For a project, a loan or a portfolio of loans, compute RAROC and interpret it, then rank business unit results using it.

FRM® Part II · Operational Risk and Resilience · Risk Capital Attribution and Risk-Adjusted Performance Measurement · FRM2.ORR.18.c

The idea

To compute RAROC, build the numerator first: revenues (interest margin and fees) less operating costs, less expected loss, less tax, plus any return credited on the risk capital. With PD the default probability, LGD the loss given default and EAD the exposure, expected loss is EL = PD × LGD × EAD. Then divide by economic capital, written EC, which for a loan is the loss at the chosen confidence level minus expected loss. Writing NI for the numerator, the ratio of NI to EC is the risk-adjusted return, so RAROC = NI ÷ EC. A project with RAROC of 15 percent earns 15 cents a year for each dollar of capital put at risk, after expected losses. Interpret the number against the hurdle rate: above it the project creates shareholder value, below it the project destroys value. To rank business units, compute each RAROC and sort from highest to lowest. Do not rank by absolute profit or by capital size, because a large low-RAROC unit can be worse than a small high-RAROC unit. The usual slip is forgetting expected loss, or using PD where PD × LGD belongs.

Formulas

RAROC

RAROC = NI ÷ EC

NI: revenues less operating costs, expected loss and tax, plus any return credited on capital
EC: economic capital, the loss at the confidence level minus expected loss

Above the hurdle the project creates value. Forgetting expected loss in NI is the usual slip.

Expected loss on a loan

EL = PD × LGD × EAD

PD: the probability of default over the horizon
LGD: loss given default, as a share of the exposure
EAD: the exposure at default

Using PD alone where PD × LGD belongs overstates expected loss and understates RAROC.

A practice question

From the question bank, with the reason each wrong option is wrong.

A loan of USD 100 million is expected to produce revenue of USD 4.5 million a year with operating costs of USD 1.5 million. Expected loss is USD 1.0 million and economic capital is USD 20 million. Ignoring tax and any return on capital, what is the RAROC?

10.0% - correct
15.0%Why not: Omits expected loss from the numerator.
17.5%Why not: Omits operating costs from the numerator.
2.0%Why not: Divides by the loan amount instead of economic capital.

Risk-adjusted return = 4.5 - 1.5 - 1.0 = 2.0 million; RAROC = 2.0 / 20 = 10.0%.

A harder one

A loan of USD 50 million earns revenue of USD 3.0 million and incurs costs of USD 0.8 million and expected loss of USD 0.4 million. Tax is 30% of revenue less costs less expected loss. Economic capital is USD 4 million and earns a risk-free return of 4% after tax, credited to the unit. What is the RAROC?

31.5%Why not: Omits the return credited on economic capital.
42.5%Why not: Omits expected loss before taxing the margin.
35.5% - correct
2.8%Why not: Divides by the loan amount, not by economic capital.

After-tax margin = (3.0 - 0.8 - 0.4) x 0.7 = 1.26. Add 4% x 4 = 0.16 to get 1.42. RAROC = 1.42 / 4 = 35.5%.

The hardest tier

A loan portfolio of USD 500 million earns a net interest margin of 2.4% of the portfolio and bears operating costs of 0.7%. Its probability of default is 1.5% and the loss given default is 40%. Economic capital is USD 35 million, which earns 3% after tax, credited to the unit. Tax is 25% of revenue less costs less expected loss. What is the portfolio's RAROC?

14.8% - correct
18.7%Why not: Leaves out the 25% tax on the risk-adjusted margin.
11.8%Why not: Leaves out the return credited on economic capital.
5.1%Why not: Uses the 1.5% default probability as the loss rate, forgetting to multiply by loss given default.

Expected loss = 1.5% x 40% = 0.6% of exposure. Margin after costs and loss = (2.4 - 0.7 - 0.6)% x 500 = 5.50; after tax 4.125; plus 3% x 35 = 1.05 gives 5.175. RAROC = 5.175/35 = 14.8%.

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