Risk Management puzzles, solved step by step

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

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006Write NPV as the product of two vectors. The cash flows are minus 100, 30, 40, 50 and 20 in years 0 to 4, and the discount rate is 10%. What is the NPV?Duration and ratesWarm upMoody'sNew York · 2018
Try it first
Which operation turns the two vectors into the NPV?
Show the worked solution
NPV is the dot product of the cash flow vector and the discount factor vector, and here it is about 11.56. The discount factors at 10% are 1, 0.909, 0.826, 0.751 and 0.683. Multiplying term by term gives minus 100, 27.27, 33.06, 37.57 and 13.66, which sum to 11.56. A positive NPV means the project earns more than 10%.
Why is NPV a dot product at all?
Think of a grocery bill. One list holds the quantity of each item, another holds each item's price, and the bill is quantity times price for each line, added up. NPV has the same shape: one vector holds the cash flows, the other holds what one rupee in each year is worth today, and the NPV is the sum of their products. Writing it this way separates the project, which is the cash flow vector, from the market, which is the discount factor vector. Change the rate and only the second vector changes.
The relationship
$\text{NPV} = \mathbf{c} \cdot \mathbf{d} = \sum_{t=0}^{4} c_t\, d_t, \qquad d_t = \frac{1}{(1+r)^t}$
c the cash flow vector, year 0 to year 4
d the discount factor vector, one entry per year
r the discount rate, 10%
What it says in wordsMultiply each cash flow by the value today of one rupee in that year, and add the results.
Multiplying the cash flows by the discount factors year by year gives present values of minus 100, 27.27, 33.06, 37.57 and 13.66, and adding them from minus 100 upward reaches an NPV of 11.56.
What does the interviewer want to hear beyond the number?
The thought process was part of the question, so say it in order. First build the discount factor vector from the rate, then take the dot product, then sanity check the sign and size. The undiscounted inflows are 140 against 100 out, so a positive but much smaller NPV is expected once four years of 10% are taken out. And in a spreadsheet the same idea is one SUMPRODUCT of two ranges, which is why the vector form is how a model is usually built.
Then give the extension that shows range. With a term structure of rates, only the discount factor vector changes: each entry uses its own year's rate. With several scenarios, stack the cash flow vectors into a matrix and one matrix multiplication gives every scenario's NPV at once. The limitation is that the vector form assumes the cash flows are known; uncertain cash flows need expected values or scenarios first.
Where candidates lose it
Candidates reach for the NPV formula and start adding fractions, which gets the number but misses the question. The interviewer asked for two vectors precisely to see whether you can separate what the project pays from what time is worth.
The other slip is discounting year 0. The first discount factor is 1; the minus 100 is already in today's money.
What the interviewer asks next
How would you write the IRR condition using the same two vectors?
The rate for year 1 is 8% and for later years 10%. What changes in the vector form?
How would you compute the NPV for 1,000 cash flow scenarios in one operation?
Asked at Moody's, Analytics, New York, 2018 (Wall Street Oasis): Construct an NPV formula using 2 vectors and show me your thought process.
018A bond callable at 102 trades at 101. If yields fall 100 basis points, does its price rise as much as an otherwise identical non-callable bond, and what happens to its duration?Duration and ratesCoreTreasury and ALMBank market risk
Try it first
When yields fall 100 basis points, roughly how does the callable bond move?
Show the worked solution
No. The callable bond rises only about 1.0 point, to about 102.0, against about 7.2 points for the straight bond, and its duration collapses. The issuer will call the bond once refinancing is cheaper, so investors will not pay much above 102. In this stylised example its effective duration is about 3.4 against 6.8 for the straight bond, and it shrinks further as yields fall.
Why can the callable bond not keep rising?
Think of a home loan with no prepayment penalty. When rates fall, the borrower refinances, and the lender who was enjoying a high rate gets the money back. A call option lets the issuer do the same: when yields fall enough, it buys the bond back at 102, so no investor will pay much more than 102 for it. The price is effectively capped at the call price, while a straight bond with the same coupon and maturity keeps gaining as yields fall.
In this stylised 10 year 8% bond, a 100 basis point fall in yield lifts the straight bond from 101.7 to 108.9 but the callable bond only from 101.0 to 102.0, because its price flattens under the 102 call price.
What happens to duration and convexity?
Duration is how much the price moves for a yield change, and near the cap the callable bond hardly moves. As yields fall towards the level where the call is exercised, the callable bond's effective duration shrinks, and its price curve bends the wrong way: negative convexityWhen a bond gains less from a fall in yields than it loses from an equal rise, because its price curve bends downward.. Here a 100 basis point rise costs both bonds about the same, roughly 6 points, while a 100 basis point fall gives the callable bond only about 1. That lopsided payoff is the price of the call the investor has sold to the issuer, and it is paid for through a higher yield.
Be clear about the model. The curve here is a stylised cap chosen for illustration, not a full option pricing model; a desk would use an interest rate model to value the call and compute effective duration by bumping the whole yield curve. The shape is what matters for the interview: gains capped, losses intact, duration that shortens exactly when you would want it long.
Where candidates lose it
The trap is applying the straight bond's duration and predicting a 7 point gain. Standard duration assumes the cash flows are fixed, and a callable bond's cash flows change when the call is exercised.
The second miss is saying duration rises because the price is near par. Effective duration falls as the call becomes more likely, because the bond starts to behave like a short bond ending at the call date.
What the interviewer asks next
Why do mortgage-backed securities show the same pattern?
How would you hedge a portfolio of callable bonds against falling yields?
What would a putable bond's price curve look like against the same straight bond?
031A floating rate note pays a coupon that resets every three months to the market rate, and it matures in 7 years. Roughly what is its interest rate duration?Duration and ratesWarm upTreasury and ALM
Try it first
Pick the duration before you reason it out.
Show the worked solution
About 0.25 years at most, and on average about an eighth of a year. Every coupon after the next one resets to the market rate, so a change in rates today is passed straight into future coupons and leaves their value unchanged. Only the next coupon is fixed, so the note behaves like a three-month bill: modified duration of about 0.25 just after a reset, falling towards zero before the next one.
Why does maturity not drive a floater's rate risk?
Think of renting a flat with the rent reset to the market every quarter. If rents jump tomorrow, your lease is not a bargain or a burden for more than a few months, because the next reset catches up. A bond loses value when rates rise only because its cash flows are fixed below the new market rate; a floater's cash flows are not fixed beyond the next reset. At each reset date the note is worth par again, provided the issuer's credit has not changed, so for rate purposes it is a claim to par plus one known coupon, three months away.
A 7-year quarterly floater has only its next coupon fixed, so its rate duration is about 0.25 years just after a reset, while a 7-year fixed 8% bond has a modified duration of about 5.32 years and loses about twenty times as much for a one point rise in rates.
How do you check the number quickly?
Right after a reset, the note is worth the present value of par plus one coupon, received in a quarter. That is a single cash flow a quarter away, so Macaulay duration is 0.25 years and modified duration is 0.25 divided by 1.02, about 0.25. A day before the next reset it is almost zero. Compare a 7-year fixed bond paying 8% quarterly at par: its modified duration is 5.32, so a one point rate rise costs it about 5.3 per 100 against about 0.25 for the floater.
Now say where the 7 years still matter. The coupon is the market rate plus a fixed quoted marginThe fixed spread over the reference rate that a floating rate note pays, set at issue and unchanged for the life of the note.. If the issuer's credit worsens and investors demand a wider spread, that fixed margin is too low for all 28 remaining quarters, so the note's spread duration is close to a fixed bond's, several years. A treasury desk that files floaters under no rate risk and forgets spread risk has the right answer to the wrong question.
Where candidates lose it
The instinct is to answer 7 years because the note matures in 7 years, or about 5 because that is what a 7-year bond usually carries. Both confuse maturity with the length of time cash flows are fixed.
The overcorrection is saying zero. Until the next reset one coupon is locked, and the credit spread is locked for the whole life. Give the rate answer, then name spread duration unprompted.
What the interviewer asks next
What is the floater's spread duration, roughly, and why?
An inverse floater pays 16% minus the market rate. What is its duration?
How would you hedge a fixed-rate loan book funded with floating-rate deposits?
042You want a curve steepener that is neutral to parallel moves: buy the 2-year bond and short the 10-year. The 10-year has a DV01 of 0.09 per 100 of face and the 2-year has 0.019. How much 2-year do you buy for each Rs 100 crore of 10-year you short?Duration and ratesCoreTreasury and ALMBank market risk
Try it first
Roughly how much 2-year face value balances Rs 100 crore of 10-year?
Show the worked solution
About Rs 474 crore of 2-year for each Rs 100 crore of 10-year. A parallel-neutral trade matches rupees per basis point, not face value. Rs 100 crore of 10-year moves Rs 9 lakh per basis point. The 2-year moves 0.019 per 100, so matching Rs 9 lakh needs 100 x 0.09 / 0.019, Rs 473.7 crore. A parallel shift then leaves the book flat; only a change in the gap between the two yields makes or loses money.
Why size by DV01 rather than by face value?
Balancing a see-saw is about weight times distance from the pivot, not the number of children on each end. DV01 is each leg's weight: the rupees it gains or loses when its yield moves one basis point. A 10-year bond is about 4.7 times as sensitive per rupee of face as a 2-year, so a parallel-neutral trade needs about 4.7 times as much 2-year face. Equal face value would leave the book mostly a bet on the 10-year, and a parallel move would swamp the curve view.
Rs 473.7 crore of 2-year bonds bought and Rs 100 crore of 10-year bonds sold each carry Rs 9 lakh of DV01, so the trade is level under a parallel move even though the 2-year face is almost five times larger.
The relationship
$N_2 = N_{10} \times \frac{DV01_{10}}{DV01_{2}} = 100 \times \frac{0.09}{0.019} = 473.7$
N face value of each leg, Rs crore
DV01 price change per 100 of face for a one basis point yield move
What it says in wordsSet the two legs' rupees per basis point equal, then solve for the face value of the hedge leg.
What does the trade make, and what can still go wrong?
If the 10-year yield rises 10 basis points while the 2-year stays put, the curve steepens and the short 10-year gains Rs 90 lakh. If both yields rise 10 basis points together, the 2-year loses Rs 90 lakh and the 10-year short gains Rs 90 lakh: flat, which is the design. Had you used equal face, that same parallel rise would have made about Rs 71 lakh, a large outright bet you did not mean to place.
Name the limits. DV01 is a local measure and drifts as yields move and time passes, so the ratio must be rebalanced. CarryThe income a position earns or pays while it is held unchanged: coupons received less the cost of funding and of shorting. differs across the two legs and can quietly dominate a slow trade. And short and long yields rarely move by the same amount; some desks weight the 2-year leg by its historical beta to the 10-year instead of one for one, which gives a different ratio.
Where candidates lose it
The fast wrong answer is equal face value, Rs 100 crore against Rs 100 crore. It leaves the trade roughly four-fifths an outright short of the 10-year, and the first parallel sell-off or rally decides the P&L, not the curve.
Candidates also invert the ratio and answer about Rs 21 crore, dividing 0.019 by 0.09. Check the direction: the less sensitive bond always needs the bigger face.
What the interviewer asks next
The trade should also be neutral to a 1-for-0.8 move between 2-year and 10-year yields. How does the ratio change?
How much does the position make if the curve steepens by 15 basis points?
Why might a treasury desk prefer futures to cash bonds for this trade?
056Two stocks both have a 10% cost of equity. One grows its dividends at 8% a year, the other at 2%. Using the Gordon growth model, how much does each price fall if the discount rate rises by 50 basis points?Duration and ratesCoreBlackRockNew York · 2026
Try it first
Which stock falls more when the discount rate rises half a point?
Show the worked solution
The 8% grower falls 20%; the 2% grower falls about 5.9%. Under Gordon growth, price is next year's dividend over r minus g. For the fast grower that gap widens from 2% to 2.5%, so the price falls to 0.02 over 0.025, or 80% of what it was. For the slow grower the gap goes from 8% to 8.5%, and the price keeps 0.08 over 0.085 of its value.
Why does the fast grower react so much more?
Think of two ways to be paid Rs 10 lakh: most of it next year, or a trickle that grows for decades. If someone doubles the rate at which you discount the future, the trickle loses far more, because most of its money is far away. A fast-growing dividend is that trickle: its value sits in cash flows many years out. A stock whose value rests on distant cash flows behaves like a long bond, so the same rise in the discount rate cuts its price far more.
Both stocks are priced at 100 with a 10% cost of equity. A rise to 10.5% takes the 8% grower to 80, a fall of 20%, and the 2% grower to 94.1, a fall of 5.9%, because the fast grower's price rests on a gap of only 2 points between r and g.
How do you turn this into a duration number?
Differentiate the price with respect to r and divide by price: the answer is 1 over (r minus g). That gives the fast grower an equity duration of 50 years and the slow grower 12.5 years. Duration times 0.5% predicts falls of 25% and 6.25%; the exact falls are a little smaller, 20% and 5.9%, because the price curve bends, the same convexity a bond has.
The relationship
$P = \frac{D_1}{r-g} \qquad D_{\text{eq}} = -\frac{1}{P}\frac{dP}{dr} = \frac{1}{r-g}$
D_1 next year's dividend
r the cost of equity, 10%
g the constant dividend growth rate, 8% or 2%
What it says in wordsAn equity's sensitivity to the discount rate is one over the gap between the discount rate and growth.
The limitation is that Gordon growth assumes growth never changes and runs for ever, which exaggerates duration for a fast grower that will slow. The direction survives any sensible model: growth stocks carry more rate risk than stocks priced on today's cash.
Where candidates lose it
The trap is answering that both fall by about the same amount because the rate change is the same. The rate change is the same; the base it lands on is not. The fast grower's r minus g is a quarter of the slow grower's, so the same half point is four times as large relative to it.
The second slip is quoting the duration answer, 25%, as exact. Give 20% and say duration overstates it because the price curve bends.
What the interviewer asks next
What happens to each price if growth expectations for the fast grower fall to 7% at the same time?
Why might a portfolio of growth stocks behave like a long-duration bond fund?
What does equity duration mean for a pension fund that holds equities against long liabilities?
Asked at BlackRock, Risk and Quantitative Analysis, New York, 2026 (Wall Street Oasis): Which equities have duration? Technical and behavioural on VaR, market views and stock valuation.
069Portfolio A holds only 5-year zero-coupon bonds. Portfolio B holds 2-year and 8-year zeros, weighted so its duration is also 5. Which portfolio does better after a large parallel move in yields, and which one loses if the curve steepens?Duration and ratesHardTreasury and ALMBank market risk
Try it first
After a 200 basis point parallel move in either direction, which portfolio is ahead?
Show the worked solution
The barbell wins a large parallel move either way; it loses when the curve steepens. Both portfolios have duration 5, but the barbell's convexity is 34 against the bullet's 25. For a 200 basis point rise the barbell loses 9.35% against 9.52%, and for a fall it gains 10.72% against 10.52%. If the curve steepens 10 basis points per year around 5 years, the bullet is flat and the barbell loses 0.88%.
If the durations match, why do the portfolios behave differently?
Two families each spend an average of Rs 50,000 a month. One spends it evenly; the other spends Rs 20,000 some months and Rs 80,000 others. A small price rise hits them alike, but a big shock hits the second family's heavy months much harder. Averages match; the spread does not. Duration is the average maturity, so it only describes small, parallel moves; the barbell spreads its money further from 5 years, which gives it more convexity and exposes it to changes in the curve's shape.
The bullet sits at 5 years and the barbell splits evenly between 2 and 8 years, both at duration 5. The barbell does better in a 200 basis point parallel move either way because its convexity is 34 against 25, and it loses 0.88% when the curve steepens around the 5-year point while the bullet is unaffected.
Why does the steepener hurt only the barbell?
In this steepener the 5-year yield does not move, so the bullet does not move. The 2-year yield falls 30 basis points and the 8-year yield rises 30. The barbell's long leg carries four times the duration of its short leg, so the loss on the 8-year bond swamps the gain on the 2-year. Roughly: half the money times 8 years times 0.30% is a 1.2% loss, and half times 2 years times 0.30% is a 0.3% gain, a net loss of about 0.9%.
The relationship
$w\cdot 2 + (1-w)\cdot 8 = 5 \Rightarrow w = 0.5 \qquad C_{\text{barbell}} = 0.5(2^2) + 0.5(8^2) = 34 > 25 = 5^2$
w the share of value in the 2-year zero
C convexity; for a zero with continuous compounding it is maturity squared
What it says in wordsMatching duration fixes the weights at half and half, and the barbell ends up with more convexity.
The limitation: the gains from convexity are small for small moves and are usually priced, since a barbell typically yields less than a bullet of the same duration. In a quiet market that yield give-up can outweigh the convexity benefit, so the choice is a view on volatility and curve shape, not a free lunch.
Where candidates lose it
The trap is saying the two are the same risk because duration matches. Duration is one number describing one kind of move, and the interviewer built the question to see if you know the two other risks it misses: convexity and curve shape.
The second slip is getting convexity right and missing the steepener. Give both halves: the barbell wins a big parallel move and loses when the long end sells off relative to the short end.
What the interviewer asks next
Which portfolio is hurt by a flattening instead?
How would you hedge the barbell's steepener exposure?
Why does a barbell usually yield less than a bullet of the same duration?
081A bond has a modified duration of 7 and convexity of 60. Yields rise 100 basis points. Estimate the price change with and without convexity.Duration and ratesCoreTreasury and ALMBank market risk
Try it first
Does convexity make the loss bigger or smaller than the duration estimate?
Show the worked solution
Duration alone gives a -7.00% fall; adding convexity gives -6.70%. Duration is the straight-line estimate: minus 7 times 1%. Convexity adds half of 60 times 1% squared, which is 0.30%. So the bond falls about 6.7%, not 7%. The gap looks small at 100 basis points, but it grows with the square of the move.
What does each number measure?
Think of a car's speedometer and its acceleration. Speed tells you how far you will go in the next minute if nothing changes; acceleration tells you how the speed itself is changing. Duration is the speedometer of a bond: the percentage price change for a small yield move. Convexity is the acceleration: how duration itself shifts as yields move. A modified duration of 7 means about 7% of price per 1% of yield, and a convexityThe second-order sensitivity of a bond price to yield; it measures how much the price curve bends away from the duration line. of 60 says the price curve bends away from that straight line.
The relationship
$\frac{\Delta P}{P} \approx -D\,\Delta y + \tfrac{1}{2} C (\Delta y)^2 = -7(0.01) + \tfrac{1}{2}(60)(0.01)^2 = -7.00\% + 0.30\%$
D modified duration, 7
C convexity, 60
Delta y change in yield, 0.01
What it says in wordsThe price change is the straight-line duration estimate plus a convexity term that grows with the square of the yield move.
The bond's price curve bows above the straight duration line whichever way yields move; at plus 100 basis points, duration predicts -7.00% and convexity adds back 0.30%, an estimated fall of -6.70%.
When does the convexity term stop being a rounding error?
Square the move and see. At 100 basis points convexity is worth 0.30%; at 300 basis points it is nine times that, 2.70%, against a duration effect of 21%. Because the convexity term grows with the square of the move, it is small for daily risk and large in a stress scenario. A treasury stress test that uses duration alone will overstate losses on a large rate rise for a plain bond and understate them for a callable bond or a mortgage book, where convexity is negative.
Close with the sign. Positive convexity is something a bond holder pays for through a slightly lower yield, and negative convexity, from options sold to borrowers, is what makes prepayable loans harder to hedge.
Where candidates lose it
The usual slip is forgetting the half in the convexity term and adding 0.60%, which gives minus 6.4%. Write the formula before the numbers so the half is on the page.
The second is subtracting convexity because yields rose. The convexity term is a square, so it is positive for a rise and a fall alike.
What the interviewer asks next
What is the price change if yields fall 100 basis points instead?
Why does a callable bond have negative convexity at low yields?
How would you hedge the convexity of a mortgage book?

c	the cash flow vector, year 0 to year 4
d	the discount factor vector, one entry per year
r	the discount rate, 10%

N	face value of each leg, Rs crore
DV01	price change per 100 of face for a one basis point yield move

D_1	next year's dividend
r	the cost of equity, 10%
g	the constant dividend growth rate, 8% or 2%

w	the share of value in the 2-year zero
C	convexity; for a zero with continuous compounding it is maturity squared

Try it first

Why is NPV a dot product at all?

What does the interviewer want to hear beyond the number?

Where candidates lose it

What the interviewer asks next

Try it first

Why can the callable bond not keep rising?

What happens to duration and convexity?

Where candidates lose it

What the interviewer asks next

Try it first

Why does maturity not drive a floater's rate risk?

How do you check the number quickly?

Where candidates lose it

What the interviewer asks next

Try it first

Why size by DV01 rather than by face value?

What does the trade make, and what can still go wrong?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the fast grower react so much more?

How do you turn this into a duration number?

Where candidates lose it

What the interviewer asks next

Try it first

If the durations match, why do the portfolios behave differently?

Why does the steepener hurt only the barbell?

Where candidates lose it

What the interviewer asks next

Try it first

What does each number measure?

When does the convexity term stop being a rounding error?

Where candidates lose it

What the interviewer asks next