In Chapter 2, we discussed risk measurement at the highest level of the bank. In the following chapters, we looked in depth at the techniques used to individually measure market, ALM, credit risk, and operating risk. With this detailed knowledge, we can now return to the bank level and discuss how all the disparate risk models fit together to give a picture of the bank’s overall risk.
The first section discusses how the risk is calculated for the bank as a whole, including diversification between different types of risks. The second section discusses how the different models fit together. Finally, we discuss a high-level implementation plan for calculating bankwide economic capital and RAROC.
In this section, we first discuss ways in which the capital can be calculated for the bank as a whole, including the correlation between risks. Then, we show ways in which the total capital for the bank is allocated to individual business units and transactions. This is similar to finding the unexpected loss for a whole credit portfolio, and then calculating the unexpected loss contribution for each loan. The difference here is that we are not only dealing with credit risk, but also market, ALM, and operating risks.
Conceptually, the most straightforward way to calculate the risk for the bank is to have a single “grand machine” that concurrently simulates all the risks for the bank, similar to Figure 21-7, but with the addition of operating risks. With such a machine, the total capital for the bank comes automatically by summing the losses in each scenario to get the loss distribution for the bank as a whole. However, such a large calculation is difficult to construct because it requires many sources of data, and requires a coordinated team that can apply all the techniques needed for market, credit, and operating risks.
It is much more common that a bank will have separate calculators for each risk: a VaR calculator for trading risks, a simulator for ALM risks, a portfolio model for credit risks, and specialized models for operating-risk. In this case, it is necessary to construct a framework and a model that can tie all these risks together.
Broadly, there are two approaches: analytical estimation of the capital, and simulation. The analytical approach is more straightforward, but the simulation is a little more accurate.
To estimate the capital, we first calculate the variance of the losses, and then estimate the capital multiplier.
We use the usual variance equation to get the variance for the whole bank based on the variance of the individual risk-types:
From the individual credit, market, ALM, and operating-risk models, we will already know the stand-alone variances: 
and 
. The difficulty is in estimating the six correlations. There are three common ways of estimating correlations: historical loss data, equity prices of monoline companies, and simulation.
The analysis of historical data simply takes the time series of losses from each of the types of risk and calculates the correlation. Ideally, the time series would be losses from each of the bank’s operations over several years. If such data are not available over a long period (e.g., due to mergers), national data can be used. For example, Table 25-1 shows bond-default rates and changes in the S&P 500 equity index over the last 20 years. The correlation between these 2 series is −40%. (When the market goes up, default rates fall.) Forty percent would therefore be a reasonable correlation to expect between losses in a portfolio of corporate loans and a trading operation that was highly correlated with the market.
An estimate of correlation can also be observed from historical equity prices of monoline companies. A monoline company is a company that only has one line of business, e.g., trading or lending. The correlation between the equity price of a monoline trading company and a monoline lending bank will give an indication of the correlation between the capital needed for a trading business and a loan business. This approach is quite crude because the credit quality of the loans and the style of trading in the monoline companies are likely to be different from the bank of interest.
TABLE 25-1 Historical Correlation Between Bond Defaults and Equity Returns
Estimation of correlation from simulation is done by using a small version of the grand machine. In this approach, a few representative loans and trades are simulated to get an example of the correlation between losses.
Once we know the variance for the bank as a whole, we can calculate the contribution for each of the risks using the usual VaRC or ULC approach:
Here, σcCredit represents the contribution of credit risk to the bank’s total standard deviation.
The ratio between the contribution and the stand-alone standard deviation is often called the inter-risk diversification factor, F:
Notice that the factor depends not only on the correlation between risks, but also on the absolute size of one risk compared with the others. For example, consider a bank with only credit and market risks with a correlation of 40%. If the variance of both market and credit risks were equal to 100, the diversification factors would both be 84%:
If the variance of credit risks was 4 times that of market risks, the diversification factor for credit risks would be 93%, and for market risks, 70%:
For an individual transaction, the capital can be allocated according to the contribution of the transaction to the stand-alone portfolio, multiplied by the portfolio’s inter-risk-diversification factor. For example, the capital allocated to a loan would be based on its ULC relative to the loan portfolio, multiplied by the inter-risk-diversification factor of the loan portfolio within the bank:
This ensures that the sum of the capital for the individual transactions equals the total capital for the bank; however, it gives a slightly different answer than if the ULC of the loan to the bank had been calculated in one step. This is because by using a single inter-risk-diversification factor for the whole credit portfolio, we are assuming that the correlation between every loan and the rest of the portfolio is the same. This point is explored further in Appendix A.
Having established the variance for the bank as a whole, it is necessary to calculate a capital multiplier for the bank. Recall from Chapter 20 that the capital multiplier (M) is the required economic capital divided by the standard deviation:
In our current discussion, we know σBank, and we want to determine MBank so we can estimate the capital.
The multiplier is determined by the shape of the probability distribution. It is difficult to calculate the shape of the probability distribution for the whole bank because it is the combination of several different distributions, such as a Normal distribution for market-risk losses and a Beta distribution for credit losses.
If we do not know the shape of the distribution for the bank as a whole, it is a reasonable approximation to use the weighted sum of the multipliers for the individual risks:
This is equivalent to the following expression for the bank’s capital:
ECBank = MCredit σcCredit + MMkt σcMkt + MALM σcALM + MOps σCOps
If this approach is used for estimating the total capital, the capital allocated to an individual transaction is calculated simply using the multiplier for the transaction’s portfolio; for example, the capital allocated to a loan is as follows:
This approximation is sufficiently accurate for most purposes, but if a more refined estimation of the multiplier is required, it can be found through a simulation that properly combines the losses from different distributions. This simulation method is detailed in Appendix B.
In the section above, we described a model to calculate the overall risk of the bank based on the risk produced from many individual models. Let us now step back and look at how all these models fit together.
In the last few chapters, we discussed the following models:
Trading Risk
• Value at risk (Chapter 6)
• Economic capital for trading based on VaR (Chapter 9)
• Simulation to estimate the likely exposure for derivatives (Chapter 17)
ALM
• Rules for transfer pricing (Chapter 15)
• Customer and interest-rate behavior models (Chapter 13)
• Simulation to estimate ALM economic capital (Chapter 13)
Credit Risk
• Models for probability of default (PD), loss given default (LGD), exposure at default (EAD), and correlation (Chapter 19)
• Portfolio models (Chapters 20 and 21)
• Loan-pricing models (Chapter 22)
Operating Risk
• Models for overall operating-risk capital (Chapter 24)
• Structural models for individual risks (Chapter 24)
Bank Level
• Models to estimate inter-risk diversification (Chapter 25)
These models fit together to calculate bank-level capital, as illustrated in Figure 25-1. We will step through each component in detail describing its function, the data that it requires, and the results that it provides.
We start with transfer pricing (shown with dashed lines). Transfer pricing in itself is not a model but a set of rules for internal trades. The internal trades move interest-rate risks from one group to another; most importantly, they remove the interest-rate risk from the lending business, leaving only credit and operating-risks. This means that the credit risk of the loan can be measured in a credit-portfolio model, and the interest-rate risk can be measured separately in the ALM model.
FIGURE 25-1 Connections Between Models to Calculate Bankwide Economic Capital
For transfers between the ALM desk and the trading desk, opposite fictitious assets and liabilities should appear in the ALM and VaR calculators to reflect each side of the internal trade.
Now let us look at the calculation of credit risk. The credit-portfolio calculator is supplied with current information on the loan portfolio and current information on the exposure to derivatives from the VaR calculator. The credit-portfolio calculator is also supplied with models that can estimate PD, LGD, EAD, and correlation, given the characteristics of the current loans. These models are typically based on regressions with historical loan data.
The credit-portfolio calculator is used to estimate the economic capital required for the portfolio as a whole, and for each transaction. The economic capital for the portfolio as a whole is fed into the bank-level capital calculator. The capital for each transaction is used to assess risk-adjusted performance, such as RAROC, and is used to create reports identifying concentrations of risk.
Importantly, the credit-portfolio calculator also provides the average correlation between each transaction and the rest of the credit-portfolio. This average correlation is used in loan-pricing models so the loan officers and traders know how much they will be charged for capital when taking on a new risk.
The information in the credit-portfolio model can be used to calculate both the economic capital and the regulatory capital for credit risk.
The calculation of trading market risks is made in the VaR calculator based on market and position information. The regulatory capital is a direct multiplication of VaR by 
The economic capital is based on VaR and a separate simulation that models trading behavior and potential losses over a year.
The ALM calculator is similar to VaR, except that the position information is largely from internal transfers from the loan and deposit groups. An important part of the ALM calculator is the customer behavior models, based on historical rate and balance data.
The operating-risk models collect information, such as key risk indicators, and try to estimate both the total capital to be set aside for operating-risks and the allocation of the capital to each business group.
Once the total economic capital for the bank is calculated, it can be allocated to the individual business units and the transactions that caused the risk. A separate exercise can be conducted to allocate all the income, interest expenses, and operating expenses to each business unit and transaction. This allows the calculation of risk-adjusted performance. The allocation of interest expenses should be made according to the rules of matched-funds-transfer pricing. The allocation of income and costs may use an approach such as activity-based costing (ABC).
As illustrated in Figure 25-2, the results of the credit-portfolio calculation are also fed to the loan-pricing model as a set of average correlations for each type of counterparty (e.g., by credit grade and industry). The loan-pricing model also has copies of the models for PD, EAD, and LGD, so it can evaluate any new customer. It contains a set of matched-funds-transfer-pricing rules and is fed with current rates to estimate the interest expense. It also has the rules for the noninterest expenses that will be allocated to the loan.
FIGURE 25-2 Data Feeds from Portfolio-Level Calculators to Desk-Level Calculators
The credit-portfolio calculator can also feed results to the trading group, so they can limit their credit exposure according to the economic capital consumed by exposure to each counterparty. Separately, at the start of the day, the VaR calculator supplies results to each portfolio that has a VaR limit, so the total VaR caused by any new intraday trade can be calculated.
We have now discussed almost all the models required to calculate RAROC. In this section, we simply pull them together to present them as a list of things to be done to calculate the bank’s profitability in terms of RAROC. In the detailed chapters, we discussed the many options available at each step. Here, we do not include every option, but just give a sense of the types of steps that need to be taken.
Implementation of RAROC needs three broad streams: capital calculation, funds-transfer pricing, and allocation of noninterest expenses.
• Decide which instruments are liquidly traded, and which should be treated as ALM instruments.
• Build pricing models for each type of security if using historical or Monte Carlo VaR. Build derivative models if using parametric VaR.
• Decide the set of market data that is required by the pricing models and collect it.
• Build a VaR calculator (Chapter 6).
• Build a model to convert from daily VaR to annual capital (Chapter 9).
• Collect historical data on the behavior of customers.
• Build models showing how balances and implicit options respond to changes in rates (Chapter 13).
• Collect data on current deposit accounts.
• Transfer-price loan accounts to bring the risk into the ALM portfolio (Chapter 15).
• Build a simulation model to estimate economic capital for interest-rate risk (Chapter 13).
• Collect data on daily flows of funds.
• Build a simulation model to estimate economic capital for liquidity risk (Chapter 14).
• Collect historical data on all companies that defaulted and at least an equal amount that did not default. (See Chapter 19 for types of data to collect.)
• Carry out regressions to relate company and loan characteristics to probability of default, loss given default, and exposure at default (Chapter 19).
• Collect information on the current portfolio.
• Calculate the expected loss.
• Estimate the asset or default correlation (Chapter 20).
• Build a portfolio model to calculate the economic capital (Chapters 20 and 21).
• Collect historical loss data both inside the bank and externally.
• Collect scale and key risk indicators.
• Calculate capital (Chapter 24).
• Collect historical data on losses.
• Estimate correlations and calculate the standard deviation of losses for the whole bank.
• Use the Merton approach (Chapter 21) or transformation simulation (Chapter 25, Appendix B) to estimate the shape of the bank’s probability distribution.
• Use the VaR contribution (Chapter 7) or unexpected loss contribution (Chapter 20), multiplied by the inter-risk-diversification factor (Chapter 25).
• Reverse any internal charges or credits for interest costs.
• Apply matched-funds-transfer pricing (Chapter 15) to charge and credit business units according to the interest-rate characteristics of their assets and liabilities.
• Either accept the bank’s current transfer pricing for noninterest expenses, or reverse all charges and apply charges according to the preferred methodology, e.g., activity-based costing.
• Use the standard RAROC equations. (Use the definition in Chapter 2 for most occasions, but use Chapter 22 for the rare occasions when you need to calculate RAROC over multiple years.)
If, in addition to calculating RAROC, we wish to calculate the shareholder value added, or price a new transaction, we need to set a hurdle rate for using economic capital. This is discussed in Appendix C.
In this chapter, we showed how the overall risk for a bank can be calculated based on the results from all the models that we discussed in the earlier chapters. You should now have a good idea of all the tools that are currently used by banks to measure their risk.
One approach to allocating capital to an individual loan is first to calculate the contribution of the loan to the risk in the credit-portfolio, and then calculate the contribution of the credit-portfolio to the risk of the whole bank. This gives a slightly different answer than directly calculating the contribution of the loan to the risk of the whole bank. This appendix explores the source of the discrepancy.
Consider a portfolio made up of two subportfolios, A and B. As usual, the variance of the total portfolio depends on the two subportfolios and their correlation:
We calculate the contributions of A and B as follows:
Here, 
is defined as the average correlation between B and the total portfolio. Now consider portfolio B to have 2 components, 1 and 2:
We could now calculate the portfolio’s standard deviation as follows:
The contribution of item 1 to the total portfolio could therefore be defined as follows:
Slightly different results for the unexpected loss contribution are obtained if the whole portfolio is considered:
In general, the unexpected loss contribution calculated in this way will not be the same as the contribution calculated in two stages, i.e.:
In matrix notation, the true correlation matrix is as follows:
But by making the calculation of ULC in two steps, we effectively use the following correlations:
This appendix discusses a method for determining the shape of the probability distribution for a bank’s combined losses, and thereby estimating the bank’s overall capital multiplier. In this approach, simulation is used to create losses from each risk source, and to ensure that the distribution of losses for each source is correct and that the losses are properly correlated.
Over the one-year horizon used for capital calculations, trading and ALM risks have probability distributions that are close to Normal. As we discussed in the chapter on Monte Carlo VaR, it is relatively easy to create correlated scenarios with Joint-Normal distributions by using Eigen-value decomposition. However, credit losses have distributions that are close to a Beta distribution.
These distributions can be combined by simulating credit events along with market-risk losses to get the loss distribution for the portfolio. We can make the credit data amenable to simulation by transforming the credit-loss data from a Beta distribution to a Normal distribution. We then carry out the simulation using a Joint-Normal distribution and transform the credit losses back to having a Beta distribution.
This process can be considered to be a stretching of the credit-loss distribution into the shape of a Normal distribution. This is done by converting the loss data from loss-amounts to probabilities, then from probabilities into a Normally distributed loss. The first step is to calculate the mean and standard deviation of the historical loss data (L). We can then calculate the a and b parameters of the Beta distribution, as in Chapter 20. We then use the Beta function (e.g., x=Betadist(L,a,b,1) in Excel) to calculate the cumulative probability of each event’s occurring. At this point the data should have a uniform distribution between 0 and 1. We then take the inverse Normal function to create a transformed series of losses (
) (=Norminv(x,0,1)). This transformed series has a Normal distribution. The transformation equation is given below and illustrated in Figure 25-B1.
At this stage, we have a time series of trading losses, ALM losses, and transformed credit-risk losses, all with Normal distributions. We can then calculate the covariance of these losses and use Eigenvalue or Cholesky decomposition to create random scenarios with the same correlation structure. The result is a number of scenarios each containing actual trading risk and ALM losses, and Normally distributed credit losses.
To calculate the actual credit-risk losses, we run the transformation in the opposite direction to produce credit losses with a Beta distribution, as illustrated in Figure 25-B2.
FIGURE 25-B1 Illustration of the Transformation of Historical Data from a Beta Distribution to a Normal Distribution
FIGURE 25-B2 Illustration of the Transformation of Simulated Losses from a Normal Distribution to a Beta Distribution
The next step is simply to add the credit losses to the trading and ALM losses to calculate the portfolio’s total loss. After running thousands of simulations, we sort the results according to the size of the loss and read off the loss for the required confidence level. For example, if we wanted to know the 10-basis-point confidence level for capital, we could run 10,000 simulations and estimate the required capital to be the tenth-worst loss.
We calculate the capital multiplier by dividing the tenth-worst loss by the standard deviation of losses. The inter-risk-diversification factors can then be defined according to the multiplier, m, required to balance the following equation:
The capital allocated to a loan is then as follows:
m should be expected to have a value close to one. The use of m forces the equations to balance and forces the sum of the economic capital allocated to individual transactions to equal the total capital of the bank. However, this is not the only way that capital can be allocated.
An alternative scheme for allocating capital is to use the results of the simulation. We can define the capital contribution of a portfolio from the total loss from that portfolio in all scenarios when the capital is exceeded, divided by the total loss for the whole bank when the capital is exceeded. For a credit-portfolio, this would be as follows:
With this definition, the capital allocated to a transaction is the capital for the portfolio multiplied by the percentage contribution of the transaction:
This allocation process also has the property that the bank’s capital equals the sum of the capital for the individual transactions, but it also directly incorporates the capital multiplier for each type of risk.
The hurdle rate determines how much the shareholders expect to be compensated for taking risk. If a transaction is expected to pay back less than the hurdle rate, it is not attractive to the bank, and the bank would not do the transaction because it would be risking shareholders’ funds without sufficient compensation.
The usual practice in banking is that the hurdle rate is determined at the beginning of the year by the CFO’s office and then set as the minimum required expected rate of return for all uses of the bank’s capital. This approach is practical in that the single hurdle rate can be easily communicated, and adherence to pricing discipline can be monitored.
The setting of the hurdle rate significantly affects the bank’s measured profitability and determines the price that loan officers and traders must charge to their customers. It is therefore highly political. There are several methods for calculating the hurdle rate, each with its own set of assumptions. Given the sensitive nature of setting the rate, it is best to use all available methods and then take an average.
One approach to setting the hurdle rate is to calculate the rate of return implied by the bank’s forecasted earnings and its current equity price. This assumes that shareholders believe the bank’s earnings estimate.
A similar approach would be to look at historical data on the returns that the bank has made on its equity, and assume that investors expect the same return in the future.
Another approach is to look at the total amount of capital that the bank has and compare it with all the different investments that it can make, starting with the most profitable, until the capital is exhausted. The least profitable investment using the last piece of capital sets the minimum hurdle rate. Given the diversification between risks, such a selection must be iterative to find the best combination of businesses. This optimization approach assumes that the bank must use all its capital. In reality, if there are investments that give poor returns, it would be better for the bank not to take the risk and instead increase its credit rating, or give the excess capital back to the shareholders (as dividends or share buy-back schemes).
The most theoretically pure approach is to set the hurdle rate based on the capital asset pricing model (CAPM). The CAPM requires the following rate of return (ri) for an investment:
ri = rf + βi(rf – rm)
Here, rf is the risk-free rate of return, rm is the average return expected on the overall market, and βi is the correlation between the return on the investment and the return on the market, weighted by the respective volatilities:
σi is the volatility of the investment, σm is the volatility of the market, and ρi;m is the correlation. When a shareholder applies this equation to investing money in a bank, σi is the volatility of the bank’s share price, ρi,m is the correlation between the share price and the market, and the required return, ri, is the hurdle rate, H:
