Defining the Yield Curve

For debt contracts, the overall duration of time of the debt security coupled with the interest rate compounded over that time frame will illustrate the overall yield of the security during its lifetime. This is referred to as a yield curve. When this is applied to U.S. treasury securities in respect to interest rates, useful information regarding projected interest rates in the future over time can be estimated. This is carefully monitored by many traders, and utilized as a point of comparison or benchmark for other investments (particularly valuation of bonds).

Yield Curve Example

This yield curve from 2005 demonstrates the projected yield over time of USD. As you can see, this is a typical yield curve shape, as the longer the contract is held out the higher the rate of return (with diminishing returns).

This yield curve from 2005 demonstrates the projected yield over time of USD. As you can see, this is a typical yield curve shape, as the longer the contract is held out the higher the rate of return (with diminishing returns).

Relationship to the Business Cycle

Through assessing the slope of a yield curve on debt instruments such as governmental treasury bonds, investors can estimate the overall health of the economy in the future (i.e. inflation, interest rates, recessions, growth). Inverted yield curves are typically predictors of recession, while positively sloped yield curves indicate inflationary growth.

The Financial Stress Index

Defined as the rate of difference between a 10-year treasury bond rate and a 3-month treasury bond rate, the Financial Stress Index is a useful tool in projected future economic well-being. In fact, each of the recessionary periods since 1970 have demonstrated an inverted yield curve when subjected to Financial Stress Test just prior to that recessionary period. 

Market Expectations (i.e. Pure Expectations)

When it comes to interest rates specifically, yield curves are useful constructs in projecting future behavior. The market expectations theory assumes that various maturities are perfect substitutes, and as a result the shape of the yield curve represents market expectations over time in relation to interest rates. In short, through investor expectations of what the 1-year interest rates will be next year, the current 2-year interest rate can be calculated as the compounding of this year's 1-year interest rate by next year's expected 1-year interest rate. Or, as an equation:

${\displaystyle (1+i_{lt})^{n}=(1+i_{st}^{{\text{year }}1})(1+i_{st}^{{\text{year }}2})\cdots (1+i_{st}^{{\text{year }}n}),}$

(i_st and i_lt are the expected short-term and actual long-term interest rates, respectively)

Heath-Jarrow-Morton Framework

When it comes to predicting future interest rates, the Heath-Jarrow-Morton framework is considered a standard approach. It focuses on modeling the evolution of the interest rate curve (instantaneous forward rate curve in particular). The equation itself is a rather evolved derivation, incorporating bond prices, forward rates, risk free rates, the Wiener process, Leibniz's rule, and Fubini's Theorem. While the details of this calculation are a bit outside the scope of discussion here, the equation can ultimately be described as:

${\displaystyle df(t,u)=\left({\boldsymbol {\Sigma }}(t,u)\int _{t}^{u}{\boldsymbol {\Sigma }}(t,s)^{T}ds\right)dt+{\boldsymbol {\Sigma }}(t,u)dW_{t}}$

For the sake of this discussion, it suffices to say that the input of existing yield curves is useful in projected future interest rates under a number of varying perspectives.