You are given:

(i) [math]\quad Z_{1}[/math] is the present value random variable for an [math]n[/math]-year term insurance of 1000 issued to [math](x)[/math]

(ii) [math]\quad Z_{2}[/math] is the present value random variable for an [math]n[/math]-year endowment insurance of 1000 issued to [math](x)[/math]

(iii) For both [math]Z_{1}[/math] and [math]Z_{2}[/math] the death benefit is payable at the end of the year of death

(iv) [math]\quad E\left[Z_{1}\right]=528[/math]

(v) [math]\operatorname{Var}\left(Z_{2}\right)=15,000[/math]

(vi) [math]\quad A_{x: n} \frac{1}{}=0.209[/math]

(vii) [math]{ }^{2} A_{x: n} \frac{1}{n}=0.136[/math]

Calculate [math]\operatorname{Var}\left(Z_{1}\right)[/math].

• 143,400
• 177,500
• 211,200
• 245,300
• 279,300

For a 2 -year deferred, 2 -year term insurance of 2000 on [65], you are given:

(i) The following select and ultimate mortality table with a 3-year select period:

(ii) [math]\quad i=0.04[/math]

(iii) The death benefit is payable at the end of the year of death Calculate the actuarial present value of this insurance.

• 260
• 290
• 350
• 370
• 410

A fund is established for the benefit of 400 workers all age 60 with independent future lifetimes. When they reach age 85 , the fund will be dissolved and distributed to the survivors.

The fund will earn interest at a rate of [math]5 \%[/math] per year.

The initial fund balance, [math]F[/math], is determined so that the probability that the fund will pay at least 5000 to each survivor is [math]86 \%[/math], using the normal approximation.

Mortality follows the Standard Ultimate Life Table.

Calculate [math]F[/math].

• 350,000
• 360,000
• 370,000
• 380,000
• 390,000

For a special whole life insurance on [math](x)[/math], you are given:

(i) Death benefits are payable at the moment of death

(ii) The death benefit at time [math]t[/math] is [math]b_{t}=e^{0.02 t}[/math], for [math]t \geq 0[/math]

(iii) [math]\quad \mu_{x+t}=0.04[/math], for [math]t \geq 0[/math]

(iv) [math]\delta=0.06[/math]

(v) [math]\quad Z[/math] is the present value at issue random variable for this insurance

Calculate [math]\operatorname{Var}(Z)[/math].

• 0.020
• 0.036
• 0.052
• 0.068
• 0.083

You are given the following extract of ultimate mortality rates from a two-year select and ultimate mortality table:

The select mortality rates satisfy the following: - [math]q_{[x]}=0.7 q_{x}[/math] - [math]q_{[x]+1}=0.8 q_{x+1}[/math]

You are also given that [math]i=0.04[/math].

Calculate [math]A_{[50]: 3]}^{1}[/math].

• 0.08
• 0.09
• 0.10
• 0.11
• 0.12

For a special whole life policy on (48), you are given:

(i) The policy pays 5000 if the insured's death is before the median curtate future lifetime at issue and 10,000 if death is after the median curtate future lifetime at issue

(ii) Mortality follows the Standard Ultimate Life Table

(iii) Death benefits are paid at the end of the year of death

(iv) [math]\quad i=0.05[/math]

Calculate the actuarial present value of benefits for this policy.

• 1130
• 1160
• 1190
• 1220
• 1250

You are given that [math]T[/math], the time to first failure of an industrial robot, has a density [math]f(t)[/math] given by

[math]f(t)= \begin{cases}0.1, & 0 \leq t\lt2 \\ 0.4 t^{-2}, & 2 \leq t\lt10\end{cases}[/math]

with [math]f(t)[/math] undetermined on [math][10, \infty)[/math].

Consider a supplemental warranty on this robot that pays 100,000 at the time [math]T[/math] of its first failure if [math]2 \leq T \leq 10[/math], with no benefits payable otherwise.

You are also given that [math]\delta=5 \%[/math].

Calculate the [math]90^{\text {th }}[/math] percentile of the present value of the future benefits under this warranty.

• 82,000
• 84,000
• 87,000
• 91,000
• 95,000

(80) purchases a whole life insurance policy of 100,000 .

You are given:

(i) The policy is priced with a select period of one year

(ii) The select mortality rate equals [math]80 \%[/math] of the mortality rate from the Standard Ultimate Life Table

(iii) Ultimate mortality follows the Standard Ultimate Life Table

(iv) [math]\quad i=0.05[/math]

Calculate the actuarial present value of the death benefits for this insurance

• 58,950
• 59,050
• 59,150
• 59,250
• 59,350

For a special fully continuous whole life insurance on [math](x)[/math], you are given:

i) [math] \mu_{x+t}=0.03, t \geq 0[/math]

ii) [math]\delta=0.06[/math]

iii) The death benefit at time [math]t[/math] is [math]b_{t}=e^{0.05 t}, t \geq 0[/math]

iv) [math]Z[/math] is the present value random variable at issue for this insurance

Calculate [math]\operatorname{Var}(Z)[/math].

• 0.300
• 0.325
• 0.350
• 0.375
• 0.400

For a two-year term insurance of 1 on (x) payable at the moment of death, you are given:

i) [math]q_{x}=0.04[/math]

ii) [math]q_{x+1}=0.06[/math]

iii) Deaths are uniformly distributed over each year of age

iv) [math]i=0.04[/math] v) [math]Z[/math] is the present value random variable for this insurance

Calculate [math]\operatorname{Var}[\mathrm{Z}][/math].

• 0.065
• 0.069
• 0.073
• 0.077
• 0.081