Show how expectations about future interest rates help determine the shape of the term structure.
The idea
Under the pure expectations hypothesis, the forward rates implied by today's curve equal the market's expected future short rates, so the slope of the curve reveals where rates are expected to go, but with premia they need not. A spot rate is the geometric average of the one-period rates along the path. Write f2 for the expected one-year rate in a year. Write Factor2 for the two-year growth factor, (1 + z2)² in terms of the spot rate. Factor2 = (1 + z1) × (1 + f2). Let m be n − 1, and write Fwd for the forward rate for year n. Fwd = (1 + zn)^n ÷ (1 + zm)^m − 1. An upward-sloping curve means forwards sit above spot rates, so short rates are expected to rise. An inverted curve means expected declines. Compound consistently with the quote and never average rates arithmetically. If investors demand a liquidity or term premium, the forward includes it. Forward = Expected + Premium. So the expected rate is the forward less the premium. Exam point: under the pure form, forward equals expected rate. Any premium must be stripped out explicitly.
Formulas
Forward rate for year n from two spot rates
Fwd = (1 + zn)ⁿ ÷ (1 + zm)ᵐ − 1
- zn
- the annually compounded spot rate for n years
- zm
- the spot rate for m years, where m = n − 1
- n
- the later maturity, in years
- m
- the earlier maturity, n − 1
Holds by no arbitrage with annual compounding. Under the pure expectations hypothesis it is also the expected one-year rate; with a term premium, subtract the premium to get the expectation.
A practice question
The one-year spot rate is 2.00% and the two-year spot rate is 5.00%, both annually compounded. Under the pure expectations hypothesis, the market's expected one-year rate one year from now is closest to:
- 2.94%Why not: Forgets to square the two-year factor: 1.05/1.02 - 1 compares a one-year growth factor with a two-year one.
- 3.00%Why not: Subtracts spot rates (5% - 2%) instead of solving 1.05² = 1.02 x (1 + f).
- 3.50%Why not: Takes the simple average of the two spot rates, which is not a forward rate.
- 8.09% - correct
1.05² = 1.02 x (1 + f), so 1 + f = 1.1025 / 1.02 = 1.0809 and f = 8.09%. Under pure expectations the forward equals the expected rate.
A harder one
The one-year spot rate is 3.00%. A forecaster expects the one-year rate to be 4.00% in one year and 5.50% in two years. Under the pure expectations hypothesis, the three-year spot rate (annual compounding) is closest to:
- 4.16% - correct
- 13.01%Why not: Uses the cumulative three-year growth, 1.03 x 1.04 x 1.055 - 1, without annualizing.
- 5.50%Why not: Uses only the last expected rate rather than the geometric average of all three years.
- 4.25%Why not: Averages the first and last rates, (3.00% + 5.50%) / 2, ignoring the middle year and compounding.
(1 + z3)³ = 1.03 x 1.04 x 1.055 = 1.1301, so 1 + z3 = 1.1301^(1/3) = 1.0416 and z3 = 4.16%.
The hardest tier
Annually compounded spot rates are 3.0% (one year), 3.8% (two years) and 4.2% (three years). Under the pure expectations hypothesis, the expected one-year rate two years from now is closest to:
- 4.61%Why not: Computes the forward for year 2, 1.038²/1.03 - 1, not the one for year 3.
- 4.81%Why not: Annualizes the two-year rate starting in one year, (1.042³/1.03)^(1/2) - 1, instead of the single year-3 forward.
- 4.20%Why not: Reports the three-year spot rate itself, which is an average over three years rather than the rate for year 3.
- 5.00% - correct
1 + f(2,3) = 1.042³ / 1.038² = 1.1314 / 1.0774 = 1.0500, so f = 5.00%. This is the expected year-3 one-year rate under pure expectations.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →