Debt Capital Markets puzzles, solved step by step

About Rs 296 crore. A 1.3x cover means only 60 divided by 1.3, Rs 46.15 crore a year, can go to interest and principal. A level payment of that size for 10 years at 9% repays a loan equal to its present value: 46.15 times the annuity factor of 6.418, about Rs 296.2 crore. Debt capacity is the present value of the cash the lender lets you spend on debt.

What does the cover ratio actually limit?

Think of a family with Rs 60,000 a month to spare applying for a home loan. The bank will not let the whole Rs 60,000 go to the EMI; it wants a cushion in case a month goes wrong. A debt service cover of 1.3x means every rupee of loan payment must be backed by 1.3 rupees of cash, so the most the project can pay each year is 60 divided by 1.3, Rs 46.15 crore. The remaining Rs 13.85 crore is the lender's buffer against a bad year. It is not money you can borrow against.

How do you turn an allowed payment into a loan amount?

The loan is whatever amount those payments exactly repay with interest. A loan is the present value of the payments that service it, discounted at the loan's own rate. Ten level payments of Rs 46.15 crore at 9% are worth 46.15 times the ten year annuity factorThe present value of receiving 1 a year for a set number of years at a given rate. For 10 years at 9% it is about 6.42.. Look at the green bars: year 1's payment is worth Rs 42.3 crore today, year 10's only Rs 19.5 crore, because money further away is worth less now.

Check it out loud. Rs 296 crore at 9% is Rs 26.7 crore of first year interest, so the first payment of Rs 46.15 crore repays only about Rs 19.5 crore of principal. With a level payment, principal is repaid slowly at first and faster later, because interest takes the biggest bite while the balance is largest. If you can say that, the interviewer knows you understand what the annuity factor is doing rather than reciting it.

What would a lender push back on?

The Rs 60 crore is a forecast, and a forecast is exactly what a sponsor is paid to be optimistic about. The cover protects the lender only if the cash flow it is applied to is honest, so the real negotiation is over which forecast the 1.3x is applied to. Lenders often size on a downside case, shorten the tenor, or ask for a reserve account that holds some months of debt service in cash. Each of those moves the answer more than a quarter point on the rate would. State the limit too: the sum assumes the project runs all ten years and that nothing needs refinancing.

About 6% a year, forever. In a steady growth model the required return equals the dividend yield plus growth. The yield is 12 divided by 240, 5%, so growth must be 11% minus 5%, which is 6%. For a lender the useful point is fragility: if the market cut its growth view to 4%, the same formula gives a price of Rs 171, and the equity cushion under the debt shrinks by 29%.

How can a price contain a growth forecast?

Think of a shop rented out for Rs 12,000 a year that sells for Rs 2,40,000. A buyer who wants 11% a year gets only 5% from the rent, so the price only makes sense if the buyer expects the rent to rise about 6% a year. A price paid today is a bet on future cash, so you can run the valuation backwards and read off the growth the buyer has assumed. The Gordon growth modelA valuation for a cash flow that grows at a constant rate forever: price equals next year cash flow divided by the required return minus growth. does exactly that for a steady dividend payer.

Why should a lender care what the equity market assumes?

Because equity is the cushion that absorbs losses before the lender does. When a price rests on a growth assumption, a small change in that assumption moves the price a lot, so the cushion is thinner than the market value suggests. Here two points less growth takes the price from Rs 240 to Rs 171 without the dividend changing at all. A lender reading leverage on market value should also look at it on a more conservative growth view.

There is a second reading. Rs 12 a share paid out and growing 6% a year is cash leaving the company ahead of any debt repayment. A lender who sees a rich implied growth rate and a generous, rising dividend asks whether the business can fund both, and may want a covenant that limits payouts if leverage rises. State the limit too: the model assumes one growth rate forever and a cost of equity that is itself an estimate, so treat 6% as the market's rough view, not a forecast.

On my assumptions, roughly Rs 175 crore to Rs 1,080 crore a year, with a base case near Rs 440 crore. Base case: 1,100 issues a year at an average Rs 400 crore is about Rs 4.4 lakh crore of issuance, and a 10 basis point fee on that is Rs 440 crore. The fee rate drives the range far more than volume does.

How do you structure it before any number?

Say the formula first: issues a year, times average size, times the fee as a share of the amount raised. A fee pool is a small percentage of a large volume, so the rate you apply matters more than how precisely you count the volume. Think of a wedding planner's income: the number of weddings and the average budget are easy to guess roughly, but whether the planner takes 2% or 10% of the budget changes the answer five times over.

Where do the assumptions come from?

Each should be defended in one line and flagged as an assumption to check. The issue count and size here are guesses for a bond market where banks, finance companies and public sector issuers place large issues with institutions; public issuance data and league tables would replace them in real work. The fee rate is the least observable input, because arranger fees on privately placed, top rated issues can be a few basis points while complex or lower rated deals pay much more. So the range runs from 5 to 20 basis points around a 10 basis point base.

What do you say after the range?

Test it against something you know. A pool of a few hundred crore rupees shared across many arrangers means each bank's domestic bond fee income is modest, which is why desks care about volume, league table rank and the other products a mandate brings. Then name what would change the view: a shift from private placements to larger public issues, or more lower rated issuance, would push the average fee up. Say clearly that every figure here is an illustrative assumption for the method, not a market statistic.

Principal of about Rs 115.76 and a year 3 coupon of about Rs 2.32. The principal is lifted by inflation each year and compounds: 100 times 1.05 cubed is Rs 115.7625. The 2% real coupon is paid on that indexed principal, so the year 3 coupon is Rs 2.315. Both the coupon and the principal keep their buying power, which is what the investor is paying for.

What exactly is being indexed?

Think of a rent agreement where the rent rises with inflation every year and the deposit is topped up by the same percentage. On an indexed bond the principal is scaled up by inflation, and the fixed real coupon rate is then applied to that scaled principal, so both the income and the repayment keep pace with prices. After one year of 5% inflation the principal is Rs 105 and the coupon is 2% of 105, Rs 2.10, not Rs 2.

Why does the principal compound instead of adding 5 a year?

Because the index is a price level, and price levels compound: 5% in year 2 is 5% of the already higher year 1 prices. The indexed principal is the original principal times the ratio of today's index to the index at issue, which after three years of 5% is 1.05 cubed. Adding 5 a year gives 115, off by Rs 0.76, small here but large over a 10 or 20 year bond.

What does that mean for the yield an investor sees?

Bought at Rs 100, the cash flows 2.10, 2.205 and 118.08 give a money return of about 7.1% a year, which is 1.02 times 1.05 minus 1: the real 2% plus inflation plus a small cross term. The investor has locked in a real return of 2% whatever inflation turns out to be, and that certainty is the product. Say the limits: actual indexed bonds use a lagged index, some protect principal from falling below par, and the tax treatment of the uplift varies, so check the specific bond's terms.

About Rs 490.2 crore of face value, recorded initially at the Rs 500 crore received. At 102, every Rs 100 of face raises Rs 102, so 500 divided by 1.02 is Rs 490.20 crore. The issuer carries the debt at the cash received and amortises the Rs 9.8 crore premium down to face over the remaining life, so its interest expense sits below the Rs 39.22 crore coupon.

Why does a premium mean less face, not more?

Think of selling gift vouchers with a face value of Rs 100 that are so popular buyers pay Rs 102 for each. To collect Rs 50,000 you need to hand out fewer vouchers than 500. When a bond trades above par, each unit of face value brings in more than its face in cash, so the issuer creates less face than the cash it raises. The bond trades at 102 because its 8% coupon is above the yield investors now demand; the extra Rs 2 is them paying today for that above-market coupon.

What goes on the balance sheet?

Under amortised cost accounting, which Ind AS and IFRS use for most issued bonds, the debt starts at the cash received, Rs 500 crore before issue costs, not at the face value. The premium is then released over the bond's remaining life, so the carrying amount falls to Rs 490.2 crore by maturity and the interest expense is the effective yield on the carrying amount, below the cash coupon. Covenants that test debt may use face value instead, so check which number the documents count; the accounting details belong to the issuer's auditors.

What else must you check on a tap?

Three practical points. Taps usually settle between coupon dates, so buyers also pay accrued interest, which is cash in hand but not new debt, and it should not be counted in the Rs 500 crore. Face is issued in set denominations, so Rs 490.2 crore rounds to what the minimum lot allows. And the issuer should compare the tap with a fresh issue: tapping an existing line at 102 adds liquidity to one bond, which investors often value, while a new bond with a lower coupon might price differently.

About 9.6%. The chance of surviving one year is 98%, and surviving five independent years is 0.98 to the power 5, or 90.39%. Default at some point is everything else: 9.61%. It is below 10% because the 2% in each later year applies only to issuers still alive, a shrinking group, so simply adding the yearly rates double counts.

Why does adding 2% five times overstate it?

Think of a phone that has a 2% chance of breaking each year. A phone that broke in year 1 cannot break again in year 2. Default is a one-time event, so each year's 2% applies only to the issuers that survived until then, and that group shrinks every year. Seen from today, the chance of defaulting in year 2 is 98% times 2%, which is 1.96%; by year 5 it is 1.845%. Those five numbers sum to 9.61%, not 10%.

What is the fastest correct route?

Go through survival. Survival over several independent years is the product of the one year survival rates, and cumulative default is one minus that product. 0.98 squared is 0.9604, times 0.98 again is 0.9412, then 0.9224, then 0.9039. Or use the shortcut that 0.98 to the fifth is about 1 minus 5 times 0.02 plus 10 times 0.0004, which is 0.904. Either way, default is about 9.6%.

Why does a credit desk care about a 0.4 point gap?

Over five years the gap is small, 10.0% against 9.6%. Over long horizons the difference becomes enormous: adding 2% a year says default is certain by year 50, while compounding survival says 63.6%. That matters for pricing long bonds, for reading a rating agency's cumulative default tables, and for turning a spread into an implied default rate. Say the limit too: real default rates are not independent from year to year, they cluster in recessions, so the 2% flat rate is a teaching simplification.

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.

A total rise of about 15.9%, or about 5.05% a year. Percentage changes multiply: 1.12 times 1.15 is 1.288, and a 10% fall from there leaves 1.1592. The steady yearly rate that gets to the same place is the cube root of 1.1592, about 1.0505. Adding the rates gives 17% and an average of 5.67%, both too high, because the fall hits a bigger base than the gains did.

Why can you not add the percentages?

A shop raises a price by 10% and later cuts it by 10%, and the tag ends below where it started, because the cut is taken on the higher price. Each percentage change is measured on the level it starts from, so a chain of changes multiplies rather than adds. Here the 10% fall comes after two good years, so it removes 12.88 points of the index, more than the 12 the first year added.

How do you do it fast under test conditions?

Break each multiplication into easy pieces. 1.12 times 1.15 is 1.12 plus 15% of 1.12, which is 1.12 plus 0.168, so 1.288; taking 10% off is 1.288 minus 0.1288, which is 1.1592. For the yearly rate, test a round guess: 1.05 cubed is 1.1025 times 1.05, which is 1.1576, just under 1.1592, so the answer is a shade above 5%. On a timed test, that is enough to pick the right option in seconds.

Why does the average of 5.67% overstate growth?

The simple average of 12, 15 and minus 10 is 5.67%, but three years at 5.67% would give 18.0%, not 15.9%. Whenever the yearly rates vary, the simple average is above the compound rate, and the gap grows with the swings. That is the same drag that makes a volatile bond fund's average return look better than what its investors actually earned. Say which average you are quoting: the compound rate describes the path, the simple average only describes the list of numbers.

105.15 today, 103.57 after one year and 101.85 after two, then 100 at maturity. The bond pays 10 a year when the market wants 8, so it trades at a premium, and the premium shrinks as the high coupons are used up. Each year the holder gets 10 of coupon but gives back part of the premium, so true income is 8.41 in year 1, exactly 8% of the 105.15 paid.

Why is the bond above 100 in the first place?

Imagine a flat rented at Rs 10,000 a month on a three year lease when similar flats rent for Rs 8,000. A buyer pays extra for that lease, but the extra is only worth the months left on it. A bond paying a coupon above the market yield trades above par, and the premium is the present value of those extra coupons still to come. Priced at 8%, three coupons of 10 and the 100 repayment are worth 105.15.

How do you get the three prices quickly?

Price the premium, not the bond. Each year the bond pays 2 more than the market rate would, so the premium is 2 times the annuity factor for the years left, at 8%. With three years left the factor is 2.577, so the premium is 5.15; with two years left it is 1.783, a premium of 3.57; with one year left, 0.926, a premium of 1.85. That is quicker than discounting every cash flow and makes the pull to par obvious.

Why does it matter that income is less than the coupon?

Because the coupon overstates what the holder earns. Of the 10 received in year 1, 1.59 is really the holder's own money coming back as the premium runs off, so the income is 8.41, exactly the 8% yield on the price paid. A bank or fund that booked the full 10 as income would show a loss of the same size in the price. This is why accounts amortise premiums and why a desk compares bonds on yield, not coupon. The limit: it all assumes the yield stays at 8%, and any move in rates adds a gain or loss on top.

The bank recovers about Rs 304.5 crore, 76.1%; the bondholders get about Rs 109.1 crore, 36.4%. The bank takes the plant, Rs 250 crore, and is still owed Rs 150 crore. That shortfall ranks as an unsecured claim beside Rs 300 crore of bonds and Rs 100 crore of trade claims. The Rs 200 crore of other assets is shared across Rs 550 crore of claims at 36.36% each.

What does security actually give the bank?

A pawnbroker holding your watch gets the watch first. If the watch sells for less than you owe, you still owe the rest, but now as an ordinary debt. Security gives a lender first call on the pledged asset, up to its value; for any shortfall the lender becomes an ordinary unsecured creditor. Here the plant covers Rs 250 crore of the Rs 400 crore loan, so the bank is undersecuredOwed more than the value of the collateral pledged against the loan, so part of the claim is effectively unsecured. by Rs 150 crore.

How is the unsecured pool shared?

Pro rata, by the size of each claim. All unsecured claims of the same rank share the unencumbered assets in proportion to what they are owed, and the bank's deficiency counts in full. The pool is 150 plus 300 plus 100, which is Rs 550 crore, against Rs 200 crore of assets, a recovery of 36.36%. The bank receives 36.36% of 150, Rs 54.5 crore, the bonds Rs 109.1 crore and trade creditors Rs 36.4 crore.

What would change the numbers in a real case?

Several things the puzzle strips out, and naming one or two shows judgement. Insolvency costs and any claims the law ranks ahead of unsecured creditors, such as some employee and statutory dues, come out of the pool first, so the real unsecured rate would be lower than 36.4%. Plant values in a forced sale are often below appraisal. And the priority rules differ by country and by process, so check the regime that applies. The core mechanics, collateral first and deficiency pari passu, hold widely.