11.5 Real and nominal rates; the yield curve

11.5.1The Fisher equation and breakevens#

Let $i$ be a nominal yield, $r$ the real yield and $\pi^e$ expected inflation over the same horizon. The Fisher equation says

The approximation drops the cross term $r\pi^e$, which is about 13 bp at $r=2.6\%,\ \pi^e=5\%$.

Inflation-indexed bonds pay coupons and principal scaled by a price index, so their yield is a market real rate. In the US these are TIPS (Treasury Inflation-Protected Securities). The breakeven inflation rate is

where IRP is an inflation risk premium (compensation for inflation uncertainty, which raises $b$) and LP is TIPS' liquidity premium relative to nominals (which lowers $b$). On FRED: DGS10 ≈ DFII10 + T10YIE.

Dated example On 22 Oct 2026 the US 10-year TIPS yield was 2.64% and the 10-year breakeven 2.32%; their sum, 4.96%, equals the nominal 10-year yield that day¹. The real yield's 2026 high was 2.95% on 5 Oct, above every daily value since 2008².

The decomposition matters for a trader. In March 2026 an oil shock raised breakevens, but the hawkish Fed repricing raised real yields more. Gold responds mainly to real yields (11.4), so it fell. A trader who watched only "inflation is rising" would have bought gold, and lost.

India India has no liquid inflation-indexed government bond curve: issuance of inflation-indexed bonds has been sporadic and trading thin. Real rates in India are therefore usually measured ex post, as the repo rate minus CPI inflation, or ex ante, as repo minus the RBI's projected inflation. With repo at 5.50% and the RBI's FY27 CPI projection of 5.2% (Oct 2026 policy [V]), the ex-ante real policy rate is about +0.3%. Against the September 2026 CPI print of 5.54%³, the ex-post real rate is slightly negative.

11.5.2Forward rates, expectations and the term premium#

Let $y_n$ be the continuously compounded yield on an $n$-year zero. The forward rate from year $n-1$ to $n$ is fixed by no-arbitrage: $e^{n y_n} = e^{(n-1)y_{n-1}} e^{f_{n-1,n}}$, so $f_{n-1,n} = n y_n - (n-1)y_{n-1}$. Then

Define the term premium $TP_n$ as the gap between the long yield and the average expected short rate:

The expectations hypothesis is the claim that $TP_n$ is constant. Two facts follow from (11.14a).

1. The curve slopes up if short rates are expected to rise, or if the term premium is positive. You cannot read expectations off the curve without a model for $TP_n$.
2. The NY Fed's ACM model (Adrian, Crump and Moench) estimates $TP_n$ with an affine term-structure model. Other estimates exist and disagree in level.

11.5.3Inversion as a recession signal#

The curve is inverted when long yields are below short yields. The usual measures are 10y–2y (FRED T10Y2Y) and 10y–3m (T10Y3M). Estrella and Mishkin (1998) showed that the 10y–3m spread was the best single financial predictor of US recessions two to six quarters ahead, in a probit model on post-1959 data⁴.

Two mechanisms support the signal:

• Expectations. Markets expect the central bank to cut, and central banks cut in recessions.
• Credit supply. Banks borrow short and lend long. A flat or inverted curve squeezes net interest margin and tightens lending.

The 2022–24 inversion⁵:

Measure	Continuously inverted from	Turned positive for good	Deepest point
10y–2y	6 Jul 2022	6 Sep 2024	−1.08 pp on 3 Jul 2023
10y–3m	25 Oct 2022	13 Dec 2024	−1.89 pp on 4 May 2023

This was the longest inversion in the modern record, and no NBER-dated recession began in 2023 or 2024. Caveats a careful trader holds:

1. Small sample. There are about eight post-war US recessions. A probit fitted on eight events cannot have tight error bands.
2. Term premium regime. After years of QE (central-bank bond buying), term premia were near zero or negative. A compressed term premium lowers long yields and makes inversion more likely without any change in expected short rates.
3. Lags are long and variable. Historically the lag from inversion to recession has been between about 6 and 24 months. The un-inversion (bull steepening as the Fed cuts) has often been closer to the recession start than the inversion itself.
4. Goodhart's law. Once a signal is widely watched, policy reacts to it.

In late 2026 the US curve is upward sloping. On 28 Oct the 2-year was 4.67%, the 10-year 5.05% and the 30-year 5.37%⁶. Long yields rose more than short ones in an inflation scare. That is a bear steepening, consistent with a rising term premium.