Risk-Return Tradeoff

Definition

The risk-return tradeoff is the proposition that expected stock returns should be positively related to their conditional variance (or volatility) — investors demand higher compensation for bearing greater risk. Empirically the relationship is elusive: different estimation strategies yield positive, negative, or insignificant estimates, a contradiction that Brandt and Kang (2004) resolve by disentangling the contemporaneous innovation correlation from the lag predictive effect.

Key Ideas

Theory (intertemporal capital asset pricing model (ICAPM), habit formation) predicts a positive relationship between $E_t[r_{t+1}]$ and $\mathrm{Var}_t(r_{t+1})$ .
The canonical econometric vehicle is the ARCH-in-Mean (ARCH-M) model of Engle-Lilien-Robins (1987), which puts the conditional variance directly into the mean equation, $r_t=\mu+\delta\,g(h_t)+\varepsilon_t$ ; the price-of-risk coefficient $\delta$ is the risk-return tradeoff, and its sign/significance is notoriously sensitive to the volatility functional form — a first hint of the contradictions below.
Empirical evidence is contradictory: generalized autoregressive conditional heteroskedasticity in mean (GARCH-M) studies (French-Schwert-Stambaugh 1987) often find negative or insignificant coefficients; Glosten-Jagannathan-Runkle (1993) find negative (their asymmetric GJR-GARCH-M is the canonical negative-coefficient result); Harvey (1989) and others find positive.
The confusion arises because the contemporaneous innovation correlation $\rho = \mathrm{Corr}(\eta_t^m, \eta_t^v)$ and the lag coefficient $a_{12}$ (volatility at $t-1$ predicting mean at $t$ ) are different objects that earlier studies conflate.
Brandt-Kang (2004) find $\rho \approx -0.56$ (strongly negative) and $a_{12} \approx 0.09$ (insignificant): the lag tradeoff is weak but the contemporaneous correlation is large.

How It Works

The Latent VAR Model (Brandt-Kang 2004)

Excess returns follow: $y_t = \bar{\mu}e^{m_{t-1}} + \bar{\sigma}e^{v_{t-1}}\varepsilon_t, \qquad \varepsilon_t \sim N(0,1)$

where $m_t$ is the log conditional mean state and $v_t$ is the log conditional volatility state. The latent state vector $s_t = (m_t, v_t)'$ follows a bivariate vector autoregression (VAR(1)): $s_t = As_{t-1} + \eta_t, \qquad \eta_t \sim MVN(0,\Sigma)$ $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}, \qquad \Sigma = \begin{bmatrix} b_{11} & \rho\sqrt{b_{11}b_{22}} \\ \rho\sqrt{b_{11}b_{22}} & b_{22} \end{bmatrix}$

with $\mathrm{Corr}(\varepsilon_t, \eta_t) = 0$ .

Key parameters:

$a_{12}$ : lag-volatility-in-mean (the "risk-return tradeoff" coefficient in lag form)
$a_{21}$ : lag-mean-in-volatility (leverage/feedback effect in lag form)
$\rho$ : contemporaneous correlation between mean and volatility innovations

Estimation: Simulated Maximum Likelihood

The likelihood integrates over the latent path $s_1,\ldots,s_T$ — analytically intractable. The simulated maximum likelihood (SML) approach uses a VAR importance sampling density (linearized Gaussian approximation) and corrects via: $\ln\hat{\mathcal{L}}(\theta) = \ln\hat{f}(\theta) + \ln E_N[\hat{f}/\hat{g}]$ where $N$ is the number of simulated paths. Finite-sample properties (Table 1, $T=636$ , $N=500$ simulations) confirm near-unbiasedness.

Empirical Results (Center for Research in Security Prices (CRSP) value-weighted monthly, 1946–1998)

Parameter	Estimate	$t$ -stat	Interpretation
$a_{11}$	0.866	11.21	Mean persistence
$a_{22}$	0.897	15.32	Volatility persistence
$a_{12}$	0.086	1.01	Lag vol $\to$ mean: insignificant
$a_{21}$	−0.089	−1.96	Lag mean $\to$ vol: marginally significant
$\rho$	−0.558	−5.80	Contemporaneous correlation: strongly negative
$\rho_\sigma$	−0.254	−4.04	Return shock $\times$ vol innovation: leverage effect

Adding the short rate, term premium, and default premium as exogenous predictors reduces $|\rho|$ to $\approx -0.45$ but it remains highly significant.

Two Correlations, Opposite Signs

The paper distinguishes:

Contemporaneous innovation correlation ( $\rho \approx -0.56$ ): at any given moment, a positive shock to expected returns is accompanied by a negative shock to volatility. This drives the short-run "negative risk-return" finding in GARCH-M studies.
Unconditional time-series correlation (positive): in long-run high-mean, high-volatility regimes (recessions), both are elevated together. This drives the positive cross-sectional and long-horizon findings.

Business Cycle Dynamics

Both mean and volatility are countercyclical:

Conditional mean peaks $\approx 1.3$ – $1.5$ percentage points (pp) above its long-run average at recession troughs.
Conditional volatility peaks $\approx 3.8$ – $4.4$ pp above its long-run average at recession troughs.
The Sharpe ratio rises heading into recessions (mean rises faster than vol) and falls as the economy recovers.

The cross-autocorrelations between mean and volatility are all negative at all lags (Figure 5): high mean today → lower volatility tomorrow, and vice versa. This is the opposite of what a positive lag risk-return tradeoff would imply.

Impulse Responses

Mean innovation: expected returns jump and revert over ~36 months; volatility drops contemporaneously (from $\rho < 0$ ), then slowly reverts. Sharpe ratio rises sharply then decays.
Orthogonalized volatility innovation: volatility jumps and reverts over ~24 months; mean barely moves; Sharpe ratio falls.

Why It Matters

Reconciles the contradictory empirical literature on the risk-return tradeoff: the sign of the relationship depends on what is being measured — contemporaneous correlation ( $\rho < 0$ ) vs. unconditional long-run co-movement (positive).
The latent VAR framework is more flexible than GARCH-M or autoregressive moving average-GARCH (ARMA-GARCH): it imposes no particular functional form on the contemporaneous relationship and does not require observable predictors.
Both moments being countercyclical implies that recession risk premia arise from elevated expected returns (not just volatility), consistent with habit formation and ICAPM mechanisms.
The negative contemporaneous $\rho$ implies the Sharpe ratio is high precisely when both moments are high — at the bottom of the business cycle — providing a rational explanation for why investors earn high risk premia in recessions.

Open Questions

The negative contemporaneous correlation $\rho$ is documented but not structurally explained; habit formation, incomplete information, and long-run risk models each provide candidate mechanisms.
The exponential functional form ( $\bar{\mu}e^{m_{t-1}}$ ) is convenient but imposes particular dynamics; alternative link functions could matter.
Extensions to allow time-varying $\rho$ or higher-order VAR dynamics for the state are not explored.

Bollerslev-Zhou (2006): Volatility Proxy Explains the Contradictions

While Brandt-Kang (2004) diagnose the sign problem through the lens of contemporaneous vs. lagged innovation correlations, Bollerslev-Zhou (2006) provide a complementary, structurally grounded explanation: the sign and magnitude of the empirical feedback coefficient depends critically on which volatility proxy is used.

Under the Heston (1993) model with parameters $(\kappa, \theta, \sigma, \rho, \lambda_v, \lambda_s)$ , the population slope in a realized-vol return regression is:

$\beta = \lambda_s + \frac{\rho\kappa}{\sigma} < \lambda_s$

Because $\rho < 0$ (leverage), $\beta$ is downward biased relative to the true risk premium $\lambda_s$ , and can even be negative when $0 < \lambda_s < -\rho\kappa/\sigma$ . This explains why autoregressive conditional heteroskedasticity in mean (ARCH-M) and GARCH-M studies — which use model-based or realized volatility proxies contaminated by the contemporaneous leverage — so often produce negative or insignificant estimates.

The implied-volatility slope, by contrast, is:

$\beta^* = \lambda_s \cdot \frac{a_\Delta}{a^*_\Delta}, \quad a_\Delta = \frac{1-e^{-\kappa\Delta}}{\kappa}, \quad a^*_\Delta = \frac{1-e^{-\kappa^*\Delta}}{\kappa^*}$

Since $\kappa^* = \kappa + \lambda_v < \kappa$ (negative vol risk premium slows risk-neutral mean-reversion), $a_\Delta/a^*_\Delta < 1$ , so $\beta^* < \lambda_s$ , but always $\beta^* > 0$ . Empirically (S&P 500 1990–2002): $\hat\beta = -0.53$ for realized vol, $\hat\beta^* = +0.24$ for implied vol — opposite signs, both consistent with $\lambda_s > 0$ .

Fiorentini-Sentana (1998): White-Noise Returns with Persistent Expected Returns

Fiorentini and Sentana (1998) provide a formal theoretical basis for why stock returns can appear as white noise in univariate tests while having a highly predictable conditional mean. Their key result is that in a bivariate VAR(1) for $(r_t, \delta_t)$ (returns, dividend yields), the marginal process for $r_t$ is an ARMA(1,1). When the VAR companion matrix $A$ is near-singular ( $|A| \approx 0$ ), the MA and AR roots nearly cancel and $r_t$ is approximately white noise. The conditional mean $\mu_{r,t+1}$ follows a separate AR(1) whose persistence is governed by $\text{tr}(A) = a_{11}+a_{22}$ .

The mechanism requires that the innovation $u_t$ to $r_t$ and the innovation $w_t$ to $\mu_{r,t+1}$ are nearly perfectly negatively correlated: a positive shock to current returns (surprise) is accompanied by a negative revision in expected future returns, leaving total return variance approximately at its unconditional level. This is the multivariate counterpart of the Campbell (1991) variance decomposition: news about future expected returns accounts for a large fraction of current return variance, but with opposite sign.

Empirical estimates (U.S. monthly, 1952:1–1994:12):

Parameter	Value	Interpretation
$\theta$ (ARMA MA root for $r_t$ )	0.9916	Returns essentially white noise
AR coefficient of $\mu_{r,t+1}$	0.9755	Highly persistent expected returns
SD of $w_t$ / SD of $r_t$	1/42	Tiny variation in expected returns
$\rho(u_t, w_t)$	−0.9466	Large negative contemporaneous correlation
$P_\infty(r_t\|w_t)$	1600	Shock to expected returns propagates into returns 1600 $\times$

The large persistence ratio (1600) means that a tiny change in expected returns has a dramatic contemporaneous impact on the actual return level — consistent with the Brandt-Kang (2004) finding that the contemporaneous innovation correlation between the mean and volatility states is strongly negative.

French-Schwert-Stambaugh (1987): Where the Puzzle Starts

French-Schwert-Stambaugh (1987) is the empirical origin of the sign puzzle the structural work above resolves. Using daily returns to build monthly volatility (an early realized-volatility construction) and separating it into predictable and unpredictable parts with ARIMA models, they find:

a weak positive relation between the expected risk premium and the predictable component of volatility (the ex-ante tradeoff), and
a strong negative relation between unexpected returns and unexpected volatility changes.

They read the second, dominant finding as indirect evidence for the first: a positive shock to expected volatility raises required returns and so depresses the current price ("volatility feedback"). They argue the negative relation is too large to be pure financial leverage. In the Brandt-Kang language this negative return–volatility-innovation relation is precisely the strongly negative contemporaneous correlation $\rho$ ; in the Bollerslev-Zhou language it is the leverage contamination $\rho\kappa/\sigma$ that biases the realized-vol slope downward. FSS thus poses, in 1987, the contradiction the later papers explain.