Optimization of a Real Estate Portfolio with Contingent Portfolio Programming

Transcription

1 Mat Independent research projects in applied mathematics Optimization of a Real Estate Portfolio with Contingent Portfolio Programming 3 March, 2005 HELSINKI UNIVERSITY OF TECHNOLOGY System Analysis Laboratory Rasmus Ahvenniemi 51026N

2 Contents 1 Introduction The investment problem The effects of macroeconomic variables on return Interdependencies between macroeconomic variables Contingent portfolio programming The state tree The decision variables The constraints The objective function A case example The state tree The decision variables The resource constraints The objective function and the deviation constraints The results Discussion and conclusions References Appendix: The parameters used in the example problem 1

3 1 Introduction Constructing an optimal portfolio is an issue of importance for any decision maker investing a large sum of money. This study focuses on the optimization of a portfolio consisting of (i) real estate, (ii) bonds, and (iii) a market portfolio of stocks. It is evident that changes in macroeconomic variables influence the return of the different components of a portfolio. In this research it is considered how changes in (i) the economic outlook, (ii) the interest level, and (iii) the general level of rent affect the return of the components of the portfolio. The interrelationships between changes in these macroeconomic variables are also taken into account. These effects and interrelationships are described in Chapter 2. The influence of macroeconomic variables on real estate investment return has been discussed e.g. by Ewing and Payne (2005). The investment problem is considered as a multi-period, stochastic, mixed asset portfolio selection problem, i.e., several time periods with uncertain returns are considered, and there are both lump-sum and continuous investments. Investments in real estate are considered as lumpsum investments (i.e., the entire real estate has to be purchased or sold) whereas investments in the market portfolio and in bonds are continuous (i.e., the amount invested is a continuous variable). An approach well suited for problems such as this, is Contingent Portfolio Programming (CPP), developed by Gustafsson and Salo (2005). CPP is developed particularly for the solving of R&D project portfolio selection problems, but it can as well be applied to other kinds of investment problems, such as the portfolio selection problem discussed here. CPP builds on the literature on decision analysis, R&D management, and portfolio optimization (Gustafsson and Salo, 2005, p.1). The method puts the portfolio selection problem in the form of a linear programming (LP) problem, and thus makes it relatively easy to solve using LP software. Gustafsson and Salo (2005) discuss earlier approaches and their deficiencies in the solving of R&D project portfolio selection problems. As the most relevant earlier approaches, they mention optimization models and dynamic programming models. Optimization models of portfolio selection can be considered as variants of capital budgeting models (Luenberger, 1998, pp ), and their main deficiency concerning project portfolio selection is that they do not usually account for risk. They can, however, involve project interdependencies and resource constrains. In contrast, dynamic programming models, such as decision trees and real options, account for 2

4 risk but do not, however, explicitly address project interactions or resource constraints. (Gustafsson and Salo, 2005, pp.2-3) Decision trees and real options have been discussed e.g. by Smith and Nau (1995). The mean-edr objective function applied in the investment problem of this study is a variant of the mean-variance model proposed by Markowitz (1952). Instead of variance, mean-edr uses expected downside risk (EDR) as risk measure. When applying EDR as risk measure, risk is modeled as the expected amount by which the return falls short of a set target. Since EDR is a linear risk measure, it is especially well suited for linear programming problems. EDR has been applied e.g. by Eppen at al. (1989). Downside-risk models in general have been discussed by Fishburn (1977). The objective of this study is to illustrate how CPP can be used in the solving of an investment problem of the kind described earlier. For this, an example problem is formulated. The parameters used in the example problem are chosen so that they reflect certain interdependencies between macroeconomic variables and the influences that changes in these variables have on the components of the portfolio. The investment problem is described in Chapter 2 and the CPP method is explained in Chapter 3. In Chapter 4, the CPP method is used to solve the example problem. 3

5 2 The investment problem The investment problem considered here is one in which the decision maker can construct a portfolio consisting of three kinds of assets: (i) real estates, (ii) the market portfolio, and (iii) bonds. The value of these assets is likely to change over time, as a result of changes in the macroeconomic environment. These changes and their probabilities should be taken into account when constructing an optimal portfolio (i.e. one having profit-risk characteristics that are in accordance with the investor s preferences). Also, a portfolio that is optimal initially is not likely to remain optimal forever. Therefore, in order to retain its optimality, the portfolio will have to be updated from time to time, as the macroeconomic variables and thereby the prospects change. 2.1 The effects of macroeconomic variables on return In this study, a model is constructed, where three macroeconomic variables influence the return of the different components of the portfolio. These variables are (i) the economic outlook, represented by the HEX all-share index 1, (ii) the interest level, represented by the EURIBOR rate 2, and (iii) the general level of rent. Qualitatively, increases in these variables are here assumed to have the following effects on the return 3 of the components of the portfolio: 1. An increase in the HEX all-share index indicates a period of economic growth, and therefore encourages people to increase their costs of living. This causes a shift from lower-cost housing into higher-cost housing, which ultimately results in an increase in the value of high-cost residential buildings and a decline in the value of low-cost residential buildings. A growing economy is also likely to increase the value of business premises. 1 The HEX all-share index represents the price and total-return development of the share series on the Main list of Helsinki Stock Exchange (OMX, 2004, p.3). 2 The EURIBOR (Euro Interbank Offered Rate) is the rate at which a prime bank is willing to lend funds in euro to another prime bank (European Central Bank, 2004, p.109). 3 The return obtained during a period is composed of the change in asset value during the period and the cash flows (such as dividends, coupon payments, rents) yielded by the asset during the period. 4

6 Naturally, an increase in the HEX all-share index also means that the value of the market portfolio increases. 2. An increase in the EURIBOR rate has a negative effect on the value of bonds, since the present value of future fixed (coupon and face value) payments declines as interest rates increase. It influences the value of real estates in two ways: First, if interest rates increase, the discounted value of future rent payments decreases, which has a negative effect on the value of all kinds of real estate. Second, high interest rates discourages people from taking mortgages and thus from buying houses. This results in a lower demand for housing and therefore in a decline in the value of residential estates. 3. An increase in the general level of rent has a positive effect on the value of real estate 4, since it increases the present value of future rent payments. Decreases in the variables have effects that are opposite to the ones described above. The effects of increases in the macroeconomic variables on the return of the components of the portfolio are summarized in Table 1. It should be noted that the causalities explained above are just one possible realization. Under certain circumstances, the economic phenomena may follow a different logic. Table 1. The effects of increases in macroeconomic variables on the change in value of real estate investments and on the return on the market portfolio and bond investments. Residential building (low-cost) Residential building (high-cost) Business premises The market portfolio Bonds The HEX allshare index The EURIBOR rate The general level of rent For simplicity, this research focuses only on the changes in value of the real estates, ignoring rent payments as sources of positive cash flows during a period. Future rent payments are, however, used in explaining the effect that a change in the level of rent has on the value of real estate. 5

7 2.2 Interdependencies between macroeconomic variables Some interdependencies are likely to exist between the macroeconomic variables, i.e., changes in them are not statistically independent. The most important of the interdependencies is caused by monetary policy, which is likely to be anti-inflationary and, to some extent, counter-cyclical. 5 Changes in the domestic economic outlook (represented here by the HEX all-share index) are likely to be positively correlated with changes in the general economic outlook of the Euro zone. Therefore, since monetary policy is anti-inflationary and counter-cyclical, it is likely that an increase in the HEX all-share index will be accompanied by rising interest rates, and a decrease in the HEX all-share index will be accompanied with decreasing interest rates. Thus, it is likely that the HEX all-share index and the EURIBOR rate will be positively correlated with each other. It should be noted, however, that this proposition does not hold under all circumstances. 3 Contingent portfolio programming The formulation and solving of a CPP problem involves the following steps, which are discussed in more detail below. The terminology used is that of Gustafsson and Salo (2005) 1. Define a state tree for the problem and determine the effect that the realization of each possible state would have on the values of the different assets 2. Define the decision variables 3. Define the constraints: a. Decision consistency constraints b. Resource constraints 5 The primary objective of the European Central Bank (ECB) is to maintain price stability. When in line with the primary objective, the ECB should avoid generating excessive fluctuations in output and employment. (European Central Bank, 2004, p.44) 6

8 c. Optional constraints d. Deviation constraints 4. Define the objective function 5. Solve the problem using linear programming An example of the use of CPP is given in Chapter 4, where a problem of the kind described in Chapter 2 is formulated and solved. 3.1 The state tree The state tree includes all the states and state transitions that are to be considered as well as the conditional probabilities associated with each state transition. The term conditional probability refers to the probability of reaching a particular state, if the parent state of that state is reached for sure. An example of a state tree is shown in Figure 1 and the associated probabilities can be found in Table The decision variables Since decisions can be made in every state of the state tree, there has to be a distinct set of decision variables associated with each possible state. For example, if investments can be made into any of n projects in each of m states, there has to be n m decision variables concerning investment decisions into these projects. The type of each decision variable also has to be defined, i.e., it has to be defined whether a particular decision variable is a continuous variable, an integer, a binary variable, and whether the variable is restricted, for example, to non-negative values. The decision variables and the respective restrictions for the example problem considered in Chapter 4 are listed and explained in Table The constraints There are four kinds of constraints to be considered. First, decision consistency constraints forbid decisions that are not compatible with decisions made in previous periods. An example of this is a constraint, which asserts that a project, which has never been started, or which has been terminated in a previous period, cannot be continued in the present period. 7

9 Second, resource constraints define a relationship between the stock of cash and the cash flows associated with each state. They can be used to ensure that the stock of resources is never negative (Gustafsson and Salo, 2005, p.11). An example is given by equations (2) and (3) in Chapter 4, which assert that the amount of cash invested in the end of each period is equal to the whole stock of resources owned in the end of that period. Third, optional constraints may include any other constraints required to realistically model a problem. Such constraints include e.g. requirements that certain other projects be finished in order to be able to start a particular project. Fourth, deviation constraints can be used to define deviation variables, which measure how much the outcome deviates from some particular value (e.g. target return) in each alternative terminal state. The deviation variables can be used to construct objective functions that take risk into account e.g. in the form of expected-downside-risk (EDR) or lower semi-absolute deviation (LSAD). As linear measures of risk, EDR and LSAD are well suited for problems such as this, where linear programming is applied. (Gustafsson and Salo, 2005, p.13) An example of deviation constraints is given by Equation (6) in Chapter The objective function The objective function is defined as a linear function of some of the variables in the problem, usually the final resource positions in alternative terminal states and the respective deviation variables (to account for risk). As noted above, if risk attitude is to be taken into account, linear risk measures, such as expected downside risk or lower semi-absolute deviation, have to be applied. The probability of each terminal state can be calculated recursively from the conditional probabilities associated with the state transitions (Gustafsson and Salo, 2005, p.9). This recursive calculation is demonstrated by equation (1) in Chapter 4. 8

10 4 A case example Contingent portfolio programming is used to solve a problem of the kind described in Chapter 2. The example problem is formulated as follows: 1. The portfolio may consist of three kinds of assets: real estate, the market portfolio, and bonds. Real estate investments are lump-sum investments, i.e., the entire real estate has to be purchased/sold. Investments in the market portfolio and in bonds are assumed to be continuous non-negative variables, i.e., any sum can be invested in them and short-selling is not allowed. 2. The following considerations determine the change in resource position during a period: the return of the market portfolio investment, the return of the bond investment, and the change in value of the real estates invested in. 3. The objective function is of the type mean-edr, which takes into account the expected value of the final resource position as well as the expected downside risk. 4. The planning horizon is two years. The portfolio can be constructed/reconstructed at two points in time: in the beginning of the first year (at time 0) and in the beginning of the second year (at time 1). At time 0, the investor has the initial resource position W. 5. The macroeconomic variables (the HEX all-share index, the EURIBOR rate and the general level of rent) may change after each decision point. A combination of these three macroeconomic variables during a year is interpreted as a state of the CPP problem. A particular state is assumed to remain in effect for one year after the decision point. The decision maker has an estimate of the probability associated with each possible state transition during future time periods, as well as their influences on the assets. 6. Transaction costs are assumed to be zero, and buy and sell prices are assumed to be equal at any point in time. The effects that changes in macroeconomic variables have on the components of the portfolio were discussed in Chapter 2, as well as the correlations between changes in the macroeconomic variables. The values of the parameters which define the example problem exhibit these 9

11 characteristics. The five real estates of the example problem, estates A, B, C, D, and E, are intended to reflect different types of real estates, and their parameter values are chosen accordingly. A and B are intended as examples of low-cost residential buildings, C and D are intended as examples of business premises, and E is intended as an example of high-cost residential buildings. The parameter values used in the investment problem are listed in the Appendix. 4.1 The state tree Since the state changes after each decision point, that is, after times 0 and 1, the state tree branches twice. At each branch point, each of the three macroeconomic variables can either go up or down. Therefore there is a total of 2 3 = 8 alternative states following each branch point. The state tree is shown in Figure 1. It can be seen that the initial state s 0 branches into 8 different states, labeled s 1, s 2,, s 8. These states determine the values of the macroeconomic variables during the time interval ]0, 1]. These 8 states branch further, into 8 states each, labeled such that for all j {1,2,, 8} state s j is followed by states s 1, s 2,, s 8. Thus, the state of time interval ]1,2] can be any of 8 8 = 64 alternative states. The possible state transitions and the respective probabilities used in the example problem are presented in Table 2. Table 2. The conditional probabilities of the state transitions. State transitions HEX EURIBOR Level of rent Probability s 0 s 1, s j s 1 UP UP UP 0.30 s 0 s 2, s j s 2 UP UP DOWN 0.30 s 0 s 3, s j s 3 UP DOWN UP 0.05 s 0 s 4, s j s 4 UP DOWN DOWN 0.05 s 0 s 5, s j s 5 DOWN UP UP 0.05 s 0 s 6, s j s 6 DOWN UP DOWN 0.05 s 0 s 7, s j s 7 DOWN DOWN UP 0.10 s 0 s 8, s j s 8 DOWN DOWN DOWN

12 s 1,1 s 1 s 1,2 s 1,8 s 2,1 s 0 s 2 s 2,2 s 2,8 s 8,1 s 8 s 8,2 s 8, Time Figure 1. The state tree. There are 8 alternative states at time 1 and 8 8 = 64 at time 2. The probabilities of the state transitions used in the example are listed in Table 2. Notice that the probability of reaching state s jk in the end equals the probability of the transition from state s 0 to s j multiplied by the conditional probability of the transition from state s j to s k, that is, p ( s k 0 j j k ) = p( s s ) p( s s ) (1) 11

13 4.2 The decision variables The variables of the problem are explained in Table 3. The real estates are considered as lumpy projects, and the decision whether to invest (=acquire/keep) or not to invest (=sell/not acquire) in a real estate is represented by a binary variable (1 = acquire/keep, 0 = sell/do not acquire ). The size of the bond investment and the size of the market portfolio investment are non-negative real variables. The final resource position variables as well as the positive and negative deviation variables are considered in Section 4.4. Table 3. The decision variables. Decision Variables Number of Type of Explanation of the decision variable variables variable A E x0,..., x0 5 Binary Investments in real estate at time 0 A E x1,..., x8 8 5 = 40 Binary Investments in real estate at time 1 b Investment in bonds at time 0 b,...,b 8 0 Investment in bonds at time m 1 0 Investment in the market portfolio at time 0 0 m,..., m 8 0 Investment in the market portfolio at time 1 1,1 8 V,...,V 8 8 = 64 0 Final resource position 8,8 1,1,..., V8, 8 V 8 8 = 64 0 Negative deviation variables + + 1,1,..., V8, 8 V 8 8 = 64 0 Positive deviation variables 12

14 4.3 The resource constraints The resource flows associated with the problem are depicted in Figure 2. At the beginning of each time period, investments can be made in bonds, the market portfolio and real estates. The resource position at the end of each time period depends on the return on the investments during the time period in question, that is, on the return on the market portfolio and bond investments, and on the change in value of the real estates. Initial resource position Real estate Market portfolio Bonds Real estate Market portfolio Bonds Final resource position Time Figure 2. Resource flows. At time 0, the investor has an initial resource position W, which he intends to invest (first decision point). It is assumed here that he invests all of it in the three available asset types. At time 1, the investments made at time 0 will have yielded return. The change in resource position during the time interval ]0,1] depends on which state (of s 1,, s 8 ) obtains. The investor will reinvest the money at time 1 (second decision point). Due to zero transaction costs, the entire resource position at time 1 can be used to construct the new portfolio. At time 2, the investments made at time 1 will have changed in value, and the updated value depends on which state obtains (of s 1,, s 8, assuming s j is the state at time 1). Based on the above, resource constraints have to be imposed on each state in the state tree. The parameters used in the constraints and in the objective function of the problem are explained in Table 4. 13

15 Table 4. The parameters. Parameter Explanation of the parameter W Initial resource position (=resource position at time 0) T Target value of the final resource position (=resource position at time 2) λ A E E0 E0 The co-efficient of the risk measure in the objective function,..., The values of the 5 real estates at time 0 (=in state s 0 ) A E E1 E8,..., The values of the 5 real estates at time 1 (=in states s 1,,s 8 ) A E E1,1 E8, 8 1,..., The values of the 5 real estates at time 2 (=in states s 1,1,,s 8,8 ) B,..., B Return on the bond investment during the period ]0,1] (=in states s 1,,s 8 ) 1,1 8 B,..., B Return on the bond investment during the period ]1,2] (=in states s 1,1,,s 8,8 ) 8,8 M 1,..., M Return on the market portfolio during the period ]0,1] (=in states s 1,,s 8 ) 8 M,..., M Return on the market portfolio during the period ]1,2] (=in states s 1,1,,s 8,8 ) 1,1 8,8 p ( s j,k ) Unconditional probabilities of the terminal states The resource constraint for state s 0 requires the initial resource position to equal the sum invested in the three types of assets at time 0. This can be written formally as follows: W b0 m0 E i 0 x i 0 = 0 (2) i { A,..., F} Investments at time 0 The resource constraints for states s 1,, s 8 require the entire resource position at time 1 to be reinvested in the three types of assets. This can be written formally as follows: { 1,...,8 }: ( 1 + B ) b + ( 1 + M ) i i i i j j 0 j m0 + E j x0 b j m j E j x j = 0 (3) i { A,..., F} i { A,..., F} Resource position at time1 Investments at time1 The resource constraints for terminal states s 1,1,, s 8,8 require the variables V k to equal the resource positions associated with the respective terminal states s k. This can be written formally as follows: j k { 1,...,8 }: ( 1 + B ) b + ( 1 + M ) E { A,..., F} i i, k j k j + k j k i m Resource position at time 2 x V = 0 (4) 14

16 Thus, the final resource position of each alternative terminal state s k can be read from the respective final resource position variable V k.. The total number of resource constraints is = 73. The values of the parameters used in the example are listed in the Appendix. 4.4 The objective function and the deviation constraints The unconditional probabilities p ) of the terminal states can be calculated with the help of ( s j,k equation (1) from the conditional state transition probabilities shown in Table 2. The expected value of the final resource position, that is, E[V], can be easily calculated as the sum of the products of the probabilities of each terminal state and the respective final resource positions: [ V ] E = p( s j, k ) V k. (5) k { 1,...,8} However, maximizing the expected value of the final resource position is unlikely to maximize the decision maker s utility, since it does not account for the investor s risk-attitude. In Chapter 3, EDR and LSAD were mentioned as suitable risk measures. The objective function considered here is of the form mean-edr. This requires deviation constraints to be introduced for every terminal state. When using EDR as a risk measure, the deviation constraints are used to determine how much the final resource position in each terminal state deviates from its target T, as well as the direction of the deviation. If the deviation is positive, i.e., the target is reached and exceeded, the positive deviation variable V, will equal the amount by which the target is exceeded in the + j k final state s k. Respectively, if the target is not reached, the negative deviation variable V, will equal the amount by which the final resource position falls short of the target in the final state s k. The deviation constraints can be expressed formally as follows: j k + { 1,...,8 }: V T V + V 0 j (6), k k k k = The mean-edr objective function for the problem is max E[ V ] λ p( s j, k ) V k, (7) k { 1,...,8} 15

17 where the expression λ p( s j, k ) V k can be interpreted as the decision maker s risk premium k { 1,..., 8} (Gustafsson and Salo, 2003, p.14). It can be seen that the risk premium equals the expected value of how much the final resource position falls short of its target, multiplied by the constant λ. When substituting E[V], from expression (5), the objective function takes the form max p( s k ) V k λ p( s k ) V k (8) k { 1,...,8} k { 1,..., 8} Risk premium resource Expected position final This is a linear function of the variables function for a LP problem. V j, k and V k, and is therefore a suitable objective 4.5 The results The LP problem has the following components: (i) 255 variables as listed in Table 3, (ii) an objective function, determined by expression (8), (iii) 73 resource constraints, determined by equations (2), (3) and (4), and (iv) 64 deviation constraints, determined by equation (6). The problem was solved using the LP software Xpress. The optimal investment strategy for the parameter values listed in the Appendix and Table 2 is shown in Table 5.. Table 5. Optimal investment decisions at each decision point. The letter X indicates that the respective estate should be invested in, either by buying or keeping it. Time State A B C D E Market Bonds portfolio 0 0 X X X X X X X X X X X X X X Using the investment strategy of Table 5., the expected final resource position is , and the expected downside risk is 0.643, while initial resource position at time 0 was 20 and the target final resource position was

18 5 Discussion and conclusions A problem of the kind described in Chapter 2 was formulated and solved in Chapter 4 using the CPP method. Depending on the state considered, different kinds of investment strategies were found optimal. In one state, for example, all the money was invested in bonds. In some other states, most of it was distributed among the real estates, while some part of it was also invested in bonds and the market portfolio. The CPP method proved to be fit for use in the solving of investment problems such as this. It provided an intuitive and easy-to-use way of doing this, especially when a modern linear optimization package such as Xpress could be utilized. There are some possible shortcomings in the way the example problem was modeled, such as the lack of explicit modeling of rent payments, dividends and coupon payments. Also, far more accurate models could have been applied to determine the interactions between the macroeconomic variables and their influences on the components of the portfolio. The objective was not, however, to give an example of a highly realistic way of modeling the economic environment but rather to show how the CPP method can be used in the solving of investment problems such as the one considered in this study. The CPP method allows for more detailed and accurate model formulation, when needed. More accurate methods could be applied in determining the probabilities of the state transitions and the influences of macroeconomic variables on the assets. First, the macroeconomic model itself could be more sophisticated, and second, the parameter values used in the problem could be based on real-world data. A larger number of investment targets could also easily be included, if needed, and the number of decision periods could be increased. Rent payments, coupon payments, and dividends could also be explicitly included in the resource constraints, if this was found appropriate. Also the coefficient λ of the risk measure could be estimated, so that it would reflect the decision maker s risk attitude. 17

19 6 References Eppen G., R. Martin, L. Schrage (1989). A Scenario Approach to Capacity Planning. Operations Research, Vol. 37, No. 4 (Jul. Aug., 1989), European Central Bank (2004). The Monetary Policy of the ECB Available on-line at: [accessed 26 February, 2005]. Ewing, B., J. Payne (2005). The response of real estate investment trust returns to macroeconomic shocks. Journal of Business Research, Vol. 58, Issue 3 (March, 2005), Fishburn, P. (1977). Mean-Risk Analysis with Risk Associated with Below-Target Returns. American Economic Review, Vol. 67, No. 2 (Mar., 1977), Gustafsson, J., A. Salo (2005). Contingent Portfolio Programming for the Management of Risky Projects. Operations Research. Forthcoming. Luenberger, D. (1998). Investment Science. Oxford University Press. New York. Markowitz,H. (1952). Portfolio Selection. Journal of Finance, Vol. 7, No.1 (March, 1952), OMX (2004). Calculation of HEX indices. Available on-line at: [accessed 26 February, 2005]. Smith, J, R. Nau (1995). Valuing Risky Projects: Option Pricing Theory and Decision Analysis. Management Science, Vol. 41, No. 5 (May, 1995),

20 Appendix: The parameters used in the example problem General parameters Initial resource position (W) = 20 Target final resource position (T) = 22 Coefficient of the risk measure (λ) = 8 State parameter table Columns Time and State : Determine the time period and state considered in the particular row. Columns Hex, Euribor, and Level of Rent : Changes in the Hex all-share index, the Euribor rate, and the general level of rent during the time period determined by column Time. It is indicated in these columns, whether the values have gone up or down. Column X value : The value of real estate X at the end of the time period determined by column Time Columns MP return / Bond return : The return of the market portfolio/bond investment during the time period determined by the column Time. Time State 6 Hex Euribor Level of Rent A value B value C value D value E value MP return (M) Bond return (B) UP UP UP % 2 % UP UP DOWN % 2 % UP DOWN UP % 8 % UP DOWN DOWN % 8 % DOWN UP UP % 2 % DOWN UP DOWN % 2 % DOWN DOWN UP % 8 % DOWN DOWN DOWN % 8 % 1-2 1,1 UP UP UP % -1 % 1-2 1,2 UP UP DOWN % -1 % 1-2 1,3 UP DOWN UP % 5 % 1-2 1,4 UP DOWN DOWN % 5 % 1-2 1,5 DOWN UP UP % -1 % 1-2 1,6 DOWN UP DOWN % -1 % 1-2 1,7 DOWN DOWN UP % 5 % 1-2 1,8 DOWN DOWN DOWN % 5 % 1-2 2,1 UP UP UP % -1 % 1-2 2,2 UP UP DOWN % -1 % 1-2 2,3 UP DOWN UP % 5 % 1-2 2,4 UP DOWN DOWN % 5 % 6 Only the subscripts of the states are listed. E.g. s 2,3 is replaced by 2,3 in the table 19

21 1-2 2,5 DOWN UP UP % -1 % 1-2 2,6 DOWN UP DOWN % -1 % 1-2 2,7 DOWN DOWN UP % 5 % 1-2 2,8 DOWN DOWN DOWN % 5 % 1-2 3,1 UP UP UP % 5 % 1-2 3,2 UP UP DOWN % 5 % 1-2 3,3 UP DOWN UP % 11 % 1-2 3,4 UP DOWN DOWN % 11 % 1-2 3,5 DOWN UP UP % 5 % 1-2 3,6 DOWN UP DOWN % 5 % 1-2 3,7 DOWN DOWN UP % 11 % 1-2 3,8 DOWN DOWN DOWN % 11 % 1-2 4,1 UP UP UP % 5 % 1-2 4,2 UP UP DOWN % 5 % 1-2 4,3 UP DOWN UP % 11 % 1-2 4,4 UP DOWN DOWN % 11 % 1-2 4,5 DOWN UP UP % 5 % 1-2 4,6 DOWN UP DOWN % 5 % 1-2 4,7 DOWN DOWN UP % 11 % 1-2 4,8 DOWN DOWN DOWN % 11 % 1-2 5,1 UP UP UP % -1 % 1-2 5,2 UP UP DOWN % -1 % 1-2 5,3 UP DOWN UP % 5 % 1-2 5,4 UP DOWN DOWN % 5 % 1-2 5,5 DOWN UP UP % -1 % 1-2 5,6 DOWN UP DOWN % -1 % 1-2 5,7 DOWN DOWN UP % 5 % 1-2 5,8 DOWN DOWN DOWN % 5 % 1-2 6,1 UP UP UP % -1 % 1-2 6,2 UP UP DOWN % -1 % 1-2 6,3 UP DOWN UP % 5 % 1-2 6,4 UP DOWN DOWN % 5 % 1-2 6,5 DOWN UP UP % -1 % 1-2 6,6 DOWN UP DOWN % -1 % 1-2 6,7 DOWN DOWN UP % 5 % 1-2 6,8 DOWN DOWN DOWN % 5 % 1-2 7,1 UP UP UP % 5 % 1-2 7,2 UP UP DOWN % 5 % 1-2 7,3 UP DOWN UP % 11 % 1-2 7,4 UP DOWN DOWN % 11 % 1-2 7,5 DOWN UP UP % 5 % 1-2 7,6 DOWN UP DOWN % 5 % 1-2 7,7 DOWN DOWN UP % 11 % 1-2 7,8 DOWN DOWN DOWN % 11 % 1-2 8,1 UP UP UP % 5 % 1-2 8,2 UP UP DOWN % 5 % 1-2 8,3 UP DOWN UP % 11 % 1-2 8,4 UP DOWN DOWN % 11 % 1-2 8,5 DOWN UP UP % 5 % 1-2 8,6 DOWN UP DOWN % 5 % 1-2 8,7 DOWN DOWN UP % 11 % 1-2 8,8 DOWN DOWN DOWN % 11 % 20