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The best-known theory regarding yield curves is based on bond investors' and issuers' expectations about future short-term interest rates. The idea is that market participants choose maturities to maximize outcomes over some known time horizon – investors maximize their expected rate of return (i.e., the horizon yield) and issuers minimize their expected cost of borrowed funds.

The conclusions of this expectations theory are quite significant: Yield curves are upward-sloping (or flat, or downward-sloping) when market participants generally expect short-term rates to be rising (or steady, or falling). In particular, the implied forward rate is an unbiased market consensus forecast for the future spot rate that will be available in the market.

That is, the calculated IFR between years A and B is equal to the expected market rate for the A × B time period. Therefore, the shape of the current yield curve tells you what bond buyers and sellers in general are expecting about future market interest rates. That's valuable information – if you can rely on it.

The problem with the expectations theory is that it rests on two very strong assumptions: Bond investors and issuers are risk neutral, and they can buy or sell bonds at any point on the yield curve. The first assumption means that decisions are made based only on the expected rates of return (and costs of borrowed funds). In sum, they care only about the mean of the probability distribution of possible outcomes and not about the standard deviation (or variance). There is no risk aversion in the expectations theory.

The second assumption means that bond buyers and sellers can move freely along the term structure, buying or issuing at whatever maturity they choose. All investors can “ride the yield curve” if that strategy maximizes the expected horizon yield. All corporate bond issuers consider rolling over commercial paper to finance a major construction project if that strategy minimizes the expected long-term cost of funds.

The segmented markets theory goes to the other extreme. Investors and issuers are assumed to be so risk averse that they buy and sell bonds only for maturities that match their underlying time horizons. Corporate treasurers managing funds until a tax payment date in three months are interested only in 3-month money market instruments. Individual investors building a fixed-income retirement portfolio buy only long-term bonds. Corporations financing the purchase of new computers issue only intermediate-term notes and never short-term commercial paper. The idea is that risk aversion becomes a barrier to entry to other maturity segments, no matter how attractive interest rates might be at those points on the yield curve.

According to the segmented markets theory, an observed yield curve is simply a collection of equilibrium interest rates based on the demand for, and supply of, money as we saw in Chapter 3. Drawing the yield curve in this theory is not a work of art; it's just connecting the dots from separate, segmented markets where in each demand and supply rule. The conclusion of this theory is that the implied forward rate that we can calculate is unrelated to the expected future short-term rate.

What about future interest rates? Proponents of this theory would contend that the best forecast for the shape and level of the future yield curve is the shape and level of the current yield curve – unless there is some reason to expect that demand and supply conditions at particular points along the curve will be changing.

The third classic theory introduces the obvious and realistic assumption that both risk and return matter to market participants. The liquidity preference theory starts with the recognition that expectations of holding-period rates of return for bond buyers and costs of borrowed funds for issuers determine what we can call the core yield curve. Then a “liquidity premium” is added to that core to capture the idea that investors, who are assumed to be risk averse, require extra compensation for longer-maturity bonds compared to shorter-maturity bonds. They know that in general longer-term bonds fall in value more if yields jump up, for example, due to higher expected inflation. Hence, there is a premium (or higher price) paid for safer, short-term bonds. Alternatively, we could say that there is an “illiquidity premium” built into yields on longer-term bonds.

An implication of the liquidity preference theory is that an implied forward rate is an upwardly biased estimate of the future short-term rate.

This bias arises because the longer-term yield (the 0 x B) contains a higher risk premium than the shorter-term yield (the 0 x A). If somehow we could extract each risk premium, we would have the core yield curve based only on expectations. That leads back to the conclusion of the expectations theory. Without that extraction, the observed yields are too high and the IFR overstates the expected market rate.

Which theory is correct? In economics, theories often are tested on their ability to explain observed data, summarized by a set of “stylized facts.” For example, yield curves are usually upwardly sloped, so much so that it is “normal” that short-term rates are lower than long-term rates. Occasionally yield curves take on other forms – flat, inverted, humped, or U-shaped – but eventually they return to the normal shape, which is upward sloping and leveling off at long maturities. The liquidity preference theory has the easiest time explaining this – it's simply because investors require a higher rate of return on longer-term bonds as compensation for greater risk.

The segmented markets theory needs the demand for long-term funds to be strong relative to short-term funds and the supply of short-term funds to be strong relative to long-term funds. That seems reasonable – borrowers prefer to lock up the source of funds for an extended time period while lenders prefer the flexibility of shorter-term investments. The expectations theory struggles with the normal shape to the yield curve because market participants would have to be expecting higher future short-term rates most of the time, presumably reflecting persistent concerns over inflation.

A second stylized fact about the term structure is that short-term yields usually are more volatile than long-term yields. That means the range of historical yields at the short-term end of the curve is greater than for longer maturities. Another way of saying this is that the term structure of volatility is downward sloping. The last few years since the financial crisis of 2007 to 2009 have reversed the usual pattern. The Fed has kept short-term money market rates persistently low. Therefore, there has been more volatility in longer-term yields. Hopefully, the traditional pattern will return one of these years.

In any case, the expectations theory explains the downward-sloping term structure of volatility by appealing to the idea that the long-term rate is an average of expected future short-term rates. Averages should have less volatility as the number of data points increases. The liquidity preference theory gets to the same conclusion after adding the risk premium to the core based on expectations.

The segmented markets theory needs more shifting in the demand and supply of short-term funds and greater stability in the curves in the long-term market. That's not at all unreasonable, given that investors often park funds in the money market while reallocating assets across sectors and currencies. That could lead to wider swings in the supply of short-term funds and greater volatility than for longer maturities that likely attract more buy-and-hold investors. For example, institutional investors, such as life insurance companies and pension funds, receive steady cash inflows that regularly are invested in long-term bonds.

Another observed pattern in yield curve data is that short-term and long-term yields usually change in the same direction, although not always in a parallel (or shape-preserving) manner. Sometimes the yield curve steepens and sometimes it flattens, but, in general, yield changes along the curve are positively correlated. Therefore, most shifts to the term structure of interest rates are more or less parallel. Factors that raise or lower expected future short-term rates – for instance, expected inflation, the business cycle and monetary policy actions, trade balances and foreign exchange rates, tax rates and fiscal policy actions – should impact the demand for and supply of long-term funds in a similar manner. All three theories are consistent with the observed pattern.

The problem with these classic term structure theories is that there is no one compelling winner among them. Perhaps the best way of assessing the theories is to look at some real data – for instance, the “hair chart” shown in Figure 5.1 produced by JP Morgan analysts. This is great visual presentation of financial data. The solid line shows the path of 6-month LIBOR between 1988 and 2004 and the corresponding LIBOR forward curve (the hair) for each date.

In the early 1990s and again in the early 2000s, the LIBOR forward curve was quite steep and upwardly sloped. Proponents of the expectations theory, who by the way are well represented among market commentators,

The LIBOR Hair Chart

FIGURE 5.1 The LIBOR Hair Chart

Source: Guy Coughlan and Nikolaos Panigirtzoglou, “Interest Rate Term Premia and the Shape of the Yield Curve over the Long Run,” JP Morgan Securities London, September 28, 2004.

would have said at the time that the market expects rates to be rising. Yet LIBOR continued to fall for several years. For a while in the late 1990s, LIBOR stabilized in the 6% range, and the forward curve flattened. Overall, there is no consistent pattern between the shape of the forward curve and subsequent market rates.

You might be asking yourself why we still care about the classical term structure theories and still teach them in academic finance courses and why I include them in a bond math book. My answer is that every finance professional making decisions about bonds relies to some extent on a forecast for future interest rates. If nothing more, these theories serve to steer us to think about what drives the shape and level of the yield curve – market participants' expectations and attitudes toward risk certainly matter, as do demand and supply factors in varying maturity segments.

Now we can move on to some bond math and topics that do have answers. Most important, we focus on applications of yield curve analysis that are theory-free. That is, these applications will not depend on the correctness of the expectations, segmented markets, or the liquidity preference theories. But first, we need to be able to get implied forward rates without continuing to commit a cardinal sin in finance – neglecting compounding.

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