Credit risk economic capital: Measure, Attribution of portfolio diversification benefit, Allocation key to portfolio components.

Emmanuel Noblet

Executive Summary
Recent years have witnessed significant advances in the design, calibration and implementation of credit risk portfolio models. [BANK X] currently uses Moody’s KMV (Kealhofer, McQuown and Vasicek) Portfolio Manager ([PM]). Models enrich management’s ability to make informed decisions to identify concentrations of risk and opportunities for diversification within a disciplined and objective framework, and thus offer a more sophisticated, less arbitrary alternative to traditional lending limit controls. It is thus essential to make sure models are in line with management’s goals and vice versa to make sure management takes some perspective to understand how the central measure it returns, namely credit risk economic capital ([EC]), is constructed and what it means. This memo aims at explaining: A) how credit risk is measured; B) what the implications of attributing or not portfolio diversification effects are; C) how this portfolio measure is then allocated back to the portfolio components.

A - How is credit risk measured?
Back to basics, a credit risk portfolio model is a function that maps a set of facility-level characteristics and market-level parameters to a distribution of potential portfolio credit losses. This definition is better known as Value-at-Risk ([VaR]). The concept of VaR has become the standard risk measure used to evaluate exposure to risk. In general terms, the VaR is the amount of capital required to ensure, with a given degree of certainty α, that the firm does not become technically insolvent, over a given time horizon tH. There are two very important notions in this definition: 1. The time horizon tH; 2. The degree of certainty α. The risk management time horizon tH should reflect the time a financial institution is committed to holding its portfolio. In other words, the time horizon should in principle reflect the length of time [BANK X] Bank is exposed to credit risk, and cannot substantially reduce this risk and / or increase capital. This time is affected by contractual and legal constraints on the one hand and by liquidity considerations on the other.

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Illiquid markets1 pose a problem for the interpretation of VaR numbers. This was first brought to the attention of professional risk managers by Lawrence and Robinson in Liquid measures. Risk, 8 July, PP52–55, 1995. To quote from their paper:
“If we ask the question: “Can we be 98% confident that no more than l would be lost in liquidating the position?” the answer must be “no”. To see why, consider what this measure of VaR implies about the risk management process and the nature of financial markets. In the liquidation scenario we are considering the following sequence of events is implied: at time t it is decided to liquidate the position; during the next 24 hours nothing is done [...]; after 24 hours of inaction the position is liquidated at prices which are drawn from a [prespecified] distribution unaffected by the process of liquidation. This scenario is hardly credible. [...] In particular, the act of liquidating itself would have the effect of moving the price against the trader disposing of a long position or closing out a short position. For large positions and illiquid instruments the costs of liquidation can be significant, in particular if speed is required.”

They conclude that “any useful measures of VaR must take into account the costs of liquidation on the prospective loss.” The events surrounding the near-bankruptcy of the hedge fund LTCM in summer 1998 clearly showed that the concerns of Lawrence and Robinson are more than justified. In fact, illiquidity of markets is regarded by many as the single most important source of risk. Ideally, we should therefore embed the effects of market illiquidity into our models. However, this turns out to be difficult for a number of reasons: first, the price impact of trading a particular amount of a security at a given point in time is hard to measure; it depends on such elusive factors as market mood or the distribution of economic information among agents; second, in illiquid markets, agents are forced to close their position gradually over time to minimize the price impact of their transactions, this in turn would lead to different time horizons tH for different positions, rendering impossible the aggregation of risk measures across portfolios. As a result, this market liquidity factor is most often ignored when computing VaR numbers or related risk measures. In principle, [BANK X]’s retained time horizon is one year. One could nevertheless imagine that EC is computed at different time horizons for different asset classes to identify “hotspots”. The degree of certainty α is arbitrary chosen, as there is no right or wrong position, as long as this choice is aligned with management’s goals. [BANK X]’s current choice (99.95%, i.e., given the chosen time horizon of one year, going bankrupt once in 2000 years) reflects [BANK X]’s willingness to provide comfort to policyholders and regulators in case of extreme losses, as well as to keep cost of debt to a decent level (AA2). A lower confidence interval, e.g.: 90%, is very close to another well-known concept, Earnings-at-Risk ([EaR]). EaR (confidence interval of 85%) might be closer to
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A market for a security is termed liquid if investors can buy or sell large amounts of the security in short time without dramatically affecting its price. Conversely, a market in which the attempt to trade has a large impact on price since there is no counterparty willing to take the other side of the trade, is termed illiquid. 2 Targeting a debt rating is nevertheless highly desirable, as it affects pricing across all types of business ranging from derivatives to general finance, and it affects access to business opportunities.

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management’s time horizon (going bankrupt once in 10 years) and thus closer to accountability vis-à-vis shareholders. A lower confidence interval thus reveals a greater concern for earnings’ volatility, as one moves down the loss distribution towards the mean. Reconciling debt holders’ and shareholders’ perspectives might therefore require using multiple risk measures, EC and EaR. Fortunately, once one has an estimate for the loss-distribution, it is no problem to compute quantiles at different confidence levels simultaneously. The loss distribution profile of [BANK X] Bank’s portfolio is obtained from KMV PM. In short, KMV PM computes a large number of potential [BANK X] Bank asset portfolio values (200.000) at the chosen time horizon tH. It then computes their net present values to match the current risk measurement date and compares them with the current [BANK X] Bank asset portfolio. Determine the default point for each company based on its PDs

Determine the correlation between asset values

Create correlated random asset values for each company

Determine DD from horizon to maturity
  If not defaulted, then value using RCV If defaulted then random draw for LGD and calculate recovery value

Sum the company losses to obtain portfolio loss

Repeat 200,000 times

Create the distribution of portfolio losses

The way KMV PM generates these scenarios can be understood as a “sell and buy-back” approach.
One must keep in mind that KMV PM is a Mark-to-Market model. In Default models, only credit defaults are modelled, variations in the value of assets due to changes in credit quality are not taken into account. Mark-to-Market models do integrate these variations, leading to higher EC figures.

These loss scenarios are then ranked, so that one can compute quantiles. In a VaR approach, the 200.000 KMV PM loss scenarios are ranked in descending (as losses are recorded as positive amounts) order, the EC figure for [Bank X] Bank portfolio is retained as being the loss amount of the 100th scenario minus the mean value of the loss

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distribution (the so-called expected loss [EL]3). One should notice that such a quantilebased approach thus ignores the 99 first worse loss scenarios.

VaR risk measurement approaches have many advantages4:  VaR provides a common measure of risk across different positions and risk factors. It can be applied to any type of portfolio and enables to compare the risks across different portfolios, such as fixed-income and equity. Traditional methods are more limited: duration and convexity measures apply only to fixed-income positions, Greek measures apply only to derivatives positions, portfolio measures apply to equity and similar positions such as commodity, and so forth. VaR enables to aggregate the risks of positions, taking account of the ways in which risk factors correlate with each other, whereas most traditional risk measures do not allow for the “sensible” aggregation of component risks. VaR is holistic in that it takes full account of all driving risk factors. Many traditional measures look at risk factors only one at a time (for example, Greek measures) or resort to simplifications that collapse multiple risk factors into one, as with durationconvexity and CAPM measures. VaR is also holistic in that it focuses assessment on a complete portfolio and not just the individual positions in it. VaR is probabilistic and gives a risk manager useful information on the probabilities associated with specified loss amounts. Many traditional measures, such as duration-convexity and Greeks, only give answers to “What if?” questions and don’t give a specific indication of loss likelihoods. VaR is expressed in the simplest and most easily understood unit of measure, namely, “lost money.” Many other measures are expressed in less transparent units, such as average period to cash flow.

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However, VaR risk measures present strong limitations. In particular, in more robust terms, VaR cannot be considered a coherent risk measure, as it is not sub-additive. What does this mean? Consider the following simple example that eliminates time horizon from consideration. Suppose there is a fair bet on a two digit wheel with equally likely possible values of 00 to 99 (each value has a 1% probability). Suppose if I bet €100 and spin the wheel without getting a value with a second digit of 9 (09, 19, 29, 39, 49, 59, …, 99  thus 10 possible outcomes out of 100 possible outcomes), I receive a €11.11 gain but if a value ending in 9 comes up, I lose my bet of €100. My 89% confidence level VAR is not a loss but a gain of €11.11 (there are 90 possible gain positions and 10 possible loss positions). Now consider an identical bet predicated on the first digit not coming up with the value of 9. Similar to the first example, my VAR at the 89% confidence level is again a
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Treatment of expected profit: Should expected net profit (after expected loss) be subtracted from EC? Net profit accrues to available capital over time. RAROC and Economic Profit calculation combine expected net profit and EC… 4 From Kevin Dowd. Kevin Dowd is professor of financial risk management at Nottingham University Business School in England and a founding partner in Black Swan Risk Advisors LLC, based in Berkeley, CA.

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gain of €11.11. Now consider a portfolio of the two bets combined. In 81 cases I make €22.22 on the two bets, in 18 cases (09, 19,…89, and 90, 91,…,98) I lose €88.89, and in one case, 99, I lose €200. The portfolio VAR at the 89% confidence level is a loss of €88.89. How can diversification result in an increase in risk?
According to Artzner, Delbaen, Eber and Heath (1999), a coherent risk measure should satisfy the four following properties for any two bounded loss random variables X and Y: Let’s denote the risk measure r(.).It is then convenient to think of r(X) as the amount of solvency capital required for the risk X. 1. Sub-additivity: r(X + Y) ≤ r(X) + r(Y). This means that the capital requirement for two risks combined will not be greater than for the risks treated separately. This is necessary, since otherwise companies would have an advantage to disaggregate into smaller companies. 2. Monotonicity: If X ≤ Y for all possible outcomes, then r(X) ≤ r(Y). This means that if one risk always has greater losses than another risk, the capital requirement should be greater. 3. Positive Homogeneity: For any positive constant λ, r(λX) = λr(X). This means that the capital requirement is independent of the currency in which the risk is measured. 4. Translation invariance: For any positive constant a, r(X + a) = r(X) + a. This means that there is no additional capital requirement for an additional risk for which there is no uncertainty. In particular, by making X identically zero, the total capital required for a certain outcome is exactly the value of that outcome.

Other authors have suggested a new coherent measure, Tail VaR (also called conditional value at risk or expected shortfall). Tail VaR is the VaR plus the expected loss in excess of VaR when such excess occurs. Coming back to our previous KMV PM loss distribution example above, Tail VaR would then be the average of the set of worst scenarios above VaR.

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Frequency

Mean = EL

VaR @ 99.95%

Tail VaR = E[L|L > VaR]

Loss VaR @ 99.95% - EL= EC

Overbeck (2000) argues that VaR is an “all or nothing” risk measure, in that if the extreme event causing ruin occurs, there is no capital to cushion losses. VaR thus reflects the point of view of a shareholder who assigns a zero value to his/her shares in the case of insolvency. However, other points of view are valid. Regulators, policyholders and debt holders may be interested in which proportion of the liabilities can be salvaged. These points of view are addressed with the Tail VaR, as, unlike VaR, it therefore also accounts for possible losses beyond the confidence level and weights them appropriately (in [Bank X]’s case, the 99 worse loss scenarios). This measure has the property that the Tail VaR calculated from a combined portfolio does not exceed the sum of the Tail VaRs calculated for its parts (sub-additivity).

B - What are the implications of attributing or not attributing portfolio diversification effects?
One can choose to attribute or not to attribute the diversification benefit to the different BUs within [Bank X] Bank. The advantages and drawbacks of both positions are summarised below:  Attribution of bank-wide diversification benefits to business units:  Allocated EC lower than stand-alone EC per BU;  Actions of other business units may impact allocated EC.  When diversification benefits are not attributed (and kept aside at Group level to create a “buffer” to potentially steer the business units):  Risk concentration within business unit does not benefit from diversification with other BUs;  Concentration effects that arise from exposures across business units are 7|Page

not recognized in EC allocated to BU. To obtain a portfolio loss distribution, KMV PM must determine the joint distribution of over credit losses at the facility level. KMV PM structures this problem by assuming that correlation across obligors can be explained through common dependencies to a set of systematic risk factors (the so-called GCorr model). These factors depict sectorial, geographical and macro-economic dimensions all firms / assets have to live with in an economy. A natural property of this GCorr model is that the marginal capital required for a loan depends on how it affects the overall portfolio diversification, and thus depends on what is already in the portfolio. Also risk of a BU (and thus of its portfolio at facility level) is actually its contribution to the overall company risk measure, not the separate measure of the BU. The remark “concentration effects that arise from exposures across business units are not recognized in EC allocated to BU” becomes especially valid in the context of a large but nevertheless multi-national financial institution (as opposed to international). One could also add that the performance measurement process of portfolio management transactions becomes very complex, as due to KMV PM correlation model, these transactions can lead to very different performances depending on the reference portfolio at stake.

C - How is EC at portfolio level allocated back to the portfolio components (i.a.: BUs, clients, facilities)?
Diversification effects cannot be controlled homogeneously throughout the organisational structure of [Bank X]. If diversification effects can easily be recognised (and thus controlled) at the [Bank X] Bank level, it becomes an entirely different story at, for example, client level, as a PAM can hardly manage what somebody else is doing in other parts of the organisation. The same is true in case tax, legal or compliance constraints apply and hinder a BU from freely defining its investment / divestment strategy. Thus allocating diversification effects (benefits… and drawbacks) at all levels of [Bank X] could be felt as biased and unfair, and might thus lead to demotivation. Using KMV PM, the total required capital is determined from the loss distribution, using a desired debt rating. This capital is then allocated to individual portfolio components based on the proportion of risk each component is contributing to the portfolio. Assuming the unexpected loss of the portfolio ([ULp]) is the risk metric we should consider, a component risk contribution, RCi, is the change in ULp for a marginal change in the component’s weight.

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Frequency

Mean = EL

VaR @ 99.95% - EL= EC

SD = UL

Loss
A AA AAA Capital required to achieve rating = Economic capital

Nevertheless, unexpected losses do not provide guidance regarding the risk of extreme losses. Capital is held in the event of low probability, extreme losses. Correlations generate skewed loss distribution. On the other hand, Tail Risk is the risk of an extreme loss which will cause insolvency. From the context of a portfolio loss distribution, a tail risk event occurs in the tail portion of the portfolio loss distribution. The tail is defined in general as an extreme probability event. Tail Risk Contribution allocates capital via a capital or probability interval. Users can calculate facility tail risk contribution to any portion of the loss distribution.

(R. Clyde (2005) For allocating marginal required economic capital of an exposure, neither Risk Contribution nor Tail Risk Contribution are wrong, unless they do not correspond with management’s goals… The key question then becomes: “What are management’s goals?”  Managing earnings or loss volatility? 9|Page

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Managing the risk of extreme losses? Managing the risk of some less-extreme loss amount?

Just as Artzner, Delbaen, Eber and Heath (1999) defined a set of necessary “good qualities” of a risk measure,, Denault (2001) suggests a set of properties to be fulfilled by a fair risk capital allocation principle. He defines a coherent allocation method as one that satisfies:     Full allocation: All of the capital is allocated to the risk(s); No undercut: Any decomposition of the total risk shall not increase the standalone capital of any given component; Symmetry: Within any decomposition, substitution of one risk for an otherwise identical risk shall not result in any change in the allocation; Riskless allocation: The capital allocation to a risk that has no uncertainty is zero.

Many allocation methods satisfy these axioms. The most prominent ones are:  Tail VaR;  Shapley;  Incremental VaR. Tail VaR is based on the idea that the capital of each BU should be based on the contribution of the specific BU to the total capital. The capital required for each BU is the expected contribution to the total company shortfall when such a shortfall occurs.
Let (1) be [BANK X] Wholesale Banking standalone EC. Let (2) be [BANK X] Retail standalone EC. Let (3) be [BANK X] Direct standalone EC. Let (1+2+3) be the total [BANK X] Bank portfolio EC. Let (1+2), (2+3), (1+3) be the pairwise EC for any combination of [BANK X] Wholesale Banking, [BANK X] Retail and [BANK X] Direct. Shapley allocation method for [BANK X] Wholesale Banking:  { (1) + [(1+2) – (2) + (1+3) – (3)]  2 + [(1+2+3) – (2+3) ] }  3 Incremental VaR allocation method for [BANK X] Wholesale Banking:  [ (1+2+3) – (2+3) ]  { (1+2+3)  [ (1+2+3) – (2+3) ] + [ ( 1+2+3) – (1+3) ] + [ (1+2+3) – (1+2) ] }

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