Debt Capital Markets puzzles, solved step by step

About Rs 7.5 crore, 1.5% of the portfolio. Portfolio modified duration is the value-weighted average: 0.6 times 4 plus 0.4 times 9 is 6.0. A 25 basis point parallel rise costs about duration times the move, 6.0 times 0.25%, or 1.5%, and 1.5% of Rs 500 crore is Rs 7.5 crore. The duration 9 block is only 40% of the money but carries 60% of the loss.

What does modified duration convert?

Think of duration as a lever length. A seesaw with a long arm moves more at the end for the same push. Modified duration tells you the approximate percentage fall in a bond's price for a one percentage point rise in its yield, so a duration of 4 means about 4% per 1% move, or 1% for 25 basis points. Once you have that, a rate shock converts straight into rupees: duration times the move times the value held.

How do you combine two blocks of bonds?

Weight each duration by the money in it. Portfolio duration is the value-weighted average of the holdings' durations, because each bond's loss is its own value times its own duration times the move. Rs 300 crore at duration 4 loses Rs 3.0 crore; Rs 200 crore at duration 9 loses Rs 4.5 crore. Together that is Rs 7.5 crore, exactly what Rs 500 crore at duration 6.0 gives. The figure makes the point visually: area is the loss.

What would make the true loss differ from Rs 7.5 crore?

Three things, and naming them is what separates a desk answer from a formula. Convexity makes the true loss slightly smaller than the duration estimate for a rise in yields, though for a 25 basis point move the difference is tiny. Curves rarely move in parallel after a surprise hike: short yields usually jump more, which would hurt the duration 4 block more than this sum assumes. And spreads on corporate bonds can move on top of the base rate. Say that Rs 7.5 crore is the first-order answer to a parallel shift, then name which of these you would check first.

Effective duration is about 3.26, and convexity is negative. Take the price gap between the two shocked prices, 102.40 less 99.10, which is 3.30, and divide by twice the price times the 0.005 yield move: 3.30 over 1.012. A fall in yields adds only 1.20 while a rise costs 2.10, so the call is capping the upside: effective convexity works out near -356.

Why do you shock the yield both ways instead of using a formula?

Think of a landlord who can ask the tenant to leave whenever a better tenant turns up. The tenant's lease is worth less when the market is hot, because that is exactly when it gets cut short. A callable bond works the same way: the issuer takes the bond back when rates fall. Because the cash flows change with the yield, no fixed schedule of payments exists to plug into the usual duration formula, so you move the yield up and down and watch what the price actually does. That is effective durationDuration measured by repricing the bond after a small yield shift each way, so that options that change the cash flows are captured.: the percentage price change per unit of yield, measured from the model rather than from a cash flow list.

What does the lopsided move say about convexity?

For an ordinary bond the price curve bends upward, so a fall in yields adds more than an equal rise takes away. Here it is the reverse. A smaller gain than loss for the same shock means the price curve bends downward, which is negative convexity. The effective convexity formula makes it a number: 102.40 plus 99.10 less twice 101.20 is minus 0.90, divided by 101.20 times 0.005 squared, about -356. The sign is the point; the size depends on the shock you chose.

Say what it means for the holder. You are short an option to the issuer: you keep the losses when rates rise but hand back most of the gains when rates fall. The extra yield a callable bond pays over a straight bond is the premium for that option. In the illustrative model behind the picture, the same bond without the call has an effective duration of about 4.5, higher than 3.26, because the call shortens the bond's expected life when yields drop.

For 2 litres: fill the 3, pour it into the 4, fill the 3 again and top up the 4; exactly 2 litres stay in the 3-litre bottle. For 5: fill the 4, pour into the 3 to leave 1, empty the 3, move the 1 into it, then fill the 4, for 1 plus 4. Every whole amount from 1 to 7 is possible, because 3 and 4 differ by 1.

What moves are you actually allowed?

Three: fill a bottle to the top, empty it, or pour from one into the other until the first is empty or the second is full. The only amounts you can know for certain are full bottles and what is left after a pour stops at a full bottle, so every measurement is built from 3s and 4s. Think of it as making change with only Rs 3 and Rs 4 coins, where you are also allowed to hand coins back.

Why can you reach every amount from 1 to 7?

Because 4 minus 3 is 1, and once you can make 1 you can make anything by adding bottles. The amounts you can measure are exactly the combinations of 3 and 4 that fit in the bottles, and since the two sizes share no common factor, every whole litre up to their total of 7 is reachable. One litre: fill the 4 and pour into the 3. Three and four: fill one bottle. Five: 1 plus a full 4. Six: 3 in each. Seven: both full.

The same rule tells you when a puzzle has no answer. With a 4-litre and a 6-litre bottle, every amount you can make is even, so 5 litres is impossible, and you can say so without trying a single pour. Interviewers like that sentence more than the pouring itself, because it shows you found the structure rather than a lucky sequence.

A naive scan takes about n K (K minus 1) comparisons; a min-heap takes about n K log2 K. For 64 investors with 1,000 orders each, that is about 4.03 million against 0.384 million, roughly 10 times fewer. The heap holds only each list's current best order, so finding the next order costs a few steps down one branch rather than a look at every list.

What is the naive way, and where does it waste effort?

Imagine 64 queues at a bank, each already in order of arrival, and you must call people one at a time in overall order. The naive clerk walks along all 64 queue fronts every time to find the earliest. Every time one order leaves the book, the naive merge re-compares all K front orders, even though only one of them changed. That is K minus 1 comparisons for each of n K orders: 64,000 orders times 63 is 4,032,000 comparisons.

The other naive route is to merge lists one at a time: merge list 1 and 2, then merge in list 3, and so on. Each merge re-reads everything merged so far, which costs about n times K squared over 2, here about 2.08 million. Better than scanning, but it still grows with the square of K.

Why does a heap fix it?

A min-heapA tree in which every parent is smaller than its children, so the smallest item is always at the top and can be removed and replaced in a number of steps equal to the tree height. keeps the K front orders only partly sorted: the best is always at the top, and the rest are arranged so that fixing the tree after a change touches one path from top to bottom. Taking the best order and inserting that investor's next one costs about log2 K comparisons instead of K, which is 6 instead of 63 at K of 64. Total work becomes n K log2 K, about 384,000 comparisons.

Say where the heap does not matter. With four or five lists, scanning is about as fast and simpler to code, and the orders arrive as fast as a person can read them anyway. The heap earns its place when K is large or the lists do not fit in memory, which is the version in the reported question: arrays read from disk, where only the front of each list is held at once. A careful heap counts about two comparisons per level on the way down, so treat log2 K as the order of the cost, not an exact count.

WACC is 8.0% before tax and 7.25% after. Read the cost of equity as the earnings yield, 1 over a P/E of 10, which is 10%. Half the capital costs 10% and half costs 6%, so the blend is 8.0%. The tax shield cuts debt to 6% x 0.75 = 4.5%, and the blend falls to 7.25%. The P/E shortcut assumes no growth; with growth the true cost of equity is higher.

Where does the cost of equity come from when you are not given a beta?

If a shop earns Rs 10,000 a year and sells for Rs 1 lakh, a buyer earns 10% on the price. Nobody would pay more unless they expected growth, so 10% is roughly what buyers of such a shop demand. A P/E of 10 means earnings are a tenth of the price, an earnings yield of 10%, and with no growth that is a quick proxy for the cost of equity. Say out loud that it is a proxy: it is the fastest honest route when the interviewer gives you only a multiple.

Why does tax only touch the debt half?

Interest is paid before tax and dividends after it. Every rupee of interest reduces taxable profit, so the government in effect pays a quarter of the interest bill at a 25% tax rate, and debt that costs 6% on paper costs 4.5% to the company. Equity gets no such relief. That is why the after-tax WACC is 0.75 points lower: half the capital times the 1.5 point shield. The difference compounds into valuation, since a lower discount rate raises the value of every future cash flow.

State the limit of the shortcut before the interviewer does. The earnings yield equals the cost of equity only for a company that does not grow and pays out all its earnings. If earnings grow at 3% a year, investors paying 10x are expecting roughly 10% plus 3%, about 13%, and the WACC rises with it. For a real company you would build the cost of equity from a risk-free rate, a beta and a market premium instead.

The last passenger gets their own seat with probability exactly 1/2. Every random choice picks seat 1, seat 100, or another seat that just passes the problem on. Seat 1 and seat 100 are always equally likely, and whichever goes first decides it. For passenger k, the same argument over the seats that still matter gives (n minus k plus 1) over (n minus k plus 2): 99 in 100 for passenger 2, falling to 1/2 for the last.

Which seats actually matter to the last passenger?

Think of a game of musical chairs where every displaced person grabs a random empty chair. It looks like a mess, but most grabs only move the mess along to someone else. For the last passenger, only two seats matter: seat 1, the drunk's own, and seat 100, their own. If a displaced passenger takes seat 1, the chain stops and everyone after sits correctly. If someone takes seat 100, the last passenger loses. Any other seat hands the same situation to a later passenger.

How do you get the answer for any passenger?

Apply the same logic to passenger k. For them, the deciding seats are seat 1 and the seats of passengers k to n, which are still unclaimed by their owners when the chain reaches them. Passenger k loses only if their own seat is picked before seat 1, and among the n minus k plus 2 seats that matter, only one of them is theirs. The chance of the chain ending well for them is therefore n minus k plus 1 over n minus k plus 2.

Check the ends of the formula, which is what an interviewer will do. Passenger 2 loses only if the drunk sits in seat 2, a 1 in 100 chance, so 99 in 100 is right. Passenger 99 has a 2 in 3 chance, and passenger 100 has 1 in 2. The answer does not depend on the size of the plane for the last passenger: with 10 seats or 1,000 it is still one half, which is the fact worth saying out loud.

At the end of year 5 the borrower owes Rs 121.67 crore, the lender has received Rs 43.33 crore of cash interest, and the IRR at par is exactly 12%. The 4% PIK is added to the balance each year, so it grows to 100 x 1.04 to the fifth. Cash interest is 8% of each opening balance, rising from Rs 8.00 crore to Rs 9.36 crore. Both pieces earn on the full balance, so the lender earns 12% a year.

What does PIK actually do to the balance?

Think of a friend who borrows from you, pays part of the interest in cash each year, and says: add the rest to what I owe. Next year you charge interest on the bigger amount. PIKPayment in kind: interest settled by adding it to the amount owed instead of paying it in cash. interest is not paid; it is added to the loan, so the balance grows every year and every later charge is worked out on the bigger number. Here the balance grows 4% a year, from Rs 100 crore to 100 x 1.04 to the fifth, Rs 121.67 crore.

Why is the IRR exactly 12% when only 8% arrives in cash?

Each year the lender earns 12% on the whole balance: 8% arrives as cash and 4% is added to the balance, which then earns 12% itself. A lender earning 12% a year on every rupee outstanding, and repaid in full, has an IRR of 12%. The cash yield on the original Rs 100 crore starts at 8% and reaches 9.36% in year five, because the cash coupon is charged on the accreted balance.

What is the catch the interviewer wants you to name?

A cash-plus-PIK loan earns the same 12% on paper as a 12% cash loan, but more of the return waits until year five. The lender collects Rs 43.33 crore in cash along the way against Rs 60 crore from a 12% cash-pay loan, and Rs 121.67 crore rides on the final repayment instead of Rs 100 crore. If the borrower defaults in year four, the PIK accrued so far is just a larger claim in the recovery, not cash in hand. That is why PIK paper is priced wider: the 12% holds only if the borrower pays at the end.

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.

Start around 10.5, say 9.5 bid and 11.5 offered, then move to about 12.2 bid, 13.8 offered once the 6 is shown. Each fair die is worth 3.5 on average, so three are worth 10.5. After the reveal the value is 6 plus 3.5 plus 3.5, which is 13.0. The market should also narrow, because one die's uncertainty has gone: the standard deviation falls from 2.96 to 2.42, so the width shrinks by about a fifth, not a third.

Where does the first quote come from?

A shop that buys and sells used phones offers to buy below what it thinks a phone is worth and to sell above it, and earns the gap. A market maker quotes around the expected value, with a bid below and an offer above, and the width reflects how uncertain that value is. Each fair die averages 3.5, so three dice average 10.5. A quote of 9.5 bid, 11.5 offered straddles that value; the sum can still land anywhere from 3 to 18, so the market cannot be tight.

What changes when the first die is shown?

Two things, and candidates usually say only one. The expected value jumps to 6 plus 3.5 plus 3.5, which is 13.0, and the uncertainty shrinks because only two dice are still hidden. Variance adds across independent dice, 35/12 for each, so it falls from 8.75 to 5.83, a third lower; the standard deviation falls from 2.96 to 2.42, about 18% lower. Scale the width by that: 2 points becomes about 1.6, so quote roughly 12.2 bid, 13.8 offered.

What does the interviewer want to hear beyond the numbers?

Move the market the moment information arrives, because a stale quote is a free option for everyone else at the table. If you stay at 9.5 to 11.5 after a 6 shows, every player buys your offer at 11.5 against a fair value of 13.0. Then say what you would do if you suspected the other side knew more than you, for example had seen a second die: widen, or lean your quote towards the risk. The limit: the width here is a choice; a real desk sets it by competition and by how much risk it can hold, not by a formula.

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.