# Yield Curves

A yield curve is a visual display of current conditions in some particular fixed-income bond market. It's a snapshot of interest rates in that market – a simple yet often informative graph that plots yields to maturity on the vertical axis and times to maturity on the horizontal axis for a homogeneous set of securities. A yield curve also is called the term structure of interest rates. Some academics distinguish the two, preserving one for zero- coupon bonds and the other for standard coupon bonds, but I use them as synonyms and specifically identify the type of securities being discussed.

Yield curves are great for the study of bond math. We see in this chapter how we can move seamlessly, albeit with some assumptions, between the commonly observed yield curve on coupon bonds and related curves that we derive – the implied spot curve and the implied forward curve. These two are hugely important in fixed-income analysis. It's no doubt an exaggeration, but I think that the implied forward curve is the single most useful line in fixed-income markets – and the implied spot curve is not far behind.

Most yield curves are based on government securities. That's so we can hold constant all the factors other than time to maturity that impact investors' required rates of return – in particular, credit risk, liquidity, and taxation. Obviously, all yields should be stated for the same periodicity. We saw in Chapter 1 that this is a problem in practice at the short-term end of the yield curve (i.e., the money market), but it can be rectified with some basic bond math. Also, the yields to maturity ideally would be for zero-coupon securities so that coupon reinvestment risk is not a factor.

In reality, there is no perfect data set for term structure analysis. Typically seen yield curves are plots of street convention yields on coupon- bearing Treasury notes and bonds instead of yields on Treasury STRIPS. In particular, the yields displayed usually are for on-the-run issues (i.e., the most recently auctioned Treasury securities). These are actively traded and typically are priced close to par value, thereby minimizing the effects of the deferral for tax purposes of the gains and losses from buying at a discount or a premium that we saw in Chapter 4. The problem is that there are gaps in the times to maturity, so some yields have to be interpolated.

We start with an intuitive look at implied forward rates in order to pay some respect to the traditional theories of the yield curve.