Second-Order Asymptotics of the Risk Concentration of a Portfolio with Deflated Risks

<div>Empirical risk concentration (full, based on <svg height="11.6425pt" id="M492" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -11.4361 17.5728 11.6425" width="17.5728pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.48,-5.741)"><path d="M453 623H65C60 580 56 530 46 472H80C103 538 110 548 175 548H389C310 380 195 168 90 0L98 -12L175 -2C273 206 366 408 462 610L453 623Z" id="g50-56"></path></g></svg> simulations) together with second-order approximation (dotted) for two i.i.d. <svg height="12.7178pt" id="M493" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 52.9992 12.7178" width="52.9992pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M372 352C427 368 507 407 507 502C507 548 486 592 451 616C420 638 376 650 301 650H44V622C124 616 128 610 128 527V123C128 40 122 34 36 28V0H257C336 0 402 12 453 40C512 72 548 124 548 191C548 291 471 335 372 350V352ZM213 362V558C213 591 216 602 226 608C235 614 258 618 282 618C377 618 415 558 415 490C415 406 369 362 260 362H213ZM213 328H258C380 328 450 283 450 181C450 76 377 35 296 34C231 34 213 53 213 125V328Z" id="g121-64"></path></g><g transform="matrix(.013,0,0,-0.013,7.553,0)"><path d="M518 50L491 51C452 54 444 60 444 110V444C429 441 405 437 377 434C348 430 314 427 287 426V403L323 397C356 392 365 386 365 333V99C331 66 293 51 257 51C212 51 169 75 169 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350V352ZM213 362V558C213 591 216 602 226 608C235 614 258 618 282 618C377 618 415 558 415 490C415 406 369 362 260 362H213ZM213 328H258C380 328 450 283 450 181C450 76 377 35 296 34C231 34 213 53 213 125V328Z" id="g121-64"></path></g><g transform="matrix(.013,0,0,-0.013,7.905,0)"><path d="M380 106C343 72 306 56 265 56C195 56 116 112 115 248C235 252 361 262 377 265C396 269 400 277 400 297C400 374 333 449 250 449H249C198 449 144 421 103 376S37 269 37 201C37 88 109 -12 232 -12C263 -12 332 6 395 84L380 106ZM225 412C281 412 315 364 314 312C314 297 308 292 290 292C232 290 176 289 120 289C135 370 180 412 225 412Z" id="g121-99"></path></g><g transform="matrix(.013,0,0,-0.013,13.43,0)"><path d="M298 36L289 62C276 55 253 45 228 45C202 45 169 60 169 141V397H276C289 405 292 426 282 437H169V574L155 576L90 509V437H45L17 408L21 397H90V107C90 28 125 -12 188 -12C198 -12 213 -8 230 1L298 36Z" id="g121-114"></path></g><g transform="matrix(.013,0,0,-0.013,17.473,0)"><path d="M433 39L423 65C413 59 399 54 387 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181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z" id="g113-68"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.619,3.132)"><path d="M698 650H468V617C551 609 555 599 536 543C506 457 432 264 379 129H377C311 297 238 488 212 560C196 604 198 609 271 617V650H21V617C80 612 98 603 127 532C183 396 268 173 335 -9H370C445 196 551 450 578 511C615 602 630 609 698 617V650Z" id="g58-84"></path></g><g transform="matrix(.0091,0,0,-0.0091,14.23,3.132)"><path d="M442 39L431 68C420 63 406 57 394 57C377 57 359 71 359 118V301C359 353 349 393 314 424C291 443 261 451 230 451C171 438 107 401 79 381C59 366 46 355 46 340C46 316 72 295 91 295C106 295 117 303 122 319C131 350 139 371 152 384C163 397 178 404 197 404C246 404 281 365 281 293V278C260 262 186 232 124 213C69 194 41 161 41 112C41 47 92 -12 163 -12C195 -12 243 25 283 51C288 35 293 22 308 7C319 -4 339 -12 354 -12L442 39ZM281 87C262 68 226 51 199 51C169 51 129 74 129 126C129 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Mathematical Problems in Engineering

fig1

Figure 1

Figure 1: Second-Order Asymptotics of the Risk Concentration of a Portfolio with Deflated Risks 