1 Introduction and Motivation Time and uncertainty are central elements in nance theory. Pricing theory, ecient market theory, portfolio selection the

Transcription

1 Stochastic Programming Tutorial for Financial Decision Making The Saddle Property of Optimal Prots Karl Frauendorfer Institute of Operations Research, University of St. Gallen Holzstr. 15, 9010 St. Gallen, Switzerland February 1996 Summary: The complexity of the interaction between time and uncertainty made nance models to one of the most important applications of probability theory and optimization theory. Stochastic programming combines those two elds with the intention to design methodologies for planning under uncertainty. This tutorial consists of two parts, written for practitioners, in particular nancial decision makers. It is to provide insights into the basic ideas of stochastic programming in an easily understandable way. This paper reveals various decision structures of investors and evaluates the prots achieved by admissible decisions. Criteria are presented which help identify the optimality of admissible decisions. Further, the optimal prot function is introduced to measure the sensitivity of optimal portfolios with respect to changes in income and term structure. In particular, the saddle property of the optimal prot function is veried with respect to dierent income and interest rate scenarios. This part concludes with a discussion on the stability of the optimal decisions and the usage of sensitivity results for analyzing the stochastic data with respect to the underlying investor's decision structure. Research of this report was supported by Schweizerischer Nationalfonds Grant Nr '575.93

2 1 Introduction and Motivation Time and uncertainty are central elements in nance theory. Pricing theory, ecient market theory, portfolio selection theory, and risk analysis provide sophisticated analytical tools for analysis on interaction eects between time and uncertainty. The complexity of this interaction made nance models one of the most important applications of optimization theory and probability theory. Remark 1.1: Practitioners usually refer to their various plans and proposed schedules as programs. This was the motivitation for the mathematician G.B. Dantzig to introduce the notion programming in The terms linear programming and mathematical programming were introduced by the economists T.C. Koopmans in 1948 and R. Dorfman in Nowadays, mathematical programming has become almost a synonym for optimization theory. Below, we shall use mathematical programming instead of optimization theory, as stochastic programming has arisen out of mathematical programmingin a natural way. (For more details on the history of mathematicalprogramming, we refer to Lenstra et al [13].) The eld of mathematical programming was initiated mainly by G. B. Dantzig in the late 40s. Dantzig focused on the ability to state objectives and then to be able to nd optimal policy solutions to practical decision problems of great complexity. Additionally, he considered the relations between the set of items being consumed or produced and the set of associated activities or production processes, which amounted to the incorporation of constraints in decision problems. Those objectives and constraints have been thought to replace ground rules, which are usually issued by experienced leaders or board members to guide the selection and planning process. Undoubtedly, Dantzig's fundamental ideas provided the basis for Markowitz's Mean- Variance Model in the early 50s. In particular, the mean-variance model may be seen as a parametric, quadratic mathematical program, whose optimal solutions represent the ecient frontier. The mean-variance model and its equilibrium version, the Sharpe-Lintner Capital Asset Pricing Model (CAPM) became one of the most important capital market models. These models are based on assumptions like uniform planning horizon for all investors, homogeneous expectations, exclusion of transaction costs, and no restrictions on short sales. In particular, the special structure of the ecient frontier, which would be destroyed if additional constraints like exclusion of short sales are imposed, provided the basis for the CAPM (see e.g., Markowitz 1987 [14], Muller 1988 [16]). These one-period models have been extended to dynamical intertemporal models (see, e.g., Ingersoll 1987 [11], Merton 1990 [15]), which led to identical portfolio rules. In the nance literature, discretetime intertemporal portfolio selection or continuous time portfolio selection are modeled as dynamic stochastic control problems over discrete or continuous time incorporating a budget equality. In those control-type models, the aim usually is to nd a stationary optimal policy, implementable for all periods of the planning horizon as a function of the current state, which is commonly represented as budget or wealth. Such stationary policy could be of the form: In each period, invest 25% of your current budget in security A, 50% in security B, and the remaining 25% in security C. The optimal policy is derived by the dynamic programming technique, 2

3 which relies on nding closed form expressions for the solution of Bellman's equation. In case the objective function is of a special analytical form, a substitution of the budget equation into the value function preserves dierentiability, and the Bellman's equation often yields an policy in closed form through dierentiation. The stochastic control problem is solved once for the entire planning horizon, yielding the optimal policy for all periods with respect to the underlying stochastic dynamics. It is NOT in the concept of those models to perform dynamical planning in dependence of information or observations becoming available in each period. Let's come back to Dantzig's approach from the early 50s, to what has been called Mathematical Programming. According to [3], Dantzig was motivated to generalize the steadystate Input-Output Model of W. Leontief to a dynamic model, one that could change over time. He had in mind planning dynamically over time, particularly planning under uncertainty. This way, stochastic programming began to emerge as one important part of mathematical programming. Contrary to stochastic control problems, stochastic programs are solved once per period, taking into account periodically updated forecasts of the dynamics (stochastic processes) with respect to the future periods. It is today's optimal policy, which is of importance, adopted with respect to the current stochastic dynamics of prices, returns, cash-ows, etc., and but also with respect to optimal policies in future periods, which in turn are adopted with respect to new information on stochastic dynamics. The main idea behind this procedure, which be decribed in more detail below, is to analyze the stochastic processes tuned to the investor's decision structure. It is this dynamic planning mechanism that characterizes stochastic programming; it has received increasing attention in nance in the U.S. and in Great Britain due to the successful and valuable contributions of W.T. Ziemba 1993 [1], M. Dempster 1994 [4], J. Mulvey 1994 [17], J.R-B. Wets 1994 [18], S. Zenios 1993 [19,20], and D'Ecclesia and S. Zenios 1994 [6] (see also Dupacova 1991 [5] and references therein). For a recent textbook in stochastic programming we refer to P. Kall and S. Wallace 1994 [12]. The concept of stochastic programming is illustrated as methodology applied to a small portfolio problem. The basic steps of the stochastic programming technique, are outlined: Herein section 2 reveals the various decision structures of an investor. Section 3 values admissible decisions with respect to the prot function given through the current term structure. Section 4 presents criteria which help identify the optimality of admissible portfolios from two viewpoints. Further, in section 5, the optimal prot function is introduced to measure the sensitivity of the optimal portfolios with respect to changes in income and in the term structure. In particular, in section 6, the saddle property of the optimal prot function is veried with respect to dierent income and interest rate scenarios. Section 7 concludes with a demonstration on the role of today's term structure and its steepness. 3

4 2 Admissible Portfolios (Investor's Decision Structures) Suppose that in each year, indexed by t = 0; 1; ; T, an investor receives a stochastic income, denoted t. He wants to place some or all of his income in bonds with a maturity of 1 and 5 years, and to maximize his prot with respect to a planning horizon of, say, 10 years. Let the associated interest rates be denoted 1Y;t ; 5Y;t : The amounts invested in 1-year bonds and 5-years bonds at the beginning of period t are denoted u 1y;t and u 5y;t. Clearly, the stochastic data t ; 1Y;t ; and 5Y;t represent stochastic processes which strongly correlate. Their dynamics are characterized by nance experts through parameters like drift and volatility which are estimated. Let's choose a period t arbitrarily and keep that t xed throughout this section. The income is t > 0, and the investor decides on how much to invest in 1-year bonds and 5-year bonds. Below, we illustrate various types of decision structures common in nancial decision making and introduce notions that should ease comprehension. The motivation is to reveal potential 'ground rules' subject to which portfolios are accepted admissible by the investor. Remark 2.1: We stress that the capital to invest depends not only on the income t, but also on what matures. In particular, for our simple example, the capital available at t equals t + u 1y;t?1 + u 5y;t?5 ; i.e., the capital equals the income plus the amount that matures. For the ease of exposition, we focus only on income t in this section, pretending that nothing matures (u 1y;t?1 = u 5y;t?5 = 0). The investor may require to place at most t in those two bonds without allowing short selling. In this case his admissible set of portfolios (u 1y;t ; u 5y;t ) is illustrated in Figure 2.1 for t = 100. Formally, it is dened D 1;t ( t ) := f(u 1y;t ; u 5y;t ) j u 1y;t + u 5y;t t ; u 1y;t 0; u 5y;t 0g: Figure 2.1 Figure 2.2 Figure 2.3 If unlimited short-selling is allowed, then nonnegativity conditions have to be omitted. The corresponding set of admissible portfolios is limited by income only, is illustrated in Figure 2.2 for t = 100, and is expressed by D 2;t ( t ) := f(u 1y;t ; u 5y;t ) j u 1y;t + u 5y;t t g 4

5 For the remaining cases, let short-selling be forbidden without loss of generality. All the gures refer to t = 100: In case the investor wants to invest exactly t in those two bonds, then his decision structure (Figure 2.3) reads as D 3;t ( t ) := f(u 1y;t ; u 5y;t ) j u 1y;t + u 5y;t = t ; u 1y;t 0; u 5y;t 0g: The investor may have a xed policy; say, he invests 40% in 1-year bonds and 60% in 5-year bonds. Then, his set of admissible portfolios (Figure 2.4) reduces to a single portfolio, namely to D 4;t ( t ) := f(u 1y;t ; u 5y;t ) j u 1y;t = 0:4 t ; u 5y;t = 0:6 t g: Figure 2.4 Figure 2.5 Less restrictive, one may place limits to enforce diversication. Requiring to invest at most t in both bonds and, additionally, at most 70 units in each of those two bonds, the set of admissible policies (Figure 2.5) is D 5;t ( t ) := f(u 1y;t ; u 5y;t ) j u 1y;t + u 5y;t t ; 70 u 1y;t 0; 70 u 5y;t 0g Summarizing the above, D 1;t ; D 2;t ; D 3;t ; D 4;t, and D 5;t represent the various sets of admissible portfolios. These sets are illustrated for a xed value t = 100. As t is stochastic and, therefore, unknown prior to period t, it is reasonable to study in period t? 1, how those sets change with varying t. Concentrating on the sets D 1;t and D 5;t, Figures 2.6 and 2.7 illustrate, how those sets look like for t = 50, t = 100, and t = 150. Studying how the other sets change is an easy task and is left to the reader. We realize that in Figure the set of admissible portfolios remains unchanged if the upper limits on each of those two bonds are not taken into account. In this case, it is said that the upper limits in D 5;t become redundant for t = 50 (Figure 2.7.1). Analogously, income becomes redundant for t = 150 in Figure Redundancy has occured as the upper limits on each of those two bonds have been stated in absolute terms. If limits are incorporated in relative terms, like, do not invest more 5

6 Figure Figure Figure Figure Figure Figure Figure Figure Figure

7 than 70% in each of the two bonds, then the associated set of admissible portfolios is D 6;t ( t ) := f(u 1y;t ; u 5y;t )ju 1y;t + u 5y;t t ; 0:7 t u 1y;t 0; 0:7 t u 5y;t 0g Figures 2.8 illustrate the shapes of those sets for t = 50, t = 100, and t = 150. Let's accept the above sets as basic shapes of an investor's decision structure. One may interpret the boundaries of such sets as ground rules. In the above, therefore, we have incorporated ground rules, like, invest at most 70 units in each of the two bonds, invest at most t units in both bonds, or, do not allow short selling. Of course, there are 'trickier ground rules', however, those above will turn out to be sucient for understanding of that what follows in the next sections. 3 Portfolio Value We focus on how to value admissible portfolios once the 'ground rules' are dened. For ease of exposition, we do not consider discounting or compounding eects, as these are only of technical nature and can be incorporated into the methodology easily once the entire concept is understood. In this section, we focus on the investor's objective, depending on which of the portfolios are valued. It is supposed that the investor's objective is prot he would like to maximize with respect to two securities, i.e., with respect to 1-year bonds and 5-year bonds. The linear case We start with the simplest objective function, a deterministic linear one. Suppose that our objective is to maximize a linear prot function of the following form f(u 1y;t ; u 5y;t ) = 1y;t u 1y;t + 5y;t u 5y;t : Let the interest rates of the 1-year bond and the 5-year bond be given at 4% and 6%, respectively. They are deterministic at this stage: 1y;t = 0:04, 5y;t = 0:06. The objective function f(u 1y;t ; u 5y;t ) = 0:04 u 1y;t + 0:06 u 5y;t is illustrated in the three-dimensional Figure 3.1. The three straight lines on the function indicate constant prot with respect to those decisions which lie on the associated three lines in the decision space (in the space spanned by the u 1y;t -axis and by the u 5y;t -axis). Those three lines in the decision space are denoted level sets, as they characterize those portfolios with the same prot. It is an obvious fact, that in our two-dimensional portfolio problem the level sets of a linear prot function are straight lines, say, they are linear. Furthermore, the slopes of the level sets make apparent, that the the 5-years bond yields more than the 1-year bond. 7

8 Profit 5Y Figure 3.1 1Y Concentrating on the 'ground rules' given by D 5;t for t = 100 in section 2, we are interested in visualizing those level sets, which are associated with prots 1:8, 2:4, 3:0, 3:6, 4:2, 4:8, 5:4, 6:0, and 6:6. This is shown in Figure 3.2. It is obvious that the level sets move to north-east with increasing prot. Note, that the term level sets refers to those portfolios which have the same prot, no matter whether those portfolios are admissible or not. For the investor, it should be clear to choose, nally, only among admissible ones. Seeking the admissible portfolio with maximum prot, the investor will choose A and will earn a prot of 5.4 units. This represents the optimal decision. In the next section, we refer to how optimality is characterized and veried. The associated level set covers exactly one portfolio within the set of admissible portfolios D 5;t. Level sets which refer to a higher prot have no portfolios in common with the admissible ones. Figure 3.2 Figure 3.3 There is one direction which is a distinguished one and which we shall need later. It is the direction orthogonal to the level sets which is called gradient. Due to the fact that the level sets of a linear function are linear and move parallel, the gradient remains unchanged, no matter from which portfolio we start (see Figure 3.3). 8

9 Remark 3.1: We stress that throughout this work we pay no attention to the length of the gradient as only the direction of the gradient is of importance. The practical meaning of the gradient is the following: Suppose that currently the investor stays with portfolio B in Figure 3.4. The gradient shows the investor the direction, which he should move his portfolio towards. This direction provides the investor with the largest rate of increase per one unit of move. (This is not surprising, as the shortest distance between two parallel lines, in our case level sets, is given by the length of that straight line which is orthogonal to the two parallel ones.) Obviously, any direction which forms an acute angle (i.e., an angle of less than 90 ) with the gradient improves the performance of the portfolio and is called ascending direction (Figure 3.4.2). Directions opposite to ascending directions are called descending directions (Figure 3.4.3). Clearly, such directions form an obtuse angle (i.e., an angle of more than 90 ) with the gradient. This classication of directions is important for realizing whether the performance of a portfolio is improving or deteriorating. Figure Figure Figure Remark 3.2: Obviously, the gradient represents an ascending direction. More precisely, it is the direction with the steepest ascent due to the above observation. Remark 3.3: Note that the set of admissible decisions D 5;t is dened by level sets L 1 ; L 2 ; L 3 ; L 4 ; L 5 (see Figure 3.5). To verify this, recall the denition of D 5;t in section 2: D 5;t ( t ) := f(u 1y;t ; u 5y;t ) j u 1y;t + u 5y;t = t ; 70 u 1y;t 0; 70 u 5y;t 0g Let's dene the following linear functions: f 1 (u 1y;t ; u 5y;t ) := u 1y;t + u 5y;t ; f 2 (u 1y;t ; u 5y;t ) := u 1y;t ; f 3 (u 1y;t ; u 5y;t ) := u 5y;t : f 1 gives the total amount invested in both bonds, f 2 the total amount invested in 1-year bonds, and f 3 the total amount invested in 5-year bonds. The corresponding 9

10 level sets read then as L 1 := f(u 1y;t ; u 5y;t )ju 1y;t + u 5y;t = 100g L 2 := f(u 1y;t ; u 5y;t )ju 1y;t = 70g L 3 := f(u 1y;t ; u 5y;t )ju 1y;t = 0g L 4 := f(u 1y;t ; u 5y;t )ju 5y;t = 0g L 5 := f(u 1y;t ; u 5y;t )ju 5y;t = 70g: Figure 3.5 The nonlinear case Now, let's pass over to a nonlinear deterministic prot function. In Figure 3.6, such a nonlinear prot function is illustrated; again, the curves (drawn on the surface horizontally) indicate constant prots, the corresponding curves in the decision space (in the space spanned by the u 1y;t -axis and u 5y;t -axis) are the associated level sets, which are now nonlinear. Profit 5Y 1Y Figure 3.6 Incorporating the 'ground rules', one observes that those portfolios with the same prot lie on nonlinear level sets, now (gure 3.7.1). For characterizing and verifying 10

11 optimality with respect to nonlinear objective functions, we refer to section 4. At this stage, we only point out that if level sets are nonlinear, the gradients change with the portfolio one refers to. The gradient, now, is the direction which is orthogonal to the line tangent in the specic point of the level set. This is shown in Figure 3.6, where dierent points have been chosen among dierent level sets. Figure Figure Optimality Criteria After introducing potential decision structures and concepts for valuing portfolios in the deterministic case, we present criteria for identifying optimality of admissible portfolios. Such criteria are called optimality criteria in literature. Below, we will focus on D 5;t. In a rst phase, we start with t = 100 and a normal term structure, where 5-years bonds yield 6% and 1-year bonds yield 4%. We state two optimality criteria, a socalled primal one and a dual one. Remark 4.1: In this deterministic case, clearly, the optimal portfolio is to invest the maximum allowable amount, namely 70, in 5-years bond and the remaining 30 in 1-year bonds. This portfolio, as already pointed out in section 3, is (u 1y;t ; u 5y;t ) = (30; 70) and yields a prot of 5:4. For ease of understanding, we consider three dierent portfolios A,B,C, which the stated optimality criteria are applied to. Optimality criterion from a primal viewpoint In order to identify optimality from the primal viewpoint, we need the set of admissible decisions, one admissible decision called candidate for optimality and its corresponding prot, the level set associated with that prot, and the gradient of the prot function at that candidate. We consider portfolio A with (u 1y;t ; u 5y;t ) = (30; 70), portfolio B with (u 1y;t ; u 5y;t ) = (70; 30), and portfolio C with (u 1y;t ; u 5y;t ) = (30; 30) and refer to Figures 4.1, 4.2, and 4.3, respectively. 11

12 We realize that portfolio A lies on the boundary of the set of admissible decisions D 5;t (100), as illustrated in Figures 4.1. Some of the ground rules are satised at their limits. In particular, the ground rules to invest at most 70 in 5-year bonds and to invest at most 100 in both bonds are satised at the limits. On the contrary, the ground rules to invest at most 70 in 1-year bonds and do not allow short-selling are satised within their limits but NOT at the limits. If a ground rule is satised at the limit, then we shall say that this ground rule is active; if a ground rule is satised within its limit but NOT at the limit, then we shall say that the ground rule is inactive. Figure Figure The prot of portfolio A is 5:4; the associated level set is represented by the dashed line in Figure 4.2 and the gradient by the dashed arrow pointed to 'north-east'. Now, the question of interest is: Figure 4.2 Is there any direction, which the portfolio may be moved into, where the ground rules are still satised and the prot is increased? The answer is: NO. Any direction from point A, which provides an admissible portfolio, is a descending direction. Admissible directions (some point to 'southwest', to 'south', and to 'south-east' in Figure 4.1.1) form an obtuse angle with the 12

13 gradient, and, hence, are descending directions. In other words, one may argue that any ascending direction from point A, (pointing to 'north' and to 'north-east' in Figure 4.1.2) provides a portfolio which violates at least one of the ground rules. This allows to conclude that portfolio A is optimal with respect to the given decision structure D 5;t (100) and with respect to the given normal term structure. Portfolio B lies on the boundary of the set of admissible portfolios D 5;t (100), illustrated in Figures 4.2. The ground rules to invest at most 70 in 1-year bonds and to invest at most 100 in both bonds are active; contrary to that, the ground rules to invest at most 70 in 1-year bonds and to do not allow short-selling are inactive. The prot of portfolio B is 4:6; the associated level set is represented by the dashed line in Figure 4.2, and the gradient by the arrow pointed to 'north-east.' Now again, the question of interest is: Is there any direction, the portfolio can be moved into, where the ground rules are still satised and the prot is increased? The answer is: YES. Directions which point into the shaded area from point B (Figure 4.2), provide admissible portfolios with an increased prot, as those directions form an acute angle with the gradient. This allows to conclude that portfolio B is NOT optimal, modications of the portfolio with regard to an admissible ascending direction are possible and recommendable. Portfolio C lies in the interior of the set of admissible decisions (Figure 4.3). No ground rules are active. The above argumentation may be applied analogously. It remains to be stressed that, in this case, contrary to the above, modications even in the direction which coincide with the gradient provide admissible portfolios (Figure 4.3). Clearly, C is NOT optimal. Figure 4.3 Summarizing, we may formulate the optimality criterion from the primal viewpoint as follows: The underlying portfolio is optimal exactly if there is no admissible ascending direction from that portfolio. 13

14 Optimality criterion from a dual viewpoint For identifying optimality from the dual viewpoint, we need the set of admissible decisions, one admissible decision, again called candidate for optimality, the gradient of the prot function at that candidate, and the gradients of the ground rules, which are active at that candidate. For portfolio A, the ground rules to invest at most 70 in 5-year bonds and to invest at most 100 in both bonds are active. The corresponding gradients are orthogonal to the level sets L 1 and L 5 (drawn in Figure 4.4 as solid arrows). The shaded area within these two gradients is called normal cone; in other words, the normal cone is spanned by the gradients of the active ground rules. The normal cone is of utmost importance as soon as the term structure changes. This becomes apparent below and in Part II of this tutorial note. Figure 4.4 The optimality criterion from the dual viewpoint may be stated as follows: The underlying portfolio is optimal exactly if the gradient of the prot function (evaluated at the underlying portfolio) lies in the normal cone, which is spanned by the gradients of the active ground rules. Applying this dual optimality criterion to portfolio A, one may conclude that A is optimal (Figure 4.4). Portfolio B is not optimal, as the gradient of the prot function (evaluated at the underlying portfolio) lies outside the normal cone, which is spanned by the gradients of the active ground rules (Figure 4.5). For portfolio C, the situation is more subtle, as no ground rules are active at C. In this case, the normal cone reduces to a set with only one single element, namely the zero-vector. As the gradient of the prot function is NOT equal to the zero-vector at C and, hence, outside of that special normal cone, C is also NOT optimal (Figure 4.6). The following might be an interpretation of the dual optimality criterion: Portfolio A incorporates the ground rules of the decision maker optimally, Portfolios B and C do NOT incorporate those ground rules optimally. 14

15 Figure 4.5 Figure 4.6 Optimality criteria in the nonlinear case For a broad class of practical problems, the primal and dual optimality criteria stated above are still valid if the prot function and/or the ground rules are of nonlinear type. In this case, one may still use the gradients in the above mentioned way. Figures 4.7 illustrate the optimality criterion from a dual viewpoint for two dierent nonlinear prot functions. In Figure 4.7.1, the gradient, which is orthogonal to the tangent line of the level set through A, lies in the normal cone spanned by the two gradients of the two active ground rules and indicates that portfolio A is optimal for the underlying rst case. In Figure 4.7.2, the gradient of the prot function in B coincides with the gradient of the solely active ground rule in B; note that the latter denes the normal cone and, hence, B is optimal for the underlying second case. Figure Figure However, one should be aware of the fact that if either objective function or ground rules are nonlinear, the gradients will change with the portfolios they refer to. This implies a major diculty, in addition to the inherent stochasticity, which will be discussed later, and which practitioners and scientists face. 15

16 5 Optimal Portfolios and Optimal Prot Functions In this section, we investigate how the optimal portfolio and its prot change if income t decreases or increases and if the term structure moves from normal to inverse with 5-years bonds yielding 4% and 1-year bonds yielding 6%. This is of particular importance for analyzing the stochastic data with respect to admissible portfolios, which is done in Part II. Income changes Let's keep the current term structure xed and suppose that income t varies between 50 and 150 in period t; formally, we allow 50 t 150. Of course, the limits 50 and 150 might depend on period t, but this is only of technical nature. In Figures 5.1, the optimal portfolios A 1 ; A 2 ; ; A 11 are illustrated for the 11 income scenarios t = 50; 60; 70; 80; 90; 100; 110; 120; 130; 140; 150 with respect to the corresponding sets of admissible portfolios and the given term structure. Optimality of those portfolios is veried easily by the gradient of the prot function (drawn as solid arrow) and the normal cone (drawn as shaded area). The associated optimal prots (in absolute terms) are listed with respect to the given term structure in Table 5.1. The function which gives the optimal prot achievable for any income t 2 [50; 150] is called optimal prot function with respect to income and is illustrated in Figure 5.2. Observe that, according to Table 5.1 and Figure 5.2, the optimal prot increases with a rate of 0:06 for each additional income unit as long as the income is less than 70. For income t 2 [70; 140) the optimal prot increases at a rate of 0:04 for each additional income unit. Finally, optimal prot stays constant if income exceeds 140; hence, the rate of prot increase is 0:0 for any additional unit of income. The values 0:06; 0:04, and 0:0 are called marginal rates of return (with respect to income) of the corresponding optimal prot function. Further, these marginal rates of return represent the slopes of the optimal prot function (Figure 5.2); They are decreasing with increasing income. This property holds for a very broad class of practical problems and will turn out to be of great advantage. Interest Income at t rates at t Table 5.1 Term structure changes Now, let's keep the income t = 100 xed and suppose that the term structure changes from normal to inverse. Formally, for ease of exposition, we allow that the interest rate of 1-year bonds and the interest rate of 5-year bonds change with the same amount but opposite in sign. In particular, it is assumed that if the 1-year rate 16

17 Figure Figure Figure Bild Figure Figure Figure Figure Figure Figure Figure

18 profit Figure 5.2 income increases by 0.2% then the 5-year rate decreases by 0.2%. This way, we consider the following 11 interest rate scenarios, denoted (a),(b),, (k) (Table 5.2). Interest rate Income at t scenarios at t 100 a b c d e f g h i j k Table 5.2 Of course, interest rate scenarios might be chosen in a much more general form; however, as in the above cases, this is only of technical nature. Each of the interest rate scenarios denes an associated prot function, which is to maximize with respect to the set of admissible portfolios, given t = 100. In Figures 5.3, the optimal portfolios are illustrated: Portfolio A is optimal for interest rate scenarios (a)-(f); portfolio B is optimal for interest rate scenarios (f)-(k). This is easily veried with the dual optimality criterion, which states that the portfolio is optimal exactly if the gradients (drawn as arrows) of the prot functions are in the normal cone of the active ground rules. Remark 5.1: Observe, that portfolios A and B and all portfolios in between are optimal with respect to interest rate scenario (f). This is due to the fact that the gradient of the associated prot function coincides with the gradient of the active 18

19 Figure Figure ground rule to invest at most your income, which spans the normal cone at A and B. The associated (optimal) prots are listed with respect to the given income t = 100 in Table 5.2. The function which gives the optimal prot to be achieved for any interest rate scenario (between (a) and (k)) is called optimal prot function with respect to interest rates and is illustrated in Figure 5.4. Observe that according to Table 5.2 and Figure 5.4, optimal prot changes at a rate of?0:08 for interest rate scenarios (a)-(f). For interest rate scenarios (f)-(k), the optimal prot changes at a rate of +0:08. The values?0:08 and +0:08 are called marginal rates of return (with respect to changes in the interest rates) of the corresponding optimal prot function. Similarly to the above, those marginal rates of return represent the slopes of the optimal prot function (Figure 5.4), and increase with changes in the underlying term structure. Again, this property holds for a very broad class of practical problems and will turn out to be of great advantage. profit a b c d e f g h i j k Figure 5.4 term structure 19

20 6 The Saddle Property of Optimal Prots As a next step, it is natural to analyze optimal prot with respect to changes in both income and term structure. To keep it simple, in Table 6.1 we list optimal prots to be achieved with respect to any combination of the above considered income and interest rate scenarios. Observe that, trivially, the rst row in Table 6.1 coincides with Table 5.1 and the column, which refers to t = 100, coincides with Table 5.2. Interest Income at t rates at t a b c d e f g h i j k Table 6.1 The function which gives the optimal prot for any combination of income and interest rate scenarios is called optimal prot function with respect to income and interest rates (Figure 6.1). 7 5 k 3 j i 150 h g f e d c 80 b 70 a Figure

21 Obviously, all the arguments of the above two subsections are valid and may be applied to any row or column of Table 6.1. Nevertheless, we should like to stress that the marginal rates of return with respect to changes in income are decreasing for any xed interest rate scenario. The associated optimal prot functions with respect to the income axis are plotted in Figure 6.2. Similarly, the marginal returns with respect to changes in the interest rates are increasing for any xed income scenario. The associated optimal prot functions with respect to the term structure axis are plotted in Figure 6.3. profit Figure 6.2 income profit income=140/150 income=130 income=120 income=110 income=100 income=90 income=80 income=70 income=60 income=50 1 a b c d e f g h i j k Figure 6.3 term structure Summarizing, we stress that the slopes (i.e., marginal rates of return) of the optimal prot functions may be positive or negative. The crucial fact is that these slopes decrease with changes in income and increase with changes in the term structure. A function like the optimal prot function in Figure 6.1 is called a saddle function. It is not straightforward to recognize Figure 6.1 as a saddle function. This is due to the fact that the slopes change very little, entailing a rather at optimal prot function. If one would draw a function whose slopes decrease drastically with changes in the income, and increase drastically with changes in the term structure, then one would get a function as illustrated in Figure 6.4. In fact, such a function very much looks like a saddle and, hence, motivates the term saddle function. 21

22 income term structure Figure Conclusions and Outlook This tutorial presented structural properties of optimal decisions and their associated optimal prots on simple (two-dimensional) portfolio problems. Once more, we stress that there is no harm if more securities and trickier ground rules are considered. For a broad class of problems, those properties have been well known for many years and today, they represent key issues within mathematical programming, being of utmost importance whenever the solvability of dicult optimization problems, deterministic or stochastic ones, is the issue. In this nal section, we stayed in the simple two-dimensional setting, summarized the outlined perception, and sketched out its role within decision making under uncertainty. General statements Potential decision structures of investors, which dene various sets of admissible portfolios, have been discussed. Such decision structures form a basis for ground rules like, do not allow short selling, invest income in a preselected set of securities, invest income according to a constant mix strategy in preselected securities, or investments in single securities must not exceed a given limit. The main motivation for investors for incorporating ground rules is to reduce risk they are exposed to, by adopting stationary policies over a certain planning horizon or setting limits according to the liquidity of the markets. However, in most classical models, like in the Markowitz approach and its dynamical extensions, ground rules are not integrated. This is due to the fact that optimal prot functions loose their dierentiability, if ground rules are incorporated, i.e., in those functions kinks are created, which renders numerical evaluations, in particular solvability and sensitivity analysis of optimal portfolios, more dicult. In Markowitz's approach, the special structure 22

23 of the ecient frontier would be destroyed if ground rules, like exclusion of short sales, are imposed. Its extension, the discrete time intertemporal portfolio selection problem is modeled as dynamic stochastic control problem over discrete time. The solvability of that problem strongly depends on the dierentiability of the optimal prot function, and, therefore, allows no ground rules to be incorporated. The stochastic control problem is solved once for the entire planning horizon, yielding the optimal policy for each period with respect to underlying dynamics. As counterparts of stochastic control-type problems there are the so called stochastic multiperiod programming problems, which focus on the incorporation of ground rules. Convexity, or more general, the saddle property, is the structural property which is exploited within stochastic programming. This opens a broad class of methodologies developed within the eld of mathematical programming. Additionally, stochastic multistage programming problems are solved once in each period, taking into account the periodically updated forecasts of the dynamics with respect to the future. This and the fact that transaction costs may be considered are in contrast to classical stochastic control type problems, that are usually used for portfolio optimization. Controlling dynamics Optimal portfolios (as illustrated in Figures 5.1 and 5.3) and optimal prots (as listed in Tables 5.1-3) are easily computable, if the interest rates ( 1y;t ; 5y;t ) of the bonds don't change within period t and if we know those rates in that period. Income t to be invested is known at the beginning of period t. At this stage, we remember that, additionally to t, we also know the maturing amounts u 1y;t?1 and u 5y;t?5. And this is responsible for the dynamics to come into play, even if income and term structure are given: Suppose that today, at t = 0, you know interest rates and income for all the periods t = 1; ; T ahead within the planning horizon; say, ( 1y;t ; 5y;t ) and t change from period to period, but their values are available for all future periods today. In the underlying two-dimensional case, our problem is to nd the sequence of admissible investment decisions u 1y;t and u 5y;t for today (t = 0) and for each period t in the future which maximize our prot. Observe that the admissibility of a decision (u 1y;t ; u 5y;t ) in period t depends on the decisions made in previous periods; in our case, the admissibility depends on decision u 1y;t?1 of period t? 1 and on decision u 5y;t?5 of period t? 5. This is the way the dynamics enter and the decisions up to period t impact the prot achievable beyond period t + 1. Clearly, if both interest rates decrease, the optimal decisions in each period are to invest admissibly (i.e., with respect to the underlying ground rules) in longterm bonds; if both interest rates increase over the entire planning horizon, the optimal decisions in each period are to invest admissibly in short-term bonds. These two cases are very obvious, and no decision maker would invoke an optimization routine for that. However, if short-term rates and/or long-term rates increased and/or decreased in some periods within the planning horizon, and if all of those rates were known today, any decision maker would certainly be grateful for an optimization tool, which would give him advice as to when to invest in long-term and when to invest in short-term securities. Computationally, due to modern computer technology, this would pose no diculty, even if thousands of securities and hundreds of ground rules incorporated and hundreds of periods were considered. 23

24 Modeling uncertainty in stochastic programming Optimization is complicated by uncertain data with respect to interest rates and income. We would like to stress, that the problem of characterizing stochastic data and the way these data evolve is one of the major issues in nancial mathematics. Therein, interest rate models, or more general, pricing models identify the stochastic evolvement through stochastic dierential equations and describe interest rates, or more general, rates of return as stochastic processes with parameters, like e.g., drift and volatility. Due to modern high-frequency data collection, such parameters can easily be estimated and the corresponding stochastic processes be determined. However, the challenge is to solve stochastic dierential equations. Solvability and goodness of the underlying model determine its applicability. Now, suppose we selected a good and solvable pricing model which characterizes stochastic data, i.e., interest rate and income through a stochastic process. This information serves as input to our portfolio problem and provides us with probabilistic statements on how interest rates and income will evolve within the planning horizon. Applied to our case, where we considered nitely many values for income and interest rate, we identify scenarios and assign probabilities to them. A small example roughly outlines the way uncertainty is taken into consideration within stochastic programming. Suppose we have to decide on how to invest today's income, 0 = 30, based on the following information: 1-year bonds and 5-year bonds are considered at today's rates ( 1y;0 = 0:04; 5y;0 = 0:06) and a planning horizonof 5 years. Let u 1y;0 denote the amount invested in 1-year bonds, and u 5y;0 the amount invested in the 5-years bonds today, at t = 0. One year from now, we might receive an income of 50,60,70, or 80 units. One of the 11 interest rate scenarios a)-k) might occur. Scenario probabilities are listed for combinations of income and interest rates in Table 7.1. One year from now, the income of next year, 1, and the amount u 1y;0, which then matures, are reinvested. Depending on the income and interest rate scenario, we are able to place total income 1 + u 1y;0 such that the prots we earn for the remaining 4 years are xed. Those yearly prots are listed in Table 7.2. We mention that the scenario probabilities shown in Table 7.1 may be seen as approximations of the future. Further, the prots listed in Table 7.2 may be due to any contracts available in the markets. However, we notice that these prots coincide with those of Table 6.1 (section 6), which are optimal with respect to the selected decision structure and which, therefore, possess the saddle property. In order to the essential steps in stochastic multistage programming, we evaluate the expected prots of four dierent strategies numerically. This helps demonstrate the importance of today's term structure and its steepness for the decision process and illustrates its impact on the optimal expected prot. 24

25 Interest Income at t = 1 rates at t = Table 7.1 Interest Total income at t = 1 rates at t = a b c d e f g h i j k Table

26 Strategy I If we decided to invest all of today's income in 5-years bonds, i.e., u 5y;0 = 30, nothing would mature next year. The prot we earn on today's income over the entire planning horizon would be 9:0. In addition to that, the expected prot we earn on next year's income of either 50,60,70, or 80 units, would be 15:2616 units totally over the remaining 4 years (see Table 7.3). Interest Total income at t = 1 rates at t = a b c d e f g h i j k Expected prot per year: Table 7.3 The total expected prot earned by this strategy is 24:2616 units over the entire planning horizon. Strategy II If we decided to invest 20 units in 5-years bond, i.e., u 5y;0 = 20, and the remaining 10 units in 1-years bond, those 10 units would mature next year, which would result in a total income of 60; 70; 80; or 90 units by the end of that year. The prot we earn on today's income over the entire planning horizon, would be 6:4. In addition to that, the expected prot we earn on next year's income, is 17:1688 units totally over the remaining 4 years (see Table 7.4). The total expected prot earned by this strategy would be 23:5688 units over the entire planning horizon. Strategy III If we decided to invest 10 units in 5-years bond, i.e., u 5y;0 = 10, and the remaining 20 units in 1-years bond, those 20 units would mature next year, which result in a total income of 70; 80; 90; or 100 units by the end of that year. The prot, we earn on today's income over the entire planning horizon, would be 3:8. 26

27 Interest Total income at t = 1 rates at t = a b c d e f g h i j k Expected prot per year: Table 7.4 Interest Total income at t = 1 rates at t = a b c d e f g h i j k Expected prot per year: Table

28 In addition to that, the expected prot we earn on next year's income would be 18:9832 units totally over the remaining 4 years (see Table 7.5). The expected prot earned by this strategy would be 22:7832 units over the entire planning horizon. Strategy IV If we decided to invest all 30 units in 1-years bond, i.e., u 1y;0 = 30, those 30 units would mature next year, which would result in a total income of 80; 90; 100 or 110 units by the end of the rst year. The prot we earn on today's income in the next year would be 1:2. In addition to that, the expected prot we earn on next year's income, would be 20:7512 units totally over the remaining 4 years (see Table 7.6). Interest Total income at t = 1 rates at t = a b c d e f g h i j k Expected prot per year: Table 7.6 The expected prot earned by this strategy would be 21:9512 over the entire planning horizon. If we had to choose one of the above strategy with respect to the current term structure ( 1y;0 = 0:04 and 5y;0 = 0:06), we would select strategy I. In order to demonstrate the importance of the current term structure and of its steepness and illustrate the associated impact on the optimal expected prots, the expected prot of strategies I-IV are listed with respect to various term structures in Table 7.7. For this small test example, it becomes apparent that diversication takes place today at t = 0 if today's term structure is not too steep. It is in the nature of stochastic multistage porgramming that major diversications take place with respect to periods of the planning horizon, which stands in opposition to stationary policies like constant mix strategies. The benets derived from solving intertemporal portfolio problems by stochastic programming are twofold: 28