3.2 The Yield Curve, the Spot Curve, and Pure Expectations

You will recall that, in studying Corporate Project Finance, we discovered that the Internal Rate of Return (the “IRR”) embeds the problem of assuming that a prospective project’s interim cash flows are reinvested at the pre-ordained IRR. This assumption is so unlikely as to be regarded as patently false. Similarly, a bond’s Yield-to-Maturity being the precise mathematical equivalent of a project’s IRR embeds this same redundancy. The YTM is a bond’s IRR. The notion of Pure Expectations addresses this problem, but does not resolve it, as we will soon see. Pure Expectations provides a conceptual basis upon which multiple discount rates for a given YTM can be discovered. In the theory of Pure Expectations, we assume that the Market Yield reflects the average of future short-term rates. Let’s first review the notion of Pure Expectations and then implement it quantitatively by calculating the Spot Curve. Let’s see what we get.

First, we assume that investors think about the future and, specifically, about the future direction of short-term interest rates. In this notion, we say that the observed Yield Curve is, in a sense, secondary to what market players believe – in their minds – future short-term yields will be. If, as in the mathematical example below, market participants believe that future short-term yields will go higher and higher, then the observed yield curve will reflect this collective belief and be positive in slope – and vice versa. In other words, the Yield Curve reflects market participants’ a priori beliefs.

What makes this interesting is that while we can directly observe the Yield Curve – after all, it is quoted in the media, and by traders and brokers – it is first the unexpressed thoughts and beliefs about the direction of future short-term rates that determine the Yield Curve’s openly expressed slope. We know what is in the public’s minds by observing the effect of their collective thought on the slope and shape of the Yield Curve! The Yield Curve expresses what people think about the future!

Similarly, the observed Yield Curve – mathematically – will express the average of market participants’ expectations about the course of future short-term rates. For example, standing here today, in the here and now, assuming first that the investor’s horizon is two years, the investor is faced with two choices: (1) Buy a two-year bond, or (2) Buy a one-year bond and “roll it over” for another year when it comes due at the end of the first year.

Given this line of thinking, if his horizon is three years, Mr. Investor can buy a three-year bond, or choose to consecutively roll it over twice. If we assume initially that the two choices in each case should be and are equivalent, we can extrapolate the investor’s beliefs about the “Spot Curve,” i.e., the market’s collective belief about future short-term rates, from the Yield Curve, working backwards.