Portfolio Management puzzles, solved step by step

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

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006What are the Macaulay duration and the modified duration of a three-year bond paying a 6% annual coupon and priced at par?Bond mathsCorePIMCOLos Angeles · 2024
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Before you calculate: where does the Macaulay duration sit?
Show the worked solution
Macaulay duration about 2.83 years and modified duration about 2.67. At par the yield equals the 6% coupon, so the cash flows discount to 5.66, 5.34 and 89.00, which add to 100. Weight each year by its share of the price: 1 x 0.0566 + 2 x 0.0534 + 3 x 0.8900 gives 2.83. Divide by 1.06 for modified duration: a one point rise in yield cuts the price by roughly 2.67%.
What is duration, if not the time to maturity?
Picture a seesaw with three children sitting at the one, two and three metre marks. If the child at three metres is much heavier, the pivot that balances the seesaw sits close to three, not at the middle. Macaulay duration is that pivot: the average time you wait for your money, with each payment weighted by its present value. Here the three weights are the discounted coupons of 5.66 and 5.34 and the discounted final payment of 89.00. The last payment is so heavy that the balance point, 2.83 years, sits only two months short of maturity.
The bond's discounted cash flows of 5.66, 5.34 and 89.00 sit at years 1, 2 and 3 and balance at 2.83 years, which is the Macaulay duration; dividing by 1.06 gives a modified duration of 2.67.
Year Cash flow Present value at 6% Share of price Year x share
1 6 5.66 0.0566 0.0566
2 6 5.34 0.0534 0.1068
3 106 89.00 0.8900 2.6700
Total 118 100.00 1.0000 2.8334
Weighting each payment date by its share of the price gives a Macaulay duration of 2.8334 years.
Why divide by 1.06 to get modified duration?
Macaulay duration is a time. Modified duration is a price sensitivity: the percentage change in price for a one point change in yield. With annual compounding the two differ by a factor of one plus the yield, so 2.833 divided by 1.06 is 2.673. Say what it means: if yields rise from 6% to 7%, the bond loses about 2.67% of its price, a little less in reality because the price-yield curve bends. That bend is convexity, and it is the natural next question.
The relationship
$D_{mac}=\sum_t t\,\frac{PV_t}{P}=\frac{1(5.66)+2(5.34)+3(89.00)}{100}\approx 2.83 \qquad D_{mod}=\frac{D_{mac}}{1+y}\approx 2.67$
PV_t the present value of the payment at year t
P the bond's price, 100 at par
y the yield, 6%
What it says in wordsMacaulay duration is the value-weighted average payment date; modified duration divides it by one plus the yield to turn it into a price sensitivity.
Where candidates lose it
The fast wrong answer is three years, which is true only of a zero coupon bond. The second is averaging the dates without weighting them, which gives two. The interviewer wants the words present value weighted before any number.
Candidates also mix up the two durations. Say which is a time and which is a sensitivity, and use modified duration for any question about how much the price moves.
What the interviewer asks next
What happens to the duration if the coupon rises to 10% and the bond still trades at par?
Estimate the price if yields jump to 7%, then say whether the true price is higher or lower.
What is the duration of a three-year zero coupon bond?
Asked at PIMCO, Product & Strategy, Los Angeles, 2024 (Wall Street Oasis): Lots of random bond math questions -- duration of this bond with x coupon sold at par
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?
064A corporate bond yields 180 basis points more than a government bond of the same maturity and has a spread duration of 5. How far can its spread widen over the next year before it earns no more than the government bond?Bond mathsCoreAQR Capital ManagementGreenwich · 2021
Try it first
How much widening does the 180 basis point spread absorb?
Show the worked solution
About 36 basis points. Over a year the bond earns 180 basis points more than the government bond. Every basis point the spread widens knocks about 5 basis points off its price, because its spread duration is 5. The extra yield is used up when widening times 5 equals 180, which is at 36 basis points. Beyond that the corporate bond does worse than the government bond.
What is the cushion, and what eats it?
Think of a shop that earns a steady margin on every sale but whose stock loses value when fashions change. The margin comes in slowly; a markdown hits all at once. A credit spread pays carry slowly over the year, while widening hits the price immediately, in proportion to spread duration. The carry is 180 basis points. A spread duration of 5 means a 1 basis point widening costs about 5 basis points of price.
The bond earns 1.80% more than the government bond if spreads do not move, loses 0.05% for every basis point of widening, and so falls behind once spreads widen more than 36 basis points.
The relationship
$\Delta s^* = \frac{\text{spread}}{\text{spread duration}} = \frac{180}{5} = 36 \text{ bp}$
Delta s* the widening at which the extra return is zero
spread the extra yield over the government bond, 180 basis points
spread duration the percentage price change for a 1 point change in spread, 5
What it says in wordsDivide the extra yield by the spread duration to find how much widening it can absorb.
What would you add to sound like a credit investor?
Two refinements, both worth a sentence. First, part of the spread pays for defaults, not risk. If expected default losses were 50 basis points a year, an illustration, only 130 is true cushion and the break-even falls to 26 basis points. The spread is not all profit, so the honest break-even uses the spread after expected losses. Second, the price loss is felt at the end of the year, when the bond is shorter; at a spread duration of about 4.2 then, the break-even is closer to 43. The 36 is the conservative, quick answer.
The ratio also compares bonds quickly. A short bond with a small spread can have a wider break-even than a long bond with a big one, because the long bond's duration magnifies every move. Credit portfolio managers often rank bonds by spread per unit of spread duration for exactly this reason.
Where candidates lose it
The trap is saying 180 basis points, as if the spread could widen by its own size before the bond loses out. Candidates forget that duration multiplies every basis point of widening into a larger price loss.
The second loss is treating the whole spread as profit. Mention expected default losses; it shows you know why the spread exists in the first place.
What the interviewer asks next
A 2-year bond at 90 bp and a 10-year at 220 bp with spread duration 8: which has the wider break-even?
How does roll-down along the credit curve change this answer?
Why might a portfolio manager hold the bond even if she expects 50 bp of widening?
Asked at AQR Capital Management, Investment Research, Greenwich, 2021 (Wall Street Oasis): Discussion on credit spreads on fixed income products and 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.

Year	Cash flow	Present value at 6%	Share of price	Year x share
1	6	5.66	0.0566	0.0566
2	6	5.34	0.0534	0.1068
3	106	89.00	0.8900	2.6700
Total	118	100.00	1.0000	2.8334

PV_t	the present value of the payment at year t
P	the bond's price, 100 at par
y	the yield, 6%

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

Try it first

What is duration, if not the time to maturity?

Why divide by 1.06 to get modified duration?

Where candidates lose it

What the interviewer asks next

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

What is the cushion, and what eats it?

What would you add to sound like a credit investor?

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