An Investment and Consumption Problem with CIR Interest Rate and Stochastic Volatility

We are concerned with an investment and consumption problem with stochastic interest rate and stochastic volatility, in which interest rate dynamic is described by the Cox-Ingersoll-Ross (CIR) model and the volatility of the stock is driven by Heston’s stochastic volatility model. We apply stochastic optimal control theory to obtain the Hamilton-Jacobi-Bellman (HJB) equation for the value function and choose power utility and logarithm utility for our analysis. By using separate variable approach and variable change technique, we obtain the closed-form expressions of the optimal investment and consumption strategy. A numerical example is given to illustrate our results and to analyze the effect of market parameters on the optimal investment and consumption strategies

Pengaruh Inflasi terhadap Strategi Optimal Investasi dan Konsumsi dengan Model Stokastik

The aim of this study is to investigate an optimal investment-consumption strategy under inflation rate with interest rate is described by Cox-Ingersol-Ross (CIR) model and volatility of the stock price is defined by Heston’s volatility model. A dynamic programming principle is used to obtain a Hamilton Jacobi Bellman (HJB) equation for the value function and choose a power utility function as utility function. The explicit solution of optimal investment and consumption are acquired with using separate variable and approach variable technique. The parameter’s values are approached by Euler-Maruyama method and Ordinary Least Square (OLS) method. Assumed that the portfolio of the investor contains a risk-free asset and a risk asset. Monthly historical data of TLK stock is used as risk asset and monthly historical data of BI 7-Day (Reverse) Repo Rate (BI7DRR) is used as risk-free asset, we obtain that the proportion of investment in stock is directly proportional to return of stock and the inflation rate does not have an impact on proportion investment in the stock. Meanwhile the optimal consumption of wealth is directly proportional to investor’s wealth and inversely proportional with inflation rate, which is the investor should consume less money of his wealth when the inflation rate increases

Risk-sensitive control for a class of non-linear systems and its financial applications

This thesis studies the risk-sensitive control problem for a class of non-linear stochastic systems and its financial applications. The nonlinearity is of the square- root type, and is inspired by applications. The problems of optimal investment and consumption are also considered under several different assumptions on the stochastic interest rate and stochastic volatility. At the beginning, we systematically investigate the nonlinearity of risk-sensitive control problem. It consists of quadratic and square-root terms in the state. Such an optimal control problem can be solved in an explicit closed form by the completion of squares method. As an application of the risk-sensitive control in financial mathematics, the optimal investment problem will be described in the Chapter 4. A new interest rate, which follows the stochastic process with mixed Cox- Ingersoll-Ross (CIR) model and quadratic affine term structure model (QATSM) is introduced. Such an interest rate model admits an explicit price for the zero- coupon bond. In Chapter 5, we consider a portfolio optimization problem on an infinite time horizon. The stochastic interest rate consists not only of the quadratic terms, but also of the square-root terms. On the other hand, the double square root process is also introduced to establish the interest rate model. Under some sufficient conditions, the unique solution of the optimal investment problem is found in an explicit closed form. Furthermore, the optimal consumption problem is considered in Chapter 6 and 7. It can be solved in an explicit closed form via the methods of completion of squares and the change of measure. We provide a detailed discussion on the existence of the optimal trading strategies. Such trading strategies can be deduced for both finite and infinite time horizon cases

Growth Options in General Equilibrium: Some Asset Pricing Implications

We develop a general equilibrium model of a production economy which has a risky production technology as well as a growth option to expand the scale of the productive sector of the economy. We show that when confronted with growth options, the representative consumer may sharply alter consumption rates to improve the likelihood of investment. This reduction in consumption is accompanied by an erosion of the option value of waiting to invest, leading to investment near the zero NPV threshold. It also has important consequences for the evolution of risk aversion, asset prices and equilibrium interest rates which we characterize in this paper. One interesting prediction of the model is that we get time varying risk aversion and equity returns by virtue of the presence of growth option. We also find that the moneyness of the growth option is the key factor which determines the extent to which the book to market ratios will influence the conditional moments of equity returns.Time varying MRS; Growth Options; General Equilibrium

Agency Conflicts, Investment, and Asset Pricing

The separation of ownership and control allows controlling shareholders to pursue private benefits. We develop an analytically tractable dynamic stochastic general equilibrium model to study asset pricing and welfare implications of imperfect investor protection. Consistent with empirical evidence, the model predicts that countries with weaker investor protection have more incentives to overinvest, lower Tobin's q, higher return volatility, larger risk premium, and higher interest rate. Calibrating the model to the Korean economy reveals that perfecting investor protection increases the stock market's value by 22 percent, a gain for which outside shareholders are willing to pay 11 percent of their capital stock.

Discounting investments in mitigation and adaptation: a dynamic stochastic general equilibrium approach of climate change

We use a dynamic stochastic general equilibrium model to determine efficient discount rates for climate (mitigation and adaptation) and non-climate investment in the face of climate change. Our main result is that the non-diversifiable risk in the economy may be related to both shocks in aggregate wealth and shocks in global average temperature. Therefore, both aggregate wealth and global average temperature will carry a risk premium reflecting their contribution to the total amount of non-diversifiable risk. We characterize both climate and non-climate investments by means of a contingent claim and show that climate and non-climate investments will in general be discounted at different rates. We discuss the conditions under which the discount rates of climate investments will be lower than the discount rate of non-climate investments. September 2013 : There is an�updated version of this paper availlable, read CPB Discussion Paper 257 .

Agency Conflicts, Investment, and Asset Pricing

Corporations in most countries are run by controlling shareholders whose cash flow rights are substantially smaller than their control rights in the firm. This separation of ownership and control allows the controlling shareholders to pursue private benefits at the cost of outside minority investors by diverting resources away from the firm and distorting corporate investment and payout policies. We develop a dynamic stochastic general equilibrium asset pricing model that acknowledges the implications of agency conflicts through imperfect investor protection on security prices. We show that countries with weaker investor protection have more overinvestment, lower market-to-book equity values, larger expected equity returns and return volatility, higher dividend yields, and higher interest rates. These predictions are consistent with empirical findings. We develop new predictions: countries with high investment-capital ratios have both higher variance of GDP growth and higher variance of stock returns. We provide evidence consistent with these hypotheses. Finally, we show that weak investor protection causes significant wealth redistribution from outside shareholders to controlling shareholdersinvestment, asset pricing, investor protection

Equilibrium Commodity Prices with Irreversible Investment and Non-Linear Technology

We model equilibrium spot and futures oil prices in a general equilibrium production economy. In our model production of the consumption good requires two inputs: the consumption good and a commodity, e.g., Oil. Oil is produced by wells whose flow rate is costly to adjust. Investment in new Oil wells is costly and irreversible. As a result in equilibrium, investment in Oil wells is infrequent and lumpy. Even though the state of the economy is fully described by a one-factor Markov process, the spot oil price is not Markov (in itself). Rather it is best described as a regime-switching process, the regime being an investment `proximity' indicator. The resulting equilibrium oil price exhibits mean-reversion and heteroscedasticity. Further, the risk premium for exposure to commodity risk is time-varying, positive in the far-from-investment regime but negative in the near-investment regime. Further, our model captures many of the stylized facts of oil futures prices, such as backwardation and the `Samuelson effect.' The futures curve exhibits backwardation as a result of a convenience yield, which arises endogenously. We estimate our model using the Simulated Method of Moments with economic aggregate data and crude oil futures prices. The model successfully captures the first two moments of the futures curves, the average non-durable consumption-output ratio, the average oil consumption-output and the average real interest rate. The estimation results suggest the presence of convex adjustment costs for the investment in new oil wells. We also propose and test a linear approximation of the equilibrium regime-shifting dynamics implied by our model, and test its empirical implication for time-varying risk-premia.

The long term equilibrium interest rate and risk premiums under uncertainty

Both the equilibrium interest rate and the equity premium, as well as risk premiums of risky investments are all important quantities in cost-benefit analyses. In the light of the current (2008 -) financial crisis, it is of interest to study models that connect the the financial sector with the real economy. The effects of climate change has, on the other hand, been the subject of extensive discussions, for example in connection with the Stern report. The paper addresses both these issues, first based on standard assumptions. In particular we investigate what is needed to have long-term interests lower than short term rates. Our model allows us to tell what happens to risk premiums in turbulent times, consistent with observations. Next we extend the pure exchange model to a production economy. As a result we obtain an equilibrium term structure of interest rates, as well as a model for the equity premium. We end by a discussion of risk adjustments of the discount factor. For projects aimed at insuring future consumption, the interest rate is smaller than the risk free rate. Mitigation can have the characteristics of such a project.Dynamic equilibrium; the Lucas model; term structure; CIR; pure exchange; production economy; equity premium puzzle; risk free rate puzzle; climate models; Stern Review