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Work out the adjusted RAROC of a project in order to decide whether it is viable.

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

The idea

RAROC divides return by total risk capital, but shareholders price only the systematic part of risk. Adjusted RAROC corrects for this using the project's equity beta, written β. Write Rf for the risk-free rate and E(Rm) for the expected market return. The adjusted measure is the project's excess return per unit of beta: Adjusted = (RAROC − Rf) ÷ β. Accept the project if it exceeds the market risk premium, E(Rm) − Rf. This is the same as requiring RAROC to exceed the CAPM hurdle, Rf + β × (E(Rm) − Rf). So two projects with the same RAROC can get different verdicts, because a low-beta project may pass where a high-beta one fails. Let ρ be the correlation of the project with the market, σP the project volatility and σM the market volatility. Beta can then be estimated from the correlation and the two volatilities: Beta = ρ × σP ÷ σM. Common errors are skipping the subtraction of Rf, multiplying instead of dividing by beta, and comparing with the market return instead of the premium. The exam point is that adjusted RAROC brings market risk into the decision.

Formulas

Adjusted RAROC

ARAROC = (RAROC − Rf) ÷ β

RAROC
the project's RAROC
Rf
the risk-free rate
β
the project's equity beta

Accept if it exceeds the market risk premium E(Rm) − Rf, not the market return. Skipping the subtraction of Rf or multiplying by beta are the usual errors.

Project beta from correlation and volatilities

β = ρ × σp ÷ σm

ρ
the correlation of the project's returns with the market
σp
the volatility of the project's returns
σm
the volatility of the market

Feeds adjusted RAROC. Using the correlation alone as beta, or inverting the volatility ratio, is the usual slip.

A practice question

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

Why is RAROC adjusted for beta in the adjusted RAROC approach?

  1. Because RAROC leaves out expected loss, which beta reintroduces into the return calculation.Why not: Expected loss is already in the RAROC numerator; beta addresses market risk.
  2. Because RAROC is stated before tax, and beta converts it to an after-tax figure for the comparison.Why not: Beta is a measure of systematic risk and has nothing to do with the tax basis.
  3. Because RAROC reflects total risk through capital, while shareholders are paid only for systematic risk. - correct
  4. Because beta turns RAROC from a one-year figure into a multi-year figure for long-dated projects.Why not: Beta does not change the horizon of the measure.

Economic capital depends on total risk, but the market prices only the part correlated with the market; adjusted RAROC uses beta to compare projects on that basis.

A harder one

A project has a RAROC of 11.0% and an equity beta of 1.5. The risk-free rate is 3.5% and the market risk premium is 6.0%. Adjusted RAROC is (RAROC - risk-free rate) / beta, and a project passes if it exceeds the market risk premium. What is the adjusted RAROC and the decision?

  1. 5.0%, reject - correct
  2. 7.5%, acceptWhy not: Subtracts Rf but never divides by beta.
  3. 7.3%, acceptWhy not: Divides RAROC by beta without first subtracting the risk-free rate.
  4. 11.25%, acceptWhy not: Multiplies by beta instead of dividing.

Adjusted RAROC = (11.0% - 3.5%) / 1.5 = 5.0%, below the 6.0% market risk premium, so reject. Equivalent check: the CAPM hurdle is 3.5 + 1.5 x 6.0 = 12.5%, above 11.0%.

The hardest tier

A project's returns have a volatility of 25% and a correlation of 0.5 with the market, whose volatility is 15% and expected return is 10%. The risk-free rate is 4% and the project's RAROC is 10%. Adjusted RAROC is (RAROC - risk-free rate) / beta, and a project passes if it exceeds the market risk premium. What is the adjusted RAROC and the decision, given that beta = correlation x project volatility / market volatility?

  1. 12.0%, acceptWhy not: Uses the correlation of 0.5 as the beta.
  2. 3.6%, rejectWhy not: Uses the ratio of volatilities (1.67) as beta and omits the correlation.
  3. 7.2%, accept - correct
  4. 20.0%, acceptWhy not: Inverts the beta formula (market volatility over project volatility).

Beta = 0.5 x 25 / 15 = 0.833. Adjusted RAROC = (10 - 4)/0.833 = 7.2%, above the market risk premium of 6%, so accept.

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