Portfolio Management puzzles, solved step by step

Each cycle wins Rs 1,000 with probability 63 in 64 and loses Rs 63,000 with probability 1 in 64, so the expected value is exactly zero. The capital covers six stakes, Rs 1,000 up to Rs 32,000, so six straight losses, a 1 in 64 chance, wipe it out. Doubling down turns many small wins into one rare, large loss without creating any edge.

Why does it feel like a sure thing?

A driver who never buys insurance saves the premium every month and feels clever for years, until the one accident. Doubling after each loss produces a long run of small wins because the one outcome that loses, six tails in a row, is rare, but when it comes it is sixty three times the usual win. Short track records of such strategies look excellent, which is exactly the danger.

What does this teach about judging a strategy?

Run it 50 times and the chance of at least one wipe-out is about 54%, yet most individual traders running it for a few weeks will show a smooth profit. A win rate says nothing on its own; what matters is the size of the losses relative to the wins, and whether the worst case can end the game. Selling far out-of-the-money options has the same shape: steady premium income, and occasionally a loss that erases years of it.

Say the limitation of the fix people suggest. More capital only pushes the wipe-out further out and makes it bigger: with infinite money and no stake limit the strategy would work, and nobody has either.

The forward is about 85.61, a 1.9% premium to spot, and hedging lifts the dollar deposit's return to about 6.5% in rupees. Converting, earning 4.5% and selling forward must give the same rupees as simply earning 6.5% at home. For an Indian investor holding dollars the hedge earns the 1.9% premium; it is a cost only for someone on the other side.

Why is the forward rate set by interest rates rather than forecasts?

If two shops sell the same phone at different prices with free delivery between them, someone will buy in one and sell in the other until the gap closes. Investing rupees at home and converting to dollars, investing and selling forward are both riskless routes from the same starting sum, so they must end at the same number of rupees; the forward rate is whatever makes that true. This is covered interest parity.

So who pays for hedging, and what does it tell the investor?

The hedged dollar deposit earns 1.045 dollars sold at 85.61, which is 6.5% in rupees: exactly the rupee rate. A fully hedged foreign bond earns roughly the home interest rate plus its own spread, so hedging removes the currency bet but cannot manufacture a higher rate. The same 1.9% is a genuine cost for a dollar investor hedging rupee bonds, which is why hedged returns on Indian debt look lower to foreign funds.

One honest limit: the forward is not a forecast. It builds in a 1.9% rupee depreciation only because of the rate gap. The unhedged investor earns 4.5% plus whatever the rupee actually does, which may be more or less than the forward implied.

About 40 basis points per rupee, and 8 times the total cost. Under a square-root law, cost per rupee rises with the square root of size: four times the size is twice the cost per rupee, 40 basis points. Total cost is size times cost per rupee, so it rises four times two, or eight times. On Rs 50 crore of daily volume, Rs 1 lakh becomes Rs 8 lakh.

Why does a bigger order cost more per rupee?

Selling one flat in a building is easy at the going price; selling ten at once means working down to buyers who would only pay less. A large order uses up the willing counterparties near the current price and has to reach further, so each extra rupee traded moves the price more than the one before. Empirical studies of trading costs across markets find that the cost per rupee grows roughly with the square root of the order's share of volume, a rule practitioners use widely as a first estimate.

What does the portfolio manager do with this?

Two things. Because total cost grows faster than size, spreading a large order over several days is usually cheaper: four days at 10% each costs about Rs 4 lakh against Rs 8 lakh in one go, provided the price impact of one day's trading has largely faded by the next and the price does not drift away meanwhile. And a strategy's capacity is limited: a fund that doubles in size pays about 2.8 times the rupee impact on each rebalance, which eats into returns quickly.

Say the limits. The square-root law is an empirical average, with a coefficient that differs by stock and market conditions, and splitting an order adds timing risk: if the information behind the trade leaks or the price moves, waiting costs more than impact would have.

About 73%, against 50% for a fair bet. This is the gambler's ruin problem. With the ratio of loss to win odds r = 0.45 / 0.55, the chance of reaching 10 units from 5 is (1 minus r to the 5) over (1 minus r to the 10), about 73.2%. Smaller bets make it higher still: with Rs 50,000 bets it is about 88%.

Why is the answer so much higher than 55%?

A slightly better tennis player might win 55% of points, yet win most matches, because a match needs many points and the small edge keeps adding up. Reaching Rs 10 lakh before zero needs a net five wins, which takes many bets, and every bet tilts the race a little in your favour, so the edge compounds into a much bigger survival advantage.

Why does bet size matter so much?

Keep the edge and the rupee distances fixed and change only the stake. With Rs 5 lakh bets the whole outcome rides on one toss, 55%; with Rs 1 lakh bets it is 73%; with Rs 50,000 bets it is 88%, because more bets give the law of large numbers room to work. That is the portfolio lesson: a real edge is only worth something if position sizes are small enough that an unlucky run cannot end the game first.

The factors explain 14.7%, leaving alpha of 2.3%. Start from the 6% risk-free rate and add each loading times its factor return: 1.1 x 8 = 8.8 for the market, 0.4 x 2 = 0.8 for size, and minus 0.3 x 3 = minus 0.9 for value. The fund's 11 points over cash are mostly priced exposure; 2.3 points is left for skill.

Why not measure the fund against cash or the market alone?

A tutor whose students all score well may simply have been given the strongest students. To judge the teaching, you first adjust for who walked in. A fund with a beta above 1 and a small-cap tilt should earn more than the market in a year when the market and small caps did well, so its exposures must be priced before anything is called skill. The Fama and French three-factor model does exactly that with market, size and value.

What does the split tell you about the manager?

Against the plain market's 14%, the fund beat by 3 points. Of those 3 points, 0.7 came from priced tilts in net terms, more beta and small caps less the growth tilt, and 2.3 is left over. The negative value loading matters too: a growth-leaning fund lost 0.9 points in a year value did well, so part of the manager's alpha was earned while swimming against a factor.

Then state the limits. One year of alpha is mostly noise; you would want a regression over many years with a standard error before calling 2.3 points skill. And the answer depends on the model: add momentum or quality factors and some of the 2.3 may turn out to be another priced exposure.

A tracking error of about 1.25% a year. Tracking error is the volatility of the active weights. A's bet contributes (5% x 30%) squared, 2.25; B's contributes (5% x 25%) squared, 1.5625; and because the bets are opposite on correlated stocks, the covariance term is minus 2.25. The sum is 1.5625, whose square root is 1.25%.

Why does adding a second bet reduce the risk?

Buying an umbrella and selling a raincoat leaves you with little net exposure to rain, because both move with the weather. Overweighting A and underweighting B, when the two stocks tend to move together, is partly a hedge: when both rise, the gain on A is partly offset by the shortfall on B. Tracking error measures the risk left after that offset.

What is the neat coincidence, and what does it hide?

Here the covariance term exactly cancels A's own variance, so the answer equals B's bet alone, 5% x 25% = 1.25%. That is a coincidence of these numbers, not a rule: the covariance term is 2 x 0.6 x 30 x 25, which happens to equal 30 squared. Change the correlation to 0.5 and the answer moves. The general lesson holds, though: tracking error depends on the size of the bets and on how much they cancel.

Say the limits. Correlations are estimated and unstable, so a pair that looks like a hedge in calm markets can decouple when a stock-specific event hits. Real tracking error also includes every other small active weight, and many small bets can add up to more than one large one.

The price implies about 10.2% growth for ever, below the 12% that reinvestment could support today. Setting 25 equal to 0.4 x (1 + g) / (0.12 - g) gives g of 2.6 / 25.4. Retaining 60% at a 20% return supports 12%, but 12% for ever equals the cost of equity and would make the stock infinitely valuable, so the market is pricing in returns fading to about 17%.

What does it mean to work backwards from a multiple?

If a flat rents for Rs 30,000 a month and sells for Rs 1.2 crore, the price tells you what buyers assume about future rents, whether or not anyone says it. A P/E is the same kind of compressed forecast: fix the payout and the cost of equity, and the multiple pins down the one perpetual growth rate the price assumes. Reverse engineering the price like this is safer than forecasting growth and seeing what multiple falls out.

Is the stock cheap because it can grow faster than the price assumes?

Not so fast. Sustainable growth is return on equity times retention, 20% x 60% = 12%. But 12% growth for ever equals the cost of equity, which would make the stock worth an infinite multiple, so no price could reflect it; the gap tells you the market expects the 20% return on new investment to fade. At the same 60% retention, the price is consistent with a long-run return on equity of about 17.1%.

That turns the question into a sharper one: will this business keep earning well above 17% on new money for a long time? That is a question about competition and moats, and it is the right place to spend the next five minutes of the interview. Say also that one-stage models are crude; a two-stage model with high growth fading to a lower rate is the usual next step.

Allocation added 0.70 points, selection cost 0.50 and interaction cost 0.30, netting to the -0.1 shortfall. Allocation prices the weight bets at benchmark returns: +10% x 12 and -10% x 5. Selection prices the return gaps at benchmark weights: 50% x (10 - 12) and 50% x (6 - 5). Interaction is the weight bets times the return gaps.

Why split the result at all, if it only lagged by 0.1?

A team can lose a match by one run after a brilliant bowling spell and a dismal batting collapse; the scoreline hides both. Attribution separates the decision of how much to hold in each asset class from the decision of what to hold within it, so each can be judged on its own. Here the allocation call was right and the equity picking was poor, which is a very different story from a flat 0.1 point miss.

What is the interaction term, and why does it bite here?

Interaction is the weight bet times the return gap. The manager overweighted equities by 10 points and underperformed within them by 2 points, so the extra equity weight magnified the poor picking by 0.2; underweighting bonds, where the picking was good, gave up another 0.1. Some firms fold interaction into selection, since the weights are the manager's choice; say which convention you are using.

One more check an interviewer may ask for. The Brinson-Fachler version measures allocation against the total benchmark return, (weight bet) x (segment return minus 8.5). It gives the same total, 0.70, here, because the weight bets sum to zero, but it splits the credit between segments differently.

A wins on IRR, 30% against about 15%; B wins on NPV at 10%, Rs 22.9 crore against Rs 18.2 crore. Take B. IRR measures a rate, NPV measures rupees of value at your actual cost of capital. The two disagree because A returns its money in one year and B keeps earning for ten. Above about 11%, the ranking flips.

How can one project have the higher return and the other create more value?

Would you rather earn 30% on Rs 100 for a day, or 15% on Rs 100 for ten years? The first is a better rate; the second makes you more money. IRR is a rate per year and says nothing about how long the money earns it, while NPV counts the rupees created at your cost of capital, so when projects differ in timing the two can rank them in opposite order.

When would A be the right choice after all?

Two cases. If the cost of capital is above about 11.2%, the crossover rate, A's NPV is the higher one; and if A's Rs 130 crore could be put straight into another project earning well above 10%, the pair of projects might beat B. Both are really statements about the reinvestment rate. IRR implicitly assumes A's proceeds are reinvested at 30%; NPV assumes the cost of capital, which is usually the more honest assumption.

In real estate and private markets this is the everyday tension between a quick flip and a long hold. Give both numbers, name the crossover, and say that for mutually exclusive projects NPV decides.

The barbell wins both ways on an immediate parallel move, by about 0.12 points if yields fall and 0.10 if they rise, but it yields about 0.15 points less a year. With duration matched, convexity is the only difference: 71 for the barbell against 49 for the bullet. Over a year the barbell needs a move of roughly 113 to 123 basis points to earn back its lower yield, and it loses if the curve steepens.

Why does spreading money to the two ends add convexity?

Balancing a plank on a see-saw with one heavy child in the middle is steady; balancing it with a child at each end makes it swing more for the same push. Price sensitivity grows faster than linearly with maturity, so a mix of short and long bonds with the same average duration as a middle bond has more curvature than the middle bond: here about 71 against 49. More convexity means gaining a little more when yields fall and losing a little less when they rise.

What does the barbell give up, and when does it lose?

First, yield. Weighting the 2-year at about 50% to match duration gives a blended yield near 6.85%, about 0.15 points below the bullet's 7.0%. That is the price of convexity: if yields sit still, the bullet simply earns more, and over a year the barbell's edge only pays for itself on a parallel move of more than about 1.2 points. Second, curve shape. If the curve steepens, with short yields falling and long yields rising, the barbell's long leg loses and the bullet wins; if the 7-year sector rallies against the two ends, the bullet wins too.

State the assumptions: zero-coupon bonds, a blended yield taken as a weighted average, which is an approximation, and an instantaneous shift. The conclusion survives all three: the barbell buys protection against large parallel moves with yield, and takes on curve-shape risk.