Hull J.C. Risk management and Financial institutions
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Economic Capital and RAROC
373
Interactions between Risks
There are interactions between the 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 (or contracts) with the counterparty.
2
If the counter-
party 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.
As the Long-Term Capital Management saga clearly shows there can
be interactions between liquidity risks and market risks (see Section 15.9).
There are also interactions between operational risks and market risks. It
is unlikely that we would know about the activities of Nick Leeson at
Barings Bank if he had guessed right about the Nikkei index (see Business
Snapshot 2.3). It is also unlikely that we would hear about a violation of
the rules for a fund (such as the one in Business Snapshot 16.1) if it had
led to a gain rather than a loss.
16.5 AGGREGATING ECONOMIC CAPITAL
Typically a financial institution calculates market, credit, operational, and
(possibly) business risk loss distributions for a number of different business
2
When calculating the expected cost of counterparty default risk in Section 12.1, we
(assumed no dependence.
Business Snapshot 16.1 The EGT Fund
In 1996 Peter Young was fund manager at Deutsche Morgan Grenfell, a
subsidiary of Deutsche Bank. He was responsible for managing a fund called
the European Growth Trust (EGT). It had grown very large and Young had
responsibilities for managing over one billion pounds of investors' money.
Certain rules applied to EGT. One of these was that no more than 10% of
the fund could be invested in unlisted securities. Peter Young violated this rul
in a way, it can be argued, that benefited him personally. When the facts were
uncovered, he was fired and Deutsche bank had to compensate investors. The
total cost to Deutsche Bank was over 200 million pounds.
374
Chapter 16
units. It is then faced with the problem of aggregating the loss distributions
to calculate a total economic capital for the whole enterprise.
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 is the total economic capital for the financial institution
facing n different risks and is the economic capital for the ith risk
considered on its own. This is what the Basel Committee does for
regulatory capital. The total regulatory capital a bank is required to keep
under Basel II is the sum of the regulatory capital amounts for credit,
market, and operational risks.
Equation (16.1) 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 that
equation (16.1), when used as a way of aggregating market, credit, and
operational risk, overstates the total capital 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 is the standard deviation of the loss from the ith source of risk
and is the correlation between risk i and risk j. The capital require-
ment 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 equation (16.2).
This approach tends to underestimate the capital requirement because
it takes no account of the Skewness and kurtosis of the loss distributions.
Economic Capital and RAROC 375
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. Copulas were discussed in Chapter 6. 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 alter-
native is to assume that they are multivariate t. This leads to the joint
probability of extreme values of two variables being higher than in the
Gaussian copula. This is discussed further in Section 6.4.
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 nonnormal, 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 16.3
Suppose that the estimates for economic capital for market, credit, and
operational risk for two business units are as shown in Table 16.2. The
correlations between the losses are shown in Table 16.3. The correlation
between credit risk and market risk within the same business unit is 0.5,
and the correlation between operational risk and either credit or market risk
within the same business unit is 0.2. (These correspond to the estimates of
Rosenberg and Schuermann mentioned above.) The correlation between two
different risk types in two different business units is zero. The correlation
between market risks across business units is 0.4. The correlation between
376 Chapter 16
Table 16.2 Economic capital estimates for
Example 16.3.
credit risk across business units is 0.6. The correlation between operational
risk across business units is zero.
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 economic capital for Business Unit 2 is
Table 16.3 Correlations between losses in Example 16.3.
MR, CR, and OR refer to market risk, credit risk, and
operational risk; 1 and 2 refer to business units.
MR-1
CR-1
OR-1
MR-2
CR-2
OR-2
MR-1
1.0
0.5
0.2
0.4
0.0
0.0
CR-1
0.5
1.0
0.2
0.0
0.6
0.0
OR-1
0.2
0.2
1.0
0.0
0.0
0.0
MR-2
0.4
0.0
0.0
1.0
0.5
0.2
CR-2
0.0
0.6
0.0
0.5
1.0
0.2
OR-2
0.0
0.0
0.0
0.2
0.2
1.0
Market risk
Credit risk
Operational risk
Business unit
1 2
30 40
70 80
30 90
Economic Capital and RAROC 377
The total enterprise-wide economic capital is the square root of
30
2
+ 40
2
+ 70
2
+ 80
2
+ 30
2
+ 90
2
+ 2 x 0.4 x 30 x 40 + 2 x 0.5 x 30 x 70
+ 2 x 0.2 x 30 x 30 + 2 x 0.5 x 40 x 80 + 2 x 0.2 x 40 x 90
+ 2 x 0.6 x 70 x 80 + 2 x 0.2 x 70 x 30 + 2 x 0.2 x 80 x 90
or 203.224.
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 two business units is
100 + 153.7 = 253.7. Both of these are greater than the total economic capital
estimate of 203.2.
16.6 ALLOCATION OF THE DIVERSIFICATION
BENEFIT
Suppose that the sum of the economic capital for each business unit,
is $2 billion and the 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 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 issues here are analogous to the issues we discussed in Section 8.5
concerned with allocating VaR. The theoretically best allocation scheme
is to allocate an amount
to the ith business unit, where E is the total economic capital and is the
investment in the ith business unit. As we pointed out in Section 8.5, a
result known as Euler's theorem ensures that the total of the allocated
capital is E.
378
Chapter 16
Define as the increase in the total economic capital when we
increase by A discrete approximation for the amount allocated
to business unit i is
where
Example 16.4
Consider again Example 16.3. The total economic capital is 203.2. The
economic capital calculated for Business Unit 1 is 100 and that calculated
for Business Unit 2 is 153.7.
A naive procedure would allocate 100/253.7 of the total economic capital to
Business Unit 1 and 153.7/253.7 of the economic capital to Business Unit 2.
This would result in 80.1 for Business Unit 1 and 123.1 for Business Unit 2.
The incremental effect of Business Unit 1 on the total economic capital is
203.2 - 153.7 = 49.5. Similarly, the incremental effect of Business Unit 2 is
203.2 - 100 = 103.2. The two incremental capitals do not add up to the total
capital (as is usually the case). We could use them as a basis for allocating the
total capital. We would then allocate 49.5/(49.5+ 103.2) of the capital to
Business Unit 1 and 103.2/(49.5 + 103.2) of it to Business Unit 2. This would
result in 65.9 for Business Unit 1 and 137.3 for Business Unit 2.
To apply equation (16.4), we could calculate the partial derivative analytic-
ally. Alternatively, we can use a numerical approximation. When we increase
the size of Business Unit 1 by 1%, its economic capital amounts for market,
credit, and operational risk in Table 16.2 increase to 30.3, 70.7, and 30.3,
respectively. The total economic capital becomes 203.906, so that =
203.906-203.224 = 0.682.
When we increase the size of Business Unit 2 by 1%, its economic capital
amounts for market, credit, and operational risk in Table 16.2 increase to 40.4,
80.8, and 90.9, respectively. The total economic capital becomes 204.577, so
that = 204.577 - 203.224 = 1.353.
In this case, because we are considering 1 % increases in the size of each unit,
= 0.01. From equation (16.4) the economic capital alloca-
tions to the two business units are 68.2 and 135.2. (These do not quite add up
to the total economic capital of 203.2 because we approximated the partial
derivative.)
16.7 DEUTSCHE BANK'S ECONOMIC CAPITAL
Deutsche Bank publishes the result of its economic capital calculation in
its annual financial statements. Table 16.4 summarizes the economic
capital and regulatory capital for 2004. Deutsche Bank calculated a
Economic Capital and RAROC
379
Table 16.4 Deutsche Bank's economic capital and regulatory
capital (millions of euros).
Credit risk 5,971
Market risk 5,476
Diversification benefit across credit and market risk (870)
Operational risk 2,243
Business risk 381
Total economic capital 13,201
Total risk-weighted assets 216,787
Tier 1 capital held (% of risk-weighted assets) 8.6%
Tier 2 capital held (% of risk-weighted assets) 4.6%
Total capital held (% of risk-weighted assets) 13.2%
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.
16.8 RAROC
Risk-adjusted performance measurement (RAPM) has become an import-
ant part of how business units are assessed. There are many different
approaches, but all have one thing in common. They compare return with
capital employed in a way that incorporates an adjustment for risk.
The most common approach is to compare expected return with
economic capital. This is usually referred to as RAROC (risk-adjusted
return on capital). The formula is
The numerator may be calculated on a pre-tax or post-tax basis. Very
often, a risk-free rate of return on the economic capital is calculated and
added to the numerator.
380
Chapter 16
Example 16.5
When lending in a certain region of the world, an AA-rated bank estimates its
losses as 1 % of outstanding loans per year on average. The 99.97% worst-case
loss (i.e., the loss exceeded only 0.03% of the time) is 5% of outstanding loans.
As shown in Example 16.1, the economic capital required per $100 of loans is
$4, which is the difference between the 99.97% worst-case loss and the expected
loss. (This ignores diversification benefits that would in practice be allocated to
the lending.) The spread between the cost of funds and the interest charged is
2.5%. Subtracting from this the expected loan loss of 1%, the expected profit
per $100 of loans is $1.50. Assume that the lending department's administrative
costs total 0.7% of the amount loan, the expected profit is reduced to $0.80 per
$100 in the loan portfolio. RAROC is therefore
An alternative calculation would add the interest on the economic capital to the
numerator. Suppose the risk-free interest rate is 2%. Then 0.02 x 4 = 0.08 is
added to the numerator, so that RAROC becomes
As pointed out by Matten, it is more accurate to refer to the approach in
equation (16.5) as RORAC (return on risk-adjusted capital) rather than
RAROC.
3
In theory, RAROC should involve adjusting the return (i.e.,
the numerator) for risk. In equation (16.5) it is the capital (i.e., the
denominator) 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-the-year bonuses, etc. The other is as a tool to decide whether a
particular business unit should be expanded or contracted. The latter
involves predicting an average RAROC for the unit and comparing it
with the bank's threshold return on capital.
When RAROC is used for the second purpose, it should be noted that
it could be low simply because the business unit had a bad year. Perhaps
credit losses were much larger than average or there was an unexpectedly
large operational risk loss. This is not necessarily an indication that the
business unit should be shut down. When RAROC is used as a forward-
looking measure, the calculation should reflect average losses. The aim is
to assess the long-term viability of the business unit, whether it should be
expanded or scaled back, and so on.
3
See C. Matten, Managing Bank Capital: Capital Allocation and Performance
Measurement, 2nd edn., Chichester, UK: Wiley, 2000.
Economic Capital and RAROC
381
SUMMARY
Economic capital is the capital that a bank or other financial institution
deems necessary for the risks it is bearing. When calculating economic
capital, a financial institution is free to adopt any approach it likes. It
does not have to use the one proposed by regulators. Typically, it
estimates economic capital for credit risk, market risk, operational risk,
and (possibly) business risk for its business units and then aggregates
these estimates to produce an estimate of the economic capital for the
whole enterprise. The one-year loss distributions for market risk, credit
risk and operational risk are quite different. The loss distribution for
market risk is symmetrical. For credit risk, it is skewed and for opera-
tional risk it is highly skewed with very heavy tails.
The total economic capital for a financial institution is allocated to
business units so that a return on capital can be calculated. There are a
number of allocation schemes. The best are those that reflect the incre-
mental impact of the business unit on the total economic capital. The
amount of capital allocated to a business unit is generally less than the
capital estimated for the business unit as a stand-alone entity because of
diversification benefits.
FURTHER READING
Dev, A., Economic Capital: A Practitioner's Guide. London: Risk Books, 2004.
Matten, C, Managing Bank Capital: Capital Allocation and Performance
Measurement, 2nd edn. Chichester, UK: Wiley, 2000.
Rosenberg, J. V., and T. Schuermann, "A General Approach to Integrated Risk
Management with Skewed, Fat-Tailed Risks," Federal Reserve Bank of New
York, Staff Report No. 185, May 2004.
QUESTIONS AND PROBLEMS (Answers at End of Book)
16.1. What is the difference between economic capital and regulatory capital?
16.2. Why do AA-rated banks use a confidence level of 99.97% when calculat-
ing economic capital for a one-year time horizon?
16.3. What is included in business risk?
16.4. In what respects are the models used to calculate economic capital for
market risk, credit risk, and operational risk likely to be different from
those used to calculate regulatory capital?
382 Chapter 16
16.5. Suppose the credit loss in a year has a lognormal distribution. The loga-
rithm of the loss is normal with mean 0.5 and standard deviation 4. What
is the economic capital requirement if a confidence level of 99.97% is used?
16.6. Suppose that the economic capital estimates for two business units are as
follows:
Market risk
Credit risk
Operational risk
Business unit
1 2
20 40
40 30
70 10
The correlations are as in Table 16.3. Calculate the total economic capital
for each business unit and the two business units together.
16.7. In Problem 16.6, what is the incremental effect of each business unit on
the total economic capital? Use this to allocate economic capital to
business units. What is the impact on the economic capital of each
business unit increasing by 0.5%? Show that your results are consistent
with Euler's theorem.
16.8. A bank is considering expanding its asset management operations. The
main risk is operational risk. It estimates that the expected operational
risk loss from the new venture in one year is $2 million and the 99.97%
worst-case loss (arising from a large investor law suit) is $40 million. The
expected fees it will receive from investors for the funds under adminis-
tration are $12 million per year and administrative costs are expected to
be $5 million per year. Estimate the before-tax RAROC?
16.9. RAROC can be used in two different ways. What are they?
ASSIGNMENT QUESTIONS
16.10. Suppose that daily gains and losses are normally distributed with a
standard deviation of (a) Estimate the minimum regulatory capital
the bank is required to hold for market risk. (Assume a multiplicative
factor of 3.0.) (b) Estimate the economic capital for market risk using a
one-year time horizon and a 99.97% confidence limit. (c) Why is the ratio
of your answer in (a) to your answer in (b) not a good indication of the
ratio of regulatory market risk capital to economic market risk capital in
practice?
16.11. Suppose that a bank's sole business is to lend in two regions of the world.
The lending in each region has the same characteristics as in Example 16.5
of Section 16.8. Lending to Region A is three times as great as lending to