利息理论公式

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