Chapter 22 - Competing Risks

Question 1

Describe an income protection insurance contract.

Answer 1

Pays an income to the policyholder during periods of sickness (with the definition of sickness being specified carefully in the policy conditions). If the policyholder recovers, the policy usually remains in force, so that subsequent periods of sickness are still covered under the policy.

Question 2

Explain what is meant by a deferred period on an income protection insurance contract.

Answer 2

A period of time while an income protection policy is in force during which a policyholder is sick, but before benefits start to be paid. A policy might typically have a deferred period of 3 months. If so, benefits, do not start to be paid until a policyholder has been continuously sick for this period of time.

Question 3

Explain the meaning of the notation:
- mu^(gh)_(x+t)
- t_p^(gh)_x
- t_p^bar(gg)_x

Answer 3

• the force of transition (or transition intensity) from state g to state h at age x + t
• P(in state h at age x + t|in state g at age x)
• P(in state g from age x to age x + t|in state g at age x)

Question 4

State the required condition for the statement t_p^bar(gg)_x = t_p^bar(gg)_x to be true.

Answer 4

If it is impossible to return to state g (within the next t years) once it has been left.

Question 5

A life assurance company uses a three-state health, sickness, death (HSD) model to calculate premiums for three-year sickness policies issued to lives aged 55. S_t denotes the state occupied by a given policyholder at age 55 + t, and S_0 = H. Given:
- p^HH_(55+t) = 0.95
- p^HS_(55+t) = 0.04
- p^SH_(55+t) = 0.95
- p^SS_(55+t) = 0.10
for t = 0, 1, 2.
Policyholders pay premiums of 500 pa annually in advance, provided they are healthy at the time the premium is due. Calculate the expected present value of the premiums paid by a single policyholder. Assume 5% pa interest.

Answer 5

500(1 + v * p^HH_55 + v^2 * 2_p^HH_55)
2_p^HH_55 = p^HH_55 * p^HH_56 + p^HS_55 * p^SH_56 = 0.95 * 0.95 + 0.04 * 0.85 = 0.9365
So EPV is 500(1 + 0.95/1.05 + 10.9365/1.05^2) = 1377.10

Question 6

State the formula for t_p^bar(gg)x in terms of the transition intensities mu^gj(x+t).

Answer 6

exp(-int^t_0[sum_j=/=g(mu^gj_(x+s)) ds])
Here, the sum relates to the total force of transition out of state g at age x + s.

Question 7

Consider the three-state health, sickness, death (HSD) model. Give an integral expression for the expected present value of a benefit of 1 payable immediately on the death of a healthy life now aged x from the healthy state or the sick state. Define all the notation that you use.

Answer 7

EPV is int^infinity_0(e^(-delta * t)[t_p^HH_x * mu_(x+t) + t_p^HS_x * nu_(x+t)] dt) where:
- t_p^gh_x = P(in state h at age x + t|in state g at age x)
- mu_(x+t) is the force of transition from healthy to dead at age x + t
- nu_(x+t) is the force of transition from sick to dead at age x + t
- delta is the force of interest

Question 8

Consider the HSD model. Give an integral expression for the expected present value of a benefit of 1 pa payable continuously while a healthy life now aged x is in the sick state.

Answer 8

int^infinity_0(e^-(delta * t) * t_p^HS_x dt)

Question 9

Give a definition of the term ‘multiple decrement model’.

Answer 9

A multiple state model that has:
- one active state
- one or more absorbing exit states.

Question 10

Explain the meaning of the symbols (ap)_x, (aq)^alpha_x and q^alpha_x.

Answer 10

• dependent probability that a life aged x remains in the population of interest for the next year. This assumes that there are multiple decrements operating on the population.
• probability that a life aged x leaves the population of interest by decrement alpha within the next year, assuming that there are other decrements operating on the population. It is an example of a dependent probability of decrement.
• probability that a life aged x leaves the population of interest by decrement alpha within the next year, assuming that there are no other decrements operating on the population. It is an example of an independent probability of decrement.

Question 11

A multiple state model has 3 states - healthy, sick and dead. The decrement of death is denoted by d and the decrement of sickness is denoted by s. The forces of transition are as follows:
- healthy to sick: sigma
- sick to healthy: rho
- healthy to dead: mu
- sick to dead: nu
State the conditions under which (aq)^d_x = p^HD_x

Answer 11

Provided that the forces of transition from sick to dead and from sick to healthy are both 0.

Question 12

For the reduced HSD model (with transitions only from healthy to sick and healthy to dead), give expressions in terms of forces of transition for:
- (d/dt)(t_(aq)^s_x)
- (d/dt)(t_(aq)^d_x)

Answer 12

• sigma * e^-(sigma + mu)t
• mu * e^-(sigma + mu)t

Question 13

For the reduced HSD model, give expressions in terms of forces of transition for:
- t_(aq)^s_x
- t_(aq)^d_x
- t_(ap)_x

Answer 13

• (sigma/(sigma + mu)) * (1 - e^-(sigma + mu)t)
• (mu/(sigma + mu)) * (1 - e^-(sigma + mu)t)
• e^-(sigma + mu)t

Question 14

For the reduced HSD model, give expressions, in terms of dependent probabilities, for the force of sickness sigma and the force of mortality mu.

Answer 14

• sigma = [(aq)^s_x / (aq)_x] * (-ln(ap)_x)
• mu = [(aq)^d_x / (aq)_x] * (-ln(ap)_x)

Question 15

For the reduced HSD model, give expressions in terms of forces of transition for:
- t_q^s_x
- t_q^d_x

Answer 15

1 - e^-(sigma * t)
1 - e^-(mu * t)

Question 16

For the reduced HSD model, give expressions, in terms of independent probabilities, for the force of sickness sigma and the force of mortality mu.

Answer 16

• sigma = -ln(1 - q^s_x)
• mu = -ln(1 - q^d_x)

Question 17

State the linking assumption between single and multiple decrement tables.

Answer 17

(a*mu)^j_x = mu^j_x for all j and all x

Question 18

Describe how the total force of decrement (a * mu)_x is related to the individual forces of decrement (a*mu)^j_x.

Answer 18

= sum_(all j)[(a * mu)^j_x]

Question 19

Suppose that a population is subject to two models of decrement, alpha and beta. Give a formula for n_(ap)_x in terms of n_p^alpha_x and n_p^beta_x.
State how this formula changes if there are three decrements alpha, beta and gamma.

Answer 19

n_p^alpha_x * n_p^beta_x
n_p^alpha_x * n_p^beta_x * n_p^gamma_x

Question 20

Give an integral formula for (aq)^j_x.

Answer 20

int^1(t=0)(t(ap)x * (a * mu)^j(x+t) dt) = int^1(t=0)(t(ap)x * mu^j(x+t) dt) assuming independence of decrements

Question 21

Explain the meaning of the symbols (al)_x and (ad)^j_x.

Answer 21

• denotes the expected number of active lives at exact age x
• denotes the expected number of decrements due to cause j over the year of age x to x + 1, given a radix of (al)_alpha lives active at age alpha

Question 22

A multiple decrement model has two decrements, death and withdrawal. For the year of age 50 to 51, the dependent probability of mortality is 0.005, and the dependent probability of withdrawal is 0.07. Given (al)_50 = 10,000, calculate the values of (ad)^d_50, (ad)^w_50 and (al)_51.

Answer 22

(ad)^d_50 = 10000 * 0.005 = 50
(ad)^w_50 = 10000 * 0.07 = 700
(al)_51 = 10000 - 50 - 700 = 9250

Question 23

Given the multiple decrement table:
x | (al)x | (ad)^d_x | (ad)^w_x
60 | 1000 | 8 | 26
61 | 966 | 9 | 35
62 | 922 | 11 | 40
63 | 871 | |
calculate (aq)^d_60, 2(aq)^w_60 and 2|_(aq)^d_60.

Answer 23

(aq)^d_60 = (ad)^d_60 / (al)60 = 8 / 1000 = 0.008
2(aq)^w_60 = ((ad)^w_60 + (ad)^w_61) / (al)60 = (26 + 35) / 1000 = 0.061
2|(aq)^d_60 = (ad)^d_62 / (al)_60 = 11 / 1000 = 0.011