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P2.T6.701. Unexpected loss and return on risk-adjusted capital (RARORAC) (De Laurentis)

Nicole Seaman

Director of FRM Operations
Staff member
Learning objectives: Explain expected loss, unexpected loss, VaR, and concentration risk, and describe the differences among them. Evaluate the marginal contribution to portfolio unexpected loss. Define risk-adjusted pricing and determine risk-adjusted return on risk-adjusted capital (RARORAC).


701.1. Consider two credit positions with identical features:
  • Adjusted exposure, EAD) = $5.0 million each
  • Probability of default, PD (aka, EDF) = 5.0%
  • Loss given default, LGD = 50.0%
  • Standard deviation of LGD, σ(LGD) = 40.0%
  • The default correlation between the positions, ρ(position #1, position #2) = 0.20
Although unexpected loss (UL) can be calibrated according to any confidence level, we decide to keep things unrealistically simple and define UL as one standard deviation, which implicitly suggests a low confidence level. As such, each position's unexpected loss (UL) is about $705,000. Which is nearest to the two-asset portfolio's unexpected loss (UL)? (Bonus questions: how was the position UL derived? What is each position's risk contribution?)

a. $1.092 million
b. $1.275 million
c. $1.410 million
d. $2.033 million

701.2. Analyst Mark is evaluating a portfolio of credit-sensitive assets. He is estimating expected loss (EL), unexpected loss (UL) and credit value at risk (CVaR) under various correlation assumptions. Each of the following is true EXCEPT which is a false dynamic?

a. An increase in the CVaR confidence level implies an increase in either the position's or portfolio's CVaR
b. An increase in (inter-position) default correlation between credit positions in a portfolio, ρ(position X, position Y), implies an increase in the portfolio's unexpected loss (UL)
c. An increase in (inter-position) default correlation between credit positions in a portfolio, ρ(position X, position Y), implies an increase in the portfolio's expected loss (EL)
d. An increase in (intra-position) correlation between a position's own default probability (PD) and its own loss given default (LGD), ρ[PD(position X), LGD(position X)], say from its typically assumed zero to a non-zero parameter, implies an increase in the position's EL

701.3. You are analyzing a $3.0 billion retail loan portfolio and you are given the following assumptions:
  • Revenue (ie, spread + fees) = $153.0 million = 5.0% of $3.0 billion portfolio assets plus (+) $3.0 million in fees
  • Expected loss, EL = $60.0 million = 2.0% of $3.0 billion portfolio assets
  • Cost of funds, COF or COC = $30.0 million = 1.0% of $3.0 billion liabilities (assume liabilities equal assets)
  • Economic capital, EC = $300.0 million = 10.0% of portfolio assets
  • Cost of operations = $23.0 million
  • Tax rate = 40.0%
Which of the following is nearest to the after-tax risk-adjusted return on risk-adjusted capital (RARORAC)? (See answer for reconciliation with T7's RAROC)

a. Zero
b. 8.00%
c. 11.50%
d. 13.33%

Answers here:
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David Harper CFA FRM

David Harper CFA FRM
Staff member
For those following along, this question is associated with the new reading (to 2017 FRM) in Topic 6: "Classifications and key concepts of credit risk" (Chapter 2) of Developing, Validating and Using Internal Ratings by Giacomo De Laurentis et al. In the answer to 701.3, I reconcile this (De Laurentis') RARORAC with Crouhy's RAROC. But candidly, this new De Laurentis Chapter 2 is not strong (in my opinion, of course). There is little in here that isn't covered in greater better detail elsewhere in FRM P2. The unexpected loss and marginal contribution, ULC(i), is particularly weak (confusing even) and lacks any numerical example. The discontinued Ong remains the most reliable text with respect to UL and UL Risk Contribution.

De Laurentis' RARORAC similarly contains no numerical example and the terms are not precisely defined. I recommend relying on well-assigned Crouhy. De Laurentis is too loose with definitions to be certain, but as far as I can tell De Laurentis's RARORAC is meant to the be the same as Crouhy's RAROC: the numerator is risk-adjusted by subtracting expected loss; the denominator is economic capital so could be RAC but is better described as ROC. I like to say the key feature of good ratios is consistency; e.g., ROE has a pretax income or net income numerator, but ROIC has EBIT/Operating Income or EBITDA numerator because you want the earnings base that flows to the associated capital base (just equity, or debt + equity?). Similarly, RAROC deducts EL in the numerator because the denominator is UL (i.e, EC), is is not (UL+EL). There is nothing "wrong" with adding EL back to both numerator and denominator, you'd just end up with an unconventional metric.

In any case, Crouhy's Essentials of Risk Management is the best shallow resource on this. It's based on an older, much deeper text (is here @ http://amzn.to/2hX2LlO ) which you don't need but is the basis for our RAROC learning spreadsheets (and is definitely consistent with FRM's somewhat established ratio approach to RAROC). Thanks!
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New Member
Is the Marginal Contribution of Marginal Risk given in this read akin to the MVaR given in the reading of Portfolio Risk: Analytical Methods(Jorion), becuase the formulas are not similar. It is pretty confusing as to what does this marginal contribution imply Marginal VaR or Componant VaR?

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @Kamakshi Good observation, I do agree! I believe this page 12 of of De Laurentis has mistakes; if he had an example, it would be easier to verify. But I do think he has confused some variables without testing them numerically. (I think the reading is generally weak as there are several formula mistakes. I've written the publisher and authors but have not received a reply).

De Laurentis's marginal contribution, ULC(i), clearly intends to be what is called by Schroeck the unexpected loss marginal contribution, ULMC(i), which is analogous to marginal VaR and is similarly unitless because it represents the first partial derivative δUL(portfolio)/δUL(position), such then when multiplying ULMC(i) by the positions's unexpected loss, UL(i,$), we retrieve the dollar-based unexpected loss contribution: ULC(i,$) = ULMC(i) * UL(i,$); ULC(i) has also be called Risk Contribution. ULC(i) is analogous to component VaR because as ULC(i)s sum to portfolio unexpected loss, component VaRs sum to portfolio VaR. To recap:
  • As Component VaRs sum to portfolio VaR and Component VaR($) = position VaR($) * marginal VaR(position w.r.t. portfolio),
  • ULC(i,$)s--aka risk contributions, RC(i,$)s--sum to portfolio unexpected loss, UL(P,$), and ULC(i,$) = position UL($) * ULMC(i).
Schroeck's Chapter 5 (assigned in P1.T4) is solid and seasoned on this, and his is informed by Michael Ong (see Chapter 6, https://www.bionicturtle.com/forum/resources/internal-credit-risk-models-by-michael-ong.131/ ) which was assigned in the FRM for several years. These are more reliable sources on this. I hope that helps!
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Nicole Seaman

Director of FRM Operations
Staff member
Hi David....Can I have the answers to the above questions as I am unable to view
Hello @Amanmarwa

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