Prices of Uncertainty and Long-Run Risk Summer Camp in Applied - PowerPoint PPT Presentation

Prices of Uncertainty and Long-Run Risk Summer Camp in Applied Econometrics Shanghai University of International Business and Economics Li Nan https://www.nanlifinance.org/teaching.html Antai College of Economics and Management, Shanghai Jiao Tong University 2018/08/12 Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 1 / 47

Outline Empirical Asset Pricing Equity Premium Puzzle Consumption Based Asset Pricing Model with Recursive Utility Measuring Long-Run Risk and Price of Long-Run Risk Long-Run Risk vs Uncertainty Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 2 / 47

Empirical Asset Pricing Main question: What risk are priced in the market? Is the market efficient? Main test: Linear Factor Model (CAPM) E t [ R i , t + 1 − R f , t ] = β i , t · λ t (conditional) E [ R i , t + 1 − R f , t ] = β i · λ (unconditional) where β i : risk exposure λ : price of risk Regression R i , t + 1 − R f , t = α i + β i · λ t + ε i , t + 1 H 0 : α i = 0 Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 3 / 47

Empirical Asset Pricing The underlying assuptions: The Market is efficient The model captures all the risk factors that are priced in the market How to measure risk? Note: "Joint-Test Problem" in Empirical Tests of the EMH: Market Efficiency per se is not testable The question whether price reflects a given piece of information always depends on the model of asset pricing that the researcher is using. It is always a joint test of market efficiency and the used pricing model. Despite the joint-test problem, tests of market efficiency, i.e. search for anomalies or "arbitrage" opportunities, improves our understanding of the behavior of returns across time and securities. It helps to improve existing asset pricing models and understanding of financial markets. Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 4 / 47

Equity Premium Puzzle Consumption-based asset pricing model with CRRA Utility: ∞ β j u ( c t + j ) ∑ E t j = 0 Euler Equation: � � u � ( c t + 1 ) � � R i E t β = 1 t + 1 u � ( c t ) which implies that � � � � t + 1 , β u � ( c t + 1 ) − 1 R ei R e E t = cov t t + 1 R f u � ( c t ) t � � u � ( c t + 1 ) �� 1 = β E t R f u � ( c t ) t Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 5 / 47

Equity Premium Puzzle With CRRA (power) utility u � ( c ) = c − γ We have � � � � t + 1 , c t + 1 R ei R e ∝ E t γ cov t t + 1 c t γσ t ( R ei t + 1 ) σ t ( ∆ c t + 1 ) ρ t ( ∆ c t + 1 , R ei = t + 1 ) � � � � � � R ei E t � � t + 1 � < γσ t ( ∆ c t + 1 ) = ⇒ � � � σ t ( R ei t + 1 ) In postwar U.S. data, the mean return of stocks over bonds is about 5 percent, with a standard deviation of about 20 percent, so the Sharpe ratio is about 0.25. Aggregate non-durable and services consumption volatility is much smaller, about 1 percent per year. We need a risk aversion of at least 25! Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 6 / 47

Equity Premium Puzzle Real Value of One Dollar Invested in 1926 (in logs) 7 Stock Index 90-Day Tbill 6 5 4 3 2 1 0 -1 1930 1940 1950 1960 1970 1980 1990 2000 2010 Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 7 / 47

Equity Premium Puzzle Real Return of Stock Index and 90-day Tbill 0.6 Stock Index Tbill 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 1930 1940 1950 1960 1970 1980 1990 2000 2010 Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 8 / 47

Equity Premium Puzzle Table: Summary Statistics of Real Return ∆ P R e (%) Rf90 Rf30 ( CPI ) Panel A: 1926-2017 Annual 8.81 0.78 0.42 2.97 std dev 20.31 4.12 3.93 4.04 Quarterly 2.66 0.19 0.11 0.72 std dev 17.45 1.32 1.28 1.31 Monthly 0.75 0.06 0.04 0.24 std dev 7.21 0.54 0.53 0.53 Panel B: 1947-2017 Annualized ∆ P R e ∆ c t Rf90 Rf30 ( PCE ) Quarterly 6.34 1.25 0.83 3.14 3.21 std dev 21.67 1.58 1.45 1.47 1.09 corr with ∆ c t 0.19 0.26 0.30 -0.28 1 Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 9 / 47

Equity Premium Puzzle Initial Reaction T-bill rate is not a good approximation for the risk-free interest rate? 1 No, as high sample Sharpe ratios are pervasive in finance and not limited to the difference between stocks and bonds Risk aversion is indeed high? 2 No, the implied risk free rate is more than 20%. Assuming log-normal distribution t = log β + γ E t ( ∆ c t + 1 ) − 1 r f 2 γ 2 σ 2 t ( ∆ c t + 1 ) — > Risk Free Rate Puzzle More information is worse, ρ = 0 . 2 , implies risk aversion of more than 3 100! Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 10 / 47

Equity Premium Puzzle Equity Premium and Risk Free Rate Puzzle Quantitative not qualitative puzzle, 1 Consumption is proportional to wealth in the derivation of the CAPM, 2 so the CAPM predicts that consumption should inherit the large 20 percent or so volatility of the stock market Implication optimal portfolio choice: 3 E ( R e ) w = 1 σ 2 ( R e ) γ 100% equity investment for γ around 3 . However, conditonal on 20% consumption std.dev.!! Consumption is much smoother that wealth, but consumption-wealth 4 ratio is stationary in the long-run = ⇒ Lettau and Ludvigon’s cay as a forecasting variable. Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 11 / 47

Equity Premium Puzzle Empirical Tests: Mehra and Prescott (1985): Calibration 1 Hansen and Singleton (1983): GMM of conditional and unconditional 2 Euler equation � � u � ( c t + 1 ) � � R i E t β = 0 t + 1 u � ( c t ) �� � u � ( c t + 1 ) � � � R i E β t + 1 − 1 z t ) = 0 u � ( c t ) where z t consists of lags of consumption and returns, which do not forecast either consumption growth or returns very well. Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 12 / 47

Equity Premium Puzzle Figure: Barillas et al.(2009) Hansen—Jagannathan volatility bound for quarterly returns on the value-weighted NYSE and Treasury Bill, 1948—2006. Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 13 / 47

Equity Premium Puzzle Risk Aversion and Intertemporal Substitution Risk aversion: measures the risk attitude 1 Intertemporal elasticity of substitution: how much consumption growth 2 changes when interest rates go up 1 percent, measures attitude towards behavior of a single asset over time and in particular to line up variation in expected consumption growth with variation in risk-free interest rates Inverse to each other in CRRA utility 3 Quite high risk aversion is required to digest the equity premium is 4 robust in consumption-based model estimation Much more debate on IES 5 Hansen and Singleton found numbers near one Hall (1988) argued the estimate should be closer to zero Campbell (2003) argues for small IES, as we observe small movements in expected consumption growth against large movements in real interest rates Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 14 / 47

Equity Premium Puzzle Questions: What utility function should one use? 1 How should one treat time aggregation and consumption data? 2 How about multiple goods? 3 What asset returns and instruments are informative? 4 Asset pricing empirical work has moved from industry or beta 5 portfolios, the use of lagged returns, and consumption growth as instruments to the use of size, book-to-market, momentum portfolios, and the dividend-price ratio, term spreads, and other more powerful instruments. How does the consumption-based model fare against this higher bar? The data may be poor enough that practitioners will still choose 6 “reduced-form” financial models, but economic understanding of the stock market must be based on the idea that people fear stocks, and hence do not buy more despite attractive returns, because people fear that stocks will fall in “bad times.” At some point “bad times” must be mirrored in a decision to cut back on consumption. Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 15 / 47

Equity Premium Puzzle Later Responses: Non-separable utility across goods: leisure (Eichenbaum, Hansen, and Singleton (1988)), durable goods (Yogo(2004), Pakos (2004)) Non-separable utility over time goods: habit (Constantinides (1991), Abel (1990), Heaton (1993, 1995) Campbell and Cochrane (2000)) � � ∞ = ∑ β t u ( k t ) = ∑ β t ( 1 − δ ) j c t + j ∑ U t t j = 0 k t + 1 = ( 1 − δ ) k t + c t + 1 or U = ∑ β t u ( c t − θ x t ) , x t = ρ x t − 1 + λ c t t SDF: � c t + 1 � − γ � s t + 1 � − γ . s t = c t − x t M t + 1 = β c t s t c t Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 16 / 47

Equity Premium Puzzle Non-separable utility across states of nature: Epstein and Zin (1989) recursive utility � � 1 / ( 1 − ρ ) � � �� 1 − ρ ( 1 − β ) c 1 − ρ U 1 − γ 1 − γ U t = + β E t t t + 1 SDF   ρ − γ � c t + 1 � − ρ U t + 1   M t + 1 = β   � � 1 / ( 1 − ρ ) c t U 1 − γ E t t + 1 � � U 1 − γ Challenges: how to compute E t ? t + 1 Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 17 / 47