ISLAMIC RISK MANAGEMENT TK 6413 / TK 5413 : ISLAMIC RISK MANAGEMENT TOPIC 7: ECONOMIC CAPITAL

(I) DEFINITION OF ECONOMIC CAPITAL Economic capital is defined as the amount of capital a bank/financial institution needs to absorb losses over a certain time horizon with a certain confidence level. A time horizon is usually chosen as one year. The confidence level depends on the bank’s/financial institution’s objectives. Capital is required to cover unexpected loss. This is defined as the difference between the actual loss and the expected loss. The economic capital for a bank that wants to maintain an AA rating is the difference between expected losses and the 99.97 percentile point on the probability distribution of losses.

Expected Loss 99.97% Worst-case loss Loss over one year

Approaches to Measurement There are two broad approaches to measuring economic capital: the “top-down” and “bottom-up” approaches. In the top-down approach the volatility of the bank’s assets is estimated and then used to calculate the probability that the value of the assets will fall below the value of the liabilities by the end of the time horizon. Using the bottom-up approach (which is most often used), the loss distributions are estimated for different types of risks and different business units and then aggregated. The first step is the aggregation can be to calculate probability distributions for total losses by risk type or total losses by business unit. A final aggregation gives a probability distribution of total losses for the whole financial institution.

TOTAL RISK Non-business risk (regulatory capital) Credit risk Market risk Operational risk Business risk (no regulatory capital) Risk from strategic decision Reputation Risk Under Basel II, regulatory capital is not required for business risk, however some banks/financial institutions do assess economic capital for business risk.

(II) COMPONENTS OF ECONOMIC RISK Market Risk Economic Capital The probability distribution of the loss or gain from market risk can be estimated using the historical simulation or model-building approach. However, this distribution is usually calculated using a one-day time horizon. When calculating economic capital we want to use time horizon (1-year) and confidence level for all risks. Therefore, some assumptions that the daily loss/gain are independent and identically distributed and central limit theorem are used as to standardized the results.

Credit Risk Economic Capital In calculating credit risk economic capital, a bank can choose to adopt a conditional or unconditional model. In a conditional (cycle-specific) model, the expected and unexpected losses take into account of current economic conditions. In an unconditional model (cycle-neutral) model, they are calculated by assuming economic conditions that are in some sense an average of those experienced through the cycle. Rating agencies aim to produce ratings that are unconditional. Moreover, when regulatory capital is calculated using the internal ratings based approach, the PD and LGD estimates should be unconditional.

Operational Risk Economic Capital Banks are given a great deal of freedom in the assessment of regulatory capital for operational risk under the advanced measurement approach. It is therefore likely that most banks using this approach will calculate operational risk economic capital and operational risk regulatory capital in the same way. Business Risk Economic Capital Business risk includes strategic risk and reputational risk. Business risk is difficult to quantify and likely to be largely subjective. It is important that senior risk managers within a financial institution have a good understanding of the portfolio of business risks being taken. This should enable them to assesses the capital required for the risks and more importantly, the marginal impact on total risk of new strategic initiatives that are being contemplated.

(III) SHAPES OF THE LOSS DISTRIBUTION GAIN LOSS Loss density distribution for market risk

Loss density distribution for credit risk

Loss density distribution for operational risk

(IV) RELATIVE IMPORTANCE RISKS The relative importance of different types of risks depends on the business mix. For a bank whose prime business is taking deposits and making loans, credit risk is of paramount importance. For an investment bank, credit risk and market risk are most important. For an asset manager, the greatest risk is operational risk. Characteristics of loss distribution for different risk types

Interactions Between Risks There are interactions between different types of risk. For example, when a derivative such as a swap is traded, there are interactions between credit and market risk. If the counterparty defaults, credit risk exists only if market variables have moved so that the value of the derivative to the financial institution is positive. Another interaction is that the probability of default by a counterparty may depend on the value of a financial institution’s contract with the counterparty. If the counterparty has entered into the contract for hedging purposes, this is not likely to be the case. However, if the contract has been entered into for speculative purposes and the contract is large in relation to the size of the counterparty, there is likely to be some dependence.

(V) AGGREGATING ECONOMIC CAPITAL The simplest approach is to assume that the total economic capital for a set of n different risks is the sum of the economic capital amounts for each risk considered separately so that Where Etotal is the total economic capital for the financial institution facing n different risks and Ei is the economic capital for the ith risk considered on its own. This is in fact what Basel Committee does for regulatory capital

The above relationship is clearly a very conservative assumption The above relationship is clearly a very conservative assumption. It assumes perfect correlation. In the context of economic capital calculations where the confidence level is 99.97%, it would mean that, if a financial institution experiences the 99.97% worst-case loss for market risk, it also experiences the 99.97% worst-case loss for credit risk and operational risk. Rosenberg and Schuermann estimate the correlation between market risk and credit risk to be approximately 50% and the correlation between each of these risks and operational risk to be approximately 20%. They estimate the above relationship, when used as a way aggregating market, credit and operational risk, overstates the total capacity required by about 40%

Assuming Normal Distributions A simple assumption when aggregating loss distributions is that they are normally distributed. The standard deviation of the total loss from n sources of risk is then Where, standard deviation of the loss from the ith source of risk, correlation between risk i and risk j The capital requirement can be calculated from this. For example, the excess of the 99.97% worst-case loss over the expected loss is 3.44 times the number calculated in the relationship above.

This approach tends to underestimate the capital requirement because it takes no account of the skewness and kurtosis of the loss distributions. Rosenberg and Schuermann estimate that, when the approach is applied to aggregating market, credit and operational risk, the total capital is underestimated by about 40%

Using Copulas A more sophisticated approach to aggregating loss distributions is by using copulas. Each loss distribution is mapped on a percentile-to-percentile basis to a standard well-behaved distribution. A correlation structure between the standard distributions is defined and this indirectly defines a correlation structure between the original distributions. Many different copulas can be defined. In the Gaussian copula the standard distributions are assumed to be multivariate normal. An alternative is to assume that they are multivariate t. This leads to do joint probability of extreme values of two variables being higher than in the Gaussian copula.

The Hybrid Approach A simple approach that seems to work well is known as the hybrid approach. This involves calculating the economic capital for a portfolio of risks from the economic capital for the individual risks using When the distributions are normal, this approach is exactly correct. When they are non-normal, the hybrid approach gives an approximate answer – but one that reflects any heaviness in the tails of the individual loss distributions. Rosenberg and Schuermann find that the answers given by the hybrid approach are reasonably close to those given by copula models.

Example: Suppose that the estimates for economic capital for market, credit and operational risk for two business units are shown below:

The correlations between the losses are shown below: MR1 CR1 OR1 MR2 CR2 OR2 1.0 0.5 0.2 0.4 0.0 0.6

We can aggregate the economic capital in a number of ways. The total market risk economic capital is The total credit risk economic capital is The total operational risk economic capital is

The total economic capital for Business Unit 1 is The total enterprise – wide economic capital is There are significant diversification benefits. The sum of the economic capital estimates for market, credit and operational risk is 58.8+134.2+94.9=287.9 and the sum of the economic capital estimates for the two business units is 100.0+153.7=253.7. Both of these are greater than the total economic capital estimates of 203.2

(VI) ALLOCATION OF THE DIVERSIFICATION BENEFIT Suppose that the sum of the economic capital for each business unit, is $2 billion and total economic capital for the whole bank, after taking less-than-perfect correlations into account, is $1.3 billion (65% of the sum of the E’s). The $0.7 billion is a diversification gain to the bank. How should it be allocated to the business units? A simple approach is to reduce the economic capital of each business unit by 35%. However, this is probably not the best approach. Suppose there are 50 business units and that two particular business units both have been an economic capital of $100 million. Suppose that when the first business unit is excluded from the calculations the bank’s economic capital reduces by $60 million and that when the second business unit is excluded from the calculation the bank’s economic capital reduces by $10 million. Arguably, the first business unit should have more economic capital than the second business unit because its incremental impact on the bank’s total economic capital is greater.

The theoretically best allocation scheme is to allocate an amount to the ith business units, where E is the total economic capital and xi is the investment in the ith business unit. Using the Euler’s theorem, ensures that the total of the allocated capital is E. Define ∆Ei as the increase in the total economic capital when we increase xi to ∆xi. A discrete approximation for amount allocated to business unit i is where, yi=

(VII) DEUTSCHE BANK’S ECONOMIC CAPITAL Deutsche Bank publishes the result of its economic capital calculation in its annual financial statements. Table below summarizes the economic capital and regulatory capital for 2004. Deutsche Bank’s economic capital and regulatory capital (millions of euros) Credit Risk Market Risk Diversification benefit across credit & market risk Operational Risk Business Risk 5,971 5,476 (870) 2,243 381 Total Economic Capital 13,201 Total risk-weighted assets 216,787 Tier - 1 capital held (% of risk-weighted assets) Tier-2 capital held (% of risk-weighted assets) Total capital held (% of risk weighted assets) 8.6% 4.6% 13.2%

Deutsche Bank calculated a diversification benefit for credit and market risk, but not for other risk type combinations. The total economic capital is about 13.2 billion euros. This is considerably less than the total regulatory capital which is 8% of 216.8 or about 17.3 billion euros. The actual capital held is about 18.6 billion euros of Tier 1 capital and 10.0 billion euros of Tier 2 capital. It would appear that Deutsche Bank is very well capitalized relative to the risks it is taking.

(VIII) RAROC Risk adjusted performance measurement (RAPM) has become an important part of how business units are assessed. They are many different approaches, but all have one thing in common. They compare return with capital employed in a way incorporate an adjustment for risk. The most common approach is to compare expected return with economic capital. This is usually referred as the risk-adjusted return on capital (RAROC). The formula is, RAROC = Revenues – Cost – Expected Losses Economic Capital

As pointed by Matten, it is more accurate to refer the approach in the above equation as RORAC (return on risk-adjusted capital) rather than RAROC. In theory, RAROC should involve adjusting the return for risk. In the above equation, it is the capital that is adjusted for risk. There are two ways in which RAROC is used. One is as a tool to compare the past performance of different business units, decide on end-of-year bonuses, etc. The other is as a tool to decide whether a particular business unit should be expanded or contracted.