Debt Capital Markets puzzles, solved step by step

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 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.