Debt Capital Markets puzzles, solved step by step

Puzzles: 100
Traced to a firm: 16
Topics: 13
Hard: 30

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014An issuer has a 2% chance of default in any year, independent of the past. What is the probability that it defaults at some point in the next 5 years, and why is it not 10%?Credit spreads and default probabilityCoreCredit researchRisk management
Try it first
Is the five year default probability above or below 10%?
Show the worked solution
About 9.6%. The chance of surviving one year is 98%, and surviving five independent years is 0.98 to the power 5, or 90.39%. Default at some point is everything else: 9.61%. It is below 10% because the 2% in each later year applies only to issuers still alive, a shrinking group, so simply adding the yearly rates double counts.
Why does adding 2% five times overstate it?
Think of a phone that has a 2% chance of breaking each year. A phone that broke in year 1 cannot break again in year 2. Default is a one-time event, so each year's 2% applies only to the issuers that survived until then, and that group shrinks every year. Seen from today, the chance of defaulting in year 2 is 98% times 2%, which is 1.96%; by year 5 it is 1.845%. Those five numbers sum to 9.61%, not 10%.
Seen from today, the chance of defaulting in each year falls from 2.00% to 1.84% as survivors shrink, summing to 9.61% over five years; over fifty years compounding survival gives 63.6% cumulative default while adding 2% a year would absurdly reach 100%.
What is the fastest correct route?
Go through survival. Survival over several independent years is the product of the one year survival rates, and cumulative default is one minus that product. 0.98 squared is 0.9604, times 0.98 again is 0.9412, then 0.9224, then 0.9039. Or use the shortcut that 0.98 to the fifth is about 1 minus 5 times 0.02 plus 10 times 0.0004, which is 0.904. Either way, default is about 9.6%.
The relationship
$P(\text{default by } n) = 1 - (1-p)^n = 1 - 0.98^5 \approx 1 - 0.9039 = 9.61\%$
p the default probability in any one year, 2%
n the number of years, 5
(1-p)^n the chance of surviving all n years
What it says in wordsCumulative default is one minus the chance of surviving every year.
Why does a credit desk care about a 0.4 point gap?
Over five years the gap is small, 10.0% against 9.6%. Over long horizons the difference becomes enormous: adding 2% a year says default is certain by year 50, while compounding survival says 63.6%. That matters for pricing long bonds, for reading a rating agency's cumulative default tables, and for turning a spread into an implied default rate. Say the limit too: real default rates are not independent from year to year, they cluster in recessions, so the 2% flat rate is a teaching simplification.
Where candidates lose it
Saying 10% is the whole trap, and it comes from treating the yearly probabilities as if they could stack. The interviewer is checking whether you see that default removes the issuer from later years.
The opposite slip is saying 2% because each year is independent. Independence means each year's odds are unchanged for a survivor, not that the risk over five years is the same as over one.
What the interviewer asks next
What yearly default probability gives a 20% chance of default over 10 years?
If recovery is 40%, roughly what credit spread compensates for a 2% annual default rate?
Why would a rating agency's cumulative default table not fit a constant yearly rate?
038Loan A is Rs 50 crore to a borrower with a 1% default probability, secured so that loss given default is 20%. Loan B is Rs 20 crore unsecured, with a 3% default probability and 75% loss given default. Which has the larger expected loss?Credit spreads and default probabilityCoreCorporate bankingCredit research
Try it first
Which loan loses more on average each year?
Show the worked solution
Loan B, with an expected loss of Rs 0.45 crore against Rs 0.10 crore for Loan A. Expected loss is exposure times default probability times loss given default. Loan A: 50 x 1% x 20% is Rs 10 lakh. Loan B: 20 x 3% x 75% is Rs 45 lakh, 4.5 times as much on a loan less than half the size. Security and borrower quality matter more than the headline amount.
What goes into expected loss?
Imagine lending your bicycle to two friends. One borrows the expensive one but always returns things and leaves his phone as a deposit; the other borrows a cheap one, loses things often, and leaves nothing. What you expect to lose depends on how much you lend, how likely it is to go wrong, and how much you get back if it does, multiplied together. Lenders write those three as EAD, PD and LGD.
The relationship
$EL = EAD \times PD \times LGD \qquad A: 50 \times 0.01 \times 0.20 = 0.10 \qquad B: 20 \times 0.03 \times 0.75 = 0.45$
EAD exposure at default, the amount owed if the borrower defaults, Rs crore
PD probability of default over the year
LGD loss given default, the share of the exposure lost after recoveries
What it says in wordsExpected loss is how much is at risk, times how likely a default is, times how much of it would be lost.
Loan A's large Rs 50 crore exposure is offset by a 1% default probability and a 20% loss given default, for an expected loss of Rs 0.10 crore, while Loan B's Rs 20 crore at 3% and 75% gives Rs 0.45 crore, 4.5 times as much.
What does the answer mean for pricing?
Express each loss as a rate on the loan. Loan A costs 0.2% a year in expected losses; Loan B costs 2.25%, so B needs roughly two points more of spread just to break even on credit losses. Two factors did that. The default probability tripled, and the lack of security nearly quadrupled the loss when things go wrong. Security is why a secured loan to a weaker borrower can be safer than an unsecured loan to a stronger one.
Say the limit. Expected loss is an average; it covers the cost of doing business, not the bad year. A credit portfolioA book of many loans, where losses depend on how many default together, not only on each loan alone. with a few large loans can lose far more than its expected loss when one of them fails. Loan A's Rs 10 lakh expected loss hides a Rs 10 crore hit if it defaults, which is why lenders hold capital for unexpected loss as well.
Where candidates lose it
The instinctive answer is Loan A because it is bigger. The interviewer chose a large, safe, secured loan against a small, risky, unsecured one precisely to see whether you multiply all three factors or anchor on size.
The second loss is multiplying by the recovery rate instead of the loss rate: 80% for A and 25% for B. Say LGD is the share lost, not the share recovered, before you multiply.
What the interviewer asks next
What spread over funding cost would Loan B need to cover expected loss and still earn 1% a year?
Loan A's collateral value halves. What happens to its LGD and expected loss?
Why is expected loss not enough to set how much capital a bank holds?
068Two loans each have a 10% chance of default. What is the probability that at least one defaults if the defaults are independent, and if they are perfectly correlated? Why does a portfolio lender care?Credit spreads and default probabilityCoreRisk managementCredit research
Try it first
If the defaults are independent, what is the chance that at least one loan defaults?
Show the worked solution
Independent: 19% that at least one defaults and 1% that both do. Perfectly correlated: 10% for at least one and 10% for both. Independent loans survive together 0.9 x 0.9 = 81% of the time, so one or more defaults 19% of the time. Perfectly correlated loans default together or not at all. The expected number of defaults is 0.2 in both cases; what changes is how defaults bunch, and the chance of losing both loans is ten times higher.
Why is the answer not simply 10% plus 10%?
A 10% chance of rain on Saturday and 10% on Sunday is not a 20% chance of a wet weekend: some outcomes rain on both days and would be counted twice. The chance that at least one of two events happens is the sum of their chances minus the chance that both happen. For independent loans both default 0.1 x 0.1 = 1% of the time, so at least one defaults 10% + 10% - 1% = 19%. The quicker route is through survival: 1 minus 0.9 squared.
With independent loans the two 10% default regions overlap in only 1% of outcomes, so at least one loan defaults 19% of the time; with perfect correlation the regions sit on top of each other, so at least one defaults only 10% of the time but both default in all of those 10%.
The relationship
$P(\text{at least one}) = 1 - (1 - 0.1)^2 = 19\% \qquad P(\text{both}) = 0.1^2 = 1\%$
0.1 each loan's chance of default
(1 - 0.1)^2 the chance both repay, if the defaults are independent
What it says in wordsAt least one default is everything except the case where both loans repay.
What does perfect correlation change?
With perfect correlation the two loans default in exactly the same states of the world, so at least one defaults 10% of the time, and whenever one does, both do. The average is unchanged: 0.2 loans default on average in both worlds. But the independent pair loses both loans only 1% of the time, while the correlated pair loses both 10% of the time, ten times as often.
Outcome Independent Perfectly correlated
Neither defaults 81% 90%
Exactly one defaults 18% 0%
Both default 1% 10%
Expected defaults 0.2 0.2
The expected number of defaults is 0.2 loans in both cases, but the chance of losing both loans rises from 1% to 10% when the defaults are perfectly correlated.
Why does a portfolio lender care?
A lender holds capital against bad years, not average years, and correlation decides how bad the bad years are. Loans to one sector, one city or one promoter group tend to default together, which is why banks cap their exposure by sector and by group. Diversification works only when defaults are not strongly linked: ten independent loans with a 10% default chance all repay 35% of the time, but ten loans to one industry can behave like one big loan. The limit: real correlations sit between zero and one and rise in downturns, exactly when they hurt most, so the two cases here are the bounds, not the answer.
Where candidates lose it
The quick wrong answer is 20%, adding the probabilities and double counting the 1% of outcomes where both loans default. Work from the chance that both repay and the error cannot happen.
The second loss is saying correlation does not matter because the expected loss is the same. The average is identical; the chance of losing both loans goes from 1% to 10%, and that tail is what a lender's capital is for.
What the interviewer asks next
With a correlation between the two extremes, is the chance of at least one default above or below 19%?
You hold 10 independent loans, each with a 10% default chance. What is the chance that none defaults?
Why do default correlations tend to rise in a recession?
072A bond trades 300 basis points over the risk-free curve, and investors expect 40% recovery if it defaults. Using the credit triangle, what annual default probability is the market implying?Credit spreads and default probabilityCoreCredit researchFixed income asset management
Try it first
What annual default probability does the spread imply?
Show the worked solution
About 5% a year. The credit triangle says the spread pays for expected loss: spread = probability of default x loss given default. With 40% recovery the loss given default is 60%, so 3.0% = PD x 60% and PD = 5.0%. Treat it as an upper-bound reading, because real spreads also pay for liquidity and for bearing risk, so the default rate the market truly expects is usually lower.
Why does a spread imply a default probability at all?
An insurer charging Rs 3,000 a year to cover a Rs 1 lakh motorbike, where a stolen bike is usually recovered at 40% of its value, is pricing in some chance of theft; work backwards and you can read that chance off the premium. A credit spread is the premium a lender demands for expected loss: roughly, the chance of default each year times the share of the money lost when it happens. Everything else equal, a lender earning 300 basis points over the risk-free curve is being paid to lose about 3% a year on average.
The 300 basis point spread equals the default probability times a 60% loss given default, which implies 5.0% a year; the same spread implies 3.0% if nothing is recovered and 10.0% if 70% is recovered.
The relationship
$s \approx PD \times (1 - R) \;\Rightarrow\; PD = \frac{3.0\%}{1 - 0.40} = 5.0\%$
s the credit spread, 3.0% a year
PD the annual probability of default
R the expected recovery rate, 40%
1 - R loss given default, 60%
What it says in wordsThe spread roughly equals the yearly chance of default times the share lost when it happens.
How does the recovery assumption change the answer?
The higher the recovery, the higher the default probability a given spread implies, because each default costs the lender less. At 40% recovery, 300 basis points implies 5%; at 0% recovery it implies only 3%; at 70% recovery it implies 10%. So the recovery assumption is not a detail. Two analysts reading the same spread with different recovery views can disagree by a factor of two on how risky the credit is.
Expected recovery Loss given default Implied annual PD
0% 100% 3.0%
40% 60% 5.0%
70% 30% 10.0%
The same 300 basis point spread implies a 3.0%, 5.0% or 10.0% annual default probability depending on whether investors expect to recover nothing, 40% or 70% in default.
What does the credit triangle leave out?
The triangle treats the whole spread as payment for expected loss, but part of any spread pays for liquidity and for the discomfort of carrying credit risk. Default probabilities read off spreads therefore tend to sit above the default rates actually seen for similar credits. Use the triangle to compare credits and to sanity-check a spread, and say out loud that it gives an upper bound, not a forecast. Held for five years, 5% a year compounds to about 23% cumulative. The limit: it is a one-period shortcut that ignores when in the year default happens.
Where candidates lose it
The quick wrong answer is 3%, reading the spread as the default probability. That assumes lenders lose everything in default; with 40% recovery they lose 60%, so the same spread implies more defaults, not fewer.
The second slip is dividing by the recovery rate instead of the loss rate and getting 7.5%. Say loss given default out loud, 1 minus recovery, before you divide.
What the interviewer asks next
A secured bond from the same issuer has expected recovery of 70%. Roughly what spread should it trade at?
Why do default probabilities implied by spreads usually sit above historical default rates?
At 5% a year, what is the cumulative chance of default over five years?
092A high yield bond portfolio earns a spread of 450 basis points and an investment grade portfolio earns 120. Recovery on high yield defaults is 30%, and investment grade defaults are negligible. What annual default rate can the high yield portfolio absorb before it earns less than investment grade?Credit spreads and default probabilityCoreCredit researchFixed income asset management
Try it first
Roughly what default rate wipes out the extra spread?
Show the worked solution
About 4.71% of the portfolio defaulting each year. High yield earns 330 basis points more than investment grade. Each default costs 70% of face, since 30% is recovered, so each 1% of defaults costs 70 basis points. 330 divided by 70 is 4.71%. Below that default rate high yield still earns more; above it, less.
Why divide by the loss, not the default rate?
A shop that sells on credit loses less than the full bill when a customer fails to pay, if it can take back some of the goods. Bonds work the same way. The annual cost of defaults is the default rate times the loss given defaultThe share of face value lost when a borrower defaults, which is one minus the recovery rate., so the extra spread covers defaults up to the extra spread divided by the loss given default. With 30% recovered, each default loses 70%, and 330 basis points of extra spread covers 4.71% a year.
Annual credit loss rises by 70 basis points for each 1% of defaults and crosses the 330 basis points of extra spread at 4.71%, so high yield earns more than investment grade only while defaults stay below that rate.
The relationship
$d^{*} = \frac{s_{HY} - s_{IG}}{1 - R} = \frac{450 - 120}{0.70} = 471\text{ bp} = 4.71\%$
s_HY, s_IG the two portfolios' spreads, in basis points a year
R recovery on high yield defaults, 30%
d* the break-even annual default rate
What it says in wordsThe default rate that exactly uses up the extra spread is that spread divided by the loss on each default.
What does the number leave out?
A break-even default rate is a cushion, not a forecast: whether 4.7% is comfortable depends on how often default rates have run above it and for how long. It also assumes investment grade loses nothing; if investment grade lost 10 basis points a year, the break-even would rise to 4.86%. Recovery is the soft number: it tends to fall in exactly the years defaults rise, so a 30% assumption can flatter the cushion. And the calculation ignores price volatility, which matters to anyone who must mark the portfolio before the bonds mature.
Where candidates lose it
The frequent slip is saying 3.3%, treating every default as a total loss. With 30% recovery, each default costs 70%, and dividing by 0.7 is the whole point of the puzzle.
The second is using the full 450 basis points rather than the 330 excess over investment grade. The question compares two portfolios, so only the difference in spread buys the cushion.
What the interviewer asks next
If recovery falls to 20% in a downturn, what is the new break-even default rate?
What default rate does the full 450 basis point spread cover if investors demanded no extra return for risk?
Why do recoveries tend to fall when default rates rise?

p	the default probability in any one year, 2%
n	the number of years, 5
(1-p)^n	the chance of surviving all n years

EAD	exposure at default, the amount owed if the borrower defaults, Rs crore
PD	probability of default over the year
LGD	loss given default, the share of the exposure lost after recoveries

0.1	each loan's chance of default
(1 - 0.1)^2	the chance both repay, if the defaults are independent

Outcome	Independent	Perfectly correlated
Neither defaults	81%	90%
Exactly one defaults	18%	0%
Both default	1%	10%
Expected defaults	0.2	0.2

Try it first

Why does adding 2% five times overstate it?

What is the fastest correct route?

Why does a credit desk care about a 0.4 point gap?

Where candidates lose it

What the interviewer asks next

Try it first

What goes into expected loss?

What does the answer mean for pricing?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer not simply 10% plus 10%?

What does perfect correlation change?

Why does a portfolio lender care?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a spread imply a default probability at all?

How does the recovery assumption change the answer?

What does the credit triangle leave out?

Where candidates lose it

What the interviewer asks next

Try it first

Why divide by the loss, not the default rate?

What does the number leave out?

Where candidates lose it

What the interviewer asks next