Debt Capital Markets puzzles, solved step by step

Puzzles: 100
Traced to a firm: 16
Topics: 13
Hard: 30

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018A 3-year bond with a 10% annual coupon is priced at an 8% yield. What is its price today, and what will it be after one and two years if the yield never moves? Why is the holder's income less than the coupon?Bond pricing and yieldCoreFixed income asset management
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The yield stays at 8% for a year. What happens to the bond's price?
Show the worked solution
105.15 today, 103.57 after one year and 101.85 after two, then 100 at maturity. The bond pays 10 a year when the market wants 8, so it trades at a premium, and the premium shrinks as the high coupons are used up. Each year the holder gets 10 of coupon but gives back part of the premium, so true income is 8.41 in year 1, exactly 8% of the 105.15 paid.
Why is the bond above 100 in the first place?
Imagine a flat rented at Rs 10,000 a month on a three year lease when similar flats rent for Rs 8,000. A buyer pays extra for that lease, but the extra is only worth the months left on it. A bond paying a coupon above the market yield trades above par, and the premium is the present value of those extra coupons still to come. Priced at 8%, three coupons of 10 and the 100 repayment are worth 105.15.
At an unchanged 8% yield the price falls from 105.15 to 103.57, 101.85 and 100, and each year's coupon of 10 splits into a fall in price and a true income of 8.41, 8.29 and 8.15, 8% of each year's opening price.
How do you get the three prices quickly?
Price the premium, not the bond. Each year the bond pays 2 more than the market rate would, so the premium is 2 times the annuity factor for the years left, at 8%. With three years left the factor is 2.577, so the premium is 5.15; with two years left it is 1.783, a premium of 3.57; with one year left, 0.926, a premium of 1.85. That is quicker than discounting every cash flow and makes the pull to par obvious.
The relationship
$P_n = 100 + (C - y \times 100) \times \frac{1-(1+y)^{-n}}{y} \qquad P_3 = 100 + 2 \times 2.577 = 105.15$
P_n price with n years left
C annual coupon, 10
y market yield, 8%
n years to maturity
What it says in wordsPrice is par plus the present value of the coupon's excess over the market rate for the years left.
Why does it matter that income is less than the coupon?
Because the coupon overstates what the holder earns. Of the 10 received in year 1, 1.59 is really the holder's own money coming back as the premium runs off, so the income is 8.41, exactly the 8% yield on the price paid. A bank or fund that booked the full 10 as income would show a loss of the same size in the price. This is why accounts amortise premiums and why a desk compares bonds on yield, not coupon. The limit: it all assumes the yield stays at 8%, and any move in rates adds a gain or loss on top.
Where candidates lose it
The common wrong answer is that the price stays put because the yield did not move. Candidates link price changes only to yield changes and forget that time alone moves a premium or discount bond towards 100.
The second slip is calling the 10% coupon the return. The return at purchase is the 8% yield; the coupon is higher only because part of it hands back the premium you paid.
What the interviewer asks next
What does the same path look like for a 6% coupon bond at an 8% yield?
If the yield falls to 7% after one year, what is the one year return?
Why do insurers sometimes prefer premium bonds with high coupons?
040A perpetual bond pays Rs 8 a year forever and trades at Rs 100. The market yield falls to 6.4%, and in a second scenario rises to 9.6%. What is the price in each case, and why is the gain bigger than the loss for the same 1.6 point move?Bond pricing and yieldCoreFixed income asset management
Try it first
What happens to the price for each move?
Show the worked solution
The price is 125 at 6.4% and 83.33 at 9.6%: a gain of 25 against a loss of 16.67. A perpetuity is worth its coupon divided by the yield, so 8 over 0.064 and 8 over 0.096. The price-yield curve is bowed, not straight, which is convexity: for the same move either way, the price rises more than it falls, and the gap grows with the size of the move.
How do you price a bond that never matures?
Think of a shop that pays you Rs 8 of rent every year forever. If you want an 8% return, you will pay Rs 100 for it; if 6.4% is enough, you will pay Rs 125, because 6.4% of 125 is 8. A perpetuity's price is simply its annual payment divided by the yield investors demand. No maturity, no final repayment, no discounting table: one division does all the work.
The relationship
$P = \frac{C}{y} \qquad \frac{8}{0.064} = 125 \qquad \frac{8}{0.096} = 83.33$
C the annual coupon, Rs 8
y the market yield as a decimal
P the price investors will pay for the stream
What it says in wordsThe price is the income divided by the return investors want on it.
On the curve 8 divided by the yield, a fall from 8% to 6.4% lifts the price by 25 to 125, while an equal rise to 9.6% cuts it by only 16.67 to 83.33, and the line joining the two points sits above 100 at 104.17.
Why is the gain bigger than the loss?
Because the price curve bends toward you. When yields fall, each extra basis point lifts the price more than the last, and when yields rise, each basis point cuts it less than the last, so equal moves produce unequal price changes in the holder's favour. That bend is convexityThe curvature of the price-yield relationship; positive convexity means prices gain more when yields fall than they lose when yields rise by the same amount.. The average of 125 and 83.33 is 104.17, above the starting 100, which is the whole idea in one number.
A perpetuity has a lot of it, because its cash flows stretch to infinity. At 8% its duration is 1 over the yield, about 12.5 years, so a straight-line estimate would say 1.6 points moves the price about 20 either way. The truth, +25 and -16.67, shows how badly a duration-only estimate does on large moves for long bonds. The limit is that convexity helps only a holder of an option-free bond; a callable perpetual would lose most of that upside.
Where candidates lose it
The fast wrong answer is plus and minus 20, from a duration estimate applied to a big move. It is a sensible first approximation, but the interviewer chose a perpetuity and a 1.6 point move precisely so that the straight line would miss by five points.
The second loss is getting 125 and 83.33 and not saying why they differ. Name convexity, and say that the gap grows with the size of the move.
What the interviewer asks next
What is the modified duration of this perpetuity at 8%, and what price change does it predict for a 10 basis point move?
The issuer can call the perpetual at 100 after five years. What happens to the upside?
Why do long-dated bonds have more convexity than short ones?
056An issuer can pay 8.00% once a year or 7.85% in two semi-annual instalments of 3.925% each. Which is more expensive for the issuer, measured as an effective annual rate?Bond pricing and yieldCoreIndian debt capital markets
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Which costs the issuer more?
Show the worked solution
The 7.85% semi-annual coupon is marginally more expensive: an effective 8.004% a year against 8.000%. Paying 3.925 every six months means the first half coupon reaches investors early and can itself earn 3.925% for six months. Compounded, 1.03925 squared minus 1 is 8.004%. The two are almost equal because the exact semi-annual equivalent of 8% annual is 7.846%, just below 7.85%.
Why can you not compare 8.00 with 7.85 directly?
A landlord offered Rs 12,000 once a year or Rs 6,000 every six months should prefer the six-monthly cheques: the first Rs 6,000 can sit in a deposit for half a year. A rate only means something together with how often it is paid, because money paid earlier starts earning sooner. 8.00% paid annually and 7.85% paid semi-annually are quoted on different bases, so they must be put on one basis before anyone can say which is cheaper.
Per Rs 100, the annual bond delivers 8.000 at the year end, while the semi-annual bond's first 3.925 grows to 4.079 by the year end and adds to the second 3.925 for 8.004, so the semi-annual structure costs the issuer 0.4 basis points more.
How do you convert both to an effective annual rate?
Take Rs 100. The annual bond pays 8.00 at the end of the year, so its effective rate is 8.00%. The semi-annual bond pays 3.925 at six months and 3.925 at twelve. Reinvest the first 3.925 at the same 3.925% for six months and it grows to 4.079, so the investor holds 8.004 per Rs 100 at the year end, an effective 8.004%. Reinvesting at the coupon's own rate is the convention that defines an effective rate.
The relationship
$\text{EAR} = \left(1 + \frac{0.0785}{2}\right)^2 - 1 = 1.03925^2 - 1 = 8.004\%$
0.0785 / 2 the rate paid each half year
squared two half years compounded into one year
EAR effective annual rate, the one-payment-a-year equivalent
What it says in wordsCompound the half-yearly rate over two periods to compare it with a rate paid once a year.
How big is the difference in money, and when does it matter?
The gap is about 0.4 of a basis point, roughly Rs 2 lakh a year on a Rs 500 crore issue: real, but tiny next to the 15 basis points the headline rates suggest. The exact semi-annual rate matching 8% annual is 2 x (square root of 1.08 minus 1), 7.846%. The comparison matters in India, where government securities usually pay semi-annually and many corporate bonds annually, so a spread between them is only fair once both sit on one basis. The limit: the conversion assumes the early coupon is reinvested at the same rate, a convention rather than a promise.
Where candidates lose it
The fast wrong answer is that 8.00% costs more because it is the bigger number. That ignores when the money moves: a coupon paid six months early is worth more to the investor and costs more to the issuer.
The opposite slip is treating compounding as a large effect. At these rates it is worth about 15 basis points in total, which is exactly why 7.85% semi-annual lands almost on top of 8.00% annual. Give the size of the gap, not just its direction.
What the interviewer asks next
What quarterly rate is equivalent to 8.00% paid annually?
A government security yields 7.10% semi-annual and a corporate bond 7.90% annual. What is the spread on a like-for-like basis?
Why does the compounding effect grow as rates rise?
088The yield on a 10-year zero-coupon bond doubles from 3% to 6%. Does its price halve? Work out the old and the new price per Rs 100 of face value.Bond pricing and yieldCoreFixed income asset management
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Roughly how much does the price fall?
Show the worked solution
No. The price falls from about 74.41 to 55.84 per Rs 100, a fall of 25.0%, not 50%. A zero's price is 100 divided by one plus the yield, compounded for ten years. Doubling the yield moves one plus the yield only from 1.03 to 1.06, so the price is divided by (1.06/1.03) to the tenth, about 1.33. Halving the price would need a yield near 10.4%.
Why is doubling the yield not doubling the discount?
A shop that raises its markup from 3% to 6% has doubled the markup, but the price tag only goes from 103 to 106. The thing that multiplies through is one plus the rate, not the rate. A bond's price depends on (1 + y) compounded over its life, so doubling y from 3% to 6% changes the base only from 1.03 to 1.06, about 2.9% a year. Ten years of that is a factor of 1.333, which takes about a quarter off the price.
A 10-year zero priced at 74.41 at a 3% yield falls to 55.84 at 6%, a 25.0% drop, well short of the 37.20 that halving would need, which would take a yield of about 10.4%.
The relationship
$P = \frac{100}{(1+y)^{10}}: \quad \frac{100}{1.03^{10}} = 74.41, \quad \frac{100}{1.06^{10}} = 55.84$
P price per Rs 100 of face value
y the annual yield
10 years to maturity, with nothing paid before then
What it says in wordsDiscount the single Rs 100 payment back ten years at each yield.
How does duration check the answer?
The modified duration of a 10-year zero at 3% is 10 / 1.03, about 9.71. Multiplied by a 3 point rise that predicts a fall of about 29.1%, against the true 25.0%. The straight-line estimate overshoots on a large move because the price curve bends: each extra point of yield takes off less than the one before. That bend is convexity, and a 3 point move is big enough to show it. For small moves duration is close; for large ones, reprice the bond directly.
Where candidates lose it
The trap is answering yes, because doubling feels proportional. Price is not inversely proportional to yield; it is inversely proportional to one plus the yield, compounded.
The second loss is getting 25% right and then being unable to say why duration gave 29%. One sentence on convexity turns a correct number into an understood one.
What the interviewer asks next
What yield would halve the price from 74.41?
If the yield falls from 3% to 0%, what happens to the price, and is the move bigger or smaller than the fall you just computed?
Would a 10-year bond paying a 6% coupon fall by more or less than the zero for the same move?

P_n	price with n years left
C	annual coupon, 10
y	market yield, 8%
n	years to maturity

C	the annual coupon, Rs 8
y	the market yield as a decimal
P	the price investors will pay for the stream

0.0785 / 2	the rate paid each half year
squared	two half years compounded into one year
EAR	effective annual rate, the one-payment-a-year equivalent

P	price per Rs 100 of face value
y	the annual yield
10	years to maturity, with nothing paid before then

Try it first

Why is the bond above 100 in the first place?

How do you get the three prices quickly?

Why does it matter that income is less than the coupon?

Where candidates lose it

What the interviewer asks next

Try it first

How do you price a bond that never matures?

Why is the gain bigger than the loss?

Where candidates lose it

What the interviewer asks next

Try it first

Why can you not compare 8.00 with 7.85 directly?

How do you convert both to an effective annual rate?

How big is the difference in money, and when does it matter?

Where candidates lose it

What the interviewer asks next

Try it first

Why is doubling the yield not doubling the discount?

How does duration check the answer?

Where candidates lose it

What the interviewer asks next