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Universiteit van Amsterdam

24-07-2014

BSc Scriptie en afstudeerseminar

Begeleider: Tim Boonen

Risk Capital Allocation for Financial

Conglomerates

Jan Uytenhout

10121234

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Table of Contents

ABSTRACT   1   1.   INTRODUCTION   2   2.   RISK  MEASURES   3   2.1      INTRODUCTION   3  

2.2   EXAMPLES  OF  RISK  MEASURES   5  

3.   THE  CAPITAL  ALLOCATION  PROBLEM   7  

3.1      INTRODUCTION   7   3.2     PROPORTIONAL  ALLOCATIONS   8   3.3       EULER  ALLOCATION   11   4.       CONTINUITY   13   5.   CONCLUSION   22   REFERENCES   23  

Abstract

This paper discusses the risk capital allocation problem and presents a unifying framework in which both risk measures and allocation methods are discussed and compared with each other. The risk measures included are Standard Deviation, Value-at-Risk and Expected Shortfall, the latter of which is highly recommended, since it is fair and coherent. Three proportional allocation methods, Haircut Principle, Quantile Principle and Covariance Principle, are discussed shortly, after which the main focus is on the Euler Allocation using Expected Shortfall. This paper shows that small continuity problems arise in certain restricted settings when using this method.

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1. Introduction

The financial crisis of 2007-2008, also known as the Global Financial Crisis, revealed some major deficiencies in financial regulation. In response to this, in 2010, the Basel Committee agreed upon the Third Basel Accord. This third installment of the Basel accords contains detailed measures to tighten supervision on financial institutions, such as banks, insurance companies and financial conglomerates. To lower the risk of default, they have to hold more risk capital of the highest quality. That means the capital should be available at all times. The risk measures used to calculate the amount of risk capital to be held should also change according to the Basel Committee (Basel Committee on Banking Supervision, 2012). Value-at-Risk, the method now commonly used, uses a confidence interval to calculate how much risk capital is needed to prevent insolvency with a certain probability. Since it does not include what happens outside that interval, it does not represent the full picture of the losses and should therefore be reconsidered. The risk measure is not only used to calculate the risk capital, but also to allocate this capital among different constituents of big financial firms. Therefore, the introduction of the Third Basel Accord has a major influence on the capital allocation of financial conglomerates.

If investment a has an expected return of 100%, whereas investment b has an expected return of only 10%, it seems more favorable to choose investment a, because it has a higher Return on Capital (RoC). However, for potential new investments, RoC often does not completely represent the value of the opportunity. It does not take the riskiness of the investment into account. An easy solution is to divide the RoC by the risk. The return !!, where m is the RoC and r is the risk, is called the Return on Risk-Adjusted Capital (RORAC). But how is this risk measure, and therefore the materiality of risk, calculated? And how can this measure of risk be used to make complicated financing decisions, like allocating risk capital? This paper explores what risk measures can be used to represent the risk financial conglomerates face, and how those risk measures can be used to calculate and allocate the risk capital needed to prevent insolvency.

Allocating the risk capital is a key issue for financial conglomerates. Wrong capital allocations can lead to wrong management decisions. Dhaene et al. (2012) discuss three major reasons to allocate capital among different lines of business. First,

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3 holding capital is expensive, and those costs need to be redistributed among the lines of business so they will be transferred back to the right depositors or policyholders. Second, allocating those costs is also necessary for financial reporting purposes. Third, it is useful for measuring and comparing the performances of the different business lines. If the capital allocation is done wrong, the performance measurement and comparison can give a distorted presentation, and based on that a firm can decide to keep running unprofitable business lines and lose profitable businesses.

An extensive amount of literature is written on this topic, in which many different methods are proposed and justified. The most common approaches are the economic approach (Tasche, 2000), the game-theoretic approach (Denault, 2001), and the axiomatic approach (Kalkbrener, 2005). Though using results of both Denaults’ and Kalkbreners’ papers, this paper mainly focusses on the economic approach. It will discuss the most commonly used risk measures using this approach and compare them using several properties of a special class of risk measures known as ‘coherent’ risk measures. It is then shown how those risk measures are commonly used to allocate the aggregate risk capital of financial conglomerates. The different methods are put in the same, common framework to enable a valid comparison.

In the next section, an introduction to risk measures is given. A few examples of commonly used risk measures follow after that in section 2. Section 3 describes the allocation problem and puts the different allocation formulae in the same framework. After that, the Euler Allocation method will be discussed using two examples. Finally, in section 7 the results will be discussed in a concluding discussion.

2. Risk Measures

2.1      Introduction  

Risk of financial conglomerates is often referred to as the uncertainty of its future net worth. To make sure this net worth does not get too low, extra capital has to be held. Risk measures are used to express this risk in terms of monetary units (Dhaene et al., 2003). With other words, a risk measure is a capital reserve for preventing insolvency (Tasche, 2000). Some authors, like Artzner et al. (1999) define risk as the variability in a future net worth, since that is only based on one value. But most authors define risk as the potential future loss, which is based on both the income and the

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4 expenditures. This paper follows the second approach. Therefore, the random variable X stands for the loss of a financial conglomerate in some point in the future, so the smaller X is, the better. The risk of a firm, in terms of monetary units, is defined as a function ! of the future loss. Then ![!] is the amount of risk capital to be held by the firm, in order to prevent solvency.

All random variables are assumed to be defined on a fixed probability space (Ω, ℱ, ℙ). That means Ω is a set of all the possible outcomes, ℱ is a σ-algebra of subsets of Ω, and ℙ is the probability measure. Let !!(Ω, ℱ, ℙ) be the space of

bounded random variables on which ! is defined. Then the function ! is given by: ! ∶   !!→  ℝ ∶ ! ∈ !! →  ![!]

Many risk measures have been developed, and they range from very basic methods to more complex and therefore more expensive methods. To analyse and compare those risk measures, Artzner et al. (1999) presented a unifying framework using axioms. This framework could not specify one unique risk measure that should be used, but instead characterized a class of risk measures called ‘coherent risk measures’. A risk measure is coherent if it satisfies the following four properties1:

Subadditivity !"#  !"#  !!, !! ∈ !!,   !  ! !+  !! ≤  !  !! + ![  !!] Positive homogeneity !"#  !"#  ! ∈ !!  !"#  ! > 0, ! !" = !"[!] Monotonicity !"#  !"#  !!, !! ∈ !!  !"#ℎ  !ℎ!"  !! ≤ !!  , !  !! ≤ ![  !!] Translation invariance !"#  !"#  !   ∈   !!  ! ∈ ℝ, ! ! + !! ! = ! ! + !,

In the last one, !!is the risk-free interest rate. These properties can be seen as necessary conditions for a risk measure to be reasonable (Denault, 2001). The first one, subadditivity, means that risks are diversifiable. Two risks put together cannot create a greater risk than the sum of those two risks. Positive homogeneity is a case of subadditivity, but when there is no diversification possible. Monotonicity implies that

1 See Artzner et al. (1999) for a full discussion on the axiomatics of these properties. 2 CTE and Tail Value-at-Risk are similar to Expected Shortfall, but use different weights.

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5 when loss !! is always smaller than !!, !! cannot be riskier that !!. The last one, translation invariance, means that if a risk free investment amount b is added to the loss (a certain loss), the same amount should be added to the risk capital. This property ensures the following relation:

!"#  !"#  ! ∈ !!,   ! ! + !  ! ! ! = 0

This matches the previous description of a risk measure. Is also ensures that the risk measure uses the same unit as the random variable X, namely currency.

2.2   Examples  of  Risk  Measures  

This section will explore three different risk measures, Standard Deviation, Value at Risk and Expected Shortfall, and check whether they satisfy all the conditions of coherent risk measures. These particular risk measures are chosen because they are the most commonly used in practice.

It would be logical to use the standard deviation to measure the fluctuations in the loss of a company. That’s why in classical portfolio theory the risk is measured by the standard deviation (Markowitz, 1952).

Example 1 (Standard Deviation)

Let c be a nonnegative real number, then

! ! = ! !"# ! + ! ! = ! ∗ !"# ! + ! !

where the constant c is often chosen as the 99%-quantile of the standard normal distribution. To the expectation and standard deviation, the following equalities apply:

! !!!+ !!! = !" !! + !"(!!) !"# !" + ! = ! !"# !  

!"# !!+ !! ≤ !"# !! + !"# !!  

This implies that the standard deviation risk measure is translation invariant, positively homogeneous and subadditive. It can also be proved that in a restricted

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6 setting it is monotone, see Kalkbrener (2005) for the full discussion on that. This all concludes to the fact that this risk measure, in the restricted setting, is coherent. However, risk only contains the positive fluctuations in the loss, whereas standard deviation measures both positive and negative ones. So even though it might be coherent, it does not fully represent the risk.

Therefore, a new risk measure emerged in the late 1980s; Value-at-Risk (VaR). Until the 2008 Global Financial Crisis it was a common industry practice, driven by banking and insurance regulations, to measure stand-alone losses by a VaR for a given probability level α. This method uses an α% confidence interval to calculate exactly how much capital is needed to prevent a default with a probability of α.

Example 2 (Value at Risk)

!"#  ! ∈ 0,1 ,  !!!! ! = !"# ! ∈ ℝ ∶  ℙ ! ≤ ! ≥ 1 − ! ! ! = !"#! =  !!!! !

It can be shown that VaR is translation invariant, positively homogeneous and monotone, however it is not subadditive, and therefore not coherent. Diversification may lead to a higher VaR, whereas it usually decreases the total risk. This can be caused by small insignificant risks that become significant when added up. For example, two firms have a 5% chance to have a loss and 95% chance to have no loss. Together, they have only a 90,25% chance of no loss at all. Another shortcoming of this risk measure is that it only looks at the probability that X is high enough to prevent insolvency, and not what happens when it is lower. For example, if a 99% confidence interval is used, what happens in that 1%, in the so-called tail? It can be a small extra loss or it can be a very significant loss. So VaR does not have all the desired properties that are needed to be a good risk measure.

A risk measure that does take the tail into account is Expected Shortfall. This method is basically the average of all the VaRs above threshold ! (Tsanakas, 2007). It reflects not only the probability of a potential default, but also the seriousness of it. It is a coherent risk measure, which is shown by Tasche (2000).

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Example 3 (Expected Shortfall)2

Let  α ∈ 0,1  be  the  significance  level, then  the  Expected  Shortfall  is  given  by !"#  !""  ! ∈ !!,

!! ! = !"! = !!! ! ! ∙ !

!!!!!!(!) − !!!!(!) ℙ ! ≥ !!!!(!) − !

Where !!!!!!!(!) is an indicator function, which is 1 if ! ≤ !!!!(!) and 0 otherwise. The second term within the parenthesis, !!!!(!) ℙ ! ≥ !!!!(!) − ! , is a correction in case X has a discrete distribution. When X has a continuous distribution, it will be 0, and therefore the risk measure will come down to the following:

! ! = !"! = ! !  |  ! ≥ !!(!)

This risk measure is often seen as a correction of the Value-at-Risk, since it also takes the ‘tail’ into account. Therefore it is also know as the Tail-Value-at-Risk. The three risk measures mentioned in this section are widely used in practice, though only one of them is coherent. Therefore, the Basel Committee is now implementing new regulations which strongly recommend financial institutions to use the Expected Shortfall (Basel Committee on Banking Supervision, 2012).

3. The Capital Allocation Problem

3.1      Introduction    

For a single cooperation, the total risk capital equals the calculated risk measure. Therefore it is said to ‘match’ the underlying risk. However, for financial conglomerates the aggregate risk of the whole firm is often smaller than the risks of the individual business lines added up together, due to diversification. This chapter will discuss how to allocate the total amount of risk capital among the individual business as fairly as possible.

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8 In the previous chapter, X was defined as the loss of a single company. Now consider a financial conglomerate that consists of a finite amount of n ‘companies’, or business lines. The aggregate loss of the firm, X, can be written by the following sum:

! = !!

! !!!

The aggregate risk capital, which is calculated before the allocation process, is given by K and has to match the risk capital allocated to every individual business line. This is called the ‘Allocation Requirement’. Lets say a financial conglomerate has n individual businesses, then the allocation requirement is given by:

!! = !

!

!!!

It is calculated with the same risk measure as the individual businesses, and is smaller than the sum of the individual risk measures due to diversification, which is also implied by the subbadditivity of the risk measure:

  ! = ! ! = ! !! ! !!! ≤ ! !! ! !!!

Though it might seem like the capital is physically shifted across the various business lines, this is not the case (Dhaene et al., 2012). The allocation is merely done on paper for bookkeeping purposes.

The allocated capital amount !! should be chosen such that it reflects the risk

of business line I. If that extra requirement is satisfied, the allocation is said to be ‘close’ to their corresponding risk (Dhaene et al., 2012). The next paragraphs will discuss some well-known formulae for allocating the risk capital.

3.2     Proportional  Allocations  

The easiest and most straightforward way of allocating the risk capital on the bases of the given risk measures, is to make proportional allocations. The amount of capital

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9 allocated to a certain subdivision i (i=1,…,n) is the corresponding risk measure, multiplied by a factor ! to make sure the full allocation requirement is satisfied:

!! = !" !! , i = 1, … , n

If one and the same ! is chosen for every subdivision, then it is defined as follows:

! = !! !

! ! !!!

Which leads to the following risk capital allocation method:

!! = ! !! ! !! ! !!!      ! = 1, … , !

This is called a proportional allocation. This method can be used with many different risk measures. The most common risk measure used is VaR, which gives rise to the so-called Haircut Allocation:

!! =  !!!! !!  !!!! !!

! !!!

!      ! = 1, … , !

Since VaR, as a quantile risk measure, is not always subadditive, it is possible that the allocated !! capitals exceed the stand-alone risk capitals ! ! (Dhaene et al., 2012). That would mean that the individual businesses would be better off alone.

Another proportional allocation method that uses VaR is the Quantile

Allocation Principle. This method uses the same risk measure, VaR, but changes the

probability level equally among the individual businesses to satisfy the allocation requirement (Dhaene et al., 2012). The allocated capitals would be calculated as follows:

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10 The factor β is chosen so that the full allocation requirement is satisfied:

 !!!!" !! = !

! !!!

In their paper, Dhaene et al. (2012) show how the exact value can be calculated. Again, this method uses VaR as risk measure, and therefore the allocated !! capitals can exceed the stand-alone risk capitals ! !! .

In these two allocation methods, the risk measure ! is law invariant, which means that it only depends on the distribution of X. There is no dependence between the individual loss !! of the different businesses. One way to add that dependence structure to the capital allocation, is to use the Covariance Allocation Principle. Let X still denote the aggregate loss of a financial conglomerate, and !! the loss of individual business unit i. Then the covariance rule uses the covariance between the individual loss !! and the aggregate loss S and divides it by the variance of X to get the right proportion.

!! =

!"# !!  , !

!"# !      ! = 1, … , !

It satisfies the full allocation principle, since the sum of all their covariances is equal to the variance of X:

!"# !!  , !

! !!!

= !"# !

This method takes into account that the losses of individual business units of financial conglomerates are often dependent on each other. Units that are more dependent on the aggregate loss, and therefore have a higher covariance, have to hold more risk capital.

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3.3       Euler  Allocation  

Though the allocation methods mentioned above are fair and close to the risk, there is a general agreement that the best way to allocate the risk capital is proportional with the partial derivatives of the risk measure and using the Euler theorem (Tasche, 2007). This chapter will introduce the Euler Theorem and present a framework in which it can be applied to risk capital allocation.

The Euler Allocation, in game theory also referred to as the Aumann-Shapley allocation (Denault, 2001), is a very old rule for allocation economic capital. To apply this method to the given risk measures, first the participation level of the subdivisions must be introduced. In this case, it is possible that subdivision participate partially (Boonen, 2014). In some literature, the level of participation can also be seen as the amount of money invested in a certain portfolio. But since this paper focuses on financial conglomerates, the participation level is merely used for the calculation of the risk capital.

Participation of subdivision i is defined by !!. This leads to the following risk capital function r:

! ! =  ! !!!!

!

!!!

The Euler allocation uses the partial derivatives of this function to calculate the allocated risk capital. The amount of risk capital allocated to a certain subdivision is equal to the derivative of the aggregate risk capital with respect to the weight, which in this case is the participation (Van Gulick et al. 2012). Therefore, this method is also known as the marginal risk contribution, or the gradient rule. First, the general Euler’s theorem is defined:

!!,! = !!!!!,!(!) !!!

! !!!

This theorem can then be applied to risk allocation theory. Tasche (2000) proves that if the risk capital function r is partially differentiable with continuous derivatives, then there is a unique and suitable risk contribution given by:

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12 !!!"#$% = !"#$% ! = !" !ℎ ! + ℎ!! |!!!= !" !!! ! |!! !,!

If the risk measure ! is positively homogenous, like the three examples given in this paper, then the risk contributions calculated with this formula are called Euler

Contributions. If that capital is then assigned to sub-portfolios or subdivisions of

financial conglomerates, it is called Euler Allocation.

This method can be used with different risk measures. Tasche (2007) simplifies the Euler contribution using the risk measures given in this paper. Using the standard deviation, it looks the same as the proportional risk contribution using that risk measure:

!"#$%!!" = !!"# !!  , !

!"# !

Like expected, subdivisions with a higher covariance with the aggregate risk have a larger marginal risk contribution and therefore have to hold more risk capital. Using the Value-at-Risk, the Euler contribution is given by:

!"#$%!!"#= ! !

!  |  ! = !"#!(!)

Again, VaR is not subadditive and therefore it is more desired to use a different risk measure. Holden (2008) shows that the Euler Allocation using VaR is not monotone, even though VaR itself is monotone. Expected Shortfall could be a better alternative. That risk measure gives the following Euler contribution:

!"#$%!!" = !!! ! ! !!!!!!!! ! + ! ∗ ! !!!!!!!!!!   !"#ℎ  ! = ! − ℙ ! ≥ !!!! ! ℙ ! = !!!! !      !ℎ!"  ℙ ! = !!!! ! > 0 0      !ℎ!"  ℙ ! = !!!! ! = 0

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13 When X has a continuous distribution, ! is 0, and it comes down to the following:

!"#$%!!" = ! !

!  |  ! ≥ !!!! !

This risk contribution formula using Expected Shortfall is the most popular in recent literature. Some writers even say it is the only possible way to allocate risk capital. For example Kalkbrener (2005) concluded that it is the only allocation principle that is compatible with diversification, Tasche (2000) says it is the only right way the allocate risk capital and Denault (2001) derives the same method using a game theoretic approach.

4. Continuity

Though the Euler allocation is often seen as the best way to allocate risk capital, it also has some downsides. It can be shown that in certain restricted settings, the Euler allocation is not continuous. That means that in certain restricted settings, it cannot be defined, which is of course not desired. This chapter researches how and when the Euler allocation can have discontinuities, and how the other methods behave in the same settings. This will be done using some numerical examples.

Throughout this chapter, the following notation is used: • ! = !!, … , !! denotes the set of losses of the subdivisions

• Ω = {!!, … , !!} denotes the set of states of the world

• X!(!) denotes the loss of X! in state ! ∈ Ω

• ℙ ! > 0 denotes the probability that ! ∈ Ω occurs

• ! = {ℙ !! , … , ℙ !! } denotes the vector of those probabilities

In this example, the Euler Allocation using Expected Shortfall for two subdivisions with discretely distributed losses is calculated. It as a basic example with only two subdivision, and therefore it is easier to calculate the Expected Shortfall first and then the Euler Allocation, instead of using the combined formula presented in the previous chapter. First, the Expected Shortfall is calculated using the formula from chapter 2. Since the loss has a discrete distribution, the general formula is used:

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14 !! ! = !"! = !!! !ℙ ! ∙ !!!!!!!(!) − !!!!(!) ℙ ! ≤ !!!!(!) − !

Then, the general Euler Theorem is applied to this risk measure: !"#$%! = !"

!!! ! |!! !,!

In the next examples, the behavior of the Euler Allocation under slightly different circumstances is researched. Both the probabilities and the outcomes vary slightly, to see the difference in the Euler Allocation. Therefore, the following values are used:

! = ! !+ ! ! !− ! !! = 1 0 !! = 1 + !0 ! = 0,05

This means subdivision 1 will have a loss of 1 with probability !!+ !, and no loss with probability !!− !. Subdivision 2 will have no loss with probability !!+ ! and a loss of 1 + ! with probability !!− !. The value for ! is not primarily relevant for these examples, so a commonly used value of 0,05 is used.

In the first example, only the outcome varies. So ! = 0. That gives the following input values:

! = ! ! ! ! !! = 1 0 !! = 0 1 + ! ! = !!+ !! = 1 + !1

With ! = 0,05 and only two different outcomes with both a probability of !

!, the

!-quantile is equal to the worst case scenario:

!!,!" ! =  !"# ! ∈ ℝ ∶  ℙ ! ≤ ! ≥ 0,95 = max  {!!, !!}

Then, the expectation of X above the 95%-quantile, ! ! ∙ !!!!!" ! , is equal to the

worst case scenario times the probability of that scenario, !!. With that information, the Expected Shortfall can be simplified to the following formula:

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15 !"!,!"= 0,05!! ! ! ∙ !!!!!" ! − !!,!" ! ℙ ! ≤ !!" ! − 0,05 = 0,05!! 1 2∗ max !!, !! − max !!, !! ∗ 1 2− 0,05 = max  {!!, !!}

This result, combined with the participation !, leads to the following risk capital function: ! ! = ! !!!! ! !!! = !"!,!" ! !! ! 1 + ! = max   !!, !! 1 + !

Again, this is simply the worst-case scenario, this time including the participation parameter. The Euler Method evaluates this function at point (1,1). Therefore, if ! is positive, !! 1 + ! is bigger than !!, so the result of this function is !! 1 + ! . If ! is negative, the result is !!. However, if ! is 0, the Euler Allocation is not defined. This gives the following results:

!"#$%!,!"!" = !"

!!! ! |!! !,! =    

0   1 + !      !"#  ! > 0 1 0      !"#  ! < 0 !"#$%&"$#      !"#  ! = 0 For slightly positive !, the worst-case scenario is that subdivision 2 has a loss of 1 + !, and therefore an amount of 1 + ! of risk capital is allocated to that

subdivision. The other subdivision gets nothing. If ! is slightly negative, the opposite happens. Subdivision 1 gets an amount of 1 of risk capital, while number 2 gets nothing. There is a gap between these two situations, at the point where ! is equal to 0. The following figures show what happens to the allocated capital for subdivision 1 (diagram 1) and subdivision 2 (diagram 2).

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Figure 1 Capital allocated to subdivision 1

Figure 2 Capital allocated to subdivision 2

At ! = 0 there is a gap for both subdivisions. The Euler Allocation is not defined for that value. This can lead to misallocations and wrong financing decisions. This problem could be solved by taking a weighted average of the allocations with ! slightly above, and slightly under 0. The result would be that both subdivisions get half the risk capital, which in this case is 0,5.

Using the existing allocation methods on this specific problem with ! = 0 seems hard, though. The Covariance Allocation Principle cannot be used, since the variance of X is 0. Either way, the loss is 1 so there is not variation possible. For slightly positive or negative values of !, the variance of X will be very small, which causes high values for the allocated capital, see figures 3 and 4. Note that because the covariance method measures both positive and negative fluctuations, the allocated capital can be negative. This is not possible, for which reason this method is not desired in these circumstances.

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Figure 3 Allocated capital to division 1 using Covariance method

Figure 4 Allocated capital to division 2 using Covariance Method

For ! = −1 the second division has no loss in both cases, so all capital is fully allocated to the first division. For other values of ! the allocation will have negative values, which is not desired.

The Quantile Allocation Principle does not give a valid allocation either. The quantiles of both !! and !! are similar, and either 1 or 0. Therefore, there is no possible value for ! for which both quantiles add up to 1. Generally, this allocation method can only be applied to continuous distributed losses.

However, the Haircut Allocation does give a fair allocation:

!! =  !!" !!

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18 Both quantiles will be 1 in this example, so the allocation will be 50/50, meaning both divisions get half the risk capital. Figures 5 and 6 show how the capital is allocated with different values of !. In this restricted setting with ! = 0, the Quantile

Allocation Principle is the only method that can be used to find an allocation.

Figure 5 Allocated capital to division 1 using Haircut method

Figure 6 Allocated capital to division 2 using Haircut method

In the second example, the outcomes are fixed, but the probabilities vary. So ! = 0. That gives the following input values:

! = ! !+ ! ! !− ! !! = 10 !! = 01 ! = !!+ !! = 11

It can easily be seen that the aggregate risk capital is 1, since there will be a loss of 1 with probability 1. The risk capital function can be derived from these input values:

! ! = ! !!!!

! !!!

= !"!,!" !!!

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19 To calculate the Expected Shortfal, the !-quantile is calculated first. It differs for different values of !:

!!,!" ! =  !"# ! ∈ ℝ ∶  ℙ ! ≤ ! ≥ 0,95 =

max !!, !!      !"#       − 0,45 ≤ ! ≤ 0,45 !!      !"#      0,45 < ! ≤ 0,5

!!      !"#     − 0,5 ≤ ! < −0,45

The previous example showed that if the !-quantile is the maximum of two values that are equal, the Euler Allocation is not defined. So for probabilities that are lower than 1- !, the Euler Allocation is not defined if the two possible outcomes are the same. When  0,45 < ! ≤ 0,5 and !! = !!,the !-quantile is !!, which gives the following risk capital function:

! ! = !"!,!" !!! ! = 0,05 !! 1 2+ ! !!+ 1 2− ! !!− !! 1 2+ ! + 1 2− ! − 0,05

This leads to the following Euler Allocation:

!"#$%!,!"!" = !!!"

! ! |!! !,! = 1 0        !"#  0,45 < ! ≤ 0,5

All the risk capital is allocated to the first subdivision. When −0,5 ≤ ! < −0,45,the !-quantile is !!, which has the same effect, but now for the second subdivision:

!"#$%!,!"!" = !"

!!! ! |!! !,! = 0 1      !"#  − 0,5 ≤ ! < −0,45

All the risk capital is now allocated to the second subdivision. This is shown in the following figures.

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Figure 7 Allocated capital to division 1 using Euler method

So only when there is at least a probability of 1 − ! that one of the outcomes is occurring, the Euler Allocation is defined. Generally, when two outcomes are exactly the opposite of each other, and the probabilities are not significantly different from each other, the Euler Allocation is not defined.

Like in the first example, the Haircut Allocation Principle has no problems with this restricted setting. But if the probability of one of the cases is smaller than ! (in this example 0,05), then the quantile can change to 0, which leads to a different allocation. This is shown in figures 9 and 10.

Figure 9 Allocated capital to division 1 using Haircut method Figure 8 Allocated capital to division 2 using Euler method

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Figure 10 Allocated capital to division 2 using Haircut method

The Covariance Principle cannot be used at all, since the variance of X is 0 for all values of !. The Quantile Allocation Principle is not defined for ! = 0, since both quantiles will be similar, which means the total allocated capital would exceed the aggregate risk capital. For other values of ! the result is show in figures 11 and 12.

Figure 11 Allocated capital to division 1 using Quantile method

Figure 12 Allocated capital to division 2 using Quantile method

The risk capital is either fully allocated to division 1, or to division 2.

Though the problems only occur in certain restricted situations, which do not occur very often, they have a big impact on the use of the allocation methods.

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22 Therefore they should be considered when making the choice of which allocation method to use. This method could also be adjusted to solve those problems, for example by taking a weighted average of two different Euler allocations to solve the discontinuity shown in example 1. Other adjustments have also been introduced, for example by Van Gulick et al. (2012) who introduce excess based allocation, to keep the excess loss to a minimum.

5.

Conclusion

This paper explores the different risk measures used by financial firms to calculate the necessary capital reserves to prevent insolvency. Three main risk measures are

discussed, and judged based on four criteria called the Coherent Risk Measure criteria. Standard deviation is coherent in a restricted setting, but takes both positive and negative fluctuations of the loss, while the real risk lies only in the positive fluctuations. Value at Risk uses a confidence interval to calculate the risk capital needed to prevent insolvency. It is not coherent since it lacks subadditivity. This means diversification can lead to a higher Value at Risk. The third risk measure is Expected Shortfall. It is proven to be fair and coherent, and is therefore highly

recommended by many authors. It does not only take into account the probability of a default, but also the seriousness of it.

After that, this paper discusses how those risk measures are used to divide the aggregate risk capital of financial conglomerates as fair as possible among the

individual subdivisions. Four different methods are compared. The Haircut Allocation uses the Value at Risk to make a proportional allocation. The Quantile Allocation Principle uses the Value at Risk as well, but changes the probability level to match the full allocation requirement. These two methods are not ideal, since they do not they the correlation among the losses of the different subdivisions into account. One method that does take that into account is the Covariance Allocation Principle. The higher the covariance of the loss of a single subdivision with the aggregate loss is, the more risk capital should be allocated to that subdivision. This is a fair allocation, yet the risk measure used is not much desired. The most popular allocation method in recent literature is the Euler Allocation. This method allocates the risk capital using

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23 the partial derivatives of aggregate risk capital with respect to the participation of the subdivision. It can be used in combination with different risk measures, yet Expected Shortfall is highly recommended.

Yet the Euler Allocation using Expected Shortfall is seen as the best way to allocate the risk capital, there are also downsides. This papers shows that in restricted settings, the Euler Allocation is not defined. When two divisions have exact opposite losses in two different outcomes, with almost the same probability, it is impossible to decide which subdivision to allocate the risk capital to using the Euler Allocation with Expected Shortfall. This continuity problem can be solved for example by using an average of two nearby allocations. This could solve some of the problems, yet further research is needed to identify more similar problems and to find good solutions.

References

Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203-228.

Basel Committee on Banking Supervision, (2012). Fundamental Review of the

Trading Book.

Boonen, T. (2014). Game-theoretic Approaches to Optimal Risk Sharing. 1st ed. Prisma Print.

Denault, M. (2001). Coherent allocation of risk capital. Journal of Risk, 4, 1-34. Dhaene, J., Goovaerts, M., & Kaas, R. (2003). Economic capital allocation derived

from risk measures. North American Actuarial Journal, 7(2), 44-56. Dhaene, J., Tsanakas, A., Valdez, E., V, Vanduffel, S. (2012). Optimal capital

allocation principles. Journal of Risk and Insurance, 79(1), 1-28.

Van Gulick, G. De Waegeraere, A. and Norde H. (2012). Excess based allocation of risk capital. Insurance: Mathematics and Economics, 50(1). 26-42.

Holden, L. (2008). Some Properties of Euler Capital Allocation. [online] Norwegian Computing Centre. Available at http://www.nr.no/ [Accessed 02 July. 2014].

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24 Kalkbrener, M. (2005). An axiomatic approach to capital allocation. Mathematical

Finance, 15(3), 425-437.

Markowitz, H. (1952). Portfolio Selection. The Journal Of Finance, 7(1), 77-91. Tasche, D. (2000). Risk contributions and performance measurement. Report Of The

Lehrstuhl Für Mathematische Statistik, TU München.

Tasche, D. (2007). Capital allocation to business units and sub-portfolios: the Euler principle. arXiv preprint arXiv:0708.2542.

Tsanakas, A. (2007). Capital allocation with risk measures. Proceedings Of The 5Th

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