var y 2 > @ 91 91 aa 1) a.jbeckley/q/wd/stat472...1 11.996 1 (11.756) 0.991473 1.06 1 1 11.756 1...
TRANSCRIPT
1)
a.
b.
c.
d.
2291 91
2
2 4 6 82
91
2 3 4
91
2
2
180 288 216 2160.82349
900
180 288 216 2160.90668
900
0.82349 (0.90668)0.96334
0.04
1.04
A AVar Y
d
v v v vA
v v v vA
Var
e.
f.
Var Yéë ùû =
2A91:3
- A91:3
( )2
d2
2A91:3
=180v +288v4 + 432v6
900= 0.83780
A91:3
=180v +288v2 + 432v3
900= 0.91488
Var =0.83780-(0.91488)2
0.04
1.04
æ
èçö
ø÷
2= 0.53197
g.
22| 91 2 91 93
432 2161 0.65715
900 432a E a v v
h.
91
9191
0.041 (0.90668)1 ln(1.04)1
1.9200ln(1.04)
iA
Aa
i.
Var Yéë ù
û =
2
A91 - A91( )2
d 2=
1+ i( )2
-1é
ëêù
ûú
2d
2A91
-i
dA
91
æ
èçö
ø÷
2
d 2
2
2
(1.04) 1 0.04(0.82349) (0.90668)
2ln(1.04) ln(1.04)
ln(1.04)Var
Var =1.03247
j.
k.
2)
a.
(straight from Table)
b.
c.
10| 60 10 60 70( ) (0.45120)(8.5693) 3.86647a E a
d.
e.
808080
11 1 (1.02971)(0.66575)5.39628
0.05827
i AAa
f.
50 20 50 70 20 5050:20
50:20
50 20 50 70 20 50
50:20
50:20
50:20
1 ( )1
1 ( ) ( )
1 (1.02971)(0.24905) (0.23047)(1.02971)(0.51495) 0.23047
0.05827
10.90248
i i
A E A EAa
A E A Ea
a
a
g.
10 60 701060:10
10
60:10
( )
1 (1.06)(1.06) (0.45120)(8.5693) 11.6682
0.06
a a E a
a
h.
13 60 731360:13
1313 73
7360:1360
13
60:13
( )
1 (1.06)(1.06) (1.06)
0.06
5,920,3949.3838 (1.06) ( )(7.7568) 12.0134
8,188,074
a a E a
la a
l
a
i.
(12)(12)
60(12) 6060 (12) (12)
(12)60
11
1 (1.02721)(0.36913)10.6800
0.05813
i
iAA
ad d
a
j.
(12)60 60(12) (12) 1.00028(11.1454) 0.46812 10.6804a a
k.
1 159 602 2
2(12)60 60 3
(12)60
ln( ) ln( ) ln(1 0.01262) ln(1 0.01376) 0.013278
12 1 12 1( )
2(12) 12
11 14311.1454 (0.05827 0.013278) 10.6811
24 1728
x
x
p p
a a
a
l.
2A60
-(A60
)2
d2=
0.17741-(0.36913)2
(0.061.06
)2=12.8443
m.
2A60:20
-(A60:20
)2
d2
2 2 20 2 2060 20 60 80 20 6060:20
2 20 201 11.06 1.0660:20
( )
0.17741 ( ) (0.14906)(0.47359) ( ) (0.14906) 0.20188
A A v E A v E
A
60 20 60 80 20 6060:20
60:20
( )
0.36913 (0.14906)(0.66575) 0.14906 0.41895
A A E A E
A
Var =0.20188-(0.41895)2
(0.061.06
)2= 8.22550
3)
50 7020 50
1 1100 400
1 (1.02971)(0.24905) 1 (1.02971)(0.51495)100 (400)(0.23047) 2019.23
0.05827 0.05827
A AE
4)
(12)5
65(12)5 60(12) (12)60:5
11(12)(800) (12)(800)
1 (1.02721)(0.4398)9600 4.34787 (0.68756) 103,990.63
0.05813
i
iAv
a Ed d
5)
a.
1 2 21 11 2 1.06 1.0654 54 54 54 54 154 :3
2
1 1 1 ( ) 1 1
0.96 (0.96)(0.945)1 2.71307
1.06 (1.06)
a v p v p q q q
b.
2 31 2 354 54 5454 :3
2 3
0.96 (0.96)(0.945) (0.96)(0.945)(0.924)2.41688
1.06 (1.06) (1.06)
a v p v p v p
6)
90 90 91
0.851 1 (3.4611) 3.77541
1.06a vp a
7)
110
10 5050:10 50:10
10 10
1
8.2066 1 (0.9195) 7.8277 0.675476
11 0.040014%
0.675476
a E a
v v
i
8)
60 60
61 61
62 62
60 60 61 60 60
61 61 62 61 61
2 60 60 61
1 11.996
1 11.756
1 11.509
11 11.996 1 (11.756) 0.991473
1.06
11 11.756 1 (11.509) 0.990647
1.06
( )(
a a
a a
a a
a vp a p p
a vp a p p
p p p ) (0.991473)(0.990647) 0.98220
9)
a.
2 31 2 340 :4
2 3
40 :4
1
1 33,485 1 33,440 1 33,3281 3.66643
1.06 33,519 1.06 33,519 1.06 33,519
x x xa v p v p v p
a
b.
2 3 41 2 3 440 1 40 1 40 1 40 140 1:4
2 3 4
40 1:4
40 1:4
1 33,440 1 33,378 1 33,309 1 33,231
1.06 33,485 1.06 33,485 1.06 33,485 1.06 33,485
3.45057
a v p v p v p v p
a
a
c.
2 3 41 2 3 440 40 40 4040 :4
2 3 4
40 :4
40 :4
( ) 2 3 4
1 33,485 1 33,440 1 33,328 1 33,309( ) 2 3 4
1.06 33,519 1.06 33,519 1.06 33,519 1.06 33,519
( ) 8.37502
Ia v p v p v p v p
Ia
Ia
d.
(IA)40éë
ùû:4
= vd
40éë
ùû
l40éë
ùû
+2v2d
40éë
ùû+1
l40é
ëùû
+3v3d
40éë
ùû+2
l40éë
ùû
+ 4v4 l43
l40é
ëùû
(IA)40éë
ùû:4
=1
1.06(
34
33,519)+2(
1
1.06)2 45
33,519+3(
1
1.06)3 62
33,519+ 4(
1
1.06)4 33,378
33,519
(IA)40éë
ùû:4
= 3.16305
e.
(1000)2(
2A41é
ëùû:4
-(A41éë
ùû:4
)2
d2)
é
ë
êê
ù
û
úú
12
2A41éë
ùû:4
= v2 39
33,467+ v4 50
33,467+ v6 69
33,467+ v8 33,309
33,4672A
41éë
ùû:4
= 0.62812
A41éë
ùû:4
= v39
33,467+ v2 50
33,467+ v3 69
33,467+ v4 33,309
33,467
A41éë
ùû:4
= 0.79251
SD =1000 0.62812-(0.79251)2
0.06
1.06
= 119.1387
f.
Pr( y < 3)= Pr(1- v K +1
d< 3)
Pr( y < 3)= Pr(1- vK +1 < 3(0.06
1.06)= 0.16981)
Pr( y < 3)= Pr(vK +1 > 0.83019)
Pr( y < 3)= Pr(K +1 <ln(0.83019)
ln( 11.06
))
Pr( y < 3)= Pr(K +1 < 3.19)
Pr( y < 3)= Pr(K < 2.19)= 1-3p
x
Pr( y < 3)= 1-l
43
l40éë
ùû
= 1-33,378
33,519= 0.00421
10)
CLYDE
11)
Now using BA-II Plus
SET BGN
IY
= 5
PV = 10.90
PMT = -1
CPT ®N = 15
12)
a.
100,000A
85=100,000( i
d)A
85=100,000(1.02971)(0.73407)= 75,587.92
b.
13)
Die in first year
b1= 15¬1st payment
qx= 0.05¬ probability
Die in year 2
b
2= 15+
20
1.06= 33.868 ¬PV of 2 payments
11q
x=
1p
x(q
x+1)=(0.95)(0.1)= 0.095
Lives 2 year
b3= 15+
20
1.06+
25
1.062= 56.118
2p
x= (0.95)(0.9)= 0.855
E yéë
ùû = 15(0.05)+(33.868)0.095+56.118(0.855)= 51.946
E y2éë
ùû = 152(0.05)+(33.868)20.095+56.1182(0.855)= 2812.81
Var yéë
ùû = E y2é
ëùû-(E yé
ëùû)2
Var yéë
ùû = 2812.81-(51.946)2 = 114.42
14)
s = 10002(12.84432) = 3583.90
Pr( y >11,145.40+3583.90)
= Pr(10001- v K +1
d>14,729.30)
= Pr(1- v K +1
d>14.73)
= Pr[1- v K +1 >(14.73)( 0.061.06
)]
= Pr[v K +1 <1-(14.73)( 0.061.06
)]
= Pr[v K +1 < 0.16627]
= Pr[K +1 >ln(0.16627)
ln( 11.06
)]
= Pr[K +1 > 30.79]
= Pr[K > 29.79]
=30
p60
=l
90
l60
= 0.12927
15)
Where K is measured in months, and v is (
1
1.06)
112
1+ i = (1-d(12)
12)-12
d(12)
12= 1-(1.06)
- 112 = 0.048440
Pr( y >12,000)
= Pr 100(1- v K +1
d(12)
12
)>12,000é
ëê
ù
ûú
= Pr (1- vK +1
d(12)
12
)>120é
ëê
ù
ûú
= Pr[1- v K +1 > 0.58128]
= Pr[v K +1 < 0.41872]
= Pr[K +1 >ln(0.41872)
ln[( 11.06
)1
12 ]]
= Pr[K +1 >179.28]
= Pr[K >178.28]
= Pr of living 14 years 11 months
=l5411
12
l40
=1
12(l
54)+ 11
12(l
55)
l40
=8,646,841
9,313,166= 0.9285