Question

1. you are given $q_{40}=0.02$, $q_{41}=0.03$, and $q_{42}=0.04$.(a) under the udd assumption in each year of age, calculate(i) $_{2.3}q_{40.2}$(ii) $e_{40:overline{1.5}|}^{\circ}$.(iii) $e_{40.5:\overline{1}|}^{\circ}$.(b) under the constant force of mortality between integer ages assumption in each year of age, calculate $e_{40:\overline{2.5}|}^{\circ}$.(c) find the probability that $(40.3)$ will die between ages 40.7 and 42.4 assuming udd between ages 40 and 41, and between ages 41 and 42, and a constant force of mortality between ages 42 and 43.(d) find $_{0.9}q_{40.2}$ under the balducci assumption between integer consecutive ages.

Explanation:

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Part (a)(i)

Step1: Define UDD mortality rate

Under UDD, $_{t}q_{x+s} = \frac{t}{1-s} q_x$ for $0<s+t<1$. Here $x=40, s=0.2, t=2.3$. Split $t$ into 0.8 (to age 41) and 1.5 (beyond).
First, $_{0.8}q_{40.2} = \frac{0.8}{1-0.2}q_{40} = \frac{0.8}{0.8} \times 0.02 = 0.02$
Then, $_{1.5}q_{41} = 1.5q_{41} = 1.5 \times 0.03 = 0.045$

Step2: Combine probabilities

$_{2.3}q_{40.2} = _{0.8}q_{40.2} + {}_{0.8}p_{40.2} \times _{1.5}q_{41}$
$_{0.8}p_{40.2} = 1 - 0.02 = 0.98$
Expression: $0.02 + 0.98 \times 0.045$

Step3: Compute final value

$0.02 + 0.0441 = 0.0641$

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Part (a)(ii)

Step1: Split term into 1 + 0.5 years

$e_{40:\overline{1.5}}^\circ = e_{40:\overline{1|}}^\circ + {}_1p_{40} \times e_{41:\overline{0.5|}}^\circ$

Step2: Calculate 1-year curtate expectation

Under UDD, $e_{x:\overline{1|}}^\circ = 1 - \frac{q_x}{2}$
$e_{40:\overline{1|}}^\circ = 1 - \frac{0.02}{2} = 0.99$

Step3: Calculate 0.5-year expectation

$e_{41:\overline{0.5|}}^\circ = 0.5 - \frac{0.5^2 q_{41}}{2} = 0.5 - \frac{0.25 \times 0.03}{2} = 0.5 - 0.00375 = 0.49625$
${}_1p_{40} = 1 - q_{40} = 0.98$

Step4: Combine terms

Expression: $0.99 + 0.98 \times 0.49625$
$0.99 + 0.486325 = 1.476325$

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Part (a)(iii)

Step1: Split term into 0.5 + 1 years

$e_{40.5:\overline{1|}}^\circ = e_{40.5:\overline{0.5|}}^\circ + {}_{0.5}p_{40.5} \times e_{41:\overline{1|}}^\circ$

Step2: Calculate 0.5-year expectation (UDD)

$e_{40.5:\overline{0.5|}}^\circ = 0.5 - \frac{0.5^2 q_{40}}{2(1-0.5)} = 0.5 - \frac{0.25 \times 0.02}{1} = 0.5 - 0.005 = 0.495$
${}_{0.5}p_{40.5} = 1 - \frac{0.5}{1-0.5}q_{40} = 1 - 0.02 = 0.98$

Step3: Calculate 1-year expectation at 41

$e_{41:\overline{1|}}^\circ = 1 - \frac{q_{41}}{2} = 1 - 0.015 = 0.985$

Step4: Combine terms

Expression: $0.495 + 0.98 \times 0.985$
$0.495 + 0.9653 = 1.4603$

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Part (b)

Step1: Split into 2 + 0.5 years

$e_{40:\overline{2.5}}^\circ = e_{40:\overline{2|}}^\circ + {}_2p_{40} \times e_{42:\overline{0.5|}}^\circ$

Step2: Calculate 1-year expectations (constant force)

Under constant force, $e_{x:\overline{1|}}^\circ = \frac{1 - e^{-\mu_x}}{\mu_x}$, where $\mu_x = -\ln(1-q_x)$
$\mu_{40} = -\ln(0.98) \approx 0.0202027$, $e_{40:\overline{1|}}^\circ = \frac{0.02}{0.0202027} \approx 0.990$
$\mu_{41} = -\ln(0.97) \approx 0.0304591$, $e_{41:\overline{1|}}^\circ = \frac{0.03}{0.0304591} \approx 0.985$

Step3: Calculate 2-year expectation

$e_{40:\overline{2|}}^\circ = e_{40:\overline{1|}}^\circ + {}_1p_{40}e_{41:\overline{1|}}^\circ \approx 0.990 + 0.98 \times 0.985 = 0.990 + 0.9653 = 1.9553$

Step4: Calculate 0.5-year expectation at 42

$\mu_{42} = -\ln(0.96) \approx 0.0408219$, $e_{42:\overline{0.5|}}^\circ = \frac{1 - e^{-0.5\mu_{42}}}{\mu_{42}} = \frac{1 - e^{-0.5 \times 0.0408219}}{0.0408219} \approx \frac{1 - 0.9798}{0.0408219} \approx 0.495$
${}_2p_{40} = 0.98 \times 0.97 = 0.9506$

Step5: Combine terms

Expression: $1.9553 + 0.9506 \times 0.495$
$1.9553 + 0.4705 = 2.4258$

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Part (c)

Step1: Split interval into 40.3-41, 41-42, 42-42.4

Probability = $_{0.4}q_{40.3} + {}_{0.4}p_{40.3} \times q_{41} + {}_{0.4}p_{40.3} \times {}_1p_{41} \times {}_{0.4}q_{42}$

Step2: Calculate UDD term (40.3-41)

$_{0.4}q_{40.3} = \frac{0.4}{1-0.3}q_{40} = \frac{0.4}{0.7} \times 0.02 \approx 0.0114$
${}_{0.4}p_{40.3} = 1 - 0.0114 = 0.9886$

Step3: Calculate constant force terms

$q_{41} = 0.03$, ${}_1p_{41} = 0.97$
Under constant force, ${}_{0.4}q_{42} = 1 - (1-q_{42})^{0.4} = 1 - 0.96^{0.4} \approx 1 - 0.9837 = 0.0163$

Step4: Combine terms

Expression: $0.0114 + 0.9886 \times 0.03 + 0.9886 \times 0.97 \times 0.0163$
$0.0114 + 0.0297 + 0.0157 = 0.0568$

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Part (d)

Step1: Balducci formula for mortality

Under Balducci, $_{t}q_{x+s} = \frac{t q_x}{1 - (1-t)q_x}$ for $0<s+t<1$. Here $x=40, s=0.2, t=0.9$.
First, adjust to the interval: $0.9q_{40.2}$ covers 0.8 years to age 41, then 0.1 years at age 41.

Step2: Calculate $_{0.8}q_{40.2}$

$_{0.8}q_{40.2} = \frac{0.8 q_{40}}{1 - (1-0.8)q_{40}} = \frac{0.8 \times 0.02}{1 - 0.2 \times 0.02} = \frac{0.016}{0.996} \approx 0.01606$
$_{0.8}p_{40.2} = 1 - 0.01606 = 0.98394$

Step3: Calculate $_{0.1}q_{41}$ (Balducci)

$_{0.1}q_{41} = \frac{0.1 q_{41}}{1 - (1-0.1)q_{41}} = \frac{0.1 \times 0.03}{1 - 0.9 \times 0.03} = \frac{0.003}{0.973} \approx 0.00308$

Step4: Combine probabilities

Expression: $0.01606 + 0.98394 \times 0.00308$
$0.01606 + 0.00303 = 0.01909$

Answer:

(a)(i) $\boldsymbol{0.0641}$
(a)(ii) $\boldsymbol{1.4763}$
(a)(iii) $\boldsymbol{1.4603}$
(b) $\boldsymbol{2.4258}$
(c) $\boldsymbol{0.0568}$
(d) $\boldsymbol{0.0191}$