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Interest Rates & Term Structureadvanced

The Term Structure & the Yield Curve

The yield curve plots yield against maturity for bonds of the same risk. The expectations theory says long rates are an average of expected future short rates. The liquidity premium theory adds a term premium that rises with maturity, which is why the curve usually slopes up. An inverted curve often signals expected rate cuts and recession.

Try it yourself

Term Structure & Implied Forward Rates

Shape the spot (zero) curve and watch the no-arbitrage forward rates respond. An upward-sloping curve implies one-year forwards above the spot rates — these are the rates an FRA locks in.

Spot curve s_nImplied 1y forwards f(n−1,n)
0.0%1.2%2.5%3.7%5.0%1y2y3y4y5yMaturity (years)Annualized rate (%)FRA 4.05%
Forward Rate Agreement — pick the period
The 1×3 forward rate f1,3 = 4.05% locks in borrowing or lending over years 1 to 3 (a 2-year period).
FRA f(1,3)
4.05%
Curve shape
Upward
1y spot
3.00%
5y spot
4.00%
1y
3.00%
spot = fwd
3.00%
2y
3.40%
f(1,2)
3.80%
3y
3.70%
f(2,3)
4.30%
4y
3.90%
f(3,4)
4.50%
5y
4.00%
f(4,5)
4.40%

Why it matters

If you can earn a known short rate now and reinvest, a long bond must offer about the average of the short rates you expect, or you would just roll over short bonds. A premium for tying up money longer tilts the curve upward on top of that.

Formulas

Expectations theory (two-year rate)
i2t≈12 (it+it+1e)i_{2t} \approx \tfrac{1}{2}\,(i_t + i^e_{t+1})
The 2-year rate is about the average of today’s 1-year rate and next year’s expected 1-year rate.

Worked examples

Scenario

The 1-year rate is 4% and markets expect the 1-year rate next year to be 6%. What is the 2-year rate under the expectations theory?

Solution

About (4% + 6%) / 2 = 5%. An upward-sloping curve here reflects expected rate increases. The liquidity premium theory would add a small term premium on top.

Common mistakes

  • ✗An upward-sloping curve just means long bonds are better. It reflects expected future short rates plus a term premium, not a free lunch.
  • ✗The yield curve never predicts anything. An inverted curve has preceded most recessions, because it signals expected rate cuts.

Revision bullets

  • •Yield curve plots yield versus maturity at equal risk
  • •Expectations theory makes the long rate average expected short rates
  • •Liquidity premium adds an upward term premium
  • •Inversion often signals recession

Quick check

Under the pure expectations theory, a steeply upward-sloping yield curve means markets expect short-term rates to

Connected topics

Bond PricingDurationRisk Structure

Sources

  1. Mishkin (2018), Ch. 6
    Mishkin, F. S. The Economics of Money, Banking, and Financial Markets. 12th ed. Pearson, 2018. ISBN 978-1-292-26885-9.
    The term structure and the expectations, segmented-markets, and liquidity-premium theories.
How to cite this page
Dr. Phil's Quant Lab. (2026). The Term Structure & the Yield Curve. Derivatives Atlas. https://phucnguyenvan.com/concept/mb-term-structure
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