Online Supplement. Wine Analytics: Fine Wine Pricing and Selection under Weather and Market Uncertainty

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1 Online Suppleent Wine Analytics: Fine Wine Pricing and Selection under Weather and Market Uncertainty Mert Hakan Hekioğlu Lally School o Manageent Rensselaer Polytechnic Institute Troy, NY 118 Burak Kazaz kazaz@syr.edu Whitan School o Manageent Syracuse University Syracuse, NY 1344 Scott Wester scott.ester@asu.edu W.P. Carey School o Business Arizona State University Tepe, AZ 8587 Details o Epirical Foundation: Data Collection and Saple Selection Wine price data is collected ro Liv-ex (.liv-ex.co), the orld s largest dataase or ine ine prices. Our saple is coposed o ive vintages o ine utures (7 to 11) and ive vintages o ottled ine (6 to 1) o 44 Bordeaux ines that aggregates the price data o 43,837 transactions (1,451 via ine utures) corresponding to a total trade volue o 5,133 ottles. We reer to the Liv-ex Bordeaux 5 index (shortly, Liv-ex 5) hen deterining the ines to e exained. This index is coposed o the 1 ost recent ottled vintages o 5 leading Bordeaux ines. Aong those 5 ines, seet Sauternes ines (Yque, Cliens, Coutet, Suduiraut, and Rieussec) are excluded ro the saple since their production process and tieline are dierent than the traditional Bordeaux ines. Another ine, Bahans/Clarence Haut Brion, is also excluded ro the analysis due to issing price data. The inal saple is coposed o 44 o the 5 leading Bordeaux ineakers. The eather inoration is gathered or the Merignac station ro TuTiepo.net. Daily axiu teperatures are collected or each groing season (i.e., May 1 August 31) or the years ro 6 to 1. We then calculate the average groing season teperature or every year. Market luctuations are captured through the Live-ex Fine Wine 1 index (shortly, Liv-ex 1). The percentage change in Liv-ex 1 index over each groing season (i.e., May 1 August 31) is otained or the years ro 8 to 1. The 1 ost sought-ater ines elong to older vintages than the vintages used in our saple, and thereore, there is no overlap o ines ith our saple. Can our Liv-ex 1 index can e replaced ith another inancial arket variale? We ind Liv-ex 1 to e a strong indicator that is distinct ro traditional inancial indices. This can e seen ro the correlation coeicients eteen the Liv-ex 1 index and the three popular inancial indicators during sae tie period ith our data involving utures and ottle prices eteen 7 and 14: The correlation coeicient ith the Standard & Poor 5 index is.3, ith the Financial Ties 1 index is.11, ith the Do Jones index is.4 hereas the correlation coeicients eteen these three inancial indices range ro.9 to.99. Thus, Liv-ex 1 is not an aritrarily chosen arket indicator. 1

2 Proos and Derivations,,,,,,, ar gax E x1, y1,,, x, y, z, z s.t. (9), (11), (1); x, y x1, i, 1 xx1, y1,,, yx1, y1,, Bx1 y1,, i,. x1, B x1 y1, x1 i, 3 Proo o Lea A1. The irst derivatives o the stage- ojective unction (8) are x, y, x,,, y, z, z ]/ x = 3 (, ) (, ) (7) Lea A1. xx1 y1 yx1 y1 = E[ 1 1 E[ x1, y1, x,,, y, z, z ]/ y = 3 () (). (8) We see that the decision that axiizes expected proit siply depends on the relative proitaility o utures and ottles or a given (, ). In Ω1, oth (7) and (8) are negative (neither utures nor ottles are proitale on expectation) hich leads to x = x 1 and y = due to (11) and (1). In Ω, (7) is nonnegative and greater than (8) (utures are ore proitale on expectation) hich leads to x = [B x 1 y 1 ]/ (, ) and y = due to (9) and (1). In Ω3, (8) is nonnegative and no saller than (7) (ottles are ore proitale on expectation) hich leads to x = x 1 and y = [B + ( (, ) 1)x 1 y 1 ]/ () due to (9) and (11). We denote the value created ro the utures liquidation option ith V l. We irst partition Ω3 into the olloing to sets: Ω3 A = {(, ): 3 ()/ () 1 > 3 (, )/ (, )},Ω3 B = {(, ): 3 ()/ () 3 (, )/ (, ) 1}. Futures do not provide a proitale return in Ω3 A, and continue to e proitale ut doinated y the returns ro ottles in Ω3 B. The distriutor ould sell utures in sets Ω1 and Ω3 A in order to avoid any urther losses. The value created ro liquidity is: Vl, 3,. (9) 1 3A In set Ω3 A utures are not proitale, and the distriutor sells the and saps the ith ottles. In Ω3 B the distriutor also eneits ro the aility to sap utures, even though they are still proitale We denote the value created ro the sapping lexiility ith V s, can express it as ollos: 3 Vs, 3, 3. (3) We next deine the value gained ro liquidation and sapping ith V ls. The distriutor eneits ro oth liquidating and sapping in set Ω3 A ; discounting the doule counting, e get: Vl s Vl Vs, 3,. (31) 3A The distriutor can eneit ro holding cash in stage 1. This oney can e used to purchase utures in Ω and ottles in Ω3. The value ro holding cash in stage 1, denoted V c, can e descried as: 3, 3 Vc 1 ( ) ( ) 1 ( ) ( ), 3 Using this notation, e can open up the expressions that appear in Proposition 1 (see the proo o Proposition 1 or supporting detail): E x, y,,, x, y, z, z y E z V. (3) =,,,,,,, l s c E x1, y1,,, x, y, z, z x1 = E 3, z 1Vc Vl = s E x1 y1 x y z z y1 V. 3 1

3 Proo o Proposition 1. Using (x, y ) (see Lea A1), e have E[( x, y,,, x, y, z, z )]/ y 1 = E[ 3 ( ) + z ] ( ) ( ) ,, ( ) ( ) 3 3 ( ) ( ) = E[ 3 ( ) + z ] 1 V c (33) hich is nonnegative ecause oth integrands are nonnegative y deinitions o Ω and Ω3. E[( x, y,,, x, y, z, z )]/ x 1 = E[ 3 (, ) + ( ) ( ) z 1 1 ] 1 3,, ( ) ( ) 3 3 ( ) ( ) 3, 3,, 3, 1 3 = E[ 3 (, ) + z ] 1 Vc + V ls (34) hich is nonnegative ecause oth integrands are nonnegative y deinitions o Ω1 and Ω3. Note that E[( x1, y1,,, x, y, z, z )] is linear in x 1 and y 1. As a consequence, (33) is positive or any (x 1, y 1 ) olloing ro (1). Moreover, olloing ro (17), E[( x, y,,, x, y, z, z )]/ x 1 E[( x, y,,, x, y, z, z )]/ y 1 = V ls hich is nonnegative or any (x 1, y 1 ). We next exaine the ipact o increasing variation in uncertainty oth in eather and arket rando variales (denoted σ and σ, respectively) on an investent strategy in stage 1. Due to the linearity o the utures and ottle price unctions, the expected prices E,, E 3, z, E, and E 3 z do not change ith dierent values o σ and σ. Moreover, in the asence o a recourse lexiility that enales a ine distriutor to change her utures and ottle positions, the expected proit ould not change ith increasing values o σ and σ. Hoever, the values ro liquidity, sapping, and coination lexiilities, and cash, denoted V l, V s, V ls, and V c in (9) (3) change ith higher values o σ and σ. Under syetric pds or eather and arket rando variales, i.e., ϕ () = ϕ ( ) ith H = L and ϕ () = ϕ ( ) ith H = L, the olloing proposition estalishes their ehavior ith respect to σ and σ. Proposition A1. When ϕ () and ϕ () ollo syetric pd, (a) the value ro liquidity V l in (9) increases in σ and σ ; () the value ro cash V c in (3) increases in σ and σ ; (c) the value ro sapping V s in (3) increases in σ ; (d) the value ro the coination o liquidity and sapping V ls in (31) increases in σ. Proo o Proposition A1. Recall that E[ 3 (, ) + z ] = 3 (, ) and E[ 3 () + z ] = 3 (). The price evolution o utures is already descried as 3 (, )/ < (, )/ < and 3 (, )/ > (, )/ >, and ottles as 3 ()/ > ()/ >. (a) With higher values o σ or a syetric pd or ϕ (), regions 1 and 3 A expand. Because 3 (, )/ < (, )/ <, V l in (9) ould e adding increasing values o (, ) 3 (, ) at each increent o H. Thus, V l in (9) increases in σ. Siilarly, ith higher values o σ or a syetric pd or ϕ (), region 1 expands. Because 3 (, )/ > (, )/ >, V l in (9) ould e adding increasing values o (, ) 3 (, ) at each reduction in L. Thus, V l in (9) increases in σ. () Increasing σ or a syetric pd or ϕ () iplies expanding region y reducing L here 3 (, )/ (, ) > 1 y deinition o the set. Because 3 (, )/ < (, )/ <, e ould e adding increasing values o [( 3 (, )/ (, )) 1]. Siilarly, increasing σ or a syetric pd or ϕ () iplies expanding region 3 y increasing H here 3 ()/ () > 1 y deinition o the set. Because 3

4 3 ()/ = ()/ = and e ould not e changing the second ter o V c in (3). The changes region is positive, and thereore, V c in (3) increases in σ. A siilar proo ollos or the ipact o σ. Increasing σ or a syetric pd or ϕ () iplies expanding region y reducing L and increasing H here 3 (, )/ (, ) > 1 y deinition o the set. Because 3 (, )/ > (, )/ >, e ould e adding increasing values o [( 3 (, )/ (, )) 1]. Siilarly, increasing σ or a syetric pd or ϕ () iplies expanding region 3 y increasing H here 3 ()/ () > 1 y deinition o the set. Because 3 ()/ > ()/ >, e ould e adding increasing values o [( 3 ()/ ()) 1] to the second ter o V c in (3). The changes in region and 3 are positive, and thereore, V c in (3) increases in σ. (c) The value ro sapping V s in (3) is deined in 3. Increasing σ or a syetric pd or ϕ () iplies expanding 1 (y reducing L ) and 3 (y increasing H ). In 3, 3 ()/ () > 1, and its value is increasing due to 3 ()/ > ()/ >. At the ne arket realization greater than H, e kno that (, )[ 3 ()/ ()] 3 (, ) > ecause o the deinition o 3 (so that the ir saps utures ith a ore proitale ottle investent). Thus, expanding the support eyond H adds value and expanding the loer support elo L does not cause any loss; thereore, V s in (3) is increasing in σ. (d) The proo ollos ro the proos o parts (a) and (c). Proo o Proposition. (a) Increasing σ or a syetric pd or ϕ () iplies reducing L and increasing H. Reducing L to L (here > ) and increasing H to H + leads to three cases or investigation. Case 1: (, L ) Ω1 and (, H + ) Ω3: Because 3 (, )/ > (, )/ > and ecause ottles are even ore proitale than utures in Ω3, the losses ro the utures investent at (, L ) Ω1 are saller in asolute value than the gains (, H + ) Ω3, and thus, the expected proit increases. Case : (, L ) Ω1 and (, H + ) Ω: I (, L ) Ω1, then ecause 3 (, )/ > (, )/ >, the losses ro the utures investent at (, L ) Ω1 are saller in asolute value than the gains (, H + ) Ω, and thus, the expected proit increases. Case 3: (, L ) Ω and (, H + ) Ω: I (, L ) Ω, the losses ro the utures investent at (, L ) Ω are recovered y the gains at (, H + ) Ω due to syetry, and thus, the expected proit does not change. Coining the results ro these three cases, the expected proit increases ith higher levels o σ. () Using the proo o Proposition 1, the expected proit or any (x 1, y 1 ) pair is E[( x, y,,, x, y, z, z )] = [E[ 3 (, ) + z ] 1 V c ]y 1 + B V c 1 1 z ] 1 Vc + V ls ]x 1 + [E[ 3 ( ) + = [E[ 3 (, ) + z ] 1 + Vls ]x 1 + [E[ 3 ( ) + z ] 1]y 1 + (B x 1 y 1 )V c. Increasing σ does not change E[ 3 (, ) + z ] and E[3 ( ) + z ]. Proposition A1() has shon that V c is increasing in σ. Thus, it is suicient to oserve that E[( x1, y1,,, x, y, z, z )]/ σ > i the coined value ro liquidity and sapping increases in σ, i.e., V ls / σ >. Lea A. [ E[( x1, y1,,, x, y, z, z )]/ y 1 (x1, y1) = (,) ]/[ E[( x1, y1,,, x, y, z, z )]/ x 1 (x1, y1) = (,)] equals to [ E[( x, y,,, x, y, z, z )]/ y 1 ]/[ E[( x, y,,, x, y, z, z )]/ x 1 ] or any (x 1, y 1 ) Proo o Lea A. Follos ro the linearity o E[( x1, y1,,, x, y, z, z )] in x 1 and y 1, as shon in the proo o proposition 1. Developent o the proo o Proposition 3 We irst deine the olloing oundary sets: E = {(, ) : <, = ()} and 3 E = {(, ) 3: = }. In the olloing analysis e exaine the value o proit unction (x 1, y 1,,, x, y, z α, z α ) at three points, and use this analysis in the proo o Proposition 3. The three points identiied in Figure A1 correspond the realizations o (, ) that yields lo values o (x 1, y 1,,, x, y, z α, z α ). Lea A3. I (3), then (x 1, y 1, -, L, x, y, z α, z α ) β or any (x 1, y 1 ). Proo o Lea A3. Note that ( -, L ) Ω E. This iplies 3 ( -, L )/ ( -, L ) = 1 y deinition o set. Thus, the realized proit at (z α, z α ) is 4

5 (x 1, y 1, -, L, x, y, z α, z α ) = [ 3 ( L ) + z α 1]y 1 + z α [B y 1 ]. (35) Note irst that (35) independent o x 1 ; ecause 3 ( -, L )/ ( -, L ) = 1 and (5) iply that 3 ( -, L ) = ( -, L ) = 1. Because L <, it ollos that 3 ( L )/ ( L ) < 1, and thus ro (6) it ollos that 3 ( L ) < ( L ) < 1. Coined ith z α < (y assuption), they iply 3 ( L ) + z α 1 <. Folloing ro (13), e have [ 3 ( L ) + z α 1]y 1 > β or any y 1 B. Furtherore, olloing ro (3), e have z α [B y 1 ] > β or any y 1 B. Figure A1. Points (1) (3) are candidates or the iniu value o (x 1, y 1,,, x, y, z α, z α ). Lea A4. I (3), then (x 1, y 1,,, x, y, z α, z α ) β or all (, ) Ω or any (x 1, y 1 ). Proo o Lea A4. We irst ocus on (, ) Ω E, or hich 3 (, )/ (, ) = 1, hich in turn iplies 3 (, ) = (, ) = 1 or all (, ) Ω E (see (5)). Thus, or any (, ) Ω E, (x 1, y 1,,, x, y, z α, z α (, ) Ω E ) = [ 3 () + z α 1]y 1 + z α [B y 1 ] [ 3 ( L ) + z α 1]y 1 + z α [B y 1 ] = (x 1, y 1, -, L, x, y, z α, z α ) β here the irst inequality ollos ro 3 () increasing in, and the last inequality ollos ro Lea A3. Note that the expression aove is independent o x 1 ecause 3 (, ) = (, ) = 1 or all (, ) Ω E. For any (, ) Ω\Ω E, e have 3 (, )/ (, ) > 1 (y the deinition o Ω). This iplies that 3 (, ) > (, ) > 1 (see (5)). Hence, the realized proit (x 1, y 1,,, x, y, z α, z α ) is increasing in x 1 or any (, ) Ω\Ω E, and thus (x 1, y 1,,, x, y, z α, z α ) β or all (, ) Ω. Note that the proit at point ( H, L ) Ω1 is (x 1, y 1, H, L, x, y, z α, z α ) = [ ( H, L ) 1]x 1 + [ 3 ( L ) + z α 1]y 1. (36) We deine x 1 H (y 1 ) hich satisies 1 (x 1 H (y 1 ), y 1, H, L, x, y, z α, z α ) = β or a given y 1, i.e., x 1 H (y 1 ) = [ [1 3 ( L ) z α ]y 1 ]/[1 ( H, L )]. (37) Lea A5. (x 1, y 1, H, L, x, y, z α, z α ) β or any y 1 B and x 1 x 1 H (y 1 ). Proo o Lea A5. We kno that ( H, L ) < 1 and 3 ( L ) < 1 (ollos ro ( H, L ) Ω1, (5), and (6)). Also, z α < y assuption. Thereore, (x 1, y 1, H, L, x, y, z α, z α ) in (36) is decreasing in x 1 and y 1. This also iplies that x 1 H (y 1 ) in (37) is decreasing in y 1. For any y 1 B (due to (13)) and x 1 x 1 H (y 1 ), (x 1, y 1, H, L, x, y, z α, z α ) (x 1 H (y 1 ), y 1, H, L, x, y, z α, z α ) = β. Lea A6. (x 1, y 1,,, x, y, z α, z α ) β or all (, ) Ω1 or any y 1 B and x 1 x 1 H (y 1 ). Proo o Lea A6. Since (, ) and 3 () are increasing in, and (, ) is decreasing in, (x 1, y 1,,, x, y, z α, z α (, ) Ω1) = [ (, ) 1]x 1 + [ 3 () + z α 1]y 1 [ ( H, L ) 1]x 1 + [ 3 ( L ) + z α 1]y 1 = (x 1, y 1, H, L, x, y, z α, z α ) β here the last inequality ollos ro Lea A5. Lea A7. (x 1, y 1, H,, x, y, z α, z α ) β or any x 1 x 1 V. Proo o Lea A7. Note that ( H, ) Ω3 E iplies 3 ( )/ ( ) = 1. This urther iplies 3 ( ) = ( ) = 1 (due to (6)). Thus, (x 1, y 1, H,, x, y, z α, z α ) = [ ( H, ) 1][1 + z α ]x 1 + z α B. (38) 5

6 Note that the expression aove is independent o y 1. It is decreasing in x 1 or to reasons: First, ( H, ) Ω iplies that 3 ( H, )/ ( H, ) < 1 hich urther iplies 3 ( H, ) < ( H, ) < 1 (due to (5)), and second, 1 + z α > (due to (13) and β < B). We deine x V 1 hich satisies (x V 1, y 1, H,, x, y, z α, z α ) = β or any y 1, i.e., x V 1 = [ + z B]/([1 ( H, τ )][1 + z ]). (39) Thereore, (x 1, y 1, H,, x, y, z α, z α ) (x V 1, y 1, H,, x, y, z α, z α ) = β or any x 1 x V 1. Lea A8. (x 1, y 1,,, x, y, z α, z α ) β or all (, ) Ω3 or any x 1 x V 1. Proo o Lea A8. We irst ocus on (, ) Ω3 E, or hich 3 ( ) = ( ) = 1 (ollos ro the deinition o Ω3 E and (6)). The realized proit can e expressed as (x 1, y 1,,, x, y, z α, z α (, ) Ω3 E ) = [ (, ) 1][1 + z α ]x 1 + z α B [ ( H, ) 1][1 + z α ]x 1 + z α B = (x 1, y 1, H,, x, y, z α, z α ) β here the irst inequality ollos ro (, ) decreasing in, and the last inequality ollos ro Lea A7. Note that the expression aove is independent o y 1 ecause 3 ( ) = ( ) = 1 or all (, ) Ω3 E. For any (, ) Ω3\Ω3 E, 3 ()/ () > 1 y the deinition o Ω3. This urther iplies that 3 () > () > 1 (due to (6)). Hence, the realized proit at (z, z ) increases in y 1 or any (, ) Ω3\Ω3 E. Thereore, (x 1, y 1,,, x, y, z α, z α (, ) Ω3\Ω3 E ) β. Lea A9. Suppose that (3) holds. Then (x 1, y 1,,, x, y, z α, z α ) β or all (, ) Ω or any y 1 B and x 1 in{x H 1 (y 1 ), x V 1 }. Proo o Lea A9. Follos ro leas A4, A6, and A8. Lea A1. Suppose that (3) holds. Then P[(x 1,,,, x, y, z, z ) < β] α or all (, ) Ω or any x 1 in{x H 1 (), x V 1 }. This eans that (x, y ) and (x 1, ) decisions such that x 1 in{x H 1 (), x V 1 } satisy oth (1) and (16). Proo o Lea A1. (x 1,,,, x, y, z, z (, ) Ω1) has neither z nor z ter. (x 1,,,, x, y, z, z (, ) Ω) has only z, and (x 1,,,, x, y, z, z (, ) Ω3) has only z. We also kno ro Lea A9 that (x 1,,,, x, y, z α, z α ) β or all (, ) Ω or any x 1 in{x H 1 (), x V 1 } hen y 1 =. Coined ith P z z = Pz z =, they iply that P[(x 1,,,, x, y, z, z ) < β] α or all (, ) Ω or any x 1 in{x H 1 (), x V 1 }. As a consequence, VaR constraints (1) and (16) are satisied y (x, y ) and (x 1, ) decisions or x 1 in{x H 1 (), x V 1 }. Lea A11. Suppose that (3) holds, and z, z ollo a ivariate noral distriution. Then P[(x 1, y 1,,, x, y, z, z ) < β] α or all (, ) Ω or any < y 1 < B and x 1 in{x H 1 (y 1 ), x V 1 }. This eans that (x, y ) and (x 1, y 1 ) decisions such that < y 1 < B and x 1 in{x H 1 (y 1 ), x V 1 } satisy oth (1) and (16). Proo o Lea A11. Note irst that y 1. (x 1, y 1,,, x, y, z, z (, ) Ω1Ω3) has only z P z =, and leas A6 and A8, it ollos that P[(x 1, y 1,,, x, y, ter. Coined ith z z, z ) < β] α or all (, ) Ω1Ω3 or any < y 1 < B and x 1 in{x H 1 (y 1 ), x V 1 }. (x 1, y 1,,, x, y, z, z (, ) Ω) has oth z and z ters. We irst consider the case here z and z are perectly positively correlated, i.e., z = k z here k >. This iplies P[ z z α & z z α ] = P[k z kz α & z z α ] = P[ z z α ] = α. Together ith Lea A4, it ollos that P[(x 1, y 1,,, x, y, z, z ) < β] α (4) or all (, ) Ω or any (x 1, y 1 ). We then consider the less-than-perect positive correlation case here ( z, z ) ollo a ivariate noral distriution. The randoness in proit can e expressed as 6

7 Z = (x1 + x ) z + (y 1 + y ) z here ρ is the correlation coeicient or ( z, z ). As a consequence o ivariate noral distriution, Z, hich is the su o noral rando variales, is a noral rando variale ith E[ Z ] = and V Z = x1 x z y1 y z z z. Fro E[ z ] = E[ z ] = and {z α, z α } <, it ollos that α.5. Thereore, P[(x 1, y 1,,, x, y, z, z ) < β] = P[ Z < β (x1, y 1,,, x, y,, )] P[ Z 1 < β (x 1, y 1,,, x, y,, )] α or all (, ) Ω or any (x 1, y 1 ). The irst inequality ollos ro α.5 and the act that variance is increasing in. The second inequality ollos ro (4), i.e., the case o perect positive correlation. As a consequence, VaR constraints (1) and (16) are satisied y (x, y ) and (x 1, y 1 ) decisions such that < y 1 < B and x 1 in{x H 1 (y 1 ), x V 1 }. Proo o Proposition 3. We egin ith relaxing (1), i.e., (x, y ) = (x, y ) is easile. We then sho that, hen (3) holds, constraint (1) is noninding at the optial solution to the prole deined in (8) (16). Fro Proposition 1, e kno that (x 1, y 1 ) = (, ) cannot e optial. Moreover, x + 1 = x H 1 () > (see (37)) due to β > and 1 > ( H, L ) (ollos ro ( H, L ) Ω1 and (5)). Part (a): When B in{x + 1, x V 1 }, then (x, y ) = (x, y ) and (x 1, y 1 ) = (B, ) satisy oth (1) and (16) olloing ro Lea A1. This iplies that (x *, y * ) = (x, y ) y deinition o (x, y ). It ollos ro Proposition 1 that (x * 1, y * 1 ) = (B, ). Part (): Note that x V 1 < x H 1 (B x V 1 ) hen x V 1 < B x + 1. Proposition 1 and Lea A11 iply that (x *, y * ) = (x, y ) and (x * 1, y * 1 ) = (x V 1, B x V 1 ). Part (c): Note that x H 1 (y 1 ) is linearly decreasing in y 1 (see (37)). As a consequence, hen x + 1 < x V 1, e have x H 1 (y 1 ) < x V 1 or any y 1. Moreover, E[( x1, y1,,, x, y, z, z )] is linear in x 1 and y 1 (see proo o Proposition 1). Thereore, de[(x H 1 (y 1 ), y1,,, x, y, z, z )]/dy 1 = E[( x1, y1,,, x, y, z, z )]/ y 1 13 Lz E[( x1, y1,,, x, y, z, z )]/ x 1. 1 H, L Part (c)(i): de[(x H 1 (y 1 ), y1,,, x, y, z, z )]/dy 1 < due to (4) and Lea A. Folloing ro Lea A1, (x, y ) = (x, y ) and (x 1, y 1 ) = (x + 1, ) satisy oth (1) and (16). Moreover, (15) is satisied due to x + 1 < B. Thereore, together ith Proposition 1, it ollos that (x *, y * ) = (x, y ) and (x * 1, y * 1 ) = (x + 1, ). Part (c)(ii): de[(x H 1 (y 1 ), y1,,, x, y, z, z )]/dy 1 due to the reversal o (4), and Lea A. Note that x H 1 (B) > (see (13) and (37)). Together ith x + 1 < B and the linearity o x H 1 (y 1 ) in y 1, it ollos that the VaR constraint (16) at ( H, L ) crosses the udget constraint at a single point, i.e., y s 1 + x H 1 (y s 1 ) = B B1 H, s L such that y B1 3 L z s H s 1 and x1 x1 3L z H, L y1 3L z H, L here {x s 1, y s 1 } > olloing ro x + 1 < B and (13). Note also that x s 1 < x + 1. Folloing ro Lea A11, (x, y ) = (x, y ) and (x 1, y 1 ) = (x s 1, y s 1 ) satisy oth (1) and (16). Thereore, together ith Proposition 1, it ollos that (x *, y * ) = (x, y ) and (x * 1, y * 1 ) = (x s 1, y s 1 ). Part (d): We no exaine the case hen x s 1 < x V 1 x + 1 < B. Part (d)(i): When x V 1 = x + V 1, it ollos ro the proo o part (c)(i). When x 1 < x + 1, x H 1 (y 1 ) linearly decreasing in y 1 iplies that there exists a single y V 1, i.e., x H 1 (y V 1 ) = x V 1 such that z 1, B H L 1 z 1 H, V y1 here x V 1 + y V 1 < B (i.e., (15) is satisied) due to x s 1 < x V 1, x + 1 < B, 1 3 Lz and (13). de[(x H 1 (y 1 ), y,,, x, y, z, z )]/dy 1 < due to (4) and Lea A. Together ith 1 7

8 Proposition 1 and Lea A11, it ollos that (x *, y * ) = (x, y ) and (x * 1, y * 1 ) = (x V 1, y V 1 ). Part (d)(ii): Since x s 1 < x V 1, it ollos ro the proo o part (c)(ii). Part (e): Note that x V 1 x H 1 (B x V 1 ) hen x V 1 x s 1. Proposition 1 and Lea A11 iply that (x *, y * ) = (x, y ) and (x * 1, y * 1 ) = (x V 1, B x V 1 ). Proo o Proposition 4. Relaxing (3) does not aect the easiility o (x, y ) in Ω1 and Ω3 (see leas A6 and A8). Hoever, (x, y ) ay no longer e easile in Ω (see Lea A4). Fro (7), the realized proit at -ractile is (x 1, y 1,,, x, y, z α, z α ) = x 1 y 1 (, )x ()y + [ 3 (, ) + z α ](x 1 + x ) + [ 3 () + z α ](y 1 + y ) hich is linear in z α. Folloing ro (11), (x 1, y 1,,, x, y, z α, z α )/ z α. Thereore, it is suicient to sho that Proposition 4 holds at the extree case such that z α. The result naturally extends to any other z α, hich ay or ay not satisy (3). z α iplies that x * = x 1 ; otherise, li zα (x 1, y 1,,, x, y, z α, z α ) =. We partition Ω into the olloing to sets: Ω A = {(, ): 3 (, )/ (, ) 1 > 3 ()/ ()}, Ω B = {(, ): 3 (, )/ (, ) > 3 ()/ () 1}. In Ω A, y * = due to 1 > 3 ()/ (). Thus, (x 1, y 1,,, x *, y *, z α, z α (, ) Ω A ) = [ (, ) 1]x 1 + [ 3 () + z α 1]y 1 [ ( H, L ) 1]x 1 + [ 3 ( L ) + z α 1]y 1 = (x 1, y 1, H, L, x, y, z α, z α ) β (41) here the irst inequality ollos ro the act that ( H, L ) and 3 ( L ) are the orst price realizations or (, ) and 3 (), respectively, and the last inequality ollos ro Lea A5. We next sho that y * = [B x 1 y 1 + (, )x 1 ]/ () given that that x * = x 1 in Ω B : (x 1, y 1,,, x *, y *, z α, z α (, ) Ω B ) = [ (, ) 1][1 + [ 3 () + z α ()]/ ()]x 1 + [ 3 () + z α 1]y 1 + [[ 3 () + z α ()]/ ()][B y 1 ] here [ (, ) 1][1 + [ 3 () + z α ()]/ ()]x 1 (4) olloing ro x 1, (, ) > 1 (due to the deinition o Ω B and (5)), and [ 3 () + z α ()]/ () > β/b > 1 (due to the deinition o Ω B, β < B, z α <, (13)); [ 3 () + z α 1]y 1 > β (43) olloing ro y 1 B and (13); and [[ 3 () + z α ()]/ ()][B y 1 ] > β (44) olloing ro y 1 B and [ 3 () + z α ()]/ () > β/b > 1 (due to the deinition o Ω B, β < B, z α <, (13)). Inequalities (4), (43) and (44) together iply that (x 1, y 1,,, x *, y *, z α, z α (, ) Ω B ) > β (45) here x * = x 1 and y * = [B x 1 y 1 + (, )x 1 ]/ (). Folloing ro (41), (45), and leas A6 and A8, x1, i, 1 A xx1, y1,,, yx1, y1,,. x1, B x1 y1, x1 i, 3 B Thus, E[( x, y,,, x, y, z, z )]/ y 1 = E[ 3 ( ) + z ] ( ) ( ) 1 1 3B 3 ( ) ( ) 1 A = E[ 3 ( ) + z ] 1 V c (46) 3 here V c 1 ( ) ( ) 3 B nonnegative y deinitions o Ω B and Ω3. Also, e have E[( x1, y1,,, x, y, z, z )]/ x 1 = E[ 3 (, ) + z ] ( ) ( ) 1 A ( ) ( ),, 3 3 3B 1 A 8

9 3, 3, 3 B = E[ 3 (, ) + z ] 1 Vc + V ls (47) here 3 V ls, 3,, 3, 1 3 A B Folloing ro the deinitions o Ω B and Ω3, ( ) = (see (18)), E[ ] =, and the syetry in 3 (), e have, 3, 3. Folloing ro the B deinitions o Ω1 and Ω A, () < or all < (see (18)), E[ ] =, and the syetry in (), e have, 3,. Thus, V ls. Folloing ro (17), 1 A E[( x1, y1,,, x, y, z, z )]/ x 1 E[( x1, y1,,, x, y, z, z )]/ y 1 = V ls. Moreover, olloing ro the deinitions o Ω and Ω3, V c V c (see (3)). Recall that (1) iplies (33) is positive (see the proo o Proposition 1). Thus, V c V c iplies that (46) is positive, i.e., E[( x, y,,, x, y, z, z )]/ y 1 >