Совершенствование валютных операций Проминвестбанка

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“нельзя поймать двух зайцев сразу”, необходимо сделать определенный выбор между эффективностью и риском (этот выбор, в конечном счете, определяется отношением ЛПР к эффективности и риску).

Модель оптимального портфеля Марковица , которая обеспечивает минимальный риск и заданную доходность имеет вид:

[pic](2)

Необходимо определить: x1,x2…xn

Оптимальный портфель Марковица максимальной доходности и заданного,

(приемлемого) риска rp можно представить в виде:

[pic](3)

Основные расчетные формулы для вычисления работы

Исходные данные для расчета является таблица доходности ценных бумаг:

|1 |2 |… |i |… |j |… |n-1 |n |
| | | |di1 | |dj1 | | | |
| | | |di2 | |dj2 | | | |
| | | |… | |… | | | |
| | | |dik | |djk | | | |
| | | |… | |… | | | |
| | | |din | |djn | | | |

Среднее арифметическое доходности i – ой ценной бумаги рассчитывается по формуле: di[pic] (4)

Kовариация или корреляционный момент доходностей ценных бумаг:

Vij=M[pic][pic][pic] (5), где [pic] и [pic][pic] - отклонения доходностей i- й и j– й бумаг от средней арифметической доходности.

Рассмотренные модели относятся к моделям нелинейного программирования.

Для решения следует применить метод сопряженных градиентов или метод

Ньютона. При решении рекомендуется использовать табличный процессор Excel и его надстройки Анализ данных и Пакет анализа.

Варианты заданий:

Виды ценных бумаг и их доходности

|вариан|Акции |Акции |Акции |Акции |Акции |Акции |mp |rp |
|т |типа 1 |типа 2 |типа 3 |типа 4 |типа 5 |типа 6 | | |
|1 |11,954 |13,907 |14,263 |14,611 |13,491 |12,424 |13 |0,0|
| | | | | | | | |88 |
| |11,913 |12,074 |11,960 |15,207 |13,367 |14,318 | | |
| |11,572 |12,654 |14,203 |11,762 |13,568 |14,922 | | |
| |12,591 |12,880 |13,333 |13,217 |14,256 |15,677 | | |
| |11,638 |12,269 |12,537 |13,624 |13,694 |13,718 | | |
| |12,536 |13,659 |13,864 |12,909 |12,687 |14,753 | | |
|2 |11,820 |13,783 |12,361 |14,612 |12,084 |13,537 |12,5 |1 |
| |11,806 |11,931 |12,132 |13,444 |13,332 |15,209 | | |
| |13,376 |13,424 |14,519 |14,817 |13,918 |16,153 | | |
| |12,175 |12,381 |14,647 |13,911 |13,596 |14,714 | | |
| |10,139 |12,112 |11,820 |10,399 |11,604 |13,345 | | |
| |11,786 |13,505 |13,440 |14,856 |12,688 |13,533 | | |
|3 |11,574 |12,955 |14,174 |14,016 |13,100 |13,394 |13,3 |0,2|
| |12,602 |13,421 |14,663 |15,042 |13,572 |15,713 | | |
| |12,012 |12,654 |12,996 |13,896 |13,463 |12,148 | | |
| |12,245 |13,043 |14,517 |15,338 |14,976 |14,063 | | |
| |12,502 |13,879 |13,744 |14,726 |13,145 |12,669 | | |
| |12,976 |13,482 |14,434 |15,594 |15,467 |14,364 | | |
|4 |14,162 |15,191 |16,511 |15,118 |14,409 |17,416 |14 |0,7|
| | | | | | | | |7 |
| |12,522 |12,995 |15,415 |15,263 |14,700 |13,072 | | |
| |13,123 |15,042 |13,177 |16,430 |14,913 |14,860 | | |
| |12,043 |12,570 |14,005 |14,864 |12,917 |12,820 | | |
| |11,472 |12,600 |11,845 |12,878 |14,097 |14,554 | | |
| |12,441 |13,296 |13,139 |15,486 |12,849 |14,172 | | |
|5 |11,265 |12,891 |11,612 |12,845 |11,401 |13,668 |12,8 |2 |
| |12,131 |12,365 |12,890 |12,283 |13,833 |15,160 | | |
| |14,480 |15,143 |17,277 |15,657 |14,960 |15,963 | | |
| |12,572 |13,837 |12,596 |13,157 |15,296 |16,424 | | |
| |10,865 |12,623 |11,558 |14,056 |10,873 |14,489 | | |
| |11,962 |12,287 |13,799 |12,841 |13,769 |15,677 | | |
|6 |10,908 |11,940 |12,719 |11,464 |12,486 |11,458 |13,123|3,2|
| | | | | | | | |3 |
| |9,766 |9,842 |10,987 |10,843 |12,003 |13,471 | | |
| |11,702 |12,463 |13,454 |12,725 |13,619 |13,226 | | |
| |11,472 |12,097 |12,784 |12,302 |14,188 |12,603 | | |
| |13,503 |13,740 |13,848 |14,758 |15,457 |13,570 | | |
| |15,456 |16,452 |16,450 |17,013 |17,052 |17,229 | | |
|7 |10,161 |12,144 |10,537 |13,397 |10,223 |12,266 |12 |1,2|
| |11,492 |12,945 |11,892 |12,298 |12,097 |11,582 | | |
| |12,428 |12,895 |15,227 |13,407 |14,462 |14,330 | | |
| |12,416 |13,050 |14,567 |15,240 |13,731 |13,284 | | |
| |10,813 |12,380 |12,008 |14,134 |10,968 |11,675 | | |
| |13,388 |14,643 |15,659 |16,734 |15,631 |16,916 | | |
|8 |9,889 |11,603 |11,612 |12,721 |11,453 |12,102 |13,25 |2 |
| |12,517 |13,256 |12,947 |12,596 |12,853 |13,036 | | |
| |12,786 |12,822 |15,447 |14,452 |15,143 |16,247 | | |
| |11,863 |12,114 |13,359 |13,437 |11,913 |15,300 | | |
| |11,444 |13,292 |13,703 |11,504 |13,406 |15,255 | | |
| |14,696 |15,946 |16,829 |17,698 |16,051 |17,140 | | |
|9 |11,999 |13,995 |13,415 |12,868 |12,339 |13,682 |13,333|0,0|
| | | | | | | | |8 |
| |12,241 |12,793 |14,227 |13,426 |12,656 |15,808 | | |
| |12,120 |13,933 |14,592 |13,354 |12,278 |14,786 | | |
| |11,506 |13,401 |12,193 |13,845 |12,406 |13,317 | | |
| |12,376 |13,710 |15,068 |13,133 |12,707 |14,716 | | |
| |12,148 |13,970 |15,119 |12,886 |14,518 |13,300 | | |
|10 |11,293 |11,493 |13,753 |12,936 |12,881 |13,820 |12 |0,3|
| |12,112 |12,919 |12,415 |14,048 |14,770 |14,310 | | |
| |11,429 |13,098 |14,277 |14,551 |11,639 |13,524 | | |
| |10,526 |11,988 |11,705 |12,466 |11,825 |10,864 | | |
| |11,467 |13,364 |12,171 |11,631 |11,923 |13,764 | | |
| |11,467 |13,334 |12,338 |14,208 |12,271 |13,324 | | |
|11 |11,954 |13,381 |14,468 |12,274 |13,094 |13,014 |12 |0,4|
| |11,913 |12,754 |14,452 |13,449 |14,079 |14,121 | | |
| |11,572 |12,623 |11,901 |12,132 |13,555 |14,708 | | |
| |12,591 |14,289 |12,943 |15,645 |15,376 |15,788 | | |
| |11,638 |12,955 |12,637 |11,702 |12,786 |13,542 | | |
| |12,536 |14,495 |14,612 |14,490 |12,852 |12,658 | | |
|12 |11,820 |12,832 |13,906 |12,432 |13,609 |12,919 |12,8 |0,5|
| |11,806 |12,724 |14,135 |14,936 |14,227 |14,873 | | |
| |13,376 |14,119 |14,326 |15,519 |15,372 |15,364 | | |
| |12,175 |12,236 |14,132 |13,943 |12,417 |13,732 | | |
| |10,139 |12,117 |10,606 |10,683 |10,995 |12,416 | | |
| |11,786 |12,572 |14,074 |15,135 |14,459 |12,269 | | |
|13 |11,574 |11,725 |11,798 |12,740 |13,207 |13,470 |13,5 |0,2|
| | | | | | | | |5 |
| |12,602 |14,100 |13,887 |14,496 |13,683 |15,434 | | |
| |12,012 |13,772 |14,191 |13,929 |13,937 |13,956 | | |
| |12,245 |12,743 |14,992 |15,045 |14,583 |12,772 | | |
| |12,502 |13,123 |15,173 |13,344 |12,592 |14,666 | | |
| |12,976 |13,812 |15,706 |15,414 |15,655 |14,494 | | |
|14 |14,162 |15,519 |16,403 |17,273 |15,211 |18,008 |12 |0,5|
| | | | | | | | |96 |
| |12,522 |12,853 |15,488 |15,031 |14,035 |14,447 | | |
| |13,123 |13,967 |13,330 |14,221 |13,849 |16,304 | | |
| |12,043 |13,658 |13,493 |13,774 |14,343 |13,151 | | |
| |11,472 |12,136 |14,103 |12,386 |12,564 |13,363 | | |
| |12,441 |13,616 |12,717 |14,347 |15,090 |15,575 | | |
|15 |11,265 |12,052 |14,016 |14,047 |12,064 |11,667 |13 |0,7|
| |12,131 |12,983 |12,296 |15,291 |14,095 |13,940 | | |
| |14,480 |16,275 |16,717 |16,194 |15,413 |15,427 | | |
| |12,572 |13,244 |14,897 |14,571 |15,340 |15,674 | | |
| |10,865 |11,659 |10,923 |11,533 |13,089 |11,709 | | |
| |11,962 |13,388 |12,492 |14,907 |13,977 |14,358 | | |
|16 |10,908 |11,114 |13,790 |11,873 |13,029 |11,012 |13,5 |2 |
| |9,766 |11,117 |12,354 |11,287 |12,546 |10,249 | | |
| |11,702 |12,455 |13,484 |12,153 |13,075 |12,310 | | |
| |11,472 |12,053 |11,617 |14,065 |11,503 |13,322 | | |
| |13,503 |14,469 |14,195 |16,018 |14,812 |15,556 | | |
| |15,456 |17,355 |16,685 |15,598 |16,603 |19,311 | | |
|17 |10,161 |10,431 |10,695 |13,393 |11,153 |11,751 |12 |0,6|
| | | | | | | | |8 |
| |11,492 |13,087 |11,889 |12,564 |13,613 |11,699 | | |
| |12,428 |14,259 |12,561 |13,100 |13,888 |14,348 | | |
| |12,416 |14,059 |12,522 |13,706 |14,389 |13,536 | | |
| |10,813 |10,818 |12,951 |12,192 |11,765 |14,450 | | |
| |13,388 |14,590 |14,628 |14,764 |14,307 |16,962 | | |
|18 |9,889 |11,198 |10,095 |12,783 |11,183 |10,835 |13,33 |1,5|
| |12,517 |13,735 |14,247 |13,208 |15,072 |15,429 | | |
| |12,786 |13,231 |15,070 |13,013 |14,133 |16,174 | | |
| |11,863 |12,183 |13,377 |13,203 |11,916 |12,421 | | |
| |11,444 |11,999 |13,243 |14,233 |13,024 |12,491 | | |
| |14,696 |14,906 |14,730 |17,126 |17,331 |15,297 | | |
|19 |11,999 |12,509 |12,361 |14,850 |14,026 |15,078 |10 |0,1|
| |12,241 |13,124 |15,153 |14,655 |15,038 |15,460 | | |
| |12,120 |12,240 |12,945 |12,701 |13,006 |13,616 | | |
| |11,506 |12,815 |13,497 |13,746 |13,218 |14,658 | | |
| |12,376 |12,808 |14,477 |15,690 |14,127 |12,427 | | |
| |12,148 |13,932 |13,771 |14,039 |14,440 |12,250 | | |
|20 |11,293 |11,563 |14,165 |14,763 |12,874 |14,426 |12 |0,2|
| | | | | | | | |8 |
| |12,112 |13,348 |13,988 |12,222 |14,409 |13,381 | | |
| |11,429 |11,680 |14,364 |11,472 |13,201 |11,925 | | |
| |10,526 |11,960 |11,740 |12,709 |12,061 |12,319 | | |
| |11,467 |11,774 |11,862 |12,466 |12,013 |12,865 | | |
| |11,467 |11,747 |12,326 |13,699 |13,968 |12,173 | | |
|21 |11,954 |13,543 |13,158 |14,299 |14,420 |12,797 |13,6 |0,0|
| | | | | | | | |6 |
| |11,913 |12,292 |13,565 |13,888 |13,886 |12,930 | | |
| |11,572 |11,745 |12,146 |12,162 |13,720 |15,055 | | |
| |12,591 |14,354 |12,829 |14,346 |13,412 |16,083 | | |
| |11,638 |13,317 |13,026 |11,899 |14,227 |13,862 | | |
| |12,536 |14,156 |14,004 |15,164 |14,165 |12,651 | | |
|22 |11,820 |12,499 |11,834 |15,220 |12,420 |12,394 |13,281|0,5|
| |11,806 |12,803 |14,115 |14,830 |12,295 |14,506 | | |
| |13,376 |13,724 |15,571 |14,055 |15,511 |14,009 | | |
| |12,175 |13,275 |14,218 |14,957 |13,936 |12,930 | | |
| |10,139 |10,635 |10,975 |12,273 |10,431 |13,031 | | |
| |11,786 |13,607 |14,470 |14,452 |13,871 |12,949 | | |
|23 |11,574 |12,615 |12,977 |14,600 |11,962 |12,629 |12,630|1 |
| | | | | | | |4 | |
| |12,602 |14,600 |13,219 |12,806 |14,657 |13,855 | | |
| |12,012 |12,482 |12,072 |13,876 |13,781 |12,645 | | |
| |12,245 |14,012 |13,432 |14,994 |14,489 |14,450 | | |
| |12,502 |12,511 |12,574 |12,581 |13,727 |13,138 | | |
| |12,976 |14,279 |15,732 |15,464 |15,332 |15,686 | | |
|24 |14,162 |15,097 |16,664 |15,119 |15,614 |16,602 |14 |0,9|
| | | | | | | | |9 |
| |12,522 |13,107 |13,259 |15,520 |14,122 |15,000 | | |
| |13,123 |13,462 |14,243 |15,733 |14,196 |13,900 | | |
| |12,043 |12,656 |14,471 |13,502 |13,883 |15,184 | | |
| |11,472 |12,402 |14,301 |11,937 |12,424 |13,662 | | |
| |12,441 |12,922 |12,501 |14,496 |14,711 |16,107 | | |
|25 |11,265 |12,585 |11,693 |11,892 |12,272 |11,771 |13,050|1,8|
| | | | | | | |3 | |
| |12,131 |13,804 |14,553 |12,888 |12,203 |13,570 | | |
| |14,480 |15,883 |16,938 |17,964 |15,827 |16,217 | | |
| |12,572 |13,682 |14,710 |13,215 |14,412 |12,883 | | |
| |10,865 |11,008 |12,093 |10,942 |11,787 |14,353 | | |
| |11,962 |12,771 |12,948 |12,553 |14,741 |14,898 | | |
|26 |10,908 |12,573 |13,457 |13,093 |13,315 |12,251 |13,476|2,5|
| | | | | | | |8 | |
| |9,766 |11,044 |11,508 |11,785 |10,927 |10,205 | | |
| |11,702 |13,345 |13,567 |14,966 |14,416 |13,449 | | |
| |11,472 |11,740 |12,569 |12,698 |13,065 |11,547 | | |
| |13,503 |14,632 |16,452 |14,309 |15,559 |16,024 | | |
| |15,456 |16,674 |17,416 |16,571 |17,512 |19,215 | | |
|27 |10,161 |12,013 |12,309 |13,480 |10,737 |10,362 |12,653|2 |
| | | | | | | |1 | |
| |11,492 |12,874 |12,130 |11,774 |12,208 |13,078 | | |
| |12,428 |14,133 |13,361 |15,555 |13,970 |12,772 | | |
| |12,416 |13,743 |12,485 |15,630 |14,477 |14,736 | | |
| |10,813 |10,866 |12,196 |10,991 |12,082 |11,145 | | |
| |13,388 |15,128 |13,703 |16,777 |13,895 |17,200 | | |
|28 |9,889 |11,361 |10,420 |10,167 |10,772 |11,689 |13,209|1,8|
| |12,517 |12,568 |13,011 |12,590 |13,770 |14,967 | | |
| |12,786 |14,656 |12,976 |13,292 |14,371 |13,211 | | |
| |11,863 |13,064 |14,263 |15,093 |13,658 |12,023 | | |
| |11,444 |12,354 |13,277 |12,915 |13,978 |15,040 | | |
| |14,696 |16,068 |17,289 |15,475 |15,921 |16,822 | | |
|29 |11,999 |13,785 |12,086 |13,075 |12,001 |14,557 |12,489|0,4|
| | | | | | | |1 |88 |
| |12,241 |13,972 |12,655 |15,409 |14,427 |15,364 | | |
| |12,120 |12,223 |13,483 |14,749 |13,943 |15,458 | | |
| |11,506 |12,741 |13,154 |14,019 |13,421 |14,352 | | |
| |12,376 |12,607 |15,165 |15,504 |12,505 |16,273 | | |
| |12,148 |13,647 |13,685 |13,575 |13,531 |13,580 | | |
|30 |11,293 |11,455 |11,496 |13,262 |12,301 |13,370 |12,126|0,5|
| | | | | | | |05 | |
| |12,112 |13,212 |14,231 |14,946 |12,130 |13,687 | | |
| |11,429 |12,990 |11,766 |14,277 |12,649 |12,901 | | |
| |10,526 |10,985 |11,664 |10,955 |12,575 |11,408 | | |
| |11,467 |13,087 |13,639 |12,660 |11,969 |14,170 | | |
| |11,467 |12,661 |13,769 |11,926 |13,023 |14,202 | | |

Задача 2. Формирование оптимального портфеля ценных бумаг по модели

Тобина.

Порядок выполнения задачи.

1. В соответствии с вариантом задания получить данные по эффективности безрисковых бумаг [pic]и допустимому риску портфеля [pic].

2. Данные по эффективности рисковых бумаг [pic] и необходимые результаты расчетов взять из предыдущей задачи.

3. Составить математическую модель портфеля Тобина минимального риска