Risk Management puzzles, solved step by step
- Puzzles
- 100
- Traced to a firm
- 17
- Topics
- 13
- Hard
- 30
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?Treasury 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?
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?Treasury 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 relationshipN 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?BlackRockNew 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 relationshipD_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.
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.Treasury 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 relationshipD 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?