The Relation between Asset Returns and Inflation at Short and Long Horizons

Summary

Engsted and Tanggaard test the Fisher hypothesis — expected nominal returns move one-for-one with expected inflation — at horizons of 1, 5, and 10 years for US and Danish stocks and bonds. They replace Boudoukh-Richardson's (1993) generalized method of moments (GMM) approach (which requires time-overlapping multi-period returns and unreliable long-lag Newey-West corrections) with a vector autoregression (VAR) framework that computes multi-period expected values from one-period variables only, avoiding overlap entirely. Results reveal large cross-country asymmetries: US stocks fail the Fisher model at long horizons (correlation weakens from N=5 to N=10), while Danish stocks come close to perfect Fisher at 10 years (corr ≈ 0.89, SD ratio ≈ 1); US bonds are good Fisher hedges at long horizons but Danish bonds are not. Neither country's stocks hedge unexpected inflation.

Key Claims

VAR multi-period forecasts avoid overlap: $E\!\left[\sum_{j=1}^N R_{t+j} \,\middle|\, H_t\right] = \bigl(e_i' \sum_{j=1}^N A^j\bigr) X_t$ where $X_t = [\text{stock return, bond return, inflation, }\log(d/p),\,\Delta\text{short rate}]'$ . Standard inference applies at any horizon $N$ .
US stocks contradict Boudoukh-Richardson (1993): Boudoukh-Richardson (BR) find near-zero correlation at N=1 but positive at N=5 (Fisher model improves). Engsted-Tanggaard (ET) find 0.47 ( $N=1$ ), 0.52 ( $N=5$ ), 0.38 ( $N=10$ ) — Fisher weakens at the longest horizon.
US bonds support Fisher at long horizons: correlation rises from 0.004 ( $N=1$ ) to 0.73 ( $N=5/10$ ); expected returns and expected inflation equally volatile at N=10.
Danish stocks: near-perfect Fisher at 10 years: correlation 0.33 → 0.78 → 0.89; standard deviation ratio approaches 1. Fisher model unambiguously improves with horizon.
Danish bonds: weakening relationship: correlation 0.29 → 0.13 → 0.06. Fisher model fails at all horizons, deteriorates.
No unexpected-inflation hedge in either country: regressions of ex post returns on VAR-generated unexpected inflation yield $R^2 \approx 0$ in all cases; one significantly negative $\beta$ for US stocks ( $N=1$ ).
GMM results mostly confirm VAR: for $N \leq 5$ the GMM (Table 2) and VAR (Table 4) results broadly agree; J-test rejects Fisher for US stocks ( $N=5$ ) and Danish bonds ( $N=5$ ).
VAR fit: US — $R^2(\text{stocks})=0.16$ , $R^2(\text{bonds})=0.16$ , $R^2(\text{inflation})=0.56$ ; Denmark — $R^2(\text{stocks})=0.18$ , $R^2(\text{bonds})=0.27$ , $R^2(\text{inflation})=0.43$ .

Concepts Introduced or Extended

Entities Mentioned

Quotes

"In contrast to the results reported by Boudoukh and Richardson (1993). The Fisher model does not perform better as the horizon increases."

"For Danish stocks, however, the relationship becomes stronger as the horizon increases."

My Take

The paper's main value is methodological: replacing overlapping-data GMM with a companion-form VAR neatly sidesteps the Richardson-Stock / Hodrick critique, and allows horizon extrapolation to 10 years that GMM cannot reach reliably. The large US/Denmark divergence is striking and somewhat hard to interpret economically — similar assets, similar time periods, opposite Fisher-model behavior. Campbell-Ammer (1993) provides one explanation via the sign of the contemporaneous news response: in the US an inflation-expectations shock immediately raises stock prices (negative discount-rate effect dominates later), while in Denmark it immediately lowers them. Whether this reflects genuine differences in monetary regimes, fiscal structures, or sample accidents is left open.