a (t ) =1+it ⎧A (1)=A (0)+A (0)i 1=A (0)(1+i 1) ⎪单利（线性积累）；⎨A (2)=A (0)(1+i 1) +A (0)i 2=A (0)(1+i 1+i 2) i i n =1+(n -1) i ⎪⎩A (n ) =A (0)(1+i 1+i 2+... +i n )
特别的：各年利率相等时，有A (t ) =A (0) +(1i t ) ≥, t ，a (t ) =(1+it ) ，i n =1+in -[1+i (n -1)]i =1+i (n -1) 1+i (n -1)
t ⎧A (1)=A (0)+A (0)i 1=A (0)(1+i 1) a (t ) =(1+i ) ⎪复利（指数积累）；⎨A (2)=A (0)(1+i 1) +A (0)(1+i 1) i 2=A (0)(1+i 1)(1+i 2) i n =i ⎪A (n ) =A (0)(1+i )(1+i )(1+i ) 12n ⎩
特别的：各年利率相等时，有A (n ) =A (0)(1+i ) n ，a (t ) =(1+i ) t ，
n （1+i ）-(1+i ) (n -1)
i n ==i (n -1) (1+i )
I (n ) ⎧期末计息——利率—第N 期实质利率i =n ⎪A (n -1) ⎪ 计息时刻不同⎨⎪期初计息——贴现率—第N 期实质贴现率d =I (n )
n ⎪A (n ) ⎩
d n =
单利场合利率与贴现率的关系I (n ) A (n ) a (n ) -a (n -1) = a (n )
i =1+in
d n =
复利场合利率与贴现率的关系I (n ) a (n ) -a (n -1) =A (n ) a (n ) i (1+i ) n -1
=(1+i ) n
i =1+i
a (t ) =1+it a -1(t ) =1-dt
积累方式不同：线形积累——单利单贴现 i d i n =d n =1+(n -1) i 1-(n -1) d
指数积累——复利
m a (t ) =(1+i ) t i n =i 复贴现a -1(t ) =(1-d ) t d n =d ⎡i (m ) ⎤i (m ) (m ) =1+i ，每一次的结算利率j =名义利率i ：⎢1+ ⎥m ⎦m ⎣
m
名义贴现率d (m ) ：⎡⎢⎣1-d (m ) ⎤
m ⎥⎦=1-d δ'(t )
t =A
A (t ) =d
dt [ln A (t ) ]
利息力=a '(t ) d ⎰t
0δs ds
a (t ) =dt [ln a (t ) ]; 一般公式a (t ) =e ; =lim i (m ) =lim d (m )
m →∞m →∞
恒定利息效力场合δ=-ln v ⇔a -1(n ) =exp{-n δ} δ=ln(1+i ) ⇔a (n ) =exp{n δ} 基本年金公式总结
等差年金
s
积累值V (n ) =Ps -n
n +Q i
a
现时值V (0)=Pa -nv n
n +Q i
等比年金
) =(1+i ) n V (0)=(1+i ) n -(1+k ) n
积累值V (n i -k
1-(1+k ) n
现时值V (0)=v +v (1+k ) ++v n (1+k ) n -1=i -k , i ≠k
a (t ) =1+it ⎧A (1)=A (0)+A (0)i 1=A (0)(1+i 1) ⎪单利（线性积累）；⎨A (2)=A (0)(1+i 1) +A (0)i 2=A (0)(1+i 1+i 2) i i n =1+(n -1) i ⎪⎩A (n ) =A (0)(1+i 1+i 2+... +i n )
特别的：各年利率相等时，有A (t ) =A (0) +(1i t ) ≥, t ，a (t ) =(1+it ) ，i n =1+in -[1+i (n -1)]i =1+i (n -1) 1+i (n -1)
t ⎧A (1)=A (0)+A (0)i 1=A (0)(1+i 1) a (t ) =(1+i ) ⎪复利（指数积累）；⎨A (2)=A (0)(1+i 1) +A (0)(1+i 1) i 2=A (0)(1+i 1)(1+i 2) i n =i ⎪A (n ) =A (0)(1+i )(1+i )(1+i ) 12n ⎩
特别的：各年利率相等时，有A (n ) =A (0)(1+i ) n ，a (t ) =(1+i ) t ，
n （1+i ）-(1+i ) (n -1)
i n ==i (n -1) (1+i )
I (n ) ⎧期末计息——利率—第N 期实质利率i =n ⎪A (n -1) ⎪ 计息时刻不同⎨⎪期初计息——贴现率—第N 期实质贴现率d =I (n )
n ⎪A (n ) ⎩
d n =
单利场合利率与贴现率的关系I (n ) A (n ) a (n ) -a (n -1) = a (n )
i =1+in
d n =
复利场合利率与贴现率的关系I (n ) a (n ) -a (n -1) =A (n ) a (n ) i (1+i ) n -1
=(1+i ) n
i =1+i
a (t ) =1+it a -1(t ) =1-dt
积累方式不同：线形积累——单利单贴现 i d i n =d n =1+(n -1) i 1-(n -1) d
指数积累——复利
m a (t ) =(1+i ) t i n =i 复贴现a -1(t ) =(1-d ) t d n =d ⎡i (m ) ⎤i (m ) (m ) =1+i ，每一次的结算利率j =名义利率i ：⎢1+ ⎥m ⎦m ⎣
m
名义贴现率d (m ) ：⎡⎢⎣1-d (m ) ⎤
m ⎥⎦=1-d δ'(t )
t =A
A (t ) =d
dt [ln A (t ) ]
利息力=a '(t ) d ⎰t
0δs ds
a (t ) =dt [ln a (t ) ]; 一般公式a (t ) =e ; =lim i (m ) =lim d (m )
m →∞m →∞
恒定利息效力场合δ=-ln v ⇔a -1(n ) =exp{-n δ} δ=ln(1+i ) ⇔a (n ) =exp{n δ} 基本年金公式总结
等差年金
s
积累值V (n ) =Ps -n
n +Q i
a
现时值V (0)=Pa -nv n
n +Q i
等比年金
) =(1+i ) n V (0)=(1+i ) n -(1+k ) n
积累值V (n i -k
1-(1+k ) n
现时值V (0)=v +v (1+k ) ++v n (1+k ) n -1=i -k , i ≠k