Mutual Fund Mastery puzzles, solved step by step

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

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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
Try it first
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
Try it first
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?
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
Try it first
What is the best single estimate of the hold-to-maturity return, per year?
Show the worked solution
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?
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?

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%

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

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

Try it first

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

Try it first

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?

Where candidates lose it

What the interviewer asks next

Try it first

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?

Where candidates lose it

What the interviewer asks next

Try it first

Why is maturity the wrong ruler?

How do you rank them without a calculator?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a fall in yields raise bond prices?

What does this approximation leave out?

Where candidates lose it

What the interviewer asks next