Основы страхования. Орлов В.П. - 12 стр.

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P (w : S(w) ≤ U) = P
Ã
w :
S(w) − E(S)
p
D(S)
≤
U − E(S)
p
D(S)
!
≈ Φ
Ã
U − E(S)
p
D(S)
!
Q
S
Q
U
Q = 1 − R R ∈ (0, 1) R
x
R
Φ(x
R
) = 1 − R
x
R
Φ(x
R
) = 1 − R R
Q = P (w : S(w) ≤ U) ≈ Φ
Ã
U − E(S)
p
D(S)
!
= Φ(x
R
).
U = x
R
p
D(S) + E(S).
U
U =
N
X
i=1
E(X
i
) = E
Ã
N
X
i=1
X
i
!
= E(S).
p
i
= ¯p
i
= E(X
i
) U =
P
N
i=1
E(X
i
) = E
³
P
N
i=1
X
i
´
= E(S)
Q ≈ Φ
Ã
U − E(S)
p
D(S)
!
= Φ
Ã
E(S) − E(S)
p
D(S)
!
= Φ(0) = 1/2.
                                      12 


òî
                        Ã                                          !        Ã              !
                               S(w) − E(S) U − E(S)                             U − E(S)
P (w : S(w) ≤ U ) = P   w:       p        ≤ p                          ≈Φ        p
                                   D(S)       D(S)                                 D(S)

â ñèëó (3.2.14) è (3.2.15).
  Âåëè÷èíà (3.2.16) îçíà÷àåò âåðîÿòíîñòü Q íåðàçîðåíèÿ êîìïàíèè.
  Èñõîäÿ èç (3.2.16), ìîæíî âñåãäà ïîäñ÷èòàòü âåðîÿòíîñòü íåðàçîðåíèÿ
êîìïàíèè, åñëè èçâåñòåí êàïèòàë êîìïàíèè è çàäàíà ñëó÷àéíàÿ âåëè÷èíà
S.
  Ìîæíî ðåøèòü äðóãóþ çàäà÷ó: äëÿ çàäàííîé âåëè÷èíû Q íåðàçîðåíèÿ
êîìïàíèè îïðåäåëèòü âåëè÷èíó U êàïèòàëà, îáåñïå÷èâàþùåãî ýòó âåðîÿò-
íîñòü íåðàçîðåíèÿ. Äëÿ ýòîãî, ïîëàãàÿ Q = 1 − R, R ∈ (0, 1) è R ìàëî,
îïðåäåëÿþò ñíà÷àëà xR êàê ðåøåíèå óðàâíåíèÿ Φ(xR ) = 1 − R. Ðåøåíèå
xR óðàâíåíèÿ Φ(xR ) = 1 − R íàçûâàåòñÿ êâàíòèëüþ ÷èñëà R.
  Çàòåì, ïîëüçóÿñü (3.2.14), âûïèñûâàþò ñîîòíîøåíèå
                                             Ã                 !
                                                 U − E(S)
            Q = P (w : S(w) ≤ U ) ≈ Φ             p                = Φ(xR ).
                                                    D(S)
 Èç ïîñëåäíåãî ðàâåíñòâà, ïðèðàâíèâàÿ àðãóìåíòû, ïîëó÷àåì, ÷òî
                                     p
                            U = xR       D(S) + E(S).                              (3.2.17)

3.3 Ïðèíöèïû íàçíà÷åíèÿ ñòðàõîâûõ ïðåìèé
Ïóñòü êàïèòàë U êîìïàíèè ñêëàäûâàåòñÿ èç ïîëó÷åííûõ îò êëèåíòîâ ïðå-
ìèé. Åñëè ñîáëþäàòü ïðèíöèï ýêâèâàëåíòíîñòè, ñóììàðíàÿ âåëè÷èíà ïðå-
ìèé, îïðåäåëåííûõ êàê íåòòî-ïðåìèÿ ïî ôîðìóëå (3.1.8), è ñîñòàâèò êàïè-
òàë êîìïàíèè, ïîñêîëüêó
                        N
                                            Ã N          !
                        X                    X
                   U=         E(Xi ) = E            Xi       = E(S).               (3.3.18)
                        i=1                   i=1

Ïóñòü ñíà÷àëà ïðåìèÿ ïî êàæäîìó äîãîâîðó ÿâëÿåòñÿ ³ íåòòî-ïðåìèåé,
                                                             ´     ò.å.
                                    PN               PN
pi = p̄i = E(Xi ). Òîãäà êàïèòàë U = i=1 E(Xi ) = E    i=1 Xi = E(S).
Èç (3.2.16) ïîëó÷àåì, ÷òî â ýòîì ñëó÷àå
               Ã              !        Ã                     !
                   U − E(S)                E(S) − E(S)
        Q≈Φ         p             =Φ         p                   = Φ(0) = 1/2.
                      D(S)                     D(S)

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