How to Calculate the Term Structure of Interest Rates
Under the pure expectations theory, a long-term bond yield is the geometric average of today's short rate and the short rates investors expect over the same span.
The Term Structure formula
Calculator
Enter today's one-year rate and the rates expected in the next two years to get the two- and three-year yields.
The yield on a one-year bond you can buy right now.
What investors expect a one-year bond to pay a year from now.
The expected one-year rate two years out, the last leg of the horizon.
The rate that compounds to the same three-year total is 5.987%, which is the point on the yield curve at three years.
- Value of $100 rolled over three years
- $119.06
- Two-year yield
- 4.995%
- Simple average of the three rates
- 6%
- Yield curve shape
- Upward sloping
Rolling one-year bonds over three times turns $100 into $119.06, and the three-year bond has to match that total.
A two-year bond has to pay 4.995% a year to tie the first two one-year bonds.
Averaging the short rates gives 6%, a shortcut that always sits just above the exact geometric answer.
The curve slopes up when the expected short rates run above today's rate, and it inverts when investors expect cuts.
How to calculate Term Structure, step by step
- 1List the short rates in order. Write today's one-year rate first, then the one-year rate investors expect for each later year, one figure per year of the horizon.
- 2Turn each rate into a growth factor. Add 1 to each rate in decimal form, so 4 percent becomes 1.04. Compounding works on factors, never on the percentages themselves.
- 3Multiply the factors together. The product is what a dollar becomes if it is rolled over in one-year bonds for the whole horizon.
- 4Take the nth root and subtract 1. Raise the product to the power 1 ÷ n and subtract 1, then multiply by 100. That single rate compounds to the same total, so it is the n-year yield.
- 5Read the shape of the curve. Compare the n-year yield with today's one-year rate. Higher means an upward sloping curve, lower means an inverted one.
Worked example: Term Structure
Suppose the one-year rate is 4%, and investors expect 6% next year and 8% the year after. Rolling over one-year bonds turns $100 into $119.06, since 1.04 × 1.06 × 1.08 = 1.190592. The two-year yield is [(1.04)(1.06)]^(1 ÷ 2) − 1 = 4.995%, and the three-year yield is (1.190592)^(1 ÷ 3) − 1 = 5.987%, a shade under the simple average of 6%. Because the three-year yield sits above today's 4% one-year rate, this curve slopes upward.
Term Structure questions
What is the expectations theory of the term structure?
It holds that a long-term rate is set by the short rates investors expect over that horizon, so a two-year bond and two consecutive one-year bonds have to offer the same expected return. Adding a premium for tying money up for longer gives the liquidity premium version.
Why does an inverted yield curve signal a recession?
An inverted curve means investors expect short rates to fall, and the usual reason for a central bank to cut is a weakening economy. The signal comes from the expected path of policy rather than from the inversion causing anything itself.
Is the long rate just the average of the short rates?
The arithmetic average is a close shortcut, but the exact answer is the geometric average, which is always slightly smaller. With rates of 4%, 6% and 8% the shortcut gives 6% while the geometric figure is 5.987%.
Get AP Econ exam tips in your inbox
Occasional emails with study tips, new interactive graphs, and exam-season reminders. Free, no spam.
No spam. Unsubscribe anytime. Read our privacy policy.
Keep track of what you have studied
A free EconLearn account adds progress tracking, your quiz history, and achievements. Studying here is free either way, and there is nothing to pay for as a student.
Create a free accountAlready have one? Sign in
Last updated