T1 MHA, C1 xdxdx lxlx d x ÷ l x =Prob. of payment νnνn PV of payment =Prob. ×νn 25318179,627318...

36
T MHA, C1 x d x l x d x ÷ l x =Prob. of payment ν n PV of payment =Prob.×νn 25 318 179,627 318 ÷ 179,627= 0.00177 0.956938 = 0.001694 26 310 179,309 310 ÷ 179,627= 0.00173 0.915730 = 0.001584 27 306 178,999 306 ÷ 179,627= 0.00170 0.876297 = 0.001490 28 304 178,693 304 ÷ 179,627= 0.00169 0.838561 = 0.001417 29 305 178,389 305 ÷ 179,627= 0.00170 0.802451 = 0.001364 =0.007549 PV of Collected Premiums = PV of Paid Benefits The NSP for a $100,000 , 5 yrs Term for a 25 yr old NSP for the amount of $100,000=100,000 × 0.007549 = $754.9

Transcript of T1 MHA, C1 xdxdx lxlx d x ÷ l x =Prob. of payment νnνn PV of payment =Prob. ×νn 25318179,627318...

T1MHA, C1

x dx lx dx ÷ lx

=Prob. of payment

νn PV of payment

=Prob.×νn

25 318 179,627 318 ÷ 179,627= 0.00177 0.956938 = 0.001694

26 310 179,309 310 ÷ 179,627= 0.00173 0.915730 = 0.001584

27 306 178,999 306 ÷ 179,627= 0.00170 0.876297 = 0.001490

28 304 178,693 304 ÷ 179,627= 0.00169 0.838561 = 0.001417

29 305 178,389 305 ÷ 179,627= 0.00170 0.802451 = 0.001364

=0.007549

PV of Collected Premiums = PV of Paid Benefits

The NSP for a $100,000 , 5 yrs Term for a 25 yr old

NSP for the amount of $100,000=100,000 × 0.007549 = $754.9

T2MHA, C1

x dx lx dx ÷ lx

= Prob. of payment

νn PV of payment

=Prob.×νn

25 318 179,627 318 ÷ 179,627= 0.00177 0.956938 = 0.001694

26 310 179,309 310 ÷ 179,627= 0.00173 0.915730 = 0.001584

27 306 178,999 306 ÷ 179,627= 0.00170 0.876297 = 0.001490

28 304 178,693 304 ÷ 179,627= 0.00169 0.838561 = 0.001417

29 305 178,389 305 ÷ 179,627= 0.00170 0.802451 = 0.001364

30 178,084 178,084÷179,627= 0.99141 0.802451

0.795558

NSP for the amount of $100,000=100,000 × $0.803107 = $80,312

The NSP for a $100,000 , 5 yrs Endowment for a 25 yr old

=$0.803107

T3MHA, C1

NSP for the amount of $1,000 = 1,000 × 4.572295 = $4,572.295

The NSP for a $1,000 , 5 yrs Annuity Due for a 25 yr old

x n lx lx +n ÷ lxProb. Of Payment

νn PV of payment=Prob.×νn

25 0 179,627 179,627÷179,627 1.000000 1 = 1.000

26 1 179,309 179,309÷179,627 0.998230 0.956938 = 0.955244

27 2 178,999 178,999÷179,627 0.996504 0.91573 = 0.912529

28 3 178,693 178,693÷179,627 0.994800 0.876297 = 0.871740

29 4 178,389 178,389÷179,627 0.993108 0.838561 = 0.832782

4.572295

T4MHA, C1

NAP for a 5 yrs Term Ins. for a 25 yrs for the

amount of $100,000 =

NSP ($754.9) ÷ NAP for an annuity due for the

same period (5yrs) for the amount of $1:

NAP for 100,000=

NSP 100,000 ÷ NAP for an annuity of $1

NAP = $754.9 ÷ 4.572295 = $165.103

The NAP for a $100,000 , 5 yrs Term for a 25 yr old

T5MHA, C1

Commutations Functions

{ x x x vlD ×=

{ x x x vlD 111 +×++ =

{ 121 _ωDxD xDxDxN ++++++=

{ nDxD xDxDnxNxN 121_ _+++++++ =

ωDnxDnxDnxDnxN 11 2 _+++++++++=+

{ nx nx nx vlD +×++ =

vlD 1_1_1_ ×=

T6MHA, C1

Cont’ Commutations Functions

{ x x x vC d 1+= ×

{ x x x vC d 211 ++=+ ×

{ nnn x x x vC d 1+++=+ ×

{ 121 _ωCxC xCxCxM ++++++=

{ ωCnxCnxCnxCnxM 121 _+++++++++=+

{ nxCxCxCxCnxM_xM 1_21 +++++++=+

{ vC d ×= 1_1_

T7MHA, C1

Pure endowment

Pays the amount of ins. If the insured reaches a specific age.

nxni)(xEn llx 1

nx nVxEn ll x +×=×

x

nx xn D

DE

v xlv xlE n nx nxxn +++× ×× =

nx xxn DDE +=×

T8MHA, C1

Ex 1: What is the net single premium for an 18-year-old man for a 20-year pure endowment of $10,000?

Examples

1139935824022329040001000010

18

38 .$..,,

D

D

T9MHA, C1

Immediate whole Life Annuity Due

provides annual payments w first payment starts at once, as long as annuitant is living:

xvlvlvlxl x xxxla _99 992211 ++++++=

99992211 vlvlvlxvxl x xxxxxxvla ++++++++=

x

xDN

xa

xDD x D x Dx D

xa 9921 ++++++=

1_321 ..... DDDDDD xxxxxxa +++++=× +++

T10MHA, C1

provides annual payments w first payment starts @ the end of first period , as long as annuitant is living,

Immediate whole Life Annuity Ordinary

xx vlvlvlvlxl x xxa _9999_98982211 ++++++=

999998982211 vlvlvlvlxvxl x xxxxa ++++++++=

xx

DN

xa 1+=

xDDD x D x D

xa 999821 ++++++=

999821 DDDDD x x xxa ++++ ++=×

T11MHA, C1

Deferred whole Life Annuity Due

provides annual payments @ beginning of each period w first payment starts after more than one period, as long as annuitant is living:

xvlvvx x mm mxmx lllam _991 991/ +++= +++++

999911/ vlvlxvx xm mxmxmxmx vlla +++= ++++++

xmx

DN

xam +=/

xDDm xDmxDm xD

xam 9921/ +++++++++=

9921/ DmxDmxDm xDxDxm a +++++++++=×

T12MHA, C1

Deferred whole Life Annuity Ordinary

provides annual payments @ end of each period w first payment starts after more than one period, as long as annuitant is living:

xvlvvx x mm mxmx lllam _992211 99/ +++= +++++++

99992211/ vlvlxvx xm mxmxmxmx vlla +++= +++ +++++

xmx

DN

xam 1/ ++=

xDDDDD mxmx

xam 999821/ ++++= ++++ 999821/ DD DDxDxm mxmxa ++++=× ++++

T13MHA, C1

Ex 4: Find the net single premium for a whole life annuity due of $10,000 per year issued to a man who is 50 years old.

Examples

00010 ,xx

DN

xa ×=

5395914800010318453

527487900010

50

50 .,,.

.,D

N =×=×=

T14MHA, C1

Ex 5: Find the net single premium for a whole life annuity ordinary (at the end of each yr) of $10,000 per year issued to a man who is 18 yrs old (or starts @19).

Examples

000101 ,x

xD

Nxa ×= +

66.97361000,55.82402

9.1604568000,10

18

19 =×=×= DN

T15MHA, C1

Ex 6: A man dies and leaves his widow a $200,000 insurance policy. If she is 60 yrs old and elects to receive annual payments for life with the first payment to be made now, what is the size of each payment?

Examples

RDN

x xxa ×=

RDN

, ×=60

60000200

R.

., ×=

33995

354147000200

37571455274913

000200.,

.

,R ==

or:

T16MHA, C1

Examples

Ex 9: A man 40 yrs old, wants a life income of $15,000 a year with the first payment to be made when he is 65. What is the net cost of this annuity?

40

65000,15D

Ncost Net ×=

63.064,40$8.299696.80048

000,15 =×=

T17MHA, C1

Ex 11: Find the net cost of a $10,000-a-year whole life annuity for a 50 - year-old woman if the first payment is to be made when she is: (a) 50. (b) 51. (c) 62.

Examples

1111620001050

5000010366485107776 ,$,,

..

DN

csot Net (a) =×=×=

1111520001050

510001036648

1101128 ,$,, ..

DN

csot Net (b) =×=×=

637396936648

463650001050

6200010 .,$.

,, DN

csot Net (c) =×=×=

T18MHA, C1

Examples

Ex 12: How much would a man aged 18 have to pay for a 5-year temporary life annuity of $10,000 per year if the first payment is to be made at the end of each year. (first payment when he is 19)?

18

2419518

_00010

D

NN, : a ×=

8666343582402

812447679160456800010 .,$

...

, =×=-

T19MHA, C1

Ex 13: How much would a 5 year temporary life annuity due of $10,000 per year cost a man who is 18 years old?

Examples

,D

N-Nn :x ä

18

231800010 ×=

0471445582402

313102764168697100010 .,$

..-.

, =×=

T20MHA, C1

Examples

Ex 14: How much would a 5-year temporary life annuity of $10,000 a year cost a man aged 18 if he is to receive the first payment when he is 28?

18

3328000109 518\D

NN,: a -

×=

0991128582402

77725623101079700010 .,$

...

, =×=

18

332800010518\10D

NN,: ä -

×=

T21MHA, C1

Ex 15: Find the net cost of a $20,000-a-year temporary life annuity for 10 years for a 50-year-old woman if the first payment is to be made when she is:

(a) 50. (b) 51. (c) 62.

Examples

T22MHA, C1

Examples

Answer:

50

6050

DNN

20,000 cost Net (a)-

×=

7$161,332.06648.3

54147.3107776.520,000 =×=

-

50

6151

DNN

20,000 cost Net (b) ×=

8$153,350.76648.3

50152101128.120,000 =×=

T23MHA, C1

Examples

Answer:

$84,205.296648.3

18373.94636520,000 =×=

_

50

7262

D

NN20,000 cost Net (c)

_×=

T24MHA, C1

Ex 16: A woman has $15,000. If she was 40 years old and used the amount to buy a temporary life annuity of 10 annual payments. What would be the size of each payment if the first payment is made when she is:

(a) 40? (b) 41? (c) 50 years old?

Examples

T25MHA, C1

Answer:

Examples

40

5040

D

NNR 15,000 (a)

-×=

5040

40

NN

D15,000 R

-×=

$1836.68107776.5195080

10689.915,000 =×=

-

T26MHA, C1

Answer:

Examples

40

5141

D

NNR 15,000 (b)

-×=

5141

40

NN

D R 15,000

-×=

$1925.83101128.1-184390.1

10689.915,000 =×=

T27MHA, C1

Answer:

Examples

40

6050

D

NNR (c) 15,000

-×=

6050 NN

D R 4015,000

-×=

$2989.9554147.3-107776.5

10689.915,000 =×=

T28MHA, C1

Ch 13

Ex 1: Find the net single premium for a $10,000 whole life policy for an 18-year-old man.

16184100010582402

801975700010

18

18 .,,.

.,

DM

xDxM

xA =×=×==

T29MHA, C1

Ch 13

Ex 2: A man retires at age 65. He has been carrying a $20,000 straight life policy that now has a cash value of $11,400. How much paid-up insurance will this buy?

xDxM

xA =

YDM

, ×=65

6540011

Y.

., ×=

57794

424434740011

07043944244347

5779440011 .,

.

.,Y =×=

T30MHA, C1

Ch 13

Ex 3: Find the net annual premium for a $10,000 whole life policy for an 18-year-old man.

$57.844.1686971

801.5757000,10000,10

18

1818 =×=×=

N

MP

T31MHA, C1

Ch 13

Ex 4: Find the net single premium & the net annual premium for a 5-year, $10,000 term policy for a man aged 18.

18

23181518

0001000010D

MM,,

: A NSP

-×=×=

6481582402

0929085801975700010 .$

.

.-., =×=

2318

23181518

0001000010NN

MM,,

: P NAP

-

-×== ×

86173131027641686971

0929085801975700010 .$

..

.-., =×=

-

T32MHA, C1

Ch 13

Ex 5: Find the net annual premium for a $10,000, 20-payment life policy issued to an 18-year-old man.

3818

181820 00010 NN

M,P

-×=

4.8858316341686971

801975700010 $

-..

, =×=

T33MHA, C1

Ch 13

Ex 6: Find the net annual premium for a $10,000 paid-up-at-age-65 policy issued to an 18-year-old man.

6518

18000,10NN

M

-×=

$60.72

6.800484.1686971801.9757

000,10 =×=-

T34MHA, C1

Ch 13

Ex 7: Find the net single premium for a $10,000, 20-year endowment policy issued to a man aged 18.

$4231.67= 82402.5

32904.2+7791.9669757.801×10,000=

-

18

3838182018

000,10D

DMMA

:

+×=

-

T35MHA, C1

Ch 13

Ex 8: Find the net annual premium for a $10,000, 20-year endowment policy issued to a man aged 18.

NN

DMM,P : 3818

3838182018 00010

-

- += ×

19.315$058316341686971

2329049667791801975700010 =

+=

..

...,

-

-

T36MHA, C1

Ch 13

Ex 9: Find the net annual premium for a $50,000 endowment policy at age 65 issued to an 18-year-old woman.

, NNDMM

: P6518

6565184718 00050

-

- +×=

613807361726608900303132714731192802

00050 .$ .-.

.-., =

+×=