Portfolio Management puzzles, solved step by step

Because the observations overlap, the honest t-statistic is close to 1, about 1.04. Ten years of monthly data give 109 rolling 12-month windows, but each shares 11 months with its neighbour. OLS assumes independent errors, so it treats 109 observations as 109 pieces of information. With the overlap, the variance of the estimate is understated about twelvefold, and 3.6 over the square root of 12 is about 1.0.

Which OLS assumption breaks?

Picture asking the same twelve people for their opinion, then swapping one person each month and asking again. You would not claim 109 independent surveys. OLS standard errors assume the regression errors are uncorrelated with each other, and overlapping windows make them strongly correlated, so the standard errors come out far too small. The alpha estimate itself is not biased; what is wrong is the claim about how precisely it is known, which is exactly what the t-statistic reports.

How do you get to about 1?

Add up the overlap. If monthly returns are independent, two 12-month windows k months apart have a correlation of 12 minus k over 12. The variance of an average of such overlapping sums is inflated by one plus twice the sum of those correlations, which here is 1 plus 2 times 5.5, or 12. The t-statistic shrinks by the square root of the inflation factor, 3.6 divided by the square root of 12. That gives about 1.04: nowhere near significant, and consistent with having only about ten independent years.

How would you fix it? Three answers, in order of simplicity. Run the regression on non-overlapping monthly returns, which uses all the data with independent errors. Or keep the overlap and use standard errors that allow for autocorrelation, such as Newey-West or Hansen-Hodrick with 11 lags. Or sample the 12-month returns once a year, which leaves only ten points but honest ones. The interviewer wants to hear that the fix is about the standard errors, not the coefficient.

Effective duration is about 1.7 and effective convexity about -200. Duration is the price difference across the two shocks over twice the price times the shock: 1.7 over 1.0, which is 1.7. Convexity is the sum of the shocked prices less twice the base, over the price times the shock squared: minus 0.5 over 0.0025, which is -200. The call caps the price, so it gains less than it loses.

Why effective duration rather than the usual formula?

Think of renting out a flat on a lease the tenant can cancel whenever cheaper flats appear. When rents fall, the tenant leaves and you do not keep the high rent; when rents rise, you are stuck. A callable bond's cash flows change with yields, because the issuer calls it when rates fall, so you measure its duration from how its price actually moves, not from a fixed schedule of coupons. That is effective duration: shock the yield both ways, reprice, and read the slope.

What does negative convexity cost the holder?

It means the bond loses more when yields rise than it gains when they fall. The holder has sold the issuer an option to refinance, and the price of that option is the upside given up when rates fall. A straight bond with a duration of about 4 would gain about 2.0 points for a 50 basis point fall; this one gains 0.6. The holder is paid for this through a higher yield than an equivalent straight bond, and the question for a portfolio manager is whether that extra yield covers the option given away.

Two more things are worth saying. The duration of 1.7 is not fixed: as yields fall towards the level where a call becomes likely, duration shrinks further, and as they rise, it lengthens towards the bond's straight duration. That shifting is why callable bonds, and mortgage securities with the same feature, need effective measures rather than the textbook formulas. The curve in the figure is a stylised fit around the three prices given; a real one would come from an option pricing model.

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.