1. Risk Measurement at the Corporate Level: Economic Capital and RAROC
CHAPTER 2
Risk Measurement at the Corporate Level: Economic Capital and RAROC

2. INTRODUCTION Economic capital
the amount of equity capital that the bank should hold for covering its risk.
The more risk a bank has taken, the more capital a bank has to prepare
RAROC (risk-adjusted return on capital) and Shareholder value added (SVA)
the profitability or return after considering risk

3. CAPITAL, RISK, AND THE PROBABILITY OF DEFAULT
There is a tight relationship between the amount of capital a bank holds, the amount of risk it takes, and the probability of the bank's defaulting

4. CAPITAL, RISK, AND THE PROBABILITY OF DEFAULT
illustrate this with an example
Consider you setting up a new bank
$5 million of capital from investors who want a share of the profits (shareholders)
borrow $95 million of debt from people who want a relatively safe return on their money of 5% interest rate

5. CAPITAL, RISK, AND THE PROBABILITY OF DEFAULT
You then buy $100 million of corporate bonds from companies like IBM. These companies promise to pay you back $106 million in one year’s time. (return rate =6%)
If none of the companies default, then in one year you will receive $106 million
You will then pay $99.8 million (95X(1+5%))) to the debt holders and pay $6.2 million to the shareholders
This gives a 25% return on equity (ROE) ((6.2-5)/5) to the shareholders

6. CAPITAL, RISK, AND THE PROBABILITY OF DEFAULT
However, what happens if some of the bond issuers default on their promises?
If at the end of the year 4% of the bonds default (with no recoveries) then the bond portfolio will pay only $101.8 million.
The debt holders still get $99.8 million, but the shareholders absorb the loss, getting only $2.0 million.

7. CAPITAL, RISK, AND THE PROBABILITY OF DEFAULT
If at the end of the year the losses are even worse, and 8% of the bonds have defaulted, the bonds will be worth $97.5 million
This forces the bank to default on its obligation to the debt holders and only give them $97.5 million instead of the $99.8 million that they were promised
In this case, the bank goes out of business and the shareholders get nothing
These three possible outcomes are given in Table 2-1.

8. CAPITAL, RISK, AND THE PROBABILITY OF DEFAULT
100x(1+6%)X(1-4%)
100X(1+6%)
Constant term regardless with borrower defaults or not

9. CAPITAL, RISK, AND THE PROBABILITY OF DEFAULT
Now let us consider the same situation with the same assets but assuming that the bank had been set up with 10% equity at the start of the year and only 90% debt.
The results in Table 2-2
ROE is lower because the profits are diluted among $10M of shareholders rather than $5M
But with $10M of initial capital, even in the worst case, at the end of the year the asset value is greater than the debt, and the bank does not fail

10. CAPITAL, RISK, AND THE PROBABILITY OF DEFAULT
To conclude, the more capital a bank prepares, the less bankruptcy risk for the bank, however, the less ROE for the owners (or shareholders) of the bank

12. The assets (namely, IBM bonds) the bank holds drop further:
(1)16% of credit assets will default
(2)In the other words, the risk of bank (credit risk) increases
In 2 of 3 cases, the bank will fails
In contrast, in Table 2-1, 1 of 3 cases, the bank will fails

13. CAPITAL, RISK, AND THE PROBABILITY OF DEFAULT
To conclude, the more risk a bank has taken,
(1) the more capital a bank has to prepare for maintaining its credit rating
(2) If the bank not increase its capital, then the credit rating of the bank will fall

14. Quick Quiz
Same with the former example, if the initial equity is $20M initial debt is $80M at the beginning of the year
What is the critical default percentage of the bank’s assets for the bank going bankruptcy?

15. Quick Quiz
If other things are constant, the more capital a bank has, the greater/smaller bankruptcy rate for the bank?
The more risk the bank has taken, the greater/smaller bankruptcy rate for the bank?
If the bank takes more risk and it wants to maintain its credit rating, the more/less capital the bank has to prepare?
If the bank takes more risk and it do not want to increase its capital preparation, then its credit rating will increase/decrease?
The risk that the bank faces and the credit rating that the bank want to maintain are two keys for determining the economic capital for the bank

16. INTRODUCTION TO PROBABILITY DISTRIBUTIONS
In reality, there are an infinite number of possible outcomes for the asset value. We represent the distribution of these possible outcomes with a probability density function
Figure 2-2 shows a typical probability density function for credit losses

17. The probability of the asset being less than the debt
y-axis gives the probability of any given asset value
(1)The equity is a random variable as well
(2)The debt is constant
The max. value of asset=106, and the max. equity=6.3
(1) The value of asset is a random variable, namely it has a distribution
(2) The value of credit asset a bank hold will change at various situations

18. The 2nd case, the bank holds more capital
Please refer to Table 2-2
(1)The probability is smaller than the case of Figure 2-2
(2) The more capital a bank has, the less probability of bank default

19. The 3rd case, the bank faces more (credit) risks but not increase its capital preparation
Please refer to Table 2-3
Same with 1st case
(1)The divergence of asset value is greater
(2) The asset risk is greater
(3) The shape of the distribution is flatter!!
The more risk a bank has, the greater default probability for the bank in a given capital

20. ECONOMIC CAPITAL
The economic capital is the net value the bank must have at the beginning of the year to ensure that there is only a small probability of defaulting within that year
The net value is the value of the assets minus liabilities, namely the equity
The small probability is the probability that corresponds to the bank's target credit rating

21. ECONOMIC CAPITAL
For example, an A-rated bank assumes a default rate of around 0.1% (0.001) over the next year.
Of course, it is impossible to actually observe the probability of default of a single bank.
Any single bank will either default or not default, but by looking at the average default rate of all banks in a given grade
it is possible to link credit ratings to the probability of default. This is illustrated in Table 2-5.

22. One basis point=0.01%
(1)AAA-rated Bank will default in the next year with probability 0.01%
(2) On average, AAA-rated bank can survive for 10,000 years(=1/0.01%)
(3) For the establishment of economic capital, the bank have to prepare enough capital for achieving the probability of 0.01%

23. Quick Quiz
What is the default probability of the A-rated (BBB-rated) bank?
In normal, the BBB-rated bank can make more/less profitability than the A-rated bank?
In normal, if you save your money in a bank, the BBB-rated will provide you the higher/lower interest payment than the A-rated bank?
What is the measure of 10 basis points?
For the A-rated (BBB-rated) bank, in average, the bank can survive for how many years?

24. Economic Capital for Credit Risks
For the credit risk of lending operations, the required economic capital (EC) depends on the probability distribution of the losses (or the distribution of asset value).
A typical probability distribution for credit losses is sketched in Figure 2-5. This sketch shows the distribution of the value of the credit asset at the end of one year

25. For single A-rated bank, the probability=0.1%
The unexpected loss or the standard error of loss:
UL=(0-EL)2X(1-4%)+(100-EL)2X4%
The expected loss
For example, in Table2-1, if we assume a bank lend out a money E=100 to a company and the default rate of the company is 4%, then EL=(1-4%)X0+4%X100
(1)The max. probable loss: if the loss is greater than it, the bank will go to bankrupt
(2)If we can calculate EL, UL and know the distribution of asset then we use EL and UL to calculate MPL
The max. value of asset=there is no loss for the credit asst

26. Economic Capital for Credit Risks
To conclude, several keys to calculate the EC for credit risk EL UL
MPL
The distribution of loss value (or asset value)
Discuss them in detail in the following chapters

27. Economic Capital for Market Risks
For market risks, we assume that the trading groups will invest in the market (buy/sell securities in the market) and that the profitability of their investment has a probability distribution
This is illustrated in Figure 2-6
It is possible that the profitability could be negative
in this case, the trading group would not be able to pay back all of the liabilities
Therefore, a bank needs to prepare economic capital to cover the potential loss
The required economic capital can be considered to be the amount of money that the shareholders put in reserve at the beginning of the year so that the trading operation can carry out its strategy and maintain the desired debt rating

28. Economic Capital for Market Risks
In the later market-risk chapters we will show how the loss distribution is calculated for a given trading strategy. This allows us to calculate a maximum probable loss, Wp, such that there is only probability p that the profitability over a year will be worse than Wp.
p = Probability [Profit < Wp]
The economic capital to be held at the beginning of the year is then the maximum probable loss, discounted back at the risk-free rate (rf) to give the amount that must be put in reserve to maintain the required target debt rating

29. (1)In general, we set up the probability =1%
(2)The loss of the investment portfolio will greater than (or equal) Wp in the next trading day with 1% probabilty
(3) The bad events with Wp loss or the loss with greater Wp will happen in 2 to 3 times for a year
Normal distribution assumption
Two parameters: variance and mean

30. Economic Capital for Operating Risks
Conceptually, the calculation of economic capital for operating risks is the same as for market risks, except that the probability distribution has a different shape, not a normal distribution
The difficulty is in finding accurate data to characterize this distribution.
This is one of the industry's main challenges in risk management
Please refer to the last chapter of this book by yourself

31. RISK-ADJUSTED PERFORMANCE
Up to this point we have discussed methods for describing risk in terms of required economic capital.
However, when deciding whether to carry out a transaction, the bank is not only concerned about the risk; it is also interested in profitability relative to that risk.
By measuring risk-adjusted performance (RAP), a bank can integrate risk measurement into the daily profitability management of the business.

32. Using Risk-Adjusted Performance to Make Business Decisions
Risk-adjusted performance can be used to support the following business decisions:
At the product level, to decide which products are profitable and how products must be priced to ensure that they are profitable.
At the relationship level, to show which customer relationships are profitable.
At the transaction level, to decide whether to enter into a transaction, and if so, at what price.

33. Using Risk-Adjusted Performance to Make Business Decisions
At the individual or group level, to compensate staff based on the profit they generate compared with the amount of the bank's capital they consume.
At the business-unit level, to decide which units are adding the greatest profit relative to the risks they are taking.
Given this information, senior management can decide which business should grow and which should shrink.
A typical finding is that high-risk, high-return businesses, such as trading and commercial lending, are less profitable on a risk-adjusted basis than the retail lending.

34. Using Risk-Adjusted Performance to Make Business Decisions
Traditionally, the banking industry relied on measurements that gave an incomplete picture of performance and its relation to risk.
The two most common measurements
return on assets (ROA)
return on equity (ROE)

35. Using Risk-Adjusted Performance to Make Business Decisions
ROA is the profit divided by the dollar value of the bank’s total asset.
The asset is less sensitive to the risk
ROE is the profit divided by either book capital or the bank’s regulatory capital.
The book capital is the net value of the bank as measured by accounting methods.
The regulatory capital is the minimum amount of capital that must be held by the bank according to government regulators
Both of them are less sensitive to the risk

36. Using Risk-Adjusted Performance to Make Business Decisions
Shortcomings of ROA and ROE
ROA takes no account of the risk of the assets
Even ROE uses the regulatory capital for considering risk. However, the regulatory capital will be very insensitive to risk
The tow risk-adjusted performance:
RAROC (risk-adjusted return on capital)
SVA (shareholder value added)
Based on economic capital, not regulatory capital
Economic capital is more sensitive to the risk a bank has taken

37. Using Risk-Adjusted Performance to Make Business Decisions
Risk-Adjusted Return on Capital (RAROC)
RAROC is the expected net risk-adjusted profit (ENP) divided by the economic capital that is required to support the transaction

38. Using Risk-Adjusted Performance to Make Business Decisions
Taking a loan transaction as an example
The revenue for the bank
the interest income on the loan: the initial loan amount (A0) multiplied by the interest rate on the loan (rA) +fees payment from borrower (F)

39. Using Risk-Adjusted Performance to Make Business Decisions
the cost for the bank
interest to be paid on debt + operating costs (OC)+ any losses (L)
The interest to be paid on the debt is the amount of debt (Do) multiplied by the interest rate on the debt (rD)
The amount of debt required is the initial loan amount (A0) minus the economic capital (EC)
Do = A0 -EC

40. Using Risk-Adjusted Performance to Make Business Decisions
ENP (expected net profit)
=A0rA+F-D0rD-OC-L (Revenue - Cost)
=A0rA+F-(A0 –EC) rD-OC-L

41. Using Risk-Adjusted Performance to Make Business Decisions

42. Using Risk-Adjusted Performance to Make Business Decisions
For a trading transaction
RAROC is the net change in the value (△V) of the position minus operating costs
in this case, any debt costs are counted as part of the change in the value of the position

43. Using Risk-Adjusted Performance to Make Business Decisions
For a prospective basis (e.g., to predict the profitability of a future deal)
we use the expected loss or change in value
For a loan, the expected RAROC is
For a trading transaction, the RAROC is

44. Using Risk-Adjusted Performance to Make Business Decisions
Hurdle rate (H)
Senior management normally sets a target for the return it expects business units to make for using capital. This minimum value for RAROC is called the hurdle rate (H)
All transactions should be expected to pass over this hurdle to be considered viable
The actual value chosen is around 12% to 20% and depends on the return that the shareholders expect for investing their capital in the bank

45. Using Risk-Adjusted Performance to Make Business Decisions
Once the hurdle has been set, it determines how much the bank must expect to make on each transaction for it to be viable.
If we replace RAROC with the minimum value for RAROC (H), we can calculate the minimum return required on a loan transaction:

46. Using Risk-Adjusted Performance to Make Business Decisions
the amount that the bank must charge the loan customer

47. Using Risk-Adjusted Performance to Make Business Decisions
Similarly, we can calculate the minimum expected change in value for a trading transaction:
Replaced with H

48. Using Risk-Adjusted Performance to Make Business Decisions
Shareholder Value Added (SVA)
The 2nd risk-adjusted performance measure
It is simply the actual or expected profitability minus the required profitability to meet the hurdle rate.
The required profitability is the hurdle rate multiplied by the economic capital required. Based on the RAROC equations

49. Using Risk-Adjusted Performance to Make Business Decisions
For a loan transaction
The actual or expected RAROC
The required RAROC
For a trading transaction

50. SUMMARY
The relationship between risk ,capital, and the probability of default.
Define the amount of capital needed to support a given level of risk and maintain a target credit rating for the bank.
Use the economic capital as a measure of risk and defined RAROC and SVA to be measures of risk-adjusted profitability
This discussion assumed that we could obtain the economic capital from loss-probability distributions.
We will now spend many chapters discussing how those distributions can be calculated