Portfolio Management puzzles, solved step by step

Reroll on 1, 2 or 3; keep 4, 5 or 6. The game is worth Rs 4,250. A reroll is worth 3.5 on average, so you keep any face above 3.5 and swap any face below it. The kept faces contribute 4, 5 and 6 each with a one-in-six chance, 2.5 in all; the other half of the time you get 3.5. The total is 2.5 plus 1.75, which is 4.25 thousand.

How do you decide whether to reroll?

Think of a job offer in hand against the option to keep interviewing. You take the offer if it beats what you expect to get by walking away. The rule is to keep a result only when it is worth more than the option you give up, and here the option is a fresh roll worth 3.5. A 3 is below 3.5, so reroll it. A 4 is above, so keep it. Nothing about how you feel about a 4 enters into it; the comparison is one number against another.

Why is this called working backwards?

Because you value the last decision first. The second roll has no choice left in it, so it is worth 3.5. Only once you know that can you decide the first roll. Every multi-stage game is solved from the final stage back to the first, because each decision depends on what the later stages are worth. Allow two rerolls and the same logic stacks: the last reroll is worth 3.5, the middle stage is worth 4.25, so on the first roll you keep only a 5 or 6, and the game is worth 4.67.

Rerolling only on 1 and 2, a common cautious answer, gives 4.17: you keep the 3 when a reroll would have averaged half a point more. The portfolio version is the same idea. A position is kept only while its expected return beats what the capital would earn redeployed, which is an opportunity cost, not a sunk cost.

Fund A, by about two points a year: roughly 8.5% against 6.5%. Long-run compound growth is approximately the average return less half the variance. For A that is 9% less half of 0.10 squared, 0.5 points. For B it is 11% less half of 0.30 squared, 4.5 points. Over twenty years a rupee in A grows to about Rs 5.11 and in B to about Rs 3.52.

Why is the average return not what you earn?

A shop that marks a price up 30% and then down 30% does not end where it started. Rs 100 becomes Rs 130 and then Rs 91, although the average move is zero. Returns compound by multiplying, so a swing up and an equal swing down always leave you behind, and the loss grows with the size of the swing. For moderate returns the loss is close to half the variance, half of volatility squared. That is why a fund's compound annual growth sits below its average annual return, and the gap widens sharply as volatility rises.

How precise is the half-variance rule?

It is an approximation, good when returns and volatility are modest, and it overstates the drag when volatility is large. A more careful calculation that treats returns as lognormal puts A at 8.5% and B at 7.2%. The size of B's handicap moves with the method, but the ranking does not: the higher-volatility fund compounds more slowly. Say that in the room. The rule is a tool for spotting the direction and rough size, not a precise forecast.

The portfolio consequence is the reason the question exists. Two funds with the same average return are not equal if one is far more volatile, and a client who holds a single volatile fund lives with the compound number, not the average in the brochure. Combining volatile assets that do not move together lowers portfolio variance, which is one of the few ways to raise compound growth without raising the average return.

About Rs 87,000 a year, and more after tax on the deposit interest. The card costs 36% of Rs 3 lakh, Rs 1,08,000. Using Rs 3 lakh of the deposit to clear it gives up 7% on that money, Rs 21,000. The difference, Rs 87,000, is a certain loss every year the household keeps both, paid for the comfort of a bigger deposit balance.

Why do people keep both?

Because money gets labels. The deposit is the emergency fund or the daughter's education money, and touching it feels like failing; the card is day-to-day spending and feels temporary. A rupee is a rupee whatever label it carries, so money earning 7% while debt costs 36% is simply borrowing at 36% to lend at 7%. The habit has a name, mental accountingTreating money differently depending on which mental category it sits in, a term from the economist Richard Thaler., and it is one of the most common and costly patterns a wealth adviser meets.

How do you get the number right?

Compare two whole choices. Keep both: the household earns Rs 35,000 and pays Rs 1,08,000, a net of minus Rs 73,000. Clear the card: it earns 7% on the remaining Rs 2 lakh, Rs 14,000, and pays nothing. The cost of a choice is the difference between the two outcomes, not the interest bill on its own. That difference is Rs 87,000, the same as 29 points of rate gap on Rs 3 lakh.

Tax widens the gap. Deposit interest is taxed at the household's slab while card interest is paid from income already taxed. At an illustrative 30% slab, confirm the current rates, the cost rises to about Rs 93,300. The honest limit is liquidity: a family with no buffer at all may need some cash on hand. Here the Rs 2 lakh left in the deposit is that buffer, so the case for keeping the debt is gone.

About 6.1%. On 100 of borrowed yen, the rupee investment grows to 107 and the loan grows to 100.5. The trade breaks even if 107 rupee-units convert back to exactly 100.5 yen-units, which means a fall of 1 minus 100.5 over 107, or 6.07%. It is a little less than the 6.5-point rate gap because the fall also eats into the interest earned.

What is the trade actually betting on?

Picture borrowing from a relative who charges almost nothing and lending the money to a friend who pays well, except the friend repays in a different currency. You pocket the gap in rates as long as the friend's currency holds its value. A carry trade earns the interest rate gap and loses whatever the high-yielding currency falls, so the gap is the only cushion. Here the cushion is 7% less 0.5%, 6.5 points a year.

Why is 6.5% not quite right, and what else should you say?

Because the fall hits the whole pot, principal and interest, and a 6.5% fall on 107 costs almost 7. The exact break-even is 6.07%, and the difference matters only for precision. The bigger point is that a currency can lose 6% in days, while the carry is earned slowly over a year. That lopsided pattern, small steady gains and occasional sharp losses, is what makes carry trades dangerous to size.

One more thing is worth one sentence. If you hedged the currency with a one-year forward, the forward rate would already build in a rupee fall of about the rate gap, and the profit would disappear. The unhedged carry trade is a bet that the rupee does better than the forward price implies. That is a view on the currency, not free money.

1.6%, not 4%. A fund with a beta of 1.3 should earn cash plus 1.3 times the market's return over cash: 6% plus 1.3 times 8%, which is 16.4%. It earned 18%, so the return its market exposure does not explain is 1.6 points. Most of the 4-point lead over the index came from simply taking more market risk in a rising year.

Why is beating the index by 4 points not the answer?

Imagine two drivers on a downhill road. One coasts, the other presses the accelerator. The second arrives first, but that says nothing about who drives better. A fund with a beta above 1 is expected to beat a rising market, so raw outperformance mixes skill with extra market exposure. Jensen's alpha strips out the part of the return that the fund's beta alone would have delivered, and asks what is left.

What would you warn about the 1.6%?

Three things. Beta is estimated from past returns, and an error of 0.1 in beta moves alpha by 0.8 points here, half the answer. A single year's alpha is mostly noise, so it describes what happened rather than proving skill. And the answer flips in a falling market: in a year when the index loses, a 1.3-beta fund is expected to lose more, and a fund that merely matches the index would show positive alpha. Always read alpha next to the market's direction and the number of years behind it.

A proposes 98 for himself, 0 for B, 1 for C, 0 for D and 1 for E. Solve from the end. With two pirates, D keeps all 100 because his own vote is half. With three, C buys E for 1 coin. With four, B buys D for 1. With five, A needs two votes and buys the two pirates who get nothing under B's plan, C and E, for one coin each, keeping 98.

Where do you start?

At the end, where there is no choice left. Think of planning a train journey with connections: you start from the time you must arrive and work back to when you must leave. A sequential game is solved backwards, because each pirate's vote depends only on what he would get if the current proposal failed. With two pirates left, D proposes 100 for himself; his own vote is half, which is enough. So E gets nothing if it ever comes to that, and E knows it.

How does each step follow from the one below?

With three pirates, C needs two votes, his own and one more. E gets 0 if C dies, so one coin buys E: C proposes [99, 0, 1]. With four, B needs two votes; under C's plan D gets 0, so B buys D for one coin: [99, 0, 1, 0]. A vote is worth exactly one coin more than that pirate's fallback, so a proposer always buys the cheapest voters, the ones left with nothing in the next round. With five, A needs three votes. Under B's plan C and E get nothing, so one coin each buys them, and A proposes [98, 0, 1, 0, 1].

State the assumptions, because the answer rests on them. If an indifferent pirate voted yes, A could buy votes for zero coins. If the rule needed a strict majority, the counts change. Interviewers often change one rule as a follow-up to see whether you rebuild the chain or reach for a memorised answer. The buy-side lesson is about incentives: what someone will accept depends on their alternative, not on fairness.

About -0.21%: the fund returns about 11.79% against the index's 12%. The fee takes 0.20 points. The 1% in cash earns 5.5 points less than the index, which costs 0.01 times 5.5, or 0.055 points. Securities lending adds back 0.05. Together: minus 0.20, minus 0.055, plus 0.05, a tracking difference of -0.205%. The expense ratio is most of the story.

What is tracking difference, and how does it differ from tracking error?

Think of a tailor copying a suit. The tracking differenceThe fund return minus the index return over a period. A steady shortfall, mostly the costs of running the fund. is how much shorter the copy comes out on average; the tracking errorThe standard deviation of the gap between fund and index returns. How unevenly the fund follows the index. is how uneven the stitching is. Tracking difference is the fund's return less the index's, and for a well-run index fund it is mostly the costs of running the fund. Investors feel it directly: it is the return they lose for holding the fund instead of the index itself.

Why is the cash drag so small?

Because the cash is not idle; it earns 6.5%. Cash drag is the cash weight times the gap between the index return and the cash return, not the cash weight times the index return. Here the gap is 5.5 points and the weight 1%, so the drag is 0.055 points. In a falling market the sign flips: if the index lost 10%, the 1% in cash would add about 0.165 points and the fund would lag by less than its fee.

Other small items sit in the same bucket and are worth naming: trading costs when the index changes its members, the timing of dividends, and taxes the index ignores but the fund pays. None usually rivals the fee. When two index funds on the same index are compared, a lower expense ratio is the first thing to check, then the reported tracking difference over several years, which the fund documents disclose; confirm the current disclosure rules for the market in question.

No. It is worth about Rs 92.7 to you, against a price of Rs 100. The expected price in a year is 0.4 times 150 plus 0.6 times 70, which is Rs 102, an expected return of only 2%. You need 10% for this risk, so you should pay no more than Rs 102 divided by 1.10. To justify Rs 100, the chance of the up case would have to be 50%.

Why is a positive expected gain not enough?

Imagine lending a friend Rs 100 for a year and expecting Rs 102 back, when a bank deposit would pay you more with no worry at all. You would not do it. An investment has to beat the return you require for its risk, not merely beat zero. Here the expected price of Rs 102 is a 2% expected return, and you have said this risk needs 10%. The shortfall is the whole answer.

What would change your mind?

Turn the question round and ask what probability makes Rs 100 fair. At a 10% required return the expected price must be Rs 110, and p times 150 plus one minus p times 70 equals 110 when p is 0.50. Stating the break-even probability turns a yes or no into a view you can argue about: do you believe the up case is at least a coin flip? That is exactly the conversation a portfolio manager has about a stock pitch, where the scenarios and their weights matter more than the single target.

Say the limits of the model too. Two outcomes are a sketch; real outcomes spread across a range. The required return itself is a judgement, and a lower one, 2% or less, would make the same stock look fair. The method survives those caveats: expected value first, then compare it with the return the risk demands.

About 18.6% a year. After a 40% fall, 100 has become 60, and getting back to 100 needs a gain of 40 on 60, which is 66.7%. Spread over three years with compounding, the annual return is the cube root of 100 over 60, about 1.186, so 18.6% a year. Dividing the loss or the gain by three gives the wrong answer both ways.

Why does a 40% loss need more than 40% to repair?

A tree that loses 40% of its branches does not regrow them at the rate it lost them; it regrows from what is left. A loss is measured against the old, larger value, but the recovery is measured against the new, smaller one, so the gain needed is always bigger than the loss. Here 100 falls to 60. Getting back to 100 needs 40 on a base of 60, which is 66.7%. The same effect grows quickly: a 60% loss needs 150%.

What should you say to a client after the number?

That 18.6% a year for three years is a demanding return for most portfolios, and chasing it by taking more risk can deepen the hole. The honest conversation is about the time it takes to recover, not about finding a return high enough to recover quickly. Allow five years and the requirement falls to about 10.8% a year. The deeper lesson for portfolio construction is that avoiding large drawdowns is worth more than it looks, because the repair costs rise so steeply.

A's expected shortfall is Rs 6.0 crore; B's is about Rs 12.0 crore. With 100 days, the worst 5% are the five worst days, and expected shortfall is their average. A's five add to 30, an average of 6.0. B's add to 60.2, an average of 12.04. Both portfolios cross Rs 5 crore on the same number of days, but when B has a bad day it is, on average, twice as bad.

What does value at risk leave out?

Think of a river's flood mark. Saying the river tops the bank five days a year tells you nothing about whether it rises a hand's width or floods the town. Value at risk tells you the loss that is exceeded on the worst 5% of days, and nothing about how big those losses are. Both portfolios breach Rs 5 crore on five days in a hundred, so their VaR is identical. What happens beyond that line is where they differ, and it is exactly the part the measure does not describe.

Why do risk managers prefer expected shortfall?

Because it cannot be gamed as easily and it adds up sensibly. A trader can sell deep out-of-the-money options, which earn small premiums almost every day and lose heavily now and then. That book can show a low VaR while hiding a large tail. Expected shortfall looks inside the tail, so strategies that pile risk just past the cut-off show up in it. It also behaves well when books are combined: diversification never makes it worse, which VaR cannot promise.

Be honest about the limits. Five observations are a thin sample for estimating anything, and one Rs 25 crore day dominates B's figure. A longer history, or a model of the tail, would be needed before setting limits on it. The comparison is still the right first read: same VaR, and one portfolio's bad days are twice as painful.