Var Y 2 > @ 91 91 AA 1) a. b.jbeckley/q/WD/STAT472/F20...October 13, 2020 Copyright Jeff Beckley 2018, 2020 f. > @ 2 · ¨¸ ©¹ 2 2 91:3 91:3 2 46 91:3 23 91:3 2 2 180 288 432 0.83780

October 13, 2020 Copyright Jeff Beckley 2018, 2020

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.

22

91:3 91:3

2

4 62

91:3

2 3

91:3

2

2

180 288 4320.83780

900

180 288 4320.91488

900

0.83780 (0.91488)0.53197

0.04

1.04

A AVar Y

d

v v vA

v v vA

Var

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.

22

22291 9191 91

2 2

1 1

2

i iA AA A

Var Y

2

2

(1.04) 1 0.04(0.82349) (0.90668)

2ln(1.04) ln(1.04)

ln(1.04)Var

1.03247Var

j.

k.


2)

a.

60 14.9041a (straight from Table)

b.

60 60 1 13.9041a a

c.

10| 60 10 60 70( ) (0.57864)(12.0083) 6.94848a E a

d.

60 20 60 8060:20( ) 14.9041 (0.29508)(8.5484) 12.38164a a E a

e.

808080

11 1 (1.02480)(0.59293)8.04192

0.04879

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 (1.02480)(0.18931) (0.34824)(1.02480)(0.42818) 0.34824

0.04879

12.51409

i i

A E A EAa

A E A Ea

a

a

g.

10 60 701060:10

10

60:10

( )

1 (1.05)(1.05) (0.57864)(12.0083) 15.05630

0.05

a a E a

a

h.


13 60 731360:13

1313 73

7360:1360

13

60:13

( )

1 (1.05)(1.05) (1.05)

0.05

87,916.89.8633 (1.05) (11.0081) 15.1745

96,634.1

a a E a

la a

l

a

i.

(12)(12)

60(12) 6060 (12) (12)

(12)60

11

1 (1.02271)(0.29028)14.4409

0.04869

i

iAA

ad d

a

j.

(12)60 60(12) (12) 1.00020(14.9041) 0.46651 14.4406a a

k.

1 159 602 2

2(12)60 60 3

(12)60

ln( ) ln( ) ln(1 0.003048) ln(1 0.003398) 0.00322822

12 1 12 1( )

2(12) 12

11 14314.9041 (0.04879 0.00322822) 14.4415

24 1728

x

x

p p

a a

a

l.

2 2 260 60

2 20.051.05

( ) 0.10834 (0.29028)10.61819

( )

A A

d


m.

2 2

60:20 60:202

( )A A

d

2 2 20 2 2060 20 60 80 20 6060:20

2 20 201 11.05 1.0560:20

( )

0.10834 ( ) (0.29508)(0.38134) ( ) (0.29508) 0.17714

A A v E A v E

A

60:20

0.41040A

2

20.051.05

0.17714 (0.41040)3.84192

( )Var

3)

50 7020 50

1 1100 400

1 (1.02480)(0.18931) 1 (1.02480)(0.42818)100 (400)(0.34824) 3254.20

0.04879 0.04879

A AE

4)

(12)5

65(12)5 60(12) (12)60:5

11(12)(800) (12)(800)

1 (1.02271)(0.35477)9600 4.44596 (0.76687) 139,022.11

0.04869

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 (4.8858) 4.9552

1.05a 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.

40 40 1 40 22 3 4 43

40 : 440 40 40 40

2 3 4

( ) 2 3 4

1 34 1 45 1 62 1 33,3782 3 4

1.06 33,519 1.06 33,519 1.06 33,519 1.06 33,519

3

d d d lIA v v v v

l l l l

.16305

e.

122 2

41 : 4 41 : 42

2

2 2 4 6 8

41 : 4

2 3 4

41 : 4

( )(1000)

39 50 69 33,3090.62812

33,467 33,467 33,467 33,467

39 50 69 33,309

33,467 33,467 33,467

A A

d

A v v v v

A v v v v

2

0.7925133,467

1000 0.62812 (0.79251)119.1387

0.06

1.06

SD


f.

11

1

11.06

433

40

1 0.06Pr( 3) Pr 3 Pr( 3) Pr 1 3 0.16981

1.06

Pr( 3) Pr( 0.83019)

ln(0.83019)Pr( 3) Pr 1 Pr( 3) Pr( 1 3.19)

ln( )

33,Pr( 3) Pr( 2.19) 1 1 1

KK

K

x

vY Y v

d

Y v

Y K Y K

lY K p

l

3780.00421

33,519

10)

0.040.96

0.040.96

1 10

1 0.6 1 (10) 0.04

0.04 and ln(1 )

1 0.04

1000 1000( ) 1000 1000 (0.6) 612.415ln(1 )

x x

x x

ix x x

a a

A da d d

i i

A A A


11)

:

:

22.9 |

8 | 20 |

| 12

10.90

xnx n

x x xx n

x

n

a a n a

a a n a n a

n a

a

Now using BA-II Plus

SET BGN

5

10.90

1

15

IY

PV

PMT

CPT N

12)

a.

85 85100,000 100,000( ) 100,000(1.02480)(0.67622) 69,299.03iA A

b.

185 84 852

185 2

85

85 85

1 185 85 852 12

1 185 2 12

85

85

ln(1 ) ln(1 )

ln(1 0.051493) ln(1 0.057665)

0.056130

1

6.7993 ln(1.05) 0.056130

6.29056

100,000 100,000 1 ln(1.05)(6.29056) 69,308.25

q q

A a

a a

a

a

A


13)

Die in first year

1 15 1

0.05

st

x

b payment

q probability

Die in year 2

2

2015 33.868

1.06b PV of 2 payments

1 111 ( ) (0.95)(0.1) 0.095x x xq p q

Lives 2 year

3 2

2

2 2 2 2

2 2

2

20 2515 56.118

1.06 1.06

(0.95)(0.9) 0.855

15(0.05) (33.868)0.095 56.118(0.855) 51.946

15 (0.05) (33.868) 0.095 56.118 (0.855) 2812.81

( )

2812.81 (51.946) 114.42

x

b

p

E Y

E Y

Var Y E Y E Y

Var Y


14)

60

2 22 60 60

2

22

20.051.05

2

1000 14,904.10

( )1000

0.10834 (0.29028)1000

( )

1000 (10.61819)

E y a

A AVar y

d

Var y

Var y

2

1

1

1 0.051.05

1 0.051.05

1

1000 (10.61819) 3258.60

Pr( 14,904.10 (0.5)3258.60)

1Pr(1000 16,533.40)

1Pr( 16.5334)

Pr[1 (16.5334)( )]

Pr[ 1 (16.5334)( )]

Pr[ 0.21270]

ln(0.21270)Pr[ 1

ln(

K

K

K

K

K

y

v

d

v

d

v

v

v

K

11.05

9131 60

60

])

Pr[ 1 31.73]

Pr[ 30.73]

0.38929

K

K

lp

l


15)

Where K is measured in months, and v is

1121

1.05

112

12(12) (12)

(12)

(12)

1 1 1 (1.05) 0.004057512 12

We can also get this by taking from our tables and dividing by 12

0.048690.0040575

12 12

d di

d

d

(12) (12)

112

1 1

12 12

1 1

11.05

Pr( 12,000)

1 1Pr 100 12,000 Pr 120

Pr[1 0.4869] Pr[ 0.5131]

ln(0.5131)Pr 1 Pr[ 1 164.11]

ln[( ) ]

Pr[ 163 .1]

K K

d d

K K

Pr of

Y

v v

v v

K K

K living

812

8 84 453 53 5412 12 12 12

40 40

( ) ( ) (98,181.8) (98,022.4)0.9873

99,33

13 8

8.3

l l l

l l

years months

Documents

Var Y 2 > @ 91 91 AA 1) a. b.jbeckley/q/WD/STAT472/F20...October 13, 2020 Copyright Jeff Beckley 2018, 2020 f. > @ 2 · ¨¸ ©¹ 2 2 91:3 91:3 2 46 91:3 23 91:3 2 2 180 288 432 0.83780