Portfolio Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 31
Topics: 13
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
047A Rs 100 crore bond portfolio holds Rs 40 crore of a bond with duration 2, Rs 35 crore with duration 5 and Rs 25 crore with duration 12. What is the portfolio's duration, and how much must move from the 2-year bond into the 12-year bond to lift it to 7?Bond mathsHardPIMCOLos Angeles · 2026
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How much must move from the duration-2 bond to the duration-12 bond to lift portfolio duration from 5.55 to 7?
Show the worked solution
Duration is 5.55, and Rs 14.5 crore must move from the 2-year bond to the 12-year bond. Portfolio duration is the value-weighted average: (40 x 2 + 35 x 5 + 25 x 12) over 100, which is 555 over 100, or 5.55. Each Rs 1 crore switched gains 10 years of duration on a hundredth of the portfolio, adding 0.1. Closing a gap of 1.45 needs Rs 14.5 crore.
Why is portfolio duration a simple weighted average?
Think of the average age of people in a room: each person counts in proportion to how many of them there are. Duration measures how much a bond's price moves for a one-point change in yield, and for a small parallel move the portfolio's rupee loss is just the sum of each bond's rupee loss, so its duration is the value-weighted average of the bonds' durations. The Rs 25 crore in the 12-year bond is only a quarter of the money but supplies more than half the duration: 3.00 of the 5.55.
Before the switch, the three bonds contribute 0.80, 1.75 and 3.00 years for a portfolio duration of 5.55. Moving Rs 14.5 crore from the 2-year to the 12-year bond changes the contributions to 0.51, 1.75 and 4.74, which total exactly 7.00.
What changes for the portfolio when duration goes from 5.55 to 7?
Solve for the switch with one line: the shift x changes duration by x times (12 minus 2) over 100, and that must equal 1.45. After the switch, a one-point parallel rise in yields costs about 7% of the portfolio, Rs 7 crore, instead of about 5.55%, Rs 5.55 crore. The portfolio gains more if yields fall and loses more if they rise. Its cash-flow profile also becomes a barbell, heavier at the long end, which gives it more convexity than a single bond with the same duration but more exposure to the long end of the curve if the curve steepens.
The relationship
$D_p = \sum_i w_i D_i = \frac{40(2) + 35(5) + 25(12)}{100} = 5.55 \qquad x = \frac{(7 - 5.55) \times 100}{12 - 2} = 14.5$
w_i each bond's share of portfolio value
D_i each bond's duration, in years
x the Rs crore switched from the 2-year to the 12-year bond
What it says in wordsDuration averages by value, so a switch moves it by the amount moved times the duration gap, over the portfolio's size.
Say the limits: a weighted average of durations describes small, parallel shifts in yields. If short and long yields move by different amounts, a portfolio at duration 7 built from a barbell behaves differently from one built from 7-year bonds, and the switch changes the portfolio's yield and credit mix as well. For large moves, convexity adds a second-order correction.
Where candidates lose it
The most common slip is taking a simple average of the three durations, 19 over 3, about 6.3, ignoring the amounts held. The other is solving for the switch but forgetting to divide by the portfolio size, which gives Rs 1.45 crore or some other scale error.
Say weighted by value first, give 5.55, then set up the switch as one equation. Close with what the higher duration means in rupees for a one-point move in yields; that sentence is the part the question actually asks about.
What the interviewer asks next
How would you reach duration 7 without selling any of the 2-year bond?
Why might the barbell at duration 7 behave differently from a bullet 7-year bond if the curve steepens?
What does a one-point parallel fall in yields do to the portfolio after the switch?
Asked at PIMCO, Generalist, Los Angeles, 2026 (Wall Street Oasis): Given a portfolio of these 3 bonds (I forgot exactly what they were) explain how the portfolio changes if duration increases.
051A two-year bond pays an 8% annual coupon and repays 100 at maturity. Market yields for this bond are 7%. What is its price, and why is it above par?Bond mathsWarm upJ.P. Morgancloumbus · 2026
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Before you discount anything: where does the price land?
Show the worked solution
About 101.81. Discount each cash flow at 7%: the year 1 coupon of 8 is worth 7.48 today and the year 2 payment of 108 is worth 94.33, which add to 101.81. The bond sits above par because it pays 8 when the market only asks for 7, and a buyer pays up for that extra coupon until the return on the price paid falls back to 7%.
What does pricing a bond actually mean?
Think of a friend who promises you Rs 8 next year and Rs 108 the year after. What would you hand over today? If you can earn 7% elsewhere, each promised rupee is worth less the further away it sits. A bond's price is every promised cash flow divided by one plus the yield, once for each year you wait, and then added up. Here that is 8 divided by 1.07, which is 7.48, plus 108 divided by 1.07 twice, which is 94.33. The total is 101.81.
The relationship
$P = \frac{8}{1.07} + \frac{108}{1.07^2} = 7.48 + 94.33 = 101.81$
8 the annual coupon on 100 of face value
108 the final coupon plus the principal
1.07 one plus the market yield
What it says in wordsDiscount each payment by the yield for as many years as you wait, then add.
Discounted at 7%, the year 1 coupon is worth 7.48 and the year 2 payment of 108 is worth 94.33, stacking to 101.81; the 1.81 above par is exactly the extra 1 a year of coupon over a 7% bond, valued today.
Why must a bond with a high coupon trade above par?
Suppose it traded at 100. A buyer would earn 8% on a bond when the market pays 7% for the same risk, so everyone would want it and the price would rise. The price climbs until the return on the price paid equals the market yield, and that happens at a premium over par. The premium is easy to see directly: compared with a 7% bond at 100, this one pays an extra rupee each year for two years, worth 0.93 plus 0.87, which is 1.81. That is a check on the long method, and it is quicker to say in the room.
The same logic runs the other way. A coupon below the market yield means a discount to par, and a coupon equal to the yield means exactly 100. If you are asked how you would price a bond in today's market, say that the yield comes from comparable bonds of the same credit and maturity; the arithmetic above is the easy part. A premium bondA bond whose price is above its face value, because its coupon is higher than the yield the market currently demands. also pulls back towards 100 as it nears maturity, so a buyer at 101.81 loses the premium slowly while collecting the fat coupon.
Where candidates lose it
The common slip is adding the extra coupons without discounting them and answering 102. The extra rupee in year 2 is worth only 0.87 today, and saying 102 tells the interviewer you know the direction but not the method.
The second loss is getting 101.81 without saying why it must be above 100. Give the one-line reason: the coupon beats the market yield, so buyers bid the price up until the return on the price paid is 7%.
What the interviewer asks next
What would the price be if yields were 9% instead?
Why does the premium on this bond shrink as it approaches maturity?
Where would you find the right yield to price a bond like this in practice?
Asked at J.P. Morgan, Generalist, cloumbus, 2026 (Wall Street Oasis): How would you price a bond in today's market
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
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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.
075A callable bond is priced at 100.0. If yields rise 50 basis points its price falls to 98.9; if yields fall 50 basis points it rises only to 100.6. What are its effective duration and effective convexity?Bond mathsHardAmundiLondon · 2018
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What sign does the convexity take?
Show the worked solution
Effective duration is about 1.7 and effective convexity about -200. Duration is the price difference across the two shocks over twice the price times the shock: 1.7 over 1.0, which is 1.7. Convexity is the sum of the shocked prices less twice the base, over the price times the shock squared: minus 0.5 over 0.0025, which is -200. The call caps the price, so it gains less than it loses.
Why effective duration rather than the usual formula?
Think of renting out a flat on a lease the tenant can cancel whenever cheaper flats appear. When rents fall, the tenant leaves and you do not keep the high rent; when rents rise, you are stuck. A callable bond's cash flows change with yields, because the issuer calls it when rates fall, so you measure its duration from how its price actually moves, not from a fixed schedule of coupons. That is effective duration: shock the yield both ways, reprice, and read the slope.
The relationship
$D_{eff} = \frac{P_- - P_+}{2P_0\Delta y} = \frac{100.6 - 98.9}{2 \times 100 \times 0.005} = 1.7 \qquad C_{eff} = \frac{P_- + P_+ - 2P_0}{P_0 \Delta y^2} = \frac{-0.5}{0.0025} = -200$
P_- price when yields fall 50 bp, 100.6
P_+ price when yields rise 50 bp, 98.9
P_0 the starting price, 100.0
Delta y the shock, 0.005
What it says in wordsDuration is the average slope across the two shocks; convexity is how much the two moved prices bend away from a straight line.
The callable bond rises only to 100.6 when yields fall 50 basis points, because the call caps it below 101, but falls to 98.9 when yields rise, so its effective duration is 1.7 and its convexity is negative, about -200.
What does negative convexity cost the holder?
It means the bond loses more when yields rise than it gains when they fall. The holder has sold the issuer an option to refinance, and the price of that option is the upside given up when rates fall. A straight bond with a duration of about 4 would gain about 2.0 points for a 50 basis point fall; this one gains 0.6. The holder is paid for this through a higher yield than an equivalent straight bond, and the question for a portfolio manager is whether that extra yield covers the option given away.
Two more things are worth saying. The duration of 1.7 is not fixed: as yields fall towards the level where a call becomes likely, duration shrinks further, and as they rise, it lengthens towards the bond's straight duration. That shifting is why callable bonds, and mortgage securities with the same feature, need effective measures rather than the textbook formulas. The curve in the figure is a stylised fit around the three prices given; a real one would come from an option pricing model.
Where candidates lose it
The trap is computing a positive convexity by habit, or dividing by the shock rather than the shock squared and getting minus 1. Write the formula, put the sign of 100.6 plus 98.9 minus 200 on the page, and the negative number is obvious.
The second loss is giving the numbers without the story. Say that the call caps the upside, which is why duration is short and convexity negative.
What the interviewer asks next
Estimate the price change for a 100 basis point fall using this duration and convexity. Why might it be wrong?
Why does a callable bond's effective duration lengthen when yields rise?
What would the price-yield curve of a putable bond look like?
Asked at Amundi, Rates, London, 2018 (Wall Street Oasis): What would your allocation be in today's market? What is effective duration?

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%

w_i	each bond's share of portfolio value
D_i	each bond's duration, in years
x	the Rs crore switched from the 2-year to the 12-year bond

8	the annual coupon on 100 of face value
108	the final coupon plus the principal
1.07	one plus the market yield

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 is portfolio duration a simple weighted average?

What changes for the portfolio when duration goes from 5.55 to 7?

Where candidates lose it

What the interviewer asks next

Try it first

What does pricing a bond actually mean?

Why must a bond with a high coupon trade above par?

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 effective duration rather than the usual formula?

What does negative convexity cost the holder?

Where candidates lose it

What the interviewer asks next