<i>CVaR</i> Robust Mean-<i>CVaR</i> Portfolio Optimization

<table>100 <svg height="11.75" id="M225" style="vertical-align:-0.23206pt" version="1.1" viewbox="0 0 44.275002 11.75" width="44.275002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5
t325.5 101.5q114 0 204 -38z" id="x1D436"></path></g><g transform="matrix(.017,-0,0,-.017,12.047,11.4)"><path d="M730 650l-8 -28q-52 -4 -72 -18t-64 -77q-79 -113 -321 -539h-33l-119 541q-13 59 -29.5 73t-66.5 20l7 28h245l-8 -28l-28 -5q-33 -6 -40.5 -15.5t-0.5 -38.5l102 -450h2q191 320 246 430q21 42 15.5 56t-43.5 19l-31 4l7 28h240z" id="x1D449"></path></g><g transform="matrix(.017,-0,0,-.017,24.66,11.4)"><path d="M483 97q-42 -50 -88.5 -79.5t-68.5 -29.5q-37 0 -17 93l22 102h-2q-54 -79 -144 -149q-59 -46 -100 -46q-24 0 -43 29t-19 86q0 78 34.5 153t94.5 120q41 31 94 51.5t98 20.5q29 0 72 -9q26 -6 39 -6l2 -4q-30 -117 -67 -323q-8 -41 2 -41q16 0 79 58zM374 387
q-32 15 -73 15q-52 0 -83 -23q-48 -36 -78 -108.5t-30 -152.5q0 -33 8.5 -50.5t20.5 -17.5q31 0 107 79t99 132q15 40 29 126z" id="x1D44E"></path></g><g transform="matrix(.017,-0,0,-.017,33.261,11.4)"><path d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" id="x1D445"></path></g>
</svg> robust mean-<svg height="11.75" id="M226" style="vertical-align:-0.23206pt" version="1.1" viewbox="0 0 44.275002 11.75" width="44.275002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5
t325.5 101.5q114 0 204 -38z" id="x1D436"></path></g><g transform="matrix(.017,-0,0,-.017,12.047,11.4)"><path d="M730 650l-8 -28q-52 -4 -72 -18t-64 -77q-79 -113 -321 -539h-33l-119 541q-13 59 -29.5 73t-66.5 20l7 28h245l-8 -28l-28 -5q-33 -6 -40.5 -15.5t-0.5 -38.5l102 -450h2q191 320 246 430q21 42 15.5 56t-43.5 19l-31 4l7 28h240z" id="x1D449"></path></g><g transform="matrix(.017,-0,0,-.017,24.66,11.4)"><path d="M483 97q-42 -50 -88.5 -79.5t-68.5 -29.5q-37 0 -17 93l22 102h-2q-54 -79 -144 -149q-59 -46 -100 -46q-24 0 -43 29t-19 86q0 78 34.5 153t94.5 120q41 31 94 51.5t98 20.5q29 0 72 -9q26 -6 39 -6l2 -4q-30 -117 -67 -323q-8 -41 2 -41q16 0 79 58zM374 387
q-32 15 -73 15q-52 0 -83 -23q-48 -36 -78 -108.5t-30 -152.5q0 -33 8.5 -50.5t20.5 -17.5q31 0 107 79t99 132q15 40 29 126z" id="x1D44E"></path></g><g transform="matrix(.017,-0,0,-.017,33.261,11.4)"><path d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" id="x1D445"></path></g>
</svg> actual frontiers (<svg height="16.6625" id="M227" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 58.587502 16.6625" width="58.587502" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M558 587q0 -32 -14 -61t-40 -53.5t-48.5 -41t-54.5 -36.5q144 -51 144 -174q0 -55 -43.5 -108t-104.5 -87q-77 -42 -131 -42q-31 0 -54 20t-31 47l11 18q48 -29 108 -29q79 0 119.5 43t40.5 109t-44.5 107.5t-119.5 50.5l22 47q34 1 65 21q96 61 96 157q0 42 -24 67.5
t-62 25.5q-24 0 -43.5 -9t-35 -29.5t-27 -44t-22.5 -63t-19.5 -75.5t-18.5 -91q-57 -294 -68 -380q-26 -190 -35 -200q-26 -31 -97 -37l-4 26q19 9 48 170l77 413q23 121 52.5 187.5t83.5 114.5q70 62 148 62q51 0 88.5 -34t37.5 -91z" id="x1D6FD"></path></g><g transform="matrix(.017,-0,0,-.017,14.647,12.162)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,29.351,12.162)"><path d="M244 635q90 0 143 -72t53 -177q0 -133 -65 -229.5t-171 -139.5q-79 -32 -140 -32l-5 30q109 18 185 91t101 186l-68 -36q-29 -16 -60 -16q-79 0 -129 51.5t-50 130.5q0 80 57 146.5t149 66.5zM228 602q-52 0 -78 -45.5t-26 -98.5q0 -69 36.5 -115.5t97.5 -46.5
q53 0 90 28q4 31 4 66q0 51 -9.5 95.5t-39 80.5t-75.5 36z" id="x39"></path></g><g transform="matrix(.017,-0,0,-.017,37.511,12.162)"><path d="M244 635q90 0 143 -72t53 -177q0 -133 -65 -229.5t-171 -139.5q-79 -32 -140 -32l-5 30q109 18 185 91t101 186l-68 -36q-29 -16 -60 -16q-79 0 -129 51.5t-50 130.5q0 80 57 146.5t149 66.5zM228 602q-52 0 -78 -45.5t-26 -98.5q0 -69 36.5 -115.5t97.5 -46.5
q53 0 90 28q4 31 4 66q0 51 -9.5 95.5t-39 80.5t-75.5 36z" id="x39"></path></g><g transform="matrix(.017,-0,0,-.017,45.67,12.162)"><path d="M594 629l-404 -661l-31 23l402 661zM194 635q70 0 107 -56t37 -125t-37.5 -125.5t-107.5 -56.5t-107 56.5t-37 124.5q0 70 37.5 126t107.5 56zM193 605q-28 0 -45 -29t-21.5 -59.5t-4.5 -62.5t4.5 -63t22 -60t45.5 -29q22 0 37.5 16t22 42t9 48t2.5 45q0 152 -72 152z
M567 351q69 0 106 -56t37 -125t-37 -125.5t-107 -56.5t-107.5 56.5t-37.5 124.5q0 43 14.5 83t49 69.5t82.5 29.5zM566 322q-28 0 -45 -28.5t-22 -59.5t-5 -64q0 -23 3 -46t10.5 -48.5t23 -41.5t36.5 -16q28 0 45 29.5t21 59.5t4 62q0 24 -2.5 46.5t-9.5 48.5t-22 42t-37 16
z" id="x25"></path></g>
</svg>).</table>

International Scholarly Research Notices

CVaR Robust Mean-CVaR Portfolio Optimization

Figure 1