Risk Management puzzles, solved step by step

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All topicsCapital and leverage6Compounding and drawdowns8Correlation and diversification8Counterparty exposure and collateral7Credit risk arithmetic10Duration and rates7Liquidity and balance sheet7Logic, estimation and brainteasers7Operational loss and fraud7Options and Greeks7Probability and base rates8Statistics and estimation10VaR and expected shortfall8
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  1. 016A swap has an expected positive exposure of about Rs 5 crore across its five-year life. The counterparty's credit spread is 200 basis points. Roughly what is the credit valuation adjustment?Counterparty exposure and collateralHardCounterparty riskQuant risk

    Try it first

    Which rough estimate is closest?

    Show the worked solution

    Roughly Rs 45 lakh. The credit spread is what the market charges each year for bearing the counterparty's default risk. Rs 5 crore of expected exposure at 2% is Rs 10 lakh a year. Over five years that is Rs 50 lakh, and discounting at an assumed 4% brings it to about Rs 44.5 lakh. That is the value you give up for trading with a counterparty that can default.

    Why does a spread times an exposure give you CVA?

    Think of a friend who owes you money and a moneylender who would charge that friend 2% more a year than a safe borrower. That 2% is the market's price for the chance your friend does not pay. The credit valuation adjustmentThe reduction in the value of a derivative to reflect the chance that the counterparty defaults while owing you money. is roughly the counterparty's credit spread charged on the amount it is expected to owe you, year by year, discounted to today. The spread already folds together the chance of default and the loss if it happens.

    CVA, roughly: each year's exposure charged at the counterparty's spread0510Rs lakh per year9.62Year 19.25Year 28.89Year 38.55Year 48.22Year 55 crore x 2% = 10 lakh before discountingExposure x spreadx discount factoreach year, at 4%Sum over 5 yearsRs 44.5 lakhabout Rs 45 lakhUndiscounted: Rs 50 lakh. With default timing modelled explicitly at 60% LGD: about Rs 41 lakh.
    Rs 5 crore of expected exposure charged at a 2% credit spread costs Rs 10 lakh a year, and discounting each year at 4% gives Rs 9.62, 9.25, 8.89, 8.55 and 8.22 lakh, a rough CVA of about Rs 44.5 lakh.
    The relationship
    CVA≈s∑t=15EPEt Dt=0.02×5×∑t=151.04−t≈0.445 crore\text{CVA} \approx s \sum_{t=1}^{5} \text{EPE}_t \, D_t = 0.02 \times 5 \times \sum_{t=1}^{5} 1.04^{-t} \approx 0.445 \text{ crore}
    sthe counterparty's credit spread, 2% a year
    EPE_texpected positive exposure in year t, Rs 5 crore
    D_tthe discount factor for year t, at an assumed 4%
    What it says in wordsCharge the spread on each year's expected exposure, discount it to today, and add the years.

    What does the rough version leave out?

    Three things, each worth a sentence. First, the rough formula ignores that a counterparty that defaults in year two cannot default again in year three, so it slightly overstates the charge. Modelled explicitly, with a 60% loss given default and a constant default rate implied by the spread, the CVA comes to about Rs 41 lakh. Second, exposure on a swap is rarely flat; it usually rises and then falls as payments are made. Third, if the counterparty's credit worsens exactly when your exposure grows, the charge is larger, which is wrong-way risk.

    The discount rate here is an assumption for the illustration, and the market spread would come from the counterparty's bonds or credit default swaps where they trade. The structure is what the interviewer wants: exposure profile, times default cost, discounted and summed.

    Where candidates lose it

    The trap is answering Rs 10 lakh, one year of the spread, or treating the spread as a one-off probability of losing the whole exposure. The spread is an annual rate, and the exposure lasts five years.

    The second loss is not naming the assumptions. Say that you used a flat exposure, a 4% discount rate and the spread as a stand-in for default probability times loss given default.

    What the interviewer asks next

    • How does a collateral agreement with a zero threshold change this CVA?
    • The exposure rises from Rs 2 crore in year one to Rs 8 crore in year five, with the same average. Is CVA higher or lower?
    • What is debit valuation adjustment, and why do some people dislike booking it as profit?
  2. 092A counterparty has a 2% probability of default. Your exposure to it is Rs 10 crore in normal states, but in the states where it defaults the exposure is Rs 40 crore. With 60% loss given default, compare the expected loss with and without that link.Counterparty exposure and collateralHardCounterparty riskQuant risk

    Try it first

    How much larger is the expected loss once exposure and default are linked?

    Show the worked solution

    Rs 12 lakh if independent, Rs 48 lakh when linked: four times as much. Independent: 2% times 60% times Rs 10 crore. Linked: the defaults happen exactly when exposure is Rs 40 crore, so 2% times 60% times Rs 40 crore. That link is wrong-way risk. Averaging exposure first gives only about Rs 12.7 lakh and misses it.

    Why does the link between exposure and default matter so much?

    Insuring a house against fire with an insurer whose only office is in the same building is poor protection: the insurer is least able to pay in exactly the event you insured against. Expected loss is probability of default times loss given default times the exposure at the moment of default, and wrong-way risk is when that exposure is largest in the states where the counterparty fails. Here the default probability and the recovery are unchanged; only the exposure in the bad states moved, and the expected loss quadrupled.

    Wrong-way risk: the exposure is biggest in the states where default happensIndependentexposure is 10 whatever happensCounterparty98%2%survivesexposure 10, loss 0defaultsexposure 10 x 60% = 6EL = 2% x 60% x 10 = Rs 12 lakhLinked (wrong-way)exposure jumps to 40 in defaultCounterparty98%2%survivesexposure 10, loss 0defaultsexposure 40 x 60% = 24EL = 2% x 60% x 40 = Rs 48 lakhSame default probability, same LGD: Rs 12 lakh against Rs 48 lakh, 4 times the loss
    With the same 2% default probability and 60% loss given default, exposure of Rs 10 crore in every state gives an expected loss of Rs 12 lakh, but exposure that jumps to Rs 40 crore in the default states gives Rs 48 lakh, four times as much.
    The relationship
    EL=PD×LGD×E[exposure∣default]0.02×0.6×40=0.48EL = PD \times LGD \times E[\text{exposure} \mid \text{default}] \qquad 0.02 \times 0.6 \times 40 = 0.48
    PDprobability of default, 2%
    LGDloss given default, 60%
    E[exposure | default]the expected exposure in the states where default happens, Rs 40 crore
    What it says in wordsUse the exposure conditional on default, not the average exposure, because only the default states produce a loss.

    Where does this happen in practice?

    Whenever the trade and the counterparty share a driver. A bank that sells a put on a country's currency to a bank in that country gains exposure just as that bank weakens. A lender taking a company's own shares as collateral sees the collateral fall as the company fails. The general signal is a trade whose value to you rises in the same scenario that damages the counterparty. A limit system that measures average exposure, about Rs 10.6 crore here, reports the position as a Rs 10 crore line and misses where the loss sits.

    Say how desks handle it. Specific wrong-way trades are flagged and either priced with the conditional exposure, collateralised more heavily or refused. General wrong-way risk, from shared exposure to a market factor, is harder to see and needs a stress scenario in which the counterparty and the market move together.

    Where candidates lose it

    The common error is averaging first: 98% of 10 plus 2% of 40 is Rs 10.6 crore, and 2% times 60% of that is about Rs 12.7 lakh, barely above the independent case. It treats the link as a small adjustment when it is the whole story.

    The other miss is changing the default probability instead of the exposure. The link is in the exposure; the default rate is the same in both trees.

    What the interviewer asks next

    • Give an example of right-way risk, where exposure falls as the counterparty weakens.
    • How would collateral posted by the counterparty change the Rs 48 lakh?
    • How do banks reflect wrong-way risk in CVA?