Mortality Studies GPV Workshop
Actuarial Experience Studies:
Mortality
By
Luc St-Amour
July 19, 2012
Mortality Experience Studies - Content
•
•
•
•
•
Use of Mortality Experience
Source of Mortality Studies
Best Estimate Assumptions
Margin for Adverse Deviation
Experience Studies
•
•
•
Exposure Measurement
Credibility
Additional / Special Considerations
2
Use of Mortality Experience
•
•
•
Valuation
Pricing
Products without mortality Risk?
•
•
•
Still needed as mortality could be a contingency with respect to persistency
Life insurance
Life annuity
3
Source of Mortality Studies
•
Industry Studies / Mortality Tables
•
•
Representative of your Company’s experience?
Indonesia vs Other Countries (e.g. US Society of Actuaries (SOA) mortality tables)
•
•
Population Mortality
•
•
•
Not representative of insured mortality (re: underwriting)
Reinsurers’ experience
Own experience
•
•
•
May not be representative due to differences in underwriting standards, socio economic factors, etc
Most Relevant
Credibility
Important to understand the experience underlying the mortality table
•
•
•
•
•
Insured mortality
Basis of the table (e.g. SOA tables are valuation tables and include “built in” margins)
Period of observation – experience may be dated and mortality improvements / deterioration may be
required to make the table current
Care needed to ensure that mortality adjustments are appropriate (for example, when using population
mortality as the source for mortality improvements)
Impact of distribution system (e.g. agency vs brokerage)
4
Best Estimate Assumption – Data Differentiation
•
•
•
•
•
•
•
Life insured’s/Annuitant’s age, sex, smoking habit, health, lifestyle
Duration since issue of policy
Plan of insurance (e.g. term, whole life) and its benefit provided
Company’s underwriting practice
Size of policy
Company’s distribution system (e.g. agency, brokerage, etc.) and marketing
practice
Each “cell” should be as homogeneous as possible
5
Margin for Adverse Deviation(MfAD)
•
•
•
•
MfAD should increase the liability – requires testing of “direction” of MfAD (i.e. increasing or
reducing the best estimate assumption)
Size of MfAD reflects the degree of uncertainty of the best estimate assumption
Uncertainty relates to misestimation of and deterioration from the best estimate assumption
Canadian Standard for life insurance: K / ex per 1,000 lives
•
•
•
Canadian Standard for annuities
•
•
•
•
High Margin: K = 15
Low Margin: K = 3.75 (i.e. 25% of High Margin)
High Margin: -20%
Low Margin: -5%
Selection of High or Low margin depends on “Significant Considerations”
In the presence of a Significant Consideration, the MfAD must be at least average of low and
high margin
6
Margin for Adverse Deviation(MfAD) - continued
•
Significant Considerations – Misestimation of best estimate assumption:
•
•
•
•
•
Significant Consideration – Deterioration of best estimate assumption:
•
•
•
•
•
Low credibility of the Company’s own experience
Lack of homogeneity
Unrefined method used to determine best estimate assumption (e.g. using single equivalent age for joint policies instead of
each individual age)
Change in underwriting practice in the Company
Anti-selection is present (e.g. re-entry products; underwriting, sales force)
Unfavourable mortality developments have emerged (e.g. AIDS)
Persistency rate of product is low
Premium structure does not recognize mortality differentials as precisely as the rest of the market (e.g. unisex rates; no
distinction between smokers/non-smokers)
Similar considerations apply to annuity business
•
But favourable mortality developments would be important consideration
7
Experience Studies – Best Estimate Assumption
•
•
qx = Deaths / Exposure
Exposure to risk
•
•
Deaths refer to Actual Claims
•
•
•
•
Critical Element of experience studies
Reconcile to Claims registry/P&L
Use annuity payments for annuitant mortality experience
Compute based on count (i.e. number of deaths) and amount
Annual Studies, but use several years’ experience
8
Exposure - Measurement
•
•
•
•
Study Period
Include all policies exposed to risk during the period
Need to determine entry point and exit point
Entry Point:
•
•
•
Exit Point:
•
•
•
End of exposure period for policies in-force at the end of the study period
Date of termination for other policies
Cause of Termination is an important consideration
•
•
•
•
Start of exposure period for policies in-force at the beginning of the study period
Issue date of policy for new business
Death claims exposed to the end of the year of death
Lapses/surrenders – exposed to the date of termination
This would be reversed if doing lapse study: lapsed policies exposed for the full period but terminations due to death/surrender
exposed only date of termination
Treatment of late reported claims?
•
•
Included in the year of study
Best to perform study a few months after close of observation period to reduce the number of late reported claims
9
Exposure – Measurement
0
r
s
New Entrants
Withdrawals
n lives, aged x+r
w lives, aged x+s
1
Beginning
Ending
A lives, age x
B lives, aged x+1
D deaths during the period
B=A+n–w-D
10
Exposure – Measurement
0
r
s
New Entrants
Withdrawals
n lives, aged x+r
w lives, aged x+s
1
Beginning
Ending
A lives, age x
B lives, aged x+1
D deaths during the period
D = A * qx + n * 1-r qx+r – w * 1-s qx+s
11
Exposure – Mortality for periods less than 1 year
•
•
•
Balducci formula:
0≤t≤1
1-t q x+t = (1-t) * q x,
Simple assumption to compute exposure
Alternate Formulas:
• Uniform distribution of deaths
0≤t≤1
• tq x = t * q x,
• 1-t q x+t = (1-t) * q x, / (1-t*qx)
•
•
Constant force of mortality
t
•
tq x = 1- (1-qx) ,
(1-t) ,
•
1-tq x+t = 1- (1-qx)
0≤t≤1
0≤t≤1
0≤t≤1
Referring to general formula “D = A * qx + n * 1-r qx+r – w * 1-s qx+s “, Balducci is simplest to use in
exposure formula
12
Exposure – Measurement
0
r
s
New Entrants
Withdrawals
n lives, aged x+r
w lives, aged x+s
1
Beginning
Ending
A lives, age x
B lives, aged x+1
D deaths during the period
D = A * qx + n * 1-r qx+r – w * 1-s qx+s
Using Balducci,
D = A * qx + n * (1-r) * qx – w * (1-s) qx
D = qx * [ A + n * (1-r) – w * (1-s) ]
With qx = D / E, we have
E = A + n * (1-r) – w * (1-s)
13
Exposure – Measurement – Example 1
0
r
s
New Entrants
Withdrawals
n lives, aged x+r
w lives, aged x+s
1
Beginning
Ending
A lives, age x
B lives, aged x+1
D deaths during the period
In the above observation period, we have:
A = 1,000
n = 40
w = 30
r = 1/4
s = 2/3
B = 990
Assuming Balducci, Solve for qx,
14
Exposure – Measurement – Example 1 - Solution
0
r
s
New Entrants
Withdrawals
n lives, aged x+r
w lives, aged x+s
1
Beginning
Ending
A lives, age x
B lives, aged x+1
D deaths during the period
Solution:
First Step: Determine Number of Deaths D
B=A+n–w–D
D=A+n–w-B
D = 1,000 + 40 – 30 - 990
D = 20
15
Exposure – Measurement – Example 1 - Solution
0
r
s
New Entrants
Withdrawals
n lives, aged x+r
w lives, aged x+s
1
Beginning
Ending
A lives, age x
B lives, aged x+1
D deaths during the period
Solution:
Second Step: Determine Exposure assuming Balducci
E = A + (1-r) * n – (1-s) * w
E = 1,000 + (1-1/4) * 40 – (1-2/3) * 30
E = 1,020
16
Exposure – Measurement – Example 1 - Solution
0
r
s
New Entrants
Withdrawals
n lives, aged x+r
w lives, aged x+s
1
Beginning
Ending
A lives, age x
B lives, aged x+1
D deaths during the period
Solution:
Third Step: Calculate qx
qx = D / E
qx = 20 / 1,020
qx ≈ 0.0196
17
Example
•
•
•
Calculate Exposure contributed for each of the following lives.
Observation period begins March 1, 2000.
Balducci hypothesis is assumed.
Case
Birth Date
Other Facts
1
Oct. 1, 1980
Withdrew Feb. 1, 2002
2
Dec. 1, 1981
Died Apr. 1, 2003
3
Apr. 1, 1979
Died May 1, 2004
4
Mar. 1, 1980
Died Apr. 1, 2000
5
Nov. 1, 1980
Withdrew Aug. 1, 2004
18
Example 2 - Solution
Case
Birth Date
Other Facts
1
Oct. 1, 1980
Withdrew Feb. 1, 2002
Exposure Period - Calendar Year
Number of
Months Description
Year
2000
10 Mar 1 - Dec 31
2001
12 Jan 1 - Dec 31
2002
1 Jan 1 - Jan 31
Total
Age at Entry
Exit
Feb 1, 2002
Entry
Mar 1, 2000
23
19 5/12
Oct 1, 1999
Oct 1, 2000
Oct 1, 2001
Oct 1, 2002
Exposure Period - By Age
Number of
Age
Months
Description
19
7 Mar 1, 2000 - Sept 30, 2000
20
12 Oct 1, 2000 - Sept 30, 2001
21
4 Oct 1, 2001 - Jan 31, 2002
Total
23
19
Example 2 - Solution
Case
Birth Date
Other Facts
2
Dec 1, 1981
Died April 1, 2003
Exposure Period - Calendar Years
Number of
Year
Months Description
2000
10 Mar 1 - Dec 31
2001
12 Jan 1 - Dec 31
2002
12 Jan 1 - Dec 31
2003
11 Jan 1 - Nov 30
Total
Age at Entry
Entry
Mar 1, 2000
Died
Apr 1, 2003
Exit
Dec 1, 2003
45
18 3/12
Dec 1, 1999
Dec 1, 2000
Dec 1, 2001
Dec 1, 2002
Dec 1, 2003
Exposure Period - By Age
Number of
Age
Months Description
18
9 Mar 1, 2000 - Nov 30, 2000
19
12 Dec 1, 2000 - Nov 30, 2001
20
12 Dec 1, 2001 - Nov 30, 2002
21
12 Dec 1, 2002 - Nov 30, 2003
Total
45
Note: Since the termination is by death, we give a full year of exposure in the year of death
20
Example 2 - Solution
Case
Birth Date
Other Facts
3
Apr 1, 1979
Died May 1, 2004
Exposure Period - Calendar Years
Number of
Months Description
Year
2000
10 Mar 1 - Dec 31
2001
12 Jan 1 - Dec 31
2002
12 Jan 1 - Dec 31
2003
12 Jan 1 - Dec 31
2004
12 Jan 1 - Dec 31
2005
3 Jan 1 - Mar 31
Total
Age at Entry
Entry
Mar 1, 2000
Died
May 1, 2004
Exit
Apr 1, 2005
61
20 11/12
Apr 1, 1999
Apr 1, 2000
Apr 1, 2001
Apr 1, 2002
Apr 1, 2003
Apr 1, 2004
Apr 1, 2005
Exposure Period - By Age
Number of
Age
Months Description
20
1 Mar 1, 2000 - Mar 31, 2000
21
12 Apr 1, 2000 - Mar 31, 2001
22
12 Apr 1, 2001 - Mar 31, 2002
23
12 Apr 1, 2002 - Mar 31, 2003
24
12 Apr 1, 2003 - Mar 31, 2004
25
12 Apr 1, 2004 - Mar 31, 2005
Total
61
Note: Since the termination is by death, we give a full year of exposure in the year of death
21
Example 2 - Solution
Case
Birth Date
Other Facts
4
Mar 1, 1980
Died Apr 1, 2000
Exposure Period - Calendar Years
Number of
Months Description
Year
2000
10 Mar 1 - Dec 31
2001
2 Jan 1 - Feb 28
Total
Age at Entry
12
20
Exposure Period - By Age
Number of
Months Description
Age
20
12 Mar 1, 2000 - Feb 28, 2001
Total
Died
Apr 1, 2000
Entry
Mar 1, 2000
Exit
Mar 1, 2001
Mar 1, 2000
Mar 1, 2001
12
Note: Since the termination is by death, we give a full year of exposure in the year of death
22
Example 2 - Solution
Case
Birth Date
Other Facts
5
Nov 1, 1980
Withdrew Aug. 1, 2004
Exposure Period - Calendar Years
Number of
Months Description
Year
2000
10 Mar 1 - Dec 31
2001
12 Jan 1 - Dec 31
2002
12 Jan 1 - Dec 31
2003
12 Jan 1 - Dec 31
2004
7 Jan 1 - Jul 31
Total
Age at Entry
Entry
Mar 1, 2000
Exit
Aug 1, 2004
53
19 4/12
Nov 1, 1999
Nov 1, 2000
Nov 1, 2001
Nov 1, 2002
Nov 1, 2003
Nov 1, 2004
Exposure Period - By Age
Number of
Months Description
Age
19
8 Mar 1, 2000 - Oct 31, 2000
20
12 Nov 1, 2000 - Oct 31, 2001
21
12 Nov 1, 2001 - Oct 31, 2002
22
12 Nov 1, 2002 - Oct 31, 2003
23
9 Nov 1, 2003 - Jul 31, 2004
Total
53
23
Credibility
•
Function of number of claims
•
•
Use to blend own experience with industry experience
•
•
•
•
CIA Educational Note sets 100% credibility at 3,007 death claims (based on Poisson distribution; 90%
confidence interval with 3% margin of error)
Assumption that Industry Tables are 100% credible
Blended qx = Z * own experience + (1-Z) * Industry experience
Z is credibility factor = minimum { (Number of Claims / 3007) ^ (1/2) , 1 }
Examples of Credibility Factors
Number 30
of
Claims
120
271
481
752
1083
1473
1924
2436
3007
Z
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0.10
24
Example 3
•
•
•
Calculate Actual-to-Expected experience by number of lives and by amounts based on the
following data
Calculate Credibility at age 22, assuming 3,007 claims for full credibility
If e52 = 28.89, calculate the Margin for Adverse Deviation (MfAD) for High Margin situations and
total “valuation” mortality at age 52 based on Canadian Actuarial Standards of Practice
Exposure
Age
Actual Claims
1000 qx
Number
Amount
Number
Amount
22
.67
471,797
17,119,826
196
6,685
37
.90
660,127
29,317,759
661
23,777
52
2.91
1,549,048
93,872,023
4,506
228,785
67
16.32
1,098,383
36,781,306
12,720
335,609
82
64.87
534,162
8,686,284
28,878
430,786
25
Example 3 - Solution
•
•
Step 1 – Calculate Expected Claims
Expected Claims = Exposure * q x
Exposure
Age
22
37
52
67
82
TOTAL
Expected Claims
1000 qx
Number
Amount
Number
Amount
0.67
0.90
2.91
16.32
64.87
471,797
660,127
1,549,048
1,098,383
534,162
4,313,517
17,119,826
29,317,759
93,872,023
36,781,306
8,686,284
185,777,198
316
594
4,508
17,926
34,651
57,995
11,470
26,386
273,168
600,271
563,479
1,474,774
26
Example 3 - Solution
•
•
Step 2 – Actual to Expected
A / E = Actual Claims divided by Expected Claims
Actual Claims
Age
22
37
52
67
82
TOTAL
Expected Claims
A/E
Number
Amount
Number
Amount
Number
Amount
196
661
4,506
12,720
28,878
46,961
6,685
23,777
228,785
335,609
430,786
1,025,642
316
594
4,508
17,926
34,651
57,995
11,470
26,386
273,168
600,271
563,479
1,474,774
62%
111%
100%
71%
83%
81%
58%
90%
84%
56%
76%
70%
27
Example 3 – Solution
•
•
Calculate Credibility at age 22, assuming 3,007 claims for full
credibility
Solution:
•
Credibility = minimum { (Actual Claims / 3007) ^ 0.5, 100%}
•
Actual Claims [22] = 196
•
Credibility = minimum { (196 / 3007) ^ 0.5 , 100%}
•
Credibility = minimum { 26% , 100%} = 26%
28
Example 3 - Solution
•
•
If e52 = 28.89, calculate the Margin for Adverse Deviation (MfAD) for
High Margin situations and total “valuation” mortality at age 52
based on Canadian Actuarial Standards of Practice
Solution:
•
•
•
MfAD [x] = K / ex , per 1,000, with k = 15 for High Margin situations
MfAD [52] = 15 / 28.89 per 1,000
MfAD [52] = 0.52 per 1,000
•
Total Valuation Assumption = Best Estimate Assumption + MfAD (*)
•
•
Total q 52 = 2.91 per 1,000 + 0.52 per 1,000
Total q 52 = 3.43 per 1,000
(*) This is applicable for life insurance under “normal condition”. For annuities and “death supported” business, the MfAD
would be deducted from the Best Estimate Assumption in order to increase the actuarial liability.
29
Best Estimate Assumption – Additional Considerations
•
Persistency
•
•
Underwriting Practices
•
•
•
Anti-selective lapses
Medical
Non-medical
Mortality Improvement
•
•
•
•
Project current mortality from dated studies
Required for annuity business
Accepted practice in Canada
Typical approach:
• qx,n = qx * (1 – improvement) n, where n is number of years since the study of the
underlying qx was performed
•
•
•
“Back-to-Back” contracts
Annuity Business
Death Supported Mortality
•
May happen with high quota share reinsurance arrangements
30
Example 4
•
•
You are given the following qx for males based on a mortality study centered in 2010
Assuming 2% annual mortality improvement, calculate the qx for a policy issued in 2012 for a
male aged 25 at issue
Age
1000 qx
25
.76
26
.78
27
.80
28
.81
29
.82
Adjusted 1000 qx
31
Example 4 - Solution
•
Step 1 – Determine number of elapsed years since the mortality study
Age
Attained Year
Elapsed Years
25
2012
2
26
2013
3
27
2014
4
28
2015
5
29
2016
6
32
Example 4 - Solution
•
Step 2 – Determine mortality improvement factor by elapsed year since study
Age
Elapsed
Years
Improvement
25
2
Improvement factor [2] = (1-2%) ^ 2 = 0.9604
26
3
Improvement factor [3] = (1-2%) ^ 3 = 0.9412
27
4
Improvement factor [4] = (1-2%) ^ 4 = 0.9224
28
5
Improvement factor [5] = (1-2%) ^ 5 = 0.9039
29
6
Improvement factor [6] = (1-2%) ^ 6 = 0.8858
33
Example - Solution
•
•
Step 3 – Calculate the adjusted q x
Adjusted qx,n = base qx * improvement factor [n], where n is the elapsed number of years since
the mortality study.
Age
Base
1000 qx
Improvement
Factor
Adjusted 1000 qx
25
.76
0.9604
.730
26
.78
0.9412
.734
27
.80
0.9224
.738
28
.81
0.9039
.732
29
.82
0.8858
.726
34
Sources of Information
•
•
Canadian Institute of Actuaries Standards of Practice
Canadian Institute of Actuaries Educational Notes:
•
•
•
Expected Mortality: Fully Underwritten Canadian Individual Life Insurance Policies
Margins for Adverse Deviation
Society of Actuaries Mortality Table Construction (Batten) , Prentice Hall, 1978
35
Mortality
By
Luc St-Amour
July 19, 2012
Mortality Experience Studies - Content
•
•
•
•
•
Use of Mortality Experience
Source of Mortality Studies
Best Estimate Assumptions
Margin for Adverse Deviation
Experience Studies
•
•
•
Exposure Measurement
Credibility
Additional / Special Considerations
2
Use of Mortality Experience
•
•
•
Valuation
Pricing
Products without mortality Risk?
•
•
•
Still needed as mortality could be a contingency with respect to persistency
Life insurance
Life annuity
3
Source of Mortality Studies
•
Industry Studies / Mortality Tables
•
•
Representative of your Company’s experience?
Indonesia vs Other Countries (e.g. US Society of Actuaries (SOA) mortality tables)
•
•
Population Mortality
•
•
•
Not representative of insured mortality (re: underwriting)
Reinsurers’ experience
Own experience
•
•
•
May not be representative due to differences in underwriting standards, socio economic factors, etc
Most Relevant
Credibility
Important to understand the experience underlying the mortality table
•
•
•
•
•
Insured mortality
Basis of the table (e.g. SOA tables are valuation tables and include “built in” margins)
Period of observation – experience may be dated and mortality improvements / deterioration may be
required to make the table current
Care needed to ensure that mortality adjustments are appropriate (for example, when using population
mortality as the source for mortality improvements)
Impact of distribution system (e.g. agency vs brokerage)
4
Best Estimate Assumption – Data Differentiation
•
•
•
•
•
•
•
Life insured’s/Annuitant’s age, sex, smoking habit, health, lifestyle
Duration since issue of policy
Plan of insurance (e.g. term, whole life) and its benefit provided
Company’s underwriting practice
Size of policy
Company’s distribution system (e.g. agency, brokerage, etc.) and marketing
practice
Each “cell” should be as homogeneous as possible
5
Margin for Adverse Deviation(MfAD)
•
•
•
•
MfAD should increase the liability – requires testing of “direction” of MfAD (i.e. increasing or
reducing the best estimate assumption)
Size of MfAD reflects the degree of uncertainty of the best estimate assumption
Uncertainty relates to misestimation of and deterioration from the best estimate assumption
Canadian Standard for life insurance: K / ex per 1,000 lives
•
•
•
Canadian Standard for annuities
•
•
•
•
High Margin: K = 15
Low Margin: K = 3.75 (i.e. 25% of High Margin)
High Margin: -20%
Low Margin: -5%
Selection of High or Low margin depends on “Significant Considerations”
In the presence of a Significant Consideration, the MfAD must be at least average of low and
high margin
6
Margin for Adverse Deviation(MfAD) - continued
•
Significant Considerations – Misestimation of best estimate assumption:
•
•
•
•
•
Significant Consideration – Deterioration of best estimate assumption:
•
•
•
•
•
Low credibility of the Company’s own experience
Lack of homogeneity
Unrefined method used to determine best estimate assumption (e.g. using single equivalent age for joint policies instead of
each individual age)
Change in underwriting practice in the Company
Anti-selection is present (e.g. re-entry products; underwriting, sales force)
Unfavourable mortality developments have emerged (e.g. AIDS)
Persistency rate of product is low
Premium structure does not recognize mortality differentials as precisely as the rest of the market (e.g. unisex rates; no
distinction between smokers/non-smokers)
Similar considerations apply to annuity business
•
But favourable mortality developments would be important consideration
7
Experience Studies – Best Estimate Assumption
•
•
qx = Deaths / Exposure
Exposure to risk
•
•
Deaths refer to Actual Claims
•
•
•
•
Critical Element of experience studies
Reconcile to Claims registry/P&L
Use annuity payments for annuitant mortality experience
Compute based on count (i.e. number of deaths) and amount
Annual Studies, but use several years’ experience
8
Exposure - Measurement
•
•
•
•
Study Period
Include all policies exposed to risk during the period
Need to determine entry point and exit point
Entry Point:
•
•
•
Exit Point:
•
•
•
End of exposure period for policies in-force at the end of the study period
Date of termination for other policies
Cause of Termination is an important consideration
•
•
•
•
Start of exposure period for policies in-force at the beginning of the study period
Issue date of policy for new business
Death claims exposed to the end of the year of death
Lapses/surrenders – exposed to the date of termination
This would be reversed if doing lapse study: lapsed policies exposed for the full period but terminations due to death/surrender
exposed only date of termination
Treatment of late reported claims?
•
•
Included in the year of study
Best to perform study a few months after close of observation period to reduce the number of late reported claims
9
Exposure – Measurement
0
r
s
New Entrants
Withdrawals
n lives, aged x+r
w lives, aged x+s
1
Beginning
Ending
A lives, age x
B lives, aged x+1
D deaths during the period
B=A+n–w-D
10
Exposure – Measurement
0
r
s
New Entrants
Withdrawals
n lives, aged x+r
w lives, aged x+s
1
Beginning
Ending
A lives, age x
B lives, aged x+1
D deaths during the period
D = A * qx + n * 1-r qx+r – w * 1-s qx+s
11
Exposure – Mortality for periods less than 1 year
•
•
•
Balducci formula:
0≤t≤1
1-t q x+t = (1-t) * q x,
Simple assumption to compute exposure
Alternate Formulas:
• Uniform distribution of deaths
0≤t≤1
• tq x = t * q x,
• 1-t q x+t = (1-t) * q x, / (1-t*qx)
•
•
Constant force of mortality
t
•
tq x = 1- (1-qx) ,
(1-t) ,
•
1-tq x+t = 1- (1-qx)
0≤t≤1
0≤t≤1
0≤t≤1
Referring to general formula “D = A * qx + n * 1-r qx+r – w * 1-s qx+s “, Balducci is simplest to use in
exposure formula
12
Exposure – Measurement
0
r
s
New Entrants
Withdrawals
n lives, aged x+r
w lives, aged x+s
1
Beginning
Ending
A lives, age x
B lives, aged x+1
D deaths during the period
D = A * qx + n * 1-r qx+r – w * 1-s qx+s
Using Balducci,
D = A * qx + n * (1-r) * qx – w * (1-s) qx
D = qx * [ A + n * (1-r) – w * (1-s) ]
With qx = D / E, we have
E = A + n * (1-r) – w * (1-s)
13
Exposure – Measurement – Example 1
0
r
s
New Entrants
Withdrawals
n lives, aged x+r
w lives, aged x+s
1
Beginning
Ending
A lives, age x
B lives, aged x+1
D deaths during the period
In the above observation period, we have:
A = 1,000
n = 40
w = 30
r = 1/4
s = 2/3
B = 990
Assuming Balducci, Solve for qx,
14
Exposure – Measurement – Example 1 - Solution
0
r
s
New Entrants
Withdrawals
n lives, aged x+r
w lives, aged x+s
1
Beginning
Ending
A lives, age x
B lives, aged x+1
D deaths during the period
Solution:
First Step: Determine Number of Deaths D
B=A+n–w–D
D=A+n–w-B
D = 1,000 + 40 – 30 - 990
D = 20
15
Exposure – Measurement – Example 1 - Solution
0
r
s
New Entrants
Withdrawals
n lives, aged x+r
w lives, aged x+s
1
Beginning
Ending
A lives, age x
B lives, aged x+1
D deaths during the period
Solution:
Second Step: Determine Exposure assuming Balducci
E = A + (1-r) * n – (1-s) * w
E = 1,000 + (1-1/4) * 40 – (1-2/3) * 30
E = 1,020
16
Exposure – Measurement – Example 1 - Solution
0
r
s
New Entrants
Withdrawals
n lives, aged x+r
w lives, aged x+s
1
Beginning
Ending
A lives, age x
B lives, aged x+1
D deaths during the period
Solution:
Third Step: Calculate qx
qx = D / E
qx = 20 / 1,020
qx ≈ 0.0196
17
Example
•
•
•
Calculate Exposure contributed for each of the following lives.
Observation period begins March 1, 2000.
Balducci hypothesis is assumed.
Case
Birth Date
Other Facts
1
Oct. 1, 1980
Withdrew Feb. 1, 2002
2
Dec. 1, 1981
Died Apr. 1, 2003
3
Apr. 1, 1979
Died May 1, 2004
4
Mar. 1, 1980
Died Apr. 1, 2000
5
Nov. 1, 1980
Withdrew Aug. 1, 2004
18
Example 2 - Solution
Case
Birth Date
Other Facts
1
Oct. 1, 1980
Withdrew Feb. 1, 2002
Exposure Period - Calendar Year
Number of
Months Description
Year
2000
10 Mar 1 - Dec 31
2001
12 Jan 1 - Dec 31
2002
1 Jan 1 - Jan 31
Total
Age at Entry
Exit
Feb 1, 2002
Entry
Mar 1, 2000
23
19 5/12
Oct 1, 1999
Oct 1, 2000
Oct 1, 2001
Oct 1, 2002
Exposure Period - By Age
Number of
Age
Months
Description
19
7 Mar 1, 2000 - Sept 30, 2000
20
12 Oct 1, 2000 - Sept 30, 2001
21
4 Oct 1, 2001 - Jan 31, 2002
Total
23
19
Example 2 - Solution
Case
Birth Date
Other Facts
2
Dec 1, 1981
Died April 1, 2003
Exposure Period - Calendar Years
Number of
Year
Months Description
2000
10 Mar 1 - Dec 31
2001
12 Jan 1 - Dec 31
2002
12 Jan 1 - Dec 31
2003
11 Jan 1 - Nov 30
Total
Age at Entry
Entry
Mar 1, 2000
Died
Apr 1, 2003
Exit
Dec 1, 2003
45
18 3/12
Dec 1, 1999
Dec 1, 2000
Dec 1, 2001
Dec 1, 2002
Dec 1, 2003
Exposure Period - By Age
Number of
Age
Months Description
18
9 Mar 1, 2000 - Nov 30, 2000
19
12 Dec 1, 2000 - Nov 30, 2001
20
12 Dec 1, 2001 - Nov 30, 2002
21
12 Dec 1, 2002 - Nov 30, 2003
Total
45
Note: Since the termination is by death, we give a full year of exposure in the year of death
20
Example 2 - Solution
Case
Birth Date
Other Facts
3
Apr 1, 1979
Died May 1, 2004
Exposure Period - Calendar Years
Number of
Months Description
Year
2000
10 Mar 1 - Dec 31
2001
12 Jan 1 - Dec 31
2002
12 Jan 1 - Dec 31
2003
12 Jan 1 - Dec 31
2004
12 Jan 1 - Dec 31
2005
3 Jan 1 - Mar 31
Total
Age at Entry
Entry
Mar 1, 2000
Died
May 1, 2004
Exit
Apr 1, 2005
61
20 11/12
Apr 1, 1999
Apr 1, 2000
Apr 1, 2001
Apr 1, 2002
Apr 1, 2003
Apr 1, 2004
Apr 1, 2005
Exposure Period - By Age
Number of
Age
Months Description
20
1 Mar 1, 2000 - Mar 31, 2000
21
12 Apr 1, 2000 - Mar 31, 2001
22
12 Apr 1, 2001 - Mar 31, 2002
23
12 Apr 1, 2002 - Mar 31, 2003
24
12 Apr 1, 2003 - Mar 31, 2004
25
12 Apr 1, 2004 - Mar 31, 2005
Total
61
Note: Since the termination is by death, we give a full year of exposure in the year of death
21
Example 2 - Solution
Case
Birth Date
Other Facts
4
Mar 1, 1980
Died Apr 1, 2000
Exposure Period - Calendar Years
Number of
Months Description
Year
2000
10 Mar 1 - Dec 31
2001
2 Jan 1 - Feb 28
Total
Age at Entry
12
20
Exposure Period - By Age
Number of
Months Description
Age
20
12 Mar 1, 2000 - Feb 28, 2001
Total
Died
Apr 1, 2000
Entry
Mar 1, 2000
Exit
Mar 1, 2001
Mar 1, 2000
Mar 1, 2001
12
Note: Since the termination is by death, we give a full year of exposure in the year of death
22
Example 2 - Solution
Case
Birth Date
Other Facts
5
Nov 1, 1980
Withdrew Aug. 1, 2004
Exposure Period - Calendar Years
Number of
Months Description
Year
2000
10 Mar 1 - Dec 31
2001
12 Jan 1 - Dec 31
2002
12 Jan 1 - Dec 31
2003
12 Jan 1 - Dec 31
2004
7 Jan 1 - Jul 31
Total
Age at Entry
Entry
Mar 1, 2000
Exit
Aug 1, 2004
53
19 4/12
Nov 1, 1999
Nov 1, 2000
Nov 1, 2001
Nov 1, 2002
Nov 1, 2003
Nov 1, 2004
Exposure Period - By Age
Number of
Months Description
Age
19
8 Mar 1, 2000 - Oct 31, 2000
20
12 Nov 1, 2000 - Oct 31, 2001
21
12 Nov 1, 2001 - Oct 31, 2002
22
12 Nov 1, 2002 - Oct 31, 2003
23
9 Nov 1, 2003 - Jul 31, 2004
Total
53
23
Credibility
•
Function of number of claims
•
•
Use to blend own experience with industry experience
•
•
•
•
CIA Educational Note sets 100% credibility at 3,007 death claims (based on Poisson distribution; 90%
confidence interval with 3% margin of error)
Assumption that Industry Tables are 100% credible
Blended qx = Z * own experience + (1-Z) * Industry experience
Z is credibility factor = minimum { (Number of Claims / 3007) ^ (1/2) , 1 }
Examples of Credibility Factors
Number 30
of
Claims
120
271
481
752
1083
1473
1924
2436
3007
Z
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0.10
24
Example 3
•
•
•
Calculate Actual-to-Expected experience by number of lives and by amounts based on the
following data
Calculate Credibility at age 22, assuming 3,007 claims for full credibility
If e52 = 28.89, calculate the Margin for Adverse Deviation (MfAD) for High Margin situations and
total “valuation” mortality at age 52 based on Canadian Actuarial Standards of Practice
Exposure
Age
Actual Claims
1000 qx
Number
Amount
Number
Amount
22
.67
471,797
17,119,826
196
6,685
37
.90
660,127
29,317,759
661
23,777
52
2.91
1,549,048
93,872,023
4,506
228,785
67
16.32
1,098,383
36,781,306
12,720
335,609
82
64.87
534,162
8,686,284
28,878
430,786
25
Example 3 - Solution
•
•
Step 1 – Calculate Expected Claims
Expected Claims = Exposure * q x
Exposure
Age
22
37
52
67
82
TOTAL
Expected Claims
1000 qx
Number
Amount
Number
Amount
0.67
0.90
2.91
16.32
64.87
471,797
660,127
1,549,048
1,098,383
534,162
4,313,517
17,119,826
29,317,759
93,872,023
36,781,306
8,686,284
185,777,198
316
594
4,508
17,926
34,651
57,995
11,470
26,386
273,168
600,271
563,479
1,474,774
26
Example 3 - Solution
•
•
Step 2 – Actual to Expected
A / E = Actual Claims divided by Expected Claims
Actual Claims
Age
22
37
52
67
82
TOTAL
Expected Claims
A/E
Number
Amount
Number
Amount
Number
Amount
196
661
4,506
12,720
28,878
46,961
6,685
23,777
228,785
335,609
430,786
1,025,642
316
594
4,508
17,926
34,651
57,995
11,470
26,386
273,168
600,271
563,479
1,474,774
62%
111%
100%
71%
83%
81%
58%
90%
84%
56%
76%
70%
27
Example 3 – Solution
•
•
Calculate Credibility at age 22, assuming 3,007 claims for full
credibility
Solution:
•
Credibility = minimum { (Actual Claims / 3007) ^ 0.5, 100%}
•
Actual Claims [22] = 196
•
Credibility = minimum { (196 / 3007) ^ 0.5 , 100%}
•
Credibility = minimum { 26% , 100%} = 26%
28
Example 3 - Solution
•
•
If e52 = 28.89, calculate the Margin for Adverse Deviation (MfAD) for
High Margin situations and total “valuation” mortality at age 52
based on Canadian Actuarial Standards of Practice
Solution:
•
•
•
MfAD [x] = K / ex , per 1,000, with k = 15 for High Margin situations
MfAD [52] = 15 / 28.89 per 1,000
MfAD [52] = 0.52 per 1,000
•
Total Valuation Assumption = Best Estimate Assumption + MfAD (*)
•
•
Total q 52 = 2.91 per 1,000 + 0.52 per 1,000
Total q 52 = 3.43 per 1,000
(*) This is applicable for life insurance under “normal condition”. For annuities and “death supported” business, the MfAD
would be deducted from the Best Estimate Assumption in order to increase the actuarial liability.
29
Best Estimate Assumption – Additional Considerations
•
Persistency
•
•
Underwriting Practices
•
•
•
Anti-selective lapses
Medical
Non-medical
Mortality Improvement
•
•
•
•
Project current mortality from dated studies
Required for annuity business
Accepted practice in Canada
Typical approach:
• qx,n = qx * (1 – improvement) n, where n is number of years since the study of the
underlying qx was performed
•
•
•
“Back-to-Back” contracts
Annuity Business
Death Supported Mortality
•
May happen with high quota share reinsurance arrangements
30
Example 4
•
•
You are given the following qx for males based on a mortality study centered in 2010
Assuming 2% annual mortality improvement, calculate the qx for a policy issued in 2012 for a
male aged 25 at issue
Age
1000 qx
25
.76
26
.78
27
.80
28
.81
29
.82
Adjusted 1000 qx
31
Example 4 - Solution
•
Step 1 – Determine number of elapsed years since the mortality study
Age
Attained Year
Elapsed Years
25
2012
2
26
2013
3
27
2014
4
28
2015
5
29
2016
6
32
Example 4 - Solution
•
Step 2 – Determine mortality improvement factor by elapsed year since study
Age
Elapsed
Years
Improvement
25
2
Improvement factor [2] = (1-2%) ^ 2 = 0.9604
26
3
Improvement factor [3] = (1-2%) ^ 3 = 0.9412
27
4
Improvement factor [4] = (1-2%) ^ 4 = 0.9224
28
5
Improvement factor [5] = (1-2%) ^ 5 = 0.9039
29
6
Improvement factor [6] = (1-2%) ^ 6 = 0.8858
33
Example - Solution
•
•
Step 3 – Calculate the adjusted q x
Adjusted qx,n = base qx * improvement factor [n], where n is the elapsed number of years since
the mortality study.
Age
Base
1000 qx
Improvement
Factor
Adjusted 1000 qx
25
.76
0.9604
.730
26
.78
0.9412
.734
27
.80
0.9224
.738
28
.81
0.9039
.732
29
.82
0.8858
.726
34
Sources of Information
•
•
Canadian Institute of Actuaries Standards of Practice
Canadian Institute of Actuaries Educational Notes:
•
•
•
Expected Mortality: Fully Underwritten Canadian Individual Life Insurance Policies
Margins for Adverse Deviation
Society of Actuaries Mortality Table Construction (Batten) , Prentice Hall, 1978
35