财务管理Chap006

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年金：等额、定期的系列收支。
如：分期付款、购房按揭、付养老金等 按收付款发生时间的不同分为： •预付年金 •递延年金 •永续年金 •普通年金
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所以，预付年金的现值=普通年金的现值×（1+r)
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递延年金的现值 （有两种方法）
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t
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A(1+r)2
2

A1 r
2
A1 r
t 1
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普通年金的终值 (1)
FVAt A A1 r A1 r A1 r
普通年金终值与
复利终值的关系
普通年金终值是 多笔复利终值之和
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普通年金的终值 0 1 A 2 A 3 A A A(1+r) A(1+r)2
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实际年利率
Hale Waihona Puke McGraw-Hill/Irwin
所以
r R 1 1 m
m
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FIGURE 6.5
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普通年金的现值
0
1
A
2
A
3
A
A(1+r)-1
A(1+r)-2
A(1+r)-3
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各年利率（不）相等
A0 A1 A2 A3 At-1 At
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1
等式两边同乘（1+r)
(t 1)
(2)
PVAt r A1 1 r
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(2)-(1)

t

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1 (1 r ) t PVAt A r
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FIGURE 6.6
4.普通年金的现值
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2
t 1
等式两边同乘（1+r) (2)
FVAt 1 r A1 r A1 r A1 r
2 t
t
(2)-(1)
FVAt r A A1 r
t
1 式中 （ r ) 1 r
（ r )t 1 1 即 FVAt A r
PVAt A（ r) A（ r) A（ r) 1 1 1
推广至t期
1 2
1
2
3
PVAt A（ r) A（ r ) A（ r ) (1) 1 1 1
t
PVAt 1 r A A（ r) A（ r) 1 1
1 (1 r ) t 式中 是 r
普通年金为1元，利率为r，经过t期的年 金现值，可据此编制“年金现值系数表” （见附表A-3）
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多笔不等额现金流量现值的计算
年金和多笔不等额现金流量混合情况下的 现值 计息期短于1年时间价值的计算
贴现率的计算
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年金的分类
•普通年金 （后付年金） •预付年金 （先付年金） •递延年金 m期 •永续年金
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0
1 A 1 A
2 A 2 A n期
3 A 3
0 A
未来
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FIGURE 6.1
先付年金的终值
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FIGURE 6.4
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1.普通年金的终值
A A(1+r)
FVAt A A1 r A1 r
推广（N次）： FVA A A 1 r
FIGURE 6.2
普通年金的终值
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FIGURE 6.3
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多笔现金流量的现值和终值
A1, A2, A3, A4,
r
t FVA
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•递延年金 m期
n期
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永续年金的现值 由普通年金的现值
1 (1 r ) t PVAt A r

当t
1 永续年金的现值 PVAt A r
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2、预付年金的终值
0 A 1 A 2 A A(1+r) A(1+r)2 A(1+r)3 所以，预付年金的终值=
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3
普通年金的终值×（1+r)
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Chapter Six
Discounted Cash Flow Valuation
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单笔现金流量的现值和终值
Pv
r
t
Fv
多笔现金流量的现值和终值
A1, A2, A3, A4,r
t FVA
所以，在其他条件相同的情况下，一年 内利息复合计算的次数越多，未来值就越大。
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令n=1,则 1年的利息
I1 FV1 PV0
m r PV0 1 1 m m r PV0 1 1 m I1 r PV0 PV0