Mutual Fund Mastery puzzles, solved step by step

Puzzles: 100
Traced to a firm: 29
Topics: 13
Hard: 30

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003A 5-year bond pays an 8% annual coupon and trades at par, so its yield is 8%. Without a calculator, bracket its modified duration, then give the exact figure.Bond maths and durationCorePIMCOLos Angeles · 2024
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Where does the modified duration sit?
Show the worked solution
Modified duration is about 3.99. Bracket first: a coupon bond's Macaulay duration sits below its 5-year maturity but not far, because the principal dominates, so somewhere in the low fours; dividing by 1.08 takes it just under 4. Exactly, the Macaulay duration is 4.312 years, and 4.312 divided by 1.08 is 3.993: a 1 point rise in yield cuts the price by roughly 4%.
How do you bracket it before doing any arithmetic?
Picture a see-saw with one heavy child at the far end and four small children spread along the plank. The balance point sits close to the heavy child but is pulled in a little by the others. A bond's Macaulay duration is the balance point of its discounted cash flows, so it can never exceed maturity and sits close to it when the final payment dominates. A zero coupon 5-year bond sits exactly at 5; this one pays 8 a year along the way, so it lands a little inside.
The final payment of 108 is worth 73.5 today, about 74% of the price of 100. The four coupons carry the rest at years 1 to 4. A balance point roughly three quarters of the way at 5 and a quarter spread between 1 and 4 lands a bit above 4, and dividing by 1 plus the yield lands just under 4.
The bond's discounted cash flows are 7.41, 6.86, 6.35, 5.88 and 73.50, which balance at 4.31 years, so its Macaulay duration sits well inside the 5-year maturity and its modified duration is 3.99.
Is there a shortcut for the exact figure?
For a bond priced at par there is a closed form. At par, Macaulay duration equals (1 + y) over y, times one minus the discount factor at maturity: 13.5 times (1 minus 0.6806), which is 4.312 years. Divide by 1.08 for modified duration, 3.993. Saying you know the par shortcut, then checking it against the bracket, is a strong answer in the room.
The relationship
$D_{mod} = \frac{D_{mac}}{1+y}, \qquad D_{mac}^{par} = \frac{1+y}{y}\left[1 - \frac{1}{(1+y)^n}\right] = 13.5 \times 0.3194 = 4.312$
y the yield, 8%, equal to the coupon because the bond is at par
n years to maturity, 5
D_{mac} Macaulay duration, the balance point in years
What it says in wordsAt par the balance point has a closed form, and modified duration is that balance point divided by one plus the yield.
Say what the number is for. A modified duration of 3.99 means a 1 percentage point rise in yield costs roughly 3.99% of price, and a 0.25 point rise roughly 1%. The estimate is linear, so it drifts for large moves; convexity handles that.
Where candidates lose it
Answering 5 is the common slip: it treats the bond as a zero coupon bond and ignores the coupons that come back early. The second is giving the Macaulay figure, 4.31, when the question asks for modified duration.
Candidates also freeze without a calculator. The interviewer wants the bracket said out loud first: below 5, above 4 for Macaulay, divide by 1.08. The exact figure is a bonus.
What the interviewer asks next
What is the duration of a 5-year zero coupon bond at an 8% yield?
If the coupon were 4% with the yield still 8%, would duration rise or fall?
Estimate the price change for a 50 basis point fall in yield.
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
014A bond has a modified duration of 7 and a convexity of 80. Yields rise by 150 basis points. Estimate the percentage price change with duration alone, then with convexity added.Bond maths and durationHardFixed income desksIndian AMCs
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With convexity added, what is the estimated price change?
Show the worked solution
About -10.5% with duration alone and about -9.6% with convexity. Duration gives -7 times 0.015, which is -10.5%. Convexity adds one half times 80 times 0.015 squared, or +0.90%. The price-yield curve bends upward, so a straight-line duration estimate overstates losses when yields rise and understates gains when they fall, and the gap grows with the square of the move.
Why is duration alone not enough for a big move?
Think of estimating how far a ball thrown upward will fall back by looking only at its speed at the start. For a moment the straight line is fine; over a longer stretch the curve of its path takes over. Duration is the slope of the price-yield curve at today's yield, so it is exact for tiny moves and drifts further from the true price the larger the move. The curve bends upward for an ordinary bond, so it always sits above the tangent.
For a 150 basis point rise, the duration tangent puts the price at 89.5 while the curve with convexity puts it at 90.4, so duration alone overstates the loss by 0.9 points, and the gap widens as the move grows.
How do you compute both estimates quickly?
Write the move as a decimal first, 0.015. The duration term is minus duration times the move, -7 times 0.015, which is -10.5%; the convexity term is half the convexity times the move squared, 40 times 0.000225, which is +0.90%. Add them: -9.60%. For a 150 basis point fall the same two terms give +10.5% plus 0.90%, a gain of 11.40%, so convexity helps in both directions.
The relationship
$\frac{\Delta P}{P} \approx -D_{mod}\,\Delta y + \tfrac{1}{2}\,C\,(\Delta y)^2 = -7(0.015) + \tfrac{1}{2}(80)(0.015)^2 = -10.5\% + 0.9\% = -9.6\%$
D_{mod} modified duration, 7
C convexity, 80
\Delta y the change in yield as a decimal, 0.015
What it says in wordsThe price change is the straight-line duration estimate plus a correction for the curve that grows with the square of the move.
The squared term is why size matters. At 50 basis points the convexity term is only +0.10%, easy to ignore; at 300 basis points it is +3.6%, too large to leave out. The limitation is that this is still a two-term approximation of a curve; for very large moves, or for bonds with call features whose convexity can turn negative, a desk reprices the bond in full rather than trusting the formula.
Where candidates lose it
The common slip is subtracting the convexity term because yields went up, giving -11.4%. For an ordinary bond the convexity term is positive in both directions; the squared move has no sign.
The other slip is a units error: using 1.5 instead of 0.015, which makes the convexity term 90 and the answer absurd, or forgetting the one half. Convert basis points to a decimal before anything else and say the half out loud.
What the interviewer asks next
Why is convexity valuable to a bondholder, and what does the market charge for it?
Why can a callable bond show negative convexity when yields fall?
Which has more convexity at the same duration, a bullet bond or a barbell of short and long bonds?
026A 5-year government bond yields 7.6% and a 4-year bond yields 7.2%. You buy the 5-year bond at par. If the yield curve does not move at all over the next year, what return do you earn for the year?Bond maths and durationCoreFixed income desksIndian AMCs
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Before you work it: roughly what does a year of holding the 5-year bond return?
Show the worked solution
About 9.0%, not 7.6%. You collect the 7.6% coupon. A year later the bond has four years left, and on an unchanged curve four-year bonds yield 7.2%, so its yield has fallen 0.40% with no market move at all. With a duration of about 3.4, that lifts the price about 1.35%, from 100 to 101.35. Coupon plus this roll-down gain is about 8.95%.
Why does the bond's yield change if the curve does not move?
Stand halfway down a slope and take one step towards the bottom. The slope has not moved, but you are lower than you were. A bond on an upward sloping yield curve is doing the same thing every day it is held. The curve stays put, but the bond's own maturity shortens by a year, so a year later it is priced off a lower point on the same curve. Nothing happened in the market; the bond simply got older.
That slide has a name, roll-downThe price gain a bond earns as it ages and its yield moves down an unchanged, upward sloping yield curve., and a fund manager counts it as part of expected return just as surely as the coupon. Today the bond is a 5-year bond at 7.6%. A year from now it is a 4-year bond, and 4-year bonds yield 7.2%. A 7.6% coupon discounted at 7.2% is worth more than par.
On an unchanged, upward sloping curve the bond starts at the 5-year point yielding 7.6% and a year later sits at the 4-year point yielding 7.2%. That 0.40% fall in its yield adds about 1.35% of price gain to the 7.6% coupon, for a one-year return near 8.9%.
How do you size the roll-down gain in your head?
Use duration. A bond's price moves by roughly its modified duration times the change in its yield, and the duration that matters is the bond's duration at the end of the year, when it is a 4-year bond. A 4-year bond with a 7.6% coupon has a modified duration of about 3.36. Multiply by the 0.40% fall and you get about 1.34%. Pricing the bond exactly, four coupons of 7.6 and 100 at maturity discounted at 7.2%, gives 101.35, a gain of 1.35%. The shortcut is within a hundredth of a per cent.
The relationship
$r_{1y} \approx y_{\text{buy}} + D_{\text{end}} \times (y_{5} - y_{4}) = 7.6\% + 3.36 \times 0.40\% \approx 8.95\%$
y_buy the yield you bought at, 7.6%
D_end modified duration of the bond a year later, as a 4-year bond
y_5 - y_4 how far the yield rolls down the curve, 0.40%
What it says in wordsOne year's return on an unchanged curve is the yield you bought plus duration times the yield you roll down.
When does the roll-down vanish or turn against you?
Roll-down is only as good as the slope. On a flat curve there is nothing to slide down and the return is the coupon. On an inverted curve, where shorter bonds yield more, the bond rolls up to a higher yield and loses price as it ages. And the curve rarely stays still: if the 4-year yield ends the year at 7.6% instead of 7.2%, the roll-down is gone and you earn roughly the coupon alone. Say this limit out loud; it shows you treat roll-down as an expected return on an assumption, not a promise.
Where candidates lose it
The common answer is 7.6%, because candidates treat yield to maturity as the return for any holding period. Yield to maturity is the return only if you hold to maturity; over one year, the price at the end of the year matters, and on a sloped curve that price has moved.
The second loss is getting the direction wrong: the yield falls, so some candidates say the return falls. A falling yield means a rising price. Say that link explicitly before you size the gain.
What the interviewer asks next
What is the one-year return if the curve is flat at 7.6%?
How much would the 4-year yield have to rise for the year's return to fall to 7.6%?
Why might a debt fund manager prefer the 5-year bond to a 4-year bond at 7.2% even with no view on rates?
035A debt portfolio holds Rs 40 crore of bonds with a duration of 1, Rs 35 crore with a duration of 4 and Rs 25 crore with a duration of 9. What is the portfolio's duration, and what happens to it and to its rate risk if the short bonds are switched into the long ones?Bond maths and durationCorePIMCOLos Angeles · 2026
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Before you work it: what is the portfolio's duration now?
Show the worked solution
A duration of 4.05, rising to 7.25 after the switch. Weight each duration by its share of the Rs 100 crore: 0.40 x 1 + 0.35 x 4 + 0.25 x 9 = 4.05. Move the Rs 40 crore of 1-year duration into the 9-year bonds and it becomes 0.35 x 4 + 0.65 x 9 = 7.25. A 1% rise in yields now costs about Rs 7.25 crore instead of Rs 4.05 crore: the rate risk is about 79% higher.
Why is portfolio duration a weighted average?
A household's average commute is not the average of the three commutes, it depends on who travels most. If the person with the one-kilometre walk makes most of the trips, the household's average is short. A portfolio's duration is the money-weighted average of its bonds' durations, because each bond's price change counts in proportion to the rupees held in it. Here the biggest holding is the shortest, so the portfolio sits at 4.05, below the simple average of 4.67.
The relationship
$D_p = \sum_i w_i D_i = 0.40(1) + 0.35(4) + 0.25(9) = 4.05 \;\to\; 0.35(4) + 0.65(9) = 7.25$
w_i the share of the portfolio's value in bond i
D_i the modified duration of bond i
D_p the portfolio's duration
What it says in wordsMultiply each bond's duration by its share of the money, and add.
Before the switch, Rs 40 crore at duration 1, Rs 35 crore at 4 and Rs 25 crore at 9 average to a duration of 4.05. Moving the Rs 40 crore into the 9-year bonds pulls the average to 7.25, and the loss from a 1% rise in yields grows from about Rs 4.05 crore to Rs 7.25 crore.
What changes in the portfolio when its duration rises?
The portfolio becomes more sensitive to interest rates in both directions: a 1% fall in yields gains about 7.25% instead of 4.05%, and a 1% rise loses as much. On Rs 100 crore, that is a swing of about Rs 7.25 crore for each 1% move instead of Rs 4.05 crore. On an upward sloping curve the portfolio's yield usually rises too, because longer bonds pay more, so the switch buys extra income with extra rate risk. Convexity also rises, which slightly cushions large moves, and the portfolio loses the cash-like buffer the short bonds gave it for meeting redemptions.
One caveat worth saying: duration is a linear estimate. For a 1% move it is close; for a 3% move, convexity makes the true loss smaller and the true gain larger than duration alone suggests. The durations here are given, and they drift as bonds age and yields move, so a portfolio's duration has to be re-measured, not set once.
Where candidates lose it
The common slip is taking the simple average, 4.67, or saying the longest bond dominates. Duration weights by money, and the portfolio's largest holding here is the shortest.
The second loss is describing the switch as only riskier. The interviewer wants the whole trade-off: more rate sensitivity in both directions, usually more yield on a normal curve, a little more convexity, and less liquidity for redemptions.
What the interviewer asks next
How much of the 9-duration bond would you sell to bring the portfolio back to a duration of 5?
Yields fall 0.5%. Roughly what does each version of the portfolio gain?
What does a debt fund's stated average maturity miss that duration captures?
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
045A target maturity debt fund has a portfolio yield to maturity of 7.4% and an expense ratio of 0.2%. What return should an investor who holds to maturity expect, and why will it not be exactly that?Bond maths and durationCoreFixed income desksIndian AMCs
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What is the best single estimate of the hold-to-maturity return, per year?
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About 7.2% a year: the portfolio's yield to maturity less the expense ratio. Holding to the fund's maturity removes most of the price risk, so the bonds earn roughly their 7.4% yield and the fund keeps 0.2% a year. It will not be exactly 7.2% because coupons are reinvested at future yields nobody knows, the fund tracks its index imperfectly, and the yield you lock in is the one on the day you invest. On these numbers the drift is about plus or minus 0.13% a year.
Why is yield to maturity a fair starting point?
A fixed deposit tells you its rate because the bank promises to hold it to maturity for you. A bond does the same if you hold it to maturity: price swings along the way wash out, and what remains is the yield to maturityThe single discount rate that makes the present value of a bond coupons and principal equal its price today; the return earned if the bond is held to maturity and coupons are reinvested at that same rate.. A target maturity fund holds bonds that mature near one date and then pays out, so an investor who stays to that date earns close to the portfolio's yield to maturity, less the fund's costs. 7.4% less 0.2% is about 7.2% a year.
The portfolio's 7.40% yield to maturity less the 0.20% expense ratio gives a best estimate of 7.20% a year, and reinvesting coupons at 6.4% to 8.4% instead of 7.4% would move the realised return to between 7.07% and 7.33% after costs, with tracking effects of a few basis points on top.
Why will the realised return not be exactly 7.2%?
Yield to maturity quietly assumes every coupon is reinvested at the same yield, and the future reinvestment rate is unknown. Take a 5-year bond bought at par with a 7.4% coupon. If its coupons can only be reinvested at 6.4%, the realised return is about 7.27% a year; at 8.4% it is about 7.53%. After the 0.2% fee that is 7.07% to 7.33%. The fund also holds bonds that do not mature on exactly one date, keeps some cash, and replicates its index imperfectly, each worth a few basis points either way.
The relationship
$r_{\text{hold}} \approx \text{YTM} - \text{TER} = 7.4\% - 0.2\% = 7.2\%$
YTM the portfolio's yield to maturity on the day you invest, 7.4%
TER the total expense ratio, 0.2% a year
r_hold the return a hold-to-maturity investor can reasonably expect
What it says in wordsFor an investor who stays to maturity, the best estimate of the return is today's portfolio yield less the yearly cost.
What else should the investor be told?
Three things. The estimate holds only to maturity: an investor who exits after two years takes the market price then, which can be well above or below the path to 7.2%. The yield locked in is the one on the day of investment, not the one quoted at launch. And the figure is before tax; how a debt fund's gains are taxed has changed in recent years, so confirm the current treatment before turning 7.2% into an after-tax number. The estimate is the honest best guess, not a promise.
Where candidates lose it
The trap is quoting 7.4% as the return, forgetting that the expense ratio is taken out of the NAV every day. The fund's yield is the bonds' yield; the investor's return is that less the cost.
The second loss is treating 7.2% as a promise because the fund holds to maturity. It is an estimate: reinvestment, tracking and timing all move it, and an exit before maturity exposes the investor to price risk.
What the interviewer asks next
Yields rise 1% the day after you invest. What happens to your NAV and to your return if you hold to maturity?
Why do coupon reinvestment effects matter less for a 2-year fund than a 10-year fund?
How would you compare this fund with a 5-year bank deposit at 7.1%?
054At an 8% yield, which has the highest duration: a 10-year zero-coupon bond, a 10-year 9% coupon bond, or a 15-year 12% coupon bond?Bond maths and durationHardFixed income desksIndian AMCs
Try it first
Gut call first: which bond carries the most interest rate risk?
Show the worked solution
The 10-year zero, with a duration of 10.0 years. A zero pays everything at maturity, so its duration equals its maturity. The 15-year 12% bond has a duration of about 8.6 years, because its large coupons bring value forward, and the 10-year 9% bond is about 7.1. The zero moves most when rates move, even though it is not the longest bond.
Why is maturity the wrong ruler?
Think of two loans to a friend. One friend repays everything in one go after ten years. The other repays a large slice every year for fifteen years. Most of the second loan is back in your hands long before the first friend pays a rupee. Duration measures when, on average, a bond's value comes back to you, weighted by the present value of each payment, and rate risk follows that average, not the final date. A zero returns all of its value at one date, so its duration is exactly its maturity.
At an 8% yield the 10-year zero balances at 10.0 years, the 10-year 9% bond at 7.1 years and the 15-year 12% bond at 8.6 years, because the large coupons of the longest bond pull its balance point inside that of the zero.
How do you rank them without a calculator?
Use two rules and one check. A zero's duration is its maturity. A coupon bond's duration is always below its maturity, and the higher the coupon, the further below. So the only real contest is between the zero at 10 and the 15-year bond, and a 12% coupon pulls hard: about 60% of that bond's present value arrives in the first ten years. The 10-year 9% bond cannot beat the 10-year zero, because it has the same final date and pays some value earlier.
The relationship
$D = \frac{\sum_t t \cdot \frac{CF_t}{(1+y)^t}}{\sum_t \frac{CF_t}{(1+y)^t}}$
t the year a cash flow arrives
CF_t the cash flow in year t, coupon plus face at maturity
y the yield, 8% here
What it says in wordsDuration is the average arrival time of a bond's cash flows, each weighted by what it is worth today.
Add the sizing, because desks price risk in rupees. Modified durationMacaulay duration divided by one plus the yield. It gives the approximate percentage price change for a one point move in yield. is Macaulay duration over 1.08, so the zero loses about 9.3% of its price for a one point rise in yield and the 15-year bond about 7.9%. The limit: this is a small-move estimate, and it ignores the curve changing shape. Even a 30-year 12% bond only reaches about 11.5 years, because a coupon bond's duration can never exceed that of a perpetuity, 13.5 years at 8%.
Where candidates lose it
The instinct is to pick the 15-year bond because it is the longest. That confuses the last payment date with the average one, and on a desk it means hedging the wrong book.
The quieter miss is treating coupon size as a detail. At 12% the coupons are large enough to pull the duration down to 8.6 years; at a 2% coupon the same 15-year bond would have a duration well above 10.
What the interviewer asks next
What coupon on the 15-year bond would make its duration equal to the zero's?
Why does a floating-rate bond have a duration close to its next reset date?
Yields rise from 8% to 10%. Which of the three loses the most in rupees per Rs 100 of face?
064The one-year rate is 7.0% and the two-year rate is 7.5% a year. What one-year rate does the curve imply for a year from now, and what would an inverted curve, 7.5% for one year and 7.0% for two, imply instead?Bond maths and durationCoreState StreetBoston · 2020
Try it first
What one-year rate does the curve imply for next year?
Show the worked solution
About 8.00% for next year, and about 6.5% if the curve is inverted. Investing for two years at 7.5% must give the same as one year at 7.0% rolled into next year's rate, or someone could profit from the gap. So 1.075 squared equals 1.07 times (1 + f), and f is 8.00%. Flip the curve to 7.5% then 7.0% and the implied rate falls to 6.50%: the market is pricing lower short rates ahead.
Why must the two paths give the same answer?
Two routes to the same railway station, one direct and one with a change, must cost about the same, or everyone would take the cheaper one until the prices matched. Money for two years can be locked in at the two-year rate, or invested for one year and rolled; if one path paid more, investors would crowd into it, so the rate the curve implies for the second year is the one that makes them equal. That rate is the forward rateThe interest rate for a future period that is locked in today by the current yield curve, found by making a long investment equal to a chain of shorter ones.. Rs 100 at 7.5% for two years becomes Rs 115.56. Rs 100 at 7.0% becomes Rs 107 after one year, so the second year must turn Rs 107 into Rs 115.56.
Rs 100 locked in for two years at 7.5% reaches Rs 115.56, so one year at 7.0% followed by one year at the forward must reach the same, which sets the forward at 8.00%, while an inverted curve of 7.5% then 7.0% implies 6.50%.
The relationship
$(1.075)^2 = 1.07\,(1 + f) \;\Rightarrow\; f = \frac{1.155625}{1.07} - 1 \approx 8.00\%$
1.075 one plus the two-year rate, applied for two years
1.07 one plus the one-year rate
f the one-year rate one year forward
What it says in wordsThe long rate is a compounded chain of the short rate now and the forward rates after it; solve the chain for the missing link.
What does an inverted curve tell you?
In your head, the forward is about twice the long rate minus the short rate: 2 x 7.5 - 7.0 = 8.0%. Turn the curve upside down, 7.5% for one year and 7.0% for two, and the same step gives 2 x 7.0 - 7.5 = 6.5%, exactly 6.50%. An inverted curve says the market expects short rates to fall, which usually happens when it expects the central bank to cut, often because it expects growth to slow. That is why inversions are watched as a recession signal.
State the limit, because it is the follow-up. Forward rates are not pure forecasts. Lenders usually want extra pay for tying money up longer, a term premium, so an upward curve partly reflects that premium rather than expected hikes. The forward rate is the break-even: if next year's actual one-year rate comes in below it, the investor who locked in two years did better.
Where candidates lose it
The fast wrong answer is 7.25%, averaging the two rates, or 7.5%, assuming next year's rate equals the two-year rate. Both forget that the two-year rate is itself an average of this year and next.
The second miss is calling the forward rate a forecast. Say it is the rate that makes the two paths equal, then add that it carries a term premium, so it overstates the expected path a little when the curve slopes upward.
What the interviewer asks next
The three-year rate is 7.8%. What is the one-year rate two years forward?
Why does a term premium make forward rates overstate expected short rates?
A debt fund manager expects rates below the forward. Should the fund extend duration or shorten it, and why?
Asked at State Street, Equity Research, Boston, 2020 (Wall Street Oasis): What is the significance of the yield curve and what does it mean for it to be inverted?
075A bond with a 7% annual coupon trades at 104. Is its yield to maturity above or below 7%, and what is its current yield?Bond maths and durationWarm upVanguardMalvern · 2023
Try it first
Which ordering is right for this bond?
Show the worked solution
The yield to maturity is below 7%, and the current yield is 6.73%. Current yield is the coupon over the price: 7 / 104 = 6.73%. The yield to maturity is lower still, because you pay 104 and get back only 100, so the Rs 4 premium is a loss spread over the bond's life. For an assumed five years left, the yield to maturity is about 6.05%.
Why must the yield be below the coupon when the price is above 100?
Pay Rs 104 for a gift voucher worth Rs 100 that also pays Rs 7 of cashback a year. The cashback is generous, but you have overpaid for the voucher by Rs 4. A bond priced above par returns less than its coupon, because the buyer pays more than the Rs 100 that comes back at maturity, and that premium is lost along the way. The coupon of 7% is set on the face value of 100 and never changes. The current yieldThe yearly coupon divided by the bond price today. It ignores any gain or loss as the price moves to face value at maturity. divides the same Rs 7 by what you actually pay: 7 / 104 = 6.73%.
A 7% coupon bond bought at 104 has a current yield of 6.73% and, with five years left, a yield to maturity of 6.05%, because its price pulls down to 100 by maturity and the Rs 4 premium is lost.
How far below 7% is the yield to maturity?
That depends on how long the bond has to run, which the question does not say, so name an assumption. With five years left, the Rs 4 premium is lost at roughly Rs 0.80 a year. A quick estimate takes the coupon less that yearly loss, Rs 6.20, over the average of the purchase and redemption prices, 102: about 6.08%. The exact yield to maturityThe single discount rate that makes all remaining coupons and the final repayment worth exactly the price paid today. is 6.05%. The longer the bond, the thinner the yearly slice of the premium and the closer the yield to maturity sits to the current yield.
The relationship
$104 = \sum_{t=1}^{5} \frac{7}{(1+y)^t} + \frac{100}{(1+y)^5} \;\Rightarrow\; y \approx 6.05\%$
104 the price paid
7 the yearly coupon per 100 of face
100 the repayment at maturity
y the yield to maturity
What it says in wordsThe yield to maturity counts the coupons and the loss of the premium, so it ends below both the coupon and the current yield.
For a debt fund this ordering is everyday arithmetic: when rates fall, older high-coupon bonds trade above par, and the fund's quoted portfolio yield sits below the coupons it receives. The limit of yield to maturity: it assumes the bond is held to maturity, never defaults and that coupons are reinvested at the same yield, none of which is guaranteed. A bond callable before maturity at par makes the premium an even bigger risk.
Where candidates lose it
The trap is quoting 7% as the yield because that is the coupon. The coupon is fixed on the face value; the yield depends on the price you pay, and above par it is lower.
The second miss is stopping at the current yield. Name the order, coupon above current yield above yield to maturity, and give the reason for the last step: the premium is lost by maturity.
What the interviewer asks next
The same bond trades at 96. Put the coupon, current yield and yield to maturity in order.
With 20 years left instead of 5, is the yield to maturity closer to or further from the current yield?
Why do debt funds holding many premium bonds report a portfolio yield below their average coupon?
Asked at Vanguard, Investments, Malvern, 2023 (Wall Street Oasis): Questions asked ranged from resume stuff, global macro/current news stuff, and simple bond math since I expressed interest in FICC.
086A client needs her money in exactly three years. Why does a bond fund with a duration of about three years protect her whether interest rates rise or fall? Show it with Rs 1 crore and yields jumping from 7% to 8% on the first day.Bond maths and durationHardPIMCOLondon · 2022
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Yields jump to 8% on day one and the fund's value falls. Where does she stand at year three?
Show the worked solution
Because a fall in price and a rise in reinvestment income cancel at a horizon equal to the duration. If yields jump to 8%, Rs 1 crore at a three-year duration drops about Rs 2.74 lakh on day one, then compounds at 8% instead of 7%. By year three it is worth Rs 122.53 lakh against Rs 122.50 lakh had nothing moved. If yields fall to 6%, the gain today offsets the lower reinvestment rate in the same way.
Why do rising rates both hurt and help a bond investor?
Think of a tenant who has locked in a lease. If market rents rise, the lease itself is worth less to sell on, because a new buyer could get a better deal elsewhere, but any new space the tenant takes from now on costs more too. A bond investor faces the mirror image. When yields rise, bonds already held fall in price, but every coupon and every maturing bond is reinvested at the new, higher yield. One effect is immediate and the other builds over time. Which one wins depends only on how long you hold.
Make it concrete with a fund that has a duration of exactly three years. Picture its Rs 100 lakh as two holdings of Rs 50 lakh each at 7%: paper that matures in one year, and paper that matures in five. The value-weighted average of one and five is three, so the fund's Macaulay durationThe average time until a bond portfolio pays back its cash, with each payment weighted by its share of present value. is 3.0 years. Left alone at 7%, both halves grow to Rs 61.25 lakh by year three, Rs 122.50 lakh in all.
A jump to 8% knocks Rs 2.74 lakh off the fund on day one, but reinvesting at 8% wins it back by about year 3.0, so at her three-year date she holds Rs 122.53 lakh against a plan of Rs 122.50 lakh; a fall to 6% gives the mirror image.
What happens to each holding when yields jump to 8%?
The one-year paper matures at Rs 53.50 lakh and is rolled for two more years at 8%, reaching Rs 62.40 lakh, Rs 1.15 lakh more than the plan. The five-year paper still has two years to run at year three and must be sold at an 8% yield, fetching Rs 60.12 lakh, Rs 1.13 lakh less than the plan. One leg carries reinvestment risk and the other carries price risk, and a duration equal to the horizon sets them against each other in equal size. Held alone, each leg would fail: all short paper with yields falling to 6% gives only Rs 119.10 lakh, and all five-year paper with yields at 8% gives only Rs 120.25 lakh.
The relationship
$V_3 = P(y)\,(1+y)^3 \qquad P(8\%) = \frac{53.50}{1.08} + \frac{70.13}{1.08^5} = 97.26 \;\Rightarrow\; 97.26 \times 1.08^3 = 122.53$
P(y) the fund's value today at yield y, Rs lakh
53.50, 70.13 the amounts the one-year and five-year paper pay at maturity
V_3 the value at year three if all cash is reinvested at y
What it says in wordsWhatever the new yield, the fund's value at year three is today's repriced value grown at that yield, and at a horizon equal to the duration the two changes offset.
Where does this protection stop working?
In three places, and naming them is what the interviewer is after. Her horizon shrinks by exactly a year every year, but the fund's duration does not keep pace on its own: maturing paper has to be reinvested and coupons and rate moves shift the average, so the fund has to be rebalanced to stay matched. The offset assumes all yields move together; if short yields rise while long yields fall, the two legs no longer cancel. And the match only protects someone whose date is fixed: an open-ended fund whose manager keeps duration at three years forever is matched to nobody's date in particular. A product that runs down towards a fixed maturity close to her date does the matching more naturally. The small surplus either way, about Rs 0.02 lakh, comes from convexity and is a bonus, not the point.
Where candidates lose it
The common answer is that a shorter fund is safer, so she should use a liquid fund. That removes price risk but leaves her fully exposed to falling rates: if yields drop to 6% and stay there, three years of rolling short paper leaves her at Rs 119.10 lakh, about Rs 3.4 lakh short of plan.
The other loss is saying duration is a measure of price sensitivity and stopping there. This question is about the second meaning of duration, a time horizon at which price risk and reinvestment risk balance. Say both meanings, then show the cancellation with one number.
What the interviewer asks next
After one year, what has happened to the fund's duration and to her horizon, and what should change?
Short yields rise 1% and long yields fall 1% on the same day. Does the protection hold?
Why does the matched fund end slightly above plan whichever way yields move?
Asked at PIMCO, Sales, London, 2022 (Wall Street Oasis): Typically the product interview was toughest with questions regarding applications of duration
096The RBI cuts the repo rate by 25 basis points and bond yields fall by the same amount. Roughly how much does a gilt fund with a modified duration of 8 gain, and a liquid fund with a modified duration of 0.1?Bond maths and durationWarm upFixed income desksIndian AMCs
Try it first
Roughly how much does the gilt fund's NAV rise?
Show the worked solution
About 2.0% for the gilt fund and about 0.025% for the liquid fund. A bond fund's NAV moves by roughly minus its modified duration times the change in yield. With yields down 0.25 points, 8 x 0.25% is 2.0% and 0.1 x 0.25% is 0.025%, about 0.03%. The same rate move hits the gilt fund 80 times harder, which is the whole difference between the two products.
Why does a fall in yields raise bond prices?
Suppose you own a bond paying 7% and new bonds now pay only 6.75%. Anyone wanting 7% must buy yours, so they pay more for it, and they keep paying more until its yield matches the market's. Bond prices and yields move in opposite directions, and modified duration tells you by how much: roughly the percentage price change for each percentage point move in yield. A fund holding bonds with an average modified duration of 8 rises about 8% for a full point fall in yields, and about 2% for a quarter point.
A 0.25 point fall in yields lifts a gilt fund with modified duration 8 by about 2.0%, a short duration fund with duration 2 by 0.5%, and a liquid fund with duration 0.1 by only 0.025%, because the price change scales with duration.
The relationship
$\frac{\Delta P}{P} \approx -D_{mod} \times \Delta y = -8 \times (-0.25\%) = +2.0\%$
\Delta P / P the percentage change in the fund's NAV
D_{mod} modified duration, 8 for the gilt fund
\Delta y the change in yield, minus 0.25 points
What it says in wordsMultiply the yield change by the duration and flip the sign to get the approximate price change.
What does this approximation leave out?
Two things, one small and one large. The small one is convexity: the price-yield curve bends, so for a fall in yields the gain is a little more than duration says. With an assumed convexity of 80, it adds only about 0.025% for a quarter-point move. The large one is the assumption that bond yields fall by exactly the cut: the repo rate is an overnight rate, and longer yields move on expectations, so if the market had already priced in the cut, long yields may barely move on the day. Gilt funds often gain before a widely expected cut and do little when it arrives.
In rupees, on Rs 10 lakh the gilt fund gains about Rs 20,000 and the liquid fund about Rs 250. That symmetry is the point to make to a client: the gilt fund that gains Rs 20,000 on a quarter-point fall loses about the same if yields rise a quarter point instead. Duration is the dial for how much rate risk the investor is buying, in either direction.
Where candidates lose it
The common slip is answering 0.25%, the size of the cut, as if bond prices moved one for one with rates. They move by the yield change times the duration, so a duration-8 fund moves eight times the yield change.
The quieter loss is ignoring which yields moved. The question assumes all yields fall by the full 25 basis points; say that this is an assumption and that long yields often move ahead of the actual cut.
What the interviewer asks next
Yields rise 0.5 points instead. What happens to each fund?
Why might a gilt fund fall on the day the RBI cuts rates?
Which of the three funds would you expect to have the largest convexity, and why?

y	the yield, 8%, equal to the coupon because the bond is at par
n	years to maturity, 5
D_{mac}	Macaulay duration, the balance point in years

D_{mod}	modified duration, 7
C	convexity, 80
\Delta y	the change in yield as a decimal, 0.015

y_buy	the yield you bought at, 7.6%
D_end	modified duration of the bond a year later, as a 4-year bond
y_5 - y_4	how far the yield rolls down the curve, 0.40%

w_i	the share of the portfolio's value in bond i
D_i	the modified duration of bond i
D_p	the portfolio's duration

Try it first

How do you bracket it before doing any arithmetic?

Is there a shortcut for the exact figure?

Where candidates lose it

What the interviewer asks next

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Why is duration alone not enough for a big move?

How do you compute both estimates quickly?

Where candidates lose it

What the interviewer asks next

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Why does the bond's yield change if the curve does not move?

How do you size the roll-down gain in your head?

When does the roll-down vanish or turn against you?

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What the interviewer asks next

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Why is portfolio duration a weighted average?

What changes in the portfolio when its duration rises?

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What the interviewer asks next

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Why is yield to maturity a fair starting point?

Why will the realised return not be exactly 7.2%?

What else should the investor be told?

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What the interviewer asks next

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Why is maturity the wrong ruler?

How do you rank them without a calculator?

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Why must the two paths give the same answer?

What does an inverted curve tell you?

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What the interviewer asks next

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Why must the yield be below the coupon when the price is above 100?

How far below 7% is the yield to maturity?

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What the interviewer asks next

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Why do rising rates both hurt and help a bond investor?

What happens to each holding when yields jump to 8%?

Where does this protection stop working?

Where candidates lose it

What the interviewer asks next

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Why does a fall in yields raise bond prices?

What does this approximation leave out?

Where candidates lose it

What the interviewer asks next