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### CHAPTER 2

Risk Measurement at the Corporate Level: Economic Capital and RAROC

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

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

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

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

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.

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.

CAPITAL, RISK, AND THE PROBABILITY OF DEFAULT

100x(1+6%)X(1-4%)

100X(1+6%)

Constant term regardless with borrower defaults or not

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

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

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

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

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?

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

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

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

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

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

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

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.

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%

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?

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

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

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

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

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

(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

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

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.

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.

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.

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)

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

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

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

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)

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

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

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

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

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

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:

Using Risk-Adjusted Performance to Make Business Decisions

the amount that the bank must charge the loan customer

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

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

Using Risk-Adjusted Performance to Make Business Decisions

For a loan transaction

The actual or expected RAROC

The required RAROC

For a trading transaction

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

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