Portfolio Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 31
Topics: 13
Hard: 30

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082Let X be the market's daily return, symmetric around zero, and let Y equal X squared, the shape of a long straddle's profit. What is the correlation between X and Y? Are they independent?Statistics and forecastingCoreQuantitative asset managementHedge funds
Try it first
What is the correlation?
Show the worked solution
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.
Plotting Y equal to X squared for returns of minus 2% to plus 2% gives a U of 4, 1, 0, 1, 4, and the best-fit straight line through those points is flat at 2, so the correlation is zero although Y is fixed exactly by X.
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.
The relationship
$\text{Cov}(X, X^2) = E[X^3] - E[X]\,E[X^2] = 0 - 0 = 0$
E[X] the average return, zero by symmetry
E[X^3] the average cubed return, also zero by symmetry
What it says in wordsFor a symmetric variable, the covariance with its own square is zero.
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.
Where candidates lose it
The trap is equating zero correlation with independence. Independence implies zero correlation; the reverse fails, and this question is the standard counter-example.
The second trap is saying the correlation is high because Y is a function of X. Correlation measures a straight-line fit, and a symmetric U has no slope.
What the interviewer asks next
What if X is skewed, with bigger falls than rises? Is the correlation still zero?
Name a measure that would detect this dependence.
Why can hedge fund returns look uncorrelated with equities and still lose money in a crash?
084A bank earns a 16% return on equity, grows at 8% a year and has a 13% cost of equity. What price to book is justified? What happens if its return on equity falls to 13%?Valuation riddlesCoreFundamental asset managementIndian equity research
Try it first
At a 13% return on equity, what is the justified price to book?
Show the worked solution
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.
With growth at 8% and a 13% cost of equity, the justified price to book is 1.0 times where return on equity equals 13% and rises to 1.6 times at 16%; below 13% the line falls under book, because growth then destroys value.
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 relationship
$\frac{P}{B} = \frac{ROE - g}{r - g} = \frac{0.16 - 0.08}{0.13 - 0.08} = 1.6$
ROE return on equity, 16%
g long-run growth, 8%
r cost of equity, 13%
What it says in wordsThe premium to book is the return spread over the cost spread, both measured from the growth rate.
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.
Where candidates lose it
The common slip is scaling: 13 is less than 16, so the multiple falls in proportion to about 1.3 times. The relationship is not proportional; it is anchored at 1.0 times where the two rates meet.
The deeper miss is not seeing that growth is only good when return on equity beats the cost of equity. Below 13%, faster growth lowers the multiple, and saying that out loud is what the question is testing.
What the interviewer asks next
The bank earns 11% on equity. What happens to the multiple if growth rises from 8% to 10%?
What return on equity does a price to book of 2.0 times imply at the same growth and cost of equity?
Why do lenders with similar returns on equity trade at very different multiples?
085An investor puts Rs 10 lakh into a fund that gains 20% in year one. Just before year two, the investor adds Rs 40 lakh, and the fund then falls 10%. What is the fund's time-weighted return, and how did the investor actually do?Performance measurementCorePerformance analysisWealth management
Try it first
Over the two years, how did the investor do in rupees?
Show the worked solution
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.
Rs 10 lakh grew to Rs 12 lakh, the investor added Rs 40 lakh, and the Rs 52 lakh then fell 10% to Rs 46.8 lakh, so the fund shows plus 8% time-weighted while the investor lost Rs 3.2 lakh, about minus 5.4% a year money-weighted.
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.
The relationship
$10(1+m)^2 + 40(1+m) = 46.8 \;\Rightarrow\; m \approx -5.4\%$
m the money-weighted return, a yearly rate
10, 40 the investor's contributions, Rs lakh, two years and one year before the end
46.8 the ending value, Rs lakh
What it says in wordsThe money-weighted return is the one rate that grows the investor's contributions into the ending value.
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%.
Where candidates lose it
The trap is reporting plus 8% as the investor's experience. The interviewer wants to hear that the fund's return and the investor's return are different quantities, and why.
The second slip is computing the investor's figure as a simple return on the total, Rs 3.2 lakh over Rs 50 lakh. That ignores that Rs 10 lakh was invested for two years and Rs 40 lakh for one. Name the internal rate of return as the correct tool.
What the interviewer asks next
Reverse it: the investor adds Rs 40 lakh after the fall instead. What changes?
Which return should appear on a fund factsheet, and which on a client statement?
Why do investors in aggregate often earn less than the funds they hold?
089A 10-year bond has a modified duration of 7 and convexity of 60. Estimate its price change for a 100 basis point fall in yield, and for a 100 basis point rise.Bond mathsCoreFixed income
Try it first
Which pair of estimates is right?
Show the worked solution
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.
The duration-only tangent predicts a 7% move each way, but the curved estimate that adds convexity sits above it on both sides, giving +7.3% for a 100 basis point fall and -6.7% for a 100 basis point rise, with the gap widening for bigger moves.
The relationship
$\frac{\Delta P}{P} \approx -D_{mod}\,\Delta y + \tfrac12\,C\,(\Delta y)^2 = -7(\pm 0.01) + 30(0.0001)$
D_mod modified duration, 7
C convexity, 60
Delta y change in yield, plus or minus 0.01
What it says in wordsThe price change is the duration effect plus a small positive bend that grows with the square of the move.
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.
Where candidates lose it
The common slip is attaching the convexity term with the sign of the yield move, giving +6.7% and -7.3%. That shows the candidate memorised a formula without seeing that the squared term cannot be negative.
The other loss is saying the answer is symmetric at 7% each way. Duration alone is a straight line; naming convexity is the point of the question.
What the interviewer asks next
Why does a callable bond have negative convexity at low yields?
Two bonds have the same duration; one has higher convexity. Which would you rather own, and what does it cost?
Estimate the price change for a 250 basis point rise.
090Two retirees each start with Rs 1 crore and withdraw Rs 10 lakh at the end of every year. One earns minus 20%, plus 10% and plus 30% over three years; the other earns the same returns in reverse order. Who ends with more, and by how much?Compounding and fee dragCoreRetirement and pensionsWealth management
Try it first
Who ends with more?
Show the worked solution
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.
Starting from Rs 1 crore with Rs 10 lakh withdrawn each year, the bad-year-first path ends at Rs 77.1 lakh and the good-year-first path at Rs 87.6 lakh, although both would reach Rs 114.4 lakh with no withdrawals.
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.
Withdrawal Bad year first: cost at the end Good year first: cost at the end
End of year 1 14.3 8.8
End of year 2 13.0 8.0
End of year 3 10.0 10.0
Total taken from the no-withdrawal Rs 114.4 lakh 37.3 26.8
Each Rs 10 lakh withdrawal costs the ending balance its own value grown at the returns it missed, which totals Rs 37.3 lakh in the bad-first order and Rs 26.8 lakh in the good-first order.
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.
Where candidates lose it
The trap is answering that order cannot matter because multiplication commutes. That is true for a lump sum left alone and exactly wrong once money moves.
The second loss is getting the two balances without explaining why. Give the one-line mechanism: each withdrawal misses the returns that follow it.
What the interviewer asks next
What happens to the gap if the withdrawal doubles to Rs 20 lakh?
Does the order matter for someone making equal yearly contributions instead?
How would you protect a new retiree against a bad first year?
091A trader bets on a fair coin, starting at Rs 1,000 and doubling the stake after every loss until a win, then starting again. The trader has Rs 63,000. What does each cycle win, how often does it wipe out, and what is the expected value?Behavioural and decision trapsCoreHedge fundsRisk management
Try it first
What is the expected value of one cycle?
Show the worked solution
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.
Six straight losses with doubling stakes of Rs 1,000 to Rs 32,000 use up the whole Rs 63,000, and the seventh stake cannot be placed; the cycle wins Rs 1,000 sixty three times in sixty four and loses Rs 63,000 once, for an expected value of zero.
The relationship
$E = \tfrac{63}{64}(1{,}000) - \tfrac{1}{64}(63{,}000) = 984.4 - 984.4 = 0$
63/64 chance of at least one head in six tosses
1/64 chance of six tails in a row
What it says in wordsFrequent small wins and a rare large loss cancel exactly on a fair coin.
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.
Where candidates lose it
The trap is being impressed by the 98.4% win rate. The interviewer is checking whether you weigh outcomes by size as well as frequency.
The second loss is saying the strategy is negative in expectation. On a fair coin it is exactly zero; the harm is in the shape, a small chance of ruin, not in the average. Say both halves.
What the interviewer asks next
The coin pays slightly less than even money. What happens to the expected value?
Where do you see this payoff shape in real portfolios?
How would you size risk so that no single loss can end the strategy?
092The dollar-rupee spot rate is 84, one-year rupee interest rates are 6.5% and one-year dollar rates are 4.5%. What is the one-year forward rate, and what does hedging the currency do to an Indian investor's return on a dollar deposit?Currency and global returnsCoreGlobal investingFixed income
Try it first
For the Indian investor holding dollars, is the hedge a cost or a gain?
Show the worked solution
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.
Rs 84 invested at 6.5% grows to Rs 89.46, and converting to one dollar, earning 4.5% and selling the $1.045 forward must reach the same Rs 89.46, which fixes the forward at 85.61, a 1.9% premium to spot.
The relationship
$F = S \times \frac{1 + r_{Rs}}{1 + r_{\$}} = 84 \times \frac{1.065}{1.045} \approx 85.61$
S spot rupees per dollar, 84
r_Rs one-year rupee rate, 6.5%
r_$ one-year dollar rate, 4.5%
What it says in wordsThe forward premium on the dollar equals, roughly, the gap between rupee and dollar interest rates.
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.
Where candidates lose it
The common slip is calling the 1.9% a cost for the Indian investor. The sign depends on which currency you are selling forward, and the interviewer asks this way round precisely to catch it.
The second slip is treating the forward as the market's forecast of the rupee. It is an arbitrage price set by interest rates, and saying so separates you from candidates who memorised the formula.
What the interviewer asks next
Rupee rates rise to 7.5%. What happens to the forward, and to the hedged return?
Why do foreign investors in Indian bonds often leave the currency unhedged?
What would you do if the quoted forward were 86.5?

E[X]	the average return, zero by symmetry
E[X^3]	the average cubed return, also zero by symmetry

ROE	return on equity, 16%
g	long-run growth, 8%
r	cost of equity, 13%

m	the money-weighted return, a yearly rate
10, 40	the investor's contributions, Rs lakh, two years and one year before the end
46.8	the ending value, Rs lakh

D_mod	modified duration, 7
C	convexity, 60
Delta y	change in yield, plus or minus 0.01

Try it first

How can a perfect relationship show zero correlation?

What does the algebra say?

Where candidates lose it

What the interviewer asks next

Try it first

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

Where does the formula come from?

Where candidates lose it

What the interviewer asks next

Try it first

How can both numbers be right?

How do you get the money-weighted return?

Where candidates lose it

What the interviewer asks next

Try it first

Why does convexity help in both directions?

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

Where candidates lose it

What the interviewer asks next

Try it first

Why does the order suddenly matter?

Where exactly does the Rs 10.5 lakh go?

Where candidates lose it

What the interviewer asks next

Try it first

Why does it feel like a sure thing?

What does this teach about judging a strategy?

Where candidates lose it

What the interviewer asks next

Try it first

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

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

Where candidates lose it

What the interviewer asks next