Debt Capital Markets puzzles, solved step by step

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

Showing 21–30 of 39 · filtered from 100Clear filters

043An issuer raises Rs 1,000 crore at 8.2% twelve months before its old bond matures and parks the cash at 6.9% until then. What does the pre-funding cost, and what risk does it remove?Compounding, PIK and feesCoreSyndicate desksCorporate banking
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What is the cost of carrying the cash for a year?
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Pre-funding costs about Rs 13 crore of negative carry, before tax. The issuer pays 8.2% on the new bond while earning 6.9% on the parked cash, a 1.3% gap on Rs 1,000 crore for twelve months. In return it removes refinancing risk: it no longer needs the market to be open, spreads to hold and its rating to survive the final year. On a five-year deal, waiting would have to cost more than about 26 basis points a year for pre-funding to lose.
What exactly does pre-funding cost?
A family renting a flat signs a new lease a year before the old one ends because rents are rising and good flats are scarce. For twelve months they pay a little extra holding the new flat, but they are not searching in a panic the week the old lease ends. Pre-funding costs the gap between what the new money costs and what the parked cash earns, for as long as both exist: the negative carryThe cost of holding a position whose income is lower than its funding cost, here cash earning less than the bond that raised it.. Here 8.2% less 6.9% is 1.3%, on Rs 1,000 crore for a year, Rs 13 crore, about Rs 1.08 crore a month.
The relationship
$\text{Carry} = 1{,}000 \times (8.2\% - 6.9\%) \times \tfrac{12}{12} = 13 \text{ crore}$
1,000 amount raised early, Rs crore
8.2% coupon on the new bond
6.9% yield earned on the parked cash
12/12 the fraction of a year the two overlap
What it says in wordsThe cost is the rate gap on the amount raised, for the time both the new bond and the parked cash exist.
Raising Rs 1,000 crore at 8.2% and parking it at 6.9% costs about Rs 1.08 crore a month, Rs 13 crore by the old bond's maturity, in exchange for removing the need to find a market on one fixed day.
What does that Rs 13 crore buy?
Certainty. Without pre-funding, the issuer must refinance in a narrow window before maturity, and if markets are shut, spreads have widened or its rating has slipped in that window, it either pays far more or cannot repay. Pre-funding is insurance with a known premium. A useful breakeven: spread over a five-year replacement bond, Rs 13 crore is about 26 basis points a year before discounting and about 33 after. If waiting risked a rise larger than that, or a missed maturity, the premium was cheap.
Say the refinements. Interest paid is usually tax deductible and interest earned is taxable, so at an assumed 25% rate on both the after-tax carry is about Rs 9.75 crore. The issuer also carries a bigger balance sheet for a year, gross debt rises before the old bond goes, which can briefly worsen reported leverage. And the parked cash must be invested safely and kept liquid; reaching for yield on it defeats the purpose of paying for certainty.
Where candidates lose it
The common miss is quoting the full coupon, Rs 82 crore, as the cost. The cash is not idle; it earns 6.9%, and only the gap is the price of pre-funding.
The second is stopping at Rs 13 crore and calling it a waste. The interviewer wants the other side of the trade named: refinancing risk, which is exactly what issuers with a large maturity and a shaky market fear most.
What the interviewer asks next
The issuer could instead buy back the old bond early. How would you compare the two?
The parked cash earns 7.5% instead. What is the carry, and does the answer change?
Why do rating agencies look favourably on issuers who pre-fund large maturities?
044A distressed bond trades at 55. It will either be repaid at 100 or restructured with a recovery of 30. Ignoring discounting, what probability of full repayment is the market pricing?Capital structure and recoveryCoreRestructuringCredit research
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The implied chance of full repayment is:
Show the worked solution
About 35.7%. Set the price equal to the probability-weighted payoff: p times 100 plus 1 minus p times 30 equals 55. The 30 is paid either way, so the price only has to cover 25 of the 70-point gap between recovery and par, and 25 over 70 is 35.7%. The market thinks restructuring is nearly twice as likely as repayment.
How do you turn a price into a probability?
Suppose a raffle ticket pays Rs 100 if you win and Rs 30 as a consolation prize if you lose, and it sells for Rs 55. Everyone gets the Rs 30, so you are really paying Rs 25 for a shot at an extra Rs 70. A price between two outcomes is a probability-weighted average of them, so you can solve for the weight. The consolation prize matters: ignore it and you would read the Rs 55 as a 55% chance.
The relationship
$p \cdot 100 + (1 - p) \cdot 30 = 55 \;\Rightarrow\; p = \frac{55 - 30}{100 - 30} = \frac{25}{70} \approx 35.7\%$
p market-implied chance of full repayment
100 payoff if repaid in full
30 recovery if restructured
55 today's price
What it says in wordsThe implied probability is how far the price sits above the recovery, as a share of the distance from recovery to par.
A bond at 55 that pays either 100 or a recovery of 30 sits 25 points up a 70-point range, which implies a 35.7% chance of full repayment and a 64.3% chance of restructuring.
What does ignoring discounting hide?
Both payoffs arrive in the future, often a year or more away, and a distressed investor wants a high return for waiting. If the payoffs arrive in a year and the buyer demands 12%, the price must equal the expected payoff divided by 1.12, which lifts the implied repayment probability to about 45.1%. So the 35.7% is a floor: part of the discount to par is a charge for time and risk, not only for default. That is why desks quote these as risk-neutral probabilities, not forecasts.
The other soft spot is the recovery of 30. It is an estimate, and the implied probability is sensitive to it: at a recovery of 40, p is 15 over 60, 25%. Before trading on the number, a credit analyst would test the recovery from the balance sheet and the claims ranking ahead of this bond.
Where candidates lose it
The instant wrong answer is 55%, reading the price as a probability. That would only be right if the bad outcome paid nothing. Here it pays 30, and the answer drops to 35.7%.
The second miss is forgetting to say the assumption. Ignoring discounting makes the probability look low; one sentence noting that a required return would raise it shows you know the difference between a market price and a forecast.
What the interviewer asks next
What price would imply a 50% chance of repayment?
If the recovery estimate falls to 20, what probability is implied at 55?
How would you set up the same calculation for a credit default swap spread?
052You invest Rs 1 lakh in a bond fund when its unit price is Rs 10 and another Rs 1 lakh when it is Rs 20. What is your average cost per unit, and why is it not Rs 15?Mental maths and numeracyCoreFixed income asset management
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Pick the average cost per unit before you calculate.
Show the worked solution
Your average cost is Rs 13.33 a unit, not Rs 15. Rs 1 lakh at Rs 10 buys 10,000 units and Rs 1 lakh at Rs 20 buys 5,000, so Rs 2 lakh buys 15,000 units. Equal money buys more units when the price is low, so the low price carries more weight. The average cost is the harmonic mean of the two prices, which always sits below the simple average when the prices differ.
Why does equal money not mean equal weight?
Suppose you fill Rs 1,000 of petrol at Rs 100 a litre one week and Rs 1,000 at Rs 125 the next. You got 10 litres and then 8, so 18 litres for Rs 2,000, about Rs 111 a litre, not the Rs 112.50 midpoint. When you spend the same money each time, you buy more units at the lower price, so the lower price counts for more in your average cost. The bond fund works the same way, only with a wider gap between the prices, so the effect is larger.
Rs 1 lakh at Rs 10 buys 10,000 units and Rs 1 lakh at Rs 20 buys 5,000, so the balance point of the 15,000 units sits at Rs 13.33, well left of the Rs 15 midpoint of the two prices.
What is the general rule, and why is it called a harmonic mean?
Divide total money by total units: 2,00,000 over 15,000 gives Rs 13.33. Written as a formula, the units are money over price, so the average cost is the number of instalments divided by the sum of one over each price. That is the harmonic mean, and for any set of unequal prices it lies below the simple average. The simple average here overstates your cost by 12.5%.
The relationship
$\bar{P} = \frac{2}{\tfrac{1}{10} + \tfrac{1}{20}} = \frac{2}{0.15} = 13.33$
2 the number of equal instalments
1/10, 1/20 units bought per rupee at each price
P-bar the average cost per unit
What it says in wordsWith equal money each time, the average cost is the harmonic mean of the prices.
This is the arithmetic behind the claim that investing a fixed sum on a schedule lowers your average cost. It is true, but only against the simple average of the prices you happened to pay. It does not make the investment itself cheaper or safer: it says nothing about whether Rs 10 or Rs 20 was the fair price. Say that limit, because the interviewer is often checking whether you oversell the result.
Where candidates lose it
The fast wrong answer is Rs 15, which averages the prices as if you had bought the same number of units twice. The question says the same money twice, and the whole puzzle turns on that difference.
The second loss is getting Rs 13.33 and then claiming the method guarantees a good outcome. It only guarantees an average cost below the average price paid; it cannot tell you whether the fund was cheap.
What the interviewer asks next
If you had bought 10,000 units at each price instead, what would your average cost be?
A yield moves between 6% and 8% and you buy equal money at each. Which average of the yields describes your purchase?
Three instalments at Rs 10, Rs 20 and Rs 40: what is the average cost?
053Nine bond certificates look identical, but one is a forgery printed on slightly heavier paper. With a balance scale and only two weighings, how do you find the forgery?Logic and brainteasersCoreSyndicate desks
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What should the first weighing be?
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Weigh three certificates against three, then one against one inside the suspect group. If the first weighing tips, the forgery is on the heavy side; if it balances, it is among the three set aside. Take that group of three and weigh one against another: the heavier one is the forgery, and if they balance, the third is. Each weighing has three outcomes, so two weighings separate nine cases.
Why split into thirds rather than halves?
Think of a quiz where each answer can be yes, no or maybe, instead of just yes or no. Every question now splits the possibilities three ways, so you get to the answer in fewer questions. A balance scale is a three-answer question: left heavy, right heavy or level, and a good weighing uses all three answers. Splitting in halves throws the level answer away, which is why it needs more weighings.
The first weighing of three against three sends each of its three outcomes to a group of three suspects, and the second weighing of one against one inside that group sends each outcome to a single certificate, so two weighings cover all nine cases.
How do you prove two weighings is the minimum, and the limit?
Count the outcomes. One weighing has 3 outcomes and two weighings have 3 x 3 = 9, so two weighings can pick out at most 9 certificates, and nine is exactly what you have. One weighing cannot do it, because 3 outcomes cannot separate 9 suspects. The same count tells you the scale for any number: three weighings handle up to 27, four handle up to 81.
The relationship
$\text{certificates you can search} = 3^{w} \qquad 3^{2} = 9$
w the number of weighings allowed
3 outcomes per weighing: left heavy, right heavy, level
What it says in wordsEach weighing multiplies the cases you can tell apart by three.
Now say why a DCM interviewer asks it. The puzzle rewards the habit of asking how much information each step gives before choosing the step. Due diligence on a bond issue works the same way: the best question to ask a management team is the one whose possible answers split the risks most evenly, not the one whose answer you already expect. The analogy is loose, so keep it to one sentence.
Where candidates lose it
Most candidates start with four against four because halving feels natural. If the scale tips, four suspects remain, and one weighing cannot finish the job, so they need three. The interviewer is waiting to see whether you notice the level outcome is information too.
The second loss is solving it but not proving two is the minimum. Have the counting argument ready: 3 outcomes per weighing, 3 x 3 = 9, and one weighing gives only 3.
What the interviewer asks next
You now have 12 certificates and do not know whether the forgery is heavier or lighter. How many weighings do you need?
What is the largest number of certificates you could search with three weighings?
If two of the nine are forged, can you still find them in two weighings?
056An issuer can pay 8.00% once a year or 7.85% in two semi-annual instalments of 3.925% each. Which is more expensive for the issuer, measured as an effective annual rate?Bond pricing and yieldCoreIndian debt capital markets
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Which costs the issuer more?
Show the worked solution
The 7.85% semi-annual coupon is marginally more expensive: an effective 8.004% a year against 8.000%. Paying 3.925 every six months means the first half coupon reaches investors early and can itself earn 3.925% for six months. Compounded, 1.03925 squared minus 1 is 8.004%. The two are almost equal because the exact semi-annual equivalent of 8% annual is 7.846%, just below 7.85%.
Why can you not compare 8.00 with 7.85 directly?
A landlord offered Rs 12,000 once a year or Rs 6,000 every six months should prefer the six-monthly cheques: the first Rs 6,000 can sit in a deposit for half a year. A rate only means something together with how often it is paid, because money paid earlier starts earning sooner. 8.00% paid annually and 7.85% paid semi-annually are quoted on different bases, so they must be put on one basis before anyone can say which is cheaper.
Per Rs 100, the annual bond delivers 8.000 at the year end, while the semi-annual bond's first 3.925 grows to 4.079 by the year end and adds to the second 3.925 for 8.004, so the semi-annual structure costs the issuer 0.4 basis points more.
How do you convert both to an effective annual rate?
Take Rs 100. The annual bond pays 8.00 at the end of the year, so its effective rate is 8.00%. The semi-annual bond pays 3.925 at six months and 3.925 at twelve. Reinvest the first 3.925 at the same 3.925% for six months and it grows to 4.079, so the investor holds 8.004 per Rs 100 at the year end, an effective 8.004%. Reinvesting at the coupon's own rate is the convention that defines an effective rate.
The relationship
$\text{EAR} = \left(1 + \frac{0.0785}{2}\right)^2 - 1 = 1.03925^2 - 1 = 8.004\%$
0.0785 / 2 the rate paid each half year
squared two half years compounded into one year
EAR effective annual rate, the one-payment-a-year equivalent
What it says in wordsCompound the half-yearly rate over two periods to compare it with a rate paid once a year.
How big is the difference in money, and when does it matter?
The gap is about 0.4 of a basis point, roughly Rs 2 lakh a year on a Rs 500 crore issue: real, but tiny next to the 15 basis points the headline rates suggest. The exact semi-annual rate matching 8% annual is 2 x (square root of 1.08 minus 1), 7.846%. The comparison matters in India, where government securities usually pay semi-annually and many corporate bonds annually, so a spread between them is only fair once both sit on one basis. The limit: the conversion assumes the early coupon is reinvested at the same rate, a convention rather than a promise.
Where candidates lose it
The fast wrong answer is that 8.00% costs more because it is the bigger number. That ignores when the money moves: a coupon paid six months early is worth more to the investor and costs more to the issuer.
The opposite slip is treating compounding as a large effect. At these rates it is worth about 15 basis points in total, which is exactly why 7.85% semi-annual lands almost on top of 8.00% annual. Give the size of the gap, not just its direction.
What the interviewer asks next
What quarterly rate is equivalent to 8.00% paid annually?
A government security yields 7.10% semi-annual and a corporate bond 7.90% annual. What is the spread on a like-for-like basis?
Why does the compounding effect grow as rates rise?
059Estimate the debt needed to build 10 GW of new solar capacity, assuming a project cost of Rs 4 to 5 crore per MW and 75% debt funding. How sensitive is the answer to the cost assumption?Estimation and market sizingCoreCorporate bankingIndian debt capital markets
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Roughly how much debt is that?
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About Rs 30,000 to 37,500 crore of debt, with Rs 33,750 crore at the midpoint. 10 GW is 10,000 MW; at Rs 4 to 5 crore per MW the build costs Rs 40,000 to 50,000 crore, and 75% of that is debt. The answer moves one for one with the cost per MW: every Rs 0.5 crore per MW shifts the debt by Rs 3,750 crore, more than the effect of moving the gearing five points.
How do you structure the estimate before any arithmetic?
Estimating a home loan starts with the price per square foot, times the floor area, times the share the bank will fund. A project debt estimate is the same chain: capacity times cost per unit of capacity gives the project cost, and the debt share of that cost gives the borrowing. Say the chain out loud first, with its units: megawatts, crore per megawatt, then a percentage. The most common slip in this question is a units error, not an arithmetic one.
The relationship
$\text{Debt} = 10{,}000\ \text{MW} \times \text{Rs } 4 \text{ to } 5\ \text{crore per MW} \times 75\% = \text{Rs } 30{,}000 \text{ to } 37{,}500\ \text{crore}$
10,000 MW 10 GW of capacity, since 1 GW is 1,000 MW
Rs 4 to 5 crore per MW the all-in project cost range given in the question
75% the share of project cost funded with debt
What it says in wordsCapacity times cost per MW gives the project cost; the debt share of that is the borrowing.
10,000 MW at Rs 4 to 5 crore per MW with 75% debt needs Rs 30,000 to 37,500 crore of borrowing, and the cost assumption swings the answer by Rs 7,500 crore against Rs 4,500 crore for a ten point swing in gearing.
Which assumption moves the answer most?
Flex each input across a sensible range while holding the other at its midpoint. Moving the cost from Rs 4 to Rs 5 crore per MW swings the debt by Rs 7,500 crore; moving the gearing from 70% to 80% swings it by Rs 4,500 crore. So the cost assumption is the one to defend, or to ask about. A lender would also want to know how much of the cost is modules against land, grid connection and construction, because those pieces move for different reasons.
Close with the range, not a single number. An estimate built on a range of inputs should come out as a range, with the midpoint named and the driver identified. At the midpoint, Rs 33,750 crore at an assumed 9% would carry roughly Rs 3,038 crore of interest a year. The costs here are the question's assumptions, not current market figures; in a real pitch you would confirm the cost per MW and typical gearing against recent project financings before using them.
Where candidates lose it
The usual loss is units. Candidates multiply 10 by Rs 4 to 5 crore and answer Rs 40 to 50 crore, or forget the 75% and give the whole project cost as the debt.
The second is a single number with false precision. The cost input is a range, so the output is a range: say Rs 30,000 to 37,500 crore, name the midpoint, and say which assumption drives the spread.
What the interviewer asks next
Roughly what annual interest bill does the midpoint debt carry at 9%, and what cash flow would cover it 1.3 times?
How would you split this debt between bank loans and bonds, and why?
Construction costs fall 20% while gearing rises to 80%. What is the new debt estimate?

064An issuer's outstanding bonds trade at 8.10% for 5 years and 8.40% for 7 years. It prices a new 6-year bond at 8.40%. What new issue premium did it pay against its own curve?Issuance and refinancing arithmeticCoreSyndicate desks

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What new issue premium did the issuer pay?

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The issuer paid a new issue premium of about 15 basis points. Its own curve runs from 8.10% at 5 years to 8.40% at 7 years, so a fair 6-year yield, interpolated halfway, is 8.25%. The new bond priced at 8.40%, 15 basis points wide of that. On a Rs 1,000 crore issue that is Rs 1.5 crore a year of extra coupon, about Rs 6.9 crore in today's money over six years.

What is fair value for a bond that does not exist yet?

If flats in one building sell for Rs 80 lakh on the fifth floor and Rs 84 lakh on the seventh, a sixth-floor flat is fairly priced near Rs 82 lakh; asking Rs 84 lakh for it is a premium. A new bond's fair value is read off the issuer's own curve at the new bond's maturity, because that curve already prices this issuer's credit. With points at 5 and 7 years, the 6-year point is the straight-line midpoint: 8.10% plus half of 30 basis points, 8.25%.

The issuer's curve runs from 8.10% at 5 years to 8.40% at 7 years, putting fair value for a 6-year bond at 8.25%, so the new 6-year bond priced at 8.40% paid a new issue premium of 15 basis points.

The relationship

y_6 = 8.10\% + \frac{6-5}{7-5}\,(8.40\% - 8.10\%) = 8.25\% \qquad \text{NIP} = 8.40\% - 8.25\% = 15\ \text{bp}

y_6	the interpolated fair yield for a 6-year bond from this issuer
(6-5)/(7-5)	how far along the gap between the 5-year and the 7-year the new bond sits
NIP	new issue premium, the new bond's yield less its fair value

What it says in wordsFair value comes from the issuer's own curve at the new maturity; the premium is what the new bond pays above it.

Why do issuers pay a premium at all, and what does it cost?

Investors want a small concession for buying a new bond rather than the existing ones: they take size, commit before the bond trades, and often sell other holdings to make room. The syndicate desk's job is to keep that premium as small as the order book allows. Here 15 basis points on Rs 1,000 crore is Rs 1.5 crore of extra coupon a year, about Rs 6.9 crore today over six years at 8.40%, which is why syndicate desks argue over five basis points.

State the assumptions behind the answer. Linear interpolation assumes the curve is straight between 5 and 7 years; a curve that bows would move fair value by a few basis points. The outstanding bonds may also be small or rarely traded, so their yields can carry a liquidity premium of their own. A desk would check the answer against a second yardstick, such as the spread over the government curve, before calling the premium high or low.

Where candidates lose it

The usual slip is comparing with the wrong bond: against the 7-year the premium looks like zero, against the 5-year it looks like 30 basis points. The new bond is a 6-year, so fair value has to come from the 6-year point on the curve.

The second loss is not saying how you interpolated. Say straight line between the two points, give 8.25%, and note that a curved line could move the answer by a few basis points.

What the interviewer asks next

The issuer's 7-year bond is illiquid and trades 10 basis points wide of fair. What is the premium now?
The book was four times covered. What does that tell you about the 15 basis points?
How would you estimate the premium for a first-time issuer with no bonds outstanding?

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
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If the defaults are independent, what is the chance that at least one loan defaults?
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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?
069Four anchor investors are each 60% likely to put in an order for your bond, independently of one another. You need at least three of them to launch. What is the probability that you launch?Probability and expected valueCoreSyndicate desks
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Before you calculate: roughly what is the chance of launching?
Show the worked solution
About 47.5%, a little under a coin flip. Exactly three anchors can happen in 4 ways, each with probability 0.6 x 0.6 x 0.6 x 0.4, so 4 x 0.0864 = 34.6%. All four come in with probability 0.6 to the fourth, 13.0%. Adding them gives 47.5%. Each anchor is more likely than not to order, yet needing most of a small group pulls the chance of launching below half.
How do you count the ways to get three of four?
Four friends each say they will probably come to dinner, 60% each, and you need three to keep the booking. The chance of any one pattern, say the first three come and the fourth does not, is 0.6 x 0.6 x 0.6 x 0.4 = 8.64%, and there are four such patterns, one for each friend who stays away. So exactly three come 4 x 8.64% = 34.6% of the time. All four come 0.6 to the fourth, 13.0% of the time. The booking survives 47.5% of the time.
The chance that three of the four anchors order is 34.6% and that all four order is 13.0%, so the deal launches 47.5% of the time even though each anchor is 60% likely to order.
The relationship
$P(X \ge 3) = \binom{4}{3} 0.6^{3}\, 0.4 + 0.6^{4} = 0.3456 + 0.1296 = 0.4752$
X the number of anchors who order
C(4,3) the four ways to choose which three anchors order
0.6, 0.4 each anchor's chance of ordering and of staying away
What it says in wordsAdd the chance of exactly three anchors, counted over every pattern, to the chance of all four.
Why is the answer below half when each anchor is likely?
Requiring most of a small group multiplies probabilities together, and products of numbers below one shrink fast. The most likely outcomes are two or three anchors, 34.6% each, so the launch hinges on which side of that line you land. If you needed only two anchors the chance would jump to 82.1%; if you needed all four it would fall to 13.0%. The threshold matters more than any single investor.
Say what a syndicate desk does with this. Either lower the threshold, by lining up a fifth anchor or sizing the deal so that fewer anchors are enough, or raise each anchor's probability before launch through early soundings. A fifth anchor at 60% lifts the chance of at least three from 47.5% to 68.3%; lifting each of the four to 70% gives 65.2%. The limit: anchors are rarely independent; they read the same market, so on a bad day they tend to drop out together, and the real chance is lower than this.
Where candidates lose it
The instinctive answer is about 60%, as if the group behaved like one anchor. Needing three of four is a joint event, and joint events are rarer than their parts.
The second loss is forgetting the four orderings and answering 8.64% plus 13.0%, or counting only the all-four case. Say the combination count out loud: four ways to choose which anchor stays away.
What the interviewer asks next
What is the chance of launching if you add a fifth anchor, also 60% likely?
If each anchor's chance rises to 70%, what is the chance of at least three of four?
Why might anchor orders be positively correlated, and what does that do to the launch probability?
070A company trades at 8x EV/EBITDA on EBITDA of Rs 100 crore. Net debt is Rs 300 crore at 10% interest, depreciation is Rs 20 crore and the tax rate is 25%. What P/E is that?Cost of capital and valuation riddlesCoreCredit researchLeveraged finance
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Which is closest to the P/E?
Show the worked solution
About 13.3x. Equity value is enterprise value less net debt: 8 x 100 = 800, less 300, is Rs 500 crore. Net income is EBITDA of 100, less depreciation of 20 and interest of 30, which leaves 50 before tax and 37.5 after 25% tax. Rs 500 crore over Rs 37.5 crore is 13.3x. The same business looks cheap at 8x EBITDA and dearer on earnings, because depreciation, interest and tax all sit between the two profit lines.
Why can the two multiples not be compared directly?
The price of a flat and the price of the owner's stake in it differ when there is a home loan: an Rs 80 lakh flat with a Rs 30 lakh loan leaves Rs 50 lakh of equity. EV/EBITDA prices the whole business against profit available to all funders; P/E prices only the equity against profit left after lenders and the tax authority are paid. To move from one to the other, bridge the value by taking off net debt, and bridge the profit by taking off depreciation, interest and tax.
Enterprise value of Rs 800 crore less Rs 300 crore of net debt leaves Rs 500 crore of equity, and EBITDA of Rs 100 crore less depreciation, interest and tax leaves Rs 37.5 crore of net income, so 8.0x EBITDA becomes 13.3x earnings.
The relationship
$P/E = \frac{8 \times 100 - 300}{(100 - 20 - 0.10 \times 300)(1 - 0.25)} = \frac{500}{37.5} = 13.3\times$
8 x 100 - 300 equity value: enterprise value less net debt
100 - 20 - 30 profit before tax: EBITDA less depreciation and interest
(1 - 0.25) what is left after 25% tax
What it says in wordsBridge the numerator from enterprise to equity value and the denominator from EBITDA to net income, then divide.
Where does the gap between 8x and 13.3x come from?
Bridge it in two moves. Depreciation and tax alone turn 8x EBITDA into 13.3x earnings: with no debt, EBIT of 80 taxed at 25% is 60, and 800 over 60 is 13.3x. So the gap here comes from the lines between EBITDA and net income, not from the debt. A business with heavy depreciation or a high tax rate always looks dearer on P/E than on EV/EBITDA.
Does the debt change the P/E at all?
With these numbers, no, and that is worth noticing. Leverage raises the P/E when the after-tax cost of debt is above the business's unlevered earnings yield, and lowers it when the cost is below. The unlevered earnings yield is 60 over 800, 7.5%; the after-tax cost of debt is 10% x 0.75, also 7.5%. Swapping equity for debt removes value and earnings in the same proportion, so the P/E stays at 13.3x. At 8% interest it would be 11.9x; at 12%, 15.2x.
Interest rate on Rs 300 crore Net income P/E on Rs 500 crore
8% 42.0 11.9x
10%, as given 37.5 13.3x
12% 33.0 15.2x
Rs crore. With the after-tax cost of debt equal to the 7.5% unlevered earnings yield, leverage leaves the P/E at 13.3x; cheaper debt lowers it and dearer debt raises it.
State the lesson for a credit desk. Never compare a P/E with an EV/EBITDA, or P/Es across companies with very different leverage, without bridging them first. The limit: this treats net debt as the only claim between enterprise and equity value; in real companies minority interests, leases and pension deficits also sit in that bridge.
Where candidates lose it
The fast wrong answer is 8x, treating the two multiples as interchangeable. They price different claims against different profit lines, so they match only by accident.
The second loss is forgetting the interest, which gives net income of 60 and a P/E of 8.3x, or dividing the whole enterprise value by net income, which gives 21.3x. Bridge the numerator and the denominator separately, and say each bridge out loud.
What the interviewer asks next
At what interest rate on the debt would the P/E be exactly 10x?
The company refinances at 12%. What happens to the P/E, and why?
Why do lenders size leverage off EBITDA rather than earnings?

p	market-implied chance of full repayment
100	payoff if repaid in full
30	recovery if restructured
55	today's price

2	the number of equal instalments
1/10, 1/20	units bought per rupee at each price
P-bar	the average cost per unit

w	the number of weighings allowed
3	outcomes per weighing: left heavy, right heavy, level

0.0785 / 2	the rate paid each half year
squared	two half years compounded into one year
EAR	effective annual rate, the one-payment-a-year equivalent

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