Economics 12

Adams (aged 40) and Brown (aged 50) are two business partners. Adams wishes to provide

for the sum of 80,000 to be paid immediately on Brown's death if Brown predeceases him

within ten years, and effects a policy providing this benefit by single premium. The life office

issuing the contract employs the following basis:

Mortality (both lives) : A1967 ? 70 ultimate

Interest : 6%

Expenses: 2% of the single premium.

Using Simpson's rule, or otherwise, estimate the single premium payable by Adams.

Estimate the value of ?q1

74:84

on the basis of A1967 ? 70 ultimate mortality.

(Assume l108 = 0, so that the integral is over a range of 24 years. Break this into 3 sub-intervals

and use Simpson's Rule over each.)

12.4 (i) Express A2

xx

in terms of ax, axx and the rate of interest.

(ii) Smith and Jones are both aged 60. A life office has been asked to issue a special joint-life

assurance policy providing 10,000 at the end of the year of death of the first to die of these

two lives. In addition, if Smith is the second to die, a further 5,000 will be payable at the

end of the year of his death. The policy is to have annual premiums payable during the joint

lifetime of Smith and Jones.

Calculate the annual premium on the following basis:

A1967 ? 70 ultimate mortality

4% interest

expenses are 5% of all premiums, with an additional initial expense of 100.

(iii) Write down (but do NOT evaluate) formulae for the reserve at duration 10 years (immediately before payment of the premium then due) on the premium basis, if

(a) both Smith and Jones are alive; and

(b) Jones has died but Smith is alive.

A policy providing the sum of 100,000 immediately on the death of (x) if she dies before (y)

is to be issued by a life office to a group of trustees.

(i) Ignoring expenses, write down an expression for the single premium in terms of an integral.

(ii) The trustees suggest that level annual premiums should be payable in advance until the

death of the last survivor of (x) and (y).

Consider a random survivorship group consisting of two subgroups: (1) the survivors of 1,600 persons joining at birth; (2) the survivors of 540 persons joining at age 10. An excerpt from the appropriate mortality table for both subgroups follows: X 0 40 10 39 70 26 If Y, and Y2 are the numbers of survivors to age 70 out of subgroups (1) and (2), respectively, estimate a number c such that Pr(Y, + Yz > c) = 0.05. As- sume the lives are independent and ignore half-unit corrections. 3.5 Let the random variable T*(x) = T(x) 0