
357 500
492 500
627 500
762 500
897 500
1 032 500
1 167 500
1 302 500
1 437 500
1 572 500
1 707 500
1 842 500

Figure 8.5 Normal and observed distributions

вЂў the class centres;
вЂў the observed frequencies relating to them;
вЂў the normal coordinates relative to each class centre.

It is therefore possible (Figure 8.5) to construct a graph, located in E2 to N32, which is
the result of the superimposition of the two types of distribution types.
We may observe an underestimation of the frequency through normal law in distribution
tails, which further conп¬Ѓrms the leptokurtic nature of the п¬Ѓnancial markets.
Part IV
From Risk Management
to Asset Management

Introduction
9 Portfolio Risk Management
10 Optimising the Global Portfolio via VaR
11 Institutional Management: APT Applied to Investment Funds
256 Asset and Risk Management

Introduction
Although risk management methods have been used п¬Ѓrst and foremost to quantify market
risks relative to market transactions, these techniques tend to be generalised especially if
one wishes to gain a comprehensive understanding of the risks inherent in the management
of institutional portfolios (investment funds, hedge funds, pension funds) and private
portfolios (private banking and other wealth management methods).
In this convergence between asset management on the one hand and risk management
on the other, towards what we term the discipline of вЂ˜asset and risk managementвЂ™, we are
arriving, especially in the п¬Ѓeld of individual client portfolio management, at вЂ˜portfolio
risk managementвЂ™, which is the subject of Chapter 9.
Next, we will look at methods for optimising asset portfolios that verify normal law
hypotheses, which is especially the case with equities.1 In particular, we will be adapting
two known portfolio optimisation methods:

вЂў SharpeвЂ™s simple index method (see Section 3.2.4) and the EGP method (see Section 3.2.6).
вЂў VaR (see Chapter 6); we will be seeing the extent to which VaR improves the optimi-
sation.

To close this fourth part, we will see how the APT model described in Section 3.3.2
allows investment funds to be analysed in behavioural terms.

Asset management
Fund management
Portfolio management

вЂў Stop loss
вЂў Portfolio risk
вЂў Asset allocation &
вЂў Credit equivalent
management
market timing
вЂў VBP
вЂў Stock picking
вЂў Fund risk вЂў VaR
вЂў Currency allocation
management вЂў MRO

Asset and risk management Risk management

Figure P1 Asset and risk management

1
In fact, the statistical distribution of an equity is leptokurtic but becomes normal over a sufп¬Ѓciently long period.
9
Portfolio Risk Management1

9.1 GENERAL PRINCIPLES
This involves application of the following:

вЂў To portfolios managed traditionally, that is, using:
вЂ” asset allocation with a greater or lesser risk proп¬Ѓle (including, implicitly, market
timing);
вЂ” a choice of speciп¬Ѓc securities within the category of equities or options (stock
picking);
вЂ” currency allocation.
вЂў To particularly high-risk portfolios (said to have a вЂ˜high leverage effectвЂ™) falling clearly
outside the scope of traditional management (the most frequent case), a п¬Ѓvefold risk
management method that allows:
вЂ” daily monitoring by the client (and intraday monitoring if market conditions require)
of the market risks to which he or she is exposed given the composition of his or
her portfolio.
вЂ” monitoring of equal regularity by the banker (or wealth manager where applicable)
of the client positions for which he or she is by nature the only person responsible.

Paradoxically (at least initially) it is this second point that is essential for the client,
since this ability to monitor credit risk with the use of modern and online tools allows
the banker to minimise the clientвЂ™s need to provide collateral, something that earns little
or nothing.

9.2 PORTFOLIO RISK MANAGEMENT METHOD
Let us take the case of the particularly high-risk portfolios, including derivatives:

вЂў linear portfolios (such as FRA, IRS, currency swaps and other forward FX);
вЂў nonlinear portfolios (options); that is highly leveraged portfolios.

In order to minimise the need for collateral under this type of portfolio wherever
possible, the pledging agreement may include clauses that provide for a risk-monitoring
framework, which will suppose rights and obligations on the part of the contractual
parties:

вЂў The banker (wealth manager) reports on the market risks (interest rates, FX, prices etc.)
thus helping the client to manage the portfolio.

1
Lopez T., Delimiting portfolio risk, Banque Magazine, No. 605, JulyвЂ“August 1999, pp. 44вЂ“6.
258 Asset and Risk Management

вЂў The client undertakes to respect the risk criteria (by complying with the limits) set out
in the clauses, authorising the bank (under certain conditions) to act in his name and
on his behalf if the limits in question are breached.

A portfolio risk management mandate generally consists of two parts:

вЂў the investment strategy;
вЂў the risk framework.

9.2.1 Investment strategy
This part sets out:

вЂў The portfolio management strategy.
вЂў The responsibilities of each of the parties.
вЂў The maximum maturity dates of the transactions.
вЂў The nature of the transactions.

9.2.2 Risk framework
In order to determine the risks and limits associated with the portfolio, the following four
limits will be taken into consideration, each of which may not be exceeded.

1. The stop loss limit for the portfolio.
2. The maximum credit equivalent limit.
3. The upper VBP (value of one basis point) limit for the portfolio.
4. The upper VaR (Value at Risk) limit for the portfolio.

For each measure, one should be in a position to calculate:

вЂў the limit;
вЂў the outstanding to be compared to the limit.

9.2.2.1 The portfolio stop loss
With regard to the limit, the potential global loss on the portfolio (deп¬Ѓned below) can
never exceed x % of the cash equivalent of the portfolio, the portfolio being deп¬Ѓned as
the sum of:

вЂў the available cash balances, on one hand;
вЂў the realisation value of the assets included in the portfolio, on the other hand.

The percentage of the cash equivalent of the portfolio, termed the stop loss, is deter-
mined jointly by the bank and the client, depending on the clientвЂ™s degree of aversion to
the risk, based in turn on the degree of leverage within the portfolio.
For the outstanding, the total potential loss on the portfolio is the sum of the differ-
ences between:
Portfolio Risk Management 259

вЂў the value of its constituent assets at the initiation of each transaction;
вЂў the value of those same assets on the valuation date;

Each of these must be less than zero for them to apply.

Example
Imagine a portfolio of EUR100 invested in п¬Ѓve equities ABC at EUR10 per share and
п¬Ѓve equities XYZ at EUR5 per share at 1 January.
If the value of ABC changes to EUR11 and that of XYZ to EUR4 on the next day,
the potential decrease in value on XYZ (loss of EUR1 on 10 equities in XYZ) will be
taken into account for determining the potential overall loss on the portfolio. The EUR5
increase in value on the ABC equities (gain of EUR1 on п¬Ѓve equities ABC) will, however,
be excluded. The overall loss will therefore be в€’EUR10.
The cash equivalent of the portfolio will total EUR95, that is, the total arising from
the sale of all the assets in the portfolio. This produces a stop loss equal to 20 % of the
portfolio cash equivalent (20 % of EUR95 or 19). See Table 9.1.

9.2.2.2 Maximum credit equivalent limit
The credit limit totals the cash equivalent of the portfolio (deп¬Ѓned in the вЂ˜portfolio stop
lossвЂ™ section). The credit liabilities, which consist of the sum of the credit equivalents
deп¬Ѓned below, must be equal to or less than the cash equivalent of the portfolio. The
credit equivalent calculation consists of producing an equivalent value weighting to base
products or their derivatives; these may or may not be linear.
The weighting will be a function of the intrinsic risk relative to each product (Figure 9.1)
and will therefore depend on whether or not the product:

вЂў involves exchange of principal (for example, a spot FX involves an exchange of prin-
cipal whereas a forward FX deal will defer this to a later date);
вЂў involves a contingent obligation (if options are issued);
вЂў involves a contingent right (if options are purchased);

Table 9.1 Stop loss

Stop loss Potential loss Use of limit

в€’EUR10
EUR19 52.63 %

Credit risk
+
Spot

Option issues

Option purchase

Forward Fx
вЂ“ FRA, IRS and currency swaps

Figure 9.1 Weight of the credit equivalent
260 Asset and Risk Management

вЂў the product price (if no exchange of principal is supposed) is linked to one variable
(interest rate for FRA, IRS and currency swaps) or two variables (interest rates and
spot in the case of forward FX).

We could for example determine credit usage per product as follows:

1. For spot cash payments, 100 % of the nominal of the principal currency.
2. For the sale of options, the notional for the underlying principal currency, multiplied
by the forward delta.
3. For the purchase of options, 100 % of the premium paid.
4. For other products, each position opened in the portfolio would be the subject of a
daily economic revaluation (mark-to-market). The total potential loss arising would
be taken (gains being excluded) and multiplied by a weighting factor (taking account
of the volatility of the asset value) equal to 100 % + x % + y % for future exchanges
and 100 % + x % for FRA, IRS and currency swaps, x and y always being strictly
positive amounts.

Example
Here is a portfolio consisting of п¬Ѓve assets (Tables 9.2 and 9.3).
The revaluation prices are shown in Table 9.4.

Table 9.2 FX products

Product P/S Currency Nom. P/S Currency Nom. Spot Forward

Spot S EUR 5m P USD 5.5 million 1.1 вЂ“
Six-month future P USD 10 m S JPY 1170 million 120 117

Table 9.3 FX derivatives and FRA

Three-month call Strike 1.1 P EUR/USD EUR11 million EUR220 000
Two-month put Strike 195.5 S GBP/JPY ВЈ5 million GBP122 000
FRA 3вЂ“6 S DKK 100 million 3.3 %

Table 9.4 Revaluation price

Product Historical price Current price Loss (currency) Potential loss
(EUR)

в€’100 000 в€’89 285.71
Spot 1.1 1.12
в€’25 million в€’189 969.60
FX forward 117 114.5
+11 000 +11 000
Long call 2.00 % nom. EUR 2.10 % nom. EUR
в€’2000 в€’3034.90
Short put 2.44 % nom. GBP 2.48 % nom. GBP
в€’25 000 в€’3363.38
FRA 3.3 % 3.4 %
в€’274653.59
Total
Portfolio Risk Management 261
Table 9.5 Credit equivalent agreements

Product Credit equivalent

Spot 100 % of nominal of principal currency
FX forward 110 % of potential loss (<0)
FX options purchase 100 % of premium paid by client
Principal dev. notional Г— forward
FX options sale
FRA 103 % of potential loss (<0)

Table 9.6 Credit equivalent calculations

Product Nominal/potential loss (EUR) Credit equivalent (EUR)

Spot 5 000 000.00 5 000 000.00
в€’189 969.60В·110 %
FX forward 208 966.56
Long call 220 000.00 220 000.00
Short put 7 587 253.41В·60 % 4 552 352.05
в€’3363.38В·103 %
FRA 3464.28
Total 9 984 782.89

Table 9.7 Outstanding in credit equivalent

Pf. cash equivalent Credit outstanding Use of limit

15 000 000 9 984 782.89 66.57 %

Given that:

вЂў The following rules have been adopted (Table 9.5).
вЂў The forward of the put totals в€’60 %, and the result is a credit equivalent calculation
shown in Table 9.6.

Suppose that in view of the cash available in the portfolio, the cash equivalent of the
portfolio is EUR15 million. Table 9.7 shows the results.

9.2.2.3 Maximum VBP of portfolio
As was shown in Section 2.1.2, the value of one basis point or VBP quantiп¬Ѓes the port-
folioвЂ™s sensitivity to a parallel and unilateral upward movement of the interest rate curve
for a unit of one-hundredth per cent (that is, a basis point).
With regard to the limit, the total VBP of the portfolio may not exceed EURx per
million euros invested in the portfolio (in cash equivalent). The total x equals:

вЂў one-hundredth of the stop loss expressed in currency;
вЂў or, 10 000 times the stop loss expressed as a percentage, which means that in a case
of maximum exposure according to the VBP criterion deп¬Ѓned here, a variation of 1 %
(100 basis points) in interest rates in an unfavourable direction in order to reach the
stop loss.
262 Asset and Risk Management

Example
Assume that the cash equivalent of the portfolio is EUR1 000 000 and the stop loss equal
to 20 % of that cash equivalent, that is EUR200 000.
The total VBP for this portfolio may not exceed:

вЂў one-hundredth of the stop loss expressed in the currency, that is, EUR200 000 divided
by 100, that is EUR2000, which equals:
вЂў 10 000 times the stop loss expressed as a percentage, that is, 10 000 multiplied by 20 %.

With regard to the calculation of the outstanding:

вЂў the total VBP per currency is equal to the sum of the VBPs of each asset making up
the portfolio in that currency, and;
вЂў the total VBP for the portfolio is equal to the sum of the VBP for each currency taken as
an absolute value. As a measure of caution, therefore, the least favourable correlations
are taken into consideration.

Example
Assume a portfolio consisting of two positions (see Tables 9.2 and 9.3), both subject to
a VBP calculation (Tables 9.8 and 9.9).
Let us п¬Ѓrst calculate the VBPs relative to the equivalent loan and borrowing for the
FX forward, and of course the VBP for the FRA.
The breakdown of the six-month FX forward gives us an equivalent loan in USD for
USD10 million and an equivalent deposit in JPY for JPY1200 million (that is, the USD
nominal of 10 million multiplied by the reference spot rate of 120). We then have:

10 000 000 В· 0.01 % В· 180/360 = 500
1 200 000 000 В· 0.01 % В· 180/360 = 60 000

The equivalent loan in USD has a VBP of в€’USD500 (with fall in rates in play), and
the equivalent deposit in JPY a VBP of +JPY60 000 (with rise in rates in play). For the
FRA, we have:
100 000 000 В· 0.01 % В· (180 в€’ 90)/360 = 2 500

Table 9.8 FX forward

Product P/S Currency Nom. P/S Currency Nom. Spot Forward

Six-month future P USD 10 million S JPY 1170 million 120 117

Table 9.9 FRA

Product P/S Currency Nominal Price

FRA 3вЂ“6 S DKK 100 million 3.3 %
Portfolio Risk Management 263
Table 9.10 Total VBP per currency

Product/currency USD JPY DKK

в€’500
Loan
+60 000
Deposit
в€’2 500
FRA
в€’500 +60 000 в€’2 500
Total

The VBP for the FRA totals в€’DKK2 500 (with fall in rates in play). The total VBP
per currency (Table 9.10) is equal to the sum of the VBP for each asset making up the
portfolio in this currency.
The total VBP of the portfolio is equal to the sum of the VBPs for each currency taken
as an absolute value, for the least favourable correlations are taken as a matter of caution.
This gives us Table 9.11.
Using the data in the previous example, that is, a portfolio cash equivalent of
EUR1 000 000 and a VBP that cannot exceed EUR2 000 per million euros invested, in
the event of maximum exposure according to the VBP criterion, a variation of 1 % (100
basis point) in interest rates in the unfavourable direction will be needed to reach the stop
loss, as the stop loss is п¬Ѓxed at 20 % of the portfolio cash equivalent at the most.
In fact, 20 % of EUR1 million totals a maximum loss of EUR200 000, a sum that will
be reached if for a VBP of EUR2000 a variation of 100 basis points in the unfavourable
direction occurs (2 000 Г— 100 is equal to 200 000).
Table 9.12 sets out the limits and outstanding for the VBP.

Table 9.11 Total VBP of portfolio

Currency/VBP VBP ABS (VBP) in EUR

USD 500 454.55
JPY 60 000 454.55
DKK 2500 336.33
Total вЂ“ 1 245.43

Table 9.12 VBP outstanding

Maximum VBP (EUR) Pf. VBP (EUR) Use of limit

2 000 1 245.43 62.27 %

9.2.2.4 Maximum VaR for portfolio
As we saw in detail in Chapter 6, VaR is a number that represents for the portfolio the
estimate of the maximum loss for a 24-hour horizon, with a probability of occurrence
of 99 changes out of 100 that the effective loss on the portfolio will never exceed that
estimate (and therefore only 1 chance in 100 of the effective loss on the portfolio exceeding
that estimate).
264 Asset and Risk Management

The VaR on the portfolio is calculated daily, in historical simulation, independently of
any (statistical) distributional hypothesis.
The VaR outstanding can never exceed the difference (limit) between

1. the stop loss for the portfolio; and
2. the potential overall loss on the portfolio taken as an absolute value on the date of
calculation of the VaR.

In fact, if the forecast of the maximum loss exceeds the total that вЂ˜canвЂ™ still be lost, that
is, the difference between the maximum acceptable loss and what is already being lost, the
tendency to move outside the limit on the maximum loss criterion becomes unreasonable.

Example
Assume the portfolio shown in the п¬Ѓrst example of this chapter, which shows a stop loss
of EUR19 and a potential overall loss on the portfolio of в€’EUR10 (Table 9.13).
The said portfolio can lose a further EUR9 before becoming out of limits for the
maximum acceptable loss criterion.
The VaR, that is the forecast of the maximum loss on the portfolio during the next 24
hours, can never exceed the difference between:

вЂў the stop loss for the portfolio, namely EUR19; and
вЂў the potential overall loss on the portfolio taken as an absolute value on the date of
calculation of the VaR, that is, EUR10.

The total of EUR9 is the total that the portfolio still has left to lose.

Table 9.13 Stop loss

Stop loss Potential loss Use of limit

в€’EUR10
EUR19 52.63 %
10
Optimising the Global
Portfolio via VaR

As explained in Section 3.2, the modern portfolio theory (MPT) produced by Markowitz1
is based on the idea that the risk2 linked to a portfolio of assets (for a given return on the
portfolio in question) can be minimised by combining risk and return so that unfavourable
variations in assets are at best compensated by the favourable variations in one or more
other assets. This is the principle of portfolio diversiп¬Ѓcation.
Although it is admitted that the distribution of the вЂ˜returnвЂ™ random variable is charac-
terised by the meanвЂ“variance pairing,3 it is easy to formulate this problem mathematically.
By considering the variance on the return as a risk measurement, the portfolio will be
optimised by minimising the variance of its return for a п¬Ѓxed expected value of it.
Variance as a measurement of risk, however, still has the disadvantage of including both
risk of loss and risk of gain; and it is here that the concept of VaR plays an important
role. VaR, unlike variance, actually measures a risk of loss linked to the portfolio and
minimisation of that loss will not therefore take account of the favourable variations in
relation to the expected yield average.
Unfortunately, VaR cannot easily be modelled mathematically and the methods of
calculating it are numerical simulation methods (historical simulation method, Monte
Carlo method). As such, they are accompanied by relatively restrictive hypotheses (esti-
mated varianceвЂ“covariance matrix method). In fact, the only case in which there is a
simple mathematical representation of VaR is where the hypothesis of normality has
been validated.
In this case, as has been seen in Section 6.2.2, VaR will be expressed as a function of
the q-quantile of the law of distribution of variations in the value pt :

V aRq = zq Пѓ ( pt ) в€’ E( pt )

We are therefore interested in asset portfolio optimisation methods that satisfy the normal
law hypothesis; this is especially the case for equities.4 In particular, we will be adapting
two recognised portfolio optimisation methods, i.e. SharpeвЂ™s simple index method (see
Section 3.2.4) and the EGP method (see Section 3.2.6), to suit VaR and will see the ways
in which it improves the optimisation process.
It must be remembered in this regard that the two methods chosen deal with the issue
of optimisation in totally different ways. SharpeвЂ™s method tends to construct an efп¬Ѓciency
limit, which is a parametric solution to the problem of optimisation,5 while the EGP

1
Markowitz H., Portfolio selection, Journal of Finance, Vol. 7, No. 1, 1952, pp. 77вЂ“91. Markowitz H., Portfolio Selection:
Efп¬Ѓcient Diversiп¬Ѓcation of Investments, John Wiley & Sons, Ltd, 1991. Markowitz H., Mean Variance Analysis in Portfolio
Choice and Capital Markets, Basil Blackwell, 1987.
2
The market risk.
3
This is the case, for example, with normal distribution.
4
The statistical distribution of a share is in fact leptokurtic but becomes normal for a sufп¬Ѓciently long period of
measurement.
5
There will be one solution for each portfolio return value envisaged.
266 Asset and Risk Management

method searches for the portfolio that will optimise the risk premium, that is, the single
solution that will maximise the relation
EP в€’ RF
=
P
ПѓP
in which RF represents the yield reckoned to be free of market risk.

10.1 TAKING ACCOUNT OF VaR IN SHARPEвЂ™S
SIMPLE INDEX METHOD
10.1.1 The problem of minimisation
The aim of the portfolio optimisation method perfected by Markowitz is to construct the
efп¬Ѓciency frontier (see Chapter 3). The optimal portfolio will therefore be a function of
the usefulness of the investorвЂ™s risk.6 From a mathematical point of view, each expected
return value has an associated Lagrangian function:
пЈ« пЈ¶
N
в€’ О»EP + m пЈ­ Xj в€’ 1пЈё
L(X1 , . . . , XN , m) = ПѓP
2

j =1

From this, the optimisation equations can be obtained:
L Xi = 0 i = 1, . . . N
Lm =0

The п¬Ѓrst series of equations expresses the minimisation7 of the portfolio variance. The
second is the constraint relative to the composition of the portfolio.
In this context, the Sharpe index expresses the assets on the basis of which the optimal
portfolio needs to be built, according to an index common to all the assets in question8
(Ri = ai + bi RI + Оµi ) and provides a quasi-diagonal form9 for the variance optimisation
equations by the introduction of an additional constraint:10
N
bi Xi = Y
i=1

Taking account of VaR in SharpeвЂ™s method will therefore consist of replacing the
variance with the VaR in the expression of the Lagrangian function
пЈ« пЈ¶ пЈ« пЈ¶
N N
L(X1 , . . . , XN , Y, m1 , m2 ) = zq ПѓP в€’ EP в€’ О»EP + m1 пЈ­ Xj в€’1пЈё + m2 пЈ­ Xj bj в€’YпЈё
j =1 j =1

referring to the expression of VaR in the normality hypotheses.

On this subject, the Lagrangian parameter l plays an important role in the matter of portfolio choice (Broquet C., Cobbaut
6

R., Gillet R. and Vandenberg A., Gestion de Portefeuille, De Boeck, 1997, pp. 304вЂ“13).
7
In fact, these equations express the optimisation of the variance, which corresponds to a minimisation because of the
convex form of the variance.
8
Methods that involve several groups of assets each dependent on one index are known as вЂ˜multi-index methodsвЂ™. SharpeвЂ™s
is a simple index method.
9
The coefп¬Ѓcients matrix for the equation system is diagonal.
10
Refer to Section 3.2.4 for notations.
Optimising the Global Portfolio via VaR 267

This form of Lagrangian function is very different from the classical form, as it involves
the standard deviation and not the portfolio variance. We will see that this leads to a
number of complications with regard to the implementation of the critical line algorithm.

10.1.2 Adapting the critical line algorithm to VaR
We have based our workings on the philosophy of the efп¬Ѓciency frontier construction
methods in order to build up a VaR minimisation method for a portfolio for a given
return on that portfolio. This has led us to adapt the Lagrangian function using SharpeвЂ™s
method. We will now adapt the critical line algorithm, used most notably in SharpeвЂ™s
method, to VaR.
The optimisation equations are written as:
пЈ±
пЈґ zq (ПѓP ) Xi в€’ ai в€’ О»ai + m1 bi + m2 = 0 i = 1, . . . , N
пЈґ
пЈґ
пЈґ z (Пѓ ) в€’ E в€’ О»E в€’ m = 0
пЈґq PY
пЈґ I I
пЈґ 2
пЈґN
пЈґ
пЈґ
пЈІ
Xj bj = Y
пЈґ j =1
пЈґ
пЈґ
пЈґN
пЈґ
пЈґ
пЈґ
пЈґ
пЈґ Xj = 1
пЈґ
пЈі
j =1

The terms
(ПѓP ) Xi i = 1, . . . , N
(ПѓP ) Y

being expressed as depending on the various assets involved:
пЈ± пЈ« пЈ¶
пЈґ
пЈґ N
Xi ПѓОµ2i
пЈґ
пЈґ (Пѓ ) = пЈ­ 2 2пЈё
пЈґ P Xi Xj ПѓОµj + Y ПѓI = i = 1, . . . , N
22
пЈґ
пЈґ ПѓP
пЈІ j =1
пЈ¶ Xi
пЈ«
пЈґ
пЈґ N
пЈґ Y ПѓI2
пЈґ (Пѓ ) = пЈ­ 2 2пЈё
пЈґ PY Xj ПѓОµj + Y ПѓI =
22
пЈґ
пЈґ
пЈі ПѓP
j =1
Y

The system of equations becomes:
пЈ±
пЈґ Xi ПѓОµi
2
пЈґ zq в€’ ai в€’ О»ai + m1 bi + m2 = 0 i = 1, . . . , N
пЈґ
пЈґ
пЈґ ПѓP
пЈґ
пЈґ
пЈґ YПѓ2
пЈґ
пЈґ
пЈґ zq I в€’ EI в€’ О»EI в€’ m2 = 0
пЈґ
пЈґПѓ
пЈґ
пЈІ P
N
пЈґ Xj bj = Y
пЈґ
пЈґ
пЈґ
пЈґ j =1
пЈґ
пЈґ
пЈґN
пЈґ
пЈґ
пЈґ
пЈґ
пЈґ Xj = 1
пЈґ
пЈі
j =1
268 Asset and Risk Management

The optimisation equations then assume an implicit and non-linear form as SP is itself a
function of XI . This problem can be resolved if we use an iterative method for resolving
the nonlinear equations on the basis of PicardвЂ™s iteration,11 as:

вЂў it shows the п¬Ѓnancial conditions that imply the presence of a solution;
вЂў it supplies a start point close to that solution (that is, the corner portfolios), thus allowing
rapid convergence.12

The resolution algorithm will therefore be the one described in Figure 10.1.

START

1st corner portfolio
construction

1st portfolio standard
deviation calculation

Start of the critical
line algorithm

Start of Picard's
iteration

Corner portfolio
constuction

Portfolio Standard
deviation calculation

no Is the convergence
acceptable?
yes

End of Picard's
iteration

Is the last corner
no
portfolio constructed?

yes

End of the critical line
algorithm

END

Figure 10.1 Algorithm for taking account of VaR in a simple index model

11
See for example Burden R. L. and Faires D. J., Numerical Analysis, Prindle, Weber & Schmidt, 1981; Litt F. X., Analyse
numВґ rique, premiВґ re partie, ULG, 1999, pp. 143вЂ“50; Nougier J. P., MВґ thodes de calcul numВґ rique, Masson, 1993.
e e e e
12
Global convergence methods, such as the bisector method, converge in all cases but relatively slowly.
Optimising the Global Portfolio via VaR 269

10.1.3 Comparison of the two methods
We have compared SharpeвЂ™s simple index method with the VaR minimisation method
described above.13 To this end, we have chosen to construct the efп¬Ѓciency frontier relative
to the equity portfolio for the Spanish market index, IBEX35, using the two methods.
The comparison of the portfolios obtained is based on VaR at 99 %, 95 % and 90 %. The
range of portfolios compared has been constructed14 so as to correspond to a range of
given values for the expected returns.
The convergence towards an optimal portfolio is of one signiп¬Ѓcant п¬Ѓgure of VaR by iter-
ation. The results obtained from the two methods are identical to three signiп¬Ѓcant п¬Ѓgures,
which implies that at least four iterations are needed to produce a difference between the
methods. Moreover, each iteration requires calculation of the standard deviation for the
portfolio. In consequence, the number of calculations required to construct the efп¬Ѓciency
frontier will be very high indeed.15
In addition, as SharpeвЂ™s method п¬Ѓxes the return value in its minimisation problem, the
minimisation of VaR will eventually minimise the variance, thus removing much of the
interest in the method.
Taking account of VaR in efп¬Ѓciency frontier construction methods does not therefore
interest us greatly, although it does provide something positive for risky portfolios.

10.2 TAKING ACCOUNT OF VaR IN THE EGP16 METHOD
Unlike SharpeвЂ™s simple index method, the EGP method does not look to construct an
efп¬Ѓciency frontier. Instead, it looks for the portfolio that will maximise the risk premium

EP в€’ RF
=
P
ПѓP
In addition, the philosophy of the EGP method is not limited merely to solving an
optimisation equation. Instead, it aims to apply a criterion of choice of securities through
comparison of the risk premiums17 of the security and of the portfolio.
Finally, remember that this method is based on the same hypotheses as SharpeвЂ™s method,
namely that the return on each security can be expressed as a function of the market return
(or of a representative index), and that the CAPM is valid.
Using the condition of optimisation of the risk premium ( P ) Xi = 0 as a basis, we
bi
look to obtain a relation of the type18 Zi = 2 (Оёi в€’ П†K ), where Оёi is a measurement of
ПѓОµ i
the risk premium on the asset (i) and П†K is a measurement of the risk premium19 of the
portfolio if it included the asset (i).
13
Explicit calculations are to be found on the CD-ROM, п¬Ѓle вЂ˜Ch 10вЂ™.
14
The basis used is the fact that a linear combination of corner portfolios is in itself optimal (Broquet C., Cobbaut R.,
Gillet R. and Vandenberg A., Gestion de Portefeuille, De Boeck, 1997).
15
In our case, the total number of operations needed for the VaR minimisation method will be about п¬Ѓve times the total
in SharpeвЂ™s simple index method.
16
Elton, Gruber and Padberg (see Section 3.2.6).
17
In fact, we are not comparing the risk premiums but the expressions of those premiums.
18
It can be shown (Vauthey P., Une approche empirique de lвЂ˜optimisation de portefeuille, eds., Universitaires Fribourg
Suisse, 1990) that the formula is valid, with or without a bear sale.
19
Also known as вЂ˜acceptability thresholdвЂ™.
270 Asset and Risk Management

From that relation, the assets to be introduced into the portfolio, and from that, the
composition of the portfolio, will be deduced.20 The merits of the method will therefore
also depends on the proper deп¬Ѓnition of the terms Оёi and П†K .

10.2.2 Adapting the EGP method algorithm to VaR
The basic idea of taking account of VaR in portfolio optimisation methods is to replace
variance (or the standard deviation) with VaR in the measurement of the risk.
As the EGP method is based on the concept of risk premium, it is at that level that we
introduce the VaR:
EP в€’ RF
P=
zq ПѓP в€’ EP
The expression of the condition of optimisation of the VaR premium, according to the
return on the assets, produces:
пЈ« пЈ¶
N
пЈ¬ Xj (Ej в€’ RF ) пЈ·
) Xi = пЈ¬ пЈ· =0
( пЈ­ пЈё
P j =1
zq ПѓP в€’ EP Xi

In other words:
Ei в€’ RF EP в€’ RF
в€’ (zq ПѓP в€’ EP )Xi = 0
zq ПѓP в€’ EP (zq ПѓP в€’ EP )2

If it is assumed that zq ПѓP в€’ EP = 0,21 and taking account of the expression of the VaR
premium, Ei в€’ RF в€’ P (zq ПѓP в€’ EP )Xi = 0 can be written:
пЈ® пЈ« пЈ¶ пЈ№
N
zq пЈ­ 2
Ei в€’ RF в€’ P пЈ° bj Xj + Xi ПѓОµ2i пЈё в€’ Ei пЈ» = 0
bi ПѓI
ПѓP j =1

By suggesting
zq P
Zi = Xi .
ПѓP
the equality becomes
N
Ei (1 + P) в€’ RF в€’ bi ПѓI2 bj Zj в€’ Zi ПѓОµ2i = 0.
j =1

or alternatively:
пЈ® пЈ№
N
1пЈ°
bj Zj пЈ» .
Zi = Ei (1 + P) в€’ RF в€’ bi ПѓI2
ПѓОµ2i j =1

20
It can be shown that the assets to be retained in the portfolio, if the portfolio is free of bear sales, are those for which
the term Оёi в€’ П†K (a measure of the additional risk premium provided for each asset considered) is positive.
21
This will always be the case if the VaR premium is bounded, a necessary hypothesis for calculating a maximum on
this variable.
Optimising the Global Portfolio via VaR 271

By multiplying the two members of the equation by bI and summarising, we arrive at:
пЈ® пЈ№
N N N
bi пЈ°
bj Zj пЈ»
Zi bi = Ei (1 + P ) в€’ RF в€’ bi ПѓI2
ПѓОµ i
2
i=1 i=1 j =1

N N N
bi2 bi
1+ ПѓI2 В· Zi bi = [Ei (1 + P) в€’ RF ]
ПѓОµ2i ПѓОµ2i
i=1 i=1 i=1

N
Zi bi into that of Zi , we п¬Ѓnally arrive at:
By introducing the expression of i=1
пЈ® пЈ№
N
bj
ПѓI2 [Ej (1 + P ) в€’ RF ] пЈє
пЈЇ
ПѓОµ2j
пЈЇ E (1 + пЈє
bi пЈЇ i P ) в€’ RF пЈє
j =1
Zi = 2 пЈЇ в€’ пЈє
ПѓОµ i пЈЇ пЈє
bi N
bj
2
пЈ° пЈ»
1 + ПѓI 2
Пѓ 2
j =1 Оµj

or:
bi
Zi = (Оёi в€’ П†)
ПѓОµ2i
This is similar to the relation used in the classical EGP method for determining the assets
to be included in the portfolio when short sales are authorised.
If short sales are forbidden, we proceed as explained in Section 3.2.6, and calculate
bi
Zi = 2 (Оёi в€’ П†K ), where П†k is the maximum value for
ПѓОµ i
k
bj
ПѓI2 [Ej (1 + P) в€’ RF ]
ПѓОµ2j
j =1
П†k = k
bj
2
1+ ПѓI2
ПѓОµ2j
j =1

after the securities have been sorted according to their decreasing Оёi values.
Note that the terms Ej and P are implicit functions of the Xi values.22 Given that P
is nonlinear in Xi , the algorithm of the EGP method must incorporate a Picard process,
for the same reasons as in Section 10.1.2 (see Figure 10.2).

10.2.3 Comparison of the two methods
We will now compare the classical EGP method with the вЂ˜EGPвЂ“VaRвЂ™ method that we
have just covered. For this purpose, we will be using a numerical example.

Example
We have chosen to determine the portfolio with optimal VaR premium relative to the
equities on the Spanish market index (IBEX35) using the two methods.

Xi represents the weighting of the security (i) in the portfolio. This is the unknown aspect of the problem.
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272 Asset and Risk Management

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Initialisation of