Portfolio Management puzzles, solved step by step

Macaulay duration about 2.83 years and modified duration about 2.67. At par the yield equals the 6% coupon, so the cash flows discount to 5.66, 5.34 and 89.00, which add to 100. Weight each year by its share of the price: 1 x 0.0566 + 2 x 0.0534 + 3 x 0.8900 gives 2.83. Divide by 1.06 for modified duration: a one point rise in yield cuts the price by roughly 2.67%.

What is duration, if not the time to maturity?

Picture a seesaw with three children sitting at the one, two and three metre marks. If the child at three metres is much heavier, the pivot that balances the seesaw sits close to three, not at the middle. Macaulay duration is that pivot: the average time you wait for your money, with each payment weighted by its present value. Here the three weights are the discounted coupons of 5.66 and 5.34 and the discounted final payment of 89.00. The last payment is so heavy that the balance point, 2.83 years, sits only two months short of maturity.

Why divide by 1.06 to get modified duration?

Macaulay duration is a time. Modified duration is a price sensitivity: the percentage change in price for a one point change in yield. With annual compounding the two differ by a factor of one plus the yield, so 2.833 divided by 1.06 is 2.673. Say what it means: if yields rise from 6% to 7%, the bond loses about 2.67% of its price, a little less in reality because the price-yield curve bends. That bend is convexity, and it is the natural next question.

About 800 tonnes a year, on stated assumptions. Weddings: about 1 crore a year at 35 grams each is 350 tonnes. Festivals and gifting: 30 crore households, a quarter buying 4 grams, is 300 tonnes. Investment coins and bars: 5% of households buying 10 grams is 150 tonnes. The total is 800 tonnes. Check it against published industry demand figures before using it anywhere.

How do you start when you have no idea of the answer?

Ask when a family actually walks into a jeweller. It is not random: a wedding, Dhanteras or Akshaya Tritiya, a birth, or a decision to save in coins. Splitting a total into the occasions that drive it turns one number nobody knows into several that anyone can picture. State the base first: about 140 crore people at four to five per household is roughly 30 crore households. That figure and every weight below are assumptions for the estimate, said out loud so the interviewer can challenge them.

How do you sanity check it, and where is it weakest?

Divide back: 800 tonnes over 30 crore households is about 2.7 grams per household per year, a small coin's worth on average, which feels plausible given that most households buy nothing in a typical year and a few buy a lot at a wedding. The weakest input is grams per wedding, because the spread is huge and the average is dragged up by a few large weddings. Moving it by 10 grams moves the total by 100 tonnes. Say that, and say you would check the result against published demand data from the industry before quoting it: an estimate is a structure plus assumptions, not a statistic.

On a multi-asset or wealth desk, the follow-up is usually why it matters: household gold is a large part of Indian savings, so its demand affects imports, the rupee and how much money is left for financial assets.

5 with one trade and 7 with any number. With one trade, buy at the lowest price that comes before a higher one: buy at 1 on day 2 and sell at 6 on day 5, for 5. With unlimited trades, collect every day-to-day rise and sit out every fall: 1 to 5 earns 4, 3 to 6 earns 3, a total of 7. The general answer is the sum of the positive daily changes.

How do you find the best single trade without checking every pair?

Walk through the prices once, like a shopper who notes the cheapest price seen so far and asks each day how much they would make selling today. The best single trade is the largest gap between today's price and the lowest price seen before today, found in one pass. At day 2 the low is 1. Day 3 offers 5 minus 1, which is 4; day 5 offers 6 minus 1, which is 5; no later day beats it. The trap is subtracting the lowest price from the highest overall: the high of 7 comes before the low of 1, so it cannot be sold after buying.

Why is the unlimited answer just the sum of the rises?

Any profitable trade from a low to a later high can be split into daily steps, and it gains only on the up days inside it while paying for every down day it sits through. With no limit on trades and no shorting, the best strategy holds the stock on every day it rises and nothing on every day it falls, so profit equals the sum of positive daily changes. Here that is 4 plus 3, which is 7. One pass through the prices gives the answer, which is the point of asking a coding-flavoured candidate.

Say the limitation as a portfolio manager would: this is perfect hindsight with no costs. Add a transaction cost per trade and the two answers move toward each other, because catching a small rise is no longer worth paying for.

Only 20%. Take 1,000 applicants. 40 will default and the model flags 36 of them. 960 are good and the model wrongly flags 15% of them, 144 people. The flagged pile holds 180, and 36 of those default: one in five. The model is good at catching defaulters, but defaulters are so rare that false alarms outnumber them four to one.

Why is 90% the wrong answer when the model catches 90% of defaulters?

Picture a smoke alarm that always sounds when there is a fire and also sounds now and then for burnt toast. Because toast burns far more often than houses do, most alarms in a year are toast. A flag's meaning depends on how common the thing it looks for is: when defaulters are 4% of applicants, even a modest false alarm rate on the other 96% produces more wrong flags than right ones. The 90% is the chance a defaulter gets flagged; the question asks the reverse, the chance a flag is a defaulter.

Why count people instead of using the formula?

Bayes' rule gives the same answer, but natural frequencies are faster to say and harder to get wrong under pressure. Turn every percentage into a count of people out of a round number, and the answer is simply the flagged defaulters over everyone flagged. The formula version is 0.04 x 0.9 over (0.04 x 0.9 plus 0.96 x 0.15), which is 0.036 over 0.18, or 20%. Offer it as the check after the counts.

Then say what a lender does with it. A flag at 20% is a reason for a closer look, not a rejection. The cleared pile, by contrast, holds only 4 defaulters in 820, about 0.5%, against 4% before the model, so the model is most useful for waving through the safe majority. Cutting the false alarm rate from 15% to 5% would lift the flagged default share to about 43%.

Roughly 12 crore 5G phones a year, on stated assumptions. Take about 60 crore smartphone users out of 140 crore people. If they replace their phone every 4 years, that is 15 crore replacements, plus about 1 crore first-time buyers, so 16 crore phones. If three quarters of new phones sold are 5G, the answer is about 12 crore. A household check, one phone every two years across 30 crore households, gives 15 crore, close enough.

Why start from users and replacement, not from the population?

Think of how many school shoes a town buys in a year. The number of children matters, but the answer is set by how often each child outgrows a pair. In a market where most people already own the product, yearly sales are the user base divided by the replacement cycle, plus a smaller flow of first-time buyers. Population only tells you the ceiling; the cycle turns the stock of users into a yearly flow.

Which assumptions should you say out loud, and how do you check them?

Every number here is an assumption to be stated, not a fact to be quoted: the user base, the cycle, the first-time flow and the 5G share. The replacement cycle deserves the most care because a one-year change in it swings the answer by between a fifth and a third; the population barely matters by comparison. A three-year cycle gives 15.75 crore and a five-year cycle 9.75 crore. The 5G share is the other moving part, because it changes fast from one year to the next; ask which year the interviewer means.

Then check from a different direction. With about 30 crore households, one new phone per household every two years gives 15 crore phones, close to the 16 crore from the user build. Two methods that land near each other are more convincing than one precise-looking number. For a real estimate, replace every assumption with published industry shipment data and confirm the current figures.

If the interviewer wants value rather than units, multiply by an assumed average selling price and say that 5G phones are skewed to the middle and upper price bands, so the price assumption needs as much care as the cycle.

35% for one stock, about 20% for ten, and a floor of 17.5% for infinitely many. Portfolio variance is the stock variance times one over n, plus the correlation times what is left. With ten stocks that is 0.1225 times 0.325, a volatility of 20.0%. As n grows the one-over-n part vanishes and only the correlation term remains: 35% times the square root of 0.25, or 17.5%.

Why does adding stocks lower risk at all?

Ten shops in ten different towns do not all have a bad week at once; ten shops in one mall often do. Some of what moves a stock is its own news and some is the market everyone shares. Stock-specific shocks cancel out as you add names, because one company's bad quarter is offset by another's good one, but the shared market shock hits every name together and does not cancel. Correlation measures how much of each stock's movement is shared.

How much of the benefit do the first ten stocks deliver?

Most of it. Going from one stock to ten cuts volatility from 35% to 20.0%, about 86% of the whole distance to the floor. Going from ten to thirty takes it only to 18.4%. Diversification pays off quickly and then almost stops, and the level where it stops is set by correlation, not by the number of holdings. That is why a manager worried about risk gains more from adding assets that are less correlated than from adding a fortieth stock of the same kind.

Say the limitation. Correlations are not fixed: in a sell-off they tend to rise together, which raises the floor exactly when diversification is needed. Real stocks also differ in volatility and correlation, so this uniform market is a teaching model; the shape of the curve survives, the exact numbers do not.

About Rs 600 crore for the operating business, before any value in the land. Next year's surplus is Rs 40 crore grown 5%, or Rs 42 crore. A surplus that grows forever at 5% and is valued at 12% is worth next year's amount divided by the 7-point gap: 42 over 0.07 is Rs 600 crore. The reason to buy is the durability of that surplus: steady demand for seats and fees that can rise with costs.

What is the question really asking?

It is a stock pitch in disguise. The interviewer wants two things in order: why this asset produces reliable cash, and what that cash is worth. Answer the why with the quality of the surplus, and the how much with a valuation you can do out loud. For a college the why is simple to say: students keep applying every year, fees are paid in advance, and a college with a good name can raise fees roughly in line with its costs. Those are the reasons the surplus can be treated as growing and durable.

Why divide by 12% less 5%?

Think of a rented flat whose rent rises every year. A buyer asking for a 12% return on a rent that grows 5% needs only 7% from the current rent; the other 5% arrives through growth. A cash flow growing at g forever, valued at a required return r, is worth next year's cash flow divided by r minus g. Here that is Rs 42 crore over 0.07, which is Rs 600 crore. Dividing this year's Rs 40 crore instead gives Rs 571 crore, a common small slip: the buyer receives next year's surplus, not this year's.

Now say what the number is sensitive to. One point on the required return moves the value by Rs 100 crore up or Rs 75 crore down, because a 7-point gap becoming 6 or 8 is a large change in proportion. The land and buildings may be worth more than the operating surplus, so a seller would also ask what the campus fetches as property. And check the structure before promising anyone the surplus: many colleges, in India among other places, are run by trusts or societies, and whether an owner can take surplus out at all is a legal question to confirm.

About 1,200 a year, within a range of roughly 1,000 to 1,500. Size the fleet first: about 100,000 flights a day, three quarters on narrow-bodies, at five flights per jet a day, needs about 15,000 jets. Yearly demand is replacement plus growth: 15,000 over a 25-year life is 600, and 4% growth adds 600. Every input is an assumption to state and then check against published fleet data.

Where do you start with a plane that lasts 25 years?

Think of refrigerators in a city. Almost every home has one, but the shops sell only the ones that break down plus the ones new homes need. For a long-lived asset, yearly sales are a small flow out of a large stock: the stock divided by its life, plus the stock times its growth rate. So the job splits into two parts. Size the fleet from how the planes are used, then turn the fleet into a yearly flow.

How do you size the fleet without knowing it?

From usage, which is easier to guess. Take about 100,000 commercial flights a day worldwide, a round number to state as an assumption. Most flights are short and medium routes, so put three quarters on narrow-bodies: 75,000. A narrow-body on short routes flies several sectors a day; call it five. That needs 15,000 jets. Each assumption is a round number said out loud, so the interviewer can challenge one without losing the structure.

Now sanity-check. 1,200 deliveries is about 8% of the fleet, which sounds right for an asset that lasts decades. If the question asks about one maker, split by an assumed market share; two main makers sharing the market would each deliver around 600. For an asset manager the useful point is the sensitivity: deliveries swing with the growth assumption far more than the fleet does, which is why aircraft makers' order books are so cyclical. Treat every number here as an assumption to check against published fleet and delivery data before using it.

About 36 basis points. Over a year the bond earns 180 basis points more than the government bond. Every basis point the spread widens knocks about 5 basis points off its price, because its spread duration is 5. The extra yield is used up when widening times 5 equals 180, which is at 36 basis points. Beyond that the corporate bond does worse than the government bond.

What is the cushion, and what eats it?

Think of a shop that earns a steady margin on every sale but whose stock loses value when fashions change. The margin comes in slowly; a markdown hits all at once. A credit spread pays carry slowly over the year, while widening hits the price immediately, in proportion to spread duration. The carry is 180 basis points. A spread duration of 5 means a 1 basis point widening costs about 5 basis points of price.

What would you add to sound like a credit investor?

Two refinements, both worth a sentence. First, part of the spread pays for defaults, not risk. If expected default losses were 50 basis points a year, an illustration, only 130 is true cushion and the break-even falls to 26 basis points. The spread is not all profit, so the honest break-even uses the spread after expected losses. Second, the price loss is felt at the end of the year, when the bond is shorter; at a spread duration of about 4.2 then, the break-even is closer to 43. The 36 is the conservative, quick answer.

The ratio also compares bonds quickly. A short bond with a small spread can have a wider break-even than a long bond with a big one, because the long bond's duration magnifies every move. Credit portfolio managers often rank bonds by spread per unit of spread duration for exactly this reason.

About Rs 16.1 lakh, a diversification benefit of about Rs 3.9 lakh. Under normal returns value at risk is a fixed multiple of standard deviation, so it combines the same way: the square root of 10 squared plus 10 squared plus 2 x 0.3 x 10 x 10, which is the square root of 260. Adding the two desks' figures would overstate the risk by Rs 3.9 lakh.

Why can you not just add the two numbers?

Two friends each walk 10 minutes from a crossing, one north and one north-east. They do not end up 20 minutes apart; the angle between their paths matters. Under normal returns, a desk's value at risk is a fixed multiple of its standard deviation, and standard deviations combine like arrows: the angle between them is set by the correlation. Only at a correlation of 1 do the arrows point the same way and add to 20.

What assumption is doing the work, and when does it fail?

The square-root rule holds only when returns are jointly normal, or close to it, so that value at risk is a clean multiple of standard deviation. With fat tails or options in the book, value at risk need not be subadditive, and the combined figure can even exceed the sum. Correlations also rise in a crisis, so the Rs 3.9 lakh benefit is thinnest on exactly the days it is needed. Firms often report both the diversified total and the sum of the parts for that reason.