Calibration and Simulation of Financial Markets--论文代写范文精选

所有这些回报率在较低的灾害强度。无风险利率是由客观的预防性储蓄所控制。为了应对这两种动机，如果一场经济危机的概率增加,预期消费增长下降，消费平滑动机增加了投资者的财富转移。下面的essay代写范文进行详述。

Abstract 
Two paths from the calibrated model are simulated for 25, 000 years at a monthly frequency and then aggregated to an annual horizon. Annual moments from the simulation are reported in table 3 along with their sample counterparts in U.S. data. Historical moments on returns, consumption, and dividends spanning the period from 1890 to 2004 are from Robert Shiller’s dataset reported in Wachter (2010).26 Throughout the empirical analysis I use returns on short term government debt in the model as a proxy for returns on U.S. treasury bills.

The Equity Premium and the Return on Government Bills 
The model with self-excitation is able to explain an equity premium of 6.08% along with a low expected return on government bills of 2.15%, both with a reasonable level of risk aversion. Furthermore, the model can address the excess volatility puzzle documented by Shiller (1981). Equity return volatility generated by the model is 18.96% per year which compares to 18.48% in U.S. data. The mechanism achieves this result without predicting counterfactually high volatility in the return on treasury bills, which is 2.79% in the simulation and 5.91% historically. The unconditional Sharpe ratio of 0.32 exactly matches its counterpart in the data. The volatility of both log consumption and log dividend growth is slightly above its historical value. 

The dependence of the risk-free rate and the yield on government bills on the disaster intensity is depicted in figure 2. Since government debt is subject to default and investors are risk averse, the risk-free rate is below the expected return on government bills, which in turn is below the promised yield of these investments. All of these rates of return are decreasing in the disaster intensity. The risk-free rate is driven by an objective to smooth consumption over time and to form precautionary savings. Both of these motives become more pronounced in response to an upwards revision of the disaster intensity. 

If the probability of a disaster increases, expected consumption growth declines and the consumption smoothing motive increases the investor’s desire to transfer wealth into the future. In addition to that, the precautionary savings objective intensifies during periods in which agents face higher odds of a disaster. Both of these effects contribute towards a negative relationship between the risk-free rate and the disaster probability. In contrast to a truly risk-free asset, government debt is itself subject to default in the event of a disaster. This systematic default risk causes risk averse investors to demand a premium that is increasing in the disaster probability. This latter effect partially counteracts the consumption smoothing and precautionary savings motives and decreases the sensitivity of the expected return on government debt with respect to the disaster intensity. 

In turn, the promised yield on these assets must be even higher than the expected yield to provide compensation for the possibility of default during economic disasters. The relationship between the risk-free rate as well as the promised and expected return on government bills are depicted in figure 2. These latter rates are less sensitive to changes in the disaster intensity compared to the risk-free rate. In times of crisis, the expected real return on government bills can be negative. This happens whenever the disaster intensity rises above 23%, which occurs in 9 out of 100 years in the simulation. In contrast to that, the realized real return on treasury bills in U.S. data has been negative in 25 out of 100 years. 

The equity premium compensates investors for two sources of risk. The compensation for diffusive risk in prices and consumption considered by Lucas (1978) is too small to account for the observed equity premium in the U.S. Its contribution towards the total equity premium of 6.07% in the calibration presented here amounts to just 0.318%. The major part of the risk premium stems from compensation for the exposure of equity investments to rare economic disasters. In the context of recursive preferences, agents care about disasters for two reasons. The occurrence of a disaster leads to a sharp contemporaneous decline in both corporate dividends and consumption which requires equity to offer a premium for this exposure to systematic risk. In addition to that, the arrival of a disaster causes an upwards revision in the conditional disaster intensity by virtue of self-excitation. 

This causes both a decline in expected future consumption growth and an increase in the uncertainty about its future realization. A shock to the persistent disaster intensity, which increases uncertainty about future consumption growth affects indirect utility from future consumption. If investors exhibit a preference for the timing of resolution of uncertainty, i.e. if ψ 6= 1/γ, indirect utility enters into the pricing kernel. In particular, if agents prefer early resolution of uncertainty, which is the case that obtains in this calibration, they dislike an increase in the uncertainty about future consumption growth and hence require additional compensation for the possibility of a positive shock to the intensity of disaster arrival in the future. Figure 3 illustrates how these individual components determine the equity premium in relation to the disaster intensity. The amount of diffusive risk equity investments are exposed to is constant with respect to the disaster intensity and its contribution to the equity pre- mium is small in magnitude. 

The compensation for the instantaneous effect of a disaster on consumption is depicted by the dashed line and represents that part of the equity premium which arises from diffusive risk as well as the direct impact of a disaster on consumption. It is the risk premium that arises in a model with power utility such as Barro (2006) if disasters take place over extended periods rather than being concentrated at a single instant. The indirect effect of disasters on the equity premium works through the forward looking component in the pricing kernel that obtains in the case of stochastic differential utility with a preference for early resolution of uncertainty. It is compensation for the upwards revision of the jump intensity that is brought about by the arrival of a disaster.

Stock Market Crashes and Macroeconomic Contractions Historically, stock market crashes have been observed even in the absence of significant declines in macroeconomic activity. Indeed, Paul Samuelson once famously remarked that ”Wall Street indexes [had] predicted nine out of the last five recessions” (Samuelson (1966). In fact, an analysis of the crisis dataset in Barro and Urs´ua (2009) indicates that in 28% of all cases, periods of negative stock market returns exceeding 25% were associated with a contemporaneous decline in macroeconomic activity of more than 10% in OECD countries. Likewise, the odds of a macroeconomic contraction exceeding 25% given a stock market decline of 25% were only 11% in this historical dataset. A standard disaster risk model, in which the entire reduction in consumption is realized instantaneously is unable to account for this phenomenon since the only cause of a significant stock market decline is an economic disaster. The possibility of a self-perpetuating economic crisis can provide an explanation for this apparent puzzle. 

In the event of a disaster, agents rationally anticipate the possibility of a more severe economic crisis which leads to an increase in the compensation required for holding risky assets. This effect on the equity premium and hence the discount rate makes stock valuations decline by more than the reduction in dividends that is due to the disaster. In most cases, a more severe economic contraction does not materialize and the downturn in financial markets appears to have been excessive ex-post. A simulation of the calibrated model demonstrates that self-excitation is consistent with this empirical fact. Table 6 illustrates the probability of observing a macroeconomic downturn in excess of a 10% and 25% threshold conditional on a stock market crash of at least 25%. In the simulation, a stock market crash exceeding 25% goes along with a macroeconomic contraction of at least 10% in roughly 4 out of 10 cases, which is slightly higher than in the historical dataset. The calibration exactly hits the mark for crises of more than 25%, which have a conditional probability of 11% both in the model and in the data.

Conclusion 
In this paper, I have demonstrated the importance of self-perpetuating economic disasters for understanding stylized asset pricing facts. A calibration of a simple model can explain the equity premium and risk free rate with a reasonable degree of risk aversion. Excess volatility and return predictability by valuation multiples arise naturally with this model. It is the first disaster risk model which can account for the empirical observation that severe economic downturns develop over extended periods of time. This feature allows predictions for the behavior of equilibrium quantities during a crisis and can account for an increase in the VIX and the dividend-yield as well as a decline in the risk-free rate in response to the occurrence of a consumption disaster. Previous disaster risk models have been criticized for their assumption that the entire extent of an economic contraction be realized in a single instant.28 This paper demonstrates the ability of self-exciting disasters to address this point which gives further justification to this important class of models.(论文代写)

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