Question $2$: Calculation question $1[17$ marks$]$

The table below gives the probability of an individual aged x$,$ surviving until the age x $+1$ i$.$e$.,$ the

probability they survive for one year for lives aged $40$ to $49$ e$.$g$.,$ there is a $97\%$ probability that someone

aged exactly $43$ will survive until at least $44.$

Age, x $40414243444546474849$

Probability of

surviving until

age x $+1$

$0\mathrm{.}990\mathrm{.}980\mathrm{.}970\mathrm{.}970\mathrm{.}960\mathrm{.}950\mathrm{.}940\mathrm{.}930\mathrm{.}920\mathrm{.}91$

The table below provides expected interest rate information for the next $10$ years. All rates relate to the

average annual interest rate from now $($time t $=0)$ until the end of year, n$.$ For example:

$$ we expect the average rate of interest over the next $4$ years to be $1\mathrm{.}75\%$ per annum $($p$.$ a$.)$

$$ the present value $($value at time t $=0)$ of a payment of $100$ due in exactly $4$ years from today is:

$100\backslash$times

$1$

$(1+1\mathrm{.}75\%)$

$4=$

$100$

$1\mathrm{.}01754=93\mathrm{.}30$

Year, n $12345678910$

Average

Interest

rate, p$.$ a$.$

$1\%1\mathrm{.}25\%1\mathrm{.}5\%1\mathrm{.}75\%2\%2\mathrm{.}1\%2\mathrm{.}5\%2\mathrm{.}8\%2\mathrm{.}9\%3\%$

a$.$ Consider an insurance product, which a customer aged exactly $40$ could purchase. It would pay

$30,000$ at the end of the year the person dies, provided they die within the next $10$ years $($this is

referred to as a term assurance product$).$

Calculate the present value of the expected death cost to the insurer of selling this policy.

$[4$ marks$]$

b$.$ Consider another insurance product, which a customer aged exactly $40$ could purchase. It would

pay $50,000$ in $10$ years, if the individual survives until then i$.$e$.,$ they will receive $50,000$ at age

$50$ if they survive until age $50($this is referred to as a pure endowment assurance$).$

Calculate the present value of the expected cost to the insurer of selling this policy.

$[2$ marks$]$

c$.$ Now consider the following insurance product which a customer aged exactly $40$ could purchase.

It would pay $30,000$ at the end of the year the person dies, provided they die within the next $10$

years, but would pay out $50,000$ if they survive to the end of the $10$ years $($this is referred to as

an endowment assurance$).$

If the customer makes annual premium payments of $$X $($at the start of each policy year$)$ for $10$

years, then calculate in terms of X$,$ the present value of the premiums the customer pays.

$[3$ marks$]$

d$.$ Using the information in parts a$-$c$,$ calculate the annual premium $$X the insurer may charge a

$40-$year$-$old for a $10-$year term assurance product which would pay $30,000$ at the end of the

year the person dies, provided they die within the next $10$ years, but would pay out $50,000$ if

they survive to the end of the $10$ years.

$[2$ marks$]$

e$.$ State additional assumptions which would help price the policy in part $($d$)$ more accurately.

$[2$ mark$]$

f$.$ How would your answers to part $($d$)$ change, for each of the following changes:

a$.$ If the death and survival payout amounts were swapped.

b$.$ If the individual was $10$ years younger.

c$.$ If the term was $20$ years rather than $10.$

d$.$ If the premium was a one$-$off rather than annual payments.

$[4$ marks$]$

Hint: The policy described in parts c and d are a combination of those in parts a and b$.$

Hint: MS Excel may help with calculations, especially for parts a $$ c$.$

Note: For part f$,$ no further working is required. Your answer should only relate to the direction $($and

magnitude$)$ of change in the premium amount paid.