Debt Capital Markets puzzles, solved step by step

Carry of 0.90%, roll-down of about 1.05%, so roughly 1.95% over funding. Carry is the bond's 7.40% yield less the 6.50% cost of funding it. Roll-down comes from the bond ageing into a 4 year bond priced at 7.10%: its yield falls 30 basis points, which at a duration of about 3.5 is a 1.05% price gain. On an unchanged steep curve, standing still earns both.

What is carry, and what is roll-down?

Carry is the simple part: you earn the bond's yield and pay your funding rate, like renting out a flat for more than its loan EMI. Roll-down is the price gain that comes purely from time passing on an upward sloping curve: the bond gets shorter, and shorter maturities are priced at lower yields. Nothing about the market has to change for either to happen; that is why desks call them the return from standing still.

How do you turn 30 basis points into a return?

Use duration. A yield change converts to a price change of roughly duration times the change, so a 30 basis point fall at a duration of 3.5 is a gain of about 1.05%. Use the duration the bond will have after the year, when it is a 4 year bond, because that is the bond being repriced; the question gave 3.5 for exactly that reason. Then total return is the yield earned, 7.40%, plus the roll gain, about 8.45% before funding.

How much can the market move before the trade loses money?

Divide the cushion by the duration. A 1.95% return cushion at duration 3.5 absorbs a rise of about 56 basis points in the 4 year yield by year end before the position breaks even. That breakeven is the number a portfolio manager actually trades on. Say the limits too: the curve rarely stands still, the roll-down depends on the curve's shape between the two points, and the duration figure is an approximation that ignores convexity.

The fair swap rate is about 6.478%, roughly 2.2 basis points below the 6.5% average. Discount factors from the path are 0.9434, 0.8858 and 0.8279. The swap rate is the average of the fixings weighted by those factors. Because the lowest fixing comes first and is discounted least, it counts for more, so the weighted average sits under the simple one.

What does it mean for a swap to be worth zero?

Picture two friends agreeing to swap rent for three years: one pays a flat amount, the other pays whatever the market rent turns out to be. The deal is fair only if, in today's money, the flat payments are worth the same as the expected market ones. A swap is worth zero at the start when the present value of the fixed leg equals the present value of the expected floating leg. Present value is the key phrase: a rupee in year one is worth more today than a rupee in year three.

Build the discount factorsThe value today of one rupee paid at a future date. A factor of 0.943 means a rupee in a year is worth 94.3 paise now. from the same path. Year one: 1 over 1.06, or 0.9434. Year two: that divided by 1.065, 0.8858. Year three: divided again by 1.07, 0.8279. Their sum, 2.6571, is the value today of receiving one rupee a year for three years.

Why does the average overshoot?

Look at the two outer years. At 6.5% fixed, the fixed payer overpays by 0.5 in year one and is overpaid by 0.5 in year three. Equal in plain rupees, but the year one amount is discounted by 0.943 and the year three amount by 0.828. The early overpayment is worth more today than the late refund, so at 6.5% the fixed payer loses about 0.058 per Rs 100, and the fair rate must come down until the discounted gaps cancel. On an upward sloping path the swap rate always sits a little under the simple average; on a downward one it sits a little over.

Check it the short way. The ratio of one less the last discount factor to the sum of factors gives the same 6.478%, because a floating leg plus the return of notional at the end is worth exactly par. The gap to 6.5% is small here, about two basis points, but on a steep ten-year curve the difference between the plain and weighted averages becomes large enough to misprice a trade.

The implied one-year rate a year from now is about 8.20%. Rs 100 at 7.6% for two years grows to 115.78. Rs 100 at 7% for one year grows to 107, so the second year must turn 107 into 115.78: a rate of 8.20%. It is above both spot rates because the two-year rate is an average, and the second year has to be high enough to pull the 7% first year up to 7.6%.

What does a forward rate actually mean?

You can reach the next town by the highway in one go or by a side road with a stop halfway. If both leave at the same time and arrive at the same time, the second leg of the side road must make up whatever speed the first leg lost. A forward rate is the rate for the second leg that makes lending for two years in one go pay exactly the same as lending for one year and rolling over. If the two routes paid differently, everyone would take the better one until they did not.

Why must the forward sit above both spot rates?

Because the two-year rate is roughly the average of the two one-year legs. If the first leg earns 7% and the average is 7.6%, the second leg must earn about 7.6% plus the 0.6 the first leg fell short, roughly 8.2%. The quick approximation is two times 7.6 less 7, which is 8.2%; compounding adds the last few hundredths to give 8.20%. On an upward sloping curve, forwards always sit above the spot rates, and the steeper the curve, the further above.

Say what the forward is not. It is not a forecast of where one-year rates will be; it is the break-even rate that makes the two routes equal today. If investors demand extra yield for locking money up longer, a term premium, the forward will sit above what the market truly expects. A desk uses it to price a forward starting loan or to judge whether rolling short or locking long is cheaper.

The two-year zero rate is about 7.035%. The one-year par bond pays 106 in a year for 100, so the one-year zero rate is 6%. The two-year par bond pays 7 and then 107. Strip the first coupon at 6%: 7 / 1.06 = 6.604. The remaining 93.396 must be the value of 107 in two years, so (1 + z) squared = 107 / 93.396, and z = 7.035%.

What does bootstrapping actually do?

A fruit seller sells one mango for Rs 20 and a bag of one mango and one papaya for Rs 70. You can price the papaya without ever seeing it sold alone: Rs 50. Bootstrapping prices each future date the same way: use the shorter bond to value the early cash flows, and whatever is left of the longer bond's price belongs to its final cash flow. A one-year par bond at 6% pays 106 in a year for 100, so the one-year zero rate is simply 6%.

Why is the zero rate above the par yield?

A par yield is one rate that blends the rates on every cash flow, so when the curve slopes up, the last cash flow must carry a rate above the blend. The two-year bond's first coupon is discounted at 6%, which makes it worth a little more than a 7% discount would say. To keep the whole bond at 100, the final 107 has to be discounted a little harder: 7.035% rather than 7%. The gap is small because only one coupon of 7 sits at the lower rate.

Say what the zero curve is for, because it is why the question is asked. With zero rates you can price any bond by discounting each cash flow at its own rate, which is how a desk checks whether a bond is fairly priced. The same two rates give the one-year rate one year forward: 1.07035 squared over 1.06, minus 1, about 8.08%. The limit: real bootstrapping uses many bonds with different coupons and settlement conventions, and a two-point curve says nothing about the shape in between.

Receiving fixed is a long fixed rate bond funded by a short floating rate note, so its DV01 is about Rs 3.95 lakh per basis point. The bond leg moves Rs 100 crore x 4.2 x 0.0001, or Rs 4.2 lakh, per basis point. A floater resetting quarterly has a duration of about 0.25, which takes off Rs 0.25 lakh. The principals cancel, and the position gains when rates fall.

Why can a swap be written as two bonds?

Suppose you lend a friend Rs 1 lakh at a fixed 7% and, the same morning, borrow Rs 1 lakh from your bank at its floating rate. The two lakhs cross in the air and cancel; what remains is that you collect 7% and pay floating. A receive-fixed swap is exactly that: the cash flows of owning a fixed rate bond and having issued a floating rate note on the same notional, with the two principals cancelling, which is why a swap needs no money upfront.

Why does the floating leg carry almost no rate risk?

A floating rate noteA bond whose coupon is reset to a market benchmark at regular dates, so its price stays close to par. resets its coupon to the market at each reset date, so on a reset date it is worth about par whatever rates have done. Its price can only drift between resets. With quarterly resets the floater behaves like a bond maturing in three months, with a duration of about 0.25, so it cancels only about 6% of the fixed leg's rate risk. The swap is, for rate purposes, nearly the whole fixed bond.

Which way does the position make money, and who uses it?

Receiving fixed gains when rates fall: a 10 basis point drop in the 5-year swap rate makes the position about Rs 39.5 lakh. That is why a desk calls receiving fixed a long duration position. It is also the everyday DCM use: an issuer that sells a 5-year fixed bond and wants floating rate debt receives fixed on a swap against its own coupon, and ends up paying floating. Treat the 4.2 and 0.25 as illustrations; the real figures depend on the coupon, the curve and the reset dates.

The real yield is about 2.39%, against 2.50% from simple subtraction. The Fisher relation divides one plus the nominal yield by one plus inflation: 1.07 / 1.045 - 1 = 2.39%. The shortcut overstates the real yield by about 11 basis points, because inflation also erodes the interest itself, and the gap widens as rates rise.

Why is subtraction only an approximation?

Lend Rs 100 for a year at 7% and get Rs 107 back. If prices have risen 4.5%, a basket that cost Rs 100 now costs Rs 104.50. Your Rs 107 buys 107 / 104.5 baskets, 1.0239 of them: a real gain of 2.39%, not 2.5%. Inflation shrinks the purchasing power of everything you get back, the interest as well as the principal, so the real yield is a ratio, not a difference.

When does the shortcut stop being good enough?

The error is roughly the real yield times inflation, divided by one plus inflation. At low rates it is a few basis points and nobody minds; at high inflation it is tens of basis points and it changes the conclusion. With nominal yields of 15% and inflation of 12%, subtraction says 3.0% but the true real yield is 2.68%. For a desk comparing an inflation-linked bond with a conventional one, a gap of that size is the whole trade. The rates here are illustrations; the relation, not the numbers, is the point.