On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks

Source: Glosten, L. R., Jagannathan, R. & Runkle, D. E. (1993) · Journal of Finance 48(5), 1779–1801 · DOI 10.1111/j.1540-6261.1993.tb05128.x

TL;DR

Introduces the asymmetric volatility model now known as GJR-GARCH — a Modified GARCH-M that lets positive and negative return innovations have different effects on next-period conditional variance, and that adds the nominal interest rate and October/January seasonal dummies as variance predictors. Using monthly CRSP value-weighted excess returns (1951:4–1989:12), the paper finds a negative relation between the conditional mean and conditional variance of excess returns, and finds that monthly conditional volatility is less persistent than previously believed: positive unanticipated returns lower future volatility, negative ones raise it.

What it models

The conditional mean and conditional variance of the monthly nominal excess return on stocks (a market index), and the intertemporal risk–return trade-off linking them. Standard GARCH-M treats positive and negative shocks symmetrically, yet equity volatility responds more strongly to bad news; the authors argue this misspecification helps explain why earlier studies disagreed on the sign of the risk–return relation.

Specification

GARCH-M mean equation (risk-return trade-off, here with seasonal/rate terms in the index): expected excess return is linear in conditional variance,

x_t = a0 + a1·v_{t-1} + ε_t, E_{t-1}[ε_t]=0, E_{t-1}[ε_t²]=v_{t-1}

so a1 is the price of conditional variance. The asymmetric ("GJR") variance equation augments GARCH(1,1) with an indicator term (the paper's Model 2):

v_{t-1} = b0 + b1·v_{t-2} + g1·ε²_{t-1} + g2·ε²_{t-1}·I_{t-1}

where, in the paper's own notation, I_{t-1}=1 when ε_{t-1} is positive and 0 otherwise. Because positive innovations were expected to lower volatility, the authors predict g2 < 0 (and find g1+g2 < 0). Equivalently — and as the model is usually written today — flipping the indicator to switch on for negative shocks gives a positive asymmetry coefficient: bad news raises next-period variance more than good news. Richer variants add the risk-free rate r_{ft} (Model 3) and October/January seasonal dummies that scale fundamental volatility; an exponential (EGARCH-style) form is also estimated to keep variance positive.

Estimation

GMM / quasi-maximum-likelihood on monthly data; t-statistics computed via Hansen (1982) GMM, robust to conditional heteroskedasticity. Sample: continuously compounded monthly excess returns on the CRSP value-weighted index, post-Treasury-Accord period 1951:4–1989:12. Diagnostics include Engle–Ng-style sign-bias, negative-size-bias and positive-size-bias tests on squared standardized residuals to check whether the asymmetry is adequately captured. Summary statistics motivate the seasonals: in months other than October/January the mean excess return is 0.48% with SD 3.83%, vs. SD 5.19% in January (1.35×) and 6.17% in October (1.61×).

What it captures

Leverage / asymmetric-volatility effect — negative shocks raise conditional variance more than equal positive shocks (the term that names the GJR model).

Seasonality in volatility — elevated October and January variance via deterministic dummies.

Interest-rate predictability of variance — the nominal short rate forecasts conditional volatility.

Lower volatility persistence — once asymmetry, seasonals and the rate are modeled, monthly conditional volatility is less persistent than standard GARCH-M implies.

Negative risk–return relation — the headline finding: support for a negative relation between conditional expected return and conditional variance, contrary to the simple "more risk → more return" intuition (consistent with theory, since risky periods can coincide with greater willingness/ability to bear risk).

Use & extensions

GJR-GARCH is one of the canonical asymmetric volatility models, used routinely in volatility forecasting, option/risk applications, and value-at-risk. It sits alongside Bollerslev's GARCH and Nelson's EGARCH (the paper also estimates an EGARCH-M variant for comparison) as a standard way to model the leverage effect; later realized-volatility and stochastic-volatility models extend the same stylized facts to higher frequencies.

Limitations

A parametric, monthly model: it cannot exploit intraday information (cf. realized volatility).

In the standard (non-exponential) GJR form, g1+g2 < 0 can let estimated conditional variance go negative for some realizations — the authors turn to an exponential form to avoid this.

The risk–return result is specification-sensitive; the broader literature it surveys reports both positive and negative signs, and the sign here is conditional on including asymmetry, seasonals, and the nominal rate.

Key references

Glosten, L., Jagannathan, R. & Runkle, D. (1993) — On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks — Journal of Finance

Bollerslev, T. (1986) — Generalized Autoregressive Conditional Heteroskedasticity — Journal of Econometrics

Nelson, D. B. (1991) — Conditional Heteroskedasticity in Asset Returns: A New Approach (EGARCH) — Econometrica

Engle, R. F. & Ng, V. K. (1993) — Measuring and Testing the Impact of News on Volatility — Journal of Finance

French, K., Schwert, G. W. & Stambaugh, R. (1987) — Expected Stock Returns and Volatility — Journal of Financial Economics

Provenance: verified/generated from the paper's full text.