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.