Portfolio Management puzzles, solved step by step

Two heads in a row takes 6 tosses on average; head then tail takes 4. Each pattern has a one-in-four chance at any pair of tosses, but after a head, a tail wrecks HH and sends you back to the start, while for HT an extra head keeps you exactly where you were. Losing progress on a miss is what costs HH the extra two tosses.

Why do two equally likely patterns take different times to appear?

Picture two ladders. On one, slipping off the second rung drops you to the ground; on the other, slipping leaves you on the second rung. Both ladders need the same two good steps, but you will climb the second one faster. For HH, a tail after a head wipes out your progress; for HT, a head after a head leaves your progress intact, so HT arrives sooner.

How do you solve it in the room?

Name the states, write one equation per state, solve. Let the expected tosses still needed be E0 from the start and E1 after a head. Every equation says: one toss, plus the expected wait from wherever that toss sends you. For HH, E1 = 1 + half of 0 + half of E0, and E0 = 1 + half of E1 + half of E0. Substituting gives E0 = 6 and E1 = 4.

For HT the second equation changes: after a head, a tail finishes and a head leaves you in the same state, so E1 = 1 + half of E1, which is 2, and E0 is 4. The same state method handles any short pattern, which is the point an interviewer at a quantitative fund wants to hear.

The correlation is zero, but X and Y are completely dependent. Because X is symmetric, a 2% fall and a 2% rise both give Y of 4, so the up-slope and the down-slope cancel and the covariance is zero. Yet Y is fixed exactly by X. Zero correlation rules out a straight-line link only, not a link.

How can a perfect relationship show zero correlation?

An umbrella seller does well when it pours and an ice cream seller when it is scorching; a stall selling both does well on any extreme day and badly on mild ones. Its takings depend completely on the weather, but not in a straight line. Correlation asks only whether a straight line fits, so a U-shaped link, rising on both sides, averages out to zero.

What does the algebra say?

Covariance is the average of X times Y, less the product of the averages. With X symmetric around zero, the average of X is zero and the average of X cubed is zero, so the covariance, and with it the correlation, is exactly zero. Using five equally likely days of minus 2, minus 1, 0, 1 and 2 per cent gives the same answer by hand: the covariance works out to 0.

Now the portfolio point. A long straddleBuying a call and a put at the same strike, so the position profits from a large move in either direction. profits roughly with the square of the move, so its returns can show near-zero correlation with the market while depending heavily on it. A risk model that reads zero correlation as no exposure will treat the straddle as a diversifier and miss that it is a pure bet on the size of market moves.

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.

1.6 times book at a 16% return on equity, falling to 1.0 times at 13%. The justified multiple is the return on equity less growth, over the cost of equity less growth: 8 over 5 at 16%, and 5 over 5 at 13%. When a bank earns exactly its cost of equity, a rupee of book is worth a rupee, and growth adds nothing.

Why does a bank earning its cost of equity trade at book?

Suppose a friend offers to take Rs 1 lakh and pay you exactly the return you could get anywhere else with the same risk. The deal is worth Rs 1 lakh; no more, however large your friend makes it. A bank reinvesting at a return equal to its cost of equity creates no value on the money it keeps, so its book is worth exactly book. Only the spread between the two rates earns a premium.

Where does the formula come from?

Start from the dividend discount model. Next year's dividend is book times return on equity times the share paid out. To grow at 8% with a 16% return, the bank must retain half its earnings, since growth is return on equity times retention; that leaves a payout of 1 minus g over ROE. Substituting gives price over book equal to (ROE minus g) over (r minus g). At 16%, retention is 50%, payout 50%, and the multiple 1.6 times.

The limit is that every input is a long-run steady state. A bank with a cyclical credit cost earns 20% in good years and 8% in bad ones, and the formula wants the through-cycle figure. Say so, then say which input you would argue about first.

The fund's time-weighted return is plus 8% over two years, but the investor lost Rs 3.2 lakh. Time-weighted return chains the yearly returns, 1.20 x 0.90, and ignores cash flows. The investor had Rs 10 lakh in for the gain and Rs 52 lakh in for the fall, so Rs 50 lakh became Rs 46.8 lakh, a money-weighted return of about minus 5.4% a year.

How can both numbers be right?

A cricket team's run rate says how well it batted; a spectator who arrived only for the collapse saw a bad match. Both are true. Time-weighted return measures what the fund did with each rupee over each period, so it judges the manager; money-weighted return weights each period by how much money was in, so it judges the investor's own timing. Here the timing was poor: most of the money arrived just before the fall.

How do you get the money-weighted return?

It is the internal rate of return on the investor's own cash flows. Find the single yearly rate at which Rs 10 lakh put in two years ago and Rs 40 lakh put in one year ago grow to exactly Rs 46.8 lakh today. That is a quadratic, and its root is a growth factor of about 0.9462, or minus 5.4% a year.

This is why fund factsheets quote time-weighted returns: the manager does not control when investors add money. It is also why many investors earn less than the funds they own, because inflows tend to follow good years. Annualised, the fund's figure is 3.9% a year against the investor's minus 5.4%.

About 26% a year for the three-year deal and 10.4% for the seven-year deal. The IRR of a single payout is the multiple to the power one over the years, less one: 2 to the one third and 2 to the one seventh. Same money back, very different speed. The multiple says how much, the IRR says how fast, and a deal needs both.

Why does the same multiple give such different returns?

Two friends each double their savings; one takes three years, the other seven. Nobody would call them equally good investors. A multiple of money ignores time entirely, while IRR is a rate per year, so the same 2.0x falls from 26% to 10.4% as the wait stretches from three years to seven. The curve is steep early: each extra year cuts the IRR most when the holding period is short.

So which number should an investor care about?

Both, because each can be gamed alone. A short, quick flip can post a high IRR on a small absolute gain, while a long hold can post a big multiple at a return below the cost of capital. Limited partners look at IRR for speed and at the multiple, often called the equity multiple or MOIC, for how much wealth was actually created. To match 26% over seven years, the second deal would need about 5.0 times the money, not 2.0.

One caution for the room: this clean formula holds only with one outflow and one inflow. With interim distributions, the IRR assumes those cash flows are reinvested at the IRR itself, which is rarely true, so a quoted IRR on a deal with early payouts flatters it.

Roughly 40 lakh new cars a year, built from stated assumptions. About 30 crore households, 8% owning a car, gives about 2.4 crore cars. Ownership rising half a point a year adds 15 lakh first-time buyers; a 12-year life means 20 lakh replacements; fleets and second cars add about 5 lakh. Then compare with published industry sales data.

How do you turn a stock of cars into yearly sales?

A school's intake each year is not its total strength; it is the students leaving that year plus any growth in the roll. New car sales work the same way: they are the growth in the number of cars owned plus the cars being replaced, so the answer needs a stock and two flows. Say this structure first; the interviewer is scoring the tree more than the final number.

Which assumption would you defend first, and which would you flag?

Take the population as roughly 140 crore and confirm the current figure; at about 4.7 people per household that is 30 crore households. The 8% ownership share is the number to flag, because ownership is concentrated in cities and in the top income bands. Replacement is the largest flow, so the assumed 12-year life matters most: at 10 years replacements rise to 24 lakh, at 15 they fall to 16.

Finish by naming what you would check. Industry bodies publish domestic sales every month, and an interviewer at a fund expects you to know the order of magnitude. If your estimate is far off, say which branch you would revisit rather than adjusting the total to fit.

n cubed minus (n minus 2) cubed, which expands to 6n squared minus 12n plus 8. Everything not on the surface forms a hidden cube with one layer peeled from each side, so its side is n minus 2. For n = 3 that is 27 minus 1, or 26; for n = 10 it is 488. If the interviewer means the coloured squares instead, the answer is 6n squared.

Why count what you cannot see?

To count the tiles round the edge of a courtyard, it is easier to measure the whole yard and subtract the inner lawn than to walk the border without counting corners twice. The cubes not on the surface form one clean block of side n minus 2, so subtracting it from n cubed avoids every double-counting trap at the edges and corners.

How do you check the formula a second way?

Count by type. There are always 8 corner cubes, 12 edges each carrying n minus 2 cubes between the corners, and 6 faces each with an (n minus 2) by (n minus 2) centre. That is 8 plus 12(n minus 2) plus 6(n minus 2) squared, which expands to 6n squared minus 12n plus 8. Two routes agreeing is the check the interviewer wants to hear.

Two edge cases show care. The formula needs n of at least 2; a single cube, n = 1, is all surface, and the formula would wrongly give 2. And the question is sometimes asked as surface area: the number of coloured squares is 6n squared, 54 for a standard cube. Ask which is meant before answering.

About +7.3% if yields fall 100 basis points and -6.7% if they rise 100. Duration alone gives 7% either way. Convexity adds half of 60 times 0.01 squared, which is 0.3%, in both directions, because the yield change is squared. So the gain is bigger than the loss for the same size of move.

Why does convexity help in both directions?

A ball rolling in a bowl rises the same way whichever side you push it. The convexity term depends on the yield change squared, so it is positive whether yields rise or fall: it adds to the price gain when yields drop and cushions the loss when they climb. Duration is the straight tangent to the price curve; convexity measures how far the real curve bends above it.

How big does convexity get, and what does it cost?

At 100 basis points the bend is only 0.3%, but it grows with the square: at 400 basis points it is 4.8%, a large share of the move. That asymmetry is valuable in volatile markets, so bonds with more convexity usually trade at a slightly lower yield: you pay for it through carry. Callable bonds and many mortgage securities have negative convexity, and for them the sign flips: losses outrun gains.

Say the limit too. This is a second-order estimate. For very large moves the true price differs again, and an interviewer may ask you to reprice the bond from its cash flows instead.

The retiree with the good year first ends with Rs 87.6 lakh, against Rs 77.1 lakh, a gap of Rs 10.5 lakh. Without withdrawals both would end at Rs 114.4 lakh, because the order of multiplication does not matter. Withdrawals break that: in the bad-first order, the money taken out early misses the strong returns that come later.

Why does the order suddenly matter?

A shop owner who must pay a fixed rent every month is hurt more by a bad first quarter than a bad last one: an early slump forces them to dip into stock just when prices are low. Without withdrawals, 0.8 x 1.1 x 1.3 gives the same answer in any order; with fixed withdrawals, each rupee taken out misses every return after it, so losses early in retirement do lasting damage.

Where exactly does the Rs 10.5 lakh go?

Think of each withdrawal as money that would otherwise have kept compounding. The ending balance equals the no-withdrawal result less each withdrawal grown at the returns it missed. In the bad-first order the first Rs 10 lakh misses a 10% and a 30% year, so it costs 14.3 lakh of ending wealth; in the good-first order it misses a 10% gain and a 20% fall, so it costs only 8.8.

This is sequence-of-returns risk, and it is why retirement planners hold a cash or short-bond buffer for the first few years of withdrawals, or vary withdrawals with markets. The same logic runs in reverse for someone still saving: early losses hurt a saver less, because little money is invested yet.