1


                                    Introduction

The principles of portfolio theory were laid more than 40 years ago....
2

related to each other. Thus, it is reasonable to take the currency and the asset allocation
decision simultaneously. Th...
3


                                        Chapter I
          Algorithms for generalized mean/variance problems



This ...
4

literature provided by Markowitz but since it is nevertheless based on the critical line algorithm
this is called exten...
5

efficient set and, in conclusion, it is shown that under certain assumptions the single period
optimization problem is ...
6

                                                                  %
mi represents the i-th moment of the probability di...
7




1.2. Mean/variance efficient frontier
From the results of the last section it can be established that expected utili...
8

portfolio theory. The aim of this section is to summarize some analytical properties of the
efficient frontier. The for...
9



Figure 1.1. depicts the graph of an arbitrary efficient frontier and its asymptotes.
    expected return




        ...
10



(1.14)                E − E z = β ( Em − E z )



Thus a linear efficient frontier is developed in the case of the e...
11

                         V −1 ( µ − re )
(1.16)           ωm =
                         e' V   −1
                    ...
12

                                    ~
                                     (       )                   ( ~
           ...
13



                        J ' (Wt )
(1.22)          ω =−                  V −1( µ − re )
                       Wt J '...
14

                       J ' (Wt )                  λ
(1.24)         ω =−                  V −1µ −                V −1e
...
15

practical implementation. A simple Lagrange approach as in (1.5) and (1.6) will not be
sufficient to find a solution. ...
16

1.   A ∈ℜ m x n denotes the matrix of m restrictions imposed on n assets fractions, b ∈ R n is the
     vector of righ...
17

 (1.25) represents a quadratic minimization problem with linear constraints. Necessary and
sufficient conditions for t...
18

                    (c)     ∀        η i > 0 ⇔ ω i = 0 and η i = 0 ⇔ ω i > 0
                          1≤ i ≤ n

     ...
19

and the vector of Lagrange multipliers λ . Each ω i > 0 is labelled base variable8, each ω i = 0
non-base variable9. T...
20

Equivalent to equation (1.33), the vector of portfolio fractions emerges from:


               ω 
(1.35)          ...
21

                                           α
                                                   i                    ...
22

Subsequently, it will be shown that adjacent segments of the efficient frontier transit
continuously into each other b...
23

Now, M −1 may be subdivided into block matrices and redefined as:


                        M1      M2
             ...
24

The purpose of this section is to show that two adjacent segments of the efficient frontier are
continuously linked (s...
25

                λ low = λ1E        and        λ high = λ4
                                                        E

I...
26

Let a and b represent two critical lines which express two adjacent segments. Segment a is
defined for greater λ E tha...
27

respectively. The variance of returns, expected return and fraction of asset n is denoted by σ 2 ,
                   ...
28

                   V( n −1)  0      A(' n −1)       µ ( n −1)     ω ( n −1)   0( n −1) 
                        ...
29

equations which is equal to (1.50)10. Thus, converging λ E to the borders of the segments a
and b, respectively, yield...
30

                  ∂σ 2                               −                      −
                                        ...
31

the Kuhn-Tucker conditions (1.29) implies restricting λ E by λ E ∈ λ low , λ high . Choosing an
expected portfolio ret...
32


       expected return




                                               restricted efficient frontier




         ...
33




2.7. Feasible base solution
Finding an initial feasible base solution is the condition for an application of the cr...
34

                        C                                    625        20


The program which has to be executed is a...
35

Using (1.35) and (1.37) provides



              αin        βin             α in              γin          δin       ...
36

    A         0.6364      -0.006061     105            0              0            -
    B            0            0  ...
37



Using (1.35) and (1.37) provides:



                αin           βin            α in           γin          δin   ...
38

                  400 150 0 1           0.0031 −0.0031                      0 0.2308 
                           ...
Algorithms for generalized mean variance problems
Algorithms for generalized mean variance problems
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Algorithms for generalized mean variance problems

  1. 1. 1 Introduction The principles of portfolio theory were laid more than 40 years ago. But although it was designed for managing private and institutional portfolios of financial assets, among the largest banks in the world there are few which systematically apply the perceptions of portfolio theory. And even if applications are implemented, there is only a small number of market participants who are able to utilize more than the basics. There are only a few, typically rather small, investment banks and institutional investors in the world who have the human resources available to exploit the superior theoretical concepts for the real investment process. However, they tend not to make their knowledge publicly available. The realization of portfolio theory and its modern developments in downside risk management necessitates the use of powerful computers and software programs. Although the theoretical work has advanced considerably, neither in the research nor in practice has there been sufficient work to provide programmable algorithms which would make approaches understandable and applicable. This thesis is intended to show how abstract theories can be applied to real portfolio management using modern financial instruments. Its purpose is to help practitioners to implement portfolio strategies based on theoretical concepts. It summarizes existing theories and expands them to frameworks which are designed to generate specific asset allocation decisions. In the first chapter the basics of portfolio theory are summarized. It follows the description of quadratic optimization algorithms. They are needed to implement and solve mean/variance optimization problems respecting various restrictions resulting from investor preferences or from laws. Mean/variance principles are the basis for asset allocation decisions and although they are extended and alternative approaches are given in this thesis, all investment decision criteria observe them. Markowitz [1987] provided a book which contains the principles of the mean/variance approach where linear restrictions are imposed. Nevertheless, this thesis extends the basic rules of Markowitz and gives comprehensive numerical examples. The second chapter is concerned with international asset allocation decisions. Investment decisions arising from international diversified portfolios are typically more complex than merely domestic investments, since they require one to respect currency movements supplementary to asset price movements. Due to the development of advanced markets for currency derivatives the currency allocation decision may be influenced by hedging decisions: selling short forward contracts on currencies provides opportunities for lowering the impact of the currency risk. Since the hedge ratio may be chosen in accordance with the expected currency development, hedging of the currency exposure can be done without avoiding potential gains on currencies. Traditionally, the hedging and the asset allocation decision are done subsequently. This is done with respect to the fact that currency and asset returns are
  2. 2. 2 related to each other. Thus, it is reasonable to take the currency and the asset allocation decision simultaneously. The approach introduced in chapter two simultaneously optimizes the hedge ratios and the fractions invested in different countries. Since in contrast to the traditional international asset allocation decisions it elaborates a model where variable hedging opportunities are allowed, it is called a variable hedging model. Apart from the analytical presentation of the model, chapter two contains a case study for an investor with the reference currency US-$ who diversifies internationally. Chapter three discusses downside risk measures. A major disadvantage of mean/variance problems is the consideration of symmetrical payoff patterns. That is, oscillations to the upside of returns series are interpreted as disadvantageous occurrences as well as downside oscillations. Obviously, investors are more concerned with the downside movements than with the upside potential. Therefore, downside risk approaches exclusively accept downside oscillations as relevant measures of risk. Although these approaches were already developed at the beginning of the fifties, they succeeded in the mid seventies with the introduction of lower partial moment frameworks. A special case of lower partial moments is the safety first principle. This allows investors to find out portfolios revealing a minimum probability for falling short of specified return level. Chapter three contains a discussion of the literature concerning lower partial moments and the safety first principle and applies it to investment allocation decisions. For identifying minimum shortfall probability when the investment opportunity set is restricted, the results from chapter one are used. It is shown that one of the major advantages of the Markowitz algorithm compared to other methods which minimize quadratic functions, is that it is easily applicable to investment problems based on the mean/variance principle. The last chapter investigates downside risk problems when derivatives, such as options, are used to influence the probability distributions of the payoffs. Using options to protect portfolios against pitfalls is a widely used technique. However, it is not clear how using options influences the probability distributions of the returns. In chapter four, a simulation approach is shown which allows their comparison. Applying downside risk measures allows comparing them. Apart from those who are interested in applied portfolio theory, this thesis is designed for programmers. Therefore, this work is accompanied by a software program which contains all the algorithms described. The figures and numerical examples contained in the following chapters are mostly calculated by this program.
  3. 3. 3 Chapter I Algorithms for generalized mean/variance problems This chapter lays the foundations for this thesis. In the first section the fundamentals of portfolio theory are summarized. These are well known from the standard literature (see for instance Ingersoll [1987] and Huang and Litzenberger [1988]). Since readers with different knowledges of portfolio theory are addressed by this thesis, this section is to provide the reader with the most important equations and interpretations of standard portfolio theory. In section one, the main results of portfolio theory without respecting practical investment restrictions are pointed out. Since with this thesis people are addressed who are concerned with the implementation of investment strategies, in the subsequent sections portfolio problems subject to investment restrictions are investigated. The algorithms provided in the sections two, three, and four are based on the critical line algorithm introduced by Markowitz [1956 and 1987] which is therefore sometimes referred to as Markowitz algorithm. Section two deals with the ordinary form of the algorithm. This allows solving portfolio problems subject to equality restrictions and with the exclusion of short selling. However, in the real investment process it is not sufficient to impose purely equality restrictions. Typically, investment situations are characterized by lower or upper boundaries for the fractions invested in specific asset classes or in combinations of asset classes. For instance, investors may be prepared to invest in foreign stocks but not more than fifty percent of the investment capital. This restriction will require to impose a quot;smaller thanquot; restriction for foreign stocks. Although these problems are briefly discussed in Markowitz's [1987] book, they are not explicitly solved. Therefore section three deals with an algorithm which allows imposing linear inequality and equality restrictions. This is, in addition to equality restrictions, quot;smaller thanquot; and quot;greater thanquot; restrictions are added. Short sales remain excluded. Since this generalizes the ordinary form of the Markowitz algorithm this is called generalized Markowitz algorithm. Some investment situations might require the exclusion of short sales for specific asset classes but not for others. Although the generalized critical line algorithm allows to solve very general investment problems the lowest fraction for each asset in the portfolio is zero: no asset can be sold short. Permitting negative portfolio fractions provides the opportunity to sell short some assets of the portfolio. Variables which are unrestricted to the downside are called unrestricted-in-sign variables. In the fourth section this problem is investigated: an algorithm is provided which allows the solution of investment problems with unrestricted-in-sign variables and with equality and inequality restrictions. Since this algorithm is not discussed in the
  4. 4. 4 literature provided by Markowitz but since it is nevertheless based on the critical line algorithm this is called extended Markowitz algorithm. All variants of the Markowitz algorithm are completed by comprehensive numerical examples. This shall enable the reader to exactly understand the algorithms in order to allow programming them. Since this chapter provides the methodical basics for solving general mean/variance problems, case studies with practical impact are not provided here but in the following chapters. To facilitate the understanding of the principles of the algorithm, the applications in this chapter deal with a hypothetical three asset case. 1. Fundamentals of Portfolio Theory Harry Markowitz [1952][1959] laid the foundations for a new view of investments in financial assets. He showed that the fundamental issue of capital investment should no longer be to pick out dominant stocks but to diversify the wealth among many different assets. His work marked a turning point in the academic literature concerning capital investments, because he showed that the success of investment does not purely depend on return, but also on the risk which has to be taken. Furthermore, he showed that the risk of investments is influenced by the correlations between different assets. Thus, the objective of investors can no longer be seen in maximizing the quot;performancequot;1 only. The performance strategies must be evaluated in the context of a certain level of risk. Investors are faced with complex optimization problems, the solution of these problems leads to a specific investment strategy, i.e. a concrete portfolio of assets. The optimization process is referred to as portfolio selection. That diversification is a crucial issue of investment, and that risk has to be considered in addition to portfolio returns, is an intuitive investment issue rather than a highly academic perception. So the main results of Markowitz' work have to be seen in his answering the question how to measure investment quot;performancesquot; and investment risks. The answers to these questions were derived from very general assumptions of utility theory. Markowitz presented quot;investment rulesquot; which maximize the expected utility of investors. The first section of this thesis summarizes briefly the main results of portfolio theory. In the first part, the mean/variance principle is elaborated from the standard assumption of utility theory. The following part contains a short survey of the analytical basics of the mean/variance 1 The term quot;performancequot; should be interpreted in its original sense, rather than according to the newer literature of portfolio performance, where it is referred to as risk adjusted portfolio returns.
  5. 5. 5 efficient set and, in conclusion, it is shown that under certain assumptions the single period optimization problem is equivalent to a multiperiod, discrete time portfolio selection model. 1.1. Utility theory and mean/variance principle The aim of this section is to show, how the quot;performancequot; and risk of any portfolio can be measured. It was assumed by von Neumann/Morgenstern [1953] that rational investors behave according to five fundamental axioms of rationality. They showed that this assumption is consistent with maximizing expected utility, given a fixed bundle of consumption goods and investment opportunities. The respective utility function has to be twice differentiable, increasing and strictly concave (see Ingersoll [1987], pp. 21-22). Thus, every rational investor endeavours to maximize his individual expected utility of consumption. First, a single period case of portfolio optimization is considered. Since in this case end-of-period-wealth equals consumption, it is admissible to maximize expected utility of end-of-period wealth instead of % consumption. Let Wt +1 denote the end of period wealth of a representative investor and U his (her) (twice differentiable, increasing and strictly concave) utility function. Then the investors' optimization problem can be stated as follows (see Huang/Litzenberger [1988], pp. 60-61): (1.1) ~ ( max EU Wt +1 ) ~ ( ) ~ ( ~ ~ ) U Wt +1 may be expressed as U EWt +1 + Wt +1 − EWt +1 , where E denotes the expectations operator. Since a single period model is considered, the time sub-index is dropped from now on. Applying a Taylor series expansion leads to: ( ) ( ) ( )( ) ( )( ) + ∑ i1!U ( EW )(W − EW ) ∞ ~ ~ ~ ~ ~ ~ ~ ~ 2 ~ ~ ~ i U W = U EW + U ′ EW W − EW + 1 2 U ′′ EW W − EW (i) i =3 (i) U is the i-th derivative of the utility function U with respect to the term in brackets. It is assumed to be at least twice differentiable. Substituting the above expression into equation (1.1) renders a restatement, where σ 2 denotes the variance of wealth: W ∞ (1.2) ~ ( ) ( ) ~ ~ ( )( ~ ~ 1 )~ 2 ( ) 1 EU W = U EW + U ′ EW E W − EW + 2 U ′′ EW σ W + ∑ U ( i ) EW mi i! ~ ( ) i=3
  6. 6. 6 % mi represents the i-th moment of the probability distribution of W . Equation (1.2) highlights the fact that the expected utility of wealth depends on the expected end of period wealth and its variance. Without making any assumptions about the probability distribution, the expected utility of wealth is furthermore influenced by the higher moments. If on the other hand it is % assumed that the end of period wealth W is normally distributed, then the moments of order greater than two can be stated as functions of the expected end of period wealth and its variance. This allows reformulating (1.2) as: (1.3) ~ ( ) ~ ( ) ~ ~ ( )(~ ) ~ 2 EU W = U EW + U ′ EW E W − EW + 1 2 U ′′ EW σ W ( ) (1.3) indicates, that investors have to concern themselves merely with the mean and the variance of the end of period wealth for normal probability distributions. It should be emphasized that equation (1.2) holds for any probability distribution whereas the mean/variance principle can be deduced out of (1.3): The higher the expected utility of wealth given a constant level of variance, the higher the expected level of end of period wealth will be. % If EW is held constant, the variance of the end of period wealth has to be minimized to maximize the expected utility. Thus, the quot;performancequot; of any portfolio according to von Neumann/Morgenstern's utility theory is measured as the expected end of period wealth assessed by a portfolio of assets. The portfolio's risk is measured by the variance of the end of period wealth. This is called a mean/variance principle of portfolio diversification. If a normal distribution for the end of period wealth cannot be assumed, the mean/variance principle can be rescued for arbitrary wealth distributions, if quadratic utility functions of the ( ) ~ ~ b ~ form U W = W − W 2 are imputed. In this case all moments of order greater than two are 2 zero and can thus be dropped. Substituting the utility function into equation (1.2) the new objective function may be expressed as: (1.4) ~ ( ) ( ) ( ) ~ b 2 ~ 2 ~ b  ~ 2 EU W = E W − EW = E W − E  EW + σ W  2  ( ) 2   ( ) Just as in equation (1.3) equation (1.4) shows that investors endeavour to minimize the variance of wealth for a given level of expected end of period wealth. Two alternative assumptions can be made to deduce the mean/variance principle out of utility theory: 1. The end of period wealth is distributed normally , or 2. investors are faced with quadratic utility functions.
  7. 7. 7 1.2. Mean/variance efficient frontier From the results of the last section it can be established that expected utility maximizers minimize the variance of the end of period wealth subject to a given level of expected end of period wealth. This is equivalent to minimizing the variance σ 2 of returns of a portfolio where the expected portfolio return E is fixed. Analytically this can be stated as: 2 min σ = min ω ' V ω ω ω (1.5) s . t .: ω' µ = E ω' e = 1 If n portfolio assets are being considered, then ω ∈ℜ n represents the vector of portfolio weights, V ∈ℜ nxn represents the variance-covariance matrix of portfolio returns, µ ∈ℜ n the vector of expected returns, and e ∈ℜ n the unity vector. The first constraint secures that the expected portfolio return equals E and the second one enforces the portfolio weights to sum up to unity. Merton [1972] showed, that the solution of problem (1.5) can be characterized by the following function2: 1 2 (1.6) σ= ( cE − 2 bE + a ) d σ represents the standard deviation of portfolio returns; a,b,c, and d are real constants which are defined as follows: (1.7) a≡ µ' V −1µ b≡ µ' V −1 e c≡ e' V −1e d≡ ac − b 2 Huang/Litzenberger [1988] (p.66) show, that equation (1.6) represents a hyperbola. Since it describes efficient portfolios, i.e. portfolios with minimum risk for any given level of expected return, the graph of function (1.6) is called efficient frontier, which is a well known result from 2 Although this was already shown by Roy [1952] the most well known reference for the following equation is Merton [1972]. In chapter three the Roy's [1952] paper will be discussed in detail.
  8. 8. 8 portfolio theory. The aim of this section is to summarize some analytical properties of the efficient frontier. The formulae are presented without proof, the interested reader may for instance refer to Huang/Litzenberger [1988] (chapter 3). The vector of portfolio weights for any efficient portfolio can be calculated as −1 −1 −1 −1 V V V µ V e (1.8) ω= ( c µ − be ) E + ( ae − b µ ) = ( cE − b ) − ( bE − a ) d d d d where first equation shows that the portfolio fraction invested in one of the n assets is a linear function of the portfolio's expected return. The second equation shows, that each efficient portfolio can be expressed as a function of two portfolios: V −1 µ and V −1 e . In equation (1.9) it is shown that the latter portfolio is the minimum variance portfolio. It is the portfolio with lowest possible variance given n portfolio assets. It will be characterized by variance σ 2 , mvp expected return E mvp , and vector of portfolio weights ω mvp . They can be calculated as: b 1 V −1e 1 (1.9) E mvp = σ2 = ω mvp = = V −1e mvp −1 c c e' V e c Any two efficient portfolios p and q have covariance c b  b 1 (1.10) σ pq =  E p −   Eq −  + , d  c  c c which determines the covariance between the minimum variance portfolio with any efficient portfolio p as: 1 (1.11) σ p,mvp = = σ2 mvp c Each hyperbola may be characterized by asymptotes; in case of the efficient frontier they are identified for large σ by: b d (1.12) E = ± σ c c
  9. 9. 9 Figure 1.1. depicts the graph of an arbitrary efficient frontier and its asymptotes. expected return standard deviation Figure 1.1: Efficient Frontier and its asymptotes The equations above briefly summarize the main results of the classical portfolio theory and lay the foundation for the more advanced applications in the following chapters. So far, it has been assumed that investments are possible only in risky assets. If the investment opportunity set is extended by a riskless asset, i.e. an asset with zero variance, the well established CAPM (see Sharpe [1964], Lintner [1965], and Mossin [1966]) equilibrium equation can be derived, where the expected return on any portfolio depends linearly on the return on the market portfolio: (1.13) E − r = β( E m − r ) The notation in equation (1.13) is as follows: r represents the riskless return, β the beta of the portfolio against the market and E m the expected return on the market portfolio. Black [1972] showed that the assumption of the existence of a riskless asset is unnecessary if instead of the riskless asset a portfolio is used, which elucidates zero covariance with the returns on the market portfolio and has a zero beta. It can easily be shown that a zero beta portfolio exists, whenever the market portfolio is an efficient risky portfolio (see e.g. Huang/Litzenberger [1988]). The Black form of the CAPM is called zero-beta CAPM. Let E z denote the expected return on the zero-beta portfolio, then (1.13) may be modified to:
  10. 10. 10 (1.14) E − E z = β ( Em − E z ) Thus a linear efficient frontier is developed in the case of the existence of a riskless asset or a zero-beta portfolio. It may be shown that it is tangent to exactly one point of the quot;risky-assets efficient frontierquot;, which is the market portfolio. The tangency is generally called Capital Market Line (CML). Figure 1.2 illustrates this phenomenon: Market portfolio expected return standard deviation Figure 1.2: Efficient Frontier and Capital Market Line Figure 1.2. clearly illustrates that the market portfolio is the only efficient risky portfolio in the case of the existence of a riskless portfolio. Given the riskless rate of interest r (or the expected return on the zero-beta portfolio E z ) the expected return on the market portfolio can be calculated as: a − br (1.15) Em = b − cr The portfolio fractions are received by substituting (1.15) into (1.8)
  11. 11. 11 V −1 ( µ − re ) (1.16) ωm = e' V −1 ( µ − re ) = 1 b − cr ( ) V −1(µ − re) , where the portfolio fractions sum up to unity. If a riskless asset exists, portfolio (1.16) represents the only risky optimal portfolio, i.e. the only risky portfolio which is chosen by investors maximizing the expected utility within a single period. 1.3. Discrete time intertemporal portfolio theory The considerations so far assumed an investor whose relevant time horizon is a single period. Of course this is not an appropriate characterization of the real decision situation of investors. This section will deal with a maximizer of expected utility in an multi-period frame. To begin with, no more is assumed than an additively separable utility function. Since the wealth at the end of all consideration periods depends on the consumption plan in the preceding periods, his ( ~ ) utility function is assumed to be U C0 , C1 ,..., CT , WT , where C represents the consumption in the respective period and T the end period of investor's life. Then the objective function of the considered investor according to the previous section may be expressed by3: (1.17) [ max Et U (C0 , C1,..., CT ) ≡ J (Wt ) Ct ,ω ] By exploiting the fact that U is additively separable, and assuming that the bequest in period T ~ ( ) is denoted by B WT , (1.17) may be expressed as: T ~  (1.18) J (Wt ) = max Et  ∑ U (Cτ ) + B WT  =   ( ) Ct ,ω τ = t    T  = max Et U (Ct ) + max Et + ∆t  ∑ U ( Cτ ) + B WT Ct ,ω  Ct + ∆t ,ω τ = t∆t ~ ( )      3 For an overview see Ingersoll [1987] pp. 235-250, the original reference is Samuelson [1969]. The intertemporal portfolio problem in a continuous time case was treated by Merton [1969] in the same issue of the Review of Economics and Statistics as Samuelson's paper.
  12. 12. 12 ~ ( ) ( ~ = U (Ct ) + Et J Wt + ∆t = U (Ct ) + Et J Wt + ∆W ) % ∆W represents the change in wealth between period t and t + ∆t . ∆t is assumed to be a very short time interval. Applying a Taylor series expansion provides:   0 = U (Ct ) + Et  J ' (Wt )∆W + J '' (Wt )∆W 2 + o( ∆t ) ~ 1 ~ (1.19)  2  The term o(∆t ) summarizes the terms dependent on ∆t of an order higher than two; they are approximately zero if ∆t is assumed to be an extremely short time interval. Then an % ∆W assumption about the changes in wealth is added. In the following part of this section is W presumed to be distributed normally, i.e. follows an arithmetic Wiener process4. Let ~W z denote a standard normal distributed random variable, µ W the expected relative change in 2 wealth, σ W its variance, and Yt the income in period t, for the relative changes in wealth holds: % ∆W (1.20) ≡ µ W ∆t + σ W z W ∆t % W C − Yt C − Yt µ W ≡ (1 − ω ' e)r + ω ' µ − t = ω '( µ − re ) + r − t Wt Wt 2 σ W ≡ ω ' Vω The definition of µ W ensures that the sum of the fractions of the riskless and the risky assets equal unity. Substituting (1.19) into (1.20) and rearranging yields:  C + Yt  (1.21) G (Wt , Ct ) ≡ U (Ct ) + Wt J ' (Wt ) ω '( µ − re ) + r − t  ∆t + 1 2 Wt J '' (Wt )ω ' Vω∆t 2  Wt  Differentiating G with respect to ω provides the following multi-period portfolio of expected utility maximization: 4 Stochastic processes were first introduced by Bachelier [1990] into the theory of capital markets.
  13. 13. 13 J ' (Wt ) (1.22) ω =− V −1( µ − re ) Wt J ' ' (Wt ) Since the fraction of wealth 1 − ω ' e is invested in the riskless assets (lending if 1 − ω ' e >0 and borrowing if 1 − ω ' e <0) the optimum risky portfolio may be normalized to: V −1(µ − re) (1.23) ω = e' V −1( µ − re) Comparing portfolio (1.22) to (1.16) proves that the holdings of the multi-period optimum portfolio are proportional to the holdings of the single period optimum portfolio. The solution of the multi-period problem (1.17) can also be obtained by solving the single period problem. Thus, in the present simple form of the multiperiod portfolio selection model, one can equivalently use a single period model. An important assumption of this model is, that the parameters of the underlying Wiener process (1.20) remain constant over time. If this assumption is given up, i.e. if it is assumed that µ W and σ 2 are dependent on a set of state W variables, then the relative changes in wealth follow a more general diffusion process instead of a Wiener process and the continuous time version of the CAPM (see Merton [1973]) has to be applied. An important result proven by this study is that for each state variable a single hedge portfolio has to be used. This guarantees protection against different sources of risk which are established by the state variables. The major property of the hedge portfolios are that they produce maximum correlation with the respective state variable. The object of this thesis is to provide algorithms for the portfolio optimization process and not the investigation of continuous time models. In this chapter attempts have been made to capture optimization solutions for the single period model, which also hold for the simple case of a discrete time multi-period model. The standard portfolio selection model of Markowitz starts with the assumption that no riskless asset is available. Thus, the multi-period optimum portfolio is considered excluding riskless assets. This is then compared to the single period optimum portfolio. In this case figure (1.17) has to be maximized subject to e' ω = 1 , since it is not possible to implicitly construct this requirement as in (1.20). Let λ denote the Lagrange multiplier belonging to the constraint e' ω = 1 , then the optimum condition (1.22) may be restated in the new context as (see Ingersoll [1987], p. 288):
  14. 14. 14 J ' (Wt ) λ (1.24) ω =− V −1µ − V −1e Wt J ' ' (Wt ) Wt J ' ' (Wt ) Equation (1.24) points out that the intertemporal discrete-time optimum portfolio equivalent to the single period optimum portfolio in (1.8) consists of two portfolios: V −1 µ and V −1 e known as the minimum variance portfolio. Thus, even when the investment opportunity set is limited to purely risky assets, the single period and the multi-period problem are solved by the same portfolio. No matter which problem is to be solved, the optimum portfolio maximizes both: the one period expected utility function (1.1) and the multiperiod expected utility function (1.17). This holds as long as the investment opportunity set is constant over time, i.e. the underlying processes are Wiener processes rather than generalized diffusion processes. Although in a discrete time intertemporal portfolio optimization framework the sum of the expected utilities of periodical consumption is maximized, without assuming intertemporal changes in µ W and σ W the solution of the single period problem does not differ from that of the multi-period problem. Consequently, the following sections will assume a one period problem. Its solution also solves the multi-period model for constant decision variables. 2. The ordinary Markowitz-algorithm The ordinary Markowitz algorithm is concerned with the solution of quadratic programs subject to equality and non-negativity restrictions. Extensions of it are given in the following sections. In addition to equality restrictions, the generalized Markowitz algorithm solves quadratic problems subject to arbitrary inequality restrictions. Unrestricted-in-sign variables are furthermore imposed to the extended Markowitz algorithm. In section one of the first chapter the fundamentals of portfolio theory were considered. The purpose of that section was to introduce the reader who may not be very familiar with portfolio theory to the main topic of this thesis. Although Markowitz's theory [1952] introduced a revolutionary understanding of capital investment at that time, it does not exactly reconstruct the decision situation of investors. A major problem which might occur is that the optimal investment policy recommended by (1.8), (1.16), (1.21), or (1.24) cannot be transposed into reality due to legal restrictions or the investment philosophy or other reasons due to the investor's attitudes. Specifically, restrictions may be imposed on some of the available assets, the vector of portfolio weights ω is restricted. Typically, one of the most common requirements to any investment policy is to exclude short sales, that is to prohibit negative portfolio fractions of one or more portfolio assets. Although in the theory the inclusion of such restrictions into equation (1.5) is not a problem, the difficulty lies in the
  15. 15. 15 practical implementation. A simple Lagrange approach as in (1.5) and (1.6) will not be sufficient to find a solution. The problem has to be solved by a quadratic program. General algorithms may be found in standard operations research literature, e.g. in Winston [1991] (pp. 658-665). Since these algorithms are not specifically fitted to the mean/variance problem, the computational efficiency can be increased by using an algorithm developed by Markowitz [1956]. Although it is not exclusively employable for solutions of the mean/variance problem, it was constructed paying special attention to such problems. The topic of the following section is the Markowitz algorithm where short selling restrictions are imposed. Since the structure of such restrictions is rather simple, this variant will be labelled ordinary Markowitz algorithm. Most aspects of his subject were covered in detail in Markowitz's book [1987]. Consequently, there is no reason to prove the respective equations. However, this section provides extensions of the original algorithm in some aspects: in part three a special case is investigated and in part five it is shown that the restricted efficient frontier, in contrast to the unrestricted efficient frontier discussed earlier, is not differentiable. Apart from short sales restrictions, the ordinary Markowitz algorithm merely includes equality restrictions. Extensions of this approach, which are not covered by Markowitz's book, will be developed in subsequent sections. A simplification of the model is to be found in Markowitz/Schaible/Ziemba [1992] where the mean/variance problem for lognormal markets and power utility functions is solved subject to arbitrary restrictions. Since this makes it more specific, this thesis concentrates on the original algorithm. The purpose of this section is to show how the Markowitz algorithm works and how it may be applied to mean/variance problems. It is subdivided in the following parts. Part one states the general model and the Lagrangian. Part two derives the Kuhn-Tucker conditions for the problem and summarizes the optimum conditions. A special case is considered in part three. The following two parts investigate whether the efficient frontier is continuous and differentiable when short selling restrictions are imposed on the problem. Part six examines concavity and the minimum variance portfolio of the restricted efficient frontier. An important issue of the model is how to find a feasible base solution; this will be investigated in part seven. A numerical application of the model is given in part eight. 2.1. Basic model The subsequent sections refer to the portfolio optimization problem (1.5). Again the variance of the portfolio returns is minimized subject to the restrictions in (1.5). Furthermore, the notation of the variables is supplemented and additional restrictions are imposed into the problem:
  16. 16. 16 1. A ∈ℜ m x n denotes the matrix of m restrictions imposed on n assets fractions, b ∈ R n is the vector of right-hand-side elements of the restrictions. It has to be emphasized that the restriction ω' µ = E is not included by A and b; i.e. it is the (m+1) st restriction. The reason for this will become obvious later on. 2. Non-negativity of the portfolio fractions is required for each asset. Further notations can be read as in part one of the present section. Thus, the extended optimization program may be stated as: 2 (1.25) ( a ) min σ = ω' V ω ω ( b ) µ' ω = E (c ) Aω = b (d ) ω ≥ 0 Let λ ' = (λ 1,..., λ m ) ∈ℜ m and λ E denote Lagrange-multipliers for the constraints (1.25 c) and (1.25 b), respectively, then the Lagrangian results as: 1 (1.26) L = ω ' Vω + λ ' ( Aω − b) − λ E (µ ' ω − E ) 2 Since b and E are constants, they are not influenced by ω , minimizing (1.26) is equivalent to minimizing a simplified Lagrangian: 1 (1.27) L = ω ' Vω + λ ' Aω − λ E µ' ω 2 Starting with equation (1.27) the Kuhn-Tucker conditions are now being constructed. 2.2. Kuhn-Tucker-conditions
  17. 17. 17 (1.25) represents a quadratic minimization problem with linear constraints. Necessary and sufficient conditions for these kinds of problems are given by the Kuhn-Tucker conditions5. Let η ∈ℜ n represent the vector of partial derivatives of the Lagrangian L with respect to the n decision variables ω i (1 ≤ i ≤ n )  ∂L ∂L  (1.28) η' ≡ (η1 ... η n ) =  ...   ∂ω 1 ∂ω n  then the following Kuhn-Tucker conditions hold6: ω  (1.29) ( a ) η = (V A')  − λ E µ ≥ 0 λ  (b ) ω ≥ 0 λ ≥ 0 (c) ∀ ηi > 0 ⇔ ω i = 0 and ηi = 0 ⇔ ω i > 0 1≤i ≤ n (d ) Aω = b ( e ) µ' ω = E Condition (1.29 c) implies that the partial derivative ηi of L with respect to ω i equals zero if and only if ω i is greater than zero, i.e. if asset i is included by the base solution. This is the well known necessary optimum condition. But however, if ω i equals zero, then the respective partial derivative ηi is positive. That means basically that the value of the Lagrangian could be improved if ω i were not restricted by a lower boundary (in the present simple case by zero). Thus, the system of equations (1.29) may be rearranged by summarizing (1.29 a) and (1.29 d), where it is assumed that all partial derivatives are equal to zero, i.e. that all available assets show non-zero portfolio fractions. To ascertain that (1.29 a) remains satisfied, (1.29 a) is maintained as optimality condition (1.30 d): V A'  ω   µ   0 (1.30) (a)    − λ E  =   A 0 λ   0   b ( b) ω ≥ 0 λ ≥ 0 5 The derivation of the Kuhn-Tucker conditions may be found in Intriligator [1971], pp. 22-36. 6 Markowitz [1956], p. 116
  18. 18. 18 (c) ∀ η i > 0 ⇔ ω i = 0 and η i = 0 ⇔ ω i > 0 1≤ i ≤ n ω  ( d ) η = (V A')  − λ E µ ≥ 0 λ  ( e ) µ' ω = E V A' The matrix M ∈ℜ ( n + m) x ( n + m) will subsequently be defined as M ≡   , so the A 0 ω  necessary optimum condition for the vector   may be stated as7: λ  ω  −1  0 −1  µ  (1.31)   = M   +λE M   λ   b 0  Furthermore, α ∈ℜ n + m and β ∈ℜ n + m are defined as:  0  µ (1.32) α = M −1  β = M −1    b 0  By adhering to the conditions laid down in (1.30 b) and (1.32), (1.31) may be simplified, such that ω  (1.33)   = α + βλ E ≥ 0 , λ  i.e. the vector of portfolio weights may be represented as a linear function of the Lagrange multiplier λ E . The major idea of the algorithm is to find an interval for λ E , such that (1.33) is satisfied, that is to obtain zero as lower boundary for both, the vector of portfolio weights ω 7 In the following rearrangements it must be assumed that M is non-singular. Markowitz [1987] (p. 137) shows, that M is singular if and only if one of the following conditions holds: a) The range of A is uncompleted b) The set of feasible mean/variance combinations is a function parallel to the horizontal axis; in this case, there are obviously no efficient portfolios.
  19. 19. 19 and the vector of Lagrange multipliers λ . Each ω i > 0 is labelled base variable8, each ω i = 0 non-base variable9. The set of base variables constitutes a portfolio. The set of all base variables is titled quot;inquot;, the set of all non-base variables is titled quot;outquot;. Since an asset of the portfolio whose fraction equals zero will not have any impact on the variance σ 2 of the portfolio, the variance/covariance matrix Vin is defined such that all columns and rows referring to variables in quot;outquot; are replaced by identity vectors. Let σ in (1 ≤ i, j ≤ n ) ij represent the element of row i and column j of Vin and σ ij (1 ≤ i, j ≤ n ) the respective element of V, then the former statements may be analytically expressed as: σ ij if i, j ∈in  ∀ σ in = 1 if i = j and i, j ∈out ij 1≤ i, j ≤ n 0 otherwise  The vector of expected returns µ in equals µ where all rows referring to an element of quot;outquot; are replaced by 0. Ain equals A, where all columns referring to an element of quot;outquot; are replaced by 0. Therefore, given a set quot;inquot; and a portfolio Pin which is defined by the indexes contained by quot;inquot;, the following may be calculated: σ 2 = ω ' Vinω P in E = ω ' µ in b = ω ' Ain ω i = 0 ⇔ ω i ∈ out Obviously, the vector b of right-hand-sides is not dependent on the sets quot;inquot; and quot;outquot;, that is why it remains unchanged. According to (1.32), further definitions may be made if non- singularity of M in is assumed: V Ain  ' −  0 −  µ in  (1.34) M in ≡  in    α in ≡ M in1   β in ≡ M in1   Ain 0   b  0  8 I.e. following equation (1.29 c) this is equivalent to ηi = 0 9 I.e. following equation (1.29 c) this is equivalent to ηi > 0
  20. 20. 20 Equivalent to equation (1.33), the vector of portfolio fractions emerges from: ω  (1.35)   = α in + β in λ E ≥ 0 λ  In addition to (1.35) equation (1.29 a) has to be satisfied, thus: (1.36) η= V( )ω  A'   − λ E µ = V λ  ( ) A' (α in + β in λ E ) − λ E µ ≥ 0 This equation can be rearranged to give: η = γ in + δ in λ E ≥ 0 with (1.37) ( A' )α in γ in = V δ in = (V A' )β in − µ Equations (1.35) and (1.37) provide the Kuhn-Tucker conditions (1.29 a,b,d) expressed as linear functions of λ E . Markowitz [1987] (pp. 156) labelled this system of equations quot;critical linesquot;, therefore the Markowitz algorithm is often referred to as the critical line algorithm. Necessary and satisfying conditions for ω and λ are solely dependent on λ E . Since E and ω have a positive linear relationship to each other, finding feasible intervals for λ E simultaneously provides feasible intervals for E. The exact relationship between λ E and E will be discussed later, where intervals for λ E are determined, which satisfy (1.35) and (1.37). Since in the case of portfolio selection, merely ω is a matter of interest not the vector of ω  Lagrange multipliers λ , only the leading n elements of   from (1.35) are investigated. λ  According to Markowitz [1987] (p. 158), let λ a , λ b , λ c , λ d ∈ℜ be scalars which are determined as follows:  α i  γ i  max − in   max − in  ∀ λ a ≡ β iin > 0 β iin ∀ λ b ≡ δ iin > 0 δ iin 1≤ i ≤ n  1≤ i ≤ n   − ∞ for β i ≤ 0  in  − ∞ for δ i ≤ 0  in (1.38)
  21. 21. 21  α i  γ in i  min −  i in  min −  i i ∀ λ c ≡ β in < 0 β in ∀ λ d ≡ δ iin < 0 δ in 1≤ i ≤ n  1≤ i ≤ n   ∞ for β i  in ≥0  ∞ for δ i ≥ 0  in The high-index i represents the i-th element of the respective vector. λ a determines a lower boundary for λ E such that (1.35) is satisfied, λ b a lower boundary such that (1.37) is satisfied. Equivalently, λ c provides an upper boundary for λ E satisfying (1.35) whereas λ d is an upper boundary for λ E to provide a feasible solution to equation (1.37). To satisfy both equations (1.35) and (1.37) simultaneously, i.e. to satisfy the Kuhn-Tucker conditions, the minimum out of λ a and λ b determines the upper boundary for λ E and the maximum out of λ c and λ d fixes the lower boundary for λ E . The respective values are titled λ low and λ high : λ low ≡ max [λ a , λ b ] (1.39) λ high ≡ min [λ c , λ d ] with λ low ≤ λ E ≤ λ high If λ low ≤ λ high is not satisfied then there is no feasible solution which may be caused by contradictionary constraints. To put it more specifically, by calculating an efficient frontier, i.e. minimizing the variance for each level of expected return with respect to the constraints (1.25 b-d), piecewise intervals for λ E have to be calculated. I.e. critical lines are calculated piecewise, the efficient frontier is said to be segmented. The portfolios located in the transition from one critical line to another are labelled corner portfolios. Starting from the lower boundary λ low for λ E and increasing it continuously up to the upper boundary λ high leads to a violation of at least one of the Kuhn-Tucker restrictions given by (1.35) or (1.37). Thus, every ω i reaches zero if (1.35) is infringed, or ηi if (1.37) is infringed. Two cases must therefore be distinguished: 1. If some ω k (1 ≤ k ≤ n ) is responsible for the violation of (1.37), it changes from the quot;inquot;- set to the quot;outquot;-set. 2. If on the other hand η k (1 ≤ k ≤ n ) does not satisfy (1.37) anymore, the partial derivative of the Lagrangian L with respect to ω k just reaches zero. As a matter of fact, η k changes from the quot;outquot;- set of non base variables to the quot;inquot;- set of base variables. The algorithm is broken up if either λ high = ∞ or λ low = − ∞ . That means that the portfolio which has to be optimized contains only one remaining asset left and therefore a change of any variable from quot;inquot; to quot;outquot; or vice versa is not possible, no matter which value λ E takes.
  22. 22. 22 Subsequently, it will be shown that adjacent segments of the efficient frontier transit continuously into each other but not differentiable. Equations (1.38) and (1.39) guarantee that the Kuhn-Tucker conditions in (1.29) are satisfied, whereas equation (1.29 e) is not explicitly satisfied. However, it can quite simply be shown that λ E and E are in a positive-linear relationship, that is the increase of λ E will lead to a linear increase in the portfolio's expected return E. This clearly implies that for each E a λ E can be found such that (1.29 e) holds, too. Therefore, it will only be necessary to consider the top n rows of α in and β in . Substituting ω as defined in (1.35) into (1.29 e) and paying attention to the results in (1.39) provides: E − µ ' α in (1.40) E = µ ' α in + µ ' β in λ E ⇔ λ E = µ' β in Thus, a segment for λ E satisfying (1.29 e) has to be chosen in such a manner that equation (1.40) is solved for a given E. A specific application of the above equation will be given in part eight of this section. 2.3. A special case In the following part a special case of the Markowitz algorithm is investigated. It is assumed that there is a segment being on the efficient frontier where all portfolio assets are base variables, i.e. where the portfolio fractions ω i (1 ≤ i ≤ n ) are non-zero for all i. Thus out = ∅. In such a case the following must hold: V = Vin A = Ain ⇒ M = M in and α = α in µ = µ in ⇒ β = β in Then using (1.36) and (1.34) yields:   0  µ  η = (V A') M −1   + M −1  λ E  − µλ E   b 0  (1.41)  0   µ  = (V A') M −1   + λ E (V A') M −1   − µ   b  0  
  23. 23. 23 Now, M −1 may be subdivided into block matrices and redefined as:  M1 M2 M −1 =   M1 ∈ℜ nxn , M 2 ∈ℜ nxm , M 3 ∈ℜ mxn , M 4 ∈ℜ mxm  M3 M4 Let E(kl) represent the identity matrix and 0(kl) the zero-matrix of dimension kxl. Then, following the definition of the inverse of a matrix M permits the following statement: V A'  M1 M 2   (V A') M −1  E( nn ) 0( nm)  (1.42) MM −1 =    =  =  A 0   M3 M 4   AM1 AM 2   0( mn )   E( mm)  ⇒ (V ( A') M −1 = E( nn ) 0( nm) ) Substituting (1.42) into (1.41)  0   µ  (1.43) ( η = E( nn ) ) ( 0( nm)   + λ E  E( nn )  b ) 0( nm)   − µ  0    and using the definitions in (1.37) yields: (1.44) γ in = γ = 0 and δ in = δ = 0 The result of this special case may be summarized such that it is sufficient to calculate λ a and λ b with equation (1.38). Equation (1.44) determines λ c and λ d as minus infinity. This result could be anticipated because it simply means that the derivatives of the Lagrangian with respect to all portfolio fractions is zero. However, the fact that this result can be derived by using the Markowitz algorithm is of great significance. 2.4. Continuity of the efficient frontier
  24. 24. 24 The purpose of this section is to show that two adjacent segments of the efficient frontier are continuously linked (see Markowitz [1987], pp. 157-166). First, a graphical example is examined to illustrate the use of critical lines. Therefore it is assumed that six assets are included in the portfolio where the quot;inquot;- and quot;outquot;-set contain the following variables: in = {ω 1 , ω 3 , ω 4 } and out = {ω 2 , ω 5 , ω 6 } Let λiE (1≤ i≤ n) represent the value for λ E given by application of (1.38) of asset i, then the following critical lines may be assumed as an example: α1 line 1: β1 > 0 ⇒ λ E > − ≡ λ1 E β1 γ2 line 2: δ 2 < 0 ⇒ λ E < − ≡ λ2 E δ2 α3 line 3: β 3 > 0 ⇒ λ E > − ≡ λ3 E β3 α4 line 4: β 4 < 0 ⇒ λ E < − ≡ λ4 E β4 γ5 line 5: δ 5 > 0 ⇒ λ E > − ≡ λ5 E δ5 γ6 line 6: δ 6 > 0 ⇒ λ E > − ≡ λ6 E δ6 Following (1.39) determines: [ λ a = max λ1E , λ3 E ] λ c = λ4 E λa = max [λ 4 6 E,λ E ] λ c = λ2 E [ [ ⇒ λ low = max max λ1E , λ3 , max λ4 , λ6 E E E ] [ ]] [ λ high = min λ2 , λ4 E E ] Under these assumptions, critical lines may be presented in the following figure. It elucidates under which circumstances a variable leaves or enters the base. From figure 1.3 it becomes clear that λ E is bounded to the bottom by λ1E and to the top by λ4 , which implies: E
  25. 25. 25 λ low = λ1E and λ high = λ4 E If λ E > λ4 the fraction of asset four ω 4 is negative and therefore leaves the base. For E 1 λ E < λ E ω 1 is negative and leaves the base. After determining the upper and lower boundaries for λ E , a new iteration is run. If λ E is increased, then in ={ω 1 , ω 3 } and out ={ω 2 , ω 4 , ω 5 , ω 6 }, whereas if λ E is decreased, asset one leaves the base and in ={ω 3 , ω 4 } and out ={ω 1 , ω 2 , ω 5 , ω 6 }. α in , β in , γ in and δ in are adapted to the new quot;inquot;- and quot;outquot;-set. Simultaneously, the intersections between the critical lines one to six and the vertical axis are changing. λ ω Critical Line 1 Critical Line 2 Critical Line 3 Critical Line 4 Critical Line 5 3 5 2 Critical Line 6 λE λE 4 λE λE 1 λE λE Figure 1.3: Critical lines - an example The efficient frontier proves to be continuous if two adjacent segments are bound by the same λ E , one of the segments to the top, the other one to the bottom. The portfolio defined by the transition point from one segment into the next is called corner portfolio. Each efficient frontier is defined segment for segment given the different levels of λ E . With (1.40) it is clear that the segmental definition of the efficient frontier is given for the different levels of the expected portfolio return E as well. Thus, different critical lines as functions of λ E are obtained, if equation (1.35) (here: (ω λ ) as a function of λ E ) or (1.37) (here: η as a function of λ E ) is employed. Since (1.40) states that λ E and E are in a positive-monotonic relationship it follows that the efficient frontier is continuous if and only if any two segments transit continuously from one to another.
  26. 26. 26 Let a and b represent two critical lines which express two adjacent segments. Segment a is defined for greater λ E than segment b. Defining λb high as λ high of segment b according to (1.39) and λa as λ low of the adjacent segment a then a sufficient condition for continuity is: low (1.45) λa = λb low high In case (1.45) the critical lines representing segment a and b, respectively, would have an intersection. Because of (1.40) the continuity of λ E required by (1.45) implies the continuity of E and (1.30 e) implies the continuity of ω along the efficient frontier. More specifically, the equality of the composition of the portfolio in segment a for λ E = λa and of the porfolio of low segment b for λ E = λb high must be proven. Obviously this implies that the equality of the expected returns and variances at the borders of segment a and b is given, respectively. Let 0(l) denote a zero-vector of dimension l the equation (1.30 a) may be expressed slightly differently as: ′ ω   Vin Ain µ in     0( n )  (1.46)     λ  =    Ain 0 0     b  λ E  Without any loss of generality a specific asset k can be assumed to leave the base. This implies ω k changes from quot;inquot; to quot;outquot;, if E and equivalently λ E moves from segment a into segment b. It therefore holds: ∃ 1≤ k ≤ n ω k λa( ) low = 0 and ( η k λb ) high = 0 ( )a where ω k λ low denotes the portfolio fraction of asset k in segment a for λ E = λa and low ( η k λb high ) the partial derivative of the Lagrangian with respect to ω k given λ E = λb . high Furthermore, it can without loss of generality be assumed that the variables are sorted such, that asset k is the element n of the vector of portfolio fractions ω . Furthermore, v' is defined as a vector of dimension n which includes the covariances of returns of all assets with asset n, V( n −1) the variance-covariance matrix of dimension n-1 including just the first n assets, A( n −1) the first n-1 columns of A, α the last column, µ ( n−1) , ω ( n−1) the first n-1 lines of µ and ω ,
  27. 27. 27 respectively. The variance of returns, expected return and fraction of asset n is denoted by σ 2 , n µ n , and ω n . The former definitions may be analytically stated as: (1.47) ( v ' ≡ σ 1n ... σ n −1, n σ2 n )  V( n −1) v  Vin ≡   v′ σ 2 n   µ ( n −1)   ω ( n −1)  µ ≡  µn   ω ≡  ωn   ( A ≡ A( n −1) α ) Substituting these definitions into (1.46) yields:  ω ( n −1)   V( n −1) v A′ ( n −1) µ ( n −1)     0( n −1)     ωn    (1.48)  v′ σ2 α′ µn  = 0  n  λ    A α 0     b   ( n −1) 0   λE  First of all, critical line b is examined. Here, ω n =ω k is non-base variable which implies: (1.49) v = 0 σ2 = 1 α = 0 µ n = 0 n In addition to (1.49), employing (1.29 a) together with (1.37) allows the following statement:  ω ( n −1)  b   ( b ) ( η n λ high = v ′ σ 2 n  ωb  α′ µn  )  λ  n b  =0  −λ b   high  b b Using this equation for λ E = λ high and summarizing it with (1.48) and (1.49) it is obtained:
  28. 28. 28  V( n −1) 0 A(' n −1) µ ( n −1)   ω ( n −1)   0( n −1)  b        0 1 0 0   ωb   0  (1.50)    n b   =   A( n −1) 0 0 0   λ   b     − λb   0   v' σ2n α' µn   high  In a further step, critical line a is considered. It is known that for λ E = λ E = λ low the asset k a a which leaves the base ω k (λ E ) equals zero. Thus, the former statement may be expressed by:  ω ( n −1)  a   (0'(n −1) 1 0'( m −1)  ωa  0   ) λ  n a  =0  −λ a   low  a a Comprising the realization from (1.48) for λ E = λ low , a further transformation may be:  V( n −1) A(' n −1) µ ( n −1)   ω ( n −1)   0 a  v    ( n −1)   ωn a   0   v' σ2 α' µn     a  = b  n    A( n −1) α 0 0   λ       −λ a   0   0 1 0 0   low  ( ) a Since ω n λ low = 0 , the second column may be replaced by arbitrary values without any changes in the result vector at the left hand side.  V( n −1) A(' n −1) µ ( n −1)   ω ( n −1)   0 a  0    ( n −1)   ωn a   0   v' σ2 α' µn     a  = b  n (1.51)    A( n −1) 0 0 0   λ       −λ a   0   0 1 0 0   low  Since in the corner portfolio λb = λa , reversing the second and the last line of the matrix high low on the left hand side and of the vector on the right hand side of (1.51) provides a system of
  29. 29. 29 equations which is equal to (1.50)10. Thus, converging λ E to the borders of the segments a and b, respectively, yields identical vectors ω , λ and scalar λ E .  ω ( n −1)   ω a b     ( n −1)   ωb   ωa  b  = n n (1.52)    λ   λa   −λ b   −λ a   high   low  From (1.52) it is clear that the values for λ E are identical at the borders. Since the portfolio ( ) ( ) ( a a b ) ( b fractions are identical and since ω n λ low = ω k λ low = ω n λ high = ω k λ high = 0 the ) expected returns in both segments are equal. The efficient frontier is continuous. 2.5. Differentiating the efficient frontier One of the properties of the ordinary efficient frontier given by (1.6) is that it is differentiable in each point and strictly concave. This part shows that in the case restrictions a non- differentiable efficient frontier is obtained. Although this was not proved by Markowitz [1987] this seems to be an important insight. The third chapter will deal with shortfall risk approaches. Since they are based on tangents to the efficient frontier, it is a matter of great interest whether the efficient frontier is differentiable or not. It is vital to bear in mind that the efficient frontier is composed of segments. Within each segment a specific portfolio is defined by the quot;inquot;-set. Thus, within segments defined by intervals λ E ∈ λ low , λ high the efficient frontier is defined by (1.6) and which implies that it is differentiable. Therefore, it is sufficient to check differentiability at the corner portfolios. Let ein represent the unity vector where elements are replaced by 0 if the respective variable is contained in quot;outquot;. Then, the derivative of the variance of the portfolio with respect to the expected portfolio return for any λ E ∈ λ low , λ high may be expressed as11: 10 Markowitz [1987], p.171 11 see Merton [1972] and (1.6)
  30. 30. 30 ∂σ 2 − − 2 (einVin 1 ein ) E − 2 (µ'inVin 1ein ) ' ( ) (1.53) = − − − − ∂E λE (µ'inVin 1µ in )(einVin 1ein ) − (µ'inVin 1ein ) (µ'inVin 1ein ) ' Obviously, the ratio changes depending on the segment which defines λ E . Since µ in , Vin and ein are dependent on the elements of quot;inquot; and quot;outquot;, for different λ E (and consequently for different E), different λ E and different slopes can be expected. To prove that the efficient frontier is not differentiable, it is sufficient to find a counter- example. With respect to the former analysed circumstances this is not very difficult, if one considers a portfolio consisting of three assets. Without loss of generality it may be assumed that asset one first of all is included in quot;inquot; but moving to quot;outquot; as soon as the adjacent −1 segment is entered. Let vij represent element of line i and column j of the matrix Vin then the ' −1 expression einVin ein of the above equation may be restated as: n n ' −1 einVin ein = ∑ ∑ vij i =1 j =1 If on the other hand asset one is excluded then −1 einVin ein = v22 + v23 + v32 + v33 ' Obviously, both expression are not equal, unless v11 + v12 = 0 . This proves that ( ) ∂σ 2 / ∂ E λ E in (1.53) differs for two adjacent segments which implies that the slopes of the efficient frontier in two adjacent segments differ. In conclusion, the efficient frontier at the corner portfolios is not differentiable. 2.6. Concavity and minimum variance portfolio The unrestricted efficient frontier following (1.6) is characterized by strict concavity (Merton [1972]). As was the case in the previous part strict concavity of the efficient frontier is a property which will be needed for implementing shortfall risk based algorithms: if the efficient frontier were not strictly concave but linear constructing tangents would be impossible. This part confirms that concavity is also a characteristic of the restricted efficient frontier. Satisfying
  31. 31. 31 the Kuhn-Tucker conditions (1.29) implies restricting λ E by λ E ∈ λ low , λ high . Choosing an expected portfolio return E, fixed and consistent with the interval for λ E , a simpler formation of optimization problem (1.25) can be stated: (1.54) a) min σ 2 = ω' Vω ω ω b) µ' =E e' ω This implies the following Lagrangian L' (1.55) L' = σ 2 - λ E E with optimality condition: ∂ L' ∂ σ2 ∂ σ2 (1.56) =0 ⇔ −λ E = 0⇔λ E = ∂E ∂E ∂E Thus, the slope of the efficient frontier decreases with λ E . Since (1.40) states that increasing λ E corresponds to increasing expected return E, the slope of the efficient frontier is negatively related to expected returns. This is usually called strict concavity. This part showed that the efficient frontier is strictly concave even if restrictions are imposed on the feasible set. The following figure illustrates that the restricted efficient frontier is composed by several efficient frontiers, one defined for each segment. Furthermore, it shows intuitively that this fact implies concavity.
  32. 32. 32 expected return restricted efficient frontier standard deviation Figure 1.4: Restricted efficient frontier composed by three efficient frontiers according to three adjacent segments From figure 1.4 it becomes clear that the restricted efficient frontier is composed of several unrestricted efficient frontiers. The unrestricted efficient frontiers are calculated by using those assets which are contained in the respective quot;inquot;-set. The complete efficient frontier is given by segment one as long as all Kuhn-Tucker conditions are satisfied. This fact is illustrated in figure 1.4 by the dotted line with points. It is illustrated that the restricted efficient frontier in each point is determined by one of the unrestricted efficient frontiers, it moves from one to another at the corner portfolios if one of the Kuhn-Tucker conditions is no longer satisfied. That is if λ E > λ high , a new segment determines the efficient frontier. Moving from one segment to another decreases the slope. Consequently, strict concavity results. From (1.56) it follows furthermore ∂ σ2 = 0 ⇔ λE = 0 ∂E which implies that the minimum variance portfolio occurs in the segment where the interval [λ low ,λ high ] includes λ E =0. If λ E is negative, following (1.56) the slope of the efficient frontier is negative12 which implies that the portfolios are inefficient. In concluding this part, it should be emphasized that the portfolio selection process should be restricted for positive λ E . ∂σ2 12 Which implies < 0. ∂E
  33. 33. 33 2.7. Feasible base solution Finding an initial feasible base solution is the condition for an application of the critical line algorithm. First, an efficient combination of portfolio assets has to be found in order to fix the initial sets quot;inquot; and quot;outquot;. In the case of the ordinary Markowitz algorithm, which is the topic of this section, a simple solution can be applied: By definition, each portfolio with maximum available expected return is efficient. Thus, a feasible solution may be found by executing the following program: (1.57) max ω' µ s. t. Aω = b ω ≥ 0 ω Then, the initial solution is in = {ω ≥ 0 and ω solves (1.57)} . By applying (1.38) and (1.39) λ low and λ high are obtained; the algorithm can then be continued as stated in the present section. In the special case of the ordinary Markowitz algorithm a trivial solution of (1.57) is obtained, if the only asset which initially enters the base is that with the highest expected return. This naturally maximizes the expected return on the portfolio. Since this is the case only when non-negativity and equality restrictions are imposed for more general inequality restrictions (1.57) has to be solved by the simplex approach. This will be comprehensively discussed in the next section. 2.8. A numerical application A portfolio optimization problem for three portfolio assets A, B and C which are defined as in table 1.1. is considered. Table 1.1: Initial data for the numerical application covariances A B C µ A 400 150 100 15 B 225 150 10
  34. 34. 34 C 625 20 The program which has to be executed is assumed to determine efficient portfolios subject to the constraints that the portfolio fractions sum up to unity and that short sales are prohibited: min σ 2 = ω' Vω ω  ω1    (1 1 1) ω 2  = 1   ω 3 ω≥0 According to the notation used thus far it may be defined:  400 150 100  15     V ≡  150 225 150 µ ≡  10 A ≡ (1 1 1)      100 150 625  20 first step: determining a feasible base solution Given the former example the solution which maximizes the portfolio's expected return is a pure investment in asset C. An initial feasible base solution may be possible: in = {ω C } and out = {ω A , ω B } . second step: determining the first segment According to equation (1.34), Min is defined as follows: 1 0 0 0 1 0 0 0      0 1 0 0 −1  0 1 0 0  M in ≡  ⇒ M in ≡ 0 0 625 1 0 0 0 1      0 0 1 0 0 0 1 −625
  35. 35. 35 Using (1.35) and (1.37) provides αin βin α in γin δin γ in Asset − − β in δ in A 0 0 - -525 5 105 B 0 0 - -475 10 47.5 C 1 0 - 0 0 - and (1.38): λa = −∞ λc = ∞ λb = 105 λd = ∞ Thus, with (1.39) λlow = 105 and λhigh = ∞ λ E cannot be increased anymore, since the . break-up-criterion λ high = ∞ holds. λ low is determined by asset A: the partial derivative of the Lagrangian L subject to ω A reaches zero. Thus, asset A enters the base in the next step. One has in = {ω A , ω C } and out = {ω B } . third step: determining the second segment Min is defined as follows (see equation (1.34)):  400 0 100 1  0.001212 0 −0.001212 0.6364      0 1 0 0 0 1 0 0 M in ≡  −1  ⇒ M in ≡   100 0 625 1  −0.001212 0 0.001212 0.3636       1 0 1 0  0.6364 0 0.3636 −290.9091 Using (1.35) and (1.37) provides αin βin α in γin δin γ in Asset − − β in δ in
  36. 36. 36 A 0.6364 -0.006061 105 0 0 - B 0 0 - -140.9031 6.8182 20.67 C 0.3636 0.006061 -60 0 0 - and (1.38): λa = -60 λc = 105 λb = 20.67 λd = ∞ Applying (1.39) yields λlow=20.67 and λhigh=105. Since λ high of the present segment equals λ low of the former one, the numerical example obviously confirms equation (1.52). Calculating the corner portfolios following (1.35) yields:  0  0.5136     ( ) ω λ E = λ high =  0   and ω (λ E = λ low ) =  0    1   0.4864 ( ) E λ E = λ high = 20 and E(λ E = λ low ) = 17.43 This illustrates that the corner portfolios for both adjacent segments are identical which supports the former proof numerically that the efficient frontier is continuous even if restrictions are imposed. λ low is determined by asset B: the partial derivative of the Lagrangian L subject to ω B reaches zero. Thus asset B enters the base in the next step. One has in = {ω A , ω B , ω C } and out = { } . All available portfolio assets are now included in the base, it can be expected that the special case in (1.44) holds, i.e. that γ in = δ in = 0. This will be examined in the next step. fourth step: determining the third segment Min is defined as follows (see equation (1.34)):  400 150 100 1  0.003088 −0.002947 −0.00014 0.2210      M in ≡  150 225 150 1 ⇒ M −1 ≡  −0.002947 0.004632 −0.001684 0.6526   100 150 625 1 in  −0.00014 −0.001684 0.001825 01263  .      1 1 1 0  0.2211 0.6526 . 01263 −198.95
  37. 37. 37 Using (1.35) and (1.37) provides: αin βin α in γin δin γ in Asset − − β in δ in A 0.2211 0.014 -15.75 0 0 - B 0.6526 -0.0316 20.67 0 0 - C 0.1263 0.0175 -7.2 0 0 - Supporting the theoretical result from (1.44) it holds γ in = δ in = 0. Further, (1.38) provides: λa = -7.2 λc = 20.67 λb = - ∞ λd = ∞ Applying (1.39) yields λlow=-7.2 and λhigh=20.67. Again, λ high of the present segment equals λ low of the former one. Giving due considerations to the previous part it should be observed that the actual interval which determines λ E includes the minimum variance portfolio. Since all efficient portfolios are identified now, the algorithm may be stopped. For didactic reasons and to demonstrate the break-up criterion it is, however, continued.  0.5136  01203 .     ( ) ω λ E = λ high =  0    and ω (λ E = λ low ) =  0.8801    0.4864  0.0  ( ) E λ E = λ high = 17.43 and E(λ E = λ low ) = 10.61 λ low is determined by asset C: the non-negativity constraint ω C reaches zero. Thus C is eliminated out of the base in the next step. One has in = {ω A , ω B } and out = {ω C } . fifth step: determining the fourth segment Min changes following equation (1.34) to:
  38. 38. 38  400 150 0 1  0.0031 −0.0031 0 0.2308      M in ≡  150 225 0 1 ⇒ M −1 ≡  −0.0031 0.0031 0 0.7692   0 0 1 0 in  0 0 1 0       1 1 0 0  0.2308 0.7692 0 −207.69 Using (1.35) and (1.37) provides αin βin α in γin δin γ in Asset − − β in δ in A 0.2208 0.01539 -14.5 0 0 - B 0.7692 -0.0153 50 0 0 - C 0 0 - -69.23 -9.615 -7.2 and with (1.38) it is obtained: λa = -15 λc = 50 λb = - ∞ λd = -7.2 Employing (1.39) yields λlow=-15 and λhigh=-7.2. As expected, λ high of the present segment equals λ low of the former one.  01203 .  0     (  ) ω λ E = λ high =  0.8801  and ω (λ E = λ low ) = 1     0.0   0 ( ) E λ E = λ high = 10.61 and E(λ E = λ low ) = 10 Now, the non-negativity constraint is violated by asset A. Hence A leaves the base. Since it emerges that B leaves the base, too, the last iteration may be skipped. The result would be λlow=- ∞and λhigh=-14.5. As demonstrated above, five iterations were needed to calculate the entire efficient frontier in this numerical example. Efficient segments were determined. Finally, all segments are characterized by both, λ E and the expected return E:

×