ÑàíêòÏåòåðáóðãñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò
Ïðèêëàäíàÿ ìàòåìàòèêà è èíôîðìàòèêà
Èññëåäîâàíèå îïåðàöèé è ïðèíÿòèå ðåøåíèé â çàäà÷àõ
îïòèìèçàöèè, óïðàâëåíèÿ è ýêîíîìèêè
Ïåòðîâà Àííà Àíäðååâíà
Ìîäåëü ïîðòôåëÿ êîëëåêöèîííûõ ìîíåò ÐÔ
Áàêàëàâðñêàÿ ðàáîòà
Íàó÷íûé ðóêîâîäèòåëü:
ê. ô.-ì. í., äîöåíò Â.Â. Áóõâàëîâà
Ðåöåíçåíò:
Ìåíåäæåð ïî ðàáîòå ñ êëþ÷åâûìè êëèåíòàìè
Ñåâåðî-Çàïàäíûé áàíê ÏÀÎ Ñáåðáàíê
À.Â. Êîâàëü÷óê
Ñàíêò-Ïåòåðáóðã
2016
SAINT-PETERSBURG STATE UNIVERSITY
Applied Mathematics and Computer Science
Operation Research and Decision Making in Optimisation, Control
and Economics Problems
Petrova Anna Andreevna
Portfolio model of Russian collection coins
Bachelor's Thesis
Scienti c supervisor:
Associate Professor V.V. Buhvalova
Reviewer:
Key Accounts Manager
NorthWest Bank ¾Sberbank of Russia¿
A.V. Kovalchuk
SaintPetersburg
2016
Ñîäåðæàíèå
1 Ââåäåíèå
5
2 Îñíîâíûå õàðàêòåðèñòèêè öåííûõ áóìàã
6
2.1
2.2
Äîõîäíîñòü . . . . . . .
Ðèñê . . . . . . . . . . .
2.2.1 Õåäæèðîâàíèå .
2.2.2 Ñòðàõîâàíèå . . .
2.2.3 Äèâåðñèôèêàöèÿ
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3 Ïîðòôåëü öåííûõ áóìàã
3.1
3.2
3.3
3.4
Ïîðòôåëü è åãî îñíîâíûå õàðàêòåðèñòèêè
3.1.1 Äîõîäíîñòü . . . . . . . . . . . . .
3.1.2 Îæèäàåìàÿ äîõîäíîñòü . . . . . .
3.1.3 Ðèñê . . . . . . . . . . . . . . . . .
Äèâåðñèôèêàöèÿ ïîðòôåëÿ . . . . . . . .
Ìîäåëè ïîðòôåëåé öåííûõ áóìàã . . . . .
3.3.1 Ìîäåëü Ìàðêîâèöà . . . . . . . . .
3.3.2 Ìîäåëü Áëýêà . . . . . . . . . . . .
3.3.3 Ìîäåëü ÒîáèíàØàðïàËèíòåðà .
Îïòèìàëüíûé ïîðòôåëü èç äâóõ àêòèâîâ
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
6
7
8
8
9
10
10
10
11
11
12
13
14
15
15
16
4 Ìåòîäû îáðàáîòêè âðåìåííûõ ðÿäîâ
20
5 Îïòèìàëüíûé ïîðòôåëü êîëëåêöèîííûõ ìîíåò
22
4.1
4.2
5.1
5.2
5.3
5.4
5.5
Ëèíåéíàÿ èíòåðïîëÿöèÿ . . . . . . . . . . . . . . . . . . . . . . . .
Ñòóïåí÷àòàÿ èíòåðïîëÿöèÿ . . . . . . . . . . . . . . . . . . . . . .
Îáðàáîòêà äàííûõ . . . . . . . . . . .
Âû÷èñëåíèå ýôôåêòèâíîãî ïîðòôåëÿ
Ïîñòðîåíèå ýôôåêòèâíîé ãðàíèöû . .
Ñðàâíåíèå ïîðòôåëåé . . . . . . . . .
Ðàñøèðåííûé ïîðòôåëü . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
20
20
22
24
26
26
28
6 Ðåàëèçàöèÿ îïòèìàëüíîãî ïîðòôåëÿ
32
7 Çàêëþ÷åíèå
39
6.1
6.2
6.3
Ïîðòôåëü èç ïÿòè ìîíåò . . . . . . . . . . . . . . . . . . . . . . . .
Ïîðòôåëü èç äåñÿòè ìîíåò . . . . . . . . . . . . . . . . . . . . . .
Âûâîä . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
33
36
38
1
Ââåäåíèå
Âî âñå âðåìåíà ëþäåé èíòåðåñîâàëè íàäåæíûå ñïîñîáû èíâåñòèðîâàíèÿ. Îäíèì èç ïåðâûõ ñïîñîáîâ èíâåñòèðîâàíèÿ áûëî êîëëåêöèîíèðîâàíèå ìîíåò. Â
íàñòîÿùåå âðåìÿ íàáëþäàåòñÿ ðîñò èíòåðåñà ê ïîäîáíîìó âèäó âëîæåíèé. Ïî
èíôîðìàöèè, ïðèâåäåííîé â [9], êîëëåêöèîíèðîâàíèåì ìîíåò â Ðîññèè çàíèìàåòñÿ íå ìåíåå 20% òðóäîñïîñîáíîãî íàñåëåíèÿ.
Èñõîäÿ èç àêòóàëüíîñòè äàííîãî ñïîñîáà èíâåñòèðîâàíèÿ, ìîèì íàó÷íûì
ðóêîâîäèòåëåì, Â. Â. Áóõâàëîâîé, áûëî ïðåäëîæåíî ðàçðàáîòàòü ìåòîäèêó, êîòîðàÿ ïîìîãëà áû èñïîëüçîâàòü ïîðòôåëüíóþ òåîðèþ Ìàðêîâèöà äëÿ èíâåñòèðîâàíèÿ â êîëëåêöèîííûå ìîíåòû.
 ðàáîòå ïðåäëîæåíû ýòàïû ôîðìèðîâàíèÿ îïòèìàëüíîãî ïîðòôåëÿ èç öåííûõ ìîíåò. Ìåòîäèêà ïðîâåðåíà íà ïîðòôåëÿõ, ñîñòîÿùèõ îò ÷åòûðåõ äî äåñÿòè
ìîíåò. Ïðîàíàëèçèðîâàíû ïðîáëåìû, âîçíèêàþùèå ïðè ðåàëèçàöèè ïîðòôåëåé.
Ðàáîòà ñîñòîèò èç 5 ãëàâ è ïðèëîæåíèé. Â ãëàâå 1 ïðèâåäåíû îïðåäåëåíèÿ
îñíîâíûõ õàðàêòåðèñòèê öåííûõ áóìàã: äîõîäíîñòü, îæèäàåìàÿ äîõîäíîñòü è
ðèñê. Â ãëàâå 2 ââåäåíî ïîíÿòèå ïîðòôåëÿ öåííûõ áóìàã, îïèñàíû åãî îñíîâíûå
õàðàêòåðèñòèêè è ðàññìîòðåíû îñíîâíûå ìîäåëè ôîðìèðîâàíèÿ ïîðòôåëÿ. Â
ãëàâå 3 ïðèâåäåíû ìåòîäû îáðàáîòêè âðåìåííûõ ðÿäîâ, íåîáõîäèìûå äëÿ ðàáîòû ñ äàííûìè î ìîíåòàõ.  ãëàâå 4 ïðåäëîæåíû ýòàïû ïîñòðîåíèÿ ýôôåêòèâíîãî ïîðòôåëÿ èç êîëëåêöèîííûõ ìîíåò, à òàêæå âû÷èñëåí è ïðîàíàëèçèðîâàí
êîíêðåòíûé ýôôåêòèâíûé ïîðòôåëü. Â ãëàâå 5 ñîñòàâëåí ðåàëüíûé ïîðòôåëü,
ïðèáëèæåííûé ê ðàíåå ðàññìîòðåííîìó ýôôåêòèâíîìó, ïîñ÷èòàíû ìèíèìàëüíûå âëîæåíèÿ äëÿ åãî ðåàëèçàöèè, à òàêæå âû÷èñëåíà äîõîäíîñòü ïîëó÷èâøåãîñÿ ïîðòôåëÿ.
5
2
Îñíîâíûå õàðàêòåðèñòèêè öåííûõ áóìàã
 ýòîé ãëàâå ïðèâåäåíû îïðåäåëåíèÿ îñíîâíûõ õàðàêòåðèñòèê öåííûõ áóìàã:
äîõîäíîñòü, îæèäàåìàÿ äîõîäíîñòü è ðèñê. [4], [6]
2.1
Äîõîäíîñòü
Äëÿ íà÷àëà îïðåäåëèì ïîíÿòèå
ïåðèîä T:
Ri =
äîõîäíîñòè àêòèâà
çà îïðåäåëåííûé
Si (T ) − Si (0)
,
Si (0)
(1)
ãäå Si (0) è Si (T ) öåíû àêòèâà â ìîìåíò âðåìåíè 0 è T ñîîòâåòñòâåííî.
ßñíî, ÷òî äîõîäíîñòü íå ÿâëÿåòñÿ ñîâåðøåííûì ïîêàçàòåëåì: äîõîäíîñòü â
10% çà ãîä çíà÷èòåëüíî ìåíüøå äîõîäíîñòè â 10% çà ìåñÿö.
Îòìåòèì, ÷òî â ôîðìóëå (1) ìû çíàåì íàâåðíÿêà ëèøü Si (0), ÷èñëî æå Si (T )
ÿâëÿåòñÿ ñëó÷àéíûì. Ýòî ïîêàçûâàåò, ÷òî äîõîäíîñòü ÿâëÿåòñÿ ñëó÷àéíîé âåëè÷èíîé.
Ââåäåì ñëåäóþùåå îïðåäåëåíèå â ïðåäïîëîæåíèè, ÷òî äîõîäíîñòü àêòèâîâ
èìååò äèñêðåòíîå ðàñïðåäåëåíèå.
Îæèäàåìàÿ äîõîäíîñòü àêòèâà îñíîâûâàåòñÿ íà ñòàòèñòè÷åñêèõ äàííûõ,
ïîñëå àíàëèçà êîòîðûõ ìîæíî âûäåëèòü íåñêîëüêî âîçìîæíûõ ñöåíàðèåâ s è
ïðåäñòàâèòü äàííûå â âèäå òàáëèöû äîõîäíîñòåé R è ñîîòâåòñòâóþùèõ âåðîÿòíîñòåé P r.
Äëÿ âû÷èñëåíèÿ îæèäàåìîé äîõîäíîñòè èñïîëüçóåòñÿ ôîðìóëà ìàòåìàòè÷åñêîãî îæèäàíèÿ:
X
E(R) =
P r(s)R(s),
s
ãäå s îäèí èç âîçìîæíûõ ñöåíàðèåâ, R(s) äîõîäíîñòü â ñëó÷àå ñöåíàðèÿ
s, P r(s) âåðîÿòíîñòü ñöåíàðèÿ s.
Ïðîèëëþñòðèðóåì ââåäåííîå ïîíÿòèå íà ïðèìåðå.
Ïðèìåð. Èìååòñÿ ñëåäóþùàÿ âåðîÿòíîñòíàÿ ñõåìà àêòèâà À ñ òðåìÿ ñöåíàðèÿìè:
Äîõîäíîñòü 5% 10% -7%
Âåðîÿòíîñòü 0.2 0.5 0.3
Ïðè òàêèõ äàííûõ îæèäàåìàÿ äîõîäíîñòü ðàâíà 3.9% :
E = 5% × 0.2 + 10% × 0.5 + (−7%) × 0.3 = 3.9%.
6
Çàìåòèì, ÷òî â íåêîòîðûõ ñëó÷àÿõ åñòåñòâåííî ñ÷èòàòü ðàñïðåäåëåíèå âåðîÿòíîñòåé äîõîäíîñòè íåïðåðûâíûì (õîòÿ íà ñàìîì äåëå öåíà öåííîé áóìàãè
íå ïðèíèìàåò ïðîèçâîëüíûå âåùåñòâåííûå çíà÷åíèÿ åñòü ìèíèìàëüíàÿ öåíà
èçìåíåíèÿ ¾òèê¿). Òîãäà îæèäàåìàÿ äîõîäíîñòü âû÷èñëÿåòñÿ ïî ôîðìóëå:
Z
∞
E(R) =
xf (x)dx,
−∞
ãäå f (x) ïëîòíîñòü ðàñïðåäåëåíèÿ âåðîÿòíîñòåé ñëó÷àéíîé âåëè÷èíû R.
Òåîðèÿ ïîðòôåëÿ Ìàðêîâèöà ïðåäïîëàãàåò, ÷òî äîõîäíîñòè öåííûõ áóìàã
ïîä÷èíåíû íîðìàëüíîìó çàêîíó.
2.2
Ðèñê
Ïîä ðèñêîì áóäåò ïîíèìàòüñÿ ðèñê îøèáèòüñÿ â ïðîãíîçå î ñðåäíåì. Ìàòåìàòè÷åñêîé ìåðîé ðàçáðîñà âîêðóã ñðåäíåãî çíà÷åíèÿ ÿâëÿåòñÿ äèñïåðñèÿ,
íàçûâàåìàÿ â òåîðèè ôèíàíñîâ âàðèàöèåé. Îäíàêî â êà÷åñòâå ðèñêà ðàññìàòðèâàåòñÿ ñòàíäàðòíîå îòêëîíåíèå îò ìàòåìàòè÷åñêîãî îæèäàíèÿ âåðîÿòíîñòíîé
ñõåìû:
p
σ(R) =
E[(R − E(R))2 ].
 äèñêðåòíîì ñëó÷àå ôîðìóëà áóäåò èìåòü âèä:
σ(R) =
s
X
P r(s)(R(s) − E(R))2 .
s
Ïðèìåð (ïðîäîëæåíèå). Â ïðèâåäåííîì âûøå ïðèìåðå ðèñê ðàâåí 7.38%
ñ òî÷íîñòüþ äî äâóõ çíàêîâ ïîñëå çàïÿòîé:
σ=
p
(5% − 3.9%)2 × 0.2 + (10% − 3.9%)2 × 0.5 + ((−5%) − 3.9%)2 × 0.3 = 7.38%.
Çàìåòèì, ÷òî ñòàíäàðòíîå îòêëîíåíèå îäèíàêîâî ïîêàçûâàåò îòêëîíåíèÿ îò
ñðåäíåãî â ìåíüøóþ è áîëüøóþ ñòîðîíó.  ñëó÷àå íîðìàëüíîãî ðàñïðåäåëåíèÿ, êîòîðîå ñèììåòðè÷íî îòíîñèòåëüíî ñðåäíåãî, ìû êàê áû ïðåóâåëè÷èâàåì
îøèáêó âäâîå. Îäíàêî, äëÿ ìèíèìèçàöèè âåëè÷èíû ðèñêà áåçðàçëè÷íî, ìèíèìèçèðîâàòü äàííóþ âåëè÷èíó èëè âäâîå ìåíüøóþ.
Äëÿ íåñèììåòðè÷íûõ ðàñïðåäåëåíèé ñêàçàííîå âûøå íåñïðàâåäëèâî. Äëÿ
íèõ ïðèíÿòî ðàññìàòðèâàòü íèæíèé ðèñê, ó÷èòûâàþùèé òîëüêî îòêëîíåíèÿ
âíèç îò ñðåäíåãî, îïðåäåëÿåìûé êàê êîðåíü èç ïîëóâàðèàöèè :
semi-σ(R)− = E[{(R(s) − E(R))− }2 ],
7
ãäå ÷åðåç ñèìâîë ¾− ¿ îáîçíà÷åíà îòðèöàòåëüíàÿ ÷àñòü ÷èñëà.
Äëÿ óìåíüøåíèÿ ðèñêà ñóùåñòâóåò ÷åòûðå îñíîâíûõ ïðèåìà óïðàâëåíèÿ
ðèñêîì [1]. Ðàññìîòðèì èõ.
•
ýòî ñîçíàòåëüíîå ðåøåíèå íå ïîäâåðãàòüñÿ îïðåäåëåííîìó âèäó ðèñêà. Ïðèìåð: îòêàç îò èíâåñòèðîâàíèÿ â ðèñêîâàííûå ïðîåêòû.
•
Ïðåäîòâðàùåíèå óùåðáà
•
Ïðèíÿòèå ðèñêà íà ñåáÿ
•
Ïåðåíîñ ðèñêà
Èçáåæàíèå ðèñêà
ñâîäèòñÿ ê äåéñòâèÿì, ïðåäïðèíèìàåìûì äëÿ óìåíüøåíèÿ âåðîÿòíîñòè ïîòåðü è äëÿ ìèíèìèçàöèè èõ ïîñëåäñòâèé. Ïðèìåð:
èçó÷åíèå ôèíàíñîâîé ñèòóàöèè íà ðûíêå äëÿ îïðåäåëåíèÿ íàïðàâëåíèÿ èíâåñòèöèîííîãî ïðîåêòà.
ñîñòîèò â ïîêðûòèè óáûòêîâ çà ñ÷åò ñîáñòâåííûõ
ðåñóðñîâ. Ïðèìåð: õðàíåíèå äåíåã â íåñòàáèëüíîé íàöèîíàëüíîé âàëþòå.
ñîñòîèò â ïåðåíåñåíèè ðèñêà íà äðóãèõ ëèö.
Çàìåòèì, ÷òî ïåðåíîñ ðèñêà çà÷àñòóþ íåâîçìîæåí. Ðàññìîòðèì ïðîñòîé ïðèìåð: ëþáîé äîì ïîäâåðæåí ðàçëè÷íûì âèäàì ðèñêà (ïîæàð, ñòèõèéíîå áåäñòâèå,
ïàäåíèå öåí è ò. ä.), î÷åâèäíî, ÷òî ïðîäàæà äîìà (òî åñòü îñâîáîæäåíèå îò âñåõ
ðèñêîâ) íå âñåãäà ÿâëÿåòñÿ æåëàåìûì âûõîäîì èç ñèòóàöèè.  òàêîì ñëó÷àå
ìîæíî óïðàâëÿòü ðèñêàìè äðóãèìè ñïîñîáàìè. Íàïðèìåð, âëàäåëåö ìîæåò çàñòðàõîâàòü ñâîé äîì, òåì ñàìûì èçáåæàòü ðèñê ïîæàðà è ñòèõèéíîãî áåäñòâèÿ
è ïðèíÿòü íà ñåáÿ òîëüêî ðèñê ïàäåíèÿ öåí íà íåäâèæèìîñòü.
Ðàçëè÷àþò òðè ìåòîäà ïåðåíîñà ðèñêà, íàçûâàåìûå òðåìÿ ñõåìàìè ïåðåíîñà
ðèñêà. Ðàññìîòðèì êàæäûé èç íèõ.
2.2.1
Õåäæèðîâàíèå
Î õåäæèðîâàíèè ðèñêà ãîâîðÿò â òåõ ñëó÷àÿõ, êîãäà äåéñòâèå, ïðåäïðèíÿòîå
äëÿ ñíèæåíèÿ ðèñêà ïîíåñòè óáûòêè, îäíîâðåìåííî ïðèâîäèò è ê íåâîçìîæíîñòè ïîëó÷èòü äîõîä.
Íàïðèìåð, åñëè âû ïîäïèñàëèñü íà æóðíàë íà íåñêîëüêî ëåò âïåðåä, âû
ñòðàõóåòåñü îò âîçìîæíîãî ïîâûøåíèÿ öåíû íà ïîäïèñêó. Òåì ñàìûì èçáàâëÿåòåñü îò ðèñêà óáûòêîâ, êîòîðûå ìîæåòå ïîíåñòè â ñëó÷àå ïîâûøåíèÿ öåíû, íî
â òî æå âðåìÿ íè÷åãî íå âûèãðàåòå, åñëè ïîäïèñêà ïîäåøåâååò.
2.2.2
Ñòðàõîâàíèå
Ñòðàõîâàíèå ïðåäïîëàãàåò âûïëàòó ñòðàõîâîãî âçíîñà (öåíû, êîòîðóþ âû
ïëàòèòå çà ñòðàõîâêó) ñ öåëüþ èçáåæàíèÿ óáûòêîâ. Ïðèîáðåòàÿ ñòðàõîâîé ïîëèñ, âû ñîãëàøàåòåñü ïîéòè íà ãàðàíòèðîâàííûå èçäåðæêè âçàìåí âåðîÿòíîñòè
ïîíåñòè ãîðàçäî áîëüøèé óùåðá, ñâÿçàííûé ñ îòñóòñòâèåì ñòðàõîâêè.
8
Ìåæäó õåäæèðîâàíèåì è ñòðàõîâàíèåì ñóùåñòâóåò ôóíäàìåíòàëüíîå ðàçëè÷èå.  ñëó÷àå õåäæèðîâàíèÿ âû óñòðàíÿåòå ðèñê ïîíåñòè óáûòêè, îòêàçûâàÿñü
îò âîçìîæíîñòè ïîëó÷èòü äîõîä. À â ñëó÷àå ñòðàõîâàíèÿ âû ïëàòèòå ñòðàõîâîé âçíîñ, óñòðàíÿÿ òåì ñàìûì ðèñê ïîíåñòè óáûòêè, íî ñîõðàíÿÿ âîçìîæíîñòü
ïîëó÷èòü äîõîä.
2.2.3
Äèâåðñèôèêàöèÿ
Äèâåðñèôèêàöèÿ âûðàæàåòñÿ âî âëàäåíèè ìíîãèìè ðèñêîâàííûìè àêòèâàìè, âìåñòî êîíöåíòðàöèè âñåõ êàïèòàëîâëîæåíèé òîëüêî â îäíîì èç íèõ. Â ñâÿçè
ñ ýòèì äèâåðñèôèêàöèÿ îãðàíè÷èâàåò âàøó ïîäâåðæåííîñòü ðèñêó, ñâÿçàííîìó
ñ îäíèì åäèíñòâåííûì âèäîì àêòèâîâ.
Ïîðòôåëüíàÿ òåîðèÿ, î êîòîðîé ïîéäåò ðå÷ü íèæå, îñíîâûâàåòñÿ íà ìåòîäå äèâåðñèôèêàöèè. Îáîñíîâàíèå åãî ýôôåêòèâíîñòè áóäåò ðàññìîòðåíî ïîñëå
ââåäåíèÿ îñíîâíûõ õàðàêòåðèñòèê ïîðòôåëÿ.
9
3
Ïîðòôåëü öåííûõ áóìàã
 ýòîé ãëàâå ââåäåíî ïîíÿòèå ïîðòôåëÿ öåííûõ áóìàã, îïèñàíû åãî îñíîâíûå
õàðàêòåðèñòèêè è ðàññìîòðåíû îñíîâíûå ìîäåëè ôîðìèðîâàíèÿ ïîðòôåëÿ. Âñå
ïîíÿòèÿ è ìîäåëè èëëþñòðèðóþòñÿ íà ïðèìåðå.
3.1
Ïîðòôåëü è åãî îñíîâíûå õàðàêòåðèñòèêè
Ïîä ïîðòôåëåì ìû áóäåì ïîíèìàòü îáúåäèíåíèå äâóõ è áîëåå öåííûõ áóìàã
(àêòèâîâ) [3].
Ïóñòü A = {a1 , . . . , an } ïåðå÷åíü àêòèâîâ ðûíêà. Çàäàäèì ïîðòôåëü è êàê
âåêòîð îòíîñèòåëüíûõ âåñîâ êàæäîãî àêòèâà:
π = {x1 , . . . , xn },
n
X
xi = 1.
i=1
Äîëÿ èñõîäíîãî êàïèòàëà, èíâåñòèðîâàííîãî â àêòèâ ai ìîæåò áû âû÷èñëåíà
ïî ôîðìóëå:
xi =
ki Si (0)
,
Sπ (0)
(2)
ãäå ki êîëè÷åñòâî àêòèâà ai , âõîäÿùåå â ïîðòôåëü π , Si (0) è Sπ (0) öåíû
àêòèâà ai è ïîðòôåëÿ π â ìîìåíò âðåìåíè 0.
 ïðîñòåéøåì ñëó÷àå âñå îòíîñèòåëüíûå âåñà íå îòðèöàòåëüíû: xi ≥ 0. Îäíàêî, åñëè âîçìîæíû êîðîòêèå (îòëîæåííûå) ïðîäàæè, òî âåñà ìîãóò áûòü ëþáîãî
çíàêà. Êîðîòêèå ïðîäàæè ïðîäàæè àêòèâîâ, êîòîðûìè èíâåñòîð â íàñòîÿùåå
âðåìÿ íå îáëàäàåò.
3.1.1
Äîõîäíîñòü
Äîõîäíîñòü ïîðòôåëÿ Rπ çà ïåðèîä T âû÷èñëÿåòñÿ ïî ôîðìóëå:
Rπ =
Sπ (T ) − Sπ (0)
.
Sπ (0)
Äîõîäíîñòü ïîðòôåëÿ, êàê ñëó÷àéíàÿ âåëè÷èíà, âûðàæàåòñÿ ÷åðåç äîõîäíîñòü åãî àêòèâîâ ïî ñîîòíîøåíèþ:
Rπ =
n
X
i=1
10
xi Ri .
3.1.2
Îæèäàåìàÿ äîõîäíîñòü
Îæèäàåìàÿ äîõîäíîñòü (ýôôåêòèâíîñòü) ïîðòôåëÿ, êàê ìàòåìàòè÷åñêîå îæèäàíèå, áóäåò ðàâíà ëèíåéíîé êîìáèíàöèè îæèäàåìûõ äîõîäíîñòåé àêòèâîâ:
mπ = E[Rπ ] =
n
X
xi mi .
i=1
3.1.3
Ðèñê
Äèñïåðñèÿ îæèäàåìîé äîõîäíîñòè ïîðòôåëÿ âû÷èñëÿåòñÿ ïî ôîðìóëå:
σπ2
2
= D[Rπ ] = E[(Rπ − mπ ) ] =
n X
n
X
νij xi xj =
i=1 j=1
n X
n
X
σi σj ρij xi xj , ãäå
i=1 j=1
νij = cov(Ri , Rj ) = M [(Ri − mi )(Rj − mj )] êîâàðèàöèÿ,
ρij êîýôôèöèåíò êîððåëÿöèè Ri è Rj ,
σi ðèñê àêòèâà ai .
Çàìåòèì, ÷òî äëÿ íåêîððåëèðóåìûõ öåííûõ áóìàã (νi,j = 0, i 6= j ) äèñïåðñèÿ
ñîñòàâèò âåëè÷èíó:
σπ2
=
n
X
σi2 x2i .
i=1
Ðèñêîì ïîðòôåëÿ íàçîâåì âåëè÷èíó σπ .
Ïðèìåð.
Ïðîèëëþñòðèðóåì ïîíÿòèÿ, ââåäåííûå â ýòîé ãëàâå, íà ïðèìåðå ðûíêà ñ äâóìÿ íåêîððåëèðóåìûìè àêòèâàìè, îäèí èç êîòîðûõ (A) ìû óæå ðàññìàòðèâàëè
ðàíåå:
Ñöåíàðèé A 1
2
3
Äîõîäíîñòü 5% 10% -7%
Âåðîÿòíîñòü 0.2 0.5 0.3
Ñöåíàðèé B 1
2
3
4
5
6
Äîõîäíîñòü 8% 5% 4% 2% -1% -5%
Âåðîÿòíîñòü 0.3 0.15 0.05 0.05 0.15 0.3
11
Ðàññìîòðèì êîíêðåòíûé ïîðòôåëü:
Àêòèâ
A
B
Ñòîèìîñòü $50 $100
Êîëè÷åñòâî 10 15
Âû÷èñëèì âåêòîð îòíîñèòåëüíûõ âåñîâ àêòèâîâ â ïîðòôåëå, èñïîëüçóÿ ôîðìóëó (2):
10 × 50
= 0.25,
10 × 50 + 15 × 100
15 × 100
xB =
= 0.75.
10 × 50 + 15 × 100
Ñëåäîâàòåëüíî, π = {0.25, 0.75}.
Äëÿ òîãî, ÷òîáû âû÷èñëèòü îæèäàåìóþ äîõîäíîñòü è ðèñê ïîëó÷èâøåãîñÿ
ïîðòôåëÿ, íåîáõîäèìî âû÷èñëèòü îæèäàåìóþ äîõîäíîñòü è ðèñê êàæäîãî àêòèâà:
xA =
Àêòèâ
A
B
Îæèäàåìàÿ äîõîäíîñòü 3.9% 1.80%
Ðèñê
7.38% 5.72%
Òàêèì îáðàçîì, îæèäàåìàÿ äîõîäíîñòü è ðèñê ïîðòôåëÿ ðàâíû:
mπ = 3.9% × 0.25 + 1.80% × 0.75 = 2.33%,
p
σπ = 7.38%2 × 0.252 + 5.72%2 × 0.752 = 4.67%.
Çäåñü óìåñòíî çàäàòü âîïðîñ, à ñóùåñòâóåò ëè äðóãîé ïîðòôåëü íà òåõ æå
àêòèâàõ ñ áîëüøåé äîõîäíîñòüþ, íî ìåíüøèì ðèñêîì? Îòâåò íà íåãî áóäåò äàí
â ñëåäóþùèõ ïàðàãðàôàõ.
3.2
Äèâåðñèôèêàöèÿ ïîðòôåëÿ
Óòâåðæäåíèå. Ïðè ðîñòå ÷èñëà íåêîððåëèðóåìûõ öåííûõ áóìàã â ïîðòôåëå
åãî ðèñê ñòðåìèòñÿ ê íóëþ.
Äîêàçàòåëüñòâî.
Ïðåäïîëîæèì, ÷òî èíâåñòîð âëîæèë ñâîè äåíüãè ðàâíûìè äîëÿìè â n àêòèâîâ, ò. å. xi = n1 .
12
Òîãäà îæèäàåìàÿ äîõîäíîñòü è ðèñê òàêîãî ïîðòôåëÿ:
n
1X
mi ,
mπ =
n i=1
v
u n
X
1u
σπ = t
σi2 .
n i=1
Îáîçíà÷èì
σmax = max σi .
i=1,...,n
Òîãäà
v
u n
X
1u
1p 2
σmax
2
t
σπ ≤
σmax =
nσmax = √ .
n i=1
n
n
Îòñþäà âèäíî, ÷òî ïðè ðîñòå ÷èñëà ðàçëè÷íûõ öåííûõ áóìàã, âõîäÿùèõ â
ïîðòôåëü (n → ∞), ðèñê ïîðòôåëÿ îãðàíè÷åí è ñòðåìèòñÿ ê íóëþ.
3.3
Ìîäåëè ïîðòôåëåé öåííûõ áóìàã
Ïåðâàÿ ìîäåëü ôîðìèðîâàíèÿ îïòèìàëüíîãî ïîðòôåëÿ ïðèíàäëåæèò Ã. Ìàðêîâèöó. Â 1952 ã. îí îïóáëèêîâàë ñòàòüþ [5], â êîòîðîé âïåðâûå ïðåäëîæèë
ìàòåìàòè÷åñêóþ ìîäåëü ôîðìèðîâàíèÿ îïòèìàëüíîãî ïîðòôåëÿ è ïðèâ¼ë ìåòîäû ïîñòðîåíèÿ ïîðòôåëåé ïðè îïðåäåë¼ííûõ îãðàíè÷åíèÿõ. Çà äàëüíåéøóþ
ðàçðàáîòêó ýòîé òåîðèè Ìàðêîâèö â 1990 ã. ïîëó÷èë Íîáåëåâñêóþ ïðåìèþ ïî
ýêîíîìèêå.
Ïåðå÷èñëèì îñíîâíûå ïîñòóëàòû êëàññè÷åñêîé ïîðòôåëüíîé òåîðèè, íà êîòîðûõ îñíîâûâàëñÿ Ìàðêîâèö:
• Ðûíîê ñîñòîèò èç êîíå÷íîãî ÷èñëà àêòèâîâ, äîõîäíîñòè êîòîðûõ äëÿ çàäàííîãî ïåðèîäà ñ÷èòàþòñÿ ñëó÷àéíûìè âåëè÷èíàìè.
• Èíâåñòîð â ñîñòîÿíèè, íàïðèìåð, èñõîäÿ èç ñòàòèñòè÷åñêèõ äàííûõ, ïîëó÷èòü îöåíêó îæèäàåìûõ çíà÷åíèé äîõîäíîñòåé, èõ ïîïàðíûõ êîâàðèàöèé è
ñòåïåíåé âîçìîæíîñòè äèâåðñèôèêàöèè ðèñêà.
• Èíâåñòîð ìîæåò ôîðìèðîâàòü ëþáûå äîïóñòèìûå (äëÿ äàííîé ìîäåëè) ïîðòôåëè. Äîõîäíîñòè ïîðòôåëåé ÿâëÿþòñÿ òàêæå ñëó÷àéíûìè âåëè÷èíàìè.
13
• Ñðàâíåíèå âûáèðàåìûõ ïîðòôåëåé îñíîâûâàåòñÿ òîëüêî íà äâóõ êðèòåðèÿõ:
îæèäàåìîé äîõîäíîñòè è ðèñêå.
• Èíâåñòîð íå ñêëîíåí ê ðèñêó â òîì ñìûñëå, ÷òî èç äâóõ ïîðòôåëåé ñ îäèíàêîâîé äîõîäíîñòüþ îí îáÿçàòåëüíî ïðåäïî÷òåò ïîðòôåëü ñ ìåíüøèì ðèñêîì.
Äëÿ ôîðìèðîâàíèÿ îïòèìàëüíîãî èíâåñòèöèîííîãî ïîðòôåëÿ íåîáõîäèìî
ðåøèòü îïòèìèçàöèîííóþ çàäà÷ó. Ñóùåñòâóåò äâà âèäà çàäà÷: ïîèñê äîëåé àêöèé â ïîðòôåëå äëÿ äîñòèæåíèÿ ìàêñèìàëüíîé îæèäàåìîé äîõîäíîñòè ïðè çàäàííîé âåðõíåé ãðàíèöå ðèñêà σreq è ìèíèìèçàöèÿ ðèñêà ïðè çàäàííîé íèæíåé
ãðàíèöå äîõîäíîñòè ïîðòôåëÿ Rreq .
3.3.1
Ìîäåëü Ìàðêîâèöà
 äàííîé ìîäåëè äîïóñòèìûìè ÿâëÿþòñÿ ïîðòôåëè, óäîâëåòâîðÿþùèå äâóì
îñíîâíûì îãðàíè÷åíèÿì:
n
X
xi = 1,
(3)
i=1
xi ≥ 0,
∀i ∈ {1, . . . , n}.
(4)
Çàìåòèì, ÷òî ïðè òàêèõ îãðàíè÷åíèÿõ äîõîäíîñòü ïîðòôåëÿ íå ïðåâûøàåò
ìàêñèìàëüíîé äîõîäíîñòè àêòèâîâ, íà êîòîðûõ îí ïîñòðîåí.
Îïòèìèçàöèîííûå çàäà÷è ìîäåëè Ìàðêîâèöà ìîãóò áûòü çàïèñàíû â òàêîì
âèäå:
• Ïðÿìàÿ çàäà÷à:
2
σπ → min,
m ≥ R ,
π
req
Pn
i=1 xi = 1,
xi ≥ 0, ∀i ∈ {1, . . . , n}.
(5)
mπ → max,
σ 2 ≤ σ 2 ,
Pπ n req
i=1 xi = 1,
xi ≥ 0, ∀i ∈ {1, . . . , n}.
(6)
• Îáðàòíàÿ çàäà÷à:
14
Çàäà÷à (5) ÿâëÿåòñÿ çàäà÷åé êâàäðàòè÷íîãî ïðîãðàììèðîâàíèÿ, òî åñòü çàäà÷åé âèäà
(
F (x) = 21 xT D x + ñT x → inf,
(7)
A[M, N ] x[N ] ≥ b[M ],
ãäå D = D[N, N ] ñèììåòðè÷íàÿ ìàòðèöà. Ïëàíîì íàçûâàåòñÿ âåêòîð x, óäîâëåòâîðÿþùèé îãðàíè÷åíèÿì çàäà÷è (7). Ìíîæåñòâî ïëàíîâ îáîçíà÷èì çà Ω.
Ïëàí x∗ íàçûâàåòñÿ îïòèìàëüíûì, åñëè F (x∗ ) = infx∈Ω F (x).
Ïðèâåäåì òåîðåìó èç [2]:
Òåîðåìà. Ïðåäïîëîæèì, ÷òî ìàòðèöà D íåîòðèöàòåëüíî îïðåäåëåíà íà Nn ,
ìíîæåñòâî Ω íåïóñòî è öåëåâàÿ ôóíêöèÿ F (x) îãðàíè÷åíà ñíèçó íà Ω. Òîãäà ó
çàäà÷è (7) ñóùåñòâóåò îïòèìàëüíûé ïëàí. Îí ìîæåò áûòü ïîëó÷åí ïóò¼ì ðåøåíèÿ êîíå÷íîãî ÷èñëà ñèñòåì ëèíåéíûõ óðàâíåíèé.
 çàäà÷å (5) ìàòðèöà êîâàðèàöèé Q âñåãäà ïîëîæèòåëüíî îïðåäåëåíà, öåëåâàÿ ôóíêöèÿ îãðàíè÷åíà, çíà÷èò, åñëè ìíîæåñòâî ïëàíîâ íå ïóñòî, ðåøåíèå
çàäà÷è ñóùåñòâóåò.
Ïðÿìàÿ çàäà÷à (5) íàõîäèò ïîðòôåëü ìèíèìàëüíîãî ðèñêà ïðè çàäàííîé
íèæíåé ãðàíèöå îæèäàåìîé äîõîäíîñòè. Òàêîé ïîðòôåëü íàçûâàåòñÿ ýôôåêòèâíûì. À ìíîæåñòâî òàêèõ ïîðòôåëåé ýôôåêòèâíîé ãðàíèöåé.
3.3.2
Ìîäåëü Áëýêà
Ìîäåëü Áëýêà îòëè÷àåòñÿ îò ìîäåëè Ìàðêîâèöà òåì, ÷òî â íåé äîïóñêàþòñÿ êîðîòêèå ïðîäàæè. Äðóãèìè ñëîâàìè âåêòîð îòíîñèòåëüíûõ âåñîâ â ìîäåëè
Áëýêà óäîâëåòâîðÿåò òîëüêî îäíîìó èç äâóõ îñíîâíûõ îãðàíè÷åíèé (3):
n
X
xi = 1.
i=1
 îòëè÷èè îò ìîäåëè Ìàðêîâèöà, â ìîäåëè Áëýêà íàëè÷èå êîðîòêèõ ïîçèöèé ïîçâîëÿåò ðåàëèçîâàòü ñêîëü óãîäíî áîëüøóþ äîõîäíîñòü çà ñ÷åò áîëüøîãî
ðèñêà.
3.3.3
Ìîäåëü ÒîáèíàØàðïàËèíòåðà
 ìîäåëè ÒîáèíàØàðïàËèíòåðà (ÒØË) ïðåäïîëàãàåòñÿ ñóùåñòâîâàíèå
áåçðèñêîâîãî àêòèâà a0 , äîõîäíîñòü êîòîðîãî íå çàâèñèò îò ñîñòîÿíèÿ ðûíêà
è èìååò ïîñòîÿííîå çíà÷åíèå R0 . Íà ðåàëüíûõ ðûíêàõ â ðîëè òàêèõ àêòèâîâ
âûñòóïàþò ãîñóäàðñòâåííûå îáëèãàöèè.
 ìîäåëè ÒØË ïîðòôåëü π = {x0 , x1 , ..., xn } ïðè x0 6= 0 ìîæíî ðàçëîæèòü â
ëèíåéíóþ êîìáèíàöèþ áåçðèñêîâîãî è ðèñêîâîãî ïîðòôåëåé. Òîãäà äîõîäíîñòü,
15
îæèäàåìàÿ äîõîäíîñòü è ðèñê ïîðòôåëÿ áóäóò ðàâíû ñîîòâåòñòâåííî:
R∗ = x0 R0 + (1 − x0 )Rπ ,
m∗π = x0 R0 + (1 − x0 )mπ = mπ + x0 (R0 − mπ ),
σπ∗ = (1 − x0 )σπ .
Èñêëþ÷àÿ x0 èç äâóõ ïîñëåäíèõ âûðàæåíèé, ïîëó÷àåì ëèíåéíóþ çàâèñèìîñòü σπ∗ è m∗π :
mπ − R0 ∗
σπ + R0 .
σπ
Íà ïðàêòèêå îïòèìèçàöèÿ ïîðòôåëÿ ÒØË îáû÷íî ñîñòîèò èç äâóõ ýòàïîâ:
m∗π =
1. âûáîð îïòèìàëüíîé êîìáèíàöèè ðèñêîâàííûõ àêòèâîâ,
2. îáúåäèíåíèå ïîëó÷åííîãî îïòèìàëüíîãî íàáîðà ðèñêîâàííûõ àêòèâîâ ñ áåçðèñêîâûìè àêòèâàìè.
Ïðîèëëþñòðèðóåì ýòîò ïðîöåññ íà ïðèìåðå ïîðòôåëÿ èç äâóõ àêòèâîâ.
3.4
Îïòèìàëüíûé ïîðòôåëü èç äâóõ àêòèâîâ
Îáîáùèì ïðèâåäåííûé âûøå ïðèìåð, ðàññìîòðåâ äâà íåêîððåëèðóåìûõ àêòèâà ñ îæèäàåìûìè äîõîäíîñòÿìè è ðèñêàìè m1 , σ1 , m2 , σ2 ñîîòâåòñòâåííî. Íàéäåì çàâèñèìîñòü îñíîâíûõ õàðàêòåðèñòèê ïîðòôåëÿ, ñîñòàâëåííîãî èç ýòèõ àêòèâîâ, ïî ìîäåëè Ìàðêîâèöà, à òàê æå ïîðòôåëü ñ íàèìåíüøèì ðèñêîì.
Íå óìîëÿÿ îáùíîñòè m1 ≤ m2 .
Ïóñòü ïîðòôåëü èìååò âèä π = {t, 1 − t}, ãäå t ≥ 0. Òîãäà mπ ∈ [m1 ; m2 ] è
âåðíû ñëåäóþùèå ôîðìóëû:
mπ = m1 t + m2 (1 − t),
σπ2 = σ12 t2 + σ22 (1 − t)2 .
Âûðàçèì èç ôîðìóëû îæèäàåìîé äîõîäíîñòè t:
t=
mπ − m2
.
m1 − m2
Òîãäà, ïîäñòàâèâ t â ôîðìóëó ðèñêà, ïîëó÷èì åãî çàâèñèìîñòü îò îæèäàåìîé
äîõîäíîñòè:
σπ2
σ12 + σ22
2(σ12 m2 + σ22 m1 )
σ12 m22 + σ22 m21
2
=
m −
mπ +
.
(m1 − m2 )2 π
(m1 − m2 )2
(m1 − m2 )2
16
(8)
Äëÿ íàõîæäåíèÿ ýôôåêòèâíîãî ïîðòôåëÿ íåîáõîäèìî ðåøèòü óðàâíåíèå
dσπ2 (t)
= 0.
dt
Ðåøèâ åãî, ïîëó÷èì ôîðìóëû:
t∗ =
σ22
,
σ12 + σ22
π ∗ = {t∗ , 1 − t∗ },
m∗π =
σπ∗2
(9)
(10)
σ12 m2 + σ22 m1
,
σ12 + σ22
(11)
σ12 σ22
.
= 2
σ1 + σ22
(12)
Èñïîëüçóåì ïîëó÷åííûå ôîðìóëû äëÿ îòâåòà íà ïîñòàâëåííûé ðàíåå âîïðîñ
èç ïðîøëîãî ïðèìåðà, ñóùåñòâóåò ëè ïîðòôåëü ñ áîëüøåé äîõîäíîñòüþ è ìåíüøèì ðèñêîì.
m1 = 3.9%, m2 = 1.8%, σ1 = 7.38%, σ2 = 5.72%
5.72%2
t =
= 0.38,
7.38%2 + 5.72%2
π ∗ = {0.38, 0.62},
∗
7.38%2 × 1.8% + 5.72%2 × 3.9%
= 2.59%,
=
7.38%2 + 5.72%2
r
7.38%2 × 5.72%2
σπ∗ =
= 4.52%.
7.38%2 + 5.72%2
Íà ðèñ. 1 ïðèâåäåí ãðàôèê çàâèñèìîñòè îæèäàåìîé äîõîäíîñòè îò ðèñêà
ïîðòôåëÿ, ïîñòðîåííîãî èç àêòèâîâ À, Â. Òî÷êà Q1 ñîîòâåòñòâóåò ïîðòôåëþ, ñîñòîÿùåìó òîëüêî èç àêòèâà A, òî÷êà Q2 ïîðòôåëþ èç àêòèâà Â, Q∗ ïîðòôåëü
ñ ìèíèìàëüíûì ðèñêîì, Qex ïîðòôåëü, îïèñàííûé â ïðåäûäóùåì ïðèìåðå, ñ
âåêòîðîì îòíîñèòåëüíûõ âåñîâ π = {0.25, 0.75}.
Çàìåòèì, ÷òî ÷àñòü ãðàôèêà îò òî÷êè Q∗ äî òî÷êè Q1 ÿâëÿåòñÿ ýôôåêòèâíîé
ãðàíèöåé ïîðòôåëÿ.
Òåïåðü ïåðåéäåì êî âòîðîìó ýòàïó ôîðìèðîâàíèÿ îïòèìàëüíîãî ïîðòôåëÿ.
Ââåäåì áåçðèñêîâûé àêòèâ ñ äîõîäíîñòüþ R0 = 1.5%. Ðàññìîòðèì åãî âîçìîæíûå êîìáèíàöèè ñ ïîðòôåëåì íàèìåíüøåãî ðèñêà Q∗ .
m∗π
17
Ðèñ. 1: Çàâèñèìîñòü îæèäàåìîé äîõîäíîñòè ïîðòôåëÿ îò åãî ðèñêà
Êàê áûëî îïèñàíî â ìîäåëè ÒØË, çàâèñèìîñòü îæèäàåìîé äîõîäíîñòè îò
ðèñêà â äàííîì ñëó÷àå ëèíåéíàÿ. Íà ðèñ. 2 ïðèâåäåí ãðàôèê ýòîé çàâèñèìîñòè,
ãäå Q0 ñîîòâåòñòâóåò ïîðòôåëþ, ñîñòîÿùåìó òîëüêî èç áåçðèñêîâîãî àêòèâà, à
ìíîæåñòâî òî÷åê îòðåçêà Q0 Q∗ ìíîæåñòâó âñåõ âîçìîæíûõ êîìáèíàöèé Q∗ ñ
áåçðèñêîâûì àêòèâîì.
Ðèñ. 2: Îæèäàåìàÿ äîõîäíîñòü ïîðòôåëÿ ñ áåçðèñêîâûì àêòèâîì è åãî ðèñê
Òàê æå èç ãðàôèêà (2) âèäíî, ÷òî ïîðòôåëè, ñôîðìèðîâàííûå èç ïîðòôåëÿ
ñ íàèìåíüøèì ðèñêîì è áåçðèñêîâîãî àêòèâà äàþò ìåíüøèé ðèñê ïðè çàäàííîé
18
îæèäàåìîé äîõîäíîñòè, ÷åì ïîðòôåëè íå ñîäåðæàùèå áåçðèñêîâûé àêòèâ.
Òåïåðü ïîñòðîèì ýôôåêòèâíóþ ãðàíèöó ïîðòôåëÿ, ñîñòàâëåííîãî èç áåçðèñêîâîãî è ðèñêîâàííûõ àêòèâîâ. Äëÿ ýòîãî íåîáõîäèìî âûáðàòü ïîðòôåëü Q∗∗ ,
ñîñòàâëåííûé èç ðèñêîâàííûõ àêòèâîâ, êîìáèíàöèè áåçðèñêîãî àêòèâà ñ êîòîðûì áóäóò ñîñòàâëåíû èç ýôôåêòèâíûõ ïîðòôåëåé. Î÷åâèäíî, ÷òî ýòîò ïîðòôåëü ÿâëÿåòñÿ òî÷êîé êàñàíèÿ ïðÿìîé, ïðîâåäåííîé èç òî÷êè (0, R0 ) ê ãðàôèêó
íà ðèñ. 1. Óðàâíåíèå ýòîé ïðÿìîé èìååò âèä:
mπ = kσπ + R0 ,
(13)
ãäå k èñêîìûé óãîë íàêëîíà ïðÿìîé. Äëÿ åãî íàõîæäåíèÿ íåîáõîäèìî ïîñëå
ïîäñòàíîâêè (13) â (8) ïðèðàâíÿòü äèñêðèìèíàíò ê íóëþ è èç äâóõ ïîëó÷èâøèõñÿ êîðíåé âûáðàòü ïîëîæèòåëüíûé.
 ðàññìàòðèâàåìîì ïðèìåðå êîýôôèöèåíò k ïîëó÷èëñÿ ðàâíûì 0.33 ñ òî÷íîñòüþ äî äâóõ çíà÷àùèõ öèôð. Íà ðèñ. 3 èçîáðàæåíà ýôôåêòèâíàÿ ãðàíèöà
ïîðòôåëÿ ñ áåçðèñêîâûì àêòèâîì, ñîñòîÿùàÿ èç îòðåçêà Q0 Q∗∗ è ÷àñòè ýôôåêòèâíîé ãðàíèöû ïîðòôåëÿ èç ðèñêîâàííûõ àêòèâîâ Q∗∗ Q1 .
Ðèñ. 3: Ïîñòðîåíèå ýôôåêòèâíîé ãðàíèöû äëÿ ïîðòôåëÿ ñ áåçðèñêîâûì àêòèâîì
Çàìåòèì, ÷òî êàæäûé èíâåñòîð èìååò ñâîå ïîíèìàíèå îïòèìàëüíîñòè ïîðòôåëÿ, ïîýòîìó â êà÷åñòâå îïòèìàëüíîãî ïîðòôåëÿ ðàçíûå èíâåñòîðû ìîãóò âûáðàòü ðàçíûå ïîðòôåëè íà ïîëó÷èâøåéñÿ ýôôåêòèâíîé ãðàíèöå.
19
4
Ìåòîäû îáðàáîòêè âðåìåííûõ ðÿäîâ
Äëÿ ïðèìåíåíèÿ ìîäåëåé ïîðòôåëüíîé òåîðèè íåîáõîäèìî ïîëó÷èòü èñïîëüçóåìûå â íèõ äàííûå. Ýòè äàííûå ìîãóò áûòü ïîëó÷åíû â ðåçóëüòàòå îáðàáîòêè
âðåìåííûõ ðÿäîâ äëÿ öåí àêòèâîâ.
 ýòîé ãëàâå ìû áóäåì ðàññìàòðèâàòü äàííûå â âèäå âðåìåííûõ ðÿäîâ
t = {tj }ni=1 ñî çíà÷åíèÿìè {zj = z(tj )}ni=1 . Ïîä îäíîðîäíûì âðåìåííûì ðÿäîì
áóäåì ïîäðàçóìåâàòü ðÿä, ó êîòîðîãî tj+1 − tj = 4t, ∀i = 1, . . . , n − 1, à ïîä
íåîäíîðîäíûì ðÿä ñ ðàçëè÷íûìè ïðîìåæóòêàìè âðåìåíè.
Êîððåêòíîå ñðàâíåíèå âðåìåííûõ ðÿäîâ ìåæäó ñîáîé âîçìîæíî òîëüêî ïðè
ñîâïàäåíèè ìíîæåñòâ t. Òàêèì îáðàçîì, âîçíèêàåò çàäà÷à ïðèâåñòè íåñêîëüêî
âðåìåííûõ ðÿäîâ ê îäèíàêîâîìó ìíîæåñòâó t. Ó÷èòûâàÿ òî, ÷òî íà ïðàêòèêå ìû
÷àùå âñåãî ñòàëêèâàåìñÿ ñ íåîäíîðîäíûìè âðåìåííûìè ðÿäàìè, ïðîùå âñåãî ýòî
îñóùåñòâèòü, íàó÷èâøèñü ïðåîáðàçîâûâàòü íåîäíîðîäíûé âðåìåííîé ðÿä â ðÿä
ñ îäíîðîäíûìè ïðîìåæóòêàìè âðåìåíè. Îïèøåì ïîñòàíîâêó çàäà÷è.
Äàí âðåìåííîé ðÿä t = {tj } ñî çíà÷åíèÿìè {zj = z(tj )}. Òðåáóåòñÿ ïðèâåñòè
åãî ê âðåìåííîìó ðÿäó t0 = {t0 +i4t}. Òî åñòü íàéòè çíà÷åíèÿ, ñîîòâåòñòâóþùèå
îäíîðîäíîìó ðÿäó t0 .  [7] ïðåäëàãàåòñÿ äâà ñïîñîáà ðåøåíèÿ ýòîé çàäà÷è.
4.1
Ëèíåéíàÿ èíòåðïîëÿöèÿ
Íàéäåì òî÷êè âðåìåííîãî ðÿäà, ìåæäó êîòîðûìè çàêëþ÷åíà òî÷êà t0 + i4t.
j 0 = min{j| tj ≤ t0 + i4t},
tj 0 ≤ t0 + i4t ≤ tj 0 +1 .
Çíà÷åíèå â òî÷êå t0 + i4t âû÷èñëèì ïî ôîðìóëå:
t0 + i4t − t0j
(zj 0 +1 − zj 0 ).
z(t0 + i4t) = zj 0 +
tj 0 +1 − tj 0
Çàìåòèì, ÷òî äëÿ ïîäñ÷åòà çíà÷åíèÿ â òî÷êå t0 +i4t ëèíåéíàÿ èíòåðïîëÿöèÿ
òðåáóåò èçâåñòíîãî çíà÷åíèÿ íå òîëüêî â òî÷êå tj 0 , íî è â òî÷êå tj 0 +1 , êîòîðàÿ
ÿâëÿåòñÿ òî÷êîé â áóäóùåì îòíîñèòåëüíî t0 + i4t.
4.2
Ñòóïåí÷àòàÿ èíòåðïîëÿöèÿ
Ñòóïåí÷àòàÿ èíòåðïîëÿöèÿ îòëè÷àåòñÿ îò ëèíåéíîé òåì, ÷òî èñïîëüçóåò òîëüêî äàííûå â òî÷êå tj 0 è âû÷èñëÿåòñÿ ïî ôîðìóëå
z(t0 + i4t) = zj 0 .
20
Ïðèìåðû ëèíåéíîé è ñòóïåí÷àòîé èíòåðïîëÿöèé ïðèâåäåíû íà ðèñ. 4, ãäå
áîëüøèìè êðóæî÷êàìè îáîçíà÷åíû òî÷êè, ïîñòðîåííûå ëèíåéíîé èíòåðïîëÿöèåé, ìàëåíüêèìè ñòóïåí÷àòîé, à t0 , t1 , t2 ÿâëÿþòñÿ ýëåìåíòàìè èñêîìîãî îäíîðîäíîãî âðåìåííîãî ðÿäà.
Ðèñ. 4: Ëèíåéíàÿ è ñòóïåí÷àòàÿ èíòåðïîëÿöèè
Çàìå÷àíèå. Ïðè âûáîðå ìåòîäà èíòåðïîëÿöèè ñëåäóåò áûòü îñòîðîæíû-
ìè. Çà÷àñòóþ íåîáðàáîòàííûå äàííûå ìîãóò èìåòü áîëüøèå ïðîìåæóòêè ïðîïóùåííîé èíôîðìàöèè, à ýòî çíà÷èò, ÷òî èñïîëüçîâàíèå ëèíåéíîé èíòåðïîëÿöèè
ìîæåò ñèëüíî ¾èñïîðòèòü¿ äàííûå.  [7] â òàêîì ñëó÷àå ðåêîìåíäóåòñÿ èñïîëüçîâàòü ñòóïåí÷àòóþ èíòåðïîëÿöèþ.
21
5
Îïòèìàëüíûé ïîðòôåëü êîëëåêöèîííûõ ìîíåò
 êà÷åñòâå ïðèìåðà ðàññìîòðèì ïîðòôåëü èç ïÿòè êîëëåêöèîííûõ ìîíåò
ÐÔ:
1. 2 ðóáëÿ. 300ëåòèå ñî äíÿ ðîæäåíèÿ Ë. Ýéëåðà
2. 3 ðóáëÿ. Àííà Ïàâëîâà
3. 1 ðóáëü. 200ëåòèå ñî äíÿ ðîæäåíèÿ Í. È. Ëîáà÷åâñêîãî
4. 3 ðóáëÿ. Ñìîëüíûé èíñòèòóò è ìîíàñòûðü â Ñàíêò-Ïåòåðáóðãå
5. 3 ðóáëÿ. 50ëåòèå ðàçãðîìà íåìåöêîôàøèñòñêèõ âîéñê ïîä Ëåíèíãðàäîì
Îñíîâíûå õàðàêòåðèñòèêè ýòèõ ìîíåò ïðèâåäåíû â ñëåäóþùåé òàáëèöå:
Íàçâàíèå
Ýéëåð
Ïàâëîâà
Ëîáà÷åâñêèé
Ñìîëüíûé
Ðàçãðîì
Äàòà âûïóñêà Òèðàæ
02.04.2007
13.12.1993
01.12.1992
05.07.1994
20.01.1994
10 000 øò.
45 000 øò.
400 000 øò.
30 000 øò.
250 000 øò.
Ìåòàëë
ñåðåáðî (ïðîáà 950/1000)
ñåðåáðî (ïðîáà 900/1000)
ìåäü, íèêåëü
ñåðåáðî (ïðîáà 900/1000)
ìåäü, íèêåëü
Ïîëíàÿ èíôîðìàöèÿ î ìîíåòàõ ïðèâåäåíà íà ñàéöå Öåòðàëüíîãî áàíêà [11].
5.1
Îáðàáîòêà äàííûõ
Ïîäãîòîâêà äàííûõ ñîñòîèò èç ïÿòè ýòàïîâ:
1. Ñáîð äàííûõ.
2. Ïðèâåäåíèå äàííûõ ê îäíîðîäíîìó âðåìåííîìó ðÿäó.
3. Âû÷èñëåíèå íåäåëüíîé äîõîäíîñòè ìîíåò.
4. Âû÷èñëåíèå îæèäàåìîé äîõîäíîñòè è ðèñêà ìîíåò.
5. Âû÷èñëåíèå ìàòðèöû êîâàðèàöèé.
22
1. Ñáîð äàííûõ.
 ñâÿçè ñ òåì, ÷òî â îñíîâíîì ìîíåòû òîðãóþòñÿ íà àóêöèîíàõ, â êà÷åñòâå
èñòî÷íèêà èíôîðìàöèè áûë âçÿò ñàéò ñî ñâîäêîé ðåçóëüòàòîâ ñàìûõ ïîïóëÿðíûõ àóêöèîíîâ Ðîññèè [10]. Äàííûå áûëè âçÿòû çà ãîä: ñ êîíöà ôåâðàëÿ
2015 ãîäà ïî ìàðò 2016.
Ïîëó÷èâøèåñÿ âðåìåííûå ðÿäû ïðîèëëþñòðèðîâàíû â ïðèëîæåíèè
íà ðèñ. 13 22.
×àñòîòà òîðãîâ êàæäîé ìîíåòû ïðèâåäåíà â ñëåäóþùåé òàáëèöå:
Ìîíåòà
1
2
3
4
5
Ñðåäíåå ÷èñëî
0.58 1.92 3.63 1.23 3.65
ïðîäàæ â íåäåëþ
2. Ïðèâåäåíèå äàííûõ ê îäíîðîäíîìó âðåìåííîìó ðÿäó.
Èç-çà òîãî, ÷òî àóêöèîíû ïðîâîäÿòñÿ êðàéíå íåðåãóëÿðíî: íåêîòîðûå èç âûáðàííûõ ìîíåò ïðîäàþòñÿ êðàéíå ÷àñòî (283 ïðîäàæè â ãîä), à äðóãèå â
ðàçû ìåíüøå (30 ïðîäàæ çà ãîä), ïîëó÷åííûå äàííûå áûëè ïðèâåäåíû ê
îäíîðîäíîìó âðåìåííîìó ðÿäó ñ t0 = 01.03.2015 00 : 00, 4t = 7 äíåé.
Äëÿ ïðèâåäåíèÿ íåîäíîðîäíîãî ðÿäà ê îäíîðîäíîìó áûë âûáðàí ïðèíöèï
ñòóïåí÷àòîé èíòåðïîëÿöèè â ñâÿçè ñ òåì, ÷òî âðåìåííûå ðÿäû ìîíåò ñîäåðæàëè áîëüøèå ïðîáåëû (ìàêñèìàëüíûé 65 äíåé). Äàííàÿ îïåðàöèÿ áûëà
ðåàëèçîâàíà ñ ïîìîùüþ ñòàòèñòè÷åñêîãî ïàêåòà SAS. Îïèñàíèå ýòîãî ïàêåòà ìîæíî íàéòè íà ñàéòå êîìïàíèè SAS Institute [12] è â ðóêîâîäñòâå ê
ïîëüçîâàíèþ [8]. Òåêñò ïðîãðàììû â ïðèëîæåíèÿõ.
Äàëåå ïîä îæèäàåìîé äîõîäíîñòüþ áóäåì ïîíèìàòü îæèäàåìóþ äîõîäíîñòü
çà íåäåëþ. À ïîä äîõîäíîñòüþ íåäåëüíóþ äîõîäíîñòü.
3. Âû÷èñëåíèå íåäåëüíîé äîõîäíîñòè ìîíåò.
Ïîñëå ïðèâåäåíèÿ äàííûõ ê îäíîðîäíîìó âðåìåííîìó ðÿäó áûëè âû÷èñëåíû
íåäåëüíûå äîõîäíîñòè ìîíåò.
4. Âû÷èñëåíèå îæèäàåìîé äîõîäíîñòè è ðèñêà ìîíåò.
Äàëåå äëÿ âû÷èñëåíèÿ îæèäàåìîé äîõîäíîñòè è ðèñêà ìîíåò áûëè âçÿòû
âåðîÿòíîñòíûå ñöåíàðèè äîõîäíîñòåé âèäà:
Ñöåíàðèé
s1
s2 ... sn
Âåðîÿòíîñòü 1/n 1/n ... 1/n
23
ãäå n = 52 êîëè÷åñòâî íåäåëü â ðàññìàòðèâàåìîì ãîäó.
Âû÷èñëåííûå îæèäàåìûå äîõîäíîñòè è ðèñêè êàæäîé ìîíåòû ïðèâåäåíû â
ñëåäóþùåé òàáëèöå:
Íàçâàíèå
Ýéëåð
Ïàâëîâà
Ëîáà÷åâñêèé
Ñìîëüíûé
Ðàçãðîì
Îæèäàåìàÿ
Ðèñê
äîõîäíîñòü
1.44%
2.41%
3.06%
3.06%
10.81%
16.96%
22.02%
23.99%
24.42%
51.00%
5. Âû÷èñëåíèå ìàòðèöû êîâàðèàöèé.
Êîâàðèàöèè Ýéëåð Ïàâëîâà Ëîáà÷åâñêèé Ñìîëüíûé Ðàçãðîì
Ýéëåð
Ïàâëîâà
Ëîáà÷åâñêèé
Ñìîëüíûé
Ðàçãðîì
5.2
2.88%
-0.86%
-0.04%
0.09%
2.15%
-0.86%
4.85%
-0.50%
-1.23%
0.26%
-0.04%
-0.50%
5.75%
0.75%
0.82%
0.09%
-1.23%
0.75%
5.96%
-1.78%
2.15%
0.26%
0.82%
-1.78%
26.01%
Âû÷èñëåíèå ýôôåêòèâíîãî ïîðòôåëÿ
Âû÷èñëèâ îñíîâíûå õàðàêòåðèñòèêè ìîíåò, ïåðåéäåì ê âû÷èñëåíèþ ýôôåêòèâíîãî ïîðòôåëÿ.
Ñôîðìóëèðóåì ïðÿìóþ îïòèìèçàöèîííóþ çàäà÷ó ïî ìîäåëè Ìàðêîâèöà ïðè
çàäàííîé íèæíåé ãðàíèöå îæèäàåìîé íåäåëüíîé äîõîäíîñòè Rreq = 3% íà ïîèñê
ïîðòôåëÿ ìèíèìàëüíîãî ðèñêà âèäà π = {x1 , x2 , x3 , x4 , x5 }.
24
2.88% x21 − 0.86% x1 x2 − 0.04% x1 x3 + 0.09% x1 x4 + 2.15% x1 x5 +
−0.86% x1 x2 + 4.85% x22 − 0.50% x2 x3 − 1.23% x2 x4 + 0.26% x2 x5 +
−0, 04% x1 x3 − 0.50% x2 x3 + 5.75% x23 + 0.75% x3 x4 + 0.82% x3 x5 +
+0.09% x1 x4 − 1.23% x2 x4 + 0.75% x3 x4 + 5.96% x24 − 1.78% x4 x5 +
+2.15% x1 x5 + 0.26% x2 x5 + 0.82% x3 x5 − 1.78% x4 x5 + 26.01% x25
1.44% x1 + 2.41% x2 3.06% x3 + 3.06% x4 + 10.81% x5
x1 + x2 + x3 + x4 + x5
∀i ∈ {1, . . . , 5} xi
→
≥
=
≥
min,
3%,
1,
0.
Ðåøåíèå ýòîé êâàäðàòè÷íîé çàäà÷è ìîæåò áûòü íàéäåíî ñ ïîìîùüþ ïîïóëÿðíîãî ïðîãðàììíîãî îáåñïå÷åíèÿ òàêîãî, êàê MS Excel, MatLab, R è äðóãèõ.
Äëÿ ñðàâíåíèÿ ðåøåíèå áûëî íàéäåíî ñ ïîìîùüþ äâóõ ðàçëè÷íûõ ïàêåòîâ íà R
è MatLab. Ðåçóëüòàòû ðàáîòû ïðîãðàìì ñîâïàëè. Òåêñò ïðîãðàìì ïðåäñòàâëåí
â ïðèëîæåíèè, ðåçóëüòàò ðàáîòû ïðîãðàìì â ñëåäóþùåé òàáëèöå:
x1
x2
x3
x4
x5
26.54% 28.95% 15.47% 21.83% 7.21%
Ðèñ. 5: Ïîðòôåëü ìèíèìàëüíîãî ðèñêà ïðè îæèäàåìîé äîõîäíîñòè íå ìåíåå 3%
Îæèäàåìàÿ äîõîäíîñòü
3%
Ðèñê
9.73%
25
5.3
Ïîñòðîåíèå ýôôåêòèâíîé ãðàíèöû
Èìåÿ ïðîãðàììó äëÿ âû÷èñëåíèÿ ïîðòôåëÿ ìèíèìàëüíîãî ðèñêà ïðè çàäàííîé íèæíåé ãðàíèöå îæèäàåìîé äîõîäíîñòè, íåñëîæíî âû÷èñëèòü ýôôåêòèâíóþ
ãðàíèöó. Ðåçóëüòàò ðàáîòû ïðîãðàììû ïî âû÷èñëåíèþ ýôôåêòèâíîé ãðàíèöû
ïðèâåäåí íà ðèñ. 6.
Ðèñ. 6: Ýôôåêòèâíàÿ ãðàíèöà ïîðòôåëÿ èç ïÿòè ìîíåò
5.4
Ñðàâíåíèå ïîðòôåëåé
Î÷åâèäíî, ÷òî åñëè èñêëþ÷èòü íåêîòîðóþ ìîíåòó èç ïîðòôåëÿ, ðèñê ëþáîãî ýôôåêòèâíîãî ïîðòôåëÿ áóäåò íå ìåíüøå ðèñêà èçíà÷àëüíîãî ýôôåêòèâíîãî
ïîðòôåëÿ ïðè òîì æå óðîâíå äîõîäíîñòè. Îäíàêî íå âñåãäà ðèñê ñîêðàùåííîãî ýôôåêòèâíîãî ïîðòôåëÿ ñòðîãî áîëüøå ðèñêà èçíà÷àëüíîãî ýôôåêòèâíîãî
ïîðòôåëÿ. Ðàâåíñòâî ðèñêîâ îçíà÷àåò, ÷òî äëÿ ôîðìèðîâàíèÿ òàêîãî ýôôåêòèâíîãî ïîðòôåëÿ èñïîëüçóþòñÿ íå âñå ìîíåòû. Èññëåäóåì êàæäûé ýôôåêòèâíûé
ïîðòôåëü íà íåîáõîäèìîñòü âõîæäåíèÿ âñåõ ïÿòè ìîíåò.
Ðàññìîòðèì ïÿòü ïîðòôåëåé, ïîëó÷åííûõ èç óæå èçó÷åííîãî ïîðòôåëÿ óäàëåíèåì îäíîé èç ìîíåò. Íàçîâåì èõ ¾Áåç Ýéëåðà¿, ¾Áåç Ïàâëîâîé¿, ¾Áåç Ëîáà÷åâñêîãî¿, ¾Áåç Ñìîëüíîãî¿, ¾Áåç Ðàçãðîìà¿.
Ñðàâíèì ïîðòôåëè ìèíèìàëüíîãî ðèñêà ïðè íèæíåì óðîâíå îæèäàåìîé äîõîäíîñòè 3%.
26
Ïîðòôåëü
5 ìîíåò
Áåç Ýéëåðà
Áåç Ïàâëîâîé
Áåç Ëîáà÷åâñêîãî
Áåç Ñìîëüíîãî
Áåç Ðàçãðîìà
Îæèäàåìàÿ äîõîäíîñòü
3%
3.31%
3%
3%
3%
3%
Ðèñê
9.73%
11.89%
12.36%
10.53%
11.46%
12.49%
Ïðè òàêîì óðîâíå äîõîäíîñòè ðèñê ïîðòôåëÿ èç ïÿòè ìîíåò ñòðîãî ìåíüøå
ðèñêà ëþáîãî äðóãîãî ïîðòôåëÿ. Òàêèì îáðàçîì, â ýòîì ñëó÷àå öåëåñîîáðàçíî âêëþ÷åíèå âñåõ ïÿòè ìîíåò. Ïîñòðîèì ýôôåêòèâíûå ãðàíèöû ïîðòôåëåé,
÷òîáû âûÿñíèòü, êàê âåäóò ñåáÿ ýôôåêòèâíûå ïîðòôåëè ïðè äðóãèõ óðîâíÿõ
ìèíèìàëüíîé îæèäàåìîé äîõîäíîñòè.
Ðèñ. 7: Ýôôåêòèâíàÿ ãðàíèöà ïîðòôåëÿ èç ïÿòè ìîíåò
Íåòðóäíî âèäåòü èç ãðàôèêà íà ðèñ. 7, ÷òî íà÷èíàÿ ñ íåêîòîðîé îæèäàåìîé
äîõîäíîñòè ãðàíèöà ïîðòôåëåé ¾Áåç Ýéëåðà¿, ¾Áåç Ïàâëîâîé¿, ¾Áåç Ëîáà÷åâ27
ñêîãî¿ î÷åíü áëèçêà â ýôôåêòèâíîé ãðàíèöå ïîðòôåëÿ ïÿòè ìîíåò.
Äëÿ âûÿñíåíèÿ òî÷íîãî çíà÷åíèÿ îæèäàåìîé äîõîäíîñòè, íà÷èíàÿ ñ êîòîðîé â ýôôåêòèâíûé ïîðòôåëü íå âõîäèò îäíà èç òðåõ ìîíåò, áûëà èñïîëüçîâàíà ïðîãðàììà ïî ïîñòðîåíèþ ýôôåêòèâíîãî ïîðòôåëÿ. Ïîëó÷åííûå ðåçóëüòàòû
ïðåäñòàâëåíû â ñëåäóþùåé òàáëèöå:
Íàçâàíèå
Ýéëåð
Ïàâëîâà
Ëîáà÷åâñêèé
Îæèäàåìàÿ
äîõîäíîñòü
4.75%
7.90%
8.22%
Ñõåìà ñîñòàâà ýôôåêòèâíîãî ïîðòôåëÿ ïðè ðàçíûõ óðîâíÿõ îæèäàåìîé äîõîäíîñòè ïðåäñòàâëåíà â ñëåäóþùåé òàáëèöå:
Îæèäàåìàÿ äîõîäíîñòü 1 2 3 4 5
2.31% < mπ < 4.75%
4.75% ≤ mπ < 7.90%
7.90% ≤ mπ < 8.22%
8.22% ≤ mπ < 10.81%
mπ = 10.81%
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Çàìå÷àíèå. Çíà÷åíèå 2.31% ÿâëÿåòñÿ ìèíèìàëüíûì çíà÷åíèåì îæèäàåìîé äî-
õîäíîñòè, äëÿ êîòîðîé ñóùåñòâóåò ýôôåêòèâíûé ïîðòôåëü.
5.5
Ðàñøèðåííûé ïîðòôåëü
Ðàñøèðèì ïîðòôåëü, äîáàâèâ ïÿòü ñëåäóþùèõ ìîíåò:
6. 2 ðóáëÿ. 225-ëåòèå ñî äíÿ ðîæäåíèÿ È. À. Êðûëîâà
7. 3 ðóáëÿ. Ìåæäóíàðîäíûé ãîä Êîñìîñà
8. 2 ðóáëÿ. 185 - ëåòèå ñî äíÿ ðîæäåíèÿ Í.Â. Ãîãîëÿ
9. 50 ðóáëåé. Ãèìàëàéñêèé ìåäâåäü
10. 25 ðóáëåé. Ýìáëåìà Èãð
28
Îñíîâíûå õàðàêòåðèñòèêè ýòèõ ìîíåò ïðèâåäåíû â ñëåäóþùåé òàáëèöå:
Íàçâàíèå Äàòà âûïóñêà Òèðàæ
Êðûëîâ
Êîñìîñ
Ãîãîëü
Ìåäâåäü
Ñî÷è
02.02.1994
09.04.1992
22.03.1994
29.09.1993
15.04.2011
250 000 øò.
600 000 øò.
250 000 øò.
300 000 øò.
9 750 000 øò.
Ìåòàëë
ñåðåáðî (ïðîáà 500/1000)
ìåäü, íèêåëü
ñåðåáðî (ïðîáà 500/1000)
ìåäü, íèêåëü
ìåäü, íèêåëü
Ïîëíàÿ èíôîðìàöèÿ î ìîíåòàõ ïðèâåäåíà íà ñàéöå Öåòðàëüíîãî áàíêà [11].
×àñòîòà òîðãîâ êàæäîé ìîíåòû ïðèâåäåíà â ñëåäóþùåé òàáëèöå:
Ìîíåòà
1
2
3
4
5 6 7
8
9
10
Ñðåäíåå ÷èñëî
0.58 1.92 3.63 1.23 3.65 3 4.62 3.48 2.17 2.60
ïðîäàæ â íåäåëþ
Äàííûå î ìîíåòàõ áûëè îáðàáîòàíû ïî òîé æå ñõåìå.
Îæèäàåìàÿ äîõîäíîñòü è ðèñê ìîíåò ïðèâåäåíû â ñëåäóþùåé òàáëèöå:
Íàçâàíèå
Êðûëîâ
Êîñìîñ
Ãîãîëü
Ìåäâåäü
Ñî÷è
Îæèäàåìàÿ
Ðèñê
äîõîäíîñòü
9.26%
5.82%
6.68%
1.41%
26.54%
47.20%
38.15%
37.52%
14.84%
89.97%
Ìàòðèöà êîâàðèàöèé ïðèâåäåíà â ïðèëîæåíèè â òàáëèöå 3.
Ñõåìà ñîñòàâà ýôôåêòèâíîãî ïîðòôåëÿ ïðè ðàçíûõ óðîâíÿõ îæèäàåìîé äîõîäíîñòè ïðåäñòàâëåíà â ñëåäóþùåé òàáëèöå:
29
Îæèäàåìàÿ äîõîäíîñòü 1 2 3 4 5 6 7 8 9 10
2.66% < mπ < 2.81%
2.81% < mπ < 3.44%
3.44% < mπ < 4.97%
4.97% < mπ < 8.68%
8.68% < mπ < 9.64%
9.64% < mπ < 12.76%
12.76% < mπ < 13.86%
13.86% < mπ < 14.51%
14.51% < mπ < 15.82%
15.82% < mπ < 22.12%
22.12% < mπ < 26.54%
mπ = 26.54%
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Çàìå÷àíèå. Çíà÷åíèå 2.66% ÿâëÿåòñÿ ìèíèìàëüíûì çíà÷åíèåì îæèäàåìîé äî-
õîäíîñòè, äëÿ êîòîðîé ñóùåñòâóåò ýôôåêòèâíûé ïîðòôåëü.
Äëÿ ñðàâíåíèÿ ðàñøèðåííîãî ïîðòôåëÿ ñ èçíà÷àëüíûì âû÷èñëèì ýôôåêòèâíûé ïîðòôåëü ïðè çàäàííîé íèæíåé ãðàíèöå íåäåëüíîé îæèäàåìîé äîõîäíîñòè
â 3%. Ðåçóëüòàò ïðèâåäåí íà ðèñ. 8:
Ðèñ. 8: Ïîðòôåëü ìèíèìàëüíîãî ðèñêà èç äåñÿòè ìîíåò ïðè îæèäàåìîé äîõîäíîñòè ≥ 3%
Çàìåòèì, ÷òî îòíîñèòåëüíûé âåñ ìîíåòû Êðûëîâà ðàâåí íóëþ, ÷òî íå ïðîòèâîðå÷èò ïðèâåäåííîé âûøå ñõåìå î ñîñòàâå ïîðòôåëÿ.
30
Íåäåëüíàÿ îæèäàåìàÿ äîõîäíîñòü è ðèñê ïîëó÷èâøåãîñÿ ïîðòôåëÿ ïðèâåäåíû â ñëåäóþùåé òàáëèöå:
Îæèäàåìàÿ äîõîäíîñòü
3%
Ðèñê
8.16%
Ðèñê ýôôåêòèâíîãî ïîðòôåëÿ èç äåñÿòè ìîíåò ìåíüøå ðèñêà ýôôåêòèâíîãî
ïîðòôåëÿ èç ïÿòè ìîíåò, êîòîðûé ðàâíÿëñÿ 9.73%.
Ñðàâíèì ýôôåêòèâíûå ãðàíèöû ïîðòôåëåé, ïðèâåäåííûå íà ðèñ. 9.
Ðèñ. 9: Ýôôåêòèâíûå ãðàíèöû ïîðòôåëåé èç ïÿòè è äåñÿòè ìîíåò
Èç ãðàôèêà âèäíî, ÷òî ïðè ëþáîì óðîâíå îæèäàåìîé äîõîäíîñòè ýôôåêòèâíûé ïîðòôåëü èç äåñÿòè ìîíåò õàðàêòåðèçóåòñÿ ìåíüøèì ðèñêîì ïî îòíîøåíèþ
ê ýôôåêòèâíîìó ïîðòôåëþ èç ïÿòè ìîíåò.
31
6
Ðåàëèçàöèÿ îïòèìàëüíîãî ïîðòôåëÿ
 ýòîé ãëàâå áóäóò ïîñòðîåíû öåëî÷èñëåííûå ïîðòôåëè, ïðèáëèæåííûå ê
ðàíåå ðàññìîòðåííûì ýôôåêòèâíûì, ïîñ÷èòàíû ìèíèìàëüíûå âëîæåíèÿ äëÿ
èõ ðåàëèçàöèè, à òàê æå âû÷èñëåíû äîõîäíîñòè ïîëó÷èâøèõñÿ ïîðòôåëåé.
Ðåøèì ïîñòàâëåííóþ çàäà÷ó â îáùåì ñëó÷àå. Ïðåäïîëîæèì, ÷òî èíâåñòîð
õî÷åò ïîñòðîèòü öåëî÷èñëåííûé ïîðòôåëü c ìèíèìàëüíûìè âëîæåíèÿìè π
e ê
∗
∗
∗
∗
ïîðòôåëþ π èç ìîíåò {ñ1 , . . . , ñm }. Ïðåäïîëîæèì, ÷òî π = {x1 , . . . , xm }, ãäå
x∗i > 0, i ∈ {1, . . . , m}.
Ôîðìèðîâàíèå öåëî÷èñëåííîãî ïîðòôåëÿ ñîñòîèò èç øåñòè ýòàïîâ:
1. Ñáîð äàííûõ.
2. Âû÷èñëåíèå âëîæåíèé â π ∗ .
3. Âû÷èñëåíèå âåêòîðà êîëè÷åñòâ ìîíåò π ∗ .
4. Âû÷èñëåíèå âåêòîðà êîëè÷åñòâ ìîíåò π
e.
5. Âû÷èñëåíèå âëîæåíèé â π
e.
6. Âû÷èñëåíèå îòíîñèòåëüíûõ âåñîâ π
e.
1. Ñáîð äàííûõ.
Ïåðâûì ýòàïîì ÿâëÿåòñÿ ñáîð äàííûõ î öåíå, ïî êîòîðîé áóäóò êóïëåíû
ìîíåòû. Îáîçíà÷èì èõ çà {S1 (0), . . . , Sm (0)}.
2. Âû÷èñëåíèå âëîæåíèé â π ∗ .
Äëÿ ôîðìèðîâàíèÿ ïîðòôåëÿ áëèçêîãî ê çàäàííîìó íåîáõîäèìî, ÷òî áû
êàæäîé ìîíåòû â ïîðòôåëå áûëî íå ìåíåå îäíîé. À äëÿ äîñòèæåíèÿ ìèíèìàëüíûõ ðàñõîäîâ íåîáõîäèìî, ÷òîáû ìîíåòà ñ ìèíèìàëüíûì îòíîñèòåëüíûì âåñîì áûëà îäíà. Âëîæåíèÿ â îñòàëüíûå ìîíåòû âû÷èñëÿþòñÿ ÷åðåç
ñòîèìîñòè ìîíåò è âåêòîð îòíîñèòåëüíûõ âåñîâ. Òàêèì îáðàçîì, âëîæåíèÿ
â π ∗ âû÷èñëÿþòñÿ ïî ôîðìóëå:
I ∗ (i) = Si (0)
xi
, i ∈ {1, . . . , m}.
x1
3. Âû÷èñëåíèå âåêòîðà êîëè÷åñòâ ìîíåò π ∗ .
Êîëè÷åñòâî ìîíåò â π ∗ âû÷èñëÿåòñÿ ïî ôîðìóëå:
I ∗ (i)
k (i) =
, i ∈ {1, . . . , m}.
Si (0)
∗
32
4. Âû÷èñëåíèå âåêòîðà êîëè÷åñòâ ìîíåò π
e.
Êîëè÷åñòâî ìîíåò â π
e ïîëó÷àåòñÿ èç êîëè÷åñòâà ìîíåò â π ∗ ïóòåì îêðóãëåíèÿ ïîñëåäíèõ:
e
k(i) = [k ∗ (i)], i ∈ {1, . . . , m}.
5. Âû÷èñëåíèå âëîæåíèé â π
e.
Âëîæåíèÿ â π
e âû÷èñëÿþòñÿ ïî ôîðìóëå:
e =e
I(i)
k(i)Si (0), i ∈ {1, . . . , m}.
6. Âû÷èñëåíèå îòíîñèòåëüíûõ âåñîâ π
e.
Ìèíèìàëüíûå âëîæåíèÿ âû÷èñëÿþòñÿ ïî ôîðìóëå:
Ie =
m
X
e
I(i).
i=1
Îòíîñèòåëüíûå âåñà π
e âû÷èñëÿþòñÿ ïî ôîðìóëå:
x
ei =
e
I(i)
, i ∈ {1, . . . , m}.
Ie
Çàìåòèì, ÷òî äàííàÿ ñõåìà ðàáîòàåò äëÿ ïîðòôåëÿ ñîñòîÿùåãî èç ëþáûõ
öåííûõ áóìàã.
6.1
Ïîðòôåëü èç ïÿòè ìîíåò
Ñôîðìèðóåì öåëî÷èñëåííûé ïîðòôåëü èç ïÿòè ìîíåò íà ìîìåíò 29 ôåâðàëÿ
2016 ã. (πe5 ).  êà÷åñòâå ýôôåêòèâíîãî ïîðòôåëÿ âîçüìåì ïîðòôåëü ìèíèìàëüíîãî ðèñêà ïðè çàäàííîì íèæíåì óðîâíå äîõîäíîñòè 3% (π5∗ ). Ðåçóëüòàòû êàæäîãî
ýòàïà îïèñàííîé ñõåìû ïðèâåäåíû â òàáëèöå 1.
Òàáëèöà 1: Âû÷èñëåíèå öåëî÷èñëåííîãî ïîðòôåëÿ èç ïÿòè ìîíåò
1
2
3
4
5
Öåíà
ïîêóïêè
1 603 ð.
2 519 ð.
545 ð.
2 460 ð.
1 049 ð.
π5∗
26.54%
28.94%
15.47%
21,83%
7,21%
Âëîæåíèÿ
Êîë-âî
â π5∗
ìîíåò â π5∗
3 859,66 ð.
4 209,32 ð.
2 250,28 ð.
3 175,16 ð.
1 049 ð.
2.41
1.67
4.13
1.29
1.00
33
Êîë-âî
ìîíåò â
πe5
2
2
4
1
1
10
Âëîæåíèÿ
â
πe5
3 206 ð.
5 038 ð.
2 180 ð.
2 460 ð.
1 049 ð.
13 933 ð.
πe5
23.01%
36.16%
15.65%
17.66%
7.53%
Ìèíèìàëüíûå âëîæåíèÿ, íåîáõîäèìûå äëÿ ôîðìèðîâàíèÿ öåëî÷èñëåííîãî
ïîðòôåëÿ ñîñòàâëÿþò 13 933 ð. Ñðàâíåíèå îòíîñèòåëüíûõ âåñîâ π5∗ è πe5 ïðèâåäåíû íà ðèñ. 10.
Ðèñ. 10: Ñðàâíåíèå îòíîñèòåëüíûõ âåñîâ π5∗ è πe5
Îñíîâíûå õàðàêòåðèñòèêè π5∗ è πe5 ïðèâåäåíû â ñëåäóþùåé òàáëèöå:
Ïîðòôåëü
Îæèäàåìàÿ äîõîäíîñòü
Ðèñê
Ýôôåêòèâíûé Öåëî÷èñëåííûé
3%
9.62%
3.03%
10.05%
Ïîëó÷èâøèåñÿ îæèäàåìàÿ äîõîäíîñòü è ðèñê öåëî÷èñëåííîãî ïîðòôåëÿ áëèçêè ê îæèäàåìîé äîõîäíîñòè è ðèñêó ýôôåêòèâíîãî ïîðòôåëÿ.
Âû÷èñëèì äîõîäíîñòü ïîëó÷èâøåãîñÿ ïîðòôåëÿ â ïðåäïîëîæåíèè, ÷òî ìîíåòû áûëè ðåàëèçîâàíû ÷åðåç íåäåëþ ïîñëå ïîêóïêè (7 ìàðòà 2016 ã.). Ïîëó÷èâøèåñÿ ðåçóëüòàòû ïðåäñòàâëåíû â ñëåäóþùåé òàáëèöå:
34
Íàçâàíèå
Ýéëåð
Ïàâëîâà
Ëîáà÷åâñêèé
Ñìîëüíûé
Ðàçãðîì
Öåíà ïîêóïêè
Öåëî÷èñëåííûé
Öåíà ïðîäàæè
29.02.2016
ïîðòôåëü
07.03.2016
1 603 ð.
2 519 ð.
545 ð.
2 460 ð.
1 049 ð.
30.09%
25.95%
17.57%
19.48%
6.91%
1 744 ð.
2 127 ð.
619 ð.
3 204 ð.
921 ð.
Äîõîäíîñòü
8.08%
-18.43%
11.95%
23.22%
-13.90%
Äîõîäíîñòü ïîðòôåëÿ ðàâíà 3.31%, ÷òî áëèçêî ê îæèäàåìîé äîõîäíîñòè
ïîðòôåëÿ.
Îäíàêî ðåàëèçîâàòü öåëî÷èñëåííûé ïîðòôåëü íå òàê ïðîñòî. Âî-ïåðâûõ, íà
àóêöèîíàõ ìîíåòû ïðîäàþòñÿ ïî îäíîé, ò.å. êóïèòü îäíîâðåìåííî äâå ìîíåòû ïî
îäèíàêîâîé öåíå ïðàêòè÷åñêè íåâîçìîæíî. Âî-âòîðûõ, ñëîæíî ñôîðìèðîâàòü
ïîðòôåëü, â êîòîðîì ìîíåòû êóïëåíû â îäèí äåíü. Â-òðåòüèõ, ïðåäëîæåíèå
ìîæåò áûòü ìåíüøå, ÷åì êîëè÷åñòâî ìîíåò, òðåáóåìîå â ïîðòôåëå. Ñôîðìèðóåì
ðåàëüíûé ïîðòôåëü, çàäàâ åãî ïåðèîä ðåàëèçàöèè.
Çàìåòèì, ÷òî ïåðèîä ðåàëèçàöèè öåëî÷èñëåííîãî ïîðòôåëÿ íå äîëæåí ïðåâûøàòü 1-2 íåäåëü. Ýòî îáóñëîâëåíî òåì, ÷òî çà áîëüøèé ïåðèîä ¾êàðòèíà¿
ìîæåò çíà÷èòåëüíî ïîìåíÿòüñÿ, à ñëåäîâàòåëüíî ìîæåò èçìåíèòüñÿ è ýôôåêòèâíûé ïîðòôåëü, íà îñíîâå êîòîðîãî ñòðîèòüñÿ ðåàëüíûé.
Óñòàíîâèì ïåðèîä ðåàëèçàöèè öåëî÷èñëåííîãî ïîðòôåëÿ 2 íåäåëè (15.02.2016
29.02.2016). Â ñâÿçè ñ òåì, ÷òî â ïåðèîä ìîíåòà Ýéëåðà òîðãîâàëàñü îäèí ðàç,
ðåàëüíûé ïîðòôåëü áóäåò ñîäåðæàòü íå äâå ìîíåòû Ýéëåðà (êàê â öåëî÷èñëåííîì), à îäíó. Îñòàëüíûå ìîíåòû òîðãîâàëèñü äîñòàòî÷íîå ÷èñëî ðàç.  êà÷åñòâå
öåíû ïîêóïêè áûëè âçÿòû öåíû ñ ïîñëåäíèõ ïðîäàæ äî 29 ôåâðàëÿ 2016 ã.. Ñðàâíåíèå îòíîñèòåëüíûõ âåñîâ öåëî÷èñëåííîãî è ðåàëüíîãî ïîðòôåëåé ïðèâåäåíû
íà ðèñ. 11.
Îæèäàåìàÿ äîõîäíîñòü ðåàëüíîãî ïîðòôåëÿ ðàâíà 4.11%, ðèñê ðàâåí 10.77%.
Âû÷èñëèì äîõîäíîñòü ïîëó÷èâøåãîñÿ ïîðòôåëÿ â ïðåäïîëîæåíèè, ÷òî ìîíåòû áûëè ðåàëèçîâàíû ÷åðåç íåäåëþ ïîñëå îêîí÷àíèÿ ñðîêà ðåàëèçàöèè (7 ìàðòà
2016 ã.). Ïîëó÷èâøèåñÿ ðåçóëüòàòû ïðåäñòàâëåíû â ñëåäóþùåé òàáëèöå:
Íàçâàíèå
Ýéëåð
Ïàâëîâà
Ëîáà÷åâñêèé
Ñìîëüíûé
Ðàçãðîì
Êîë-âî
ìîíåò
1
2
4
1
1
Öåíà
ïîêóïêè
29.02.2016
1 603 ð.
4 568 ð.
2 527 ð.
2 460 ð.
1 049 ð.
12 207 ð.
Ðåàëüíûé
ïîðòôåëü
13.13%
3.42%
20.70%
20.15%
8.59%
Öåíà
ïðîäàæè
07.03.2016
1 744 ð.
4 254 ð.
2 665 ð.
3 204 ð.
921 ð.
12 788 ð.
Äîõîäíîñòü
8.08%
-7.38%
5.18%
23.22%
-13.90%
Ìèíèìàëüíûå âëîæåíèÿ, íåîáõîäèìûå äëÿ ôîðìèðîâàíèÿ ðåàëüíîãî ïîðò35
ôåëÿ ñîñòàâëÿþò 12 207 ð. Äîõîäíîñòü ðåàëüíîãî ïîðòôåëÿ ðàâíà 2.86%.
Ðèñ. 11: Ñðàâíåíèå îòíîñèòåëüíûõ âåñîâ öåëî÷èñëåííîãî è ðåàëüíîãî ïîðòôåëåé
6.2
Ïîðòôåëü èç äåñÿòè ìîíåò
Ñôîðìèðóåì öåëî÷èñëåííûé ïîðòôåëü èç äåñÿòè ìîíåò íà ìîìåíò 29 ôåâðàëÿ 2016 ã. (πf
10 ).  êà÷åñòâå ýôôåêòèâíîãî ïîðòôåëÿ âîçüìåì ïîðòôåëü ìèíè∗
ìàëüíîãî ðèñêà ïðè çàäàííîì íèæíåì óðîâíå äîõîäíîñòè 3% (π10
). Ðåçóëüòàòû
êàæäîãî ýòàïà îïèñàííîé ñõåìû ïðèâåäåíû â òàáëèöå 2.
Ìèíèìàëüíûå âëîæåíèÿ, íåîáõîäèìûå äëÿ ôîðìèðîâàíèÿ öåëî÷èñëåííîãî
ïîðòôåëÿ ñîñòàâëÿþò 158 577 ð. Îòìåòèì, ÷òî ðîñò âëîæåíèé â ïîðòôåëü ïðè
óâåëè÷åíèè êîëè÷åñòâà ìîíåò ðàñòåò íå ïðîïîðöèîíàëüíî.
∗
Ñðàâíåíèå îòíîñèòåëüíûõ âåñîâ π10
è πf
10 ïðèâåäåíî íà ðèñ. 12.
∗
Îñíîâíûå õàðàêòåðèñòèêè π5 è πe5 ïðèâåäåíû â ñëåäóþùåé òàáëèöå:
Ïîðòôåëü
Îæèäàåìàÿ äîõîäíîñòü
Ðèñê
Ýôôåêòèâíûé Öåëî÷èñëåííûé
3%
8.16%
3.04%
8.18%
Ïîëó÷èâøèåñÿ îæèäàåìàÿ äîõîäíîñòü è ðèñê öåëî÷èñëåííîãî ïîðòôåëÿ áëèçêè ê îæèäàåìîé äîõîäíîñòè è ðèñêó ýôôåêòèâíîãî ïîðòôåëÿ.
36
Òàáëèöà 2: Âû÷èñëåíèå öåëî÷èñëåííîãî ïîðòôåëÿ èç äåñÿòè ìîíåò
1
2
3
4
5
6
7
8
9
10
Öåíà
ïîêóïêè
1 603 ð.
2 519 ð.
545 ð.
2 460 ð.
1 049 ð.
1 055 ð.
367 ð.
1 055 ð.
1 133 ð.
800 ð.
∗
π10
25.08%
27.25%
10.56%
11.10%
0.67%
0.00%
8.68%
3.24%
12.09%
1.34%
Âëîæåíèÿ
Êîë-âî
∗
â π10
∗
ìîíåò â π10
39 557.90 ð.
42 971.69 ð.
16 647.94 ð.
17 501.22 ð.
1 049 ð.
0.00 ð.
13 696.00 ð.
5 103.04 ð.
19 066.35 ð.
2 118.75 ð.
24.68
17.06
30.55
7.11
1.00
0.00
37.32
4.84
16.83
2.65
Êîë-âî
ìîíåò â
Âëîæåíèÿ
πf
10
25
17
31
7
1
0
37
5
17
3
143
â
πf
10
40 075 ð.
42 823 ð.
16 895 ð.
17 220 ð.
1 049 ð.
0 ð.
13 579 ð.
5 275 ð.
19 261 ð.
2 400 ð.
158 577 ð.
πf
10
25.27%
27.01%
10.65%
10.86%
0.66%
0.00%
8.56%
3.33%
12.15%
1.51%
Âû÷èñëèì äîõîäíîñòü ïîëó÷èâøåãîñÿ ïîðòôåëÿ â ïðåäïîëîæåíèè, ÷òî ìîíåòû áûëè ðåàëèçîâàíû ÷åðåç íåäåëþ ïîñëå ïîêóïêè (7 ìàðòà 2016 ã.).
Íàçâàíèå
Ýéëåð
Ïàâëîâà
Ëîáà÷åâñêèé
Ñìîëüíûé
Ðàçãðîì
Êðûëîâ
Êîñìîñ
Ãîãîëü
Ìåäâåäü
Ñî÷è
Öåíà ïîêóïêè
Öåëî÷èñëåííûé
Öåíà ïðîäàæè
29.02.2016
ïîðòôåëü
07.03.2016
1 603 ð.
2 519 ð.
545 ð.
2 460 ð.
1 049 ð.
1 055 ð.
367 ð.
1 055 ð.
1 133 ð.
800 ð.
25.27%
27.00%
10.65%
10.86%
0.66%
0.00%
8.56%
3.33%
12.15%
1.51%
1 744 ð.
2 127 ð.
619 ð.
3 204 ð.
921 ð.
1 071 ð.
816 ð.
312 ð.
1 112 ð.
209 ð.
Äîõîäíîñòü
8.08%
-18.43%
11.95%
23.22%
-13.90%
1.49%
55.02%
-238.14%
-1.89%
-282.78%
Äîõîäíîñòü ïîðòôåëÿ îòðèöàòåëüíà è ðàâíà −6.95%.
Çàìå÷àíèå. Ïîëó÷èâøèéñÿ öåëî÷èñëåííûé ïîðòôåëü ðåàëèçîâàòü íåâîç-
ìîæíî. Ýòî ñâÿçàíî ñ òåì, ÷òî â ïåðèîä 01.02.2016 15.05.2016 ìîíåòà Ýéëåðà
òîðãîâàëàñü âñåãî 6 ðàç, â òî âðåìÿ êàê ïîðòôåëü òðåáóåò 25 ìîíåò.
37
∗
è πf
Ðèñ. 12: Ñðàâíåíèå îòíîñèòåëüíûõ âåñîâ π10
10
6.3
Âûâîä
 ñâÿçè ñ òåì, ÷òî êîëëåêöèîííûå ìîíåòû îáëàäàþò íèçêîé ëèêâèäíîñòüþ,
èõ òîðãè íåðåãóëÿðíû, à ñðîê ðåàëèçàöèè ïîðòôåëÿ íå äîëæåí ïðåâûøàòü
1-2 íåäåëü, ïîðòôåëü äîëæåí ñîñòîÿòü èç íåáîëüøîãî ÷èñëà ÷àñòî òîðãóþùèõñÿ ìîíåò. Òî åñòü äëÿ ýôôåêòèâíîñòè ïðåäëîæåííîé ìåòîäèêè ôîðìèðîâàíèÿ
îïòèìàëüíîãî ïîðòôåëÿ îí äîëæåí ñîñòîÿòü èç ïîðÿäêà ïÿòè ìîíåò ñî ñðåäíèì
÷èñëîì ïðîäàæ â íåäåëþ íå ìåíåå òðåõ.
38
7
Çàêëþ÷åíèå
Ïîñòàâëåííàÿ â íà÷àëå ðàáîòû öåëü äîñòèãíóòà, à èìåííî:
• Âûÿâëåíà è îöåíåíà ìåòîäèêà ôîðìèðîâàíèÿ îïòèìàëüíîãî ïîðòôåëÿ èç
êîëëåêöèîííûõ ìîíåò.
• Ìåòîäèêà ïðîâåðåíà íà ïîðòôåëÿõ, ñîñòîÿùèõ îò ÷åòûðåõ äî äåñÿòè ìîíåò.
• Ðåàëèçîâàí ïîäõîä ê ïîñòðîåíèþ ýôôåêòèâíîãî ïîðòôåëÿ è ýôôåêòèâíîé
ãðàíèöû.
• Ïðîâåäåí àíàëèç ýôôåêòèâíîé ãðàíèöû ðàññìàòðèâàåìûõ ïîðòôåëåé, â ðåçóëüòàòå êîòîðîãî ïîñòðîåíà ñõåìà âõîæäåíèÿ ìîíåò â ýôôåêòèâíûé ïîðòôåëü.
• Ïîñòðîåí ïðèìåð öåëî÷èñëåííîãî ïîðòôåëÿ, âû÷èñëåíû ìèíèìàëüíûå çàòðàòû íà åãî ðåàëèçàöèþ.
• Ïîñòðîåí ïðèìåð ðåàëüíîãî ïîðòôåëÿ.
• Âûÿâëåíû ïðîáëåìû ðåàëèçàöèè ïîðòôåëåé.
• Ñîñòàâëåíû ðåêîìåíäàöèè ïî âûáîðó ìîíåò ïðè ôîðìèðîâàíèè ïîðòôåëÿ.
39
Ñïèñîê ëèòåðàòóðû
[1] Áîäè Ç., Ìåðòîí Ç. Ôèíàíñû. Èçäàòåëüñêèé äîì ¾Âèëüÿìñ¿, 2007.
[2] Ãàâóðèí Ì. Ê., Ìàëîç¼ìîâ Â. Í. Îñíîâû òåîðèè êâàäðàòè÷íîãî ïðîãðàììèðîâàíèÿ Âåñòíèê ËÃÓ. 1980. 1. Ñ. 916.
[3] Êî÷åòûãîâ À.À. Ôèíàíñîâàÿ ìàòåìàòèêà. Ðîñòîâ-íà-Äîíó: Ôåíèêñ, 2004.
[4] Î'Áðàéåí Äæ., Øðèâàñòàâà Ñ. Ôèíàíñîâûé àíàíëèç è òîðãîâëÿ öåííûìè
áóìàãàìè. Ì.: ¾Äåëî ËÒÄ¿, 1995.
[5] Markowits Harry M. Portfolio Selection Journal of Finance, Vol. 7, No. 1, 1952,
pp. 7191.
[6] Zvi Bodie, Alex Kane, Alan J. Marcus. Investments. McGraw-Hill Education,
2010.
[7] Michel M. Dacorogna at al. An Introduction to High-Frequency Finance.
Academic Press, 2001.
[8] Andy Ravenna, Stacey Syphus. Querying and Reporting Using SAS Enterprise
Guide. Course Notes. Copyright, 2006 by SAS Institute Inc., Cary, NC 27513,
USA
[9] Èíôîðìàöèîííûé ïîðòàë. http://www.banki.ru/news/columnists/?id=7797341
[10] Ïîèñê ìîíåò ïî àóêöèîíàì. http://www.fcoins.ru/
[11] Ñàéò Öåíòðàëüíîãî Áàíêà ÐÔ. http://www.cbr.ru/
[12] Îôèöèàëüíûé ñàéò ïðîäóêöèè êîìïàíèè SAS. http://www.sas.com/
40
8
Ïðèëîæåíèÿ
 ïðèëîæåíèÿõ ñîäåðæèòñÿ:
• Òàáëèöà 3 ñ ìàòðèöåé êîâàðèàöèé äåñÿòè ìîíåò.
• Ïðîãðàììà íà R äëÿ âû÷èñëåíèÿ ýôôåêòèâíîãî ïîðòôåëÿ è ýôôåêòèâíîé
ãðàíèöû.
• Ïðîãðàììà íà Matlab äëÿ âû÷èñëåíèÿ ýôôåêòèâíîãî ïîðòôåëÿ.
• Ïðîãðàììà äëÿ ïðèâåäåíèÿ íåîäíîðîäíîãî âðåìåííîãî ðÿäà ê îäíîðîäíîìó
ñ èíòåðâàëîì 1 íåäåëÿ (SAS).
• Èñõîäíûå äàííûå äåñÿòè ìîíåò íà ðèñ. 13-22.
• Èëëþñòðàöèè äåñÿòè ìîíåò íà ðèñ. 23-32.
Òàáëèöà 3: Ìàòðèöà êîâàðèàöèé
1
1
2
3
4
5
6
7
8
9
10
2.88%
-0.86%
-0.04%
0.09%
2.15%
-1.05%
0.52%
-0.75%
0.72%
0.32%
2
-0.86%
4.85%
-0.50%
-1.23%
0.26%
4.04%
-3.46%
2.32%
-0.10%
-2.33%
3
-0.04%
-0.50%
5.75%
0.75%
0.82%
0.66%
0.16%
0.27%
0.84%
-0.28%
4
0.09%
-1.23%
0.75%
5.96%
-1.78%
-0.84%
1.61%
-0.82%
0.79%
3.19%
5
2.15%
0.26%
0.82%
-1.78%
26.01%
1.02%
-0.34%
4.45%
1.57%
-5.51%
6
-1.05%
4.04%
0.66%
-0.84%
1.02%
22.28%
-3.49%
9.49%
2.36%
-3.04%
7
0.52%
-3.46%
0.16%
1.61%
-0.34%
-3.49%
14.56%
-1.54%
0.26%
9.28%
8
-0.75%
2/32%
0/27%
-0.82%
4.45%
9.49%
-1.54%
14.08%
0.47%
-0.77%
9
0.72%
-0.10%
0.84%
0.79%
1.57%
2.36%
0.26%
0.47%
2.20%
-1.81%
10
0.32%
-2.33%
-0.28%
3.19%
-5.51%
-3.04%
9.28%
-0.77%
-1.81%
80.94%
Âû÷èñëåíèå ýôôåêòèâíîãî ïîðòôåëÿ è ýôôåêòèâíîé
ãðàíèöû (R)
library("quadprog")
library("rJava")
library("xlsx")
require(xlsx)
book <- loadWorkbook("E:\\Course\\Petrova_Anna_10_05.xlsx")
sheets <- getSheets(book)
file <- "E:\\Course\\Petrova_Anna_10_05.xlsx"
book <- loadWorkbook(file)
41
sheets <- getSheets(book)
sheet <- sheets[[8]]
rows <- getRows(sheet)
cells <- getCells(rows)
t <- lapply(cells, getCellValue)
# ÷òåíèå äàííûõ èç ôàéëà
Dmat1 <- C(t$'10.1',t$'11.1',t$'12.1',t$'13.1',t$'14.1')
Dmat2 <- c(t$'10.3',t$'11.3',t$'12.3',t$'13.3',t$'14.3')
Dmat3 <- c(t$'10.5',t$'11.5',t$'12.5',t$'13.5',t$'14.5')
Dmat4 <- c(t$'10.7',t$'11.7',t$'12.7',t$'13.7',t$'14.7')
Dmat5 <- c(t$'10.9',t$'11.9',t$'12.9',t$'13.9',t$'14.9')
Dmat <- cbind(Dmat1, Dmat2, Dmat3, Dmat4, Dmat5)
dvec <- c(0,0,0,0,0)
A <- matrix(1,1,5)
r <- c(t$'17.1',t$'17.3',t$'17.5',t$'17.7',t$'17.9')
A <- rbind(A, -A, r, diag(5))
Amat <- t(A)
bvec <- c(t$'16.12', -t$'16.12',t$'17.12',t$'18.12',
t$'19.12',t$'20.12',t$'21.12',t$'22.12')
# ðåøåíèå
sol <- solve.QP(Dmat,dvec,Amat,bvec, meq=1)$solution
addDataFrame(sol,sheets[[8]],col.names=FALSE,row.names=FALSE,
startRow=24,startColumn=3)
# ðèñê
sol <- t(sol)
solt <- t(sol)
risk <- sol %*% Dmat %*% solt
addDataFrame(risk,sheets[[8]],col.names=FALSE,row.names=FALSE,
startRow=32,startColumn=3)
# îæèäàåìàÿ äîõîäíîñòü
expret <- r %*% solt
addDataFrame(expret,sheets[[8]],col.names=FALSE,row.names=FALSE,
startRow=31,startColumn=3)
#---------------------------------------------------------------------------------# ÝÔÔÅÊÒÈÂÍÀß ÃÐÀÍÈÖÀ
m <- 10000 # ðàíã äðîáëåíèÿ
k <- 0 # êîëè÷åñòâî òî÷åê
for (R in seq (min(r) , max(r), by = 1/m)){
k <- k+1
bvec <- c(t$'16.12',-t$'16.12',R,t$'18.12',t$'19.12',
t$'20.12',t$'21.12',t$'22.12')
# ðåøåíèå
sol <- solve.QP(Dmat,dvec,Amat,bvec, meq=1)$solution
for (i in 1:5){
addDataFrame(sol[i],sheets[[8]],col.names=FALSE,row.names=FALSE,
startRow=40+k,startColumn=4+i)}
42
# ðèñê
sol <- t(sol)
solt <- t(sol)
risk <- sol %*% Dmat %*% solt
addDataFrame(risk,sheets[[8]],col.names=FALSE,row.names=FALSE,
startRow=40+k,startColumn=3)
# îæèäàåìàÿ äîõîäíîñòü
expret <- r %*% solt
addDataFrame(expret,sheets[[8]],col.names=FALSE,row.names=FALSE,
startRow=40+k,startColumn=4)
}
saveWorkbook(book,"E:\\Course\\Petrova_Anna_10_05.xlsx")
Âû÷èñëåíèå ýôôåêòèâíîãî ïîðòôåëÿ (MatLab)
clear all
close all
clc
H = xlsread('Petrova_Anna_10_05.xlsx','Ðåøåíèå 5','G2:K6');
G = xlsread('Petrova_Anna_10_05.xlsx','Ðåøåíèå 5','M2:Q2');
B = [0.03];
Aeq = [1 1 1 1 1];
beq = [1];
lb = zeros(5,1);
f = zeros(5,1);
A = -G;
b = -B;
[x,fval] = quadprog(H,f,A,b,Aeq,beq,lb,[]);
Ïðèâåäåíèå íåîäíîðîäíîãî âðåìåííîãî ðÿäà ê
îäíîðîäíîìó ñ øàãîì 1 íåäåëÿ (SAS)
/* -------------------------------------------------------------------×òåíèå äàííûåõ èç âðåìåííîãî ôàéëà, ñîçäàííîãî
ìàñòåðîì èìïîðòà äàííûõ. Çíà÷åíèÿ âðåìåííîãî ôàéëà áûëè ïîëó÷åíû
èç èñõîäíîãî ôàéëà Excel.
-------------------------------------------------------------------- */
DATA WORK.raz;
LENGTH
'date'n 8
'price'n 8 ;
FORMAT
'date'n
DATETIME18.
'price'n
BEST12. ;
INFORMAT
43
'date'n
DATETIME18.
'price'n
BEST12. ;
INFILE '/u0/work/SAS_work35B500004828_ds-sasfront02p/#LN00245'
LRECL=15
ENCODING="WCYRILLIC"
TERMSTR=CRLF
DLM='7F'x
MISSOVER
DSD ;
INPUT
'date'n
: BEST32.
'price'n
: BEST32. ;
RUN;
/* --------------------------------------------------------------------*/
data week;
array week{53} week1-week53;
run;
PROC SQL;
CREATE VIEW WORK.SORTTempTableSorted AS
SELECT T.week1, T.week2, T.week3, T.week4, T.week5, T.week6, T.week7,
T.week8, T.week9, T.week10, T.week11, T.week12, T.week13, T.week14,
T.week15, T.week16, T.week17, T.week18, T.week19, T.week20, T.week21,
T.week22, T.week23, T.week24, T.week25, T.week26, T.week27, T.week28,
T.week29, T.week30, T.week31, T.week32, T.week33, T.week34, T.week35,
T.week36, T.week37, T.week38, T.week39, T.week40, T.week41, T.week42,
T.week43, T.week44, T.week45, T.week46, T.week47, T.week48, T.week49,
T.week50, T.week51, T.week52, T.week53
FROM WORK.WEEK as T
;
QUIT;
PROC TRANSPOSE DATA=WORK.SORTTempTableSorted
OUT=WORK.TRNSTransposed(LABEL="Òðàíñïîíèðîâàííûé WORK.WEEK")
PREFIX='p'n
NAME='week'n
LABEL='ßðëûê'n
;
VAR week1 week2 week3 week4 week5 week6 week7 week8 week9 week10 week11 week12
week13 week14 week15 week16 week17 week18 week19 week20 week21 week22 week23
week24 week25 week26 week27 week28 week29 week30 week31 week32 week33 week34
week35 week36 week37 week38 week39 week40 week41 week42 week43 week44 week45
week46 week47 week48 week49 week50 week51 week52 week53;
/* ------------------------------------------------------------------Ðåøåíèå çàäà÷è
------------------------------------------------------------------- */
RUN; QUIT;
%_eg_conditional_dropds(WORK.SORTTempTableSorted);
44
TITLE; FOOTNOTE;
DATA TRNSTransposed (drop=p1);
SET TRNSTransposed;
RETAIN K 1740268800;
K = K+604800;
RUN;
proc sql;
create table ann as
select week,K format=DATETIME18. as date
from TRNSTransposed;
quit;
proc sql;
create table raz as
select date as date_0,DHMS(floor((floor(date/86400)+4)/7)*7+3,00,00,00)
format=DATETIME18. as date,price
from raz;
quit;
proc sort data=raz;
by date_0 date;
run;
data raz;
set raz;
by date;
if last.date;
run;
proc sql;
create table result as
select a.*,b.price
from ann a left join raz b on (a.date=b.date)
;quit;
45
Ðèñ. 13: Èñõîäíûå äàííûå ìîíåòû Ýé- Ðèñ. 14: Èñõîäíûå äàííûå ìîíåòû Ïàâëåð
ëîâà
Ðèñ. 15: Èñõîäíûå äàííûå ìîíåòû Ëî- Ðèñ. 16: Èñõîäíûå äàííûå ìîíåòû
Ñìîëüíûé
áà÷åâñêèé
Ðèñ. 17: Èñõîäíûå äàííûå ìîíåòû Ðàç- Ðèñ. 18: Èñõîäíûå äàííûå ìîíåòû
Êðûëîâ
ãðîì
46
Ðèñ. 19: Èñõîäíûå äàííûå ìîíåòû Êîñ- Ðèñ. 20: Èñõîäíûå äàííûå ìîíåòû Ãîãîëü
ìîñ
Ðèñ. 21: Èñõîäíûå äàííûå ìîíåòû ÌåäÐèñ. 22: Èñõîäíûå äàííûå ìîíåòû Ñî÷è
âåäü
Ðèñ. 23: 2 ðóáëÿ. 300 - ëåòèå ñî äíÿ
ðîæäåíèÿ Ë. Ýéëåðà
Ðèñ. 24: 3 ðóáëÿ. Àííà Ïàâëîâà
47
Ðèñ. 25: 1 ðóáëü. 200 - ëåòèå ñî äíÿ
ðîæäåíèÿ Í. È. Ëîáà÷åâñêîãî
Ðèñ. 26: 3 ðóáëÿ. Ñìîëüíûé èíñòèòóò
è ìîíàñòûðü â Ñàíêò-Ïåòåðáóðãå
Ðèñ. 27: 3 ðóáëÿ. 50-ëåòèå ðàçãðîìà
íåìåöêî-ôàøèñòñêèõ âîéñê ïîä Ëåíèíãðàäîì
Ðèñ. 28: 2 ðóáëÿ. 225-ëåòèå ñî äíÿ
ðîæäåíèÿ È. À. Êðûëîâà
Ðèñ. 29: 3 ðóáëÿ. Ìåæäóíàðîäíûé ãîä
Êîñìîñà
Ðèñ. 30: 2 ðóáëÿ. 185 - ëåòèå ñî äíÿ
ðîæäåíèÿ Í.Â. Ãîãîëÿ
Ðèñ. 31: 50 ðóáëåé. Ãèìàëàéñêèé ìåäâåäü
Ðèñ. 32: 25 ðóáëåé. Ýìáëåìà Èãð
48
Отзывы:
Авторизуйтесь, чтобы оставить отзыв