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New insights on the mean-variance portfolio selection from de Finetti s suggestions Flavio Pressacco and Paolo Serafini, Università di Udine Abstract: In this paper we offer an alternative approach to

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New insights on the mean-variance portfolio selection from de Finetti s suggestions Flavio Pressacco and Paolo Serafini, Università di Udine Abstract: In this paper we offer an alternative approach to the standard portfolio selection problem following the mean-variance approach to financial decisions under uncertainty originally proposed by de Finetti. Beside being a simple and natural way, alternative to the classical one, to derive the critical line algorithm, our approach throws new light on the efficient frontier and a clear characterization of its properties. We also address the problem with additional threshold constraints.. Introduction It has been recently recognized (Rubinstein (2006)) that Bruno de Finetti (940) was the first to apply the mean-variance approach in the theory of modern finance in order to solve a proportional reinsurance problem. In de Finetti s approach, as early signaled by Pressacco (986), the reinsurance problem has much in common with the problem of asset portfolio selection, which some years later was treated in a mean-variance setting by H. Markowitz (952, 956). Indeed both problems aim at minimizing the same quadratic function subject to slightly different sets of linear constraints. To solve the portfolio selection problem, but also to face the reinsurance problem in his recent critical review of de Finetti s paper, Markowitz (2006) makes recourse to the technical tools suggested by the Karush-Kuhn-Tucker optimality conditions (Karush (99), Kuhn and Tucker (95)), from which he derives the so called critical line algorithm. In his paper de Finetti uses a different approach based on intuitive yet powerful ideas. In more detail he looks at the optimal set as a path in the n dimensional set of feasible retentions. The path starts from the point of largest expectation and goes on along a direction aiming to realize the largest advantage, properly measured by the ratio decrease of variance over decrease of expectation. In order to pass from this vision to an operational procedure, de Finetti introduces a set of key functions designed to capture the benefits obtained from small feasible basic movements of the retentions. A basic movement is simply a movement of a single retention. We showed elsewhere (Pressacco and Serafini (2007), shortly denoted as PS07) that a procedure coherently based on the key functions is able to generate the critical line algorithm and hence the whole optimal mean-variance set of a reinsurance problem. De Finetti did not treat at all the asset portfolio selection problem. Then it is natural to raise the question whether his approach may work as an alternative way also to solve the portfolio selection problem. As a matter of strategy, the idea of looking at the efficient set as a path connecting the point of largest expectation to the point of smallest variance still maintains its validity. However, in the new portfolio setting small movements of single assets cannot be accepted as feasible basic movements. Hence we make recourse to a new type of feasible basic movements characterized by a small bilateral feasible trading between pairs of assets. Accordingly, we define a new set of key functions designed to capture the consequences (still measured by the ratio decrease of variance over decrease of expectation) coming from such basic movements. On the basis of this set of key functions, we are able to build a procedure for the portfolio selection problem which turns out to be the exact counterpart of the one ruling the search for the mean-variance efficient set in the reinsurance case. Beside being the natural way to generate the critical line algorithm, this approach offers also a surprisingly simple and meaningful characterization of the efficient set in terms of the key functions. Precisely, a feasible portfolio is efficient if and only if there is a non-negative value of an efficiency index such that any feasible basic bilateral trading involving two active assets has the same efficiency, while any basic bilateral trading involving one active and one non-active asset has a lower efficiency. It is interesting to note that the efficiency index of de Finetti s approach is nothing but the Lagrange multiplier of the expectation constraint in the constrained optimization setting. In our opinion these results, that we have obtained as a proper and coherent extension of de Finetti s approach, throw new light on the intrinsic meaning and the core characteristics of the mean-variance optimum set in classical portfolio selection problems. The same approach works also in case there are additional upper and/or lower threshold constraints, through a modification of the key functions defined now on an extended space in order to capture beside the direct also the indirect benefit coming from active constraints. The plan of the paper is as follows. In Section 2 we briefly recall the essentials of de Finetti s approach to the reinsurance problem. In Section we discuss the logic behind the introduction of the new key functions suited for the portfolio selection problem. Section 4 is devoted to the application of the new key functions approach to the portfolio selection problem in the unrealistic but didascalic case of no correlation. This opens the way to Section 5 where the application of the new approach to the general case with non-null correlation is discussed. In Section 6 we introduce the threshold constraints. Connections of our approach with classical mathematical programming is discussed in Section 7 for the standard case and in Section 8 for the case with additional constraints. Two examples follow in Section 9 before the conclusions in Section 0. An appendix is devoted to prove some formulas. 2. A recall of de Finetti s approach to the reinsurance case We briefly recall the essentials of de Finetti s approach to the reinsurance problem. An insurance company is faced with n risks (policies). The net profit of these risks is represented by a vector of random variables with expected value m := {m i 0 : i =,..., n} and a non-singular covariance matrix V := {σ ij : i, j =,..., n}. The company has to choose a proportional reinsurance or retention strategy specified by a retention vector x. The retention strategy is feasible if 0 x i for all i. A retention x induces a random profit with expected value E = x m and variance V = x V x. A retention x is by definition mean-variance efficient or Pareto optimal if for no feasible retention y we have both x m y m and x V x y V y, with at least one inequality strict. Let X be the set of optimal retentions. 2 The core of de Finetti s approach is represented by the following simple and clever ideas. The set of feasible retentions is represented by points of the n dimensional unit cube. The set X is a path in this cube. It connects the natural starting point, the vertex of full retention (with the largest expectation E = i m i), to the opposite vertex 0 of full reinsurance (zero retention and hence minimum null variance). De Finetti argues that the optimum path must be the one which at any point x of X moves in such a way to get locally the largest benefit measured by the ratio decrease of variance over decrease of expectation. To translate this idea in an operational setting de Finetti introduces the so called key functions F i (x) := 2 V x i = E x i n j= σ ij m i x j, i =,..., n Intuitively these functions capture the benefit coming at x from a small (additional or initial) reinsurance of the i-th risk. The connection between the efficient path and the key function is then straightforward: at any point of X one should move in such a way to provide additional or initial reinsurance only to the set of those risks giving the largest benefit (that is with the largest value of their key function). If this set is a singleton the direction of the optimum path is obvious; otherwise x should be left in the direction preserving the equality of the key functions among all the best performers. Given the form of the key functions, it is easily seen that this implies a movement on a segment of the cube characterized by the set of equations F i (x) = λ for all the current best performers. Here λ plays the role of the benefit parameter. And we continue on this segment until the key function of another non-efficient risk matches the current decreasing value of the benefit parameter, thus becoming a member of the best performers set. Accordingly, at this point the direction of the efficient path is changed as it is defined by a new set of equations F i (x) = λ, with the addition of the equation for the newcomer. A repeated sequential application of this matching logic defines the whole efficient set. De Finetti (940) offers closed form formulas in case of no correlation and gives a largely informal sketch of the sequential procedure in case of correlated risks. As pointed out by Markowitz (2006) in his recent critical review, de Finetti overlooked the (say non-regular) case in which at some step it is not possible to find a matching point along an optimum segment before one of the currently active variables reaches a boundary value (0 or ). We showed elsewhere (PS07) that a natural adjustment of the key functions procedure offers, also in the non-regular case, a straigthforward approach to solve the problem. Precisely, if a boundary event happens, the direction of the optimum is changed simply by forcing the boundary risk to leave the set of actively reinsured risks and freezing it at the level of full reinsurance (in case of boundary at level 0) or, less likely, at the level of full retention (in case of boundary at level ). It is instructive to look carefully at what happens to the key function of a risk having reached the boundary level 0. In this case of course no additional reinsurance may take place and the meaning of the key function F i (x), if x i = 0, should be seen as the burden (measured by increase of variance over increase of expectation) obtained from coming back from full reinsurance. Even if this makes no sense in isolation, it should be kept in consideration in a mixture of reinsurance moves involving more than one risk. Intuitively, in order to stay frozen at level 0 and not being involved in reinsurance movements, the risk ought to be inefficient. In other words its key function value should remain at least for a while at a level of burden larger than the current level of the efficiency parameter λ. Moreover, this makes clear that the proper economic characterization of a mean-variance efficient reinsurance choice is resumed by the following motto: internal matching coupled with boundary dominance. Precisely, a point x is optimal if and only if there is a non-negative value of an efficiency parameter λ such that for internal risks (0 x i ) the key functions of all internal variables match the value of the efficiency parameter, while boundary variables are dominated, that is exhibit less efficiency. More precisely, this means F i (x) λ for risks for which F i (x) represents a benefit, (lower benefit lower efficiency for x i = ), while F i λ if F i (x) captures a burden (higher burden lower efficiency for x i = 0). The dominance is driven by strict inequalities for any boundary variable in the internal points of the piecewise linear efficiency path, while (under a proper mild non-degeneracy condition) at the corner points of the path just one boundary variable satisfies the strict equality F i (x) = λ. Indeed at such corner points x, such a risk either has just matched from below (from F i (x) λ) or from above (from F i (x) λ) the set of the other previously efficient risks and is just becoming (at the corner) an efficient variable, or rather it has just reached a boundary value from an internal one and thus it is just leaving at the corner point the efficiency set becoming henceforth strongly dominated or inefficient. The distinction between those two possibilities needs a dynamic analysis of the efficient set and not merely the observation of conditions prevailing at the corner point. An exception to this simple rule is to be found for vertex points of the unit cube where all variables are boundary variables (or the set of internal variables is empty). Then in some sense they are all inefficient, because they keep a bigger burden (if at level 0) or a lower benefit (at level ) in comparison with (an interval of) values of the efficiency parameter λ. For further details on this point see Section 5 of PS07.. Key functions in the asset portfolio problem Let us recall the essentials of the standard asset portfolio problem we first investigate in this paper. An investor is faced with n assets. The net rate of return of these assets is represented by a vector of random variables with expected value m := {m i : i =,..., n} and a non-singular covariance matrix V := {σ ij : i, j =,..., n}. The investor has a budget to invest in the given assets. It is convenient to normalize the budget to. Let x i be the fraction of budget invested in the asset i. If short positions are not allowed, the portfolio strategy is feasible if x i 0 for all i and i x i =. A portfolio x induces a random rate of return with expected value E = x m and variance V = x V x. A portfolio x is by definition mean-variance efficient or Pareto optimal if for no feasible portfolio y we have both x m y m and x V x y V y, with at least one inequality strict. Let X be the set of optimal portfolios. De Finetti did not treat at all (neither in his 940 paper nor even later) the asset portfolio problem. This was later analyzed and solved by Markowitz by making recourse to technical tools provided by constrained optimization techniques meanwhile developed mainly by Kuhn and Tucker (95). We think that de Finetti s idea of moving along the feasible set, starting from the point of largest expectation and following the path granting at any point the largest efficiency until an end point of minimum variance is reached, is valid also in the asset portfolio problem. However, we cannot use the same set of key functions exploited in the reinsurance problem. Indeed, beyond individual constraints, the collective constraint i x i = implies that benefits driven by movements 4 of single assets, as captured by the key functions F i (x) of the reinsurance case, do not make any sense. At first glance this seems to exclude the possibility of applying de Finetti s key function approach to the asset portfolio selection problem. On the contrary, we will show that, through a proper reformulation of key functions, it is possible to build a procedure mimicking the one suggested by de Finetti for the reinsurance case and to obtain in a natural and straightforward way something analogous to the critical line algorithm to compute the meanvariance efficient set. Furthermore, from this idea we get a simple and meaningful characterization of the efficient set. Henceforth we will assume a labeling of the assets coherent with a strict ordering of expectations, namely m m 2 ... m n. In our opinion this is not a restrictive hypothesis; after all the event of finding two assets with precisely the same expectation could be considered as one of null probability. Recall that we want to capture through the key functions the benefit (or burden) coming from a small portfolio movement on the feasible set. In order to reach this goal let us consider for the moment simple basic feasible movements, that is portfolio adjustments coming from small tradings between asset i (decreasing) and asset j (increasing). More formally and with reference to a portfolio x with positive quotas of assets i and j, let us call bilateral trading in the i-j direction an adjustment of the portfolio obtained through a small exchange between i decreasing and j increasing. If the benefit (burden) measure is given by the ratio decrease (increase) of variance over decrease (increase) of expectation the key functions ought to be defined as: F ij (x) := 2 V x V i x j E x E = i x j n h= σ ih σ jh m i m j Note that the sign of the denominator is the same as (j i); and also that, as both i and j are active assets (that is with positive quotas) in the current portfolio x, a feasible bilateral trading may happen also in the j-i direction. Then both F ij (x) and F ji (x) describe the results of a feasible bilateral trading at x. Moreover it is immediate to check that F ij (x) = F ji (x). Yet it is convenient to think that the economic meaning of the two functions is symmetric: precisely, if without loss of generality i is less than j, F ij (x) describes a benefit in algebraic sense, while F ji (x) describes a burden. If in the current portfolio x i is positive and x j is null, then the only feasible bilateral trading may be in the direction i-j. And F ij (x) describes a benefit if i is less than j or a burden in the opposite case. Obviously, if both x i and x j are at level 0 no feasible trade between i and j may take place. x h 4. The standard case with null correlation Armed with these definitions of key functions and their economic meaning, let us look at the meanvariance efficient path starting from the case of n risky asset with no correlation. This case is clearly unrealistic in an asset market, but it is nevertheless didascalic. Then the key functions simplify to F ij (x) = σ ii x i x j m i m j 5 The starting point of the mean-variance path is (, 0, 0,..., 0), the point with largest expectation (recall the ordering convention). Let us denote this starting point as ˆx 2. The choice of the index 2 may sound strange, but it is convenient because it indicates that at ˆx 2 the second asset starts to be active (see below). We leave ˆx 2 in the direction granting the largest benefit that is the largest value over j of σ F j (ˆx 2 ) = m m j (note that σ ii 0 by hypothesis). It is trivial to check that it is j = 2. Then the initial benefit is ˆλ 2 := F 2 (ˆx 2 ) = m m 2 σ This means that the bilateral trading of the type -2 gives the largest benefit and dictates the efficient path leaving ˆx 2 in the direction ( ε, ε, 0,..., 0). The bilateral trading -2 remains the most efficient until we find a point on the above segment where the benefit granted by this trade is matched by another bilateral trade, that is until the nearest point, let us label it as ˆx, where F 2 (ˆx ) = F j (ˆx ) for some j 2. It is easy to check that j = and that where the benefit is ˆx = m m σ + m2 m σ 22 ( m m σ, ˆλ := F 2 (ˆx ) = F (ˆx ) = m 2 m m m σ σ 22, 0,..., 0 + m2 m σ 22 Going back to the reinsurance terminology, ˆx is nothing but a corner point of the matching type. Now the matching of the two key functions F 2 and F signals that at ˆx the bilateral trade -2 has the same efficiency (the same benefit) of the trade -. Let us comment on this point. Remark a): at ˆx also the bilateral trade 2- matches the same benefit ˆλ. Indeed the following result holds also in case of non-null correlation: for any triplet i, j, h of assets and any portfolio x such that at least x i and x j are strictly positive F ij (x) = F ih (x) = λ implies F jh (x) = λ. If i j h the key functions value describes a matching of the benefits given by any bilateral trade i-j, i-h, j-h. The proof is straightforward (see the Corollary in Section 6). Remark b): small feasible changes in a portfolio composition may come as well from joint movements of more than two assets, to be seen as a multilateral trading. But of course any multilateral feasible trade may be defined as a proper combination of feasible bilateral trades. For example δx i = 0.000, δx j = , δx h = comes from combining δx i = , δx j = and δx i = , δx h = ; and the multilateral benefit is the algebraic combination of the bilateral benefits and if all the bilateral trades share the same benefit index then the benefit of the multilateral trade matches any combination of benefits f

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