Chapter 2

Question 1

What are the key variables that influence mortality?

Answer 1

• sex/gender
• age
• smoking habits
• lifestyle
• status/wealth
• education
• marital status

Question 2

What is the definition of t-year survival probability for an x-year old person?

Answer 2

tpx = P(X > x + t | X > x)

Question 3

What does tqx represent?

Answer 3

tqx = P(x < X < x + t | X > x)

Question 4

How is the u-year mortality probability deferred by t-years for an x-year old person defined?

Answer 4

t|uqx = P(x + t < X < x + t + u | X > x)

Question 5

What is the relationship between tpx and tqx?

Answer 5

tpx + tqx = 1

Question 6

What does the cumulative distribution function F(x) represent?

Answer 6

F(x) := P(X < x)

Question 7

What property does the survival function s(x) have?

Answer 7

s(x) := 1 - F(x) = P(X > x)

Question 8

What is the mortality intensity µx defined as?

Answer 8

µx := lim (P(X < x + ε | X > x) / ε) as ε approaches 0

Question 9

True or False: The mortality intensity can be interpreted as the death probability within an infinitesimal time interval.

Answer 9

True

Question 10

What is the future lifetime K(x)?

Answer 10

K(x) = k for k = 0, 1, 2, … is the future lifetime in years or curtate future lifetime

Question 11

How is lx defined in the context of mortality tables?

Answer 11

lx := l0 · s(x), where l0 > 0 is a constant

Question 12

What does the notation lx represent?

Answer 12

The expected number of newborns surviving to age x

Question 13

What is the relationship between lx and qx?

Answer 13

qx = lx / lx+1

Question 14

What is the mortality table composed of?

Answer 14

• discrete values of age x
• lx
• additional derived quantities

Question 15

Define the notation 1px and 1qx.

Answer 15

1px = px, 1qx = qx

Question 16

What is the interpretation of the mortality table DAV 2008 T?

Answer 16

It provides mortality probabilities for males and is used in products like term life insurance.

Question 17

How is the mortality intensity µx modeled in the Gompertz model?

Answer 17

µx = a · exp(b · x), where a, b > 0

Question 18

Fill in the blank: The survival function s(x) is ______.

Answer 18

monotone decreasing and continuous

Question 19

What does the notation dx signify in mortality tables?

Answer 19

dx = number of deaths between ages x and x + 1

Question 20

What relationship does the mortality table establish between lx and the probability of surviving?

Answer 20

lx = l0 · xp0

Question 21

What is the significance of the constant l0 in mortality tables?

Answer 21

It represents the initial population size, commonly set at 100,000.

Question 22

What does the term ‘actuarial age’ refer to?

Answer 22

The age used in calculations based on the nearest birthday.

Question 23

What is the primary risk associated with the use of mortality tables?

Answer 23

Mortality risk leading to higher benefits that exceed expectations

Calculations based on expected survivors may differ from actual observed deaths.

Question 24

What are the two types of mortality risk?

Answer 24

• Idiosyncratic (unsystematic) mortality risk
• Aggregate (systematic) mortality risk

Idiosyncratic risk can be diversified by a larger pool size, while aggregate risk cannot.

Question 25

Define idiosyncratic mortality risk.

Answer 25

Risk stemming from the uncertainty of an individual’s lifetime ## Footnote It can be mitigated by increasing pool size.

Question 26

Define aggregate mortality risk.

Answer 26

Risk arising from the uncertainty in determining the true underlying mortality law ## Footnote This risk cannot be diversified by a larger pool size.

Question 27

What safety measures are included in mortality probabilities?

Answer 27

* Increase probabilities for death risk * Decrease probabilities for longevity risk ## Footnote Safety margins (Sicherheitszuschläge) are added to manage risks.

Question 28

How are mortality assumptions affected by medical progress?

Answer 28

Mortality probabilities may decrease over time due to unexpected medical advancements ## Footnote Examples include changes in mortality rates from 1930 to 1955.

Question 29

What is the formula connecting qx, lx, and dx?

Answer 29

qx = lx / lx+1, lx = dx, lx+1 = lx(1 - qx) ## Footnote This formula helps in understanding the relationship between different mortality metrics.

Question 30

What is the limitation of mortality tables regarding demographic data?

Answer 30

Mortality tables typically have a limiting age ε, defined as the lowest age where qω = 1 ## Footnote This indicates that all individuals at or above this age are expected to die.

Question 31

What is the assumption made for non-integer values in mortality tables?

Answer 31

Uniform distribution of deaths over a year ## Footnote tqx = t · qx for t, x where t is an integer.

Question 32

What is the purpose of using unisex tables in mortality calculations?

Answer 32

To avoid discrimination between genders in pricing ## Footnote Unisex tables are denoted as qunisex.

Question 33

What is the cohort effect in mortality tables?

Answer 33

Mortality rates depend on the calendar year and generational factors ## Footnote This effect is significant for products with longevity risk, such as annuities.

Question 34

What is the method of age shifting in mortality tables?

Answer 34

Adjusting cohort tables for different calendar years using a given function ## Footnote This method allows for the estimation of mortality probabilities for different years.

Question 35

What distinguishes cohort tables from period tables?

Answer 35

Cohort tables consider generational effects, while period tables are independent of the calendar year ## Footnote Examples include DAV 2004 R (cohort) and DAV 2008 T (period).

Question 36

Fill in the blank: Mortality tables include safety margins to account for _______.

Answer 36

[death risk and longevity risk]

Question 37

True or False: Mortality probabilities are constant over time.

Answer 37

False ## Footnote Mortality probabilities can change due to various factors, including medical advancements.