An Illustrated Guide to the Yield Curve

AT SIXTY TRILLION DOLLARS, the market value of US fixed-income securities is twice as large as the market capitalization of all US corporations and thrice the size of the American economy as measured by GDP. These debt securities consist of some $12tn in corporate bonds, $15tn in mortgages, another $15tn in interest-rate swaps; all of it anchored on the $16tn market for US Treasuries. When one talks about “the” yield curve one means the term structure of US Treasuries.

The yield curve does not actually exist. It is estimated from market prices of on-the-run Treasury securities using something called a quasi-cubic hermite spline function. (Don’t ask.) You don’t want to know how the sausage is made. What you want to learn above all is how to read the damn thing.


Figure 1. US yield curve as of 15 February 2018. (Source: NYFed.)

Figure 1 displays the yield curve as of 15 February 2018. Here’s how you should read this graph. Think of the bonds as paying $1 in n years time (called the maturity or term). What the graph tells you is that the prevailing market prices of bonds are such that buying one today and holding it to maturity to get that $1 yields the displayed return via the simple time-value-of-money formula:

bond price formula

The relationship between the price and yield of a zero-coupon bond.

Table 1 displays the prices of these hypothetical bonds whose yields are displayed in Figure 1.

Bond prices and yields
Maturity 1 2 3 4 5 6 7 8 9 10
Yield 2.00 2.22 2.41 2.55 2.65 2.74 2.80 2.86 2.90 2.94
Price $0.98 $0.96 $0.93 $0.90 $0.88 $0.85 $0.82 $0.80 $0.77 $0.75

These hypothetical, plain vanilla bonds that promise to pay $1 exactly n years in the future are called zero-coupon bonds. They are the building blocks of all bonds; any bond whatsoever can be mathematically represented as a portfolio of zero-coupon bonds. In other words, all the information contained in the messy markets for default-remote bonds can be read off the yield curve for zero-coupon bonds. And that is what is displayed in Figure 1.

The expectations theory of the yield curve says that the yield curve reflects the expected path of the short rate. More precisely, that the yield at time t on a zero-coupon bond of maturity n equals the market expectation at time t of short rate prevailing at time t+n.

expectations theory


The expectations theory is wrong. Bond prices are dramatically more volatile than that implied by the theory and they routinely move in opposite direction that that implied by the expected path of policy rates. The basic reason why the theory fails is that default-remote bonds, even the obligations of a global military hegemon, are not in fact risk-free. The most obvious risk is that inflation may erode the real value of the bond. But beyond inflation risk, a bond holder faces duration risk: The risk that interest rates may rise faster than expected thereby reducing the market value of the bond in her portfolio. Only bonds of very short maturity are truly risk-free which is why the risk-free rate is approximated by the 3-month Treasury bill. Figure 2 displays the time-variation in the yield curve and the expected path of the Fed’s policy rate since 2010. Note how the former is much more volatile than the latter and has utterly distinct dynamics.


Figure 2. The yield curve and the expected path of short-term rates.  (Source: NYFed.)

Bond yields have two basic components. They reflect on the one hand the expected path of short-term rates and on the other the compensation for the risk they pose to the bondholder’s balance sheet. The latter component is called the term risk premium. In order to extract the expected path and risk premium from bond yield, we have to use a term structure model. We use estimates from the gold standard of term structure models, the ACM model developed by economists at the NY Fed.

Figure 3 displays the evolution of the term structure of the term risk premium, ie the difference between the two “sheets” displayed in Figure 2.


Figure 3. Term structure of the term risk premium. (Source: NYFed.)

Comparing Figure 2 and Figure 3, observe that the bulk of the variation in the yield curve since 2010 has been driven by variation in the term risk premium. Figure 4 displays the two components of the yield on the 5-year note.


Figure 4. Components of the yield on the five-year note. (Source: NYFed.)

We can see that, until 2014, most of the variation in the yield on the 5-year note was driven by variation in the risk premium. The big shock in 2013 was, of course, the taper tantrum induced by Bernanke’s announcement that the Fed would slow down its bond buying. Since 2014, the expected rate has gone up but the risk premium has fallen. Note that the term risk premium can, has episodically been, and at present is, negative. The collapse of risk premiums is not confined to the market for US Treasuries. Risk premiums (ie, expected returns in excess of the risk-free rate) get bid away across asset classes by investors’ increasingly desperate search for yield. Put another way, we are in an asset price boom—asset prices are too high. For high quality collateral like US Treasuries, this translates into negative risk premiums.

The expected path of the policy rate reflects the market’s expectation of the Fed’s reaction function on the one hand and the trajectory of core macroeconomic variables on the other. A steeper path implies that the market expects the Fed to tighten faster, say in order to counter inflation. So what has been going on over the past few weeks? Figure 5 zooms in on the components of the 5-year yield in 2018.


Figure 5. Recent movements in the components of the 5-year yield.

Both expected rates and the risk premium have gone up since the year began. Medium term rate expectations (5-year) went up 13 basis points—a basis point is hundredth of a percentage point—while the term risk premium went up 26 basis points. Rate expectations fell dramatically with the return of market volatility in early February. But that has since been priced out. Figure 6 displays the term structure of rate expectations. We see that rate expectations rose uniformly, fell together, and are now priced back in.


Figure 6. Term structure of rate expectations.

What all this means is that the market expects the Fed to hike faster, perhaps because it expects inflation to surprise on the upside, thereby forcing the Fed to hike faster than hitherto expected. Since there is no reason to believe that inflation has returned, this should get priced out soon enough.



Bonus chart: The expected path of policy rates.




2 thoughts on “An Illustrated Guide to the Yield Curve

  1. Pingback: Best of the Web: 18-02-20 nr 1836 | Best of the Web

  2. Pingback: Best in Economics this week: February 23 | Best of the Web

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