Portfolio Management puzzles, solved step by step

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

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019The three-year government yield is 7.0% and the two-year yield is 6.6%. You buy a three-year bond with a 7% coupon at par. If the yield curve does not move over the next year, roughly what does the bond return?Bond mathsCoreFixed income
Try it first
Pick the closest one-year return.
Show the worked solution
About 7.7%. You collect the 7 coupon, and a year later the bond is a two-year bond. If the curve has not moved, it is priced at the two-year yield of 6.6%, which makes it worth 100.73. Coupon 7.00 plus a price gain of 0.73 is 7.73%. A shortcut gives the same: 7% plus the two-year duration of about 1.82 times the 0.4 point fall in yield.
Why does an unchanged curve still move the bond's price?
Think of walking down a gentle slope while standing still relative to the hillside: the ground stays put, but you end up lower because you moved along it. A bond ages along the curve; if the curve slopes up and stays still, the bond's yield falls as its maturity shortens, and its price rises. Today it is a three-year bond at 7.0%. In a year it will be a two-year bond, and two-year bonds yield 6.6%. That 0.4 point fall in yield is the roll downThe price gain a bond earns as it ages into a shorter maturity with a lower yield on an upward sloping curve..
The three-year bond bought at 7.0% becomes a two-year bond priced at 6.6% a year later, so an unchanged curve returns the 7.00 coupon plus a 0.73 price gain, 7.73% against 7.00 on a flat curve.
How do you check it without a calculator?
Use duration. A two-year bond with a 7% coupon has a modified duration of about 1.82, so a 0.4 point fall in yield lifts its price by about 1.82 x 0.4, which is 0.73. Add the 7 coupon and the one-year return is about 7.73%, within a hair of the exact 7.73%. Say the limitation: the answer depends entirely on the curve staying put. If two-year yields rise to 7.0% by next year, the roll down disappears and the return is the coupon alone.
The relationship
$P_1=\frac{7}{1.066}+\frac{107}{1.066^2}\approx 100.73 \qquad r=\frac{7+(100.73-100)}{100}\approx 7.73\%$
P_1 the bond's price in a year, as a two-year bond at a 6.6% yield
7 the annual coupon
100 the price paid today, at par
What it says in wordsThe one-year return is the coupon plus the price gain from repricing at the lower two-year yield, over the price paid.
Where candidates lose it
The common answer is 7%, the yield at purchase, which is right only if the curve is flat. The interviewer asked about an unchanged curve precisely to see whether you notice that the bond moves along it.
The overshoot is adding the whole slope of the curve or forgetting that the price gain depends on duration. The gain is duration times the yield change, not the yield change alone.
What the interviewer asks next
What would the return be if the curve were inverted, with the two-year at 7.4%?
Which point on this curve gives the most roll down per unit of duration?
How much must the two-year yield rise over the year to wipe out the roll down?
034A five-year bond pays a 7% annual coupon and trades at 95. Without a calculator, estimate its yield to maturity.Bond mathsCoreFixed income
Try it first
Which is closest to the yield to maturity?
Show the worked solution
About 8.2%. The bond pays 7 a year and also rises from 95 to 100 by maturity, which is about 1 point a year over five years. That is 8 a year of return on money that averages about 97.5 invested, and 8 over 97.5 is 8.21%. The exact yield with annual coupons is 8.26%, so the shortcut is within a few hundredths of a point.
Where does the return on a discount bond come from?
Buy a Rs 100 gift voucher for Rs 95 that also pays you Rs 7 each year until it can be cashed at full value in five years. You collect the Rs 7 every year, and you also pocket the Rs 5 discount at the end. A discount bond's yield is its coupon plus the discount it recovers as the price is pulled to par, spread across the years to maturity. Spread evenly, that is 1 point a year, so the bond earns roughly 8 a year.
The five-year 7% bond is priced at par when its yield is 7%, and at 95 its yield is 8.26%. The shortcut of coupon plus yearly pull to par, divided by the average price, gives 8.21%, while the current yield of 7.37% misses the pull to par entirely.
Why divide by the average price rather than 95?
The amount you have invested is not fixed at 95: in this rough picture the bond's value drifts up towards 100 over the five years, so the capital at work averages about 97.5. Dividing by the average price corrects most of the error from spreading the discount in a straight line. Dividing 8 by 95 instead gives 8.42%, too high. As a check, a bond priced at 8.21% comes out at 95.21, within a fraction of 95.
The relationship
$y \approx \frac{C + (F - P)/n}{(F + P)/2} = \frac{7 + 5/5}{97.5} = 8.21\%$
C the annual coupon, 7
F the face value repaid at maturity, 100
P today's price, 95
n years to maturity, 5
What it says in wordsYearly income plus the yearly share of the discount, divided by the average amount invested.
Know when the shortcut drifts. It is close for short bonds near par and gets worse for long maturities or deep discounts, where the true discounting curve bends away from a straight line. For a quick answer in the room, 8.2% with the one-line derivation is what the interviewer is after; then say that the exact figure is a touch higher.
Where candidates lose it
The common wrong answers are 7%, which confuses coupon with yield, and 7.37%, the current yield, which forgets that the bond will be repaid at 100. A few candidates add the whole 5-point discount to a single year and say 12%.
Say the two sources of return aloud, coupon and pull to par, then give the formula. The direction check helps too: the bond trades below par, so its yield must be above the coupon.
What the interviewer asks next
The same bond trades at 105. Estimate its yield.
Why does the shortcut get worse for a 20-year bond at 80?
If yields rise one point from here, roughly what happens to the price?
076A Rs 100 crore bond portfolio has a modified duration of 5. What is its DV01, and roughly what does a 25 basis point rise in yields cost?Bond mathsWarm upFixed incomeRisk management
Try it first
Say the DV01 before you work it through.
Show the worked solution
DV01 is Rs 5 lakh, so a 25 basis point rise costs about Rs 1.25 crore. Modified duration of 5 means the value moves about 5% for each percentage point of yield, or 0.05% per basis point. 0.05% of Rs 100 crore is Rs 5 lakh, and 25 basis points is 25 times that.
Why turn duration into rupees at all?
A shopkeeper who says the rent went up 4% has told you something; one who says it went up Rs 2,000 a month has told you what to do about it. Duration is the percentage version. DV01, the rupee change for a one basis point move, is the version a desk can add across positions, compare with a limit and put in a risk report. Two portfolios with the same duration but different sizes carry very different rupee risk, and only DV01 shows it.
Rs 100 crore times a modified duration of 5 times 0.0001 is Rs 5 lakh per basis point, so a 25 basis point rise in yields takes Rs 1.25 crore off the portfolio, one Rs 5 lakh notch at a time.
The relationship
$\text{DV01} = V \times D_{mod} \times 0.0001 = 100 \times 5 \times 0.0001 = 0.05 \text{ crore}$
V portfolio value, Rs 100 crore
D_mod modified duration, 5
0.0001 one basis point written as a decimal
What it says in wordsThe rupee move for one basis point is the value times the duration times one hundredth of one per cent.
Where does the straight-line answer stop being right?
The modified durationThe percentage change in a bond price for a one percentage point change in its yield, taken at the current yield. estimate is a tangent: it treats the price and yield relationship as a straight line. For 25 basis points the straight line is close enough, but for a 200 basis point shock the curve bends away from it and the true loss is smaller than DV01 times 200. That bend is convexity. Mention it in one sentence; the interviewer will often ask for it next.
Say also that DV01 assumes every yield in the portfolio moves by the same amount. If short rates rise and long rates stay still, a single DV01 number misses it, which is why desks also keep DV01 by maturity bucket.
Where candidates lose it
The common slip is out by a factor of 100: treating duration 5 as 5% per basis point and saying Rs 5 crore. The interviewer hears that you have not held a real rate position, where the size of one basis point is the first thing you learn.
The second loss is stopping at the number. Say that DV01 is linear, so it overstates the loss for large rises, and that it assumes a parallel move.
What the interviewer asks next
The portfolio doubles in size and duration falls to 2.5. What is DV01 now?
How would you cut DV01 by half without selling any bonds?
Why might a desk set a limit in DV01 rather than in duration?
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.
100You can hold a 7-year bullet yielding 7.0%, or a barbell of 2-year bonds at 6.5% and 12-year bonds at 7.2% with the same duration. Which does better if yields shift 100 basis points in parallel, either way? What does the barbell give up?Bond mathsHardFixed incomeInstitutional asset management
Try it first
On an immediate 100 basis point parallel move, up or down, which wins?
Show the worked solution
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.
The bullet and barbell price changes almost coincide across parallel shifts, but the barbell's advantage grows with the square of the move, from about 0.10 to 0.12 points at 100 basis points to over 0.7 at 300; it only clears the 0.15 points of yield it gives up each year beyond moves of roughly 113 to 123 basis points.
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.
Bullet Barbell
Yield, blended 7.00% 6.85%
Modified duration 6.54 6.54
Convexity 48.9 70.6
Yields fall 100 bp +6.79% +6.91%
Yields rise 100 bp -6.30% -6.20%
Priced as zero-coupon bonds with modified duration matched at 6.54, the barbell gains 6.91% against 6.79% on a 100 basis point fall and loses 6.20% against 6.30% on a rise, while yielding 0.15 points less.
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.
Where candidates lose it
The common slip is saying same duration, same result. Duration matching removes only the first-order difference, and the question is about what is left over.
The second loss is praising the barbell without the cost. It gives up yield and carries curve risk, and a candidate who names both sounds like someone who has run a bond book.
What the interviewer asks next
How would you build the barbell to match both duration and yield? What would you give up?
Which does better if the curve flattens with no change in the 7-year yield?
Why do liability-driven investors often prefer bullets matched to their payment dates?

P_1	the bond's price in a year, as a two-year bond at a 6.6% yield
7	the annual coupon
100	the price paid today, at par

C	the annual coupon, 7
F	the face value repaid at maturity, 100
P	today's price, 95
n	years to maturity, 5

V	portfolio value, Rs 100 crore
D_mod	modified duration, 5
0.0001	one basis point written as a decimal

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

Try it first

Why does an unchanged curve still move the bond's price?

How do you check it without a calculator?

Where candidates lose it

What the interviewer asks next

Try it first

Where does the return on a discount bond come from?

Why divide by the average price rather than 95?

Where candidates lose it

What the interviewer asks next

Try it first

Why turn duration into rupees at all?

Where does the straight-line answer stop being right?

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 spreading money to the two ends add convexity?

What does the barbell give up, and when does it lose?

Where candidates lose it

What the interviewer asks next