Debt Capital Markets puzzles, solved step by step

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  1. 017A one-year rating transition matrix says an A-rated issuer stays A with 92% probability, moves to BBB with 7.5% and defaults with 0.5%. A BBB issuer defaults with 2% in a year. Using the same matrix each year, what is the A issuer's cumulative two-year default probability?Credit spreads and default probabilityHardRisk managementCredit research

    Try it first

    Before working the tree: is the two year figure above or below 1.0%, twice the one year rate?

    Show the worked solution

    About 1.11%. There are three ways to be in default by the end of year 2. Default in year 1: 0.5%. Stay A, then default: 92% times 0.5%, which is 0.46%. Fall to BBB, then default: 7.5% times 2%, which is 0.15%. They add to 1.11%, more than twice the one year rate, because downgrades move issuers into riskier states.

    Why can you not just double 0.5%?

    Think of a student who might fail an exam this year or slip into a weaker class and fail next year from there. Multi-year default risk comes from migration as well as direct default: an issuer that is downgraded in year 1 faces a higher default rate in year 2. Doubling 0.5% assumes every survivor is still A, and the independent-survival answer, 1.00%, makes the same assumption. Both miss the 7.5% who drift to BBB.

    Two roads to default in year 2: from A directly, or via a downgrade to BBBA todayA92%BBB7.5%Default0.5%92%7.5%0.5%Year 10.5%Defaultyr 22%Defaultyr 2Year 20.92 x 0.5% = 0.46%0.075 x 2% = 0.15%year 1 = 0.50%Two year default = 1.11%
    An A issuer can default in year 1 (0.5%), stay A and default in year 2 (0.46%), or fall to BBB and then default (0.15%); the three paths add to a two year default probability of 1.11%.

    How do you set it up so nothing is missed?

    List where the issuer can be after year 1, then ask for each state what happens in year 2. Default is an absorbing state, so a year 1 default stays counted; the other states each carry their own year 2 default rate. The A branch contributes 0.92 times 0.005 and the BBB branch 0.075 times 0.02. A small matrix multiplication does the same thing: square the one year matrix and read off the A to default cell.

    The relationship
    P2(A→D)=pAD+pAA pAD+pAB pBD=0.5%+0.92×0.5%+0.075×2%=1.11%P_2(A \to D) = p_{AD} + p_{AA}\,p_{AD} + p_{AB}\,p_{BD} = 0.5\% + 0.92 \times 0.5\% + 0.075 \times 2\% = 1.11\%
    p_ADone year default rate from A, 0.5%
    p_AAchance of staying A for a year, 92%
    p_ABchance of moving from A to BBB, 7.5%
    p_BDone year default rate from BBB, 2%
    What it says in wordsAdd every path that ends in default within two years, each path's probability being the product of its steps.

    What does the migration path tell a credit investor?

    The downgrade path is only 7.5% likely but supplies 14% of the two year default risk. For a high rated issuer, most of the medium-term risk is the chance of becoming a weaker credit, which is why investors watch outlooks and migration as closely as default rates. Downgrades also cost money before any default, through wider spreads. Say the limits: real matrices are estimated from history, the same matrix is unlikely to hold every year, and defaults cluster in downturns, so treat this as the arithmetic, not a forecast.

    Where candidates lose it

    The fast wrong answer is 1.0%, doubling the one year rate, or 0.9975%, compounding it as if the issuer could only ever be A. Both are the same error: ignoring that the issuer's rating can change before it defaults.

    The second loss is forgetting the year 1 default path, adding only the two year 2 paths to get 0.61%. Default is absorbing: once in, the issuer stays counted.

    What the interviewer asks next

    • BBB issuers can also fall to BB, which defaults at 8% a year. What else would you need to add a third year?
    • How would you compute the same answer with a matrix multiplication?
    • Why might a transition matrix estimated from a calm decade understate this risk?
  2. 054A 3-year floating rate note pays the benchmark plus 100 basis points, resetting quarterly. The issuer's market spread widens to 150 basis points. Roughly what price does the note fall to, and why does a floater still lose money?Credit spreads and default probabilityHardSyndicate desksFixed income asset management

    Try it first

    Before any maths: what happens to the note's price?

    Show the worked solution

    The note falls to about 98.7. The benchmark part of the coupon resets every quarter, so benchmark moves barely touch the price. The 100 basis point margin is fixed, and the market now wants 150. That 50 basis point shortfall runs for 12 quarters; discounted, it is worth about 1.32 per Rs 100, so the price is roughly 100 minus 0.50 x 2.64, about 98.7.

    Which part of a floater's coupon resets, and which does not?

    Picture a shop lease with rent set at inflation plus a fixed Rs 5,000. When inflation rises, the rent rises with it, so the landlord is protected. If rents on the street jump to Rs 8,000 over inflation, the landlord is still stuck at Rs 5,000 until the lease ends. A floating rate note resets its benchmark every quarter, but its credit margin is fixed at issue for the life of the note. Interest rate risk is almost gone; credit spread risk is not.

    The benchmark resets; the fixed 100 bp margin falls short of the new 150 bpWhat the note paysBenchmark 6.50%: resets each quarter+1.00What the market now wantsBenchmark 6.50%: resets each quarter+1.500.50% shortEach quarter the note pays 0.125 less per Rs 100 than the market wants. Today's value of each shortfall:Q1Q2Q3Q4Q5Q6Q7Q8Q9Q10Q11Q120.1230.099Sum of the twelve1.32Price 100 - 1.32= 98.68
    The benchmark part of the coupon resets, but the fixed 100 basis point margin now falls 50 basis points short of what the market wants; twelve quarterly shortfalls of 0.125, discounted, are worth 1.32 per Rs 100, so the note trades near 98.68.

    How much is a 50 basis point shortfall worth today?

    The note pays benchmark plus 1.00%; a new note from the same issuer would pay benchmark plus 1.50%. The holder is short 0.50% a year, 0.125 per Rs 100 each quarter, for 12 quarters, and the price falls by today's value of that stream. Assume the benchmark sits flat at 6.50%, so the discount rate is 8.00% a year, 2% a quarter. Twelve payments of 0.125 at 2% a quarter are worth 1.32, so the note trades near 98.68.

    The relationship
    P≈100−Δs×Ds=100−0.50×2.64≈98.68P \approx 100 - \Delta s \times D_s = 100 - 0.50 \times 2.64 \approx 98.68
    Delta sthe widening in the issuer's spread, 0.50 percentage points
    D_sspread duration, the price change per point of spread, about 2.64 years here
    100the price at a reset when the margin is fair
    What it says in wordsA floater loses the spread change times its spread duration, and a floater's spread duration is close to its remaining life.

    Say the shortcut first, then the refinement. A floater's spread duration is a little under its remaining life, so a 50 basis point widening on a 3-year note costs about 1.3 to 1.5 points. Using 3 years gives 98.5; discounting each shortfall gives 98.7, because the later shortfalls are worth less today. Either is a good answer if you say which one you did. The limit: this assumes a flat 6.50% benchmark, and a different curve moves the second decimal, not the story.

    Where candidates lose it

    The trap is saying a floater always trades near par because it resets. That is true for rate moves and false for credit moves: the margin is locked at issue, so a wider market spread leaves the note underpaying every quarter until maturity.

    The second loss is using the maturity for both risks. Rate duration runs only to the next reset, about 0.25 years; spread duration runs to maturity, about 2.6 years here. Keep the two apart out loud.

    What the interviewer asks next

    • The same 150 basis point widening hits a 7-year floater from the same issuer. Roughly how far does it fall?
    • What is this note's interest rate duration, and why is it so much smaller than its spread duration?
    • If the issuer's spread tightens back to 100 basis points after a year, where does the note trade?
  3. 084A 5-year credit default swap trades with a standard running coupon of 100 basis points, but the market spread for the name is 400 basis points and the risky duration is 4.0. What upfront payment changes hands, and who pays it?Credit spreads and default probabilityHardSyndicate desksCredit research

    Try it first

    Who pays the upfront amount, and roughly how much?

    Show the worked solution

    The protection buyer pays about 12% of notional upfront. The contract's running coupon is 100 basis points but fair protection costs 400, so the buyer underpays by 300 basis points a year. Multiply by the risky duration of 4.0, the present value of one basis point a year until default or maturity, and the shortfall is 1,200 basis points, or 12 points: Rs 12 crore on Rs 100 crore of protection.

    Why does a standard coupon create an upfront payment?

    Think of a gym that charges every member the same Rs 1,000 a month but costs Rs 4,000 a month to run for some members. It can only take them if they pay the Rs 3,000 a month difference as a lump sum on joining. Standardised credit default swaps work that way. The running coupon is fixed by the contract rather than set to the market, so the gap between the market spread and the coupon, for every year the protection is expected to run, is settled as one payment on the first day.

    The coupon is fixed, so the gap to the market spread is paid upfrontProtection buyershort the creditProtection sellerlong the creditrunning coupon: 100 bp a yearupfront on day one: 12 pointspays the loss if the name defaultsSpread, bp a yearMarket spread400Paid as coupon300 bp missing each year100Upfront = (400 - 100) bp x risky duration 4.0 = 1,200 bp= 12 points, Rs 12 crore on Rs 100 crore, paid by the protection buyer
    Paying a running coupon of 100 basis points for protection the market prices at 400 leaves 300 basis points a year missing, and 300 times a risky duration of 4.0 is 12 points, which the protection buyer pays upfront.
    The relationship
    Upfront≈(S−C)×RD=(400−100) bp×4.0=1,200 bp=12%\text{Upfront} \approx (S - C) \times RD = (400 - 100)\text{ bp} \times 4.0 = 1{,}200\text{ bp} = 12\%
    Sthe market spread, 400 bp a year
    Cthe contract's fixed running coupon, 100 bp a year
    RDrisky duration: the present value of 1 bp a year, paid until default or maturity
    What it says in wordsThe upfront payment is the yearly shortfall in the coupon multiplied by how many risky years it is paid for.

    Why is the risky duration less than five?

    Premiums on a credit default swap stop the moment the name defaults, so the expected number of payments is lower than five, and each one is discounted. A risky duration of 4.0 sits below the 4.33 of a plain five-year annuity at 5% because some of those payments will never be made. The riskier the name, the lower the risky duration, which is why a desk re-computes it rather than assuming it stays fixed as the spread moves. The formula above is the market's first approximation; the full calculation also nets accrued premium and uses a model of default timing.

    What happens when the spread is below the coupon?

    The sign flips. If the market spread were 50 basis points against a 100 coupon, the buyer would overpay by 50 a year and the seller would pay the buyer about 2 points upfront. Whoever is getting the better running deal pays the difference on day one.

    Where candidates lose it

    The common mistake is the sign. Candidates say the seller pays because the seller is taking the risk, but the seller is being underpaid at 100 basis points, so it is the buyer who owes the difference.

    The second is forgetting the multiplication: 300 basis points is a yearly shortfall, and it has to be turned into a present value with the risky duration before it is a payment.

    What the interviewer asks next

    • The spread widens to 500 the next day. Roughly what is the buyer's gain on Rs 100 crore?
    • Why would a market move to fixed coupons with upfront payments instead of quoting a running spread?
    • What default probability does a 400 basis point spread imply if you expect 40% recovery?