Par yield

There are a multitude of terms in finance, one such concept is the par yield. Often seen as one of the more difficult ideas within the realm of yield metrics, the par yield is simply a type of yield to maturity. Specifically, it is the yield to maturity that prices a bond at par.

Let's delve into this intricate concept by making some assumptions. For this illustration, we will assume that the face value of the bonds is 100. The crucial input assumption here is the spot rate curve, also known as the zero rate curve. This is a steep curve, exaggerated for our understanding, and is the fundamental building block in academic finance.

The spot rate curve is an upward sloping line, with rates increasing from 3% to 9% over four years. We use this curve to price cash flow instruments. It's worth noting that there's an infinite number of these curves, one for each gradation of risk. The theoretical spot rate curve is often bootstrapped using treasuries to create the zero rate or zero coupon yield curve for riskless instruments.

Corresponding to the spot rate curve are the discount factors. These are the values by which future cash flows are multiplied to determine their present value. For example, with annual compounding, the three-year discount factor is 1 divided by 1 plus the three-year spot rate of 7%, giving us 0.816.

The par yield is one of the yield to maturities. To illustrate, let's consider a 4% coupon bond. This bond pays $4 annually for three years, and in the fourth year, it pays a $4 coupon plus the return of par. We can use the discount factors to price this bond. The sum of the present value of these cash flows is the price of the bond, which in this case is 84.45.

We can also compute the yield, or the yield to maturity. Although there isn't a straightforward formula for this, if you're using Excel, you can use the rate function. For our 4% coupon bond, the yield is 8.78%. This means that we can discount the bond's cash flows at the yield of 8.78% to get the same price of the bond. It's worth noting that all cash flows are discounted at the same yield, unlike the spot rate curve.

The yield is primarily a function of the spot rate curve, but it's also a function of the instrument. So, technically, we could have an infinite number of yields depending on the coupon. This is why we use a par yield, which is less arbitrary.

The par yield is the one yield to maturity that would price the bond at par, and therefore would also equal the coupon. This is the sweet spot where the yield equals the coupon, and the bond is priced at par. So, when we specify the par yield, we know that it represents the coupon rate that would price the bond at par and would therefore be the same as the yield.

In conclusion, the par yield is an essential tool in bond pricing and yield calculations. By understanding how it's calculated and what it represents, investors and financial professionals can make more informed decisions about the pricing and purchase of bonds. Remember, the par yield is simply the yield to maturity that prices a bond at par, making it an invaluable measure in the complex world of yield metrics.