SOA Exam ALTAM Sample Question 15

You are using the following model to price a 10-year Disability Income Insurance policy issued to $(x)$ .

The product has the following features.

(i) Policyholders are healthy at the issue date.

(ii) A benefit of 20,000 is payable immediately on death within 10 years.

(iii) A continuous benefit of 1000 per year is paid while the policyholder is in the Sick state, during the 10-year term.

(iv) Premiums are payable continuously while the life is in the Healthy state.

(v) Maintenance expenses are paid at a continuous rate of 60 per year while the insured is alive.

(vi) Commissions are payable continuously at a rate of 5% of gross premium.

(vii) No benefits, premiums or expenses are payable beyond 10 years.

You are also given:

(viii) $\mu_{x+t}^{01} = 0.03$ ; $\mu_{x+t}^{02} = 0.02$ ; $\mu_{x+t}^{10} = 0.01$ ; $\mu_{x+t}^{12} = 0.05$ , for $t > 0$

(ix) $\delta = 0.07$ .

(x) $\bar{a}_{x:\overline{10}|}^{00} = 5.844$ ; $\bar{a}_{x:\overline{10}|}^{01} = 0.684$ ; $\bar{A}_{x:\overline{10}|}^{01} = 0.175$ ; $\bar{A}_{x:\overline{10}|}^{02} = 0.151$ .

Part a (2 points)

Show that the gross premium rate, calculated using the equivalence principle, is 738 to the nearest 1. You should calculate the value to the nearest 0.1.

Part b (2 points)

Calculate the probability that the life dies without becoming sick, within the 10-year term.

Part c (1 point)

The insurer offers a new product with the following features:

For any period of sickness that starts within the 10-year term, the benefit of 1000 per year is paid until death or recovery, even if that occurs after the 10-year term.
There is an elimination period of 1 year.
All other policy features and assumptions are the same as given above.

Define the term elimination period in the context of a disability income insurance product.

Part d (2 points)

Show that the expected present value of a 1-year continuous sojourn annuity in State 1 is $\bar{a}_{x:\overline{1}|}^{\overline{11}} = 0.94$ to the nearest 0.01. You should calculate the value to the nearest 0.0001.

Part e (2 points)

Calculate the premium for this new policy, using the model and parameters given above.

(LTAM, Spring 2020, Question 4)

Answer this question and see the worked solution