Pdf Pablo Kurlat Dynamic Optimization in Continuous Time

Endogenous savings and extensions of the baseline model

Klaus Prettner , David E. Bloom , in Automation and Its Macroeconomic Consequences, 2020

5.2.3 Dynamic optimization in discrete time in the case of two time periods

With the standard method of Lagrange, we can also solve simple dynamic optimization problems, which we encounter later in this chapter when we discuss the OLG model. The trick is to assume that the choice variable at different points in time is actually a different variable (e.g., consumption at time $t$ is $c_t$ and consumption at time $t+1$ is $c_{t+1}$ and so on). Then, one can follow the recipe/cookbook procedure for the method of Lagrange with $n$ choice variables and $m$ constraints. However, for advanced problems with many time periods and in which strategic interactions occur between agents or in which the future is uncertain, more sophisticated methods of dynamic programming are required. Dixit (1990), Léonard and van Long (1992), Sorger (2015), Stokey, Lucas, and Prescott (1989), and Sydsaeter et al. (2008), among others, provide good descriptions of these methods. Example 2 applies the method of Lagrange in case of a two-period optimization problem such that $n=2$ and $m=1$ .

Example 2

Assume that individuals live for two time periods, $t$ and $t+1$ , in which they earn wages $w_t$ and $w_{t+1}$ , respectively. Individuals have an isoelastic utility function over consumption in the two time periods, as given by

$U(c_t,c_{t+1})=\frac{c_t^{1-\theta }-1}{1-\theta }+\beta \frac{c_{t+1}^{1-\theta }-1}{1-\theta }.$

In this formulation, $\beta \in (0,1)$ is the discount factor measuring how much weight an individual attaches to future consumption (i.e., it measures impatience). The discount factor is determined by the discount rate, denoted by $\rho >0$ via the relationship $\beta =1/(1+\rho )$ . The parameter $\theta$ measures the risk aversion of the individual (i.e., the concavity of the utility function). ⁴ Its inverse, $1/\theta$ , is the elasticity of intertemporal substitution measuring an individual's willingness to depart from consumption smoothing to earn higher interest income. ⁵ The individual's choice is how much of her income to save in the first period of life to consume more in the second period of life. The savings carried over from the first to the second period earn an asset income at the going interest rate $r$ , and we assume that the individual starts the first period without any assets.

1.

Write down the budget constraint that the individual faces.

2.

Derive optimal individual consumption in periods $t$ and $t+1$ and use the result to characterize the rule of optimal individual consumption growth.

Solution

Denoting savings by $s_t$ and assuming that individuals start their lives without assets, the choice of individuals has to satisfy

$c_t+s_t=w_t$

in the first period such that consumption and savings add up to wage income in the first period. Note that savings can also be negative such that individuals go into debt. In the second period, the choice of consumption has to satisfy

$c_{t+1}=w_{t+1}+(1+r)s_t$

such that consumption in the second period equals wage income in the second period, plus savings from the previous period on which interest is paid. Substituting $s_t=w_t-c_t$ obtained from the first equation and plugging it into the second equation yields

$c_{t+1}=w_{t+1}+(1+r)(w_t-c_t).$

Rearranging all terms involving consumption to appear on the left-hand side leads to the lifetime budget constraint

$c_t+\frac{c_{t+1}}{1+r}=w_t+\frac{w_{t+1}}{1+r}.$

The left-hand side comprises total discounted lifetime consumption expenditures, while the right-hand side comprises total discounted lifetime income. By using this equation as the constraint, we can solve the given simple dynamic optimization problem by following our recipe/cookbook procedure.

1.

The Lagrangian is

$\mathcal{L}\mathcal{=}\frac{c_t^{1-\theta }-1}{1-\theta }+\beta \frac{c_{t+1}^{1-\theta }-1}{1-\theta }+\lambda \left(w_t+\frac{w_{t+1}}{1+r}-c_t-\frac{c_{t+1}}{1+r}\right).$

2.

The FOCs are

$L_{c_t}=c_t^{-\theta }-\lambda \stackrel{!}{=}0,$

$L_{c_{t+1}}=\beta c_{t+1}^{-\theta }-\frac{\lambda }{1+r}\stackrel{!}{=}0,$

$L_{\lambda }=w_t+\frac{w_{t+1}}{1+r}-c_t-\frac{c_{t+1}}{1+r}\stackrel{!}{=}0.$

3.

Using $c_t^{-\theta }=\lambda$ , as implied by the first FOC; plugging it into the second FOC; and reformulating yields the so-called consumption Euler equation:

(5.11) $\frac{c_{t+1}}{c_t}={[(1+r)\beta ]}^{\frac{1}{\theta }}.$

In this equation, the left-hand side is the growth factor of consumption at the optimum, while the right-hand side is a function of the interest rate, the discount factor, and individuals' risk aversion. Optimal consumption grows over time if the right-hand side is larger than one. This is the case if the interest rate is high in comparison with the discount factor such that the product $(1+r)\beta$ is larger than one. In this case, the financial market overcompensates for individual impatience by paying a sufficiently high interest rate to induce individuals to save. Saving, in turn, is tantamount to having positive consumption growth, $c_{t+1}/c_t>1$ . Note that consumption growth (saving) is higher if

•

The interest rate $r$ is higher.

•

Households are more patient ( $\rho$ is lower such that $\beta$ is higher).

•

Households are less risk averse ( $\theta$ is lower). If households are less risk averse, the elasticity of intertemporal substitution is higher, and they are more willing to shift consumption over time to take advantage of the higher interest rate.

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Calculus of variations

Giovanni Romeo , in Elements of Numerical Mathematical Economics with Excel, 2020

Abstract

It is highly suggested reading this chapter together with the classical textbook by Alpha Chiang, Elements of Dynamic Optimization , which would represent a necessary theoretical prerequisite. This chapter is divided into three parts: 1. From Sections 9.1–9.4, you will see what is a functional, the simplest problem of Calculus of Variations (CoV) and how this can be discretized (simple examples will be provided) and solved via the Lagrange multipliers technique. The Excel Generalized Reduced Gradient Solver will be used to solve these problems. 2. The second part goes from Sections 9.5–9.10 and includes some of the most used applications of CoV to the economic problems. 3. The last part of this chapter covers more advanced topics, like the applications of contour lines, CoV involving two independent functions in the functional, the constrained problems (ordinary differential equation and isoperimetric constraints) and the checking of the sufficient conditions.

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RESOURCES

M.R. Caputo , in Encyclopedia of Energy, Natural Resource, and Environmental Economics, 2013

Summary and Conclusion

Optimal control theory is one of the most important mathematical tools used by natural resource economists to analyze continuous-time dynamic optimization problems. Having said that, it is important to understand that even though the theorems stated in this article are applicable to many of the typical optimal control problems encountered in natural resource economics, they are not the most general available. Indeed, there are several complications that can and do occur when formulating such problems. For example, some natural resource economics problems are linear in the control variables, whereas others have equality or inequality constraints on the control and state variables, or include equality or inequality constraints on just the state variables, the last being the most difficult complication to address. In each of these cases, more general theorems are required in order to find and characterize the solution.

It is also worthwhile to remember that the two examples were not chosen because of their generality or realism, but to demonstrate how to (1) solve for the explicit solution of an optimal control problem and conduct a qualitative analysis and (2) derive a qualitative characterization of the solution of an optimal control problem when the functional forms of the integrand and transitions functions are not given.

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RESOURCES

J.A. Roumasset , C.A. Wada , in Encyclopedia of Energy, Natural Resource, and Environmental Economics, 2013

Abstract

This article reviews economic concepts relevant to groundwater management. It begins by discussing how groundwater resource management has evolved from sustainable yield to dynamic optimization. General management principles developed through the solution of a single-aquifer optimization problem are then extended to the management of multiple resources including additional groundwater aquifers, surface water, recycled wastewater, and upland watersheds. However, in many parts of the world, especially in agriculture, groundwater is characterized as a common-pool resource. Therefore, the open-access equilibrium for groundwater and the conditions under which the Gisser–Sanchez effect (the result that the present value generated by competitive resource extraction and that generated by optimal control of groundwater are nearly identical) is valid are also discussed. From the models and examples discussed, one can conclude that optimization across any number of dimensions (e.g., space, time, and quality) is driven by a system shadow price, and augmenting groundwater with any number of alternatives lessens scarcity and increases welfare if timed appropriately.

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VOLATILITY PROCESSES

Michel M. Dacorogna , ... Olivier V. Pictet , in An Introduction to High-Frequency Finance, 2001

8.4.2 Performance of ARCH-Type Models

In Table 8.6 , the results for the different performance measures are presented for the most traded FX rate, USD-DEM, for the static and dynamic optimizations. In parentheses, the results for the scaled forecasts are presented. For all measures, three parameter models perform better than the benchmark and the EMA-HARCH performs the best. The forecast accuracy is remarkable for all ARCH-type models. In more than two-thirds of the cases, the forecast direction is correctly predicted and the mean absolute errors are smaller than the benchmark errors for all models. The realized potential measure shows that the forecast of volatility change is accurate not only for small | s _r | but also for large ones. The condition expressed in Equation 8.42 is always satisfied for all models. Neither the scaled forecast nor the dynamic optimization seems to significantly improve the forecasting accuracy. The realized potential Q _r is the only measure that consistently improves with dynamic optimization. Examining the model coefficients computed in moving samples shows that they oscillate around mean values. No structural changes in the coefficients were detected. The accuracy improvement in Q _r together with the loss in Q _f in the case of dynamic optimization indicates that the prediction of large movements is improved at the cost of the prediction of direction of small real movements. From the point of view of forecasting short-term volatility, the EMA-HARCH is the best of the models considered here and compares favorably to HARCH. Similar conclusions can be drawn from the results for four other FX rates. ¹¹ The cross rate JPY-DEM presents results slightly less accurate than the other currencies, but it should be noted that the early half of the sample has been synthetically computed from USD-DEM and USD-JPY. This may lead to noise in the computation of hourly volatility and affect the forecast quality.

Table 8.6. Forecasting performance for USD-DEM.

Forecasting accuracy of various models in predicting short-term market volatility. The performance is measured every hour over 5 years, from January 1, 1992, to December 31,1996, with 43,230 observations. In parentheses, the accuracy of rescaled forecasts is shown.

USD-DEM	Q d	Q r	Q f
Static Optimization
Benchmark	67.7% (67.6%)	54.2% (54.3%)	0.000
GARCH(1,1)	67.8% (67.3%)	58.5% (59.7%)	0.085 (0.072)
HARCH(7c)	69.2% (68.7%)	58.3% (59.2%)	0.134 (0.129)
EMA-HARCH(7)	69.4% (68.8%)	60.7% (62.5%)	0.140 (0.128)
Dynamic Optimization
Benchmark	67.7% (67.4%)	54.2% (54.6%)	0.000
GARCH(1,1)	67.0% (66.0%)	59.5% (59.8%)	0.074 (0.057)
HARCH(7c)	67.7% (66.8%)	60.1% (60.8%)	0.113 (0.102)
EMA-HARCH(7)	68.8% (67.7%)	62.4% (62.9%)	0.133 (0.117)

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Theory of optimal control

Giovanni Romeo , in Elements of Numerical Mathematical Economics with Excel, 2020

10.5 Consumption model

Let us see now some economic models applied with the OC theory starting with a simple consumption model. ⁶

The consumption model is expressed as follows:

$\underset{\left\{u\right\}}{\max }J\left(u\right)={\int }_0^1\log \left(u4y\right)dt$

$s.t.\dot{y}=4y(1-u)\text{and y}\left(0\right)=1,y\left(1\right)=e^2$

where u(t) is the consumption and y(t) is the total output of the economy over the time interval [0, 1], while $\dot{y}=\frac{dy}{dt}$ is the variation in the total output (i.e., the aggregate investment expenditures).

The objective is to maximize the economy utility function.

i.

The Hamiltonian is given by:

$\mathcal{H}=\log \left(u4y\right)+\lambda \left[4y(1-u)\right].$

Let us check as usual the first-order condition to find the maximum for ℋ:

$\frac{\partial \mathcal{H}}{\partial u}=\frac{4y}{4yu}-\lambda 4y=\frac{1}{u}-\lambda 4y=0\Rightarrow {\text{u}}^{\ast }\left(\text{t}\right)=\frac{1}{\lambda 4y}.$

The second derivative is:

$\frac{{\partial }^2\mathcal{H}}{\partial u^2}=-\frac{1}{u^2}<0.$

In this problem, we find that current consumption is inversely related to the total aggregate output and the costate variable.

ii.

From the equation of motion for $\lambda \left(t\right)$ we obtain:

$\frac{d\lambda }{dt}=-\frac{\partial \mathcal{H}}{\partial y}=-\left[\frac{1}{y}+4\lambda \left(1-u\right)\right].$

As $u\left(t\right)=\frac{1}{\lambda 4y}$ the above equation of motion becomes:

$\frac{d\lambda }{dt}=-\frac{\partial \mathcal{H}}{\partial y}=-\left[\frac{1}{y}+4\lambda \left(1-\frac{1}{\lambda 4y}\right)\right]=-4\lambda$

which has general solution:

${\lambda }^{\ast }\left(t\right)=k{e}^{-4t}$

iii.

Now, from the equation of motion $\dot{y}=4y-4yu$ we have the following ordinary linear differential equation to solve:

$\frac{dy}{dt}=4y-4y\frac{1}{\lambda 4y}$

which becomes:

$\frac{dy}{dt}-4y=-\frac{1}{\lambda }.$

As ${\lambda }^{\ast }\left(t\right)=k{e}^{-4t}$ we have:

$\frac{dy}{dt}-4y=-\frac{1}{k}{e}^{4t}$

which is a first-order differential equation with general solution:

$y\left(t\right)=e^{4\int dt}\left[c-\frac{1}{k}\int \left(e^{-4\int dt}{e}^{4t}\right)dt\right]$

$y\left(t\right)=e^{4t}\left[c-\frac{1}{k}\int dt\right]=e^{4t}\left[c-\frac{t}{k}\right]=c{e}^{4t}-t\frac{e^{4t}}{k}.$

iv.

From the boundary conditions y(0)   =   1, y(1)   = e ² we can now determine the two constants of integration as:

$y\left(0\right)=1\Rightarrow c=1$

$y\left(1\right)=e^2\Rightarrow e^4-\frac{e^4}{k}=e^2\Rightarrow k=\frac{e^4}{e^4-e^2}\cong \mathrm{1.1565.}$

The optimal y ^∗(t) is determined then as follows:

$y^{\ast }\left(t\right)=e^{4t}-t\frac{e^{4t}}{1.1565}$

while the costate variable is:

${\lambda }^{\ast }\left(t\right)=1.1565e^{-4t}.$

We finally obtain u ^∗(t) as:

${\text{u}}^{\ast }\left(\text{t}\right)=\frac{1}{\lambda 4y}=\frac{e^{4t}}{4k\left(e^{4t}-\frac{t}{k}{e}^{4t}\right)}=\frac{1}{\left(4·1.1565-4t\right)}=\frac{1}{\left(4.626-4t\right)}.$

The Excel worksheet for this model is built as follows (Fig. 10.5-1):

Figure 10.5-1. Consumption model Excel worksheet setup.

The Solver Parameters window is set up as in Fig. 10.5-2. We maximize Cell K24 by changing $CellH2=\lambda \left(0\right)$ subject to the boundary condition y(1)   = e ². Table 10.5-1 is the final computed numerical solution for the variables involved in the problem.

Figure 10.5-2. Consumption model Solver parameters.

Table 10.5-1. Consumption model complete numerical solution.

It is worthwhile to notice that for completing the Excel numerical solution is enough for us to identify the theoretical Hamiltonian function and its maximum point ${\text{u}}^{\ast }\left(\text{t}\right)=\frac{1}{\lambda 4y}$ which has been inserted in Column E. The Excel Solver will compute then the solutions for the state and control variable, passing through the costate variable (Figs.10.5-3 and 10.5-4).

Figure 10.5-3. Consumption model exact versus numerical aggregate output y∗(t).

Figure 10.5-4. Consumption model approximate numerical versus exact solutions for u∗(t) and λ∗(t).

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RESOURCES

H.J. Albers , in Encyclopedia of Energy, Natural Resource, and Environmental Economics, 2013

Future Directions for Spatial Resource Management

As mentioned earlier, solving resource management models that address both dynamic and spatial characteristics poses significant challenges but is critically important to improving resource management. The relatively small body of literature that combines spatial and dynamic optimization finds that the resulting policy suggestions and resource outcomes rely on the interactions between space and time; purely spatial models and purely dynamic models generate different management prescriptions than those formed by integrated spatial–dynamic models. Because so many natural resources require spatial and temporal management, further development of spatial–dynamic optimization frameworks and solution methods will enhance the efficiency of resource management.

As spatial aspects of resource management come to the fore, policies and institutions for resource management may need to evolve in order to address these issues. For example, conservation's policy tool of establishing permanent parks and reserves fits within current land ownership or property rights regimes in many countries but may not adequately address issues of changes over space in time in the conserved resources. More time and space flexible designations of conservation areas could, potentially, better serve conservation goals but would require different property rights institutions than are currently common. Similarly, spatial considerations often lead to policy suggestions that are heterogeneous across seemingly similar land and landowners, which might also be difficult to implement given current institutions. Research into policies that provide differential incentives or that target specific segments of a landscape should prove particularly useful in guiding a landscape of landowners whose uncoordinated actions add up to landscape management. Interplay between institutional economics and spatial natural resource economics could prove useful in moving from optimization models to implementing spatial policy.

Although access to spatial data and GIS has revolutionalized empirical spatial resource economics, scale gaps and information gaps inhibit some potentially useful analysis. In terms of scale, often spatial data for one aspect of the system, such as forest cover, exist at a different scale from the spatial data for another part of the system, such as income. A lack of correspondence between the spatial data limits the power of the analysis and can contribute to the choice of scale for the problem. Similarly, socioeconomic data may correspond to political or institutional boundaries such as counties while the relevant biological data correspond to ecological boundaries such as watersheds. In terms of spatial information itself, resource management can be constrained by a lack of information to describe spatial processes and interactions in both the socioeconomic and ecological/resource realms of the problem. Together, these information and data scale issues form constraints on current work that could be assuaged by future data collection efforts across disciplinary lines. However, in some cases, spatial data and empirical tools for analyzing those data have outstripped the development of models of human behavior that reflect spatial decisions beyond simple distance decisions. The future of spatial resource management requires advances in the models of both human and ecological behavior at relevant spatial scales in addition to augmenting spatial data and information.

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Univariate and multivariate calculus

Giovanni Romeo , in Elements of Numerical Mathematical Economics with Excel, 2020

Abstract

This chapter introduces the essential techniques for the numerical standard calculus that will be used throughout the whole book. It represents therefore the basis, to understand the way the static optimization in Part II, the dynamic modeling, as well as the dynamic optimization in Part III have been applied with Excel. This chapter will try to execute and grasp with Excel the geometric interpretation of the differentiation and integration, so that the way calculus is applied to economics will be somehow easier to understand. This chapter has been divided into numerical differentiation, partial differentiation, and numerical integration. The last paragraph describes some applications to economics, even though the whole book will be covering examples where the univariate and multivariate calculus is applied. The univariate and multivariate calculus has a vast range of applications in economics and essentially the whole book is going to apply, one way or another, all that has been discussed within this chapter. Chapter 4, dedicated to the differential equations, uses the similar techniques for the numerical integration. We will see again the differential calculus to find the extreme points of a function, within Chapter 5, dedicated to the optimization techniques for univariate and multivariate functions. Chapter 6 will show all these concepts applied to the theory of microeconomics. In Part II of the book, within the calculus of variations and the optimal control theory, we will always resort to numerical differentials and numerical integration, in a context of finding an optimizing function, also called the optimal path or optimal trajectory.

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Mathematics for dynamic economic models

Giovanni Romeo , in Elements of Numerical Mathematical Economics with Excel, 2020

Abstract

Mathematics for economic dynamic models is a chapter of key importance for the whole book, and it concludes the section dedicated to the fundamental mathematics for the economic analysis. Within the dynamic analysis the dimension of time is added, and the dependent variables are analyzed considering how they evolve over time and whether they converge to a stationary value (the equilibrium value) or not, as time increases. Both the differential equations and the difference equations are used, depending if we work in a continuous or discrete framework, together with the systems. This chapter is a prerequisite to understand the dynamic optimization section, to which this chapter is strictly related. Here the problem is to find the general time path solution, while in the dynamic optimization the objective is also to understand whether the time path optimizes a given performance measure (i.e., the functional) or not. The chapter will begin showing two important methods for solving numerically, in Excel, the differential equations (i.e., Euler and Runge-Kutta), and it will continue showing some economic models that use the ordinary, first-order differential equations, which represent the main type of equations used in the book. Then, the chapter will continue with the difference equations and the way the phase diagrams are applied in Excel. An important model of microeconomics, the cobweb model and a basic Keynesian model are taken as applications of the difference equations. The chapter will also cover the systems of linear differential equations, which are solved resorting both to the eigenvalue problem and to the Euler's method. The phase diagrams for the systems of differential equations will be also shown. Some economic applications (e.g., the modified model for the Walrasian price adjustment) of the systems of differential equations will be developed as well at the end of the chapter.

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Portfolio Optimization and Management of Default Risk

Morton Glantz , Johnathan Mun , in Credit Engineering for Bankers (Second Edition), 2011

Procedure

1.

Start Excel and open the example file Risk Simulator | Example Models | 11 Optimization Continuous.

2.

Start a new profile with Risk Simulator | New Profile (or click on the New Profile icon) and give it a name.

3.

The first step in optimization is to set the decision variables. Select cell E6, set the first decision variable (Risk Simulator | Optimization | Set Decision) or click on the D icon. Then click on the Link icon to select the name cell (B6), as well as the lower bound and upper bound values at cells F6 and G6. Then, using Risk Simulator Copy, copy this cell E6 decision variable and paste the decision variable to the remaining cells in E7 to E15.

4.

The second step in optimization is to set the constraint. There is only one constraint here, which is that the total allocation in the portfolio must sum to 100%. So click on Risk Simulator | Optimization | Constraints…, or click on the C icon, and select ADD to add a new constraint. Then, select the cell E17 and make it equal (=) to 100%. Click when done.

•

Exercise Question: Would you get the same results if you set E7 = 1 instead of 100%?

•

Exercise Question: In the constraints user interface, what does the Efficient Frontier button mean and how does it work?

5.

The final step in optimization is to set the objective function. Select cell C18 and click on Risk Simulator | Optimization | Set Objective or click on the O icon. Check to make sure the objective cell is set for C18 and select Maximize.

6.

Start the optimization by going to Risk Simulator | Optimization | Run Optimization or click on the Run Optimization icon and select the optimization of choice (Static Optimization, Dynamic Optimization, or Stochastic Optimization). To get started, select Static Optimization. You can now review the objective, decision variables, and constraints in each tab if required, or click OK to run the static optimization.

•

Exercise Question: In the Run Optimization user interface, click on the Statistics tab and you see that there is nothing there. Why?

7.

Once the optimization is complete, you may select Revert to revert back to the original values of the decision variables as well as the objective, or select Replace to apply the optimized decision variables. Typically, Replace is chosen after the optimization is done. Then review the Results Interpretation section below before proceeding to the next step in the exercise.

8.

Now reset the decision variables by typing 10% back into all cells from E6 to E15. Then, select cell C6 and Risk Simulator | Set Input Assumption and use the default Normal distribution and the default parameters. This is only an example run and we really do not need to spend time to set proper distributions. Repeat setting the normal assumptions for cells C7 to C15.

•

Exercise Questions: Should you or should you not copy the first assumption in cellC6 and then copy and paste the parameters to cells C7:C15? And if you copy and paste the assumptions, what is the difference between using Risk Simulator Copy and Paste functions as opposed to using Excel's copy and paste function? What happens when you first hit Escape before applying Risk Simulator Paste?

•

Exercise Question: Why do we need to enter 10% back into the cells?

9.

Now run the optimization Risk Simulator | Optimization | Run Optimization, and this time select Dynamic Optimization in the Method tab. When completed, click on Revert to go back to the original 10% decision variables.

•

Exercise Question: What was the difference between the static optimization run in Step 6 above and dynamic optimization?

10.

Now run the optimization Risk Simulator | Optimization | Run Optimization a third time, but this time, select Stochastic Optimization in the Method tab. Then notice several things:

•

First, click on the Statistics tab and see that this tab is now populated. Why is this the case and how do you use this statistics tab?

•

Second, click on the Advanced button and select the checkbox Run Super Speed Simulation. Then click OK to run the optimization. What do you see? How is Super Speed integrated into stochastic optimization?

11.

Access the advanced options by going to Risk Simulator | Optimization | Run Optimization and clicking the Advanced button. Spend some time trying to understand what each element means and how it is pertinent to optimization.

12.

After running the stochastic optimization, a report is created. Spend some time reviewing the report and try to understand what it means, as well as reviewing the forecast charts generated for each decision variable.

Figure 15E.B shows the screen shots of the procedural steps just given. You can add simulation assumptions on the model's returns and risk (columns C and D) and apply the dynamic optimization and stochastic optimization for additional practice.

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