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.

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.

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 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.