Forward-Looking Market Risk Premium Weiqi Zhang National University of Singapore Dec 2010

Background Estimating risk premium Historical average of realized excess returns –Backward-looking –The risk premium estimate can be negative even using an estimation period of 10 years (from 1973 to 1984) Forward-looking risk premium –Our new approach is based on option prices (Relate forward-looking market risk premium to (1) investors’ risk aversion implied by the option market, and (2) forward-looking physical moments – variance, skewness and kurtosis)

Notation (over the time period t to t+τ) Continuously compounded risk-free rate: r t (τ) Dividend yield of the market portfolio: δ t (τ) Market portfolio’s cumulative return: R t (τ)=ln(S t+τ /S t ) Mean, standard deviation, skewness and kurtosis: –under the physical measure P: μ Pt (τ), σ Pt (τ), θ Pt (τ), κ Pt (τ) –under the risk neutral measure Q: μ Qt (τ), σ Qt (τ), θ Qt (τ), κ Qt (τ) Forward-Looking Risk Premium Theory

The equilibrium risk-free interest rate can be expressed as Idea: Expand and impose the fact that risk- neutral expected return equals risk-free rate minus dividend yield. –Can we express it in terms of physical moments? Forward-Looking Risk Premium Theory

Assume the form of stochastic discount factor: Rely on an approximate expression moment generating function of R t * (τ) =R t (τ) - μ Pt (τ) under measure P: Uses the role of stochastic discount factor to link MGF under probability Q and P Express risk neutral moments in terms of physical moments. Forward-Looking Risk Premium Theory

Substitute the derived risk-neutral moment expressions into the risk-free rate equation and obtain a new market risk premium expression entirely based on physical return moments: Proposition 1 Under Assumption 1, the τ-period market risk premium can be expressed as a function of investors’ risk aversion, physical return variance, skewness and kurtosis: To apply, one needs to estimate γ, σ Pt (τ), θ Pt (τ), κ Pt (τ). Forward-Looking Risk Premium Theory

Econometric Formulation Estimate γ using GMM The risk-neutral moment expressions can also be used to derive a volatility spread formula similar to that of Bakshi and Madan (2006): In order to implement, one needs to have estimates for (1) the risk-neutral return volatility and (2) the physical return volatility, skewness and kurtosis.

Econometric Formulation A model-free risk-neutral volatility can be derived via the typical mimicking approach using an option portfolio: where

Econometric Formulation For the physical return moments, we use forward-looking physical return moments deduced from an estimated NGARCH(1,1) model. Estimate by QMLE with a moving window of 5 years of daily S&P500 index returns. Obtain σ t+1 and 5 years of standardized residuals for the bootstrapping usage later.

Econometric Formulation The cumulative physical return volatility can be analytically computed using the formula: The physical skewness and kurtosis are computed by bootstrapping (the smooth stratified bootstrap method of Pitt 2002 and generating 100,000 sample paths)

Empirical Analysis Data source: OptionMetrics for option prices, S&P500 index values, risk-free yield curves. Data period: daily from January 1996 to October Set the target return horizon to 28 calendar days, i.e., τ = 28. The risk-free rate for 28 calendar days is obtained by interpolating the risk-free yield curve. Set the observation date to 28 calendar days before each monthly option expiration date. Use a moving window of 60 monthly data points.

Empirical Analysis Risk aversion None of the 106 rolling GMM over-identification tests of the model is rejected. (The instruments are: constant and risk-neutral return variance being lagged one, two and three periods.) Range of γ: 1.8 to 7.1 Smallest t(γ): 2.62

Empirical Analysis

Asset Pricing Implications The relationship between the change in the forward- looking risk premium and the excess holding period return –Price equals the future cash flows discounted at the cost of capital (risk free rate + risk premium). –Holding period return (change in price) should thus be affected by a change a change in the discount rate and/or in the expected cash flows. An empirical test: –predictions: β 1 0.

Asset Pricing Implications Proxy for EPS: (1) current EPS as expectation (2) analyst forecasted EPS in I/B/E/S

Asset Pricing Implications Liquidity and the forward-looking risk premium Amihud (2002) used data from 1964 to 1996 to find –A positive relationship between lagged illiquidity and excess return. –A negative relationship between unexpected illiquidity and contemporaneous excess return. –The presence of illiquidity risk premium in the stock market Is illiquidity risk premium also reflected in FLRP?

Asset Pricing Implications Replicate the Amihud (2002) study using our data from Jan 2001 to Dec 2008.

Asset Pricing Implications How about FLRP and illiquidity?

Conclusion Propose a new approach for estimating market risk premium on a forward-looking basis. Empirically, the estimates were all positive and were higher during the recession and/or crisis periods. The forward-looking risk premium estimate is consistent with the asset pricing implications such as the holding period return behavior and the illiquidity risk premium.