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
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
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
Main question: What risk are priced in the market? Is the market efficient? Main test: Linear Factor Model (CAPM) Et[Ri,t+1 − Rf ,t] = βi,t · λt (conditional) E[Ri,t+1 − Rf ,t] = βi · λ (unconditional) where βi : risk exposure λ : price of risk Regression Ri,t+1 − Rf ,t = αi + βi · λt + εi,t+1 H0 : αi = 0
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 3 / 47
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
Consumption-based asset pricing model with CRRA Utility: Et
∞
j=0
βju(ct+j) Euler Equation: Et
u(ct+1) u(ct)
t+1
which implies that Et
t+1
− 1 Rf
t
covt
t+1, βu(ct+1)
u(ct)
Rf
t
= Et
u(ct+1) u(ct)
Uncertainty and Long-Run Risk 2018/08/12 5 / 47
With CRRA (power) utility u(c) = c−γ We have Et
t+1
γcovt
t+1, ct+1
ct
γσt(Rei
t+1)σt(∆ct+1)ρt(∆ct+1, Rei t+1)
= ⇒
t+1
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
1930 1940 1950 1960 1970 1980 1990 2000 2010
1 2 3 4 5 6 7 Real Value of One Dollar Invested in 1926 (in logs) Stock Index 90-Day Tbill
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 7 / 47
1930 1940 1950 1960 1970 1980 1990 2000 2010
0.2 0.4 0.6 Real Return of Stock Index and 90-day Tbill Stock Index Tbill
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 8 / 47
Table: Summary Statistics of Real Return
(%) Re Rf90 Rf30
∆P (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 Re Rf90 Rf30
∆P (PCE )
∆ct 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 ∆ct 0.19 0.26 0.30
1
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 9 / 47
Initial Reaction
1
T-bill rate is not a good approximation for the risk-free interest rate? No, as high sample Sharpe ratios are pervasive in finance and not limited to the difference between stocks and bonds
2
Risk aversion is indeed high? No, the implied risk free rate is more than 20%. Assuming log-normal distribution rf
t = log β + γEt(∆ct+1) − 1
2γ2σ2
t (∆ct+1)
—> Risk Free Rate Puzzle
3
More information is worse, ρ = 0.2, implies risk aversion of more than 100!
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 10 / 47
Equity Premium and Risk Free Rate Puzzle
1
Quantitative not qualitative puzzle,
2
Consumption is proportional to wealth in the derivation of the CAPM, so the CAPM predicts that consumption should inherit the large 20 percent or so volatility of the stock market
3
Implication optimal portfolio choice: w = 1 γ E(Re) σ2(Re) 100% equity investment for γ around 3. However, conditonal on 20% consumption std.dev.!!
4
Consumption is much smoother that wealth, but consumption-wealth 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
Empirical Tests:
1
Mehra and Prescott (1985): Calibration
2
Hansen and Singleton (1983): GMM of conditional and unconditional Euler equation Et
u(ct+1) u(ct)
t+1
E
u(ct+1) u(ct)
t+1 − 1
where zt 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
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
Risk Aversion and Intertemporal Substitution
1
Risk aversion: measures the risk attitude
2
Intertemporal elasticity of substitution: how much consumption growth 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
3
Inverse to each other in CRRA utility
4
Quite high risk aversion is required to digest the equity premium is robust in consumption-based model estimation
5
Much more debate on IES
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
Questions:
1
What utility function should one use?
2
How should one treat time aggregation and consumption data?
3
How about multiple goods?
4
What asset returns and instruments are informative?
5
Asset pricing empirical work has moved from industry or beta 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?
6
The data may be poor enough that practitioners will still choose “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
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)) U = ∑
t
βtu(kt) = ∑
t
βt
∑
j=0
(1 − δ)jct+j
= (1 − δ)kt + ct+1
U = ∑
t
βtu(ct − θxt), xt = ρxt−1 + λct SDF: Mt+1 = β ct+1 ct −γ st+1 st −γ .st = ct − xt ct
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 16 / 47
Non-separable utility across states of nature:
Epstein and Zin (1989) recursive utility Ut =
t
+ β
t+1
1−ρ
1−γ
1/(1−ρ) SDF Mt+1 = β ct+1 ct −ρ Ut+1 Et
t+1
1/(1−ρ)
ρ−γ
Challenges: how to compute Et
t+1
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 17 / 47
Two ways:
Campbell (1996), Bansal and Yaron (2004) and etc: Use return on wealth to proxy and log-linearize around stationary consumption-wealth ratio mt+1 = a − bRW
t+1
Bansal, Dittmar, and Lundblad (2005) also argue that average returns
with long-run consumption growth in this framework Hansen, Heaton, and Li (2008): Log-linearize around ρ = 1. (Et+1 − Et)mt+1 = −γ(Et+1 − Et)∆ct+1 +(1 − γ)(Et+1 − Et)
∑
j=1
βj(∆ct+1+j)
small changes in consumption growth ∆ct+j Piazzesi and Schneider (2006) apply the framework to bonds. They generate risk premia in the term structure by the ability of state variables to forecast future consumption growth.
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 18 / 47
Questions:
Is the elasticity of intertemporal substitution really that different from the coefficient of risk aversion? Are there really important dynamics in consumption growth? Hansen, Heaton, and Li (2008) sensitivity analysis show that long-run properties of anything are hard to measure, Bansal, Dittmar, and Lundblad’s finding of a strong beta to explain value premia depends crucially on the inclusion of a time trend in the regression of earnings
Large risk aversion is still needed.
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 19 / 47
Recursive utility: Vt = {(1 − β)C 1−ρ
t
+ β[Rt(Vt+1)]1−ρ}
1 1−ρ
where Vt+1 is the continuation value of a consumption plan from time t + 1, Rt(Vt+1) = [Et(V 1−γ
t+1 )] 1 1−γ is certainty equivalent of
random future utility governed by risk preferences γ > 0 is RRA for static wealth gamble; 1/ρ > 0 is the IES Budget Constraint: Wt+1 = RW
t+1(Wt − Ct)
This recursion is homogeneous of degree 1 in Wt, hence when consumption is chosen optimally, V ∗
t =
t
Wt 1/(1−1/ρ) Wt Note: This solution is not the full solution
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 20 / 47
This recursion is homogeneous of degree 1 in (Ct, Vt+1) hence Vt = (MCt)Ct + Et(MVt+1Vt+1) MCt = ∂Vt ∂Ct = (1 − β)V ρ
t C −ρ t
MVt+1 = ∂Vt ∂Vt+1 = β
Rt(Vt+1) ρ Vt+1 Rt(Vt+1) −γ Stochastic discount factor: St+1 = MVt+1MCt+1 MCt = β Ct+1 Ct −ρ Vt+1 Rt(Vt+1) ρ−γ (1) Euler Equation: Et [St+1Rt+1] = 1
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 21 / 47
Problem: How to measure Vt+1 or Wt+1
Epstein and Zin (1991) and Campbell (1996) use the link between return on wealth and continuation value. Give a well-specified stochastic process governing consumption and avoid the need to construct a proxy to the return on wealth.
Restoy and Weil (1998) and Bansal and Yaron (2004): Loglinearize around a constant consumption-wealth ratio Hansen, Heaton and Li (2008): linearize around ρ = 1.
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 22 / 47
Return on wealth portfolio: Rw
t+1 =
Wt+1 Wt − Ct = ⇒ RW
t+1 = 1
β Ct+1 Ct ρ Vt+1 Rt(Vt+1) 1−ρ (2) from (2),we have the continuation value could be written in term of RW
t+1 and Ct+1/Ct,
( Vt+1 Rt(Vt+1))1−ρ = β Ct+1 Ct −ρ RW
t+1
from (1)and (2), we have the SDF written in term of RW
t+1 and
Ct+1/Ct St+1 =
Ct+1 Ct −ρ 1−γ
1−ρ
RW
t+1
ρ−γ
1−ρ
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 23 / 47
Euler equation for wealth portfolio, Et
Ct+1 Ct −ρ RW
t+1
1−γ
1−ρ
= 1 (3) Plug in any asset return Ri
t+1 in the Euler equation we have
Etri
t+1 − rf t + 1
2vart(ri
t+1)
= (1 − θ)covt(rw
t+1, ri t+1)
+θρcovt(ri
t+1, ∆ct+1)
Special cases:
θ = 1, i.e. ρ = γ, power utility θ = 0,i.e. γ = 1, static CAPM pricing formula ρ = 1 then Ct
Wt is constant.
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 24 / 47
How to measure rw ?
Epstein and Zin (1991): use stock market return as proxy Campbell (1996): include human capital return, which has assigned market or shadow value, assume the share of financial wealth and labor income are stable and obtain a three-factor model: rw
t+1 = (1 − ν)rm t+1 + νry t+1 + ˆ
k Santos and Veronesi: shares varies over time, which is an important explanatory variable (ν− > νt) Letteau and Ludvigson (2001): cointegration relationship is important. (cay)
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 25 / 47
Under the assumption of linear dynamics of consumption and dividend growth, we explicitly solve the value function and stochastic discount factor when ρ = 1. In the more general case of ρ = 1, we approximate the solution around ρ = 1. Solve the model for ρ = 1
Take logs of value function vt ≡ log(Vt Ct ) = 1 1 − ρ log ((1 − β) + β exp ((1 − ρ)Qt (vt+1 + ∆ct+1))) where Qt (vt+1 + ∆ct+1) ≡ log Rt Vt+1 Ct+1 Ct+1 Ct
1 1 − γ log Et [exp((1 − γ) (vt+1 + ∆ct+1))] Stochastic Discount Factor: st+1 ≡ log St+1 st+1 = log β − ρ∆ct+1 + (ρ − γ)(vt+1 + ∆ct+1 − Qt (vt+1 + ∆ct+1))
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 26 / 47
Specification of dynamics: state vector xt follows VAR, dividend growth and consumption growth is linear in states. xt+1 = Gxt + Hwt+1 (4) ∆ct+1 = µc + Uc · xt + λ0 · wt+1 dt+1 − dt = µd + Ud · xt + ι0 · wt+1 where wt+1˜N(0, I) is a i.i.d normally distributed vector.
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 27 / 47
We are interested in the discounted response of consumption growth and dividend growth to the shocks, namely, λ(β) and ι(β) defined as following λ(β) =
∞
j=0
βj ∂∆ct+1+j ∂wt+1 = λ0 + Uc
∞
j=1
βjG j−1H (5) = λ0 + βUc(I − βG)−1H λ(1) = λ0 + Uc(I − G)−1H (6) ι(β) =
∞
j=0
βj ∂∆dt+1+j ∂wt+1 = ι0 + Ud
∞
j=1
βjG j−1H (7) = ι0 + βUd(I − βG)−1H ι(1) = ι0 + Ud(I − G)−1H (8)
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 28 / 47
Dividend dynamics is given as Dt = D∗
t f (xt), D∗ t = exp
t
j=1
πwj
t is a geometric random walk (governs long-run growth rate)
and f (xt) is the transitory component, let dt = log(Dt) dt+1 − dt = ζ + πwt+1 + log f (xt+1) − log f (xt) (9) and we need to link it to the VAR system (4). We do this by performing martingale extraction dt+1 − dt = µd + Udxt + ι0wt+1 = µd + ι(1)wt+1 − U∗
d (xt+1 − xt)
where ι(1) = ι0 + Ud(I − G)−1H, U∗
d ≡ Ud(I − G)−1
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When ρ = 1, the recursion of value function and SDF became vt ≡ log Vt − log Ct = β 1 − γ log Et[exp(1 − γ)(vt+1 + ∆ct+1)] ≡ βQt(vt+1 + ∆ct+1) st+1 = log β − ∆ct+1 + (1 − γ)[vt+1 + ∆ct+1 − Qt (vt+1 + ∆ct+1)] Guess and Verify: value function and SDF is log-linear in the states
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 30 / 47
Value function vt = µv + Uvxt Uv = βUc(I − βG)−1 µv = β 1 − β
2
st+1,t = µs + Usxt + ξ0εt+1 µs ≡ log β − µc − (1 − γ)2|λ(β)|2 2 Us ≡ −Uc ξ0 = (1 − γ)λ(β) − λ0
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 31 / 47
Long-Run Pricing Operator and Growth Operator Given the dynamics of dividends, the pricing vector is obtained by finding the dominant eigenvalue and eigenvector of the pricing
Pt D∗
t
= E
D∗
t
|xt
= E [exp(st+1,t + ζ + πwt+1)f (xt+1)|xt] ≡ Pf (xt) Et Dt+1 D∗
t
|xt
(11) We can show that for any function f (x) that is log-linear in the Markov state x, the function Pf and Gf (xt)is also log-linear in the Markov state x, that is, P exp(−ωx + κ) = exp(−ωx + κ) G exp(−ωgx + κg ) = exp(−ωgx + κg)
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 32 / 47
Pjf is also log-linear in x and satisfies the recursion ω(j) = ω(j−1)G − Us (12) κ(j) = κ(j−1) + µs + ζ + |ξ0 + π − ω(j−1)H|2 2 with ω0 = ω, κ0 = κ As j − → ∞, the limit of ω(j) is the fixed point of (12), and ¯ ω = lim
j→∞ ω(j) = −Us(I − G)−1 = Uc (I − G)−1
(13) −ν = lim
j→∞
= µs + ζ + |ξ0 + π − ¯ ωH|2 2 (14) The price of a claim to a single cash flow Dt+j = D∗
t+jf (xt+j) decays
at a rate which is asymptotically constant (ν).
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 33 / 47
Plug in the solution of SDF and dividend dynamics we have1 −ν = log β − µc + µd + |ι(1) − λ(1)|2 2 + (1 − γ)λ(β) · [ι(1) − λ(1)] In fact −ν and exp( ¯ ωx) are the dominant eigenvalue and eigenfunction of pricing operator, that is P exp( ¯ ωx) = exp(−ν) exp( ¯ ωx) Similarly, the dominant eigenvalue and eigenfunction of growth
G exp( ¯ ωgx) = exp(η) exp( ¯ ωgx) ¯ ωg = 0, i.e. exp( ¯ ωgx) = 1 η = ζ + |π|2 2 = µd + |ι(1)|2 2 As j gets large, we can show that for any f (x) Pjf exp(−jν) exp( ¯ ωx), Gjf exp(jη) exp( ¯ ωgx) = exp(jη)
1Use the fact that
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 34 / 47
Long-Run Return: Consider the expected return to holding a claim to a single cash flow Dt+j Et(Rt,t+j) = Et Dt+j Pt
t ]
Pt/D∗
t
= Gjf Pjf = ((η + ν)j) exp(− ¯ ωx) Hence the APR as j gets large is then lim
j− →∞
1 j log Et(Rt,t+j) = η + ν The long-run rate of return can be rewrite as ν + η = ς∗
(long-run risk free rate)
+ π∗
(long-run price of risk) ·
π
(long-run risk exposure)
π∗ = λ(1) − (1 − γ)λ(β) ς∗ = − log β + µc − |λ(1)|2 2 − (1 − γ)λ(β) · λ(1)
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 35 / 47
In this case we find approximating solution around ρ = 1. st+1,t = s1
t+1,t + (ρ − 1)Ds1 t+1,t
Ds1
t+1,t = 1
2w
t+1Θ0wt+1 + w t+1Θ1xt + ϑ0 + ϑ1x1 + ϑ2wt+1
Long-run Return η + ν = ς∗ + ππ∗ + 1 2πΠ∗π In this case, the return not only depends linearly on the risk of the cash flows (π) but also depends on the quadratic of the risk of the cash flows.
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 36 / 47
First estimate VAR xt+1 = Gxt + Hwt+1 (15) ∆ct+1 = µc + Uc · xt + λ0 · wt+1 dt+1 − dt = µd + Ud · xt + ι0 · wt+1 We impose the dividends growth does not Granger cause the state variables x. Note: Cointegration test is very sensitive to the model specification (lags, with or without trend)
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Next, find suitable state variables to identify important shocks that matters for the long-run:
In Hansen, Heaton and Li (2008): aggregate consumption and earnings (corporate profit) are cointegrated = ⇒ one permanent shock yt =
et − ct
In Li (2018): aggregate investment, consumption and relative price of investment goods, with nominal consumption and investment are cointegrated = ⇒ two permanent shock, one aggregate tech shock, one investment-specific technological shock yt = it − it−1 ct − ct−1 it + pit − ct yt = µy + B1yt−1 + ... + Blyt−l + V εt
Identification of shocks: Sims (1972) or Blanchard and Quah (1989)
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 38 / 47
EW mean ret
mean grwoth
crspmkt 1.23% 5.28% 2.46% 1.38% mkt 1.47% 6.67% 2.59% 2.46% RDNA 1.22% 5.56% 2.27% 2.28% Rdzero 1.23% 5.88% 2.49% 4.09% 1 1.32% 5.38% 2.42% 1.80% 2 1.39% 5.53% 2.46% 2.25% 3 1.58% 6.65% 2.78% 4.13% 4 1.67% 7.83% 2.66% 7.08% 5 1.39% 9.58% 2.12% 14.47%
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16.2 8.1 5.4 4.1 3.2 2.7 2.3 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 C
e r e n c e w i t h ∆ l
C Equipment Intellectual Earnings
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0.1 0.2 0.3 0.4 Market Portfolio Cash Flow 10.7 5.4 3.6 2.7 2.1 10.7 5.4 3.6 2.7 2.1 0.2 0.4 0.6 0.8 0.05 0.1 0.15 0.2 Portfolio 1 10.7 5.4 3.6 2.7 2.1 10.7 5.4 3.6 2.7 2.1 0.2 0.4 0.6 0.8 Investment 0.1 0.2 0.3 0.4 Portfolio 2 Cash Flow 10.7 5.4 3.6 2.7 2.1 10.7 5.4 3.6 2.7 2.1 0.2 0.4 0.6 0.8 0.05 0.1 0.15 0.2 Portfolio 3 10.7 5.4 3.6 2.7 2.1 10.7 5.4 3.6 2.7 2.1 0.2 0.4 0.6 0.8 Investment 0.1 0.2 0.3 0.4 Portfolio 4 Cash Flow 10.7 5.4 3.6 2.7 2.1 10.7 5.4 3.6 2.7 2.1 0.2 0.4 0.6 0.8 0.1 0.2 0.3 0.4 Portfolio 5 10.7 5.4 3.6 2.7 2.1 10.7 5.4 3.6 2.7 2.1 0.2 0.4 0.6 0.8 Investment
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50 100 150 200 0.005 0.01 0.015 C
ption to Shock
50 100 150 200 2 3 4 5 6 x 10 -3 50 100 150 200
5 x 10 -4 50 100 150 200 2 4 6 8 x 10 -3 C
ption to Shock
s 50 100 150 200 2 4 6 8 10 x 10 -3 50 100 150 200
0 x 10 -3 C
F irst D ifference Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 42 / 47
0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1
2 4 6 8 10 x 10
λ(β) Sims C Shock Inv Shock Pinv Shock 0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1
2 4 6 8 10 x 10
λ(β) BQ Permanent Shock 1 Permanent Shock2 T em porary Shock
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 43 / 47
In the long-run risk model, a large risk aversion is still needed to generate sizable risk premium
No more risk free rate puzzle But as Lucas said "No one has found risk aversion parameters of 50 or 100 in the diversification of individual portfolios, in the level of insurance deductibles, in the wage premiums associated with occupations with high earnings risk, or in the revenues raised by state-operated lotteries. It would be good to have the equity premium resolved, but I think we need to look beyond high estimates of risk aversion to do it."
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 44 / 47
Barillas, Hansen, and Sargent (2009) show that a max—min expected utility theory lets us reinterpret risk-aversion parameter in EZ preference when IES =1 as measuring a representative consumer’s doubts about the model specification. Hence, reinterprete the (log-run) market price of risk as a price of model uncertainty.
Doubts about the model specification can be substantial Prices of model uncertainty contains information about the benefits of removing model uncertainty, instead of the "consumption risk" that Lucas studied. Allow us to estimate the welfare cost of increasing model uncertainty due to policy uncertainty
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Agent with ambiguity aversion W (x0) = min
[gt+1] ∞
t=0
E0
= gt+1Gt, Et[gt+1] = 1, gt+1 ≥ 0 xt+1 = Axt + Bεt+1 ct = Hxt where θ indicates fear of model misspecification as measured by how much the minimizing agent gets penalized for raising entropy. Observationally equivalent to V for ρ = 1 with θ = 1 (1 − β)(1 − γ)
Li Nan (SJTU, ACEM) Uncertainty and Long-Run Risk 2018/08/12 46 / 47
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Nan Li, Department of Finance,ACEM,SJTU
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Nan Li, Department of Finance,ACEM,SJTU