Study of asset returns in SGM

Naik, V. 1994. "Asset Prices in Dynamic Production Economies with Time-Varying Risk." Review of financial studies, 7(4), 781-801.

Model
(i)   time-varying uncertainty of the marginal product of capital (productivity shocks),
(ii)  the presence of capital adjustment costs,
(iii) the separation of intertemporal substitution and risk aversion (EZ preferences), and
(iv) the allowance for first-order risk aversion in investors' preferences.

Results
(i)   the sensitivity of the price of the aggregate capital stock to shifts in risk is crucially dependent on the level of capital adjustment costs.  The effect of shifts in uncertainty is significant only when capital adjustment costs are substantial.
(ii)  Intertemporal substitution governs the direction of the effect of changes in risk on the value ofcapital stock.  Risk aversion is important in the determination of the magnitude of this effect.


Campanale, C.; R. Castro and G. L. Clementi. 2010. "Asset Pricing in a Production Economy with Chew-Dekel Preferences." Review of Economic Dynamics, 13(2), 379-402.

<quote>
Tallarini (2000) showed that the first of the two issues just outlined can be addressed by disentangling risk aversionfrom the elasticity of intertemporal substitution. By assuming Epstein–Zin preferences, he was able to raise risk aversion at arbitrarily high levels, while keeping the elasticity of substitution anchored at 1. Tallarini went on to show the existence of RRA coefficients such that the market price of risk is consistent with the empirical evidence. However, the price of capital being constant at 1, his model essentially generates no equity premium. This issue was dealt with successfully by Jermann (1998) and Boldrin et al. (2001), who assumed that the allocation of capital cannot adjust immediately or costlessly to productivity shocks. Impediments to the smooth adjustment of capital imply that its price may vary away from 1. However, without limiting households’ willingness to substitute consumption intertemporally, this innovation results mainly in a higher volatility of consumption growth. Both Jermann (1998) and Boldrin et al. (2001) lower the IES by introducing habit formation. The resulting increase in the curvature of the Bernoulli utility function, by increasing risk aversion, also takes care of increasing the volatility of the stochastic discount factor.
<unquote>

SGM; Stochastic Growth Model

EZ Two Country Recursive Formulation

https://files.nyu.edu/ht477/public/files/Tretvoll_anomaly.pdf

Asset price leads business cycles

Backus, David K.; Routledge, Bryan R.; and Zin, Stanley E., "Asset Prices in Business Cycle Analysis" (2007). Tepper School of Business. Paper 414.
http://repository.cmu.edu/cgi/viewcontent.cgi?article=1414&context=tepper

Unit Roots vs. Trend Stationary

Christiano, L. J. and M. Eichenbaum. 1990. "Unit Roots in Real Gnp: Do We Know, and Do We Care?" Carnegie-Rochester Conference Series on Public Policy

---<quote>---
Macroeconomists have traditionally viewed movements output as representing temporary fluctuations about a deterministic trend. According to this view, innovations to real gross national product (GNP) should have no impact on long-run forecasts of aggregate output. Increasingly, however, this view of aggregate fluctuations has been challenged, Following the provocative work of Nelson and Plosser (1982), numerous economists have argued that real GNP is best characterized as a stochastic process that does not revert to a deterministic trend path. Under these circumstances, innovations to real GNP should affect output forecasts into the indefinite future. In pursuing this interpretation of the data, various researchers have tried to measure the long-run response of real GNP to a shock. Estimates of this response are often referred to as the persistence of shocks to real GNP.
---<unquote>---

Recursive Preferences

http://www.princeton.edu/bcf/newsevents/events/lectures-in-finance/2010larshansen1.pdf

Role of EIS and Risk Aversion

http://www.lse.ac.uk/finance/prospectiveStudents/phdFinance/files09/Job_Market_Paper_Aytek_Malkhozov.pdf
Malkhozov and Samloo (2009) "Asset Prices in a News Driven Real Business Cycle Model"

<quote>
Lets first consider a Lucas-tree economy. A positive shock to expected consumption growth (or a negative shock to uncertainty) increases wealth to consumption ratio, which adjusts through movements in wealth since consumption is exogenous. This adjustment depends on the size of the elasticity of intertemporal substitution. If the substitution effect dominates the wealth effect, i.e. elasticity of intertemporal substitution is greater than one, the agent would like to hold more of the asset, thus driving prices up. Otherwise (when elasticity of intertemporal substitution is less than 1) the agent prefers bringing the increase in consumption forward, depressing prices.

How does this matter for risk premia? Shocks to expected consumption growth affect expected future returns to wealth. The agent with relative risk aversion greater than 1 wants to hedge against these changes in the investment opportunity set (and bet on them if relative risk aversion is less than one). Notice that relative risk aversion and inverse of the elasticity of intertemporal substitution are comparable measures of propensity to smooth consumption across states and time respectively. Therefore if the two are equal (CRRA case) the changes in wealth-consumption ratio exactly offset the hedging demand. With Epstein-Zin preferences there can be a wedge between relative risk aversion and the inverse of the elasticity of intertemporal substitution which will translate into premia. As an example, consider an agent with both elasticity of intertemporal substitution and relative risk aversion greater than 1, exposed to a positive shock to expected consumption growth. The intertemporal substitution effect drives up asset prices. The hedging demand effect would imply that the agent wants his portfolio to depreciate. Therefore a premium is required for the agent to hold the asset in equilibrium. If consumption and dividends are correlated the results for the pricing of aggregate risk carry forward to the risk premium for the claim on aggregate dividends. Recursive preferences are crucial for this mechanism.
<unquote>

Stochastic Volatility/ Higher-Order Approximation

http://www.aytekmalkhozov.com/papers/news_vol_asset_prices_article.pdf
see p.14

Habit Formation

Abel1990AER
<quote>
This paper introduces a utility function that nests three classes of utility functions: 1) time-separable utility functions; 2) "catching up with the Joneses" utility functions that depend on the consumer's level of consumption relative to the lagged cross-sectional average level of consumption; and 3) utility functions that display habit formation. Incorporating this utility function into a Lucas (1978) asset pricing model allows calculation of closed-form solutions for the prices of stocks, bills and consols under the assumption that consumption growth is i.i.d. Then equilibrium asset prices are used to examine the equity premium puzzle.

Panel C presents the unconditional expected rates of return under habit formation. The expected rates of return on both long-lived assets (stocks and consols) are extremely sensitive to the value of ALPHA (coefficient of risk averesion). Under logarithmic utility (ALPHA = 1), the expected rates of return are the same as under time-separable preferences and relative consumption. However, with ALPHA = 1.14, the expected rates of return on stocks and consols are both greater than 35 percent.
<unquote>

Catching up with the Joneses

Abel(1990)AER
<quote>
"catching up with the Joneses" utility functions that depend on the consumer's level of consumption relative to the lagged cross-sectional average level of consumption.
<unquote>

Abel(1999)JME
<quote>
The catching up with the Joneses feature of the utility function was originally
introduced to help account for the high average value of the equity premium
observed empirically. However, when this form of the utility function was
specified to imply a realistic value of the equity premium in Abel (1990), the
model produced a riskless rate of return that was far too volatile. Campbell and
Cochrane (1994) developed a form of catching up with the Joneses preferences
that yielded, as in actual data, a large equity premium and low variability of the
riskless rate. They achieved this low variability of the riskless rate by specifying
a complicated recursive function for the determination of the benchmark level of
consumption.  Here I adopt a simpler formulation of the benchmark level of
consumption that produces, with the inclusion of leverage, low variability of the
riskless rate along with a large equity premium.

The analysis of leverage arises naturally from the formulation of the canonical
asset introduced in this paper.  The payoff in period t on the canonical asset is
specified to be proportional to yt^LAMBDA, where yt is an observable random variable and LAMBDA is a constant. I use this formulation in order to include fixed-income securities and equities as special cases.

Rather than proceed with separate derivations for the prices and rates of
return for different assets such as equity, short-term bills and long-term bonds,
I will introduce a canonical asset that includes all of these assets as special cases.

The canonical asset introduced here includes equities and fixed-income securities
of all maturities.
<unquote>

Roll's critique

http://en.wikipedia.org/wiki/Roll%27s_critique

Epstein-Zin(1991)JPE
<quote>
The nominal return on the optimal portfolio is measured with the value-weighted index of shares traded on the New York Stock Exchange. A number of issues arise from the use of this measure, but the primary concern for our purposes is whether it is sufficiently broad to capture the relevant part of actual holdings of wealth; that is, Roll's (1977) critique of CAPM is relevant here. If stochastic wages are a large factor in the wealth constraint of the typical agent, then, as discussed in Section II, the return on the optimal portfolio of the agent should reflect the shadow return of the agent's human capital. Rather than attempt a lengthy analysis of this issue at this time, we shall simply assume that factors that may not be properly measured by the value- weighted index of stock returns do not affect the empirical analysis in an appreciable way. The appropriateness of this assumption, vis-a-vis the empirical results below, remains an open question.
<unquote>

Campbell(1996)JPE
<quote>
In response to the Roll (1977) critique, I extend the Campbell (1993) model to allow for human capital as a component of wealth. I impute the return on human capital from data on aggregate labor income and asset returns. Finally, I develop an econometric frame- work in which the model can be confronted with historical data.

The asset pricing model developed in Section II is empirically testable only if one can measure the return on the market portfolio. Financial economists commonly proxy the market portfolio by a value-weighted index of common stocks, but this practice is questionable. Even if the stock index return captures the return on financial wealth, as argued by Stambaugh (1982), it may not capture the return on human wealth. Approximately two-thirds of gross national product goes to labor and only one-third to capital, so human wealth is likely to be about two-thirds of total wealth and twice financial wealth. This suggests that the omission of human wealth may be a serious matter.

Increases in expected future labor income cause a positive return on human capital, but increases in expected future asset returns cause a negative return on human capital because the labor income stream is now discounted at a higher rate and is therefore worth less today.
<unquote>

Elasticity of intertemporal substitution

Elasticity of intertemporal substitution (or intertemporal elasticity of substitution) is a measure of responsiveness of the growth rate of consumption to the real interest rate.

http://en.wikipedia.org/wiki/Elasticity_of_intertemporal_substitution

Multi-Factor Model

http://web.cenet.org.cn/upfile/76975.pdf
p.10

Campbell-Shiller decomposition

http://people.bu.edu/fgourio/ec745ln_F2010.pdf

International Equity Premium Puzzle

Colacito and Croce (2010) "Risks For The Long Run And The Real Exchange Rate"

<quote>
i) risk aversion has to be large to reconcile the low volatility of consumption growth rates with highly volatile stochastic discount factors (see equity premium puzzle)

ii) consumption is poorly correlated across countries at annual or higher frequencies
<unquote>

International Equity Premium Puzzle
Since Δe = m* - m ⇨ V(Δe) = V(m*) + V(m) - 2Cov(m*,m), if Cov(m*,m) is small as is observed in the data (consumption correlation), the volatility of exchange rate in the data is too small (V(m*) and V(m) are large).

Hansen-Jagannathan Bounds

see p.12

http://people.bu.edu/fgourio/lecturenotes741Fall2006.pdf

absolute / relative risk aversion

準備

u: 効用関数

π: リスクプレミアム（期待値ゼロのくじに対して、その期待値を確実に貰えるなら払っても良いと考える保険料）

ε〜（0,σ^2）

リスクプレミアムの定義から

u(x - π) = E[u(x + ε)]

が成り立つ。

左辺をπについてゼロ周りで１次テイラー展開、右辺をεについてゼロ周りで２次テイラー展開すると、

u(x) - u'(x)π = E[ u(x) + u'(x)ε+ u''(x) ε^2 ]

この式より、

π = - (1/2)(u''(x)/u'(x))σ^2 　…（＊）

絶対的危険回避度(absolute risk aversion)

- u''(x)/u'(x)

絶対的危険回避度が一定の時(CARA)、消費者は所得水準(ここでは消費水準x)に関わらず、そのくじの分散に比例した保険料を設定する。

相対的危険回避度(relative risk aversion)

- x u''(x)/u'(x)

相対的危険回避度が一定の時(CRRA)、消費者は所得水準で標準化した分散に比例した保険料を設定する（同じくじなら所得水準が高い人ほど低い保険料を設定する）。これは、（＊）が、

π = - (1/2)(x u''(x)/u'(x))(σ^2/x)

と変形できることからわかる。

http://www.econ.hit-u.ac.jp/~makoto/PDF/Appendix3JIS.pdf

Fisher Premium

F≡(1+n)E(p(t)/p(t+1))-(1+r)

n: nominal interest rate
p: price level
r: real interest rate

LeRoy, 1984, "Nominal Prices and Interest Rates in General Equilibrium: Endowment Shocks"

Special Markov Chain (IID Case)

Let P be the transition matrix of the two state Markov chain, where
p11=1-a, p12=a, p21=b, p22=1-b. In other words,



If a=1-b, then the state X1, X2, ... are independently identically distributed random variables with P{Xn=0}=b and P{Xn=1}=a.

http://www.rslabntu.net/Random_Processes/Chapter_2.pdf

Recursive Competitive Equilibrium

"Formally, a Recursive Competitive Equilibrium (RCE) is characterized by time invariant functions of a limited number of ‘state variables’, which summarize the effects of past decisions and current information.

These functions (decision rules) include

(a) a pricing function, 

(b) a value function,

(c) a period allocation policy specifying the individual’s decision,

(d) period allocation policy specifying the decision of each firm and

(e) a function specifying the law of motion of the capital stock."

Mehra 2005 "Recursive Competitive Equilibrium"
www.academicwebpages.com/preview/mehra/pdf/REC Nov 9.pdf

Equity Premium Puzzle/ Risk Free Rate Puzzle/ Lucas Cost of Business Cycles

1. Equity Premium Puzzle

With standard time separable and constant relative risk averse preferences, the consumption based asset pricing model is not consistent with the differences in average return between stocks (equity return) and bonds (risk free return).

Equity return >> Risk Free Return

To be data consistent, the relative risk aversion (RRA) parameter must be very high.

According to the US data: the return on the S&P500 from 1889 to 1978 and the yield on government bonds,

Equity Return ー Bond (Risk Free) Return 〜 6%

http://www.econ.yale.edu/smith/econ510a/book9.pdf

2. Risk Free Rate Puzzle

When RRA parameter is high, the risk free rate must be also high. This is not consistent with the real data.

<Mehra Prescott Model 1985>
qt=(1+r)^(-1)=βE[1/(λ^γ)]

λ: Consumption growth rate, γ: Relative risk aversion parameter

When γ is high, the right hand is low (λ>1), and r must be high.


3. Lucas Cost of Business Cycles

Based on Mehra Prescott model, he found that business cycle costs are very low because agents would pay a very small amount to insure their consumption against the business cycle volatility.

In the Mehra Prescott model, there must be some problem in preference specification. Hence, it is not surprising that Business cycle risk is quite small.