We have spent the last few chapters discussing methods for calculating the expected loss and economic capital required for credit risks. Economic capital is useful for identifying large risks and setting the total amount of capital to be held by the bank, but 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. In Chapter 2, we discussed the concepts of risk-adjusted return on capital (RAROC) and shareholder valued added (SVA). In this chapter, we briefly review those concepts for loans. In Chapter 2, we implicitly only discussed RAROC for a transaction that happens over a single time period, e.g., a year. However, banks often grant multiyear loans, so the second half of this chapter discusses the application of RAROC over multiple years.
In Chapter 2, we defined RAROC to be the expected net risk-adjusted profit divided by the economic capital required to support the transaction:
The return on the transaction is the net increase in value. For a loan, the expected net profit is the interest income on the loan, plus any fees (F), minus interest to be paid on debt, minus operating costs (OC), and minus the expected loss (EL).
The interest income on the loan asset is the initial loan amount (A0), multiplied by the interest rate on the loan (rA). The interest to be paid on the debt is the amount of debt (D0), multiplied by the interest rate on the debt (rD). The amount of debt required is the loan amount minus the economic capital. The RAROC equation for a loan is therefore as follows:
The term EC rD is sometimes called the “capital benefit” because it is a reduction in the debt interest charge because part of the loan is funded by capital.
The equation above determines the RAROC based on the risk and the expected income. Alternatively, for a new loan, we can fix the RAROC equal to the hurdle rate (H), and calculate the required fees and spread:
A0rA + F = (A0 − EC)rD + OC + EL + H × EC
Another twist to the RAROC equation is to calculate the SVA, which is defined as the expected net profit, minus the required return on capital:
SVA = ENP − H × EC
= [A0rA + F − (A0 − EC)rD − OC − EL] − H × EC
To clarify these concepts, let us consider an example. Consider our previous example of a loan of $100 for 1 year to a company rated BBB with the following assumptions:
• The collateral is such that the loss in the event of default (LIED) is 30% with no uncertainty.
• The average default correlation with the rest of the portfolio is 3%.
• The capital multiplier for the portfolio is 6.
Using the covariance-portfolio approach, we calculate the EL for this loan to be $0.066:
EL = $100 × 30% × 0.22% = $0.066
The UL is $1.41, and the UL contribution is $0.24.
With a capital multiplier of 6, the economic capital for this loan is $1.46:
EC = 6 × $0.24 = $1.46
We now know the risk characteristics of the loan, but to calculate RAROC, we also need the income and costs. Let us assume the following:
• The interbank rate for one-year debt is 5% (rD).
• The customer is being charged 6.5% interest (rA).
• The operating costs are $1 (OC).
We can now calculate that the RAROC is 35%:
If we assume a pretax hurdle rate of 25%, the SVA is $0.13:
SVA = [$100 × 6.5% − ($100 − $1.46) × 5% − $1 − $0.066] − 25% × $1.46
In the analyses above, we defined RAROC to be simply the expected net profit divided by the economic capital:
For a one-year loan, this is quite adequate, and as we saw, it is relatively easy to calculate. The multiyear case is more complicated for several reasons:
• Over time, the probability of the company defaulting changes, as predicted by the migration matrices in Chapter 18. This affects the required economic capital, and therefore the required debt.
• After one or more years, there is the possibility that the loan will have defaulted, thus slightly reducing the expected value of the costs and interest income.
• The outstanding amount may vary over time, for example, if the loan is amortizing.
• Over time, the collateral value may change as a percentage of the exposure amount, thereby changing the LIED.
To deal with these effects, we need a more subtle definition of RAROC.
We can define RAROC to be the internal rate of return on the set of expected cash flows. The internal rate of return (irr) is the discount rate that sets the net present value of a series of cash flows (Ct) to equal zero:
To see the connection with the earlier definition of RAROC, consider that at the beginning of the period we invest the economic capital. Thus, the cash flow at the t = 0 equals the economic capital. At the end of the period (t = 1), we expect to receive the capital back, plus the additional net expected profit. The irr equation is then as follows:
We can rearrange this expression to take us back to the original definition of RAROC for a one-year loan:
As a further step toward calculating multiyear RAROC, let us again look at the one-year case but more carefully decomposing the cash flows. This is a complex way to get to the same result we already have, but it gives a good introduction into the process we use to calculate RAROC over multiple years.
At the beginning of the year, the bank pays the customer the amount of the loan (A). To fund this loan, the bank will invest some of its economic capital, and will raise some debt (D). The net cash flow to the bank at the beginning of the year is D − A. This is a negative amount equal to the economic capital that has effectively been paid to the customer.
At the end of the year, the bank will pay back the debt (D) with interest (DrD), and it will have paid some operating costs (OC). If the customer does not default, the bank will receive the principle amount (A) with interest (ArA). If the customer does default, the bank can sell the defaulted loan and receive the recovery amount, R. The expected cash flow from the customer is calculated from these two possibilities weighted by the probability of default (p):
Expected Payment = A(1 + rA)(1 − p) + Rp
These cash flows are summarized in Table 22-1 and illustrated in Figure 22-1. In this figure, the central line represents the bank, and the arrows indicate flows into or out of the bank. The flows to the customer are above the line, and the flows to the capital markets and internal expenses are below the line. The cash flows from the customer at the end of the year are dashed because they are alternatives depending on default.
We put these expected cash flows into the equation for the internal rate of return:
We can replace D with A − EC:
TABLE 22-1 Expected Cash Flows for a One-Year Loan
FIGURE 22-1 Illustration of Cash-Flows for a One-Year Loan
And rearrange as follows to get the original one-year equation for RAROC:
The final step in this arrangement recognizes that (A − R)p is the probability of default multiplied by the loss given default (in dollars), which is the definition of EL.
Compared with the original definition of RAROC, we have a small extra term: −ArAp. This is the amount of interest income that is expected to be lost due to default and was neglected in the original calculation of EL.
Using this framework, we are now ready to get into the algebra for the case of a two-year loan. For a two-year loan, two effects are important: the changing amount of economic capital, and survivorship.
For a two-year loan, the percentage of capital that we expect to hold will change in the second year due to the changing marginal probability of default from p1 in the first year to p2 in the second, as predicted by the migration matrices in Chapter 18. The change in capital has the effect of changing the amount of debt required from one year to the next. For clarity, we assume that all the first-year debt (D1) is paid back at the end of the first year, and then a new amount of debt (D2) is raised at the same interest rate, but with an amount depending on the new capital.
We also need to include the survivorship effects. The survivorship effects arise because if the loan defaults in the first year, there will be no need to raise debt or pay operating costs in the second year. We therefore multiply the second year’s debt and operating costs by one minus the probability of a default in the first year. Similarly, if the loan defaults in the first year, it cannot default in the second year.
Let us use X to denote cash received by the bank, and Y for cash paid out as in Table 22-2 and Figure 22-2. The cash flows at the start of the first year are the same as before.
The cash flows at the end of the first year do not have a repayment of the principal A, but do have the possible payment of the amount that would be recovered if the loan defaulted in the first year, R1. At the end of the first year, there is also an amount D2(1 − p1) that is the debt to be raised to cover the loan in the second year, multiplied by the probability of surviving through the first year. The cash flows at the end of the second year are almost the same as the cash flows at the end of the one-year example in Table 22-1, except that all the terms are multiplied by (1 − p1). If the loan was amortizing, we would include the expected repayments of principal and the consequent changes in capital and debt.
Once we have established the set of expected cash flows, the RAROC is simply found by solving for the internal rate of return.
TABLE 22-2 Expected Cash Flows for a Two-Year Loan
FIGURE 22-2 Illustration of Cash Flows for a Two-Year Loan
(In Microsoft’s Excel spreadsheet you can use the command “irr()”).
The internal rate of return found from this equation is the RAROC for the loan. If we are dealing with a new loan and want to find the spread to be charged, the internal rate of return is fixed equal to the hurdle rate, and we solve the equation for rA, which is lengthy but not difficult.
As an example, let us reconsider the loan to a BBB company. Let us keep all other factors the same in the second year as in the first except for the probability of default and the LIED. Assume that the probability of default increases from 0.22% in the first year to 0.32% in the second year (as predicted by the migration matrix in Table 18-5), and the LIED increases from 30% to 40% because of some deterioration in the collateral value. The calculation of the cash flows are shown in Table 22-3 and Table 22-4. In this case, a loan with a 6.5% interest-rate will have a RAROC of 26%. We would need to charge 6.65% to bring the RAROC back up to 34%.
In this chapter, we reviewed the concepts of risk-adjusted return on capital (RAROC) and shareholder value added (SVA), and applied them to measuring the profitability of loans. Now, we shift focus slightly and move from a discussion of economic capital to a discussion of regulatory capital.
TABLE 22-3 Calculation of Parameters for a Two-Year Loan
TABLE 22-4 Calculation of Cash Flows for a Two-Year Loan
