Back-to-Basics: The Yield Curve

Teri Geske
Senior Vice President, Product Development

Any discussion of fixed income securities is likely to include some mention of "the yield curve". While this term is generally understood at a basic level, it is important to recognize there are various forms of a yield curve that play a role in valuing and analyzing fixed income securities and portfolios. So, in this Back-to-Basics column, we thought it would be useful to take a look at this fundamental building block of the fixed income analysis.

In the U.S. market, when we say "the yield curve" we are likely referring to the U.S. Treasury yield curve, constructed from the yield-to-maturity (YTM) of U.S. Treasury securities ranging from short to intermediate to long maturities¹. Right away, however, we introduce some potential ambiguity into the discussion, as we could be referring to the "on-the-run" curve, based on the YTM's of the most recently issued Treasury securities, or to a theoretical curve (sometimes called an "All Treasury" curve) constructed by fitting a curve through the yields of all or most of the outstanding Treasury issues across all maturities. Since the on-the-run Treasuries are in highest demand, their yields are almost always lower than the yields of all other Treasuries, and thus there is usually a difference between the on-the-run curve and an All Treasury curve at a given maturity point - sometimes the difference is only a couple of basis points, at other times it is much more. The All Treasury curve is similar, but not identical to a Constant Maturity Treasury (CMT) curve. A CMT curve is built from the YTMs that hypothetical Treasury instruments of selected constant maturities would have, interpolated from specific Treasury instruments that mature around the constant maturity dates.

There is a theoretical difficulty with using a yield curve constructed from the yields-to-maturity of coupon-bearing Treasury securities in analytical models. Yield-to-Maturity implicitly assumes that all of the cash flows for that security are discounted at that single yield. Thus, if a three year Treasury has a YTM of 3% and a two year Treasury has a yield of 2.75%, the coupon rate paid in Year Two from the 3 year bond would be discounted at different yield than the principal and interest paid on the two year bond at maturity if we use the Treasury yield curve to determine the discount rates applied in a valuation model. To avoid this inconsistency, we can "strip out" the value of the coupons from the prices of the Treasury securities and then solve for the rates that would produce the prices of the resulting theoretical zero coupon bonds - these rates are called "spot rates", which are important because they represent the single set of rates used to discount all (risk-free) cash flows that occur at a point in time. This allows us to construct another type of Treasury curve, called the "spot curve", which is used as the basis for most fixed income valuations and other analyses.

Whether we are looking at a "yield curve" or a "spot curve", the Treasury curve is usually upward-sloping, meaning the yields (or spot rates) on short-term maturities are lower than the yields on intermediate and long-term maturities. However, short-term yields can be higher than yields on longer maturities, producing an inverted Treasury curve (which was the prevailing shape of the curve for much of the year 2000). There are a number of theories to explain why the Treasury curve has any slope at all: the "Liquidity Preference" theory states that investors demand a premium to entice them to invest for longer terms - this theory is only consistent with a positively sloped curve. The "Market Segmentation" or "Clientele" theory says that the yield at each maturity is driven by the supply and demand from investors within that particular segment of the market - so, short-term investors determine short-term rates, while long-term investors determine long-term rates. Under this theory, the yield curve could have any shape. The "Expectations" theory says that investors expect to earn the same return regardless of the holding period; thus, the Treasury yield curve reflects expectations about where interest rates and inflation are headed. So, if the two year Treasury yield is higher than the one year yield, this theory says that the market is expecting inflation between years one and two such that an investor who purchased the one year Treasury today, then reinvested the proceeds one year from now into the then-prevailing one year Treasury to earn that rate from Year One to Year Two, would realize the same return as the investor who purchased the two year Treasury today and held it to maturity.

This Expectations theory leads us to the notion of implied forward rates. Implied forward rates are the Treasury rates that would be realized in the future if today's investors were to all earn the same return from Treasury securities, regardless of their holding period. So, continuing with the example above, the one-year forward rate is that rate that would cause the two consecutive one-year Treasury investments to earn the same return as an investment in the two year security. In simplified terms, (1+r_2yr) = (1+r_1yr) X (1+₁ƒ_1yr); and therefore, (1+₁ƒ_1yr) = [(1+r_2yr) ÷ (1+r_1yr)] –1; where ₁ƒ_1yr represents the one year Treasury rate one year from now, and r represents the spot rate for a given maturity.

It is well known that implied forward rates are generally unreliable predictors of where rates will actually be at a given future date. We reviewed monthly Treasury yield curves over eight years (1995 - 2002) and compared the actual one year yields each month to the one year implied forward rate that prevailed one year earlier. The implied forward rates differed from the actual, realized rates by an average of 103 bps, with a maximum difference of 365 bps. The one year implied forward rate was within 10bps of the realized rate for only 8 of the 96 months in the sample period. Rather than viewing implied forward rates as "predictors" of future interest rates, we can instead view them as the rates that must evolve within a theoretical modeling environment, given the current prices of Treasury securities today, to avoid the possibility of a riskless arbitrage within the model.

One question that often comes up in discussions about the Treasury yield curve is, how volatile are Treasury yields over time? On a proportional basis, meaning the change in rates as a percentage of the level of rates, we can see that volatility has varied considerably over the past five years:

However, on an absolute basis, we find that over 80% of weekly changes in the 10 year Treasury yield over the past five years have been 20 bps or less, regardless of the level of rates, as summarized in the following table:

We hope this discussion of the Treasury curve has been useful. If you have any questions, comments or suggestions for future "Back-to-Basics" articles, please contact marketing at fia.marketing@interactivedata.com.

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¹Investors can also use the swap curve as the benchmark yield curve, but unless otherwise specified, the term “yield curve” refers to the Treasury curve.