Volatility, Valuation Ratios, and Bubbles: An Empirical Measure of Market Sentiment Can Gao Ian Martin October, 2019 Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 1 / 22
Two views of the equity premium Based on valuation ratios (Campbell and Thompson, RFS , 2008) and on index option prices (Martin, QJE , 2017) 20 SVIX 15 Black Monday 10 5 CT 0 1980 1990 2000 2010 Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 2 / 22
Outline Very roughly, think of D / P as revealing E R − E G , and interest rates and option prices as revealing E R ; then the gap between the two reveals E G Specifically, today: Relate dividend yields to expected returns and dividend growth using a twist on the 1 Campbell–Shiller methodology Introduce a bound on expected returns based on interest rates and option prices 2 Derive a bound on expected dividend growth by playing off (1) against (2) 3 Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 3 / 22
Campbell–Shiller decomposition (1) Notation: log dividend yield dp t = log ( D t / P t ) ; log return r t + 1 ; log dividend growth g t + 1 Campbell and Shiller (1988) famously showed that, up to a linearization, � ∞ k ρ i E t [ r t + 1 + i − g t + 1 + i ] dp t = 1 − ρ + where ρ ≈ 0 . 97 i = 0 These are expected log returns, not expected returns Low expected log returns may be consistent with high expected returns if returns are volatile, right-skewed, or fat-tailed All three plausibly true in late 1990s, so the distinction between log returns and simple returns matters Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 4 / 22
Campbell–Shiller decomposition (2) ρ i � � 2 � ∞ � ∞ k ρ i ( r t + 1 + i − g t + 1 + i ) − ρ ( 1 − ρ ) dp t = 1 − ρ + dp t + 1 + i − dp 2 i = 0 i = 0 � �� � second order term ≈ − 0 . 145 in late ’90s In the late ’90s dp t was 2.2 sd below its mean (using CRSP data 1947–2017) Ignoring the second order term is equivalent to understating E t r t + 1 + i − g t + 1 + i by 14.5 pp for one year, 3.1 pp for five years, or 1.0 pp for 20 years ◮ In long sample, 1871–2015, numbers are even bigger: 25.3 pp for one year, 5.5 pp for five years, 1.8 pp for 20 years, or 1.0 pp for ever Thus the CS decomposition may “cry bubble” too soon Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 5 / 22
An alternative approach (1) Campbell and Shiller loglinearize � � 1 + e − dp t + 1 r t + 1 − g t + 1 = dp t + log We start, instead, from � 1 − e − y t � � 1 − e − y t + 1 � r t + 1 − g t + 1 = y t + log − log where � � 1 + D t y t = log P t y t , unlike dp t , is in natural units: if D t / P t = 2 % then y t = 1 . 98 % whereas dp t = − 3 . 91 Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 6 / 22
An alternative approach (2) Result We have the loglinearization � ∞ ρ i ( r t + 1 + i − g t + 1 + i ) y t = ( 1 − ρ ) i = 0 where ρ = e − y ≈ 0 . 97 . On average, this relationship holds exactly —no linearization needed: y = r − g Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 7 / 22
An alternative approach (3) We have already seen that the Campbell–Shiller approximation may lead one to conclude too quickly that the market is bubbly, as � ∞ k ρ i E t ( r t + 1 + i − g t + 1 + i ) dp t < − 1 − ρ + i = 0 Our variant is a conservative diagnostic for bubbles. If y t is far from its mean then � ∞ ρ i E t ( r t + 1 + i − g t + 1 + i ) y t ≥ ( 1 − ρ ) i = 0 � ( y t + i − y ) 2 � ≤ ( y t − y ) 2 for all i ≥ 0 ◮ Far from its mean : E t ◮ In AR(1) case, “ far ” means “one standard deviation” Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 8 / 22
Information in valuation ratios (1) If y t follows an AR(1) with autocorrelation φ y , E t ( r t + 1 − g t + 1 ) = constant + 1 − ρφ y 1 − ρ y t In the unit root case φ y = 1, we have y t = E t ( r t + 1 − g t + 1 ) So we use y t to forecast r t + 1 − g t + 1 We estimate the regression freely, but results are almost identical if we estimate ρ and φ y from time series, then use the formula above AR(1) is not critical: key is that we have a forecast of E t y t + 1 . Will show AR( k ) later Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 9 / 22
Information in valuation ratios (2) R 2 RHS t LHS t + 1 � s . e . � s . e . a 0 a 1 r t + 1 − g t + 1 − 0 . 067 [0.049] 3.415 [1.317] 7.73% y t r t + 1 − 0 . 018 [0.050] 3.713 [1.215] 10.51% − g t + 1 − 0 . 049 [0.028] − 0 . 298 [0.812] 0.32% r t + 1 − g t + 1 0.417 [0.146] 0.107 [0.042] 7.58% 0.500 [0.138] 0.114 [0.041] 9.92% dp t r t + 1 − g t + 1 − 0 . 083 [0.085] − 0 . 007 [0.024] 0.19% Table: S&P 500, annual data, 1947–2017, dividends reinvested monthly at CRSP 30-day T-bill rate. Hansen–Hodrick standard errors. Relative importance of r and g is sample specific: g more important in long sample. But coefficient estimates for r − g are stable Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 10 / 22
Information in options (1) We start from an identity 1 E t r t + 1 = E ∗ t ( R t + 1 r t + 1 ) − cov t ( M t + 1 R t + 1 , r t + 1 ) R f , t + 1 1 M t + 1 is an SDF. Risk-neutral E ∗ t satisfies R f , t + 1 E ∗ t ( X t + 1 ) = E t ( M t + 1 X t + 1 ) We assume that cov t ( M t + 1 R t + 1 , r t + 1 ) ≤ 0 ◮ Similar to the negative correlation condition of Martin (2017) ◮ Loosely, requires that investors are sufficiently risk-averse wrt R t + 1 ◮ Holds in Campbell–Cochrane (1999), Bansal–Yaron (2004), Barro (2006), Wachter (2013), Bansal et al. (2014), Campbell et al. (2016), . . . We then have 1 E ∗ t ( R t + 1 r t + 1 ) E t r t + 1 ≥ R f , t + 1 Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 11 / 22
Information in options (2) 1 E t r t + 1 ≥ E ∗ t ( R t + 1 r t + 1 ) R f , t + 1 Doesn’t require that the market is complete Doesn’t require any distributional assumptions (eg lognormality) Allows for the presence of constrained and/or irrational investors Holds with equality for a log investor who chooses to hold the market This investor’s perspective works well empirically for forecasting ◮ the market as a whole (Martin, QJE , 2017) ◮ individual stocks (Martin and Wagner, JF , 2019) ◮ currencies (Kremens and Martin, AER , 2019) Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 12 / 22
Information in options (3) option prices put t ( K ) call t ( K ) K F t Using the result of Breeden and Litzenberger (1978), we show �� F t � � ∞ 1 t ( R t + 1 r t + 1 ) = r f , t + 1 + 1 put t ( K ) call t ( K ) E ∗ dK + dK R f , t + 1 P t K K 0 F t � �� � LVIX t This gives the lower bound E t r t + 1 − r f , t + 1 ≥ LVIX t Bootstrapped p -value for the mean of r t + 1 − r f , t + 1 − LVIX t being negative is 0.097 Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 13 / 22
A sentiment index Putting the pieces together, � � E t g t + 1 = r t + 1 − r f , t + 1 + r f , t + 1 − E t ( r t + 1 − g t + 1 ) E t LVIX t + r f , t + 1 − E t ( r t + 1 − g t + 1 ) ≥ � �� � B t We replace E t ( r t + 1 − g t + 1 ) by the forecast based on y t : B t = LVIX t + r f , t + 1 − ( � a 0 + � a 1 y t ) with � a 0 and � a 1 calculated on a rolling basis so B t is observed at t The bound E t g t + 1 ≥ B t relies on two key assumptions: ◮ the modified NCC ◮ a stable statistical relationship between valuation ratios and r − g Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 14 / 22
The sentiment index 0.15 0.10 0.05 0.00 2000 2005 2010 2015 B t B dp , t Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 15 / 22
The three components of the sentiment index, B t 0.08 0.06 0.04 0.02 0.00 - 0.02 - 0.04 2000 2005 2010 2015 LVIX t E t [ g t + 1 - r t + 1 ] r f,t + 1 Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 16 / 22
Allowing y t to follow an AR( k ) 0.12 0.10 0.08 0.06 0.04 0.02 0.00 - 0.02 2000 2005 2010 2015 AR ( 1 ) AR ( 2 ) AR ( 3 ) Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 17 / 22
Sentiment index vs. detrended volume (1) 2 1 0 - 1 2000 2005 2010 2015 B t v t Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 18 / 22
Sentiment index vs. detrended volume (2) corr ( B t + k , v t ) 0.9 0.8 0.7 0.6 0.5 0.4 30 k - 30 - 20 - 10 0 10 20 Figure: Correlation between B t + k and detrended volume at time t . Gao & Martin (Imperial College & LSE) Volatility, Valuation Ratios, and Bubbles October, 2019 19 / 22
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