Financial Markets & Expectations

The Yield Curve & Term Structure

coreExam · highTA · TA4-Q1, TA4-Q2

Under the expectations hypothesis, a long-term interest rate equals the average of expected future short-term rates. Two-year rate ≈ (i_{1,t} + i^e_{1,t+1})/2. With risk aversion, long rates include a term premium on top. The shape of the yield curve reveals market expectations about future monetary policy.

Derivation

Step 1 / 7

1Press Space or click Reveal next
(hidden)
2Press Space or click Reveal next
(hidden)
3Press Space or click Reveal next
(hidden)
4Press Space or click Reveal next
(hidden)
5Press Space or click Reveal next
(hidden)
6Press Space or click Reveal next
(hidden)
7Press Space or click Reveal next
(hidden)

The Expectations Hypothesis

An n-year bond must earn the same as rolling one-year bonds n times (in expectation):

(1 + i_{n,t})^n = (1 + i_{1,t})(1 + i^e_{1,t+1}) \cdots (1 + i^e_{1,t+n-1})

Log-linearising for small rates:

i_{n,t} \approx \frac{1}{n} \sum_{k=0}^{n-1} i^e_{1,t+k}

The long rate is (approximately) the average of expected future short rates. The yield curve is a picture of the market's expected policy path.

Adding a Term Premium

In reality, long bonds carry price risk — their value is more sensitive to rate changes. Risk-averse investors demand a term premium:

i_{n,t} = \underbrace{\frac{1}{n} \sum_{k=0}^{n-1} i^e_{1,t+k}}_{\text{expectations}} + \underbrace{x_n}_{\text{term premium}}

Reading the Curve

| Shape | Implied expectation | Historical signal | |-------|--------------------|--------------------| | Upward-sloping | Rising future short rates | Normal expansion | | Flat | Stable rates | Late cycle | | Inverted | Falling future short rates | Recession incoming |

TA4 Example (Alfa)

$i_{1,t} = 5\%$ , $i^e_{1,t+1} = 5\%$ → $i_{2,t} = 5\%$ .
Rates expected to fall by 1 pp after $t+2$ : $i^e_{1,t+2} = 4\%$ → $i_{3,t} = (5 + 5 + 4)/3 \approx 4.67\%$ .
Anchor at $3\%$ by $t+3$ : $i_{4,t} = (5 + 5 + 4 + 3)/4 = 4.25\%$ .

The curve flattens then inverts as expected cuts enter the average.

Worked Example

Alfa: i_1 = 5%, expected next-period i^e_{1,t+1} = 5%. Starting from t+2, short rates expected to fall by 1 pp per year. (a) Compute i_{2,t} and i_{3,t}. (b) If at t+3 markets expect the CB to anchor at 3%, compute i_{4,t}.

(a) i_{2,t} = (i_{1,t} + i^e_{1,t+1})/2 = (5 + 5)/2 = 5%.
i_{3,t} = (i_{1,t} + i^e_{1,t+1} + i^e_{1,t+2})/3 = (5 + 5 + 4)/3 ≈ 4.67%.
(b) i_{4,t} = (5 + 5 + 4 + 3)/4 = 4.25%. The curve inverts as the anchor enters the average.

i_{2,t} = 5%, i_{3,t} ≈ 4.67%, i_{4,t} = 4.25%. Expected rate cuts flatten then invert the yield curve.

Common Mistakes

—Using simple averages for high rates — the product formula (1+i_1)(1+i^e) is exact; arithmetic averages are approximate.
—Forgetting the term premium — the expectations hypothesis rarely fits data precisely; risk aversion adds a wedge.
—Confusing yield (i) with price — bond prices move inversely, so inverted yield curve = long-bond prices high relative to short-bond prices.
—Interpreting an inverted curve as recession certainty — it is a signal, not a mechanism.

Exam Cues

→Two-year rate ≈ (i_1 + i^e_{1,t+1})/2. Three-year ≈ average of current + two expected.
→Inverted curve → market expects cuts → often precedes recessions.
→Term premium: x_n (weakly) increasing in n. Adds to the pure expectations component.
→Policy signal: flattening after hike surprise = hawkish; steepening after cut surprise = dovish.