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