Основы страхования. Орлов В.П. - 17 стр.

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i
(m)
∗
1/m
t
C(t) = (1 + i)
t
C
1
(t)
1
m
i
(m)
∗
[0, t]
[0,
1
m
] tm t
C
1
(mt) = (1 + i
(m)
∗
)
mt
i
(m)
∗
C
1
(mt) =
C(t) (1 + i)
t
= (1 + i
(m)
∗
)
tm
t = 1 (1 + i) = (1 + i
(m)
∗
)
m
i
(m)
∗
= (1 + i)
1
m
− 1.
i
(m)
∗
= (1+i)
1
m
−1
m
C
1
(τ) = e
δ
(m)
∗
τ
,
τ = m C
1
(m) = e
δ
(m)
∗
m
= e
δ
= C(1) δ
(m)
∗
= δ/m
m
δ
(m)
∗
= δ/m.
d
[0, 1/m]
[0, 1/m]
d
(m)
∗
=
i
(m)
∗
1 + i
(m)
∗
i
(m)
= mi
(m)
∗
.
                                    17 

                                          (m)
ñ ýôôåêòèâíîé ïðîöåíòíîé ñòàâêîé i∗ è ïåðèîäîì êîíâåðñèè 1/m. Åñëè
ìû âîçüìåì âðåìÿ t, òî ïî áàíêîâñêîìó ñ÷åòó (4.2.1) ñîâðåìåííàÿ ñòîèìîñòü
1 ó.å. ðàâíà C(t) = (1 + i)t . Åñëè ìû âîçüìåì íîâûé ñ÷åò C1 (t) ñ ïåðèîäîì
              1            (m)
êîíâåðñèè m     è ñòàâêîé i∗ , òî íà ïðîìåæóòêå [0, t] ïðîìåæóòîê êîíâåðñèè
[0, m1 ] óêëàäûâàåòñÿ tm ðàç, è ñòîèìîñòü 1 ó.å. â ìîìåíò t ïî íîâîìó ñ÷åòó
                          (m)                (m)
ðàâíà C1 (mt) = (1 + i∗ )mt . Îïðåäåëèì i∗ òàê, ÷òîáû íà÷èñëåííàÿ ïî
íîâîìó è ñòàðîìó ñ÷åòó ñóììû ñîâïàäàëè. Òîãäà äîëæíî áûòü C1 (mt) =
                            (m)                                        (m)
C(t) èëè (1 + i)t = (1 + i∗ )tm . Ïîëàãàÿ t = 1, èìååì (1 + i) = (1 + i∗ )m .
Îòñþäà ëåãêî ñëåäóåò, ÷òî
                                                 1
                             i(m)
                              ∗   = (1 + i) m − 1.                          (4.2.14)

  Îïðåäåëåíèå 4.10. ×èñëî i(m)
                                                          1
                           ∗   = (1 + i) m − 1 íàçûâàåòñÿ ýôôåêòèâíîé
ïðîöåíòíîé ñòàâêîé, âûïëà÷èâàåìîé ñ ÷àñòîòîé m.

 Ïåðåïèñàâ (4.2.13) â âèäå
                                                (m)
                               C1 (τ ) = eδ∗          τ
                                                          ,                 (4.2.15)
                                      (m)                            (m)
è ïîëàãàÿ τ = m, èìååì C1 (m) = eδ∗         m
                                                = eδ = C(1), îòêóäà δ∗     = δ/m.

  Îïðåäåëåíèå 4.11. Èíòåíñèâíîñòüþ ïðîöåíòíîé ñòàâêè, âûïëà÷è-
âàåìîé ñ ÷àñòîòîé m, íàçûâàåòñÿ ÷èñëî

                                 δ∗(m) = δ/m.                               (4.2.16)

 Àíàëîãè÷íî ñòàâêå äèñêîíòèðîâàíèÿ d, îïðåäåëåííîé ÷åðåç (4.2.12) íà
ïðîìåæóòêå [0,1], ââîäèòñÿ ñòàâêà äèñêîíòèðîâàíèÿ íà [0, 1/m].

  Îïðåäåëåíèå 4.12. Ýôôåêòèâíîé ñòàâêîé äèñêîíòèðîâàíèÿ íà
ïðîìåæóòêå [0, 1/m] íàçûâàåòñÿ ÷èñëî
                                                (m)
                                            i∗
                               d(m)
                                ∗     =           (m)
                                                                            (4.2.17)
                                          1 + i∗
  Â ñòðàõîâîé ìàòåìàòèêå äëÿ óäîáñòâ çàïèñè ôîðìóë èñïîëüçóþò íå òîëü-
êî ýôôåêòèâíûå, ò.å. ðåàëüíûå, íî è íîìèíàëüíûå ñòàâêè.

   Îïðåäåëåíèå 4.13. Íîìèíàëüíîé ïðîöåíòíîé ñòàâêîé íàçûâàåòñÿ
÷èñëî
                                i(m) = mi(m)
                                         ∗ .                                (4.2.18)

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