Annals of Operations Research 97 (2000) 131–141
131
Optimal portfolio choice under a liability constraint Leszek S. Zaremba and Włodzimierz H. Smole´nski † Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warsaw, Poland E-mail: {lzaremba; smolensk}@ibspan.waw.pl
The problem of characterizing the least expensive bond portfolio that enables one to meet his/her liability to pay C dollars K years from now is dealt with in this article. Bond prices are allowed to be either overpriced or underpriced at the purchase time, while at the sale time the bonds are suppose to be fairly priced. Assuming shifts in spot rates to occur instantly after the acquisition of a bond portfolio Z and to follow fairly general type of behavior described by the condition (2), we give both necessary and sufficient conditions for Z to solve the immunization problem above. Our model is general enough to cover situations with twists in the yield curve. Making use of the K-T conditions, we explain in remark 7 why we focus on search of an optimal portfolio in the class of barbell strategies. Finally, by means of the K-T conditions we find an optimal bond portfolio which solves the immunization problem.
1.
Introduction
The immunization problem (P ) of meeting the liability to pay C dollars K years from now irrespective of unknown shifts ht in spot rates yt (yt → yt + ht ) is dealt with in this paper. Problem P has to be solved by means of a portfolio consisting of treasury bills and fixed coupon bonds. Here, K is any number between the shortest and the longest maturity of the fixed income instruments we are concerned with. The shifts are supposed to occur just once in the period of K years, instantly after the acquisition of an immunization portfolio. In the classical approach one assumes that spot rates yt , as well as expected shifts ht in the latters, were identical, that is, yt ≡ y, ht ≡ h, so that one dealt with parallel shifts (y → y + h). It is known since the pioneering work of Macaulay, Redington and Fisher [5,7,10] that any bond with duration of K years and an appropriate investment value solves this basic immunization problem in the case when the yield curve is flat and the shifts are parallel. In Zaremba [11–13] a more general model was dealt with by assuming unknown changes ht in arbitrary spot rates yt1 , yt2 , . . . , ytn to occur in a future to be proportional to their values plus one, e.g., ht ht1 = , 1 + yt 1 + yt1 †
Deceased.
J.C. Baltzer AG, Science Publishers
t ∈ {t1 , t2 , . . . , tn }.
(1)
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The immunization problem has attracted a lot of attention, resulting in an extensive literature. Good survey of known results is presented in Priesman and Shores [9]; see also [1–3,6,8]. In this paper, we are looking for a solution to the immunization problem P in a general context. First of all, we suppose that the bond market under consideration is temporarily unbalanced. It means, we suppose that at the time of bond purchases some bonds are either overpriced or underpriced, while after changes in spot rates, the bond’s prices will adjust accordingly to reach their investment value levels. For a given bond A, let P A stand for the current price of A, while P (A) denotes the investment value of A, that is, the present value of the cash flow to be generated by A in future. Since one can always express P A in terms of P (A) by means of the formula P A = P (A)[1 + m(A)], the bond A is said to be overpriced (underpriced) if m(A) > 0 (m(A) < 0). In addition, we allow spot rate shocks ht to satisfy the general type condition ht ht1 = gt , 1 + yt 1 + yt1
t ∈ {t1 , t2 , . . . , tn },
(2)
where the coefficients gt are known a priori (usually estimated empirically by an investor based on historical data). The assumption (2) is general enough to cover situations with twists in the yield curve. Clearly, the coefficients gt characterize changes in a term structure of interest rates under consideration (such as the one in Poland, Canada, Singapore, etc.). The only place where no knowledge about gt is needed by the investor is theorem 2 stating that the least amount of money needed to acquire a portfolio Z solving problem P equals C ∗ 1 + m(Z) , (3) where C ∗ stands for the present value of the liability, that is, C ∗ = C[1 + yk ]−k with yk meaning the appropriate spot rate, and m(Z) being a suitable defined misvaluation coefficient for Z (see (9b) and (6)). Assuming knowledge of the coefficients gt to an investor, we give in subsequent theorems a necessary condition for Z to be the least expensive portfolio solving the above immunization problem P and a sufficient condition for Z to solve P . Finally, by means of the Kuhn–Tucker conditions, we prove a result stating that the least expensive immunization portfolio consists of just two types of bonds. In addition, we construct it. 2.
Unanticipated rate of return
To simplify the notation, we shall write in what follows y = (yt1 , yt2 , . . . , ytn ) and h = (ht1 , ht2 , . . . , htn ) to denote the vector of all spot rates (representing the current term structure of interest rates) and the vector of all shifts in the spot rates that have taken place recently. To start with let us recall some basics. Each bond A will
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be identified with a sequence (CtA1 , CtA2 , . . . , CtAn ), where CtA1 , CtA2 , . . . , CtAn represent fixed coupons to be paid by A at instances t1 , t2 , . . . , tn , respectively; for simplicity of notation, a face value payment has been added to the last nonzero coupon (if any). Since in our presentation t1 , t2 , . . . , tn represent all instances in future when some bond promise to pay coupons, therefore for a concrete bond A many of Ct ’s are equal to zero. If a particular bond generates just one payment (equal to its face value), it is said to be a zero-coupon bond. By the intrinsic (investment) value of a fixed coupon bond A, P (A), one understands the present value of the cash flow to be generated by A, that is, P (A) =
t=t Xn
CtA (1 + yt )−t .
(4)
t=t1
We shall sometimes write P (A, y) instead of P (A) to indicate dependence of P (A) on the current term structure of interest rates y = (yt1 , yt2 , . . . , ytn ). It is well known (see, for example, Zaremba [11,12]) that when a market is in an equilibrium (m(A) = 0 for each bond A) and (1) holds then the unanticipated rate of return on each bond A due to proportional shifts in spot rates (for the definition of this concept see, for example, Elton and Gruber [4]) dP (A) P (A, y + h) − P (A, y) = P (A, y) P (A) is given by 2 dP (A) ˜ , ˜ 2+O h ˜ h ˜ + V (A) h = −D(A)h P (A)
˜= h
ht1 1 + yt1
(5)
˜ = 0, where O(h) with lim h→0 ˜ D(A) =
t=t Xn
txA t ,
V (A) =
t=t1
t=t Xn t=t1
1 t(t + 1)xA t 2
stand for the duration and convexity of the bond A, respectively. Above xA t is the weight of the coupon CtA to be paid by A, that is, xA t =
CtA (1 + yt )−t . P (A)
Clearly, the formula (5) results immediately from the Taylor expansion of P (A, y). It is well known (see, for example, Zaremba [11, theorem 3]) that (5) extends to all portfolios Z with Z = (x1 , x2 , . . . , xr ) denoting the portfolio consisting of x1 copies of bond A1 , x2 copies of bond A2 , etc. If by Pi we denote the intrinsic value of a single bond Ai , that is, Pi = P (Ai ), then zi =
xi Pi P (Z)
(6)
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will represent the proportion of money spent P on the purchase of all bonds Ai present in Z to the price of Z, which implies zi = 1, zi > 0. Clearly, a portfolio Z = (x1 , x2 , . . . , xr ) can thus be also represented by the pair (P (Z), z), where z = (z1 , z2 , . . . , zr ) is the vector of weights (6) with r denoting the number of all types of bonds available on a given financial market. What’s more, it is straightforward to see that the relationship (5) is also valid in case (2) with the duration and convexity given by D(A) =
t=t Xn
t·
gt xA t ,
t=t1
t=t 1 Xn V (A) = t(t + 1)gt2 xA t . 2 t=t
(7)
1
By the same reasons as in case (1), the relationship (5) can be further generalized to cover the case of all portfolios Z. Suppose for the remainder of the paper that the financial market under consideration is temporarily unbalanced, that is, for some bonds A, the price P A (y) 6= P (A, y) and that the bond prices after shifts in spot rates will coincide with their intrinsic values, that is, the equality P A (y + h) = P (A, y + h) will hold for all bonds A. Taking this into account and the relationship (5), we conclude that the unanticipated rate of return on a bond A due to shocks in spot rates satisfies the relationship ˜ + V (A)(h) ˜ 2 + O(h)( ˜ h) ˜ 2 − m(A) e P (A, y + h) − P A (y) −D(A)h dP (A) = = , (8) P A (y) P A (y) 1 + m(A) where, clearly, P (A, y + h) stands for both the new investment value of A and the new price of A (after the shocks in spot rates). It is straightforward to notice that formula (8) is a generalization of formula (5) to the case when the market is temporarily unbalanced and (2) holds. Moreover, (8) extends to all portfolios Z = (P (Z), z), e.g., e dP (Z) P (Z, y + h) + P Z (y) = P Z (y) P Z (y) ˜ 2 + O(h)( ˜ h) ˜ 2 − m(Z) −D(Z)h + V (Z)(h) = , 1 + m(Z) where m(Z) =
r X
zi m(Ai ),
z = (z1 , z2 , . . . , zr ),
(9a)
(9b)
i=1
while the duration D(Z) and convexity V (Z) are given by (see(7)) D(Z) =
t=t Xn t=t1
t
· gt xzt ,
t=t 1 Xn V (Z) = t(t + 1)gt2 xzt 2 t=t 1
(10)
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with the weights xzt and the investment value of Z given respectively by Pr r X C m (1 + yt )−t z xt = m=1 t xi P (Ai ), Z = (x1 , x2 , . . . , xr ). , P (Z) = P (Z)
135
(11)
i=1
3.
Necessary and sufficient conditions for Z to solve the immunization problem
Let us start with the observation that m(Z) = m[P (Z), z] does not depend on P (Z). In fact, if P (Z1 ) = λP (Z), λ 6= 1, and the portfolios Z, Z1 have the same vectors of weights, then based on (6), Z1 is a “copy” of Z, that is, Z1 = λZ = (λx1 , λx2 , . . . , λxr ) whenever Z = (x1 , x2 , . . . , xr ) and consequently m(Z) = m(Z1 ). Remark 1. P z (y) = P (Z, y)[1 + m(Z)]. Proof.
It follows from (6), (9b) and (11) that P z (y) =
r X k=1
r X xk P (Ak , y) 1 + m(Ak ) = P (Z, y) + xk P (Ak ) · m(Ak )
"
= P (Z) 1 +
r X k=1
#
k=1
zk m(Ak ) = P (Z) 1 + m(Z) .
Let O stand for a zero-coupon bond (real or imaginary) paying C dollars K years from now. Thus, P (O) = P (O, y) = C(1 + yk )−k = C ∗ can be thought of as the intrinsic value of bond O. The following fact is shown below without assuming the coefficients gt are known to an investor. Theorem 2. Assume that shocks in spot rates are going to happen just once in the period of K years, instantly after the acquisition of a portfolio Z. If Z solves the immunization problem P to pay C dollars K years from now on a temporarily unbalanced market (some bonds are overpriced or underpriced), then its price P z (y) > C ∗ [1 + m(Z)]. What’s more, if P z > C ∗ [1 + m(Z)], then one can find a less expensive portfolio Z1 = λZ = (λx1 , λx2 , . . . , λxn ), λ < 1, solving problem P for which P Z1 = C ∗ [1 + m(Z1 )] = C ∗ [1 + m(Z)]. Proof. We first observe that a portfolio Z = (x1 , x2 , . . . , xr ) solves the immunization problem P if and only if P Z, y + h > P O, y + h , (12) the latter meaning, according to our notation, the investment value of bond O after the shifts h = (ht1 , ht2 , . . . , htn ) in the term structure of interest rates y = (yt1 , yt2 , . . . , ytn ). Assuming P (O, y + h) > P (Z, y + h) we see that the value of Z after the shocks in
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spot rates, P (Z, y + h), will grow up to the amount P (Z, y + h)(1 + yk + hk )k at time K, which is less than P (O, y + h)(1 + yk + hk )k = C whatever yk and hk mean. This contradicts the assumption that Z solves the problem P . Having proved (12), we deduce from (5) that 2 P (O, y + h) ˜ ˜ h ˜ 2−O h ˜ + 1 K(K + 1)g2 h = 1 − Kgk h k P (O, y) 2
(13a)
because D(O) = K, V (O) = 12 K(K + 1) and, by our assumption, m(O) = 0. Similarly, based on (9a), we infer that ˜ h) ˜2 ˜ + V (Z)(h) ˜ 2 − O(h)( P (Z, y + h) 1 − D(Z)h = . P z (y) 1 + m(Z)
(13b)
Since P (Z, y + h) > P (O, y + h) and P (O, y) = C ∗ , the following inequality results P (O, y + h) C∗ P (Z, y + h) > · . P z (y) P (O, y) P z (y) Making use of (13a) and (13b) we finally get 2 ˜ 2 ˜ 1 ˜ ˜2 z ∗ [1 + m(Z)] 1 − Kgk h + 2 K(K + 1)gk (h) − O(h)(h) P (y) > C ˜ + V (Z)(h) ˜ 2 − O(h)( ˜ h) ˜2 1 − D(Z)h ˜ Passing to the limit with h ˜ tending to zero, the inequality for all small enough h. z ∗ P (y) > C [1 + m(Z)] results. If, however, P (Z, y + h) were equal to P (O, y + h) then we would have P z = C ∗ [1 + m(Z)] by the same reasoning as the one presented above. To complete the proof, suppose that P z (y) > C ∗ [1+m(Z)]. It follows from the last observation and (12) that P (Z, y + h) > P (O, y + h), what implies that for some λ satisfying 0 < λ < 1 the portfolio Z = λZ = (λx1 , λx2 , . . . , λxr ) solves the problem P with P (Z, y + h) + P (O, y + h), the latter implying P Z (y) = C ∗ [1 + m(Z)] = C ∗ [1 + m(Z)] (see (6) and (9b) for the proof of the equality m(Z) = m(Z)). Corollary 3. If a bond Z solves the immunization problem P on a temporarily unbalanced market then P (Z, y) = P (Z) > C ∗ = P (O) = P (O, y), that is, the investment value of Z is at least the same as the present value of the liability C to be paid at time K. If Z is the least expensive immunization portfolio, then P (Z) = C ∗ = P (O). The proof readily follows from theorem 2 and remark 1. Theorem 4. If Z is the least expansive portfolio solving the immunization problem above, then P z (y) = C ∗ 1 + m(Z) , P (Z, y) = C ∗ , D(Z) = Kgk , V (Z) > V (O). (14) When, in addition to (14), V (Z) > V (O), that is, the convexity of Z is greater than the convexity of the bond O (paying C dollars K years from now), then Z solves
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the immunization problem for “small” shifts h = (ht1 , ht2 , . . . , htn ) and is the least expensive portfolio with this property. Proof. We already know from the proof of theorem 2 that Z satisfies the inequality (12) and consequently, based on corollary 3, we infer that P (O, y + h) P (Z, y + h) > P (Z, y) P (O, y)
(15)
which by virtue of (13a) and (13b) is equivalent to 2 2 ˜ . ˜ + V (Z) − 1 K(K + 1)g2 h ˜ h ˜ >O h (16) Kgk − D(Z) h k 2 ˜ in such a way that h[Kg ˜ If D(Z) were different from Kgk , then choosing h k− D(Z)] < O, we would arrive at a contradiction with (16). In this way we have shown D(Z) = Kgk . We can see from (16) that V (Z) > V (0). It follows from theorem 2 that the least expensive portfolio Z solving problem P satisfies P z (y) = C ∗ [1 + m(Z)]. Combining this result with corollary 3, we see that (14) is a necessary condition. The fact that (14) is also a sufficient condition for Z ∗ satisfying V (Z ∗ ) > V (O) = 1 2 2 K(K + 1)gk to solve problem P for small shifts h readily follows from (16). In fact, under the assumptions made, (16) is obviously valid and consequently (15) holds. Taking into account that P (Z ∗ , y) = C ∗ = P (O, y), the inequality (12) results which, in turn, implies that Z ∗ solves the problem P . The fact that P (Z ∗ , y) = C ∗ means Z ∗ is the least expensive portfolio solving problem P . 4.
Construction of the least expensive immunization portfolio It follows from theorem 4 that the least expensive immunization portfolio Z ∗ = is a solution to the following constrained problem: m(Z ∗ ) = min m(Z): P (Z, y) = C ∗ , D(Z) = Kgk , (17)
(P (Z ∗ )z ∗ )
provided V (Z ∗ ) > V (0), where, as we know (see (6), (10), (11)) P (Z, y) = P (Z) =
r X
xi Pi =
i=1
r X i=1
zi P (Z),
D(Z) =
t=t Xn
t · gt xzt .
t=t1
It follows from the observation made at the very beginning of chapter 3 that minimizing m(Z) over all Z’s satisfying P (Z) = C ∗ , D(Z) = Kgk is the same as minimizing m(Z) over all Z’s satisfying D(Z) = Kgk . Therefore, our minimization problem (18) can be replaced with the problem of finding a portfolio Z ∗ = (P (Z ∗ ), z ∗ ), z ∗ = (z1∗ , z2∗ , . . . , zr∗ ) satisfying ) ( r r X X X X m(Z ∗ ) = zi∗ mi = min zi mi : zi = 1, zi > 0, zi Di = Kgk , (18) i=1
i=1
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with mi = m(Ai ), Di = D(Ai ). We are going to demonstrate that a solution to problem (18) can be found among such portfolios Z whose weight vectors z = (z1 , z2 , . . . , zr ) have just two coordinates, say zi , zj , different from zero. Such portfolios Z will be denoted in what follows by Z = {zi , zj } for short, to indicate that Z contains two types of bonds only, namely Ai and A j. P Since the minimum of the continuous function zi mi is sought on a compact r (closed and bounded) set in R , at least one optimal solution z ∗ = (z1∗ , z2∗ , . . . , zr∗ ) must exist by the Weierstrass theorem. The Lagrangian associated with this problem takes the form r X zl ml − λl zl + µ1 zl + µ2 zl Dl . (19) L(z1 , z2 , . . . , zr ) = l=1
It is well known that the Kuhn–Tucker conditions are here necessary and sufficient for (z1∗ , z2∗ , . . . , zr∗ ) to be an optimal solution to problem (18). They take the form ∂L(z ∗ ) = ml − λl + µ1 + µ2 Dl , ∂zl λl zl∗ = 0, λl > 0, zl∗ > 0, 0=
(20) (21)
that is, (21) holds and λl = ml + µ1 + µ2 Dl .
(22)
Proposition 5. If at least there coordinates zi∗ , zj∗ , zk∗ are different from zero, then (i)
mj Di − mi Dj mk Dj − mj Dk mk Di − mi Dk = = = µ1 , Dj − Di Dk − Dj Dk − Di
(ii)
mi − mj mj − mk mi − mk = = = µ2 . Dj − Di Dk − Dj Dk − Di
Proof. It follows from (21) that λi = λj = λk = 0 and consequently µ1 + µ2 Di = −mi , µ1 + µ2 Dj = −mj , µ1 + µ2 Dk = −mk . Solving the first two equations for µ1 , µ2 one obtains mj Di − mi Dj = µ1 , Dj − Di
mi − mj = µ2 . Dj − Di
Solving the second and the third equations for µ1 , µ2 and next the first and the third equations also for µ1 , µ2 one obtains the remaining part of the proposition. Remark 6. The conditions (i), (ii) of proposition 5 are equivalent to the seemingly weaker ones: (iii)
mk Dj − mj Dk mj Di − mi Dj = = µ1 , Dj − Di Dk − Dj
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mj − mk mi − mj = = µ2 . Dj − Di Dk − Dj
(iv)
A straightforward proof is left for the reader. Remark 7. The conditions (iii) and (iv) of remark 6 can be reformulated as follows: the situation when an optimal solution (z1∗ , z2∗ , . . . , zr∗ ) has at least three of its coordinates different from zero is very rare. It takes place when for given misvaluation coefficients mi , mj and known apriori durations Di , Dj , Dk the misvaluation coefficient mk attains exactly that single value for which the relationships (iii) and (iv) are satisfied. Taking remark 7 into account, we shall be striving to find a solution to problem (18) among portfolios Z = {zi , zj } for which we will have zi + zj = 1,
zi > 0,
zj > 0,
zi Di + zj Dj = Kgk .
(23)
Changing the numbering of bonds A1 , A2 , . . . , An if necessary one may assume that the durations of bonds Ai increase with the index i, so that, we will have Di < Kgk < Dj
(24)
for bonds Z = {zi , zj } satisfying (23). Denote by Z the class of such portfolios. Theorem 8. A portfolio Z ∗ solving the minimization problem (18) can be found among the portfolios Z = {zi , zj } ∈ Z, provided V (Z ∗ ) > V (0) . Strictly speaking, if Z ∗ = {zc∗ , zd∗ } ∈ Z satisfies the condition (25) zc∗ mc + zd∗ mc = min zi mi + zj mj : {zi , zj } ∈ Z , then Z ∗ solves the problem (18) and (17) of finding the least expensive portfolio solving the immunization problem P of meeting the liability to pay C dollars K years from now irrespective of unknown shifts ht in spot rates yt provided the shifts in the term structure of interest rates obey the key assumption (2) with known coefficients gt . Proof. We are going to justify our claim by demonstrating the relationships (21) and (22). Since for each portfolio {zi , zj } ∈ Z we have zc∗ mc + 1 − zc∗ md 6 zi mi + (1 − zi )mj , (26) where zc∗ Dc + (1 − zc∗ )Dd = Kgk and zi Di + (1 − zi )Dj = Kgk , we infer from (23) that Dj − Kgk Dd − Kgk , zi = (27) zc∗ = Dd − Dc Dj − Di and consequently zd∗ = 1 − zc∗ =
Kgk − Dc , Dd − Dc
zj = 1 − zi =
Kgk − Dj . Dd − Di
(28)
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Clearly, we silently suppose Dd 6= Dc and Dj 6= Di , which is fully justifiable. Taking these relationships into account, we can rewrite (26) in the form of the inequality Dj − Kgk Dd − Kgk Kgk − Dc Kgk − Di mc + md 6 mi + mj . Dd − Dc Dd − Dc Dj − Di Dj − Di
(29)
In order to demonstrate that z ∗ = {zc∗ , zd∗ } is a solution to the problem (18), we want to make sure that the Kuhn–Tucker conditions (21), (22) are met for appropriate parameters µ1 , µ2 , that is to say, µ1 + µ2 Dc = −mc ,
µ1 + µ2 Dd = −md
(λc = 0 = λd )
(30)
and λl = ml + µ1 + µ2 Dl > 0 for all indices l = 1, 2, . . . , r different from c and d. Solving (30) for µ1 , µ2 we arrive at md Dc − mc Dd (mc − md ) µ1 = , µ2 = , Dd = 6 Dc . (31) Dd − Dc Dd − Dc What remains to prove is the following set of inequalities: λl = ml +
md Dc − mc Dd (mc − md ) + Dl > 0, Dd − Dc Dd − Dc
l 6= c, l 6= d.
(32)
To justify (32), let us first assume Dl < Kgk . It follows from (25) that zc∗ mc +zd∗ md 6 zl ml + zd md whenever {zc , zd } ∈ Z. Taking into account (27) and (28) we conclude that Kgk − Dc Dd − Kgk Kgk − Dl Dd − Kgk mc + md 6 ml + md , (33) Dd − Dc Dd − Dc Dd − Dl Dd − Dl which after simple although rather tedious calculations can be rewritten as Dd (mc − ml )(Dd − Kgk ) + Dl (md − mc )(Dd − Kgk ) + Dc (ml − md )(Dd − Kgk ) 6 0. Since Dd − Kgk > 0, inequality (33) can be rewritten in the form Dd (mc − ml ) + Dl (md − mc ) + Dc (ml − md ) 6 0, and next as ml (Dd − Dc ) − mc (Dd − Dl ) + md (Dc − Dl ) > 0. Dividing by (Dd − Dc ) > 0, we arrive at ml +
md (Dc − Dl ) − mc (Dd − Dl ) > 0, Dd − Dc
l = 1, 2, . . . , r,
which is the same as inequality (32). It therefore remains to consider the case Dl > Kgk > Dc . Since, by (25), zc∗ mc + zd∗ md 6 zc mc + zl ml whenever {zc , zl } ∈ Z, we arrive at Kgk − Dc Dl − Kgk Kgk − Dc Dd − Kgk mc + md 6 mc + ml , (34) Dd − Dc Dd − Dc Dl − Dc Dl − Dc
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which is equivalent to Dd (ml − mc )(Dc − Kgk ) + Dl (md − mc )(Kgk − Dc ) + Dc (ml − md )(Kgk − Dc ) 6 0. Dividing the both sides of the last inequality by Kgk − Dc and noting that ml − mc = (ml − md ) + (md − mc ) we obtain the inequality (mc − md )(Dl − Dd ) + (ml − md )(Dd − Dc ) > 0. Dividing once again the both sides of the last inequality by positive number Dd − Dc , we arrive at (32), as required. It all shows that the necessary and sufficient conditions (21), (22) hold with µ1 , µ2 given by (31). The proof is complete. Corollary 9. The minimal value of the misvaluation coefficient m(Z), achieved at the optimal portfolio (zc∗ zd∗ ), is equal to m(Z ∗ ) =
Dd − Kgk Kgk − Dc mc + md . Dd − Dc Dd − Dc
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