expected returns and risk in the stock market
TRANSCRIPT
Expected Returns and Risk in the Stock Market
Michael J. Brennan
Alex P. Taylor
Q- Group Conference
San Diego
October, 2019
SUMMARYโข Two new models for predicting expected return
โข Pricing kernel modelโข Constrains predictors with discipline of asset pricing modelโข Predicts market return as function of estimated covariances with portfolio returns โ riskโข Quarterly (annual) R2 of 8.3% (17.1%)โข Out of sample reduces 1 year naรฏve forecast error by 13%
โข Discount rate modelโข Exploits accounting identity of log-linear present value model (no economic content)โข Consistent with sentiment, illiquidity etcโข Identifies shocks to the discount rate and then sums them up to get current discount rateโข Predicts market return as a function of past portfolio returnsโข Quarterly (annual) R2 of 5.6% (9.9%)โข Does not improve on naรฏve forecast out of sample
โข Provide independent evidence on predictability
Models for predicting expected returns
โข Yield based modelsโข Dividend yield, market to book ratios, T-Bill rate
โข Information based modelsโข Lettau, Ludwigson (2001)โข Rapach et al. (2016)
โข Sentiment based modelsโข Baker and Wurgler (2006)โข Huang et al. (2015)
โข Risk- based modelsโข Merton (1980), Ghysels et al.(2005), Scruggs(1998), Guo et al (2009).
Our primary model is a risk-based model โ pricing kernel modelsupplementary analysis with a (changing) yield based model โ discount rate model
Principal Findings
โข Pricing kernel modelโข In sample R2 1954-2016, 15-18% for 1 year returnsโข Out of sample R2 1965-2016, 9-16%
โข Rapach et al (2016) around 13% for short interest predictor
โข Discount rate modelโข In sample R2 1954-2016 18% for 1 year returnsโข Out of sample R2 essentially zeroโข Allows us to separate cash flow news from discount rate news
โข Interpretation
โข The time varying risk that is driving time varying returns istimeโvarying risk of cash flow news
Two structural models of expected returns
Both assume that expected returns follow an AR1 process:
โข Model 1: pricing kernel model
โข Constrains predictors with discipline of asset pricing model
โข Not consistent with sentiment/liquidity explanations
โข Model 2: discount rate model: a factor model of returns based on cash flow and discount rate news
โข Purely statistical model relying on accounting identity of Campbell Shiller (1988)
โข Consistent with time variation due to sentiment, liquidity etc.
The models are intimately related โ but not equivalent
The model of time-variation in expected excess returns
๐ ๐,๐ก+1 = ๐๐ก + ๐๐ก
๐๐ก follows an AR1 (mean-reverting) process with innovation ๐ง๐ก
๐๐ก = าง๐ + ๐ ๐๐กโ1 โ าง๐ + ๐ง๐ก = าง๐ +
๐ =0
โ
๐๐ ๐ง๐กโ๐
Pricing kernel model
๐๐ก driven by changing risk or covariance with pricing kernel which we estimated directly
Discount rate model
๐๐ก estimated by aggregating past changes in discount rate, ๐ง๐กโ๐
Pricing kernel โ general model of asset pricing
โข Marginal utility of representative investor, mt+1
โข Risk premium on any security (including the market portfolio) is given by (negative of) its covariance with the pricing kernel, mt+1
๐ธ๐ก ๐ ๐,๐ก+1 โ ๐ ๐น๐ก โ ๐๐๐ก = โ๐๐๐ฃ๐ก(๐ ๐,๐ก+1, ๐๐ก+1)
โข Example 1: CAPM: ๐๐ก+1 = ๐ฟ0 โ ๐ฟ1๐ ๐,๐ก+1
๐๐๐ก = ๐ฟ1๐๐๐ฃ๐ก ๐ ๐,๐ก+1, ๐ ๐,๐ก+1 , proportional to ๐ฝ๐
For Market ๐๐๐ก = ๐ฟ1๐๐๐ฃ๐ก ๐ ๐,๐ก+1, ๐ ๐,๐ก+1 = ๐ฟ1๐๐,๐ก2 - Merton (1980)
Pricing kernel โ general model of asset pricing - continued
โข Example 2: Fama-French 3 factor model: ๐๐ก+1 = ๐ฟ0 โ ๐ฟ1๐ ๐,๐ก+1 โ ๐ฟ2๐ป๐๐ฟ๐ก+1 โ ๐ฟ3๐๐๐ต๐ก+1
Market risk premium: ๐๐๐ก = ๐ฟ1๐๐๐ฃ๐ก ๐ ๐,๐ก+1, ๐ ๐,๐ก+1 + ๐ฟ2๐๐๐ฃ๐ก ๐ ๐,๐ก+1, ๐ป๐๐ฟ๐,๐ก+1 + ๐ฟ3๐๐๐ฃ๐ก ๐ ๐,๐ก+1, ๐๐๐ต๐ก+1
โข We can generalize this to any pricing kernel written as a linear function of portfolio returns:
๐๐ก+1 = ๐ฟ0 โ ๐ฟ1๐ 1,๐ก+1- ๐ฟ2๐ 2,๐ก+1- ๐ฟ3๐ 3,๐ก+1โฆ.. - ๐ฟ๐๐ ๐,๐ก+1
๐๐๐ก =
๐=1
๐
๐ฟ๐๐๐๐ฃ๐ก ๐ ๐,๐ก+1, ๐ ๐,๐ก+1
The market risk premium is a weighted sum of the conditional covariances of market return with the returns on the portfolios that span the pricing kernel. How do we estimate the conditional covariances ??
Estimating conditional covariances
We assume that covariances can be written as geometrically weighted sum of past products of excess returns:
๐๐๐ฃ๐ก(๐ ๐,๐ก+1, ๐ ๐,๐ก+1) = ฯ๐ =0โ ๐ฝ๐ (๐ ๐,๐กโ๐ ๐ ๐,๐กโ๐ )
Then
๐๐,๐ก = ๐ฟ1
๐ =0
โ
๐ฝ๐ (๐ ๐,๐กโ๐ ๐ 1,๐กโ๐ ) + ๐ฟ2
๐ =0
โ
๐ฝ๐ (๐ ๐,๐กโ๐ ๐ 2,๐กโ๐ ) + โฆ + ๐ฟ๐
๐ =0
โ
๐ฝ๐ (๐ ๐,๐กโ๐ ๐ ๐,๐กโ๐ )
๐๐,๐ก = ๐ฟ1 ๐ฅ1,๐ก ๐ฝ + ๐ฟ2 ๐ฅ2,๐ก ๐ฝ + โฆ + ๐ฟ๐ ๐ฅ๐,๐ก ๐ฝ
where ๐ฅ๐,๐ก ๐ฝ = ๐๐๐ฃ๐ก(๐ ๐,๐ก+1, ๐ ๐,๐ก+1)
Estimation equation:
๐ ๐,๐ก+1 โ ๐ ๐น,๐ก = ๐0 + ๐1๐ฅ1,๐ก ๐ฝ + ๐2๐ฅ2,๐ก ๐ฝ + โฆ + ๐๐๐ฅ๐,๐ก ๐ฝ
Estimation procedure:
1. Choose pricing kernel e.g. CAPM, FF3
2. Choose value for ฮฒ
3. Calculate covariance estimates ๐ฅ๐๐ก ๐ฝ = ฯ๐ =0โ ๐ฝ๐ (๐ ๐,๐กโ๐ ๐ ๐,๐กโ๐ )
4. Run OLS regression of market excess return on covariance estimates ๐ฅ๐๐ก ๐ฝ
5. Calculate R2
6. Iterate over ฮฒ to find maximum R2
7. Compute significance levels, s.eโs, bias adjustment using bootstrap
Estimated covariances with market return
The return interval
โข Increasing prediction horizon tends to increase predictive R2
โข But more precise estimates of covariances possible with short return intervals
โข However theory implies relevant return interval for covariances of returns should be same as prediction horizonโข NB covariances of arithmetic returns do not scale with return intervalโข Returns not iid and lagged cross-correlations of returns
โข As compromise, compute covariances using lagged 1โquarter returns to predict 1-quarter return
โข Persistence in ฮผ means this covariance will also predict 1 year returns (estimates of 1-year covariances unreliable)
โข Compare with results using 1-month lagged returns to calculate covariances
Data
FF aggregate market factor
๐ ๐น๐ก - 1-month T-bill rate compounded
๐ ๐๐ก difference between market factor and ๐ ๐น๐ก
Pricing kernel portfoliosโข Market
โข FF3F
โข Zero โ market + zero dividend portfolio
โข Growth โ market + average return on (sl, bl)
โข Sample 1946 โ 2016โข Predictions: 1954.1 โ 2016.4
Predicting 1- quarter market excess returns
KernelPortfolios
a0 RM SMB HML ฮฒ R2 R2c p-value
M 0.00(0.01)
1.34(0.44)
0.45(0.26)
0.04 0.04 0.01
FF3F -0.00(0.01)
1.65(0.51)
-1.18(0.97)
1.50(0.82)
0.61(0.23)
0.07 0.06 0.00
Predicting 1- year market excess returns
M 0.04(0.09)
3.00(1.36)
0.35(0.30)
0.04 0.03 0.21
FF3F 0.03(0.12)
6.11(2.15)
-10.02(3.82)
5.52(3.02)
0.52(0.22)
0.19 0.17 0.00
s.e. in parens.
17% R2 for 1 year predictions compares with the following R2 for prior predictors
Divyld Earnings yield
Book/Mkt
Value Spread
Glamor Stock Variance
T-Billrate
Long term yld
0.04 0.01 0.01 0.03 0.05 0.01 0.01 0.00
TermSpread
Inflation DefaultSpread
Kelly-Pruitt
Guo &Savickas
cay hjtzSentiment
Shortinterest
0.04 0.02 0.02 0.05 0.03 0.07 0.08 0.08
Constant 0.00(0.00)
0.17(1.59)
0.28(1.31)
Div yield 0.04(1.82)
0.03(1.47)
Glamor -0.03(1.9)
-0.03(1.90)
Kelly-Pruitt 0.00(0.02)
0.01(0.24)
cay 0.06(2.76)
0.04(2.37)
FF3F prediction 1.00(5.94)
0.82(4.52)
R2 0.19 0.17 0.29
Throwing all the major predictors into the predictive regression:
Only cay is significant but much less strong than FF3F (t-stats in parentheses)
Inspection of the FF portfolios reveals we can get strong results with just two portfolios in the pricing kernel: the market portfolio and either
โข Growth = (sl + bl)/2
Or
โข Zero = the zero dividend yield portfolio
Predicting 1- year market excess returns
KernelPortfolios
a0 RM Growth Zero ฮฒ R2 R2c p-value
Growth 0.03(0.01)
25.2(5.99)
-18.6(4.79)
0.51(0.29)
0.16 0.15 0.01
Zero 0.03(0.07)
16.8(4.25)
-9.90(2.73)
0.58(0.26)
0.19 0.18 0.00
s.e. in parens.
Kernel Portfolios Sample a0 RM SMB HML ฮฒ R2 p-value
FF3F 1955-85 0.01 8.72 -12.82 2.99 0.47 0.21 0.03(0.32) (3.18) (4.97) (4.18)
1986-2010 0.07 3.30 -6.78 7.3 0.57 0.17 0.08(0.75) (2.43) (5.22) (4.55)
a0 RM GrowthGrowth 1955-85 0.01 30.51 -21.00 0.42 0.20 0.03
(0.26) (8.12) (6.20)
1986-2010 0.07 24.50 -19.95 0.54 0.14 0.15(0.64) (8.22) (7.11)
a0 RM ZeroZero 1955-85 0.01 20.78 -11.04 0.57 0.22 0.01
(0.16) (6.22) (3.79)
1986-2010 0.07 18.18 -12.6 0.53 0.20 0.04(0.61) (5.68) (4.16)
The reduced form models, Growth and Zero, have much more stable coefficients across subperiods
Subperiod Analysis
s.e. in parens.
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1 year expected returns from pricing kernel model
PKM(FF3) PKM(G) PKM(Zero)
Correlations: FF3F, Growth 0.96; FF3F, Zero 0.87
Expected Return Series
โข Mean 1 year expected return 7.4%
โข Peaks
โข 1955.2 27% (21%) Formosa crisis
โข 1962.4 26% (20%) Cuban missile crisis
โข 1974.3 32% (40%) First oil crisis
โข 1987.4 22% (32%) Lehman Bros.
โข Negative
โข Around the millennium (dot.com)โข 1973-4 (Cf. Boudoukh et al. 1993)
From FF3F (Zero) model
Pseudo R2 for 1-year out of sample forecasts - 1965-2016
FF3F Growth Zero
9.4% 13.8% 16.1%
โข And the reduced form models perform better in out of sample forecasts
โข Previous resultsโข Kelly-Pruitt (2013): 3.5-13.1% for 1980-2010โข Rapach et al. (2016) short interest predictor: 13.2% for 1990-2014.
Discount Rate Model
โข Provides further check on Pricing Kernel Model
โข Allows us to distinguish cash flow news from discount rate news
โข We shall show that it is changing covariance with cash flow news that is driving changing expected returns
Discount Rate Model โ intuition
For a bond, if we know:its initial yield, and the time series of returns (changes in yield)
Then we know its current yield and expected return
100 100
Tt
The problem is more difficult for stocks โbecause returns affected by news about cash flows as well as about discount rates.
We shall use returns on several portfolios to soak up cash flow news and isolate discount rate news
Discount Rate Modelโข Expected log (excess) return follows AR1
๐ ๐,๐ก+1 = ๐๐ก + ๐ ๐ก+1
๐๐ก = าง๐ + ๐ ๐ ๐กโ1 โ าง๐ + ๐๐
= าง๐ +
๐ =0
โ
๐๐ ๐ง๐กโ๐
โข There exists set of well diversified portfolios that span innovations in cash flow (yjt )and discount rate (๐๐) factors:
๐ ๐,๐ก = ๐ฝ ๐0 +
๐=1
๐
๐ฝ๐๐๐๐๐ก + ๐พ๐๐๐
Choose portfolio, ฮดd, so that ฯ๐=1๐ ๐ฟ๐
๐ ฮฒ๐๐ = 0, ๐๐๐ ๐๐๐ ๐, ฯ๐=1๐ ฮด๐
๐ฮณ๐ = 1.
Then ๐๐ก = ฯ ๐๐กโ1 + ๐ฟ0๐ + ฯ๐=1
๐ ๐ฟ๐๐ ๐ ๐๐ก
Successive substitution for ๐๐กโ1 yieldsโฆ. ๐ ๐,๐ก+1 = ๐0 + ฯ๐=1๐ ๐ฟ๐
๐ ฯ๐ =0โ ๐ฝ๐ ๐ ๐,๐กโ๐ + ํ๐ก+1
Discount rate news
๐ฅ๐๐ก๐ ๐ฝ
Empirical discount rate model:
๐ ๐,๐ก+1 = ๐0 +๐=1
๐
๐ฟ๐๐๐ฅ๐๐ก
๐ ๐ฝ + ํ๐ก+1
where ๐ฅ๐๐ก๐ ๐ฝ = ฯ๐ =0
โ ๐ฝ๐ ๐ ๐,๐กโ๐ .
Estimation as for pricing kernel model
RHS `spanning portfoliosโ:
Market and pricing kernel portfolios for FF3F, Growth and Zero
a0 RM m_FF3F m_Growth m_Zero ฮฒ R2 p-value
0.08 -0.42 1.88 0.98 0.10 0.05
(0.03) (0.16) (1.89)
0.1 -0.51 2.66 0.99 0.14 0.01
(0.03) (0.16) (2.00)
0.07 -0.50 2.11 0.82 0.09 0.07
(0.03) (0.18) (2.04)
Discount Rate Model 1954-2016
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1 year Expected Returns from Discount Rate Model
DRM(FF3) DRM(Growth) DRM(Zero)
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1-year Expected Returns from Pricing Kernel and Discount Rate Models
PKM(FF3) DRM(FF3)
Correlations of PKM and DRM estimates around 0.4 โ 0.5 โ but both have separate information about expected returnsโฆ..
Regression of market returns on PKM and DRM estimates of expected return
1 2 3
constant -2.85
(1.11)
PKM(FF3) 1.00 0.78
(5.94) (3.65)
DRM (FF3) 1.00 0.60
(4.32) (2.41)
R2 0.19 0.14 0.23
Both estimatesadd information
The BIG issue
The market moves because of
โข new information about future cash flows โ cash flow news
โข new information about future expected returns โ discount rate news
Our discount rate model gives us a direct estimate of cash flow news - zt
This enables us to ask:
Is it time-varying risk of cash flow news or time varying risk of discount rate news that is driving time-varying expected returns?
A Cash Flow News Pricing Kernel Model
โข Define cash flow news as the component of ๐ ๐ that is orthogonal to discount rate news, zt
โข We have three estimates of zt corresponding to our 3 different discount rate models, FF3, Growth, Zero.
โข So we have three different estimates of cash flow news obtained by regressing ๐ ๐on zt (FF3), zt (Growth), zt (Zero).
โข Call them CFN(FF3), CFN(Growth) and CFN(Zero)
Then consider the following cash flow news version of the pricing kernel model:
๐ ๐,๐ก+1 โ ๐ ๐น,๐ก = ๐0 + ๐1๐ฅ๐ก ๐ฝ
where ๐ฅ๐ก ๐ฝ = ฯ๐ =0โ ๐ฝ๐ (๐ ๐,๐กโ๐ ๐ถ๐น๐๐กโ๐ )
We shall estimate it for the three different estimates of cash flow news
a0 CFN(FF3F) CFN(Growth) CFN(Zero) ฮฒ R2 p-value
0.04 22.3 0.60 0.21 0.00
(0.05) (6.44)
0.04 19.1 0.60 0.18 0.00
(0.06) (13.23)
0.03 23.0 0.65 0.21 0.00
(0.05) (6.78)
Cash Flow News Pricing Kernel Model
s.e. in parens.
, ๐ฅ๐ก ๐ฝ = ฯ๐ =0โ ๐ฝ๐ (๐ ๐,๐กโ๐ ๐ถ๐น๐๐กโ๐ )
RM SMB HML ฮฒ R2 p-value
M 0.04(0.09)
3.00(1.36)
0.35(0.30)
0.03 0.21
FF3F 0.03(0.12)
6.11(2.15)
-10.02(3.82)
5.52(3.02)
0.52(0.22)
0.17 0.00
๐ ๐,๐ก+1 โ ๐ ๐น,๐ก = ๐0 + ๐1๐ฅ๐ก ๐ฝ
For comparison
Summary
โข Pricing Kernel Model: ๐ฅ๐๐ก ๐ฝ = ฯ๐ =0โ ๐ฝ๐ (๐ ๐,๐กโ๐ ๐ ๐,๐กโ๐ )
โข FF3F, Growth and Zero R2 = 17-18%
โข OOS R2 = 9 -16%
โข Discount Rate Model: ๐ฅ๐๐ก๐ ๐ฝ = ฯ๐ =0
โ ๐ฝ๐ ๐ ๐,๐กโ๐
โข FF3F, Growth and Zero R2 = 9-14 %
โข Not significant OOS
โข Identifies discount rate news
โข Cash Flow News Pricing Kernel Model: ๐ฅ๐ก ๐ฝ = ฯ๐ =0โ ๐ฝ๐ (๐ ๐,๐กโ๐ ๐ถ๐น๐๐กโ๐ )
โข FF3F, Growth and Zero R2 = 18-21 %
โข Time Varying Risk of Cash Flow News drives time-varying expected returns.