Wzory teorii oprocentowania
edytuj
1
=
δ
a
¯
t
¯
|
+
v
t
{\displaystyle 1=\delta {\bar {a}}_{{\overline {t}}|}+v^{t}}
α
(
m
)
=
i
⋅
d
i
(
m
)
⋅
d
(
m
)
β
(
m
)
=
i
−
i
(
m
)
i
(
m
)
⋅
d
(
m
)
{\displaystyle \alpha (m)={\frac {i\cdot d}{i^{(m)}\cdot d^{(m)}}}\qquad \beta (m)={\frac {i-i^{(m)}}{i^{(m)}\cdot d^{(m)}}}}
Wzory do modelu demograficznego
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Współczynnikiem umieralności (central death rate ) nazywamy
m
x
=
q
x
∫
0
1
t
p
x
d
t
{\displaystyle m_{x}={\frac {q_{x}}{\int _{0}^{1}{}_{t}p_{x}dt}}}
Egzaminy aktuarialne w sieci
edytuj
Gotowce z angielskiej wikipedii
edytuj
q
x
=
d
x
/
l
x
{\displaystyle \,q_{x}=d_{x}/l_{x}\!}
t
p
x
=
l
x
+
t
l
x
{\displaystyle \,{}_{t}p_{x}={\frac {l_{x+t}}{l_{x}}}}
n
q
x
=
n
d
x
/
l
x
{\displaystyle \,_{n}q_{x}=_{n}d_{x}/l_{x}}
n
p
x
=
l
x
+
n
/
l
x
{\displaystyle \,_{n}p_{x}=l_{x+n}/l_{x}}
e
x
=
∑
t
=
1
∞
t
p
x
{\displaystyle \,e_{x}=\sum _{t=1}^{\infty }\ _{t}p_{x}}
l
x
+
t
=
(
1
−
t
)
l
x
+
t
l
x
+
1
{\displaystyle \,l_{x+t}=(1-t)l_{x}+tl_{x+1}}
l
x
+
1
=
l
x
⋅
(
1
−
q
x
)
=
l
x
⋅
p
x
{\displaystyle \,l_{x+1}=l_{x}\cdot (1-q_{x})=l_{x}\cdot p_{x}}
l
x
+
1
l
x
=
p
x
{\displaystyle \,{l_{x+1} \over l_{x}}=p_{x}}
d
x
=
l
x
−
l
x
+
1
{\displaystyle \,d_{x}=l_{x}-l_{x+1}}
t
|
k
q
x
=
t
p
x
⋅
k
q
x
+
t
=
l
x
+
t
−
l
x
+
t
+
k
l
x
{\displaystyle \,{}_{t|k}q_{x}={}_{t}p_{x}\cdot {}_{k}q_{x+t}={l_{x+t}-l_{x+t+k} \over l_{x}}}
a
n
|
¯
i
=
v
+
v
2
+
⋯
+
v
n
=
1
−
v
n
i
{\displaystyle \,a_{{\overline {n|}}i}=v+v^{2}+\cdots +v^{n}={\frac {1-v^{n}}{i}}}
a
¨
n
|
¯
i
=
1
+
v
+
⋯
+
v
n
−
1
=
1
−
v
n
d
{\displaystyle {\ddot {a}}_{{\overline {n|}}i}=1+v+\cdots +v^{n-1}={\frac {1-v^{n}}{d}}}
a
n
|
¯
i
(
m
)
=
1
−
v
n
i
(
m
)
{\displaystyle a_{{\overline {n|}}i}^{(m)}={\frac {1-v^{n}}{i^{(m)}}}}
a
¨
n
|
¯
i
(
m
)
=
1
−
v
n
d
(
m
)
{\displaystyle {\ddot {a}}_{{\overline {n|}}i}^{(m)}={\frac {1-v^{n}}{d^{(m)}}}}
a
¯
n
|
¯
i
=
1
−
v
n
δ
{\displaystyle {\overline {a}}_{{\overline {n|}}i}={\frac {1-v^{n}}{\delta }}}
a
n
|
¯
i
<
a
n
|
¯
i
(
m
)
<
a
¯
n
|
¯
i
<
a
¨
n
|
¯
i
(
m
)
<
a
¨
n
|
¯
i
{\displaystyle a_{{\overline {n|}}i}<a_{{\overline {n|}}i}^{(m)}<{\overline {a}}_{{\overline {n|}}i}<{\ddot {a}}_{{\overline {n|}}i}^{(m)}<{\ddot {a}}_{{\overline {n|}}i}}
m
|
a
¨
x
=
m
p
x
v
m
a
¨
x
+
m
=
A
x
:
n
¯
|
1
a
¨
x
+
m
{\displaystyle {}_{m|}{\ddot {a}}_{x}={}_{m}p_{x}v^{m}{\ddot {a}}_{x+m}=A_{x:{\overline {n}}|}^{\;\;\;1}{\ddot {a}}_{x+m}}
