Transcript Slide 1

TK 6413 / TK 5413 :
ISLAMIC RISK
MANAGEMENT
TOPIC 7:
ECONOMIC CAPITAL
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(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.
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Expected Loss
99.97%
Worst-case loss
Loss over one year
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a) 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.
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TOTAL RISK
Non-business risk
(regulatory capital)
Business risk
(no regulatory capital)
Credit risk
Market risk
Operational risk
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.
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(II) COMPONENTS OF ECONOMIC RISK
a) 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.
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b) 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.
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c) 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.
d) 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
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are being contemplated.
(III) SHAPES OF THE LOSS DISTRIBUTION
GAIN
LOSS
Loss density distribution for market risk
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LOSS
Loss density distribution for credit risk
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LOSS
Loss density distribution for operational risk
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(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.
Type of risk
Second moment
(Standard
deviation)
Third moment
(Skewness)
Fourth moment
(Kurtosis)
Market risk
High
Zero
Low
Credit risk
Moderate
Moderate
Moderate
Low
High
High
Operational risk
Characteristics of loss distribution for different risk types
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a) 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.
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(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
Etotal 
n
E
i 1
i
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
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•
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%
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a) 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
 total 
n
n
i 1
j 1

 i j  ij
Where,  i  standard deviation of the loss from the ith source
of risk, ij  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.
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• 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%
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b) 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.
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c) 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
Etotal 
•
n
n
i 1
j 1

E i E j  ij
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.
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Example:
Suppose that the estimates for economic capital for
market, credit and operational risk for two business units
are shown below:
Type
Business Unit
Market risk
1
30
2
40
Credit risk
70
80
Operational risk
30
90
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The correlations between the losses are shown below:
MR1
CR1
OR1
MR2
CR2
OR2
MR1
1.0
0.5
0.2
0.4
0.0
0.0
CR1
0.5
1.0
0.2
0.0
0.6
0.0
OR1
0.2
0.2
1.0
0.0
0.0
0.0
MR2
0.4
0.0
0.0
1.0
0.5
0.2
CR2
0.0
0.6
0.0
0.0
1.0
0.2
OR2
0.0
0.0
0.0
0.0
0.2
1.0
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We can aggregate the economic capital in a number of ways.
The total market risk economic capital is
30  40  2  0.4  30 40  58.8
2
2
The total credit risk economic capital is
70  80  2  0.6  70 80  134.2
2
2
The total operational risk economic capital is
30  90  94.9
2
2
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The total economic capital for Business Unit 1 is
302  702  302  2  0.5  30 70  100.0
 2  0.2  30  30  2  0.2  70  30
The total economic capital for Business Unit 2 is
402  802  902  2  0.5  40 80  153.7
 2  0.2  40  90  2  0.2  80  90
The total enterprise – wide economic capital is
302  402  702  802  302  902  2  0.4  30 40  2  0.5 30 70  203.224
 2  0.2  30  30  2  0.5  40  80  2  0.2  40  90
 2  0.6  70  80  2  0.2  70  30  2  0.2  80  90
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
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these are greater than the total economic capital estimates of 203.2
(VI) ALLOCATION OF THE DIVERSIFICATION BENEFIT
n
•
E

Suppose that the sum of the economic capital for each business unit,
i 1
i
is $2 billion and total economic capital for the whole bank, after taking lessthan-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.
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•
The theoretically best allocation scheme is to allocate an amount
xi
E
xi
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
Ei
yi
where, yi=  xi xi
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(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.
Credit Risk
Market Risk
Diversification benefit across credit & market risk
Operational Risk
Business Risk
Total Economic Capital
Total risk-weighted assets
Tier - 1 capital held (% of risk-weighted assets)
Tier-2 capital held (% of risk-weighted assets)
Total capital held (% of risk weighted assets)
5,971
5,476
(870)
2,243
381
13,201
216,787
8.6%
4.6%
13.2%
Deutsche Bank’s economic capital and regulatory capital (millions
of euros)
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• 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.
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(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 riskadjusted return on capital (RAROC). The formula is,
RAROC
=
Revenues – Cost – Expected Losses
Economic Capital
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• As pointed by Matten, it is more accurate to refer the
approach in the above equation as RORAC (return on riskadjusted 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.
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