When we abruptly stopped last time, we had just finished figuring out the Present Value of the variable side of the swap I offered you:

PV of variable side = 1 – P(t_{N})

…where t_{N} is the time of the final payment at the swap’s maturity, and P(t) is the discount factor derived from the zero-coupon yield curve:

P(t) = 1/(1 + y(t))^{t}

Now you get to offer me a fixed rate in exchange. Call that fixed rate “C” — you will see why soon — and express it as a simple annual rate. For example, if C were 5% and you had to pay it for 1 year, you would owe me 5%. If you had to pay it for 2 years, you would owe me 10%. If you had to pay it for six months, you would owe me 2.5%. In general, to compute the fixed payment for a simple annual rate C over an interval of Δt years, multiply C by Δt.

Recall that I am offering to pay you variable interest at specific times t_{1}, t_{2}, …, t_{N}, with each interest payment calculated over the latest interval (and with no payment at t_{0} = 0). You offer me payments on the same schedule, but at the simple fixed rate C. So at time t_{1}, you will pay me (t_{1} – t_{0})*C. At time t_{2}, you will pay me (t_{2}-t_{1})*C. At time t_{3}, you will pay me (t_{3}-t_{2})*C. And so forth.

What is the Present Value of your series of payments? To compute the present value of a future payment at time t, we multiply that payment by the discount factor P(t). So do that for each of your payments and add them up:

PV of fixed side = (t_{1} – t_{0})*C*P(t_{1}) + (t_{2}-t_{1})*C*P(t_{2}) + … + (t_{N}-t_{N-1})*C*P(t_{N})

For the price of the swap to be “fair”, the Present Value of the fixed side must equal the Present Value of the variable side. So you just set:

(t_{1}-t_{0})*C*P(t_{1}) + (t_{2}-t_{1})*C*P(t_{2}) + … + (t_{N}-t_{N-1})*C*P(t_{N}) = 1 – P(t_{N})

…and solve for C. And that is how you price a swap.

Bear with me.

I have to back up a bit now and correct some of my earlier exaggerations / errors. Throughout these posts, I have tried to maintain a consistent approach. In particular, I have only used “yield” in the sense of yield-to-maturity and “yield curve” in the sense of zero-coupon yield curve. I did this because yield-to-maturity and the zero-coupon yield curve are the concepts we need to calculate the Present Value of future money, and that is how I think.

But not everyone thinks this way. In particular, the U.S. Treasury does not think this way. The Treasury does not say to itself: “Let’s auction off some 10-year notes with $10 billion total face value and a coupon of 2%”. The Treasury says to itself: “Let’s auction off some 10-year notes with $10 billion total face value **at par**“.

Definition: The face value of a bond is also called its *principal* or its *par value*. (Yes, we have three terms for the same thing.) If a bond’s current market price — that is, its total Present Value — is less than its face value, it is said to be trading *below par*. If its current market price is greater than its face value, it is said to be trading *above par*. And if its current market price equals its face value, it is said to be trading *at par*.

Treasury notes and bonds always have coupons that pay semi-annually. When the Treasury auctions off, say, $10 billion worth of 10-year notes, they set the coupon sufficiently high to attract enough bidders so that the bonds sell for exactly $10 billion today. The technical process is called a “Dutch auction”, and “exactly” is an overstatement, but this description is close enough for government work, as it were.

In other words, the Treasury focusses on the interest it will have to pay on the principal it borrows, not on the yield-to-maturity that the investors will enjoy. So the Treasury is interested in the following question: If it tried to sell a bond today with $1 face value maturing in (say) t_{N} years, what coupon C would make the Present Value of that bond equal $1?

The Treasury — and we — can answer this question given the zero-coupon yield curve and its associated discount factor P(t). A $1 t_{N}-year Treasury bond with coupon C pays interest of ½*C every six months, followed by payment of the $1 principal at the maturity time t_{N}. We just need to add up the Present Value of all of those future payments and make the total equal $1:

½*C*P(0.5) + ½*C*P(1) + ½*C*P(1.5) + … + ½*C*P(t_{N}) + P(t_{N}) = 1

(The terms involving C are the present values of the interest payments. The final term on the left side is the present value of the $1 principal payment at maturity. The right-hand-side is the Present Value we are targeting.)

Solve for C, and we find the coupon that will let Treasury sell the bond at par.

Every day, the Treasury monitors market prices of Treasury bonds, calculates C for each of several maturities, and publishes the results. Note that C depends on the maturity — in our notation, t_{N}. For a particular maturity t_{N}, C is called the *par bond yield at maturity t _{N}*. A graph of par bond yields by maturity is called the

*par bond yield curve*.

Aside: You might think these should be called the “par bond coupon” and “par bond coupon curve”. That is what I thought at first, anyway. But notice that for a bond trading at par, the coupon equals the yield. (What APY would a savings account have to pay on $1 to give you 5 cents per year and eventually return the $1?) So calling this a “yield curve” is correct, if confusing. There are yield curves, and then there are yield curves…

The par bond yield curve is what Bloomberg et. al. show you when you click on “yield curve”. It can be derived from the zero-coupon yield curve as we just showed. Going the other direction — i.e., deriving a zero-coupon yield curve from a par bond yield curve — is called “bootstrapping” the zero-coupon yield curve, and it is (much) more complicated.

We are now very close to our destination. Look again at the equation you used to calculate the fair fixed rate to offer for our swap. In particular, see what happens if I suggest we exchange interest payments at regular six-month intervals. That is, set t_{0}=0, t_{1}=0.5, t_{2}=1, t_{3}=1.5, …, t_{N} = N/2. Go ahead, try it.

…

Lo and behold, you get exactly the **same equation** as the one for computing par bond yields. So for a fixed/floating swap exchanging payments every six months for t_{N} years referenced to Treasuries, **the fair market rate for the fixed side is equal to the par bond yield of maturity t _{N}**. As it turns out, you did not need to calculate anything; you could have just read off the answer from the par bond yield curve provided by Treasury! Well, for swapping periodic six-month payments, anyway.

Now, go back through these last two posts and substitute “LIBOR” for “Treasury bonds”. That is, assume I offer to swap short-term LIBOR at certain points in the future. And assume we do not trust each other completely, but only as much as large financial insitutions trust each other. And so on.

All of the derivations are identical, except *there is no LIBOR yield curve* beyond a maturity of one year. But there is a market for swaps! So by examining LIBOR swap rates in the marketplace, we can infer what the LIBOR yield curve would look like if it existed.

Swap rates **are** the par bond yield curve for LIBOR. And therefore they can be compared directly to par bond yields of Treasuries at the same maturity to provide some measure of systemic credit risk. At a given maturity, the difference between the LIBOR market swap rate and the Treasury par bond yield is called the *swap spread* at that maturity.

Next time: Swap spreads conclusion, references, and odds and ends.

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