Debt Capital Markets puzzles, solved step by step

At Rs 700 crore, senior secured recovers 100%, senior unsecured 66.7% and the subordinated debt nothing, so the senior unsecured layer is the fulcrum. Value is paid down the stack in order: Rs 500 crore to the secured lenders leaves Rs 200 crore for Rs 300 crore of unsecured claims, and nothing below. The equity is worth something only once enterprise value exceeds total debt of Rs 950 crore.

How is value shared out when it falls short of the debt?

Picture water poured into a stack of glasses, each filling completely before any spills into the next. In a recovery, value pays each layer of the capital structure in full, in order of priority, before anything reaches the next layer down. The senior secured lenders' glass holds Rs 500 crore, the senior unsecured Rs 300 crore, the subordinated Rs 150 crore, and the equity takes whatever is left. Pour in Rs 700 crore and watch where it stops.

Why does the fulcrum matter to a credit investor?

The fulcrum securityThe most senior layer of the capital structure that the company value does not cover in full, so it is likely to receive the equity in a restructuring. is the layer where the value runs out. It usually ends up owning the business in a restructuring, because the layers above are paid in full and the layers below are wiped out. Here the senior unsecured lenders would likely exchange their Rs 300 crore of claims for most of the new equity. Distressed investors buy the fulcrum because its value moves most with the enterprise value: between Rs 500 and Rs 800 crore, every extra Rs 1 crore goes straight to it.

At what value does the equity come back to life?

The equity is out of the money until enterprise value covers every debt claim, Rs 500 plus 300 plus 150 crore, which is Rs 950 crore. At the original Rs 1,000 crore it was worth only Rs 50 crore, 5% of enterprise value, which is why a 30% fall wiped it out and went on through the subordinated layer too. The limit: this is a strict priority waterfall; in real restructurings junior classes often receive a small share to win their votes, and claims include accrued interest and fees, so treat these recoveries as the starting point of a negotiation.

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%.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Short about Rs 63.2 crore face of the government bond. The corporate position moves Rs 100 crore x 0.043%, Rs 4.3 lakh, for each basis point. Each Rs 1 crore of the 10-year government bond moves Rs 6,800 per basis point. Matching the two takes 4.3 lakh over 6,800, about 63.2 lots of Rs 1 crore. The hedge ratio is the ratio of the DV01s, 0.043 over 0.068, not one for one on face value.

What exactly are you trying to balance?

Two children on a seesaw balance when weight times distance from the pivot matches on both sides, not when they weigh the same. A rate hedge balances when the rupee change per basis point matches on both sides, so you hedge DV01 against DV01, not face value against face value. The corporate bond's DV01The change in a bond position value for a one basis point change in yield, in rupees. is 0.043 per Rs 100, so Rs 100 crore of it gains or loses Rs 4.3 lakh for every basis point. That is the number the short has to offset.

Why do you need less of the government bond than you hold?

The 10-year government bond has a longer duration, so each rupee of it moves more when rates move: 0.068 per Rs 100 against 0.043. A smaller position in the more sensitive bond carries the same rupee risk. Check the balance: Rs 63.2 crore x 0.068% is Rs 4.3 lakh a basis point, the same as the corporate position. If rates fall 20 basis points in parallel, the corporate bond gains about Rs 86 lakh and the short loses about Rs 86 lakh.

Then say what the hedge does not cover, because that is the follow-up. A DV01 hedge removes the risk of a parallel move in rates, but it leaves you exposed to the curve twisting between 5 and 10 years and to the corporate bond's credit spread moving. If 5-year yields rise while 10-year yields stay put, the hedge does nothing for you. Desks often hedge with a bond or swap closer in maturity for exactly this reason, and rebalance as DV01s drift with time and rates.

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.

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.

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.

Unlevered free cash flow is about Rs 61.8 crore. EBITDA is 18% of Rs 800 crore, Rs 144 crore. Less Rs 30 crore of depreciation gives EBIT of Rs 114 crore, and tax at 25% leaves NOPAT of Rs 85.5 crore. Add back depreciation, which is not cash, to reach Rs 115.5 crore, then take off capex of Rs 45 crore and the working capital build of 12% of the Rs 72.7 crore revenue increase, Rs 8.7 crore.

What order do you walk the lines in?

Think of a shopkeeper's year: sales come in, stock and staff are paid, the taxman takes a share, some money goes on a new fridge, and more stock sits on the shelves because the shop is busier. What is left is what the owner could take out. Unlevered free cash flow follows the same order: operating profit, less tax on that profit, plus non-cash charges, less investment in fixed assets and in working capital. It is unlevered because interest is left out: the cash belongs to lenders and shareholders together.

Why tax EBIT rather than profit after interest?

Unlevered cash flow is the cash the business makes before any financing choice, so tax is charged as if there were no debt: 25% of EBIT, Rs 28.5 crore. Real tax is lower when there is interest to deduct, and that saving is counted elsewhere, in the discount rate or in a separate tax shield line. Taxing profit after interest and then discounting at a rate that already includes the shield counts the same benefit twice.

Where does the cash go between EBITDA and free cash flow?

Of Rs 144 crore of EBITDA, only Rs 61.8 crore, about 43%, becomes unlevered free cash flow: tax takes 28.5, capex 45 and working capital 8.7. Capex is 1.5 times depreciation, so the company is investing to grow, and working capital rises with sales because more revenue means more receivables and stock. A lender sizing debt on EBITDA should look at this conversion, because interest and repayments are paid from the Rs 61.8 crore, not the Rs 144 crore. The limit: one year of cash conversion can mislead when capex is lumpy.

Rs 30.5 per share with the discount and Rs 38.0 without it. The parts add to Rs 2,500 crore. A 15% discount takes off Rs 375 crore, leaving Rs 2,125 crore of enterprise value. Less Rs 600 crore of net debt gives equity of Rs 1,525 crore over 50 crore shares. With no discount equity is Rs 1,900 crore, or Rs 38.0. A 15% discount costs shareholders 19.7%.

Why does the discount come off before the debt?

Think of a house worth Rs 1 crore with a Rs 60 lakh home loan on it. If buyers suddenly pay 15% less for houses on that street, the house is worth Rs 85 lakh, the bank is still owed Rs 60 lakh, and the owner's share drops from Rs 40 lakh to Rs 25 lakh. A conglomerate works the same way. The holding company discount is a haircut on what the whole group is worth, and the lenders' claim is a fixed number that sits ahead of the shareholders, so the haircut is taken from enterprise value and passes through to equity untouched.

So the order is: value each business, add them to get the sum of the partsValuing each business on its own, usually against its own peers, and adding the values together., apply the holding company discountThe gap between what a group trades at and what its businesses would be worth separately, often blamed on head office costs, capital allocation or tax leakage. to the total, subtract net debt, and divide by shares. Rs 2,500 crore less Rs 375 crore is Rs 2,125 crore; less Rs 600 crore is Rs 1,525 crore; over 50 crore shares is Rs 30.5.

Why does a 15% discount cost the shareholders more than 15%?

Equity is the thin slice left after debt, so any fall in enterprise value is a bigger fall as a share of equity. The Rs 375 crore haircut is 15% of Rs 2,500 crore but 19.7% of Rs 1,900 crore. Push net debt to Rs 1,500 crore and the same haircut takes 37.5% of the equity. This is why a credit analyst reading a holding company cares about the discount even though no lender ever sees it on a statement: it is a measure of how much equity cushion the market will actually pay for.

What should you say about where the discount is applied?

Some desks apply the discount to equity value instead of enterprise value. On equity of Rs 1,900 crore, 15% gives Rs 1,615 crore and Rs 32.3 a share. The two conventions differ by Rs 1.8 a share here, so name your convention in the first sentence rather than let the interviewer find it. The enterprise value version is the more common reading when the discount is described as a discount to the sum of the parts, and it is the one that treats the lenders' claim as fixed.

Put one white ball alone in bowl A and the other 99 balls, 59 white and 40 black, in bowl B. Bowl A gives white every time and bowl B gives white 59 times in 99. Each bowl is chosen half the time, so the chance is 0.5 x 1 + 0.5 x 59/99, about 79.8%, against 60% for an even split.

Why does a bowl with one ball count as much as a bowl with 99?

Picture two teams tossing a coin to bat first: the coin does not care that one side has eleven players and the other has one. The bowls are chosen the same way. The coin picks a bowl, not a ball, so a bowl holding a single white ball gets the same half of the draws as a bowl holding ninety nine. That makes one lone white ball the cheapest way to buy certainty for half of all outcomes, and it costs bowl B only one white ball out of sixty.

How do you show that nothing beats it?

Bowl A cannot do better than 100%, and one white ball is the least that gets it there. Every further white ball moved into bowl A is wasted there and missed in bowl B, and every black ball moved into bowl A drags it below 100%. So bowl A is one white ball and bowl B takes what is left. Checking all 2,499 possible splits by computer agrees: the best is 79.80%, and only the one-ball split reaches it.

The formula also shows how the answer moves. With 50 white and 50 black, the version a candidate reported from an interview, it gives 74.7%. More white in the pool lifts bowl B and the total; bowl A is already at its ceiling. Here the split adds about 20 percentage points because one ball is turned into half of the probability.