Risk Management puzzles, solved step by step

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100
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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. 055A Rs 80 crore credit line is 60% drawn. The credit conversion factor on the undrawn part is 50%, the probability of default is 2% and the loss given default is 45%. What is the expected loss?Credit risk arithmeticWarm upBank credit riskNBFC credit risk

    Try it first

    Which exposure number goes into the expected loss formula?

    Show the worked solution

    About Rs 57.6 lakh, or Rs 0.576 crore. The line is Rs 48 crore drawn and Rs 32 crore undrawn. Exposure at default adds half the undrawn part, Rs 16 crore, for Rs 64 crore. Expected loss is exposure times default probability times loss given default: 64 x 2% x 45% = Rs 0.576 crore.

    Why is today's drawn balance the wrong exposure?

    Think of a friend with a credit card who is quietly losing his job. In the months before he stops paying, the card balance climbs, because he is using the card to cover what his salary no longer does. Companies behave the same way with bank lines. Borrowers draw down their undrawn limits on the way to default, so exposure at default is the drawn balance plus a credit conversion factorThe share of an undrawn limit a bank assumes will be drawn by the time the borrower defaults. share of what is still undrawn.

    Exposure at default counts what the borrower draws on the way downCredit lineDrawn 48Undrawn 32= 80Exposure at default48+16= 6450% of the undrawn 32EAD64x PD2%x LGD45%Expected loss0.576 cr= Rs 57.6 lakh
    The Rs 80 crore line has Rs 48 crore drawn and Rs 32 crore undrawn. Adding half the undrawn part gives exposure at default of Rs 64 crore, and 64 times 2% times 45% gives an expected loss of Rs 0.576 crore, about Rs 57.6 lakh.

    How much does the conversion factor change the answer?

    A lot. Using only the drawn balance gives Rs 43.2 lakh; using the full limit gives Rs 72 lakh. The 50% conversion factor moves the expected loss by a third over the drawn-only figure, with no change in the borrower's credit quality. That is why the conversion factor is estimated from the bank's own default history on committed lines, not guessed.

    The relationship
    EL=PD×LGD×(drawn+CCF×undrawn)=0.02×0.45×64=0.576EL = PD \times LGD \times (\text{drawn} + CCF \times \text{undrawn}) = 0.02 \times 0.45 \times 64 = 0.576
    PDprobability of default over the year, 2%
    LGDthe share of exposure lost if default happens, 45%
    CCFthe share of the undrawn limit assumed drawn at default, 50%
    What it says in wordsExpected loss is the chance of default times the share lost times the amount out on the day of default.

    Say the limitation plainly. Expected loss is an average, the amount the bank should price into the loan's spread and provide for, not the loss it would face if this borrower actually defaulted, which would be 45% of Rs 64 crore, or Rs 28.8 crore.

    Where candidates lose it

    Most candidates multiply the three inputs by Rs 48 crore, the drawn balance, and give Rs 43.2 lakh. They have read the conversion factor in the question and not used it.

    The other trap is the unit. The answer is Rs 0.576 crore, which is Rs 57.6 lakh; saying Rs 57.6 crore is off by a factor of a hundred and loses the room fast.

    What the interviewer asks next

    • If the conversion factor were 75%, what would the expected loss be?
    • What is the loss if this borrower actually defaults tomorrow?
    • Why might a bank's conversion factor for a working capital line differ from one for a project finance facility?
  2. 080A corporate bond trades at a 300 basis point spread over the government curve. If investors expect to lose 60% of face value on default, what annual default probability does the spread imply?Credit risk arithmeticWarm upBank credit riskRating agency

    Try it first

    Which default probability does a 300 bp spread imply at 60% loss given default?

    Show the worked solution

    About 5% a year. The spread roughly pays for expected loss, which is default probability times loss given default. So default probability is the spread divided by the loss: 300 basis points over 60% is 5%. That is an upper bound for the real-world rate, because part of every spread pays for risk and illiquidity, not expected loss.

    Why is spread roughly default probability times loss?

    Suppose you lend Rs 100 to each of 100 shopkeepers for a year. If 5 of them fail, and you get back only 40 paise in the rupee from each, you lose Rs 300 across the group. To break even you need to charge 3% more than a loan to the government. The spread is the extra yield that pays for expected loss, and expected loss is how often borrowers default times how much you lose when they do. This is sometimes called the credit triangle.

    Spread = how often it defaults x how much you lose when it does100 bonds, Rs 100 face eachlost 60% of facerecovered 40%5 of 100 default in a yearthe default probability, PD = 5%each loses 60% of facethe loss given default, LGD = 60%5 x 60% = 3 lost per 1003% a year = 300 bp of spreadBackwards: PD = 300 bp / 60% = 5% a year
    Out of 100 bonds, 5 default in a year and each loses 60% of face value, which is 3 lost per 100, the 300 basis point spread; run backwards, 300 basis points over 60% implies a 5% annual default probability.
    The relationship
    s≈PD×LGD⇒PD≈sLGD=0.030.60=5%s \approx PD \times LGD \quad\Rightarrow\quad PD \approx \frac{s}{LGD} = \frac{0.03}{0.60} = 5\%
    scredit spread over the government curve, 300 bp
    PDannual probability of default
    LGDloss given default, 60% of face
    What it says in wordsDivide the spread by the share of face value lost in default to get the default probability the market is pricing.

    Why is 5% probably too high as a real forecast?

    Because bond investors demand more than their expected loss. A spread also pays a risk premium for bearing uncertain losses and a liquidity premium for holding a bond that is hard to sell, so the implied probability is a risk-neutral figure that sits above the real-world default rate. For investment grade bonds, historical default rates are often a small fraction of what the spread implies. Say this as a limitation, and add that a rating agency would compare the implied 5% against the default history of similar ratings before drawing any conclusion.

    Also check the recovery assumption. If recovery were 20% rather than 40%, LGD would be 80% and the implied default probability would fall to 3.75%; the answer moves a lot with a number that is itself a guess.

    Where candidates lose it

    The fast wrong answer is 3%: reading the spread straight as the default rate. That assumes a default wipes out the whole bond, and a risk interviewer will ask where the recovery went.

    The quieter miss is presenting 5% as a forecast. Call it the market-implied rate and say that risk and liquidity premia push it above the real-world rate.

    What the interviewer asks next

    • The spread widens to 500 bp with no change in the company. What might explain it?
    • How would you convert a five-year spread into a cumulative default probability?
    • Why do rating agencies and bond markets often disagree about the same issuer?