Risk Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 17
Topics: 13
Hard: 30

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091Asset A has 10% volatility. Asset B has 30% volatility and a correlation of minus 0.2 with A. What weight in B minimises the portfolio's volatility, and what is that minimum?Correlation and diversificationHardAsset manager riskQuant risk
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Can adding some of the 30% volatility asset make the portfolio less risky than A alone?
Show the worked solution
About 14.3% in B, giving a minimum volatility of about 8.78%. The minimum variance weight in B is A's variance less the covariance, over the sum of the variances less twice the covariance: 160 over 1,120. The portfolio is less risky than A alone for any weight in B up to 28.6%, even though B is three times as volatile.
How can a riskier asset lower total risk?
An umbrella seller and an ice cream seller at the same beach each have bad days, but rarely the same bad days. A shopkeeper who sells mostly ice cream and a few umbrellas has steadier takings than one who sells only ice cream, even if umbrella sales swing wildly. For a small weight, the risk an asset adds depends on how it moves with what you already hold, and a negatively correlated asset offsets part of the existing risk. B's own variance enters with the square of its weight, so at small weights it barely registers, while the offset enters in proportion to the weight.
The relationship
$w_B^* = \frac{\sigma_A^2 - \rho\sigma_A\sigma_B}{\sigma_A^2 + \sigma_B^2 - 2\rho\sigma_A\sigma_B} = \frac{100 + 60}{100 + 900 + 120} = 14.3\%$
sigma_A, sigma_B volatilities 10% and 30%, so variances 100 and 900 in per cent squared
rho correlation, minus 0.2
rho sigma_A sigma_B the covariance, minus 60
What it says in wordsThe minimum-risk weight in B is A's variance less the covariance, divided by the variance of the difference between the two assets.
Portfolio volatility falls from A's 10% to a minimum of 8.78% at a 14.3% weight in the 30% volatility asset, stays below 10% until B's weight reaches 28.6%, and only then climbs toward B's own 30%.
How do you check the minimum volatility?
Plug the weight back in: 0.857 squared times 100, plus 0.143 squared times 900, plus twice 0.857 times 0.143 times minus 60. That is 73.5 plus 18.4 less 14.7, about 77.1, whose square root is 8.78%. A shortcut confirms it: the minimum variance for two assets is the product of the variances times one minus rho squared, over the same denominator, 90,000 times 0.96 over 1,120, which is 77.1. Two routes agreeing is the check the interviewer is listening for.
Then give the limit. The answer rests on a correlation estimate, and correlations drift. If the correlation moved to zero, the best weight would fall to 10% and the minimum would rise to about 9.5%; if it turned positive in a stress, the benefit would shrink further. A risk manager treats the minimum variance weight as a range, not a point.
Where candidates lose it
The instinct that a 30% volatility asset must add risk is the trap. For small weights, correlation dominates and B's own volatility barely enters.
The mechanical slip is the sign of the covariance: with a negative correlation, minus rho sigma sigma becomes plus 60 in the numerator and minus two rho sigma sigma becomes plus 120 in the denominator. Write the covariance as a number, minus 60, before substituting.
What the interviewer asks next
What is the minimum-variance weight if the correlation is zero?
At what correlation does adding any B increase risk from the start?
Why might a risk manager distrust a correlation estimated from calm years?
092A counterparty has a 2% probability of default. Your exposure to it is Rs 10 crore in normal states, but in the states where it defaults the exposure is Rs 40 crore. With 60% loss given default, compare the expected loss with and without that link.Counterparty exposure and collateralHardCounterparty riskQuant risk
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How much larger is the expected loss once exposure and default are linked?
Show the worked solution
Rs 12 lakh if independent, Rs 48 lakh when linked: four times as much. Independent: 2% times 60% times Rs 10 crore. Linked: the defaults happen exactly when exposure is Rs 40 crore, so 2% times 60% times Rs 40 crore. That link is wrong-way risk. Averaging exposure first gives only about Rs 12.7 lakh and misses it.
Why does the link between exposure and default matter so much?
Insuring a house against fire with an insurer whose only office is in the same building is poor protection: the insurer is least able to pay in exactly the event you insured against. Expected loss is probability of default times loss given default times the exposure at the moment of default, and wrong-way risk is when that exposure is largest in the states where the counterparty fails. Here the default probability and the recovery are unchanged; only the exposure in the bad states moved, and the expected loss quadrupled.
With the same 2% default probability and 60% loss given default, exposure of Rs 10 crore in every state gives an expected loss of Rs 12 lakh, but exposure that jumps to Rs 40 crore in the default states gives Rs 48 lakh, four times as much.
The relationship
$EL = PD \times LGD \times E[\text{exposure} \mid \text{default}] \qquad 0.02 \times 0.6 \times 40 = 0.48$
PD probability of default, 2%
LGD loss given default, 60%
E[exposure | default] the expected exposure in the states where default happens, Rs 40 crore
What it says in wordsUse the exposure conditional on default, not the average exposure, because only the default states produce a loss.
Where does this happen in practice?
Whenever the trade and the counterparty share a driver. A bank that sells a put on a country's currency to a bank in that country gains exposure just as that bank weakens. A lender taking a company's own shares as collateral sees the collateral fall as the company fails. The general signal is a trade whose value to you rises in the same scenario that damages the counterparty. A limit system that measures average exposure, about Rs 10.6 crore here, reports the position as a Rs 10 crore line and misses where the loss sits.
Say how desks handle it. Specific wrong-way trades are flagged and either priced with the conditional exposure, collateralised more heavily or refused. General wrong-way risk, from shared exposure to a market factor, is harder to see and needs a stress scenario in which the counterparty and the market move together.
Where candidates lose it
The common error is averaging first: 98% of 10 plus 2% of 40 is Rs 10.6 crore, and 2% times 60% of that is about Rs 12.7 lakh, barely above the independent case. It treats the link as a small adjustment when it is the whole story.
The other miss is changing the default probability instead of the exposure. The link is in the exposure; the default rate is the same in both trees.
What the interviewer asks next
Give an example of right-way risk, where exposure falls as the counterparty weakens.
How would collateral posted by the counterparty change the Rs 48 lakh?
How do banks reflect wrong-way risk in CVA?
094A bank has assets of Rs 1,000 crore with a duration of 4 and liabilities of Rs 900 crore with a duration of 1. What is the duration of its equity, and what happens to its equity if rates rise 1%?Liquidity and balance sheetHardTreasury and ALMBank market risk
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If rates rise 1%, roughly how much of the bank's Rs 100 crore of equity is lost?
Show the worked solution
The duration of equity is about 31, so a 1% rate rise costs about Rs 31 crore, 31% of equity. Assets fall 4% of Rs 1,000 crore, Rs 40 crore. Liabilities fall 1% of Rs 900 crore, Rs 9 crore. Equity absorbs the difference, Rs 31 crore on a base of Rs 100 crore. Leverage of ten to one turns a modest duration gap into a large equity sensitivity.
Why is equity so much more sensitive than either side?
A family buys a Rs 1 crore flat with Rs 10 lakh down and a Rs 90 lakh loan. If the flat's value falls 4%, they lose Rs 4 lakh, 40% of their stake, even though the flat barely moved. Equity is the thin difference between two large numbers, so any gap in how assets and liabilities respond to rates is magnified by the leverage. A bank that funds four-year assets with one-year deposits has that gap built in.
A 1% rate rise cuts assets of Rs 1,000 crore by Rs 40 crore and liabilities of Rs 900 crore by Rs 9 crore, so the Rs 100 crore equity slice falls Rs 31 crore to Rs 69 crore, a duration of equity of 31.
The relationship
$D_E = \frac{A \cdot D_A - L \cdot D_L}{E} = \frac{1000 \times 4 - 900 \times 1}{100} = 31$
A, L, E assets 1,000, liabilities 900, equity 100, in Rs crore
D_A, D_L asset duration 4 and liability duration 1
D_E duration of equity
What it says in wordsEquity's duration is the rupee duration of assets less that of liabilities, spread over the small equity base.
How do you say this in the language a treasury uses?
Two equivalent ways. The leverage-adjusted duration gap is 4 less 0.9 times 1, which is 3.1; multiply by the 1% move and by total assets, Rs 1,000 crore, and you get the same Rs 31 crore. Either way the answer is an economic value of equity loss, the number supervisors ask banks to measure for interest rate risk in the banking book. Say too that the income view is different: in the first year, deposits reprice faster than loans, so net interest income is squeezed as well.
Name the assumptions. The calculation treats the rate rise as a parallel shift, uses first-order duration without convexity, and takes the deposit duration of 1 at face value. Deposits that customers leave in place for years behave as longer-dated funding; deposits that can leave overnight behave as shorter. That behavioural assumption can move the answer more than the arithmetic.
Where candidates lose it
The common error is answering 3%, the simple gap between 4 and 1, and applying it to equity. The gap has to be weighted by the balance sheet sizes and then divided by equity, which multiplies it roughly ten times.
The second is forgetting that liabilities fall in value too. Subtract the Rs 9 crore gain on liabilities, or the answer comes out at 40% instead of 31%.
What the interviewer asks next
What liability duration would make the equity immune to a parallel rate move?
How would you use interest rate swaps to shorten the asset duration?
Why might the economic value of equity fall while net interest income rises?
096An options book on a Rs 1,000 stock has a gamma of 500 shares per rupee and theta of minus Rs 40,000 a day. How large a daily move does the book need to break even, and what annual volatility does that imply?Options and GreeksHardBank market riskQuant risk
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Roughly what daily move breaks even?
Show the worked solution
About Rs 12.65 a day, a 1.26% move, which is roughly 20% annual volatility. A delta-hedged long gamma book earns half of gamma times the move squared each day and pays theta. Setting 250 times the move squared equal to Rs 40,000 gives a move of the square root of 160. Times the square root of 250 trading days, 1.26% a day is about 20% a year.
What does a long gamma book earn, and what does it pay?
Picture a shopkeeper who pays a fixed daily rent for a stall that earns money only when a crowd passes, and earns much more from a big crowd than a small one. A quiet day loses the rent; a busy day covers it many times. A delta-hedged long option book pays theta every day and earns from gamma, which pays in proportion to the square of the stock's move, so small moves lose and large moves win. Rebalancing the hedge after each move locks in that gamma profit: buying low after a fall and selling high after a rise.
The relationship
$\tfrac{1}{2}\,\Gamma\,(\Delta S)^2 = |\Theta| \quad\Rightarrow\quad \Delta S = \sqrt{\frac{2 \times 40{,}000}{500}} = \sqrt{160} \approx 12.65$
Gamma change in delta per rupee move, 500 shares
Delta S the stock's daily move in rupees
Theta time decay paid each day, Rs 40,000
What it says in wordsThe book breaks even on the day when the gamma profit from the move equals the theta paid.
Gamma profit of 250 times the move squared crosses the Rs 40,000 daily theta at moves of plus and minus Rs 12.65, so smaller moves lose money, larger moves make it, and the breakeven corresponds to about 20% annual volatility.
Why does the breakeven turn into a volatility?
Because that is how the market priced the options. A Rs 12.65 move on a Rs 1,000 stock is 1.26% a day, and 1.26% times the square root of 250 is 20.0% a year. For a delta-hedged option book, the breakeven daily move is the implied volatility in disguise: the book profits if realised volatility beats about 20% and loses if it falls short. That is the one-line view a market risk manager wants: this book is long volatility at about 20%.
State the limits. Gamma and theta change as the stock moves and time passes, so the breakeven drifts; the relation holds day by day, not for a month in one go. And the average of squared moves is what counts, so one large day can pay for several quiet ones, which is why realised volatility, not the number of up days, decides the result.
Where candidates lose it
The common slip is dropping the half, setting 500 times the move squared equal to 40,000 and getting Rs 8.94. Write the Taylor term, one half gamma times the move squared, before any numbers.
The second miss is stopping at Rs 12.65. The interviewer wants the translation into volatility, because that is how a risk manager describes the position: long volatility at about 20%.
What the interviewer asks next
Realised volatility comes in at 15% for a month. Roughly what does the book lose?
How do gamma and theta change as expiry approaches?
Why is a short gamma book said to pick up coins in front of a steamroller?
097A client offers you a game: a fair coin is flipped until the first tail, and you are paid Rs 2 raised to the number of flips. The client can pay at most Rs 1 crore. What is the fair price of the game?Probability and base ratesHardQuant riskCounterparty risk
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Roughly what is the capped game worth?
Show the worked solution
About Rs 24.19. A game that ends on flip n pays 2 to the n with probability one half to the n, so each flip count adds exactly Rs 1. Uncapped, that sum is infinite. The client can pay only Rs 1 crore, which 2 to the 23 stays below, so 23 flip counts add Rs 23 and every longer game pays Rs 1 crore, adding about Rs 1.19.
Why is the uncapped game worth an infinite amount?
Picture a prize that doubles every time you survive another round, while the chance of surviving halves. Each round's prize times its chance is always the same Rs 1. In this game every possible length contributes exactly Rs 1 to the expected value, one Rs 1 for each flip count, forever, so the sum never stops. This is the St Petersburg paradox: the arithmetic says pay anything, yet nobody would pay even Rs 100.
Each flip count from 1 to 23 adds exactly Rs 1 to expected value, but beyond the payer's Rs 1 crore cap the contributions halve each flip and add only Rs 1.19 in total, so the capped game is worth about Rs 24.19.
How does the cap turn infinity into Rs 24?
Find where the cap bites: 2 to the 23 is Rs 83,88,608 and 2 to the 24 is over Rs 1 crore, so the first 23 flip counts pay in full. Those contribute Rs 23; every longer game pays the capped Rs 1 crore, and the chance of lasting past 23 flips is one half to the 23, so the tail adds Rs 1 crore divided by 2 to the 23, about Rs 1.19. The fair price is about Rs 24.19.
The relationship
$EV = \sum_{n=1}^{23} 2^{-n} \cdot 2^{n} + 10^{7} \cdot 2^{-23} = 23 + 1.19 = 24.19$
2^{-n} the chance the first tail comes on flip n
2^n the payout if it does
10^7 the payer's cap, Rs 1 crore
What it says in wordsEvery flip count below the cap adds one rupee; the capped tail adds the cap times the chance of reaching it.
Why is this a counterparty risk question?
Because the value came entirely from payouts the client could never make. A promised payoff is worth only what the payer can actually pay, and most of this game's theoretical value sat in states where the payer would default. Raising the cap a thousandfold to Rs 1,000 crore only lifts the value to about Rs 34.16, because each doubling of capacity adds just one rupee. The same logic prices protection bought from a seller who could not survive the event it insures.
Where candidates lose it
The tempting answer is infinity, or a large number near the cap, because the uncapped maths is famous. The question gives the cap precisely to see whether you use it.
The other slip is stopping at Rs 23 and forgetting the capped tail, or adding Rs 1 crore for every long game rather than weighting it by the chance of getting there.
What the interviewer asks next
What would you pay if the client could pay at most Rs 1,000 crore?
Why might a risk-averse person pay far less than the expected value even with the cap?
Where do you see payoffs that are only as good as the payer's capacity in real markets?
100A firm's assets are worth Rs 100 crore with 25% volatility and 5% expected growth; Rs 70 crore of debt is due in one year. What is the distance to default, and what default probability does it imply?Credit risk arithmeticHardQuant riskBank credit risk
Try it first
Roughly what one-year default probability does the Merton model give?
Show the worked solution
A distance to default of about 1.50, implying a one-year default probability of about 6.7%. In the Merton model the firm defaults if assets end below the debt. The log of 100 over 70 is 0.357; add growth less half the variance, 0.019; divide by 25% volatility to get 1.50 standard deviations. The normal tail beyond that is 6.66%.
What is the model saying in plain words?
A homeowner with a Rs 70 lakh loan on a Rs 1 crore house is in trouble at repayment only if the house is then worth less than Rs 70 lakh. How likely that is depends on the cushion, Rs 30 lakh, and on how much house prices swing. The Merton model treats a firm the same way: default happens if the value of its assets at the debt's maturity falls below what it owes, so the default probability is the share of possible asset values that land below the debt. Distance to default is that cushion measured in standard deviations.
With assets of Rs 100 crore, 25% volatility and 5% growth, the one-year asset value centres near Rs 101.9 crore and only the tail below the Rs 70 crore of debt, 6.66% of outcomes, ends in default, a distance to default of 1.50.
The relationship
$DD = \frac{\ln(V/D) + (\mu - \tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} = \frac{0.357 + 0.019}{0.25} = 1.50 \qquad PD = N(-1.50) = 6.7\%$
V asset value today, Rs 100 crore
D debt due, Rs 70 crore
mu expected asset growth, 5%
sigma asset volatility, 25%
N the standard normal distribution
What it says in wordsDistance to default is the log cushion plus expected drift, in units of asset volatility; the default probability is the normal tail beyond it.
Why not just divide the cushion by volatility?
The simple version, Rs 30 crore of cushion over Rs 25 crore of one-year volatility, gives 1.2 standard deviations and a default probability of about 11.5%. It ignores two things that both favour the lender: assets are expected to grow 5%, and a lognormal asset value cannot fall as easily in rupees as it can rise, so the log cushion is wider than the simple ratio suggests. The simple form is still a useful first number in the room, as long as you say which way it errs.
Then give the model's limits. Asset value and asset volatility are not observed; in practice they are backed out from the equity price and equity volatility. The model assumes default happens only at maturity, a single debt payment, and normal log returns, and it tends to produce very low default probabilities for safe firms over short horizons. Commercial versions map distance to default onto historical default rates rather than trusting the normal tail.
Where candidates lose it
The common error is using the simple ratio, 1.2 standard deviations, and quoting 11.5% as the Merton answer. It is a quick estimate, not the model; show the log form or say that you are approximating.
The second is forgetting the minus half sigma squared term. It is small here, 0.031, but leaving it out, or adding it, moves the distance to default and shows the interviewer the formula is memorised rather than understood.
What the interviewer asks next
Asset volatility rises to 35%. What happens to the default probability?
How would you estimate asset value and asset volatility from the share price?
Why does the Merton model tend to understate short-term default risk?

sigma_A, sigma_B	volatilities 10% and 30%, so variances 100 and 900 in per cent squared
rho	correlation, minus 0.2
rho sigma_A sigma_B	the covariance, minus 60

PD	probability of default, 2%
LGD	loss given default, 60%
E[exposure \| default]	the expected exposure in the states where default happens, Rs 40 crore

A, L, E	assets 1,000, liabilities 900, equity 100, in Rs crore
D_A, D_L	asset duration 4 and liability duration 1
D_E	duration of equity

Gamma	change in delta per rupee move, 500 shares
Delta S	the stock's daily move in rupees
Theta	time decay paid each day, Rs 40,000

Try it first

How can a riskier asset lower total risk?

How do you check the minimum volatility?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the link between exposure and default matter so much?

Where does this happen in practice?

Where candidates lose it

What the interviewer asks next

Try it first

Why is equity so much more sensitive than either side?

How do you say this in the language a treasury uses?

Where candidates lose it

What the interviewer asks next

Try it first

What does a long gamma book earn, and what does it pay?

Why does the breakeven turn into a volatility?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the uncapped game worth an infinite amount?

How does the cap turn infinity into Rs 24?

Why is this a counterparty risk question?

Where candidates lose it

What the interviewer asks next

Try it first

What is the model saying in plain words?

Why not just divide the cushion by volatility?

Where candidates lose it

What the interviewer asks next