Portfolio Management puzzles, solved step by step

About 45%. One rupee at 15% grows to 2.66 after seven years and 4.05 after ten. The total gain is 3.05, and the last three years add 4.05 minus 2.66, which is 1.39. That is 45.5% of the decade's gain in 30% of the time, because each year's 15% is earned on a larger base than the year before.

Why is the gain not spread evenly across the years?

Think of a snowball rolled down a long slope. In the first few metres it picks up a little snow; near the bottom each turn picks up much more, because the ball itself is bigger. Compounding earns the same rate on a growing base, so every year's rupee gain is 15% larger than the year before. Year 1 adds 0.15 on each rupee. Year 10 adds 0.53, three and a half times as much. The first three years together add only 0.52; the last three add 1.39.

What does this mean for an investor who leaves early?

Someone who exits after seven years has sat through 70% of the time but collected only 55% of the decade's gain. Because compounding back-loads the reward, leaving a long plan early costs far more than the fraction of time given up. The same arithmetic runs against the investor with fees: a charge taken every year compounds too, and its cost is also concentrated at the end. Say the limitation plainly: 15% is an assumed rate for the arithmetic, not a forecast, and real returns arrive unevenly, so the actual last three years could be the worst three.

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.

About 19.9% at plus 0.3 and 15.6% at minus 0.3. The unhedged return is roughly the local return plus the currency return, so the variances add with a correlation term: 16 squared plus 8 squared, plus or minus 2 x 0.3 x 16 x 8. That is 396.8 or 243.2, with square roots of 19.9% and 15.6%. With negative correlation the unhedged position is less volatile than the hedged one at 16%.

How can adding a second risk reduce the total?

Think of a shop that sells umbrellas and sunglasses. Each product's sales swing a lot with the weather, but in opposite directions, so the till is steadier than either product alone. When two sources of return tend to move against each other, combining them lowers risk even though each one is volatile on its own. For an Indian investor holding foreign shares, the currency is the second source. If the foreign currency tends to strengthen against the rupee when that market falls, it cushions the loss, and the unhedged position is steadier: 15.6% instead of 16%.

So should a global portfolio hedge its currency?

The puzzle gives the risk side of the answer, not the whole decision. A hedge removes the currency's volatility, but it also removes the currency's correlation with the market, and when that correlation is negative the hedge adds risk rather than cutting it. The rest of the decision is cost, which depends on the interest rate gap between the two currencies, and the investor's own liabilities in rupees. Say the limitation: correlations are measured on history and tend to shift in a crisis, so the minus 0.3 that makes the unhedged position look safer is the number least likely to hold when it matters. The formula also ignores the small cross term from multiplying the two returns.

About 64%, and a per-test threshold of about 0.25%. Each worthless strategy passes by luck 5% of the time, so all 20 fail with chance 0.95 to the 20th, 35.8%, and at least one passes 64.2% of the time. To hold the overall rate at 5%, test each at 5% divided by 20, which is 0.25%; the exact version, 1 minus 0.95 to the power of one twentieth, is 0.256%.

Why does testing more ideas throw up a false winner?

Ask a room of 20 people to each flip a coin five times, and there is a fair chance someone gets five heads. Nobody in the room has a lucky hand; there were simply enough tries. A 5% test lets one worthless idea in twenty through by chance, so a researcher who tests twenty ideas should expect about one false winner, not be impressed by it. The expected number of false positives here is 20 x 0.05, exactly 1, and the chance of at least one is 64.2%.

What does a quant desk actually do about it?

Dividing the threshold by the number of tests is called the Bonferroni correctionA rule that divides the significance level by the number of tests run, so the chance of any false positive across all of them stays near the original level.. The real discipline is counting every test you ran, including the ones you dropped quietly, because the correction is only as honest as that count. A researcher who tried 200 variants and reports the best 20 has a far bigger multiple testing problem than the 20 suggest. Desks also hold out data the research never touched and demand a reason for the edge before the backtest. Say the limitation: the correction assumes independent tests, and for correlated strategies it is too strict, which costs real ideas.

Sell Rs 35 crore of index futures notional. At a beta of 1.2 the portfolio moves like Rs 60 crore of the index. At the target of 0.5 it should move like Rs 25 crore. The difference, (1.2 minus 0.5) x Rs 50 crore, is Rs 35 crore, assuming the futures move one for one with the index. At an assumed Rs 10 lakh a contract, that is about 350 contracts.

Why is the portfolio's market exposure not simply Rs 50 crore?

Think of a car that goes 1.2 km for every km a reference car goes. Holding Rs 50 crore of it is like holding Rs 60 crore of the reference. Beta converts a portfolio's value into index-equivalent exposure, so a Rs 50 crore book at beta 1.2 carries Rs 60 crore of market risk. Once you see the exposure in index rupees, the hedge is a subtraction: you want Rs 25 crore left, so you take away Rs 35 crore by selling futures, which carry a beta of one to the index.

What does the hedge not do?

It removes market risk, not stock risk. After the hedge the portfolio still carries every stock-specific bet it had; only its sensitivity to the index has been cut. That is often the point: a manager who likes the stocks but not the market can keep the stock picks and trim the market bet. Say the limitations: beta is estimated from history and drifts, so the hedge is right only on average; futures need margin and must be rolled at expiry, and the futures price can move slightly differently from the index, which is called basis risk.

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.

People refuse because a loss hurts more than an equal gain pleases, but ten plays lose money only about 17% of the time. Each play is worth plus Rs 25,000 on average. Over ten plays, four wins and six losses break even, so only three or fewer wins lose: 176 of 1,024 sequences, 17.2%. Judging each bet alone makes a good repeated bet feel bad.

Why does a bet with a positive expected value get refused?

Ask someone whether they would bet their weekend on a coin flip to win a second weekend, and most say no, because losing the one they have feels worse than gaining a new one feels good. This is loss aversionThe tendency, documented by Kahneman and Tversky in prospect theory, to feel a loss more strongly than a gain of the same size.. If a loss feels twice as bad as an equal gain feels good, this bet's felt value is 0.5 x 1.5 minus 0.5 x 2 x 1, which is minus 0.25 lakh, so refusing it once is consistent with how the person feels, even though its expected value is plus Rs 25,000. The factor of two is an assumption for the arithmetic, not a measured constant.

How do you get the chance of a loss over ten plays?

Count wins. With k wins and 10 minus k losses the total is 1.5k minus (10 minus k), which is 2.5k minus 10 lakh. That is negative only when k is three or less. Because each win outweighs each loss, you can lose six flips out of ten and still break even, which is why pooling the bets makes a loss so much rarer than a single flip suggests. The number of ways to get 0, 1, 2 or 3 wins is 1 plus 10 plus 45 plus 120, which is 176, out of 1,024 equally likely sequences: 17.2%.

What does this mean for how clients see a portfolio?

A client who checks a portfolio every day sees each day as a separate bet and feels every loss. Looking at the same holdings less often pools the bets, and the pooled result loses far less often than any single period does. The economist Paul Samuelson told the story of a colleague who refused one such bet but would take a hundred; the lesson for a wealth desk is to frame decisions at the horizon the money actually has. State the limitation: pooling helps only when the bets are independent and the investor can survive the bad runs, and ten plays still lose 17% of the time.

Interview and pass on the first 3, then hire the first manager better than all of them. You get the best one about 40% of the time. The first three set the bar at no cost but the chance the best is among them. Passing on 4 gives almost the same, 39.8%. As the number of candidates grows, the rule becomes: pass on about 37%, one over e, and the chance of success tends to 37% too.

Why pass on anyone at all?

Think of house hunting in a fast market where every flat is gone the moment you walk away. If you sign the first one, you have no idea whether it was good. If you look at every flat before deciding, the best has already been taken. A short look-only phase buys you a benchmark; after that, the first candidate who beats the benchmark is likely to be the best overall. The cost is that the best might be inside the look-only phase, which happens with chance r in 10 if you pass on r. Hiring the first manager blindly succeeds only 10% of the time.

How do you compute the chance for a given rule?

Suppose you pass on r and the best manager sits at position i, after r. You hire them only if nobody between r and i beat the first r, which happens when the best of the first i minus 1 candidates is among the first r: a chance of r over i minus 1. Averaging over where the best one sits gives r over 10 times the sum of 1 over (i minus 1), for i from r plus 1 to 10. For r equals 3 that is 0.3 times (1/3 + 1/4 + ... + 1/9), which is 0.3987. The curve is flat near the top: 3 and 4 differ by less than half a point.

Say the limitation, because allocators hear this puzzle and then ask about real selection. The rule maximises the chance of the single best and scores a second-best hire as a total failure. Real allocators care about hiring someone good, can often revisit a manager, and rarely see candidates in random order, and each of those changes the rule.

About 7.7%. You collect the 7 coupon, and a year later the bond is a two-year bond. If the curve has not moved, it is priced at the two-year yield of 6.6%, which makes it worth 100.73. Coupon 7.00 plus a price gain of 0.73 is 7.73%. A shortcut gives the same: 7% plus the two-year duration of about 1.82 times the 0.4 point fall in yield.

Why does an unchanged curve still move the bond's price?

Think of walking down a gentle slope while standing still relative to the hillside: the ground stays put, but you end up lower because you moved along it. A bond ages along the curve; if the curve slopes up and stays still, the bond's yield falls as its maturity shortens, and its price rises. Today it is a three-year bond at 7.0%. In a year it will be a two-year bond, and two-year bonds yield 6.6%. That 0.4 point fall in yield is the roll downThe price gain a bond earns as it ages into a shorter maturity with a lower yield on an upward sloping curve..

How do you check it without a calculator?

Use duration. A two-year bond with a 7% coupon has a modified duration of about 1.82, so a 0.4 point fall in yield lifts its price by about 1.82 x 0.4, which is 0.73. Add the 7 coupon and the one-year return is about 7.73%, within a hair of the exact 7.73%. Say the limitation: the answer depends entirely on the curve staying put. If two-year yields rise to 7.0% by next year, the roll down disappears and the return is the coupon alone.

About 25% a year for ten years. With no dividend, all of the 12% return must come from price, so the price in ten years must be 60 x 1.12 to the tenth, about 186 times today's earnings. If the stock then trades at 20 times, earnings must be 186 divided by 20, or 9.32 times today's. That is growth of 25.0% a year. At 15% growth the investor would earn only about 3% a year.

How do you turn a multiple into a growth forecast?

Think of paying 60 years of a shop's current profit for the shop. That only makes sense if the profit is going to be far bigger soon. A high multiple is a forecast you can read: work forward from the return investors want, and back from the multiple the stock should end at, and the growth in between is what the price assumes. Here the investor wants 12% a year with no dividend, so the price must grow 3.11 times in ten years, to about 186 times today's earnings. At a mature 20 times, earnings must reach 9.32 times today's level.

What do you do with the number once you have it?

You ask how often a company sustains that growth for a decade, which is rarely, and you say so. The question the interviewer wants answered is not whether the company is good but whether the price has already paid for more than the company is likely to deliver. At a still strong 15% a year, earnings reach 4.05 times today's, the price at 20 times is about 81, and the investor earns about 3.0% a year rather than 12%. Say the limitations: the exit multiple of 20 is an assumption, and buybacks, dividends or a higher exit multiple would lower the growth needed.