Question:

Assume the following: a loan portfolio with a principal of $1 billion paying an annual rate of 10%. Economic capital (EC) held against the loan is 8% of the loan; it is invested in securities returning 8%. Therefore, $920 million ($1 billion - 8%) in deposits fund the loan. The deposits cost the bank 7%. Operating cost is $10 million (i.e., the portion allocated to the loan). The expected loss (EL) is 2% of the loan.

(i) What is the loan’s risk-adjusted return on capital (RAROC)?
(ii) If the bank’s hurdle rate is 12%, will the loan “project” be approved/accepted?
(iii) If the equity beta is 2.0, the riskfree rate is 5%, and the market’s expected return is 11%, what is the loan “project’s” adjusted RAROC?

Answer:

See this spreadsheet for the calculations.

(i) RAROC = 15% (see cell D23)

If you noticed that funding the loan with $920 million implies $80 million in equity capital plus another $80 million in cash-like assets creates an “unbalanced balance sheet,” we can alternatively fund the assets with $1 billion in deposits. (see column E for the alternative assumption). Under this scenario, interest on more deposits creates a lower RAROC (8%). For exam purposes, you should not worry about this nuance: the question should give you the amount of deposits (liabilities) which fund the loan (assets).

(ii) Yes, because 15% > 12% hurdle. But this is a “first generation” RAROC application that does not consider risk.

(iii) Adjusted RAROC (ARAROC) = (15% - 5%)/2 = 5%. As the market premium is 6% (11%-5%), the project would be rejected by adjusted RAROC.

Hi David,

I think there is a mismatch between the values in the question and the values used in the spreadsheet for actual calculation . I am getting a RAROC of 15% using the values in the question . Wanted to confirm if my calculations are correct

Ravi

Hi Ravi,

Yes, you are right (I had Crouhy’s example data in there from the reading). It is fixed. I get 15% too.

Thank you!

David

Hi David

Please refer to the balance sheet placed in the spreadsheet. If we are required to keep only $80 as economic capital, why do I need to raise $160 as equity?  I might as well raise $1000 as deposit and make the loan. In that case the RAROC calculations will change. Alternatively, if we have $160 as equity, should not that be the denominator for calculating RAROC? Please clarify.

Regards

Rajiv

Hi Rajiv,

Right, that’s actually why i showed the balance sheet at the bottom. Please note, to your point, I just updated the XLS at the link above. Please take another look.

See our discussion last year. We saw that Crouhy originally missed the circularity, but I wanted to keep the question faithful to the way it has been asked on the exam (purely practical).

Then I got hold of the latest Crouhy print, and he actually changed the assumption, to your point (p 533). And, lo and behold, Crouhy changed the problem (with no note or anything) Crouhy used to fund $1 BB with $925 and the latest print funds with $1 BB. (see Crouhy v1 and v2 to the right above)

So, the bottom line is, you are probably right. I would argue your view. But I could take the other side (See the updated.  I’d start with: the RAROC is about the instrument not the whole balance sheet, and the problem with the second Crouhy approach is you are costing $1,075 [75 EC + 1,000 deposits] against only $1,000 is assets), but that’s just to float that i think it’s tougher than it looks. Ultimately, I think you are correct, and the reason is, under the 2nd approach, see row 25, RAROC * equity = RAROC numerator (i.e., seem to be the correct return on equity). But the devil’s advocate is, if the bank already has EC, why can’t the bank fund the $1 BB with only $925 in deposits…

It’s a long way to say: I agree. I updated the question above, see what you think.

Thanks,
David

Hi David,

Thanks for very prompt response. However, I notice that the asset size on the balance sheet remains the same in both cases i.e. loan plus EC invested in govt securities.

In my opinion, the moot point is whether the equity is required to be invested in Govt. securities or it can be utilized to make the loan.

I am also interested to know what RAROC calculation will fetch me the point in the exam?

Regards

Rajiv

Hi Rajiv,

“the moot point is whether the equity is required to be invested in Govt. securities or it can be utilized to make the loan”
Not quite, because in both scenarios the EC is invested in cash like securities, so it’s not going to fund the loans. If the EC goes to fund loans, then it should come out of the numerator. I added Crouhy v3 to illustrate.

“I am also interested to know what RAROC calculation will fetch me the point in the exam?”
The question will give you the deposits to fund the loan. Only a bad question will make you grapple with the circularity.

David

Dear David,
Could you please confirm the EC($80mil) already had the EL of $20mil deducted. When I do the calculation, I don’t have to minu $20mil from the $80mil. Thanks.

Philip Wong

Philip - I don’t follow: the EC is an input assumption in the question. Did you see the spreadsheet for the calculations Row 21 is numerator, it deducts EL. David

David,
I thought the EL has to be deducted from both the numerator and denominator.

Philip Wong

Philip - It’s already implicitly excluded from the denominator. Removing EL in numerator is a way to match the EC in the denominator; e.g., you could add EL back to numerator (not subtract it), then add the corresponding loan reserve account (a contra asset) to the denominator. Then it would be a sort of “gross loan revenue” per total capital (EL + UL). But also, the EL is a flow measure (like income/revenue) and EC is a stock measure so it would be an awkward subtraction anyway.

but i mean you are right in the sense: EC doesn’t include EL.

- David