Investment Banking puzzles, solved step by step

About Rs 92.4 per Rs 100 of face value. Discount the five Rs 8 coupons and the Rs 100 repayment at 10%: the coupons are worth Rs 30.33 today and the repayment Rs 62.09. It sits below par because it pays Rs 2 a year less than a new bond, and a buyer wants those five years of shortfall, worth Rs 7.58 today, taken off the price.

How do you price it from first principles?

Imagine you let a flat on a five-year lease at Rs 8,000 a month, and every similar flat now fetches Rs 10,000. Someone buying your lease will pay less for it, by roughly the rent you miss over the remaining term. A bond is the same lease on money. A bond's price is every future payment discounted at the yield the market now demands, so when that yield rises above the coupon, the price has to fall below par. At 10%, each Rs 8 coupon and the final Rs 100 are worth less today the further away they sit.

Is there a shortcut that skips most of the discounting?

Yes, and it is the one to use out loud. Price equals par minus the yearly coupon shortfall times the annuity factor: 100 minus 2 x 3.79 is 92.42. The annuity factorThe present value of Rs 1 a year for a set number of years. At 10% for five years it is about 3.79. is the only number that needs any work, and at 10% for five years it is close to 3.8. The same rule runs the other way: a bond paying above the market yield trades above par by the extra coupon times the annuity factor.

Two limits are worth one sentence each. The yield must come from bonds of the same maturity and credit quality, which is what similar bonds means; in today's market you would read it off comparable issues. Real bonds also pay coupons twice a year in many markets and trade with accrued interest, which shifts the answer by small amounts. For an interview answer, annual discounting is the expected level.

About Rs 8. With near-zero rates, an at-the-money call is worth roughly 0.4 times the price times the volatility times the square root of time: 0.4 x 100 x 0.20 x 1 = 8.0. The full Black-Scholes formula gives Rs 7.97. The intuition: the option collects the upside of a bell-shaped spread of outcomes and nothing on the downside, and the average of that upside half is about 0.4 of one standard deviation.

What is Black-Scholes actually doing?

Picture a ticket that pays you whatever a stock gains over the next year and nothing if it falls. You would pay something for it, even though an at-the-moneyAn option whose strike price equals the current stock price, so exercising it today would pay nothing. ticket is worth nothing if cashed in today. Black-Scholes prices that ticket as the average payoff across all the ways the stock could move, with the spread of those moves set by volatility and time. Its inputs are the stock price, strike, time, volatility and interest rate, and with zero rates the formula reduces to S times N(d1) minus K times N(d2), where the two N terms come from the bell curve.

Where does the 0.4 come from?

Over a year at 20% volatility, the stock's likely outcomes form a bell curve around Rs 100 with a standard deviation of about Rs 20. The call keeps the right half and treats the left half as zero. The average of the positive half of a bell curve, counting the negative half as zero, is the standard deviation divided by the square root of 2 pi, about 0.4 of it. So the call is worth about 0.4 x Rs 20, which is Rs 8. Halve the time to six months and the spread shrinks by the square root of 2, so the option is worth about Rs 5.6, not Rs 4.

What does the shortcut leave out?

It holds for at-the-money options with modest volatility and low rates. Away from the strike, the payoff is no longer the clean right half of the curve, and with meaningful rates the forward price drifts above the spot, which raises a call's value. The rule is a sanity check on a quoted price, not a replacement for the formula. Black-Scholes itself assumes constant volatility and smooth price moves, which is why traders quote options in implied volatility and adjust for the skew they see in the market. Doubling volatility to 40% roughly doubles this call's value, to about Rs 15.9.

The lottery ticket has a beta of zero, and the half-and-half portfolio has a beta of 0.5. Beta is the covariance of an asset's return with the market's return divided by the market's variance. A lottery draw does not move with the index at all, so its covariance is zero, however wild its payoff. Betas combine in proportion to weights, so half an index fund at beta 1 and half tickets at beta 0 give 0.5 x 1 + 0.5 x 0 = 0.5.

Why is a wildly risky asset given a beta of zero?

Think of two shops on one street. One sells umbrellas, whose takings rise and fall with the weather; the other sells birthday cards, whose takings have nothing to do with it. The card shop's sales swing plenty, but not with the rain. Beta measures only the part of an asset's movement that follows the market, so an asset whose payoff is decided by a draw of numbered balls has a beta of zero no matter how large its swings. On a Rs 10 ticket with an assumed one in a crore chance of a Rs 5 crore prize, the standard deviation of the return is about 1,600 times the stake and the beta is still zero. Volatility is total riskThe full spread of an asset's returns, usually measured by standard deviation, including both the part linked to the market and the part unique to the asset.; beta is only its market-linked part.

How does the portfolio beta follow?

Covariance is linear in the weights, so the beta of a portfolio is the weighted average of the betas inside it. An index fund tracking the market has a beta of one by construction. Half at beta 1 and half at beta 0 gives 0.5, the same beta as a portfolio that is half the index and half cash. That comparison is the point: in the capital asset pricing model the lottery half and a cash half demand the same return, the risk-free rate, because neither adds market risk; the lottery's enormous extra risk is diversifiable in theory and simply unrewarded.

What does the security market line say about the blend?

Plot required return against beta. With an assumed 6% risk-free rate and 6% equity premium, the line runs from 6% at beta 0 to 12% at beta 1, and the blend at beta 0.5 would be required to earn 9%. Its expected return is nothing like that: if a ticket pays out an assumed half of its price in prizes, its expected return is -50%, and the blend's is 0.5 x 12% plus 0.5 x -50%, which is -19%. Beta tells you where an asset sits along the line, and the lottery ticket sits about 56 points below it: a zero-beta asset with an appalling expected return. That is the limitation to state: beta says nothing about expected return outside the model's assumptions, and nothing about risk the market does not share.

Why a banker asks it: the same logic decides what goes into a cost of equity. A project whose cash flows do not move with the economy, a regulated pipeline or a contracted toll road, carries a low beta and a low cost of equity even if its engineering risk is large, because that risk is specific and diversifiable. Candidates who confuse volatility with beta tend to over-discount exactly those assets, and to under-discount a cyclical business that happens to look calm.

Your personal beta is 0.8. Salary does not move with the market, a beta of 0; the bonus moves twice as much, a beta of 2. Weighted by share of pay, 0.6 x 0 plus 0.4 x 2 gives 0.8. If the market falls 20%, the bonus falls 40% and total pay falls 16%. Your earnings already carry most of the market risk of an index fund before you invest a rupee of savings.

How do you get a beta for a person?

Think of a household with two earners, a schoolteacher on a fixed salary and a salesperson paid on commission. In a downturn the teacher's pay holds while the commission falls sharply, and the household's income falls by a blend of the two. The beta of a mix is the weighted average of the betas of its parts, so a pay packet's beta is each component's beta weighted by its share of pay. Salary has a beta of 0 and the bonus a beta of 2, so the blend is 0.8.

What does 0.8 look like in rupees?

Take pay of Rs 25 lakh: Rs 15 lakh of salary and Rs 10 lakh of bonus. If the market falls 20%, the bonus falls 40% to Rs 6 lakh, and total pay falls from Rs 25 lakh to Rs 21 lakh, a 16% drop, which is 0.8 times the market's fall. The salary cushions the swing, but the bonus is exactly the part that shrinks in the years the market is weakest.

Why does it matter for your savings?

Your future earnings are an asset, often the largest one you own early in a career, and finance calls it human capitalThe value today of the earnings a person expects to make over the rest of a working life.. If that asset already has a beta of 0.8, your total wealth carries market risk before your savings add any, and the worst years for markets are the years your bonus, and perhaps your job, are most at risk. This is a way of counting risk, not a recommendation for anyone's savings: the point the interviewer wants is that risk is measured across everything you own, including your pay.

Roughly 34,050 to 41,950, with about two-thirds confidence. Four months is a third of a year, and volatility scales with the square root of time, so 18% a year becomes 18% times 0.577, about 10.4%, over four months. One standard deviation is about 3,949 points either side of 38,000. If returns are roughly normal, about 68% of outcomes land inside that band and about 95% inside 30,100 to 45,900. Give the range and the confidence, never a single number.

Why does volatility shrink with the square root of time?

Picture someone leaving a party and taking steps at random, forward or back. After four steps they are not four steps from the door; some steps cancelled, and on average they are about two away. Random moves partly cancel, so the spread of where you end up grows with the square root of the number of moves, not with the number itself. Four months is a third of a year, the square root of a third is 0.577, and 18% times 0.577 is 10.4%. Dividing 18% by three to get 6% would be right only if every month moved the same way, which is the opposite of random.

How do you turn 10.4% into a range?

10.4% of 38,000 is about 3,949 points. One standard deviationA measure of how far outcomes typically land from the centre. Under a normal distribution about 68% of outcomes fall within one of the centre and 95% within two. either side is 34,050 to 41,950; two is 30,100 to 45,900. A one standard deviation band is a two-in-three bet, so the confidence attached to the width is the answer, and a range without one is just a guess with margins. Say the band, say the two in three, and offer the wider nineteen-in-twenty band if the interviewer wants more certainty.

What do you add to sound like a trader rather than a textbook?

Three refinements, each one sentence. Returns compound, so the band is slightly lopsided: 38,000 grown and shrunk by 10.4% continuously is about 34,250 to 42,150, a little more room on the upside. The expected drift over four months is a percent or two, small next to 10.4%, so ignoring it is fair. Real markets have fatter tails than the bell curve, so the two standard deviation band gets breached more often than one time in twenty. Say the range, say the confidence, then say what would make you wrong: the 18% itself is a historical figure, and the volatility the options market implies for the next four months may be higher or lower. This is a way of describing uncertainty, not a forecast, and the interviewer is testing whether you think in distributions.

The price falls by about 3.5%. Modified duration says how many per cent the price moves for a one-percentage-point move in yield, in the opposite direction. A 50 basis point rise is half a point, so 7 x 0.5 = 3.5% down. On a Rs 100 crore holding that is a loss of about Rs 3.5 crore. The bend in the price curve makes the true fall slightly smaller.

Why do bond prices fall when yields rise?

A bond pays fixed cheques. If new bonds start paying more, nobody will pay the old price for the old, smaller cheques, so the old bond's price drops until its yield matches the market. Think of a flat let at a fixed rent when rents nearby jump: a buyer now pays less for it. Price and yield move in opposite directions, and duration measures how hard the price reacts. A longer bond reacts more, because more of its cash arrives far in the future, where a change in the discount rate bites hardest.

How accurate is the duration answer?

Duration draws a straight line, the tangent, along a price curve that actually bends. For small moves the line and the curve are almost the same, and for large moves the curve sits above the line, so the true loss is smaller than duration says. Take a zero-coupon bond yielding 6% with 7.42 years to run, which has a modified duration of exactly 7. A 50 basis point rise cuts its price by 3.43%, against 3.5% from the shortcut. A 300 basis point rise cuts it by 18.7%, against 21.0% from the line.

Add one sentence that marks you out. The bend is convexityHow much the price curve of a bond bends, which is the change in duration as yields move., and for an ordinary bond it works in the holder's favour both ways: the price falls less than duration predicts when yields rise and gains more than predicted when they fall. For a 50 basis point move the effect is a few hundredths of a per cent, so 3.5% is the answer to give.

About 4%, below the unlevered 7%. Per Rs 100 of property, a 7% cap rate means Rs 7 of net operating income. A 60% loan is Rs 60, and at 9% interest-only it costs Rs 5.4 a year. That leaves Rs 1.6 of cash for the Rs 40 of equity, which is 4%. Because the loan costs more than the property yields, this is negative leverage: borrowing lowers the equity return rather than raising it.

Why is the cash-on-cash below the cap rate?

Suppose you borrow money at 9% to put in a deposit paying 7%. Every rupee borrowed loses you 2 paise a year, and the more you borrow the worse it gets. Leverage lifts the equity return only when the asset yields more than the debt costs; when the cap rate sits below the interest rate, every rupee of debt subtracts from the equity's return. Here the Rs 60 of debt earns 7% inside the property, Rs 4.2, but costs 9%, Rs 5.4, so it drags Rs 1.2 off the Rs 7 that the equity would otherwise keep on Rs 100 unlevered.

How do you say the answer in a way the interviewer can follow?

Fix the value at Rs 100 so every input becomes a rupee figure. Cap rate times value is the income; loan-to-value times value is the debt; debt times the rate is the interest; the rest is cash on the equity, and the equity is value minus debt. That is four multiplications and two subtractions, and saying each one aloud keeps you honest. Then say the comparison that matters: 4% levered against 7% unlevered, so the loan has cost the equity 3 points of yield.

Why would anyone buy with negative leverage?

Because cash-on-cash is a snapshot of year one. A buyer accepting 4% today is betting on rent growth lifting the income, on refinancing when rates fall, or on selling at a lower cap rate, none of which is in this calculation. Say that the figure ignores principal repayment, because the loan is interest-only, and ignores any appreciation. A total return would add both, and a lender would also look at the debt service coverage, which here is 7 over 5.4, about 1.3x, thin for most lenders.

About Rs 5.24. A call plus cash worth the strike's present value pays the larger of the share price and 100 at expiry, and so does a put plus the share. Identical payoffs must cost the same today: 10 + 100/1.05 = P + 100. The present value of 100 is 95.24, so the put is 5.24. No pricing model or volatility estimate is needed.

Why can you price the put without knowing volatility?

If two combo meals contain exactly the same food, a restaurant cannot charge different prices for long: everyone would buy the cheaper one. Options work the same way. Two portfolios that pay the same in every outcome must cost the same today, so if you know the price of one, the other follows without any view on where the share goes. Volatility matters for what a call is worth, but the call's price already contains it. The put inherits it through the parity relationship.

Build the two portfolios. A: buy the call and put aside 95.24 at 5%, which grows to exactly 100. B: buy the put and buy one share. If the share ends at 130, A exercises the call for 30 and adds its 100 cash, B holds a share worth 130 and lets the put expire: both have 130. If the share ends at 70, A lets the call expire and holds 100 cash, B exercises the put to sell the share for 100: both have 100.

What does the formula say, and what if the put traded at Rs 7?

If the put traded at 7, portfolio B would cost 107 against 105.24 for A, so you would sell B and buy A, pocketing 1.76 today with payoffs that cancel at expiry. Selling B means writing the put and short selling the share. Traders arbitraging that gap is what holds parity in place, which is why quoted put and call prices sit close to it in practice.

Say the assumptions: European options, so no early exercise; no dividend before expiry; and annual compounding. With continuous compounding the strike's present value is 100 x e to the minus 0.05, about 95.12, and the put comes out near 5.12. Give the convention, not just the number.

About 84.58 rupees per dollar. Rs 83 invested at 7% grows to 88.81. Converted to 1 dollar and invested at 5% it grows to 1.05 dollars. Both routes are riskless once the forward is agreed, so they must end equal: 1.05 x F = 88.81, giving F of about 84.58. If the quoted forward sat far above that, you could borrow rupees, invest in dollars and sell them forward to lock in a gain.

Why is the forward set by interest rates and not by a forecast?

Imagine two banks next door, one paying 7% on rupee deposits and one paying 5% on dollar deposits, and a counter that promises today to swap your dollars back into rupees in a year at a fixed rate. If that promised rate were 83, everyone would ignore the dollar bank. The forward rate must exactly cancel the interest difference, otherwise one route earns more than the other with no risk, and the gap would be traded away. That rule is called covered interest parity, covered because the forward removes the currency risk.

What would you actually do if the forward were quoted at 86?

At 86, dollars are expensive in the forward market relative to parity, so sell dollars forward and build the dollars cheaply today. Borrow Rs 83 at 7%, owing 88.81 in a year. Convert to 1 dollar at spot and deposit it at 5%, which grows to 1.05 dollars. Sell those 1.05 dollars forward at 86 for Rs 90.30. After repaying the rupee loan you keep about Rs 1.49 per dollar, without taking any view on where the rupee goes. Traders doing exactly this pushes the quote back toward 84.58.

The forward rupee is about 1.9% weaker than spot, roughly the 2 point rate difference. A quick approximation is spot times one plus the difference, 83 x 1.02 = 84.66, close enough to sanity-check. The limitation: in practice, capital controls, transaction costs and limits on who can borrow in each currency let small gaps persist, so the arbitrage is cleaner on paper than on a dealing desk.