ńņš. 11 |

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 ā˜pipā™, upwards or downwards

on the result for the position. The calculations are simple:

75 000 000 Ć— 1.0267 = 77 002 500

75 000 000 Ć— 1.0268 = 77 010 000

77 010 000 ā’ 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 (ā˜Returnsā™ sheet, cell AO4) is ā’1.0497 %. This

means that: 0.9336 Ć— (1 + (ā’1.0497)) = 0.9238.

By applying Position 3 of the treasury portfolio to the current rate, we have: 1.0267 Ć—

(1 + (ā’1.0497)) = 1.01592273.

The investorā™s position is ā˜longā™, so a fall in the EUR/USD rate will be unfavourable

for him, and the loss (in USD) will be:

75 000 000 Ć— ((1.0267 Ć— (1 + (ā’1.0497)) ā’ 1.0267))

= 75 000 000 Ć— (1.01592273 ā’ 1.0267) = ā’808 295.31

This result is displayed in cell F3 of the ā˜Treasury Revalā™ 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 ā˜Treasury Revalā™ sheet.

248 Asset and Risk Management

For each of the positions, from 1ā“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 proļ¬t (positive total) as calculated above taking

account of past returns.

Let us take as an example the ļ¬rst revaluation (corresponding to the 26 January 2001

return) for Position 1 of the portfolio (cell D3 in the ā˜Treasury Revalā™ sheet).

The formula that allows the loss or proļ¬t 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 Ā· R Ā· (1 + HR) Ā· ā’ CĀ·RĀ·

DIV DIV

ND ND

=CĀ· R Ā· (1 + HR) Ā· ā’ RĀ·

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 simpliļ¬ed3

it here:

ND ND

L=CĀ· R Ā· (1 + HR) Ā· ā’ RĀ·

DIV DIV

ND

=CĀ·RĀ· Ā· ((1 + HR) ā’ 1)

DIV

ND

=CĀ·RĀ· Ā· HR

DIV

If the investor is ā˜longā™, 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 (proļ¬t). This is the case for Position 1 of the portfolio on

26 January 2001.

On the other hand, if the investor is ā˜shortā™, 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 ā˜longā™) must be multiplied by ā’1.

For Position 2 of the portfolio, the investor is ā˜shortā™ and we have (cell E3 of the

ā˜Treasury Revalā™ sheet:

ND ND

L = (ā’1) Ā· C Ā· R Ā· (1 + HR) Ā· ā’ RĀ·

DIV DIV

3

We have not simpliļ¬ed 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 proļ¬t 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 ā˜Treasury Revalā™ 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 ā˜Treasury Revalā™ sheet) a

series of 100 losses or proļ¬ts 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 ļ¬nd column G of the ā˜Treasury Revalā™ sheet classiļ¬ed in increasing

order on the ā˜Treasury VaRā™ sheet, in column B. To the right of this column, 1 ā’ q appears.

A. Numerical interpretation

We think it important to state once again that when 1 ā’ q corresponds to a plateau of the

loss distribution function, we have chosen to deļ¬ne VaR as the left extremity of the said

section (see Figure 6.7).

We therefore say that:

ā¢ There are 66 chances out of 100 that the actual loss will be ā’EUR360 822 or less

(1 ā’ q = 0.34), or VaR 0.66 = ā’360 822.

ā¢ There are 90 chances out of 100 that the actual loss will be ā’EUR1 213 431 or less

(1 ā’ q = 0.10), or VaR 0.90 = ā’1 213 431.

ā¢ There are 99 chances out of 100 that the actual loss will be ā’EUR2 798 022 or less

(1 ā’ q = 0.01), or VaR 0.99 = ā’2 798 022.

B. Representation in graphical form

If the forecast of losses is shown on the x-axis and 1 ā’ q is shown on the y-axis, the

estimated loss distribution will be obtained. Figure 8.1, also appears on the ā˜Treasury

VaRā™ 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

ā“3 000 000 ā“2 000 000 ā“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 ļ¬rst stage once again consists of determining the past returns (in this case, weekly).

8.2.2.1 Past variations to be applied

The main difļ¬culty 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 ļ¬rst 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 simpliļ¬ed 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 ā˜yield to maturityā™ or YTM, can be calculated as shown in Section 4.1.2.

Columns H3 to H9 of the ā˜Portfoliosā™ sheet show the relative yields for the bond in

our ļ¬ctitious portfolio. As not all versions of Excel contain the ā˜yieldā™ ļ¬nancial 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 reļ¬‚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 ā˜Ratesā™ sheet the returns to be applied to the yield to maturity;

the remaining periods are in fact broken.

We have determined (in the ā˜Bonds Interpā™ 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 ļ¬nd the value of the rates to be interpolated (taken from the ā˜Ratesā™ sheet)

in the ā˜Variation Bondsā™ 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

ā˜Variation Bondsā™ sheet.

8.2.2.2 Composition of portfolio

The ā˜bondā™ portfolio is found on the ā˜Portfoliosā™ sheet in CH8.XLS. This sheet, like the

rest of the portfolio, is purely ļ¬ctitious. The investorā™s national currency is the euro.

Setting Up a VaR Methodology 251

The portfolio is ā˜longā™, with six bonds, for which the following are given:

ā¢ currency;

ā¢ coupon;

ā¢ maturity date;

ā¢ ISIN code;

ā¢ last known price (this is the ā˜bidā™, because if the position is closed, one should expect

to deal at the bid price);

ā¢ yield on maturity (in formula form in column H, in copied value in column I);

ā¢ basis of calculation (current/current or 30/360);

ā¢ frequency of payment of the coupon.

8.2.2.3 Portfolio revaluation

In Table B2ā“H9, the ā˜Losses Bondsā™ 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 ā˜historicalā™ yields to maturity in

order to reconstruct a synthesised price (column N). The ā˜Priceā™ 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 ā˜Priceā™

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 proļ¬t or loss (column P for bond 1). As indicated in cell B11, we assume that

we are holding a nominal of Ā¤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

Ā¤100 million. In fact, rather than expressing the nominal in PLN, calculating the loss or

proļ¬t 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 sufļ¬cient (column AP) to summarise the six losses and/or proļ¬ts 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 proļ¬ts (ā˜Losses

Bondsā™ 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

ā“1 600 000 ā“1 100 000 ā“600 000 ā“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 ļ¬nd column

AP in the ā˜Losses Bondsā™ sheet classiļ¬ed in ascending order on the ā˜Bonds VaRā™ sheet

in column B. 1 ā’ q is located to the right of that column.

A. Numerical interpretation

We say that:

ā¢ There are 66 chances out of 100 that the actual loss will be ā’EUR917 or less (1 ā’ q =

0.34), or VaR 0.66 = ā’917.

ā¢ There are 90 chances out of 100 that the actual loss will be ā’EUR426 740 or less

(1 ā’ q = 0.10), or VaR 0.90 = ā’426 740.

ā¢ There are 99 chances out of 100 that the actual loss will be ā’EUR1 523 685 or less

(1 ā’ q = 0.01), or VaR 0.99 = ā’1 523 685.

B. Representation in graphical form

If the loss estimates are shown on the x-axis and 1 ā’ q is shown on the y-axis, the esti-

mated loss distribution will be obtained. Figure 8.2 also appears on the ā˜Bonds VaRā™ 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 speciļ¬cally in extreme values) will report the

observed historical frequencies poorly, as they will be too ļ¬‚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 ā˜Calc Nā™ 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

ā˜Bonds VaRā™ sheet). We have produced 26 categories (from ā’1 600 000 to ā’1 465 000,

from ā’1 465 000 to ā’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 ā’1 523 685.01 (cell D4) will belong

to the ļ¬rst class (from ā’1 600 000 to ā’1 465 000, column G).

The table G2ā“AF103 on the ā˜Calc Nā™ sheet contains one class per column (lines

G2ā“AF3) and 100 lines, that is, one per loss (column D4ā“D103). Where a given loss

intersects with a class, there will be a ļ¬gure of 0 (if the loss is not in the category in

question) or 1 (if otherwise).

By ļ¬nding the total of 1s in a column, we will obtain the number of losses per class,

or the frequency. Thus, a loss of between ā’1 600 000 and ā’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 AH2ā“AJ29 carry the category centres (ā’1 532 500 for the class ā’1 600 000 to

ā’1 465 000), and the frequencies as a ļ¬gure 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 xā’Āµ 2

1

Section 3.4.2 ā˜Continuous modelā™), that is, f (x) = ā exp ā’ , to each

Ļ

2

2ĻĻ

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 ā˜Graph

Nā™ 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

ā˜Graph Nā™ sheet):

Observed distribution (Bonds Pf.)

14 %

12 %

10 %

8%

6%

4%

2%

0%

ā“1 532 500

ā“1 397 500

ā“1 262 500

ā“1 127 500

ā“992 500

ā“857 500

ā“722 500

ā“587 500

ā“452 500

ā“317 500

ā“182 500

ā“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

ā“1 600 000 ā“850 000 ā“100 000 650 000 1 400 000

Figure 8.4 Normal distribution

Observed and normal distribution (Bonds Pf.)

2.E ā“ 06

Observed dist.

1.E ā“ 06 Normal dist.

8.E ā“ 07

6.E ā“ 07

4.E ā“ 07

2.E ā“ 07

0.E + 00

ā“1 532 500

ā“1 397 500

ā“1 262 500

ā“1 127 500

ā“992 500

ā“857 500

ā“722 500

ā“587 500

ā“452 500

ā“317 500

ā“182 500

ā“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

ā¢ the class centres;

ā¢ the observed frequencies relating to them;

ā¢ 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 conļ¬rms the leptokurtic nature of the ļ¬nancial 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 ļ¬rst 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 ā˜asset and risk managementā™, we are

arriving, especially in the ļ¬eld of individual client portfolio management, at ā˜portfolio

risk managementā™, 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:

ā¢ Sharpeā™s simple index method (see Section 3.2.4) and the EGP method (see Section 3.2.6).

ā¢ 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

ā¢ Stop loss

ā¢ Portfolio risk

ā¢ Asset allocation &

ā¢ Credit equivalent

management

market timing

ā¢ VBP

ā¢ Stock picking

ā¢ Fund risk ā¢ VaR

ā¢ Currency allocation

management ā¢ 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 sufļ¬ciently long period.

9

Portfolio Risk Management1

9.1 GENERAL PRINCIPLES

This involves application of the following:

ā¢ To portfolios managed traditionally, that is, using:

ā” asset allocation with a greater or lesser risk proļ¬le (including, implicitly, market

timing);

ā” a choice of speciļ¬c securities within the category of equities or options (stock

picking);

ā” currency allocation.

ā¢ To particularly high-risk portfolios (said to have a ā˜high leverage effectā™) falling clearly

outside the scope of traditional management (the most frequent case), a ļ¬vefold risk

management method that allows:

ā” 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.

ā” 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 clientā™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:

ā¢ linear portfolios (such as FRA, IRS, currency swaps and other forward FX);

ā¢ 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:

ā¢ 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, Julyā“August 1999, pp. 44ā“6.

258 Asset and Risk Management

ā¢ 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:

ā¢ the investment strategy;

ā¢ the risk framework.

9.2.1 Investment strategy

This part sets out:

ā¢ The portfolio management strategy.

ā¢ The responsibilities of each of the parties.

ā¢ The maximum maturity dates of the transactions.

ā¢ 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:

ā¢ the limit;

ā¢ 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 (deļ¬ned below) can

never exceed x % of the cash equivalent of the portfolio, the portfolio being deļ¬ned as

the sum of:

ā¢ the available cash balances, on one hand;

ā¢ 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 clientā™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

ā¢ the value of its constituent assets at the initiation of each transaction;

ā¢ 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 ļ¬ve equities ABC at EUR10 per share and

ļ¬ve 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 ļ¬ve equities ABC) will, however,

be excluded. The overall loss will therefore be ā’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 (deļ¬ned in the ā˜portfolio stop

lossā™ section). The credit liabilities, which consist of the sum of the credit equivalents

deļ¬ned 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:

ā¢ 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);

ā¢ involves a contingent obligation (if options are issued);

ā¢ involves a contingent right (if options are purchased);

Table 9.1 Stop loss

Stop loss Potential loss Use of limit

ā’EUR10

EUR19 52.63 %

Credit risk

+

Spot

Option issues

Option purchase

Forward Fx

ā“ FRA, IRS and currency swaps

Figure 9.1 Weight of the credit equivalent

260 Asset and Risk Management

ā¢ 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 ļ¬ve 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 ā“

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

Table 9.3 FX derivatives and FRA

Product P/S Currency Nominal Price/premium

Three-month call Strike 1.1 P EUR/USD EUR11 million EUR220 000

Two-month put Strike 195.5 S GBP/JPY Ā£5 million GBP122 000

FRA 3ā“6 S DKK 100 million 3.3 %

Table 9.4 Revaluation price

Product Historical price Current price Loss (currency) Potential loss

(EUR)

ā’100 000 ā’89 285.71

Spot 1.1 1.12

ā’25 million ā’189 969.60

FX forward 117 114.5

+11 000 +11 000

Long call 2.00 % nom. EUR 2.10 % nom. EUR

ā’2000 ā’3034.90

Short put 2.44 % nom. GBP 2.48 % nom. GBP

ā’25 000 ā’3363.38

FRA 3.3 % 3.4 %

ā’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 Ć— 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

ā’189 969.60Ā·110 %

FX forward 208 966.56

Long call 220 000.00 220 000.00

Short put 7 587 253.41Ā·60 % 4 552 352.05

ā’3363.38Ā·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:

ā¢ The following rules have been adopted (Table 9.5).

ā¢ The forward of the put totals ā’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 quantiļ¬es the port-

folioā™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:

ā¢ one-hundredth of the stop loss expressed in currency;

ā¢ 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 deļ¬ned 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:

ā¢ one-hundredth of the stop loss expressed in the currency, that is, EUR200 000 divided

by 100, that is EUR2000, which equals:

ā¢ 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:

ā¢ the total VBP per currency is equal to the sum of the VBPs of each asset making up

the portfolio in that currency, and;

ā¢ 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 ļ¬rst 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 Ā· 0.01 % Ā· 180/360 = 500

1 200 000 000 Ā· 0.01 % Ā· 180/360 = 60 000

The equivalent loan in USD has a VBP of ā’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 Ā· 0.01 % Ā· (180 ā’ 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 3ā“6 S DKK 100 million 3.3 %

Portfolio Risk Management 263

Table 9.10 Total VBP per currency

Product/currency USD JPY DKK

ā’500

Loan

+60 000

Deposit

ā’2 500

FRA

ā’500 +60 000 ā’2 500

Total

The VBP for the FRA totals ā’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 ļ¬xed 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 Ć— 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 ā“ 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 ā˜canā™ 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 ļ¬rst example of this chapter, which shows a stop loss

of EUR19 and a potential overall loss on the portfolio of ā’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:

ā¢ the stop loss for the portfolio, namely EUR19; and

ā¢ 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

ā’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 diversiļ¬cation.

Although it is admitted that the distribution of the ā˜returnā™ random variable is charac-

terised by the meanā“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 ļ¬xed 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 varianceā“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 Ļ ( pt ) ā’ 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. Sharpeā™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. Sharpeā™s method tends to construct an efļ¬ciency

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. 77ā“91. Markowitz H., Portfolio Selection:

Efļ¬cient Diversiļ¬cation 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 sufļ¬ciently 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 ā’ RF

=

P

ĻP

in which RF represents the yield reckoned to be free of market risk.

10.1 TAKING ACCOUNT OF VaR IN SHARPEā™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

efļ¬ciency frontier (see Chapter 3). The optimal portfolio will therefore be a function of

the usefulness of the investorā™s risk.6 From a mathematical point of view, each expected

return value has an associated Lagrangian function:

ļ£« ļ£¶

N

ā’ Ī»EP + m ļ£ Xj ā’ 1ļ£ø

L(X1 , . . . , XN , m) = ĻP

2

j =1

From this, the optimisation equations can be obtained:

L Xi = 0 i = 1, . . . N

Lm =0

The ļ¬rst 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 + Īµi ) 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 Sharpeā™s method will therefore consist of replacing the

variance with the VaR in the expression of the Lagrangian function

ļ£« ļ£¶ ļ£« ļ£¶

N N

L(X1 , . . . , XN , Y, m1 , m2 ) = zq ĻP ā’ EP ā’ Ī»EP + m1 ļ£ Xj ā’1ļ£ø + m2 ļ£ Xj bj ā’Yļ£ø

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. 304ā“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 ā˜multi-index methodsā™. Sharpeā™s

is a simple index method.

9

The coefļ¬cients 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 efļ¬ciency 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 Sharpeā™s

method. We will now adapt the critical line algorithm, used most notably in Sharpeā™s

method, to VaR.

The optimisation equations are written as:

ļ£±

ļ£“ zq (ĻP ) Xi ā’ ai ā’ Ī»ai + m1 bi + m2 = 0 i = 1, . . . , N

ļ£“

ļ£“

ļ£“ z (Ļ ) ā’ E ā’ Ī»E ā’ m = 0

ļ£“q PY

ļ£“ I I

ļ£“ 2

ļ£“N

ļ£“

ļ£“

ļ£²

Xj bj = Y

ļ£“ j =1

ļ£“

ļ£“

ļ£“N

ļ£“

ļ£“

ļ£“

ļ£“

ļ£“ Xj = 1

ļ£“

ļ£³

j =1

The terms

(ĻP ) Xi i = 1, . . . , N

(ĻP ) Y

being expressed as depending on the various assets involved:

ļ£± ļ£« ļ£¶

ļ£“

ļ£“ N

Xi ĻĪµ2i

ļ£“

ļ£“ (Ļ ) = ļ£ 2 2ļ£ø

ļ£“ P Xi Xj ĻĪµj + Y ĻI = i = 1, . . . , N

22

ļ£“

ļ£“ ĻP

ļ£² j =1

ļ£¶ Xi

ļ£«

ļ£“

ļ£“ N

ļ£“ Y ĻI2

ļ£“ (Ļ ) = ļ£ 2 2ļ£ø

ļ£“ PY Xj ĻĪµj + Y ĻI =

22

ļ£“

ļ£“

ļ£³ ĻP

j =1

Y

The system of equations becomes:

ļ£±

ļ£“ Xi ĻĪµi

2

ļ£“ zq ā’ ai ā’ Ī»ai + m1 bi + m2 = 0 i = 1, . . . , N

ļ£“

ļ£“

ļ£“ ĻP

ļ£“

ļ£“

ļ£“ YĻ2

ļ£“

ļ£“

ļ£“ zq I ā’ EI ā’ Ī»EI ā’ m2 = 0

ļ£“

ļ£“Ļ

ļ£“

ļ£² P

N

ļ£“ Xj bj = Y

ļ£“

ļ£“

ļ£“

ļ£“ j =1

ļ£“

ļ£“

ļ£“N

ļ£“

ļ£“

ļ£“

ļ£“

ļ£“ Xj = 1

ļ£“

ļ£³

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 Picardā™s iteration,11 as:

ā¢ it shows the ļ¬nancial conditions that imply the presence of a solution;

ā¢ 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

numĀ“ rique, premiĀ“ re partie, ULG, 1999, pp. 143ā“50; Nougier J. P., MĀ“ thodes de calcul numĀ“ 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 Sharpeā™s simple index method with the VaR minimisation method

described above.13 To this end, we have chosen to construct the efļ¬ciency 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 signiļ¬cant ļ¬gure of VaR by iter-

ation. The results obtained from the two methods are identical to three signiļ¬cant ļ¬gures,

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 efļ¬ciency

frontier will be very high indeed.15

In addition, as Sharpeā™s method ļ¬xes 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 efļ¬ciency 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

10.2.1 Maximising the risk premium

Unlike Sharpeā™s simple index method, the EGP method does not look to construct an

efļ¬ciency frontier. Instead, it looks for the portfolio that will maximise the risk premium

EP ā’ RF

=

P

ĻP

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 Sharpeā™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 (Īøi ā’ ĻK ), where Īøi is a measurement of

ĻĪµ i

the risk premium on the asset (i) and ĻK 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, ļ¬le ā˜Ch 10ā™.

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 ļ¬ve times the total

in Sharpeā™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 lā˜optimisation de portefeuille, eds., Universitaires Fribourg

Suisse, 1990) that the formula is valid, with or without a bear sale.

19

Also known as ā˜acceptability thresholdā™.

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 deļ¬nition of the terms Īøi and ĻK .

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 ā’ RF

P=

zq ĻP ā’ EP

The expression of the condition of optimisation of the VaR premium, according to the

return on the assets, produces:

ļ£« ļ£¶

N

ļ£¬ Xj (Ej ā’ RF ) ļ£·

) Xi = ļ£¬ ļ£· =0

( ļ£ ļ£ø

P j =1

zq ĻP ā’ EP Xi

In other words:

Ei ā’ RF EP ā’ RF

ā’ (zq ĻP ā’ EP )Xi = 0

zq ĻP ā’ EP (zq ĻP ā’ EP )2

If it is assumed that zq ĻP ā’ EP = 0,21 and taking account of the expression of the VaR

premium, Ei ā’ RF ā’ P (zq ĻP ā’ EP )Xi = 0 can be written:

ļ£® ļ£« ļ£¶ ļ£¹

N

zq ļ£ 2

Ei ā’ RF ā’ P ļ£° bj Xj + Xi ĻĪµ2i ļ£ø ā’ Ei ļ£» = 0

bi ĻI

ĻP j =1

By suggesting

zq P

Zi = Xi .

ĻP

the equality becomes

N

Ei (1 + P) ā’ RF ā’ bi ĻI2 bj Zj ā’ Zi ĻĪµ2i = 0.

j =1

or alternatively:

ļ£® ļ£¹

N

1ļ£°

bj Zj ļ£» .

Zi = Ei (1 + P) ā’ RF ā’ bi ĻI2

ĻĪµ2i 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 Īøi ā’ ĻK (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:

ļ£® ļ£¹

N N N

bi ļ£°

bj Zj ļ£»

Zi bi = Ei (1 + P ) ā’ RF ā’ bi ĻI2

ĻĪµ i

2

i=1 i=1 j =1

N N N

bi2 bi

1+ ĻI2 Ā· Zi bi = [Ei (1 + P) ā’ RF ]

ĻĪµ2i ĻĪµ2i

i=1 i=1 i=1

N

Zi bi into that of Zi , we ļ¬nally arrive at:

By introducing the expression of i=1

ļ£® ļ£¹

N

bj

ĻI2 [Ej (1 + P ) ā’ RF ] ļ£ŗ

ļ£Æ

ĻĪµ2j

ļ£Æ E (1 + ļ£ŗ

bi ļ£Æ i P ) ā’ RF ļ£ŗ

j =1

Zi = 2 ļ£Æ ā’ ļ£ŗ

ĻĪµ i ļ£Æ ļ£ŗ

bi N

bj

2

ļ£° ļ£»

1 + ĻI 2

Ļ 2

j =1 Īµj

or:

bi

Zi = (Īøi ā’ Ļ)

ĻĪµ2i

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 (Īøi ā’ ĻK ), where Ļk is the maximum value for

ĻĪµ i

k

bj

ĻI2 [Ej (1 + P) ā’ RF ]

ĻĪµ2j

j =1

Ļk = k

bj

2

1+ ĻI2

ĻĪµ2j

j =1

after the securities have been sorted according to their decreasing Īøi 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 ā˜EGPā“VaRā™ 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.

22

272 Asset and Risk Management

START

Initialisation of

VaR's premium

Start of Picard's

process

Optimal portfolio

construction

ńņš. 11 |