金融数学引论答案第一章__北京大学出版[1] 下载本文

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第一章习题答案

1.解: 把t = 0 代入得A(0) = 3 于是:a(t) =A(t)/A(0)=(t2 + 2t + 3)/3 In = A(n) ? A(n ? 1)

= (n2 + 2n + 3) ? ((n ? 1)2 + 2(n ? 1) + 3)) = 2n + 1

2. 解:?1?I?A(n)?A(t)?In?In-1?????It?1?n(n? 1)/2?t(t? 1)/2 (2)I?A(n)?A(t)? 3.解: 由题意得

a(0) = 1, a(3) =A(3)/A(0)= 1.72? a = 0.08, b = 1 ∴ A(5) = 100

A(10) = A(0) ? a(10) = A(5) ? a(10)/ a(5)= 100 × 3 = 300.

4. 解:(1)i5 =(A(5) ? A(4))/A(4)=5120≈ 4.17% i10 =(A(10) ? A(9))/A(9)=5145≈ 3.45% (2)i5 =(A(5) ? A(4))/A(4)

k?t?1?Ink? 2n?1?2t?1

100(1 ? 0.1)5?100(1 ? 0.1)4?? 100(1 ? 0.1)4i10?(A?10??A?9?)/A?9??100(1 ? 0.1)?100(1 ? 0.1)? 100(1 ? 0.1)9109

5.解:A(7) = A(4)(1 + i5)(1 + i6)(1 + i7) = 1000 × 1.05 × 1.06 × 1.07 = 1190.91

6.解: 设年单利率为i

500(1 + 2.5i) = 615 解得i = 9.2%

设500 元需要累积t 年 500(1 + t × 7.8%) = 630 解得t = 3 年4 个月

7.解: 设经过t 年后,年利率达到2.5%

1 ? 4%?t? (1 ? 2.5%)t t ≈ 36.367 8. 解:(1 + i)11 = (1 + i)5+2*3 = XY 3 9. 解: 设实利率为i 600[(1 + i)2 ? 1] = 264 解得i = 20%

∴ A(3) = 2000(1 + i)3 = 3456 元 10.解: 设实利率为i

11??1 n2n(1?i)(1?i)解得(1 + i)-n =5?1 25?1?23?5 )?22所以(1 + i)2n = (11.解:由500×(1 + i)30 = 4000 ? (1 + i)30 = 8 于是PV =

100001000010000??

(1 ?i)20(1 ?i)40(1 ?i)60?23?43 = 1000 × (8?8?8?2) = 3281.25

12解:(1 + i)a = 2 (1)

3(1 + i)b = (2)

2(1 + i)c = 5 (3)

3(1 + i)n = (4)

2(4) ? n ? ln (1 + i) = ln 5 ? ln 3 (3) ? ln 5 = c × ln (1 + i)

(1) × (2) ? ln 3 = (a + b) ? ln (1 + i) 故n = c ? (a + b) 13.解:

A ? i = 336

A ? d = 300 i ? d = i ? d ? A = 2800 14.解: (1) d5 ==

a?5??a?4?

a?5?10%

1 ? 5?10%= 6.67%

(2)a-1(t) = 1 ? 0.1t ? a(t) = =

1

1?0.1t? d5 =

a?5??a?4?

a?5?= 16.67%

15.解:由

i(3)3d(4)(?4)(1?)?(1?)34 3(3)?i?d(4)?4?[1?(1?)4]3由

i(6)6d(12)(?12)(1?)?(1?)612 (12)d?i(6)?6?[(1?)?2?1]1216.解: (1) 终值为100 × (1 + i(4)/ 4 )4*2 = 112.65元

(2) 终值为100 × [(1 ? 4d( 1/4 ))1/4 ]-2 = 114.71元 17.解: 利用1/d(m)? 1/i(m) = 1/m? m = 8 18. 解:aA(t) = 1 + 0.1t ? δA(t)

?

a'A(t)0.1?aA(t)1?0.1t(a?1B(t))'0.05?1aA(t)?1?0.05t??B??1?aB(t)1?0.05t

由δA(t) = δB(t)得 t = 5

19.解: 依题意,累积函数为a(t) = at2 + bt + 1a(0.5) = 0.25a + 0.5b + 1 = 1.025 a(1) = a + b + 1 = 1.07 ?a = 0.04 b = 0.03

于是δ0.5 =

a'(0.5)? 0.068 a(0.5)22t?(t) ?, B21?t1?t20.解: 依题意,δA(t) =由?A(t)??B(t) ?

2t2? 1 ?t21 ?t? t > 1

21.解: d?4?? 8%,设复利下月实贴现率为d,单利下实利率为d0。 __________全部采用复利:

(1?d)3? 1?8% 2PV? 5000(1?d)25? 4225.25前两年用复利:

1?3d0? 1?8% 2PV? 5000(1?d)24(1?d0) ? 4225.46

4 22.解: i??? 6%,则i? (1 ?6%4)?1 ? 6.14% 4设第3年初投入X,以第3年初为比较日,列价值方程

2000(1 ?i)2? 2000(1 ?i) ?X? 2000v2? 5000v8解得X = 504.67 元 23.解: 对两种付款方式,以第5年为比较日,列价值方程:

200 ? 500v5? 400.94解得v5? 0.40188 所以

P? 100(1 ?i)10? 120(1 ?i)5? 917.762

24.解:1000?1 ? 6%?? 2?1000?1 ? 4%?解得: t = 36 年

tt25.解: 列价值方程为100vn? 100v2n? 100解得n = 6.25 26.解:?t?1,得基金B的积累函数为 6t