Forward-Looking Market Risk Premium Weiqi Zhang National University of Singapore Dec 2010
Feb 24, 2016
Forward-Looking Market Risk Premium
Weiqi ZhangNational University of Singapore
Dec 2010
BackgroundEstimating 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: rt(τ)• Dividend yield of the market portfolio: δt(τ)• Market portfolio’s cumulative return:
Rt(τ)=ln(St+τ /St)• Mean, standard deviation, skewness and kurtosis:
– under the physical measure P: μPt(τ), σPt(τ), θPt(τ), κPt(τ)– under the risk neutral measure Q: μQt(τ), σQt(τ), θQt(τ),
κQt(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
2
3 4
1( ) ( ) ( ) ( )2
1 1( ) ( ) ( )[ ( ) 3]6 24
t t Qt Qt
Qt Qt Qt Qt
r
exp( ( ) ( ))t QtR
• Assume the form of stochastic discount factor:
• Rely on an approximate expression moment generating function of Rt
*(τ) =Rt(τ) - μPt(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
exp( ( ))tR
*2 3 4
( ) 2 3 4 4 4( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( )2 6 24
tRPt t Pt Pt Pt Pt Pt PtC E e
* **
*
( ) ( ) ( ) ( )( )
( ) ( )
( ) ( ) ( )( )( ) ( )( )
t t t Ptt
t t Pt
R R RP PRQ t t t
t RP RPt tt
E e e E e e CE eE e CE e e
• 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, σPt(τ), θPt(τ), κPt(τ).
22 3
3 24
1 3 3 1( ) ( ) ( ) ( ) ( ) ( )2 6
4 6 4 1 ( ) ( ) 324
Pt t t Pt Pt Pt
Pt Pt
r
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.
2 2 2
22
( ) ( )( ) ( ) ( ) ( ) 3 0
( ) 2Qt Pt
Pt Pt Pt Pt tPt
E I
Econometric Formulation
• A model-free risk-neutral volatility can be derived via the typical mimicking approach using an option portfolio:
where
22 2( ) [ ( )] [ ( )]Q QQt t t t tE R E R
( ) ( )2 20
( ) ( ; , ) ( ; , )[ ( )] ln t
t t
t
Kr rQ t t t t tt t K
t t
K F K C K S P K SE R e dK e dK
S K K K
2
2
( ) ( )2 20
( )[ ( )] ln 2 ln
2 1 ln 2 1 ln( ; , ) ( ; , )t
t t
t
Q t t t tt t
t t t
Kt tr rt tK
K F K KE RS K S
K KS S
e C K S dK e P K S dKK K
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 σt+1 and 5 years of standardized residuals for the bootstrapping usage later.
11 1
2 2 21 0 1 2
2
ln 0,1,...
( )
. . . ( ) 0 & ( ) 1
tt t
t
t t t t
P Pt t t
S for tS
i i d E E
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)1
2 2 0 01 2
21 2
( 1) (1 )1( )1 1 (1 )
(1 )
Pt t
where
Empirical Analysis
• Data source: OptionMetrics for option prices, S&P500 index values, risk-free yield curves.
• Data period: daily from January 1996 to October 2009.
• 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: β1 < 0 and β2 > 0.
1 2( ) * emt ft t t tR R FLRP EPS
Asset Pricing Implications
Proxy for EPS: (1) current EPS as expectation (2) analyst forecasted EPS in I/B/E/S
Asset Pricing ImplicationsLiquidity 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.
0 1 1 2
3
ln( ) ln( )Umt ft t t
t t
R R g g MILLIQ g MILLIQ
g JANDUM
Asset Pricing Implications• How about FLRP and illiquidity?
1 0 1 1 2( ) ln( )t t t tFLRP MILLIQ JANDUM
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.