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Explain how the RAROC (risk-adjusted return on capital) approach works and how capital budgeting draws on it.

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

The idea

RAROC measures the return a business earns per unit of risk capital. In its simplest form it is the ratio RAROC = risk-adjusted after-tax return ÷ economic capital. The numerator is expected revenue, less operating costs, less expected loss and taxes, plus the return earned on the economic capital itself. Credit is given for that return because capital is invested in safe assets rather than borrowed. The denominator is economic capital, not book equity or loan notional. In capital budgeting the bank compares each project's RAROC with a hurdle rate, the return shareholders require. Projects above the hurdle add value and earn capital. Those below destroy value. When capital is scarce, rank projects and fund them in order until the capital budget is used. A common error is to compare RAROC with the average RAROC of the unit instead of the hurdle rate. The exam point: RAROC compares return with risk capital, and the decision rule is RAROC above hurdle.

Formulas

RAROC with tax and a return on capital

RAROC = (revenue − costs − EL − tax + rf × EC) ÷ EC

revenue
expected revenue
costs
operating costs
EL
expected loss
tax
taxes on the risk-adjusted income
rf
the rate earned on the capital, for instance the risk-free rate
EC
economic capital

The denominator is economic capital, not book equity or the loan amount. Compare with the hurdle rate, not the unit's average RAROC.

Value added by a project

value added = (RAROC − hurdle) × EC

RAROC
the project's risk-adjusted return on capital
hurdle
the shareholders' required return
EC
the economic capital the project uses

Ranking by RAROC alone can favour a small project over a larger one that adds more value in total when capital is not binding.

A practice question

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

Which expression describes RAROC?

  1. Net income divided by the book value of the equity on the balance sheet.Why not: Describes accounting ROE, which does not deduct expected loss or use risk-based capital.
  2. Expected loss divided by economic capital.Why not: Uses expected loss as the numerator; RAROC takes expected loss out of the return and measures the remainder.
  3. Risk-adjusted after-tax return divided by economic capital. - correct
  4. Total revenue divided by risk-weighted assets.Why not: Ignores costs, expected loss and tax, and uses regulatory exposure instead of economic capital.

RAROC = (revenue - costs - expected loss - tax + return on capital) / economic capital, so the return is net of expected loss and measured per unit of risk capital.

A harder one

Many RAROC formulas add a return on economic capital to the numerator. What is the reason?

  1. The capital is invested in safe assets, so the unit is credited with the interest it earns rather than debt funding. - correct
  2. Regulators require the return on capital to be included in any reported risk-adjusted return of a bank unit.Why not: Invents a regulatory requirement; the credit comes from the funding logic of the capital.
  3. Capital cannot lose value in a bad year, so its return is certain and should be treated as part of profit.Why not: Capital absorbs losses in the tail; the credit is the risk-free earnings on it, not a guarantee.
  4. It raises RAROC above the hurdle rate so that more projects can be accepted.Why not: A formula element is justified by funding economics, not by a wish to pass more projects.

Because economic capital is equity, not borrowed money, the unit does not pay interest on it; it is credited with the risk-free return that the capital earns, which is added to the numerator.

The hardest tier

A unit currently earns a RAROC of 18% and the bank's hurdle rate is 12%. A new project would need USD 30 million of economic capital and has an expected RAROC of 14%. What should the bank do?

  1. Reject: its RAROC of 14% is below the unit's current 18% and would dilute the unit's average RAROC over time.Why not: Compares the project with the unit's average RAROC instead of the hurdle rate.
  2. Accept only if the project's RAROC also exceeds the risk-free rate plus the unit's credit spread.Why not: Invents a second test; the standard test is RAROC against the shareholders' hurdle rate.
  3. Accept: the project returns 2 points over the hurdle, adding about USD 0.6 million a year despite lowering the average. - correct
  4. Be indifferent, because any RAROC that is positive adds the same amount of value to shareholders.Why not: Ignores the hurdle: value is added only by the excess of RAROC over the hurdle times capital.

Value added = (14% - 12%) x 30 = 0.6 million a year. Capital budgeting uses the hurdle rate, not the average return of existing business.

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