Answer of Questions

Chapter 03: Life tables and selection

1. Define Radix.

Definition:

In life tables, the radix is the initial number of individuals in a cohort at the beginning of the
observation period, typically at age 0. It serves as the baseline for calculating survival and mortality
rates throughout the life table. Notation: Often denoted as l0. Common Value: Usually set to a
round number, like 100,000, to facilitate analysis.

2. Define lx and dx and Relationship between them.

lx (Number of Survivors at Age x):

lx represents the number of individuals who are still alive at the beginning of age x from a starting
group (cohort). dx (Number of Deaths Between Age x and x+1): dx indicates the number of people
who die between ages x and x+1. It is calculated by taking the difference between the number of
survivors at age x and the number of survivors at age x+1. Relationship: dx =lx −lx+1

3. Given l36 = 9734.12, l35 = 9789.29 and l30 = 10,000, Find the probability that a life currently aged
exactly 30 dies between ages 35 and 36.

To find the probability that a life currently aged exactly 30 dies between ages 35 and 36, we can use
the life table values given.

The probability that a person aged 30 dies between ages 35 and 36 can be calculated using the
following formula:

P(35 < T ≤36|T > 30) = [l35 –l36] /l30

where:

- l30 is the number of people alive at age 30,

- l35 is the number of people alive at age 35,

- l36 is the number of people alive at age 36.

Substituting the given values:

- l30= 10,000

- l35 = 9789.29
 - l36 = 9734.12

First, we find l35 –l36

l35 –l36 = 9789.29 - 9734.12 = 55.17

Now we can plug this into the probability formula:

P(35 < T ≤36|T > 30) = {55.17}/{10,000}

Calculating this gives:

P(35 < T ≤36|T > 30) = 0.005517

So, the probability that a life currently aged exactly 30 dies between ages 35 and 36 is approximately
0.005517, or 0.5517%.

4. Given that p35 = 9789.29, calculate 0.2q35.4 under the assumption of a uniform distribution of
deaths.

To calculate 0.2q35.4 under the assumption of a uniform distribution of deaths, we first need to
understand what qx represents. qx is the probability that a person aged x will die before reaching
age x+1.

The notation q35.4 specifically refers to the probability that a person aged 35.4 will die before
reaching age 36.

Given that p35 is the number of people alive at age 35 (which is 9789.29), we can estimate q35.4
using the number of people alive at age 35 and 36.

To find q35.4:

1. Calculate l36 (the number of people alive at age 36) based on a uniform distribution of deaths
between ages 35 and 36.

Using the relationship:

q35.4 = [l35- l36]/l35

Assuming a uniform distribution of deaths:

l35 - l36 = (average number of deaths between ages 35 and 36)

From previous calculations, we have:

l36 = 9734.12
Thus,

l35 - l36 = 9789.29 - 9734.12 = 55.17

Now, the total number of deaths between ages 35 and 36 can be assumed to be evenly distributed,
so we can estimate the deaths that occur before age 36.4:

q35.4 = [Deaths between ages 35 and 36\times (0.4)]/l35 = [55.17× 0.4]/9789.29

Calculating this:

q35.4 approx 22.068/9789.29approx 0.00225

Finally, we calculate 0.2q35.4

0.2q35.4 = 0.2×0.00225 approx 0.00045

So, 0.2q35.4 approx 0.00045 or 0.045\% .

5. Define UDD1.

UDD1 stands for **Uniform Distribution of Deaths (UDD) over the interval of one year**. It is a
common assumption in actuarial science used to model mortality between two ages.

### Key Features of UDD1:

1. Uniformity: It assumes that deaths occur uniformly throughout the year. This means that the
likelihood of death is constant over the age interval.

2. Application: UDD1 is typically used to estimate probabilities of death and survival for individuals
aged between two specific ages (e.g., from age 35 to 36).

3. 2WSMathematical Representation: Under UDD1, if a life aged x survives to age x + t (where t is a

fraction of a year), the probability of dying within that interval is directly proportional to t.

4. Implication: This assumption simplifies calculations in life tables and is particularly useful when
there is limited data on mortality patterns.

Overall, UDD1 is a foundational concept in actuarial modeling, particularly in life insurance and
pension calculations.
6. Given that q70 = 0.010413andq71 = 0.011670, calculate 0.7q70.6 assuming a uniform distribution
of deaths.

To calculate 0.7q70.6 under the assumption of a uniform distribution of deaths, we can use the
values of q70 and q71.

Steps to Calculate q70.6:

1. Understand the Interval: Since q70.6 represents the probability that a person aged 70.6 dies
before reaching age 71.6, we can derive it based on the uniform distribution of deaths between ages
70 and 71.

2. Calculate q70.6: The relationship between q70, q71, and q70.6 is based on the uniform
distribution assumption. We can find q70.6 by interpolating between q70 and q71.

Using the formula:

q70.6 = q70 + 0.6 × (q71 - q70)

Substituting the values:

- q70 = 0.010413

- q71 = 0.011670

Calculating q70.6:

q70.6 = 0.010413 + 0.6 × [0.011670 - 0.010413]

Calculating the difference:

q71 - q70 = 0.011670 - 0.010413 = 0.001257

Now substituting back:

q70.6 = [0.010413 + 0.6] × 0.001257

q70.6 = 0.010413 + 0.0007542 approx 0.0111672

Calculate 0.7q70.6:

Now, multiply q70.6 by 0.7:

0.7q70.6 = 0.7× 0.0111672 approx 0.00781704

Thus, 0.7q70.6 approx 0.00782 or 0.782%.

 7. What’s the difference between Force of Mortality and Constant Force of Mortality?

Force of Mortality: The force of mortality, denoted as μ(x), represents the instantaneous rate of
mortality at a specific age x. It indicates the likelihood of an individual dying at that exact moment,
given that they have survived to age x.

Constant Force of Mortality: The constant force of mortality assumes that the mortality rate is
constant over time. This means that the likelihood of dying remains the same for any individual at
any age, regardless of how old they are.

8. Given that q70 = 0.010413 and q71 = 0.011670, calculate 0.7q70.6 under the assumption of a
constant force of mortality.

To calculate 0.7q70.6 under the assumption of a constant force of mortality, we can derive q70.6
using the given values of q70 and q71.

Steps to Calculate q70.6:

1. Understand the Relationship: Under the assumption of a constant force of mortality, the
probability of dying in a fraction of a year can be calculated using the exponential function.

2. Calculate q70.6: The formula for qx when using a constant force of mortality u is:

qx = 1 - e-u

However, we can interpolate q70.6 using the values of q70 and q71 as follows:

q70.6 = q70 + 0.6 × [q71 - q70]

Given:

- q70 = 0.010413

- q71 = 0.011670

Calculate the difference:

q71 - q70 = 0.011670 - 0.010413 = 0.001257

Now substituting back into the formula:

q70.6 = 0.010413 + 0.6 × 0.001257

q70.6 = 0.010413 + 0.0007542 approx 0.0111672

Calculate 0.7q70.6:
 Now, multiply q70.6 by 0.7:

0.7q70.6 = 0.7 × 0.0111672 approx 0.00781704

Thus, 0.7q70.6 approx 0.00782 or 0.782%.

9. Calculate 0.2q52.4 assuming constant force of mortality (fractional age assumption).

To calculate 0.2q52.4 under the assumption of a constant force of mortality, we need to derive
q52.4.

Steps to Calculate q52.4:

1. Understanding qx : Under the constant force of mortality assumption, the probability of dying
between ages x and x + t is given by:

qx = 1 - e-ut

Where u is the force of mortality. For fractional ages, we can use the following relationship:

qx+t = qx + t {DOT} (qx+1 - qx)

However, in this case, we will directly calculate q52.4 using interpolation based on standard
mortality values, assuming we have q52 and q53.

2. Assuming Values for q52 and q53 : If we assume hypothetical values (since they weren't
provided), let’s say:

- q52 = 0.005

- q53 = 0.006

The difference:

q53 - q52 = 0.006 - 0.005 = 0.001

3.Calculate q52.4:

q52.4 = q52 + 0.4 × q53 - q52

q52.4 = 0.005 + 0.4 × 0.001 = 0.005 + 0.0004 = 0.0054

Calculate 0.2q52.4:

Now multiply q52.4 by 0.2:

0.2q52.4 = 0.2 × 0.0054 = 0.00108

 Thus, under these hypothetical values, 0.2q52.4 approx 0.00108 or 0.108%.

Note: Please replace the hypothetical values with actual q52 and q53 if they are available for a more
accurate calculation.

10. Define Life Insurance Underwriting.

**Life Insurance Underwriting** is the process used by insurance companies to evaluate the risk
associated with insuring an individual’s life. It involves assessing the applicant’s health, lifestyle, medical
history, and other relevant factors to determine:

1. **Eligibility**: Whether the applicant qualifies for life insurance coverage.

2. **Premium Rates**: What premium amount should be charged based on the assessed risk level.

3. **Policy Terms**: Any specific conditions or exclusions that might apply to the policy.

Key Components of Life Insurance Underwriting:

- **Risk Assessment**: Underwriters analyze information provided in the application, such as age,
gender, occupation, health conditions, and family medical history.

- **Medical Exams**: Depending on the policy and the applicant’s risk profile, underwriters may require
medical examinations or additional tests to gather more detailed health information.

- **Lifestyle Evaluation**: Factors such as smoking, alcohol consumption, and participation in high-risk
activities (e.g., extreme sports) are considered.

- **Use of Underwriting Guidelines**: Insurers typically have established guidelines that help
underwriters categorize applicants into different risk classes (e.g., preferred, standard, substandard).

- **Data Analysis**: Underwriters may use statistical data and mortality tables to evaluate the
likelihood of death and set premiums accordingly

Importance of Underwriting:

- **Financial Stability**: Proper underwriting helps insurers maintain financial stability by accurately
pricing the risk of policies.

- **Fairness**: It ensures that individuals are charged premiums that are commensurate with their level
of risk, promoting fairness in the insurance market.

- **Fraud Prevention**: Thorough underwriting processes can help identify and prevent fraudulent
applications.
Overall, life insurance underwriting is a critical step in the insurance process that ensures both the
insurer and the insured have a clear understanding of the risks involved.

11. Learning Outcomes of the Chapter on Lifetable and Selection

- **Understanding Lifetables**: Grasp the construction and interpretation of lifetables, including key
concepts like mortality rates, survival functions, and life expectancy.

- **Selection Principles**: Learn about the principles of risk selection in insurance, including how
demographics and health factors influence underwriting.

- **Application of Models**: Apply lifetable data to real-world scenarios in insurance and actuarial
science, such as calculating premiums and reserves.

- **Statistical Analysis**: Develop skills in using statistical methods to analyze mortality and longevity
data.

12. Primary Parameters of the Gompertz Model

- **Baseline Mortality Rate (α)**: Represents the initial mortality rate at birth or the starting point of
mortality in a population.

- **Rate of Increase in Mortality (β)**: Reflects how quickly the mortality rate increases with age. A
higher β indicates a steeper rise in mortality as age increases.

- **Age (x)**: This parameter represents the chronological age of the individuals in the population being
modeled.

13. Difference Between UDD1 and UDD2

- **UDD1 (Uniform Distribution of Deaths 1)**: Assumes that deaths are uniformly distributed over the
age interval, meaning that the number of deaths is evenly spread throughout the period.

- **UDD2 (Uniform Distribution of Deaths 2)**: This version assumes that deaths occur uniformly
throughout the age interval but adjusts for the actual survival rates at the start of the interval, which
provides a more refined estimate of deaths.

14. Main Types of Life Insurance Policies That Require Underwriting

- **Term Life Insurance**: Provides coverage for a specified period and typically requires underwriting
to assess risk.
- **Whole Life Insurance**: Offers lifetime coverage with a cash value component, requiring
comprehensive underwriting.

- **Universal Life Insurance**: A flexible policy that combines insurance and investment, also requiring
underwriting.

- **Variable Life Insurance**: Allows for investment in various options, necessitating underwriting to
evaluate risk.

15. Underwriting Process: Term Life vs. Whole Life Insurance

- **Term Life Insurance**:

- **Focus**: Primarily assesses the applicant's current health and lifestyle.

- **Process**: Usually involves a simplified application and may require medical exams for higher
coverage amounts.

- **Duration**: Underwriting is often quicker, as it’s for a specific period.

- **Whole Life Insurance**:

- **Focus**: Evaluates long-term health, family medical history, and lifestyle factors more
comprehensively.

- **Process**: Typically involves a more detailed application, medical examinations, and possibly
additional tests.

- **Duration**: Underwriting may take longer due to the complexity of the policy and the lifetime
coverage aspect.

17. 5.7𝑝52.4 assuming UDD.

To calculate 5.7p52.4 assuming a uniform distribution of deaths (UDD), we can break it down as follows:

Calculation Steps

1. Understand p52.4 :

- p52.4 represents the probability that a person aged 52.4 survives to the next age (53.4).

- Under UDD, px can be calculated as:

 p52.4 = 1 - q52.4

- Here, q52.4 is the probability of dying between ages 52.4 and 53.4.

2. Using UDD:

- Under UDD, deaths are uniformly distributed across the age interval.

- If n is the number of deaths in the age interval, the probability of dying in the interval is given by:

q52.4 = d/l52.4

- Where d is the number of deaths and l52.4 is the number of people alive at age 52.4.

3. Example Calculation:

- If we assume q52.4 = 0.01 (1% chance of dying), then:

p52.4 = 1 - q52.4 = 1 - 0.01 = 0.99

4. Final Calculation:

- Therefore:

5.7p52.4 = 5.7 × 0.99 = 5.643

Summary

Assuming a probability of dying of 1% at age 52.4, 5.7p52.4 equals approximately 5.643. If you have a
specific mortality rate or more data, you can adjust this calculation accordingly!

16. Given that 𝑝40 = 0.999473, calculate 0.4𝑞40.2 under the assumption of a uniform distribution of
deaths.

To calculate 0.4q_40.2 under the assumption of a uniform distribution of deaths, we can follow these
steps:

Step 1: Understand the Notation

- p40: The probability that a person aged 40 survives to age 41.

- q40.2: The probability that a person aged 40.2 dies before reaching age 41.2.

Step 2: Calculate q40.2

From the relationship between survival and mortality:

px = 1 - qx

Thus, we can find q40.2 using the information about p40.

Since we know p40 = 0.999473:

q40 = 1 - p40 = 1 - 0.999473 = 0.000527

Assuming a uniform distribution of deaths, we can approximate q40.2 as:

q40.2 approx q40× [0.4/1] = q40×(age interval)

Where the age interval from 40 to 41 (which is 1 year) can be used to adjust for the interval from 40.2 to
41.2.

Step 3: Adjust q40 for the Interval

Using the UDD assumption:

q40.2 approx [0.4/1]×q40 = 0.4 × 0.000527 = 0.0002108

Step 4: Calculate 0.4q40.2

Now we can find 0.4q40.2:

0.4q40.2 = 0.4 × 0.0002108 = 0.00008432

Summary

Thus, 0.4q40.2 under the assumption of a uniform distribution of deaths is approximately 0.00008432.

----------------------------------------------------