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