Debt Capital Markets puzzles, solved step by step

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

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022Debt is Rs 500 crore and EBITDA Rs 100 crore, growing 10% a year. Free cash flow is 40% of EBITDA and all of it repays debt. In which year does leverage, debt over EBITDA, first fall below 3.0x?Leverage, coverage and cash flowHardLeveraged financePrivate credit
Try it first
Your first guess: in which year does leverage drop below 3.0x?
Show the worked solution
Year 3, at about 2.66x. EBITDA grows to 110, 121 and 133.1, and 40% of each repays debt: 44, 48.4 and 53.2. Debt falls to 456, 407.6 and 354.4, so leverage runs 4.15x, 3.37x and 2.66x. Deleveraging comes from both ends: repayment shrinks debt while growth enlarges EBITDA, and either lever alone would leave leverage near 3.8x in year 3.
Why do both repayment and growth matter?
A family with a home loan feels less stretched both when it prepays the loan and when its salary rises. Leverage is a ratio, so it falls when the debt on top shrinks and when the EBITDA below grows, and here both happen every year. The growth also feeds the repayment: a bigger EBITDA means more cash, so each year's repayment is larger than the last. That compounding is why the answer arrives sooner than a straight-line guess.
Year EBITDA Cash repaid Debt at year end Debt / EBITDA
0 100.0 500.0 5.00x
1 110.0 44.0 456.0 4.15x
2 121.0 48.4 407.6 3.37x
3 133.1 53.2 354.4 2.66x
4 146.4 58.6 295.8 2.02x
Rs crore. EBITDA grows 10% a year and 40% of it repays debt at year end; leverage first drops below 3.0x in year 3, at 2.66x, and reaches 2.02x in year 4.
With growth and repayment together leverage falls from 5.0x to 2.66x by year 3, crossing the 3.0x line, while growth alone leaves it at 3.76x and repayment alone at 3.80x in the same year.
How do you show the interviewer which lever did the work?
Run each lever on its own. With flat EBITDA and 40 a year of repayment, leverage is 3.80x in year 3 and only reaches 3.0x in year 5; with growth but no repayment it is 3.76x in year 3. Neither alone gets there, and together they reach 2.66x, because each lever makes the other stronger. That split is the follow-up in most leveraged finance interviews, and it is how a lender judges how much of a deleveraging story depends on the growth forecast.
What would a lender worry about in this path?
The growth assumption carries much of the result. If EBITDA stalls, the same repayments leave leverage near 3.8x in year 3, so a covenant set to step down to 3.0x by year 3 would be breached. The model also assumes all free cash flow repays debt, with nothing for dividends, acquisitions or a working capital squeeze. Close with the sensitivity: a lender would size the covenant step-downs on a cash flow case below the base case, and say so.
Where candidates lose it
The common slip is running only one lever: repaying 40 a year against flat EBITDA, which says year 5, or growing EBITDA without increasing the repayment. The question built in 10% growth so that cash flow grows too.
The second is a timing error: dividing year-end debt by the opening year's EBITDA. Match the two: debt at the end of year 3 against year 3 EBITDA.
What the interviewer asks next
EBITDA grows only 3% a year. When does leverage fall below 3.0x now?
Half the free cash flow goes to dividends instead. Which year does it cross?
How would you set covenant step-downs for this borrower?
023You roll a fair die repeatedly. What is the expected number of rolls to see a six, and the expected number of rolls to see two sixes in a row?Probability and expected valueHardSyndicate desks
Try it first
Expected rolls to see two sixes in a row?
Show the worked solution
6 rolls for one six, and 42 rolls for two sixes in a row. A six comes up one time in six, so the wait averages 6. For two in a row, define E0 as the expected rolls from the start and E1 from one six showing. From E1 the next roll ends it with 1/6 or sends you back with 5/6. Solving E0 = 1 + (5/6)E0 + (1/6)E1 and E1 = 1 + (5/6)E0 gives E0 = 42.
Why is the first answer 6?
If a bus comes with probability one in six each minute, on average you wait six minutes. For a repeated trial with success probability p, the expected number of tries until the first success is 1 over p. With p equal to 1/6, that is 6. Say it quickly; the interviewer is only using it to set up the second part.
Why is two in a row not 12?
Twelve assumes the progress you make is kept. In a row means a single miss after a first six erases it, so you keep paying the six-roll wait again and again, and the answer is driven by those resets. Think of climbing two steps on a slippery staircase where any slip on the second step sends you to the ground: most attempts end on step one. Two states capture this: E0 with no six showing, E1 with one six showing.
From the start a six moves you to state E1 with probability 1/6; from E1 a second six finishes, but any other roll, 5/6 of the time, sends you back to the start, so the expected rolls solve to 42 from the start and 36 from one six showing.
The relationship
$E_1 = 1 + \tfrac{5}{6}E_0 \qquad E_0 = 1 + \tfrac{5}{6}E_0 + \tfrac{1}{6}\left(1 + \tfrac{5}{6}E_0\right) \;\Rightarrow\; \tfrac{1}{36}E_0 = \tfrac{7}{6} \;\Rightarrow\; E_0 = 42$
E_0 expected further rolls from the start, no six showing
E_1 expected further rolls with one six showing
5/6, 1/6 chance of a non-six and a six on any roll
What it says in wordsEach state's expected rolls equal one roll plus the expected rolls from wherever that roll sends you.
Is there a quick check you can say out loud?
Yes: for k in a row with success probability p, the expected wait is 1/p plus 1/p squared, and so on up to 1/p to the k. For two sixes that is 6 plus 36, which is 42; for three sixes in a row it is 6 plus 36 plus 216, which is 258. The pattern shows why streaks get expensive fast. The desk version of the same idea: requiring several conditions to hold consecutively, a covenant tested on two quarters in a row, or a run of clean prints, is far rarer than requiring them separately.
Where candidates lose it
The trap answer is 12, doubling the single six, or 36, reading two in a row as a single one in 36 event. Both ignore that failure after the first six throws away progress.
The second loss is setting up E1 wrongly, sending a non-six from E1 back to E1 instead of E0. Say where each roll sends you before writing the equations.
What the interviewer asks next
What is the expected number of rolls to see a six followed immediately by a five?
How many rolls on average to see three sixes in a row?
A game pays Rs 100 when you first roll two sixes in a row and each roll costs Rs 2. Is it worth playing?
024A 5-year bond yields 7.40%, the 4-year point on the same curve is 7.10%, and you fund the position at 6.50%. If the curve does not move for a year, what are the carry and the roll-down, and what is the rough total return over funding? Assume a modified duration of about 3.5 after a year.Yield curve and forward ratesCoreFixed income asset managementSyndicate desks
Try it first
Roughly what excess return over funding does the position earn if nothing moves?
Show the worked solution
Carry of 0.90%, roll-down of about 1.05%, so roughly 1.95% over funding. Carry is the bond's 7.40% yield less the 6.50% cost of funding it. Roll-down comes from the bond ageing into a 4 year bond priced at 7.10%: its yield falls 30 basis points, which at a duration of about 3.5 is a 1.05% price gain. On an unchanged steep curve, standing still earns both.
What is carry, and what is roll-down?
Carry is the simple part: you earn the bond's yield and pay your funding rate, like renting out a flat for more than its loan EMI. Roll-down is the price gain that comes purely from time passing on an upward sloping curve: the bond gets shorter, and shorter maturities are priced at lower yields. Nothing about the market has to change for either to happen; that is why desks call them the return from standing still.
Over a year the 5 year bond ages into a 4 year bond and is repriced from 7.40% to the curve's 7.10%, a 30 basis point fall worth 1.05% at duration 3.5; added to 0.90% of carry over 6.50% funding, the position earns about 1.95% if the curve does not move.
How do you turn 30 basis points into a return?
Use duration. A yield change converts to a price change of roughly duration times the change, so a 30 basis point fall at a duration of 3.5 is a gain of about 1.05%. Use the duration the bond will have after the year, when it is a 4 year bond, because that is the bond being repriced; the question gave 3.5 for exactly that reason. Then total return is the yield earned, 7.40%, plus the roll gain, about 8.45% before funding.
The relationship
$\text{excess} \approx (y_5 - f) + D \times (y_5 - y_4) = (7.40 - 6.50) + 3.5 \times 0.30 = 0.90 + 1.05 = 1.95\%$
y_5, y_4 today's 5 year and 4 year yields on the curve
f funding rate, 6.50%
D modified duration after one year, about 3.5
What it says in wordsExcess return is carry plus the duration-weighted fall in yield as the bond rolls down the curve.
How much can the market move before the trade loses money?
Divide the cushion by the duration. A 1.95% return cushion at duration 3.5 absorbs a rise of about 56 basis points in the 4 year yield by year end before the position breaks even. That breakeven is the number a portfolio manager actually trades on. Say the limits too: the curve rarely stands still, the roll-down depends on the curve's shape between the two points, and the duration figure is an approximation that ignores convexity.
Where candidates lose it
The common miss is quoting carry alone, 0.90%, and forgetting that on a steep curve the bond also gains in price as it ages. The question gave the 4 year yield specifically to see whether you use it.
The second slip is applying the 30 basis points without duration, calling the roll-down 0.30%. A yield move is not a return until it is multiplied by duration.
What the interviewer asks next
The curve between 4 and 5 years flattens so both yield 7.40%. What is the excess return now?
Funding rises to 7.50% the day after you buy. Is the trade still attractive?
Why is roll-down usually largest at the steepest part of the curve?
025There are 100 closed doors in a row. On pass 1 you open every door. On pass 2 you toggle every second door, on pass 3 every third door, and so on, until pass 100 toggles only door 100. How many doors end open, and which ones?Logic and brainteasersCoreSyndicate desks
Try it first
Quick instinct: how many doors are open at the end?
Show the worked solution
Ten doors end open: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Door n is toggled once on each pass whose number divides n, so it is toggled as many times as n has divisors. Divisors come in pairs, 2 with 6 for 12, so most numbers have an even count and end closed. Only a perfect square has a divisor paired with itself, giving an odd count, so only the squares stay open.
What decides whether one door ends open?
Think of a light switch pressed by everyone who walks past: whether the light ends on depends only on whether an odd or even number of people pressed it. Door n is toggled on pass k exactly when k divides n, so the number of toggles equals the number of divisors of n, and the door ends open only if that number is odd. This changes a hundred passes into one question about numbers.
Why do only perfect squares have an odd number of divisors?
Divisors come in pairs that multiply to the number: for 12 they are 1 and 12, 2 and 6, 3 and 4, six in all. Every divisor has a partner, so the count is even, unless a divisor is its own partner, which happens only when the number is a perfect square. For 16, the pairs are 1 and 16, 2 and 8, and 4 on its own, five in all. Squares up to 100 run from 1 times 1 to 10 times 10, so ten doors.
Of the 100 doors only the ten perfect squares end open, because door 12 is toggled six times by its three divisor pairs and closes, while door 16 is toggled five times, since 4 pairs with itself, and stays open.
The relationship
$\text{toggles}(n) = d(n) \qquad d(n) \text{ odd} \iff n = m^2 \qquad \#\{m^2 \le 100\} = \lfloor \sqrt{100} \rfloor = 10$
d(n) the number of divisors of n
m^2 a perfect square
floor of root 100 how many squares fit below 100, which is 10
What it says in wordsToggles equal divisors; the count is odd only for perfect squares, and there are ten of those up to 100.
How do you show your working in the room?
Start by tracking one small door aloud, say door 6: passes 1, 2, 3 and 6, four toggles, closed. Moving from a single example to the rule about divisor pairs is what the interviewer is really scoring, more than the final count. Then generalise: with N doors, the open ones are the squares up to N, so the count is the whole part of the square root of N. For 1,000 doors, 31 stay open. The limit is simply that the puzzle is pure logic; its value on a desk is the habit of reducing a big process to one property.
Where candidates lose it
The common wrong answers are 50, guessing that alternate doors survive, and the primes, because primes feel special in divisor puzzles. Primes have exactly two divisors, so they end closed.
The other loss is simulating pass by pass and running out of time. Track one door, find the rule, then count.
What the interviewer asks next
With 1,000 doors, how many end open?
Which doors are toggled exactly twice, and what are they called?
If you only did passes 1 to 50, how many of the 100 doors would be open?
027Expected floating fixings for the next three years are 6.0%, 6.5% and 7.0%, and discount factors come from the same path. What fixed rate makes a 3-year interest rate swap worth zero at the start, and why is it not the simple average of 6.5%?Yield curve and forward ratesHardSyndicate desksFixed income asset management
Try it first
Is the fair swap rate above, at, or below the 6.5% average?
Show the worked solution
The fair swap rate is about 6.478%, roughly 2.2 basis points below the 6.5% average. Discount factors from the path are 0.9434, 0.8858 and 0.8279. The swap rate is the average of the fixings weighted by those factors. Because the lowest fixing comes first and is discounted least, it counts for more, so the weighted average sits under the simple one.
What does it mean for a swap to be worth zero?
Picture two friends agreeing to swap rent for three years: one pays a flat amount, the other pays whatever the market rent turns out to be. The deal is fair only if, in today's money, the flat payments are worth the same as the expected market ones. A swap is worth zero at the start when the present value of the fixed leg equals the present value of the expected floating leg. Present value is the key phrase: a rupee in year one is worth more today than a rupee in year three.
Build the discount factorsThe value today of one rupee paid at a future date. A factor of 0.943 means a rupee in a year is worth 94.3 paise now. from the same path. Year one: 1 over 1.06, or 0.9434. Year two: that divided by 1.065, 0.8858. Year three: divided again by 1.07, 0.8279. Their sum, 2.6571, is the value today of receiving one rupee a year for three years.
The relationship
$S = \frac{\sum_i f_i\,DF_i}{\sum_i DF_i} = \frac{1 - DF_3}{DF_1 + DF_2 + DF_3} \approx 6.478\%$
f_i the expected floating fixing for year i
DF_i the discount factor for year i, from the same path
S the fixed swap rate that makes both legs equal in present value
What it says in wordsThe swap rate is a weighted average of the expected fixings, and the weights are the discount factors.
At a fixed rate of 6.478%, year one's shortfall of 0.478 per Rs 100 and year three's excess of 0.522 cancel once discounted, at -0.451 and +0.432; at the 6.5% average they leave -0.058, so the average is too high.
Why does the average overshoot?
Look at the two outer years. At 6.5% fixed, the fixed payer overpays by 0.5 in year one and is overpaid by 0.5 in year three. Equal in plain rupees, but the year one amount is discounted by 0.943 and the year three amount by 0.828. The early overpayment is worth more today than the late refund, so at 6.5% the fixed payer loses about 0.058 per Rs 100, and the fair rate must come down until the discounted gaps cancel. On an upward sloping path the swap rate always sits a little under the simple average; on a downward one it sits a little over.
Check it the short way. The ratio of one less the last discount factor to the sum of factors gives the same 6.478%, because a floating leg plus the return of notional at the end is worth exactly par. The gap to 6.5% is small here, about two basis points, but on a steep ten-year curve the difference between the plain and weighted averages becomes large enough to misprice a trade.
Where candidates lose it
Most candidates say 6.5% at once, because averaging the three fixings feels like the obvious answer. It ignores that the fixed and floating payments arrive at different times and are worth different amounts today.
The second loss is knowing the answer is weighted but using equal weights anyway when doing the arithmetic. Build the three discount factors out loud first, then weight; the direction of the gap should be stated before its size.
What the interviewer asks next
If the fixings were 7.0%, 6.5% and 6.0% instead, would the swap rate be above or below 6.5%?
What is the value of this swap to the fixed payer one year later if the path is realised exactly?
Why does a floating rate note reset to par on each fixing date?
How would you get the discount factors from a set of par bond yields instead?
028Size the annual flow of new two-wheeler loans in India that a securitisation desk could buy. Build it from households, ownership, the replacement cycle, the financed share and the ticket size, state every assumption, and do not rely on a remembered industry figure.Estimation and market sizingHardStructured creditIndian debt capital markets
Try it first
Which link in the chain moves the answer most if you get it wrong?
Show the worked solution
About Rs 1 lakh crore of new two-wheeler loans a year, of which perhaps Rs 31,000 crore could reach a securitisation desk. Assume 30 crore households, 55% owning 1.2 vehicles each: 19.8 crore in use. Replacing them every 10 years plus 0.3 crore first-time buyers gives 2.28 crore sales. At 60% financed and Rs 75,000 a loan, that is about Rs 1,02,600 crore; the 30% pool-eligible share is an assumption.
Where do you start a sizing question without a known number?
Start from people and the things they own, then ask how often those things are bought. A family that runs one scooter for ten years buys a scooter every tenth year, so a street of a hundred such families buys about ten a year. Annual sales of a durable good are roughly the number in use divided by how many years each one lasts, plus whatever first-time buyers add. That structure lets you build the answer from facts you can defend instead of a figure half remembered from a newspaper.
Say each assumption out loud as an assumption. Households: 30 crore, and say you would confirm the current census estimate. Ownership: 55% of households, with 1.2 vehicles each where they own one, which gives 19.8 crore two-wheelers on the road. Life: 10 years, so replacement demand is 1.98 crore a year. New owners: if ownership climbs one point a year, 0.3 crore households buy their first. Total: 2.28 crore a year.
Thirty crore households at 55% ownership and 1.2 vehicles each give 19.8 crore two-wheelers in use, a 10-year cycle plus first-time buyers gives 2.28 crore sales, and 60% financed at Rs 75,000 gives about Rs 1,02,600 crore of loans a year, within a range of Rs 74,100 to Rs 1,35,660 crore.
How do you turn units into a rupee flow the desk can buy?
Two more links. Share financed: assume 60% of buyers borrow. Ticket: an average vehicle of Rs 1 lakh with a 75% loan-to-valueThe loan as a share of the price of the asset it buys. A 75% loan-to-value on a Rs 1 lakh scooter is a Rs 75,000 loan. is a Rs 75,000 loan. So 1.37 crore loans at Rs 75,000 each is about Rs 1,02,600 crore a year. Then cut to what a securitisation desk can actually buy: only loans made by lenders who sell pools, which at an assumed 30% is about Rs 30,780 crore. Banks that keep loans on their books never reach the desk.
How do you show the interviewer the number is sane?
Give a range and name the link that drives it. Moving the financed share between 50% and 70% and the ticket between Rs 65,000 and Rs 85,000 spreads the answer from Rs 74,100 crore to Rs 1,35,660 crore. A range with a named driver is more credible than a single precise figure. Then say the check you would run outside the room: compare 2.28 crore units with the industry body's published domestic sales, and the rupee total with the originators' disclosed disbursements. The chain is only as good as its weakest assumption, which here is the replacement cycle.
Where candidates lose it
The fast failure is quoting a figure you think you read somewhere and defending it. The interviewer asked for the build precisely to take memory out of it; a remembered number with no structure scores lower than a transparent chain that lands a little off.
The second failure is stopping at total loan volume. The question asked what a securitisation desk could buy, so the last link, the share of loans made by lenders who actually sell pools, is part of the answer and deserves its own stated assumption.
What the interviewer asks next
Which of your assumptions would you test first with one phone call, and to whom?
How does the answer change if electric two-wheelers push the average ticket to Rs 1.2 lakh?
Why might a desk prefer these pools to a single corporate bond of the same size?
029An investor asks for Rs 300 crore of 4.5-year paper at 8.35%. The issuer's curve is 8.10% at 3 years and 8.40% at 5 years. Is the reverse enquiry at, above or below fair value for the issuer, and by how much?Issuance and refinancing arithmeticCoreSyndicate desksIndian debt capital markets
Try it first
Where does the issuer's curve sit at 4.5 years?
Show the worked solution
The ask is about 2.5 basis points above the issuer's fair level, a small concession. Interpolating between 8.10% at 3 years and 8.40% at 5 years puts 4.5 years at 8.325%. Paying 8.35% costs the issuer 2.5 basis points on Rs 300 crore, about Rs 7.5 lakh a year and Rs 33.75 lakh over the life, which is often less than a public deal would need to clear.
What do you compare a reverse enquiry against?
A buyer walks into a tailor and asks for a jacket in a size the shop never stocks. The tailor prices it from the sizes either side, not from a different shop's rate card. A reverse enquiry, where an investor asks the issuer for a tailored bond, is judged against the issuer's own curve at that exact tenor. The issuer's 3-year and 5-year yields bracket the request, so the fair yield at 4.5 years comes from reading between them.
The relationship
$y_{4.5} = 8.10\% + \frac{4.5 - 3}{5 - 3}\,(8.40\% - 8.10\%) = 8.325\%$
8.10%, 8.40% the issuer's yields at 3 and 5 years
(4.5 - 3)/(5 - 3) how far along the gap the requested tenor sits, three quarters
What it says in wordsWalk three quarters of the way from the 3-year yield to the 5-year yield.
The issuer's curve reads 8.325% at 4.5 years, so the investor's 8.35% ask sits 2.5 basis points above it, worth about Rs 7.5 lakh a year on Rs 300 crore.
Is 2.5 basis points a lot?
Put it in rupees. One basis point on Rs 300 crore is Rs 3 lakh a year, so 2.5 basis points is Rs 7.5 lakh a year, about Rs 33.75 lakh over 4.5 years before discounting. Against that, a reverse enquiry saves the issuer the cost of a public deal: no roadshow, no marketing, and no new issue concessionThe extra yield a new bond usually offers over the issuer existing bonds so that investors take it up.. If a public 4.5-year deal would need a concession wider than 2.5 basis points to clear, the enquiry is the cheaper route.
Say the assumption and its limit. Linear interpolation treats the curve as a straight line between 3 and 5 years. Real curves usually bend, flattening as tenor lengthens, so the true 4.5-year point may sit a fraction above 8.325%, which would shrink the gap. An odd tenor also leaves the issuer with a bond that matches nothing else outstanding and trades less easily, which is one reason investors pay a little for the tailoring and issuers are willing to give a little back.
Where candidates lose it
The common error is comparing 8.35% with the nearest point, 8.40% at 5 years, and calling the ask cheap for the issuer. The tenor is 4.5 years, and the right benchmark is the curve at 4.5 years, not the closest liquid bond.
The second loss is giving the answer only in basis points. A syndicate desk speaks in rupees to the issuer's treasurer: Rs 7.5 lakh a year is the number that makes the decision concrete.
What the interviewer asks next
The issuer's 4-year bond trades at 8.28%. Does that change your fair value?
The investor wants a call option at year 3. Should the yield go up or down?
Why might an issuer accept a reverse enquiry that is slightly above its curve?
030Senior lenders are owed Rs 400 crore and junior lenders Rs 200 crore; enterprise value is Rs 380 crore. The juniors threaten a fight that would take six months and cost 10% of enterprise value. Should the seniors give the juniors 5% of the new equity to settle?Capital structure and recoveryHardRestructuringCredit research
Try it first
The juniors are out of the money. What do the seniors keep if they settle, against if they fight and win?
Show the worked solution
Yes: settling leaves the seniors Rs 361 crore against Rs 342 crore by fighting, Rs 19 crore better. A fight that costs 10% of Rs 380 crore shrinks value to Rs 342 crore before anyone is paid. Giving the juniors 5% of Rs 380 crore costs Rs 19 crore. The seniors should pay anything up to 10% of the equity to avoid the fight, before counting six months of delay.
Why pay anything to a class that is out of the money?
Two heirs argue over a house worth Rs 38 lakh. One has the clear legal claim; the other can drag the case through court for a year and run up lawyers' fees of Rs 3.8 lakh. The strong heir who hands over Rs 1.9 lakh to end it keeps more than one who wins in court. A settlement payment to a junior class is rational whenever it costs less than the value the fight would destroy. Being right about priority does not make fighting free: the costs come out of the same pot.
Here the seniors are owed Rs 400 crore against Rs 380 crore of value, so under absolute priorityThe rule that each class of creditor is paid in full before the class below it gets anything. the juniors get nothing and the seniors take the whole company. The juniors' only lever is delay and cost. Fighting burns 10% of value, Rs 38 crore, leaving Rs 342 crore.
Settling gives the seniors 95% of Rs 380 crore, Rs 361 crore, while fighting and winning gives them 100% of a value cut to Rs 342 crore, so the 5% gift leaves the seniors Rs 19 crore better off.
How big a gift would still make sense?
The relationship
$(1 - s) \times 380 \ge 342 \;\Rightarrow\; s \le 10\%$
s the share of new equity handed to the juniors
380 enterprise value if the deal settles now
342 enterprise value after a fight costing 10%
What it says in wordsThe seniors are ahead as long as what they give away is smaller than what the fight would burn.
The seniors break even at a gift of 10%, the same as the share of value the fight would destroy. At 5% they are comfortably inside that line, and the six months of delay tips it further, because a recovery received now is worth more than the same recovery received later and a distressed business often loses customers and staff while it waits. Senior recovery rises from 85.5% to 90.25% of their claim.
Say the limits too. The answer assumes the seniors would win the fight outright; if there is any chance the juniors win something, settling looks better still. It also assumes the 10% cost estimate is honest. And a gift to one class can invite others to threaten the same thing, which is why seniors often pair the equity with warrants or tie it to the juniors voting for the plan.
Where candidates lose it
The instinctive answer is no: the juniors are out of the money, so giving them anything breaks priority and rewards a threat. That answers a fairness question the interviewer did not ask.
The question is what maximises the seniors' recovery. Compare 95% of the bigger value with 100% of the smaller one, then give the breakeven gift. Candidates who stop at priority rules miss that the fight is paid for with the seniors' own money.
What the interviewer asks next
What if the juniors have a 30% chance of winning a Rs 60 crore share in court?
How would warrants rather than equity change the juniors' incentive?
Why do courts and plans sometimes allow this kind of gift, and when is it challenged?
031Without a calculator, estimate 1.07 to the power 5. Then check how close the rule of 72 gets to the doubling time at 7%.Mental maths and numeracyCoreFixed income asset management
Try it first
Your first instinct for 1.07 to the power 5?
Show the worked solution
1.07 to the fifth is about 1.40, 1.4026 exactly, and the rule of 72 gives a doubling time of 10.29 years against a true 10.24. Expand it: 1 plus 5 x 0.07 plus 10 x 0.0049 is already 1.399. The rule of 72 is out by about 15 days over a decade, well inside any rounding a desk cares about.
How do you get 1.07 to the fifth in your head?
A savings account paying 7% a year gives you Rs 7 on every Rs 100 in the first year. In the second year you earn Rs 7 on the original Rs 100 again, plus 49 paise on the first year's Rs 7. Compounding is simple interest plus interest on interest, and for small rates over a few years the first correction does almost all the work. That is why the expansion below settles so quickly.
The relationship
$1.07^5 = 1 + 5(0.07) + 10(0.07)^2 + 10(0.07)^3 + \ldots = 1 + 0.35 + 0.049 + 0.0034 + \ldots \approx 1.4026$
5, 10, 10 the number of ways to pick one, two or three of the five years, from the binomial expansion
0.07^2 = 0.0049 interest earned on one year's interest
What it says in wordsAdd simple interest, then the pairs of years, then the triples; each layer is about a tenth the size of the last.
A second route checks the first. Square 1.07 to get 1.1449, square again to get about 1.3108, then multiply by 1.07: 1.3108 plus 7% of it, which is 0.0918, gives 1.4026. Two methods landing on the same number is the thing to say out loud; it tells the interviewer you do not trust a single chain of mental arithmetic.
One rupee at 7% compounds to 1.4026 after five years and to 2 at 10.24 years, a whisker before the rule of 72's 10.29 years, while simple interest reaches only 1.70 by year ten.
How good is the rule of 72 at 7%?
The exact doubling time is the log of 2 over the log of 1.07, 10.245 years. The rule of 72 gives 72 over 7, 10.286 years. The error is about 15 days in ten years, because 72 is tuned to rates near 8% and 7% is close enough. At 2% the rule of 70 does better; at 20% the rule of 72 starts to drift, since the true figure is 3.8 years against its 3.6. Knowing where a shortcut breaks is worth as much as the shortcut.
The desk use is fast sanity checking. If a 7% bond's coupons are reinvested at 7%, Rs 100 grows to about Rs 140 in five years and Rs 200 in a little over ten. Simple interest would reach only Rs 170 by year ten, so the gap between the two lines in the picture is what reinvestment is worth.
Where candidates lose it
The common wrong answer is 1.35: five times 7% with no compounding. Candidates who know better sometimes jump to 1.5 because it feels bigger, without a method behind it.
The other loss is reciting the rule of 72 without checking it. The interviewer wants the exact figure or a reason the rule is close here; say that 72 over 7 is 10.29 and the truth is about 10.24.
What the interviewer asks next
Estimate 1.07 to the power 10 from your answer.
At what rate does money double in exactly 8 years, roughly?
Why is the rule of 69.3 exact for continuous compounding?
032The leverage covenant is 4.0x. Debt is Rs 800 crore and EBITDA has dropped to Rs 180 crore, so leverage is 4.44x. The sponsor can inject equity either as a cure counted in EBITDA or as a prepayment of debt. How much does each route need?Leverage, coverage and cash flowHardLeveraged financePrivate credit
Try it first
Which route needs less sponsor cash, and by roughly how much?
Show the worked solution
The EBITDA cure needs Rs 20 crore; the prepayment needs Rs 80 crore. To reach 4.0x with debt of 800, EBITDA must be 200, so the cure adds 20. To reach 4.0x with EBITDA of 180, debt must be 720, so the prepayment is 80. The cure is four times cheaper because each rupee of EBITDA supports four rupees of debt, which is why lenders cap how often and how much a cure can count.
Why does the same covenant give two very different cheque sizes?
A bank lets you borrow up to four times your monthly salary. If you are over the limit, you can either repay some of the loan or show a higher salary. Showing Rs 1,000 more salary makes room for Rs 4,000 of loan, so it takes a quarter as much money to fix. A leverage ratio has debt on top and EBITDA underneath, and at a 4.0x test a rupee added below does the work of four rupees removed above. An equity cureA right in a loan agreement letting the owner inject cash to fix a covenant breach, with the cash treated as if it were extra EBITDA for the test. exploits exactly that.
The relationship
$\text{Cure} = \frac{800}{4.0} - 180 = 20 \qquad \text{Prepay} = 800 - 4.0 \times 180 = 80$
800 debt, Rs crore
180 EBITDA after the drop, Rs crore
4.0 the maximum leverage the covenant allows
What it says in wordsSolve the ratio for the EBITDA that passes at today's debt, and for the debt that passes at today's EBITDA.
Adding Rs 20 crore of cure to EBITDA takes leverage from 4.44x to 4.0x with debt unchanged, while prepayment needs Rs 80 crore of debt repaid to reach the same 4.0x, four times the cash.
Why do lenders limit equity cures?
Because the cure is cosmetic for the covenant. Rs 20 crore of one-off equity counted as EBITDA does not make the business earn more; it makes the ratio pass for the test periods it is counted in. The lenders still hold Rs 800 crore against a business earning Rs 180 crore. So documents typically cap how many cures are allowed over the life, forbid them in consecutive quarters, cap the amount at what is needed to pass, and stop the cure cash counting as EBITDA for anything else, such as the room to pay dividends.
One wrinkle is worth saying. Some documents also let the cure cash sit on the balance sheet and be netted against debt. Then you solve 800 minus x over 180 plus x equals 4.0, and x falls to Rs 16 crore. Lenders resist that double count for the same reason: one rupee should not fix both halves of the ratio.
Where candidates lose it
The common answer is Rs 80 crore either way, from candidates who only think about paying down debt. They miss that the question offered a cure counted in EBITDA precisely to see whether you notice the ratio's multiplier.
The second loss is getting Rs 20 crore and stopping. The interviewer wants to hear why lenders cap cures: the business still earns Rs 180 crore, and a cure hides that for a quarter.
What the interviewer asks next
EBITDA falls again to Rs 170 crore next quarter. What cure is needed, and is a second cure likely to be allowed?
The cure cash can also be netted from debt. What is the smallest cheque now?
Why might a sponsor choose to prepay even though the cure is cheaper?

Year	EBITDA	Cash repaid	Debt at year end	Debt / EBITDA
0	100.0		500.0	5.00x
1	110.0	44.0	456.0	4.15x
2	121.0	48.4	407.6	3.37x
3	133.1	53.2	354.4	2.66x
4	146.4	58.6	295.8	2.02x

E_0	expected further rolls from the start, no six showing
E_1	expected further rolls with one six showing
5/6, 1/6	chance of a non-six and a six on any roll

y_5, y_4	today's 5 year and 4 year yields on the curve
f	funding rate, 6.50%
D	modified duration after one year, about 3.5

d(n)	the number of divisors of n
m^2	a perfect square
floor of root 100	how many squares fit below 100, which is 10

Try it first

Why do both repayment and growth matter?

How do you show the interviewer which lever did the work?

What would a lender worry about in this path?

Where candidates lose it

What the interviewer asks next

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Why is the first answer 6?

Why is two in a row not 12?

Is there a quick check you can say out loud?

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How do you get 1.07 to the fifth in your head?

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Where candidates lose it

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Try it first

Why does the same covenant give two very different cheque sizes?

Why do lenders limit equity cures?

Where candidates lose it

What the interviewer asks next