ACTL10001 Introduction to Actuarial Studies

Problem Sets

� The problem sets correspond to each week of the subject with theexception of Problem Set 0 which is for discussion at your tutorialclass in the �rst week of semester.

� You should attempt all numerical questions using a scienti�c calculator�you can only use a scienti�c calculator in exams.

� Problem Sets 1�12 contain three types of question:

� tutorial problems for discussion in the tutorial class; these shouldbe attempted before you attend your tutorial and you should beprepared to present your solutions in class.

� past exam questions (from mid-semester or end of semester ex-ams); these are not for discussion in tutorial classes. After eachtutorial class, you should attempt these questions.

� problems from An Introduction to Actuarial Studies, 2nd edition(AITAS2E hereafter).

� Model solutions to tutorial problems and past exam questions will bemade available on the LMS with a two week time lag. You shouldreview these model solutions when they appear as part of your learningprocess.

� Solution to problems in AITAS2E are available in the book.

� By the end of semester you will have been presented with about 150problems of varying length and of varying degrees of di¢ culty. If youunderstand the solutions to these questions you are adequately pre-pared for the end of semester examination. If you do not understandthe solutions, extra problems will not help you.

Problem Set 0

The following questions are NOT based on lectures. Rather, they test pre-requisite knowledge that you should have coming into this subject.

1. You should know that

1 + 2 + 3 + : : :+ n =n(n+ 1)

2:

(a) Prove this result from �rst principles.

(b) Prove this result by induction.

2. Show from �rst principles that

nXi=1

xi = x1� xn1� x :

Hence give an expression forPn�1

i=0 xi.

3. Letg(t) = expf�+ �tg.

Given that g(10) = 8:1882 and g(20) = 60:3403, calculate g(15).

4. Consider a function f such that f(11) = 1:234 and f(12) = 2:345.Assuming that the function is linear over the interval from 11 to 12,calculate f(11:36).

5. Let g(x) = log(1 + x2). (We will use log to denote natural logarithm,so that log ex = x.)

(a) Find and expression for ddxg(x).

(b) Explain why g(x) is an increasing function of x for x > 0.

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Problem Set 1

1. An investor can earn simple interest at 6% per annum.

(a) Calculate the accumulated amount at time 3.5 years of an invest-ment of $2,000.

(b) How much should be invested now to secure an accumulation of$5,000 six years from now?

2. Calculate the accumulation for three years of an investment of $5,000under a rate of simple discount of 5% per annum.

3. An investor is choosing between banks A and B. In Bank A, moneyaccumulates at 6% per annum simple interest. Bank B o¤ers a rateof simple discount d per annum. The investor (correctly) decides thatBank A provides a greater accumulation of money over a half-yearperiod. What does this tell you about the value of d?

4. An investor purchased a $100,000 bill 100 days before maturity at 6%per annum simple discount. 40 days before maturity, the investor re-ceived an o¤er to sell the bill at 4% simple discount. The proceeds willbe deposited and will earn interest at 4.5% per annum simple interestuntil the maturity date of the bill.

(a) Determine whether the investor should sell the bill or hold it untilmaturity.

(b) Calculate the rate of simple interest which an investor would earnif he holds the bill from purchase for 100 days until maturity.

5. According to the Australian O¢ ce of Financial Management, the price,P per $100 face value, of Treasury Notes is

P =100

1 + ( f365)i

where f is the number of days from the date of settlement to thematurity date and i is the annual yield (per cent) to maturity dividedby 100.

(See http://www.aofm.gov.au/content/pricing_formulae.asp?NavID=58if you are interested.)

If the price on 20 June of a Treasury Note with maturity date of 26November is $98, what is the annual yield?

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Exam questions:You will not yet be able to give full answers yet to these questions, but

have a look at the simple interest calculations.

1. An investor has $10,000 to invest for 6 months. Bank A o¤ers a rateof simple interest of 8% per annum on investments. Bank B o¤ers arate of compound interest of 8% per annum e¤ective on investments.With which bank would you advise the investor to deposit her money?Justify your answer.

2. (a) Find the accumulation of $1,000 for 90 days under simple interestat 6% per annum. Give your answer to the nearest cent.

(b) Find the force of interest per annum that gives the same accumu-lation as in part (a). Give your answer to 4 decimal places.

Problems from AITAS2E:Chapter 2: 1, 2 and 3

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Problem Set 2

1. Consider an investment of $1,000 made for three years.

(a) Calculate the accumulation of this investment at 6% per annumsimple and at 6% per annum compound.

(b) Explain why they are di¤erent.

(c) Calculate the di¤erence when the investment was made for 10years.

2. How long does it take an investment to double in value under:

(a) simple interest at 5% per annum,

(b) compound interest at 5% per annum.

3. How much does an investor need to deposit today to have $10,000 inan account in 6 years� time if interest is earned at 5.5% per annumcompound?

4. To the nearest $1, calculate the present value of $100,000 due in 8.25years using:

(a) a rate of interest of 0.05 per annum e¤ective,

(b) a force of interest of 0.06 per annum,

(c) a rate of interest of 8% per annum convertible quarterly,

(d) a rate of interest of 6% per annum convertible monthly, and

(e) a rate of interest of 7.2% per annum convertible half-yearly.

5. Correct to 5 decimal places, calculate:

(a) i(2) given � = 0:05;

(b) � given i(4) = 0:05;

(c) i(12) given i = 0:12, and

(d) i given i(12) = 0:063:

6. Calculate the total present value of two payments of $1,000 each duein 3.5 and 5 years�time assuming a rate of interest of 6% per annume¤ective for the �rst two years and a force of interest of 5% per annumthereafter.

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7. Investments of $1,000 are made at time 0 and time 2 (measuring inyears). At time 4 the accumulated amount of these investments is$2,692.96. If these investments were accumulated under a nominal rateof interest i(2) per annum convertible half-yearly, �nd i(2).

Exam questions:

1. Calculate i(6) when � = 0:05: Give your answer to 4 decimal places.

2. Calculate to the nearest cent the accumulation at time 10 years of aninvestment of $1,000 now assuming that the interest rate for the �rst�ve years will be 6.2% per annum e¤ective, and for the next �ve yearswill be a nominal rate of 6.4% per annum convertible quarterly.

3. Find the amount of money required now to provide payments of $100two years from now, $200 seven years from now and $300 twelve yearsfrom now, assuming an e¤ective rate of interest of 5.1% per annum for�ve years, followed by a nominal rate of interest of 5.2% per annumconvertible quarterly. Give your answer to the nearest cent.

Problems from AITAS2E:Chapter 2: 4, 5(a)-(c), 6, 7

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Problem Set 3

1. Suppose that interest rates in odd numbered years are 8% per annume¤ective, and in even numbered years 10% per annum e¤ective. Findthe present value (i.e. at the start of year 1) of payments of $1,000 atthe end of every two years until the total amount of the payments is$10,000.

Hint : use a result from Problem Set 0.

2. Calculate the present value of an annuity of $1,000 per annum payableannually in arrear for 10 years, deferred for 3 years (i.e. the �rst pay-ment is received in 4 years�time). Interest rates are as follows: 0.10 perannum e¤ective for the �rst 4 years, and 0.08 per annum convertiblequarterly thereafter.

3. Calculate the accumulated value at the end of 10 years of a series ofpayments as follows:

� for the �rst �ve years, $100 per annum half-yearly in arrear, and

� for the next �ve years, $125 per annum annually in arrear.

Interest rates are 8.4% per annum convertible half-yearly for the �rstseven years, and 7.3% per annum e¤ective for the next three years.

4. Show thatsm+n = sm + (1 + i)

msn

and explain in words what this equality means.

5. Suppose that interest rates in odd numbered years are 8% per annume¤ective, and in even numbered years are 10% per annum e¤ective.Payments of $1,000 are made at the end of every two years. The numberof payments is the least integer such that the present value of thepayments exceeds 5,000. How many payments are made?

Exam questions:

1. Using an e¤ective rate of interest of 6.2% per annum, compute 10ja10and �a(4)

15: Give your answers to 4 decimal places.

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2. An annuity is payable in arrear for 20 years under which paymentsoccur at times 1,2,3,. . . ,20 years from now. The annuity payment attime t is t for t = 1; 2; 3; : : : ; 10 years, and each subsequent annuitypayment is 10.

Let S denote the present value of the payments at times 1,2,3,. . . ,10 ate¤ective rate i per annum.

(a) Write down an expression for S in terms of v, where v = (1+ i)�1.

(b) By writing down an expression for (1 + i)S or otherwise, �nd anexpression for S:

(c) Find the present value of the 20 year annuity when i = 0:08:

Problems from AITAS2E:Chapter 2: 5(d)-(e), 8, 9, 10, 11, 12, 13, 14, 15, 17

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Problem Set 4

1. A �xed coupon bond paying interest half-yearly at 6% per annum with8 years to maturity, is priced at $98 for $100 of nominal value of thebond.

(a) Set out the equation of value (in the form P (j) = 0) for calculatingthe half-yearly e¤ective rate of interest, j, that corresponds to theyield per annum.

(b) Estimate, roughly, the solution to this equation of value to thenearest 0.1% per half-annum.

(c) Evaluate your equation of value at your estimate under (b).

(d) Is the solution higher or lower than this estimate? Suggest asecond suitable test value whose evaluation of the equation ofvalue will have the opposite sign to your evaluation under (c).Evaluate the equation of value at this second test value.

(e) If necessary, repeat (d) until there are two values for j for whichthe evaluations of the equation of value have opposite signs.

(f) Use linear interpolation to solve for j.

(g) State the annual yield involved in the purchase at $98 per $100 offace value.

2. Consider a loan of $100,000 repayable by equal monthly instalments ofprincipal and interest.

(a) Calculate the monthly instalment required to repay this loan over25 years when interest is charged on the outstanding balance at thebeginning of each month at 8.4% per annum convertible monthly.

(b) Calculate the balance outstanding immediately after the 120thinstalment has been paid.

(c) Calculate the interest charged during the 10th year of the loan.

(d) If the rate of interest is reduced, after 10 years, to 7.2% per annumconvertible monthly, calculate the level monthly instalment nowrequired to repay the loan over the remaining term of 15 years.

3. Consider a loan of $200,000 repayable by equal annual instalments. Theterm of the loan is 25 years and interest is at 10% per annum e¤ective.Set up a loan schedule in Excel, showing the loan outstanding at thestart of year, the capital and interest components of each instalment,and the capital outstanding at the end of each year.

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4. Consider the loan of the previous question. Suppose that after 12 years,the rate of interest is reduced to 9% per annum e¤ective. If the borrowerdoes not change the amount of the annual repayments (except for the�nal repayment), how many years does it take to repay the loan andwhat is the amount of the �nal instalment?

5. Suppose that a loan is repayable by annual instalments at an e¤ectiverate of i per annum. In the notation of lectures, show that Ct+1 =Xvn�t. What does this result tell us about the behaviour of the capitalcomponent of instalments over time?

Exam questions

1. A government bond has a 20 year term, a face value of $100 and acoupon of 9% per annum, payable half yearly. An investor purchasedthis bond at its issue date for $95. Calculate the yield (as an e¤ectiverate of interest per annum) to the investor assuming the investor holdsthe bond for 20 years.

2. Six years ago, Peter borrowed $500,000 from his bank to purchase ahouse. The loan has a twenty-�ve year term with equal monthly re-payments of principal and interest calculated using a nominal interestrate of 6.6% per annum convertible monthly.

(a) Calculate the monthly repayment.

(b) Calculate the amount of the loan outstanding after �ve years.

(c) Calculate the amount of interest paid in the 6th year.

(d) Peter now wishes to borrow a further $20,000 to renovate hiskitchen. He does not want to change his monthly repayment,and so extends the term of the loan. How many extra repaymentsare made beyond the original twenty-�ve year term of the loan?

Give your answers to parts (a) to (c) to the nearest cent.

Problems from AITAS2E:Chapter 2: 16, 18, 19, 20, 21, 22, 23, 24, 25

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Problem Set 5

1. In a certain country, the population size at the start of a year was 16.23million and at the end of the year it was 16.42 million. During the yearthere were 0.3 million deaths in the population, there were 0.12 millionimmigrants and 0.03 million emigrants

(a) Calculate the natural increase in the population during the year.

(b) Calculate the net migration in the population during the year.

(c) Estimate the crude birth rate for the population.

(d) Estimate the crude death rate for the population.

2. Use the spreadsheet to calculate the following ratios for the state ofVictoria:

(a) child-woman ratio,

(b) dependency ratio,

(c) age dependency ratio, and

(d) youth dependency ratio.

3. Use the spreadsheet to calculate the sex ratio in Victoria

(a) for the state overall, and

(b) at each individual age from 0 to 105.

Plot the results for (b) in a graph and comment.

4. The table below gives information about two towns in a certain state,and about the state itself.

Town A Town B StateAge Population Number of Population Number of Populationgroup size deaths size deaths size0-15 38,064 6 42,822 8 317,20116-44 67,753 89 76,222 104 564,61145-64 39,985 410 32,715 328 363,48965+ 24,143 1,859 48,286 3,645 268,256All 169,945 2,364 200,045 4,085 1,513,557

Population sizes were estimated mid-year and each number of deathsrefers to that year.

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(a) Calculate the crude death rate for each town.

(b) Calculate the standardised crude death rate for each town, takingthe state�s population as the standard population. (You may �ndit easier to set out your calculations in a spreadsheet.)

(c) By reference to your answers to (a) and (b), comment on theusefulness of the crude death rate as a measure of mortality.

5. You are a tutor in actuarial studies and have arrived at a tutorial witha population pyramid of Australia to discuss with students. Unfor-tunately it does not show which side represents males and which siderepresents females. Explain two ways in which you might identify whichside is which.

6. Visit the US Census Bureau website athttp://www.census.gov/ipc/www/idb/informationGateway.php.

(a) Consider the population pyramids for Australia in 2000 and 2050.In which year is

i. the age dependency ratio higher?ii. the youth dependency ratio higher?

Give reasons for your answers.

(b) Repeat this question for (i) China and (ii) Malaysia.

Exam questions

1. The population pyramid below shows the population of Indonesia in2005. For this population, state which is the largest and which is thesmallest of

(a) the child-woman ratio,

(b) the age dependency ratio,

(c) the number of males aged 80 and above per 100 females aged 80and above.

You are not required to justify your answers.

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2. The population pyramid below shows the population of a particularcountry in 2005. State with reasons whether this country is (a) India,(b) Kenya, or (c) Italy

Problems from AITAS2E:Chapter 3: 1, 2 and 3

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Problem Set 6

1. Let F (x) = 1� expf�0:01xg for x > 0.

(a) Calculate the probability that a newborn life survives to age 60.

(b) Calculate the probability that a newborn life dies before age 40.

(c) Calculate the probability that a newborn life dies between theages of 30 and 50.

(d) Calculate the probability that a life aged 40 survives 10 years.

(e) Calculate the probability that a life aged 50 survives 10 years.

(f) Calculate the probability that a life aged 30 dies between the agesof 70 and 80.

(g) Calculate the force of mortality at age 50.

(h) Explain why this survival function is unsuitable as a model ofhuman mortality.

2. Let s(x) = (1� x=!)4, for 0 � x � !.

(a) Find an expression for �x.

(b) Calculate the probability of survival from birth to ages 10, 30 and60 when ! = 105.

(c) Consider two independent lives, aged 20 and 25, each subject tothis survival model. Again assuming ! = 105, calculate the prob-ability that

i. both are alive in 10 years�time,ii. only (20) is alive in 10 years�time, andiii. at least one of them is alive in 10 years�time.

(d) Comment on the suitability of this survival function for a humanpopulation.

(Remember, for two independent events A and B, Pr(A and B) =Pr(A)� Pr(B):)

3. Construct a life table showing values of lx, dx and px for ages 0; 1; 2 and3 given the following mortality rates.

x qx0 0.000611 0.000582 0.000543 0.00051

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Use a radix of 100,000.

4. In lectures it was shown that

�x =�1s(x)

d

dxs(x):

(a) Give an expression for �x in terms of lx.

(b) From Australian Life Tables 2000�02 Females, you are given thatl60 = 93; 919, l61 = 93; 440 and l62 = 92; 920: Let us suppose thatover the interval [60; 62], lx is a quadratic function, i.e. that for0 � t � 2 we can write

l60+t = l60 � at� bt2:

Use the values of l61 and l62 to �nd the values of a and b.

(c) Using the assumptions from part (b), estimate �61. (The valuegiven in Australian Life Tables is 0.00534).

5. A nursing home accepts patients on their 90th birthday and has nowreached a stationary population. Their intake is 100 patients per year.Assuming that 3 times as many women as men are accepted and thatthe patients su¤er mortality in accordance with the mortality tables inAITAS2E, calculate

(a) the total population of the nursing home,

(b) the number of individuals reaching their 100th birthday each year,and

(c) the number of deaths per year of those aged over 105.

Exam Questions

1. The survival function in a population is given by s(x) = 1 � x=!, for0 � x � !. You may assume that ! > 20: Find expressions in termsof !, simpli�ed as far as possible, for:

(a) the probability that (50) survives to age 60, but not to age 70,

(b) the probability that two independent lives aged !=2 both survivefor 10 years,

(c) ex. (Attempt this part next week.)

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2. The survival function in a population is given by s(x) = expf�0:02x1=2g,for x > 0. Calculate

(a) the probability that (40) dies after age 45,

(b) the probability that of two independent lives aged 30 and 50; only(30) survives for 10 years, and

(c) the force of mortality at age 50.

Give your answers to 4 decimal places.

3. Suppose that both males and females in a country experience the mor-tality of the life tables in AITAS2E. The population of this countryhas reached a stationary condition. There are 10,400 male births eachyear, and the sex ratio at birth is 104.

(a) How many female births are there each year?

(b) How many male children die before age 5 each year?

(c) How many deaths are there in the whole population each year?

(d) The government wants to instigate a vaccination program for allchildren on their �rst and second birthday. The cost of the �rstvaccination is $5 and the second is $10. How much will the pro-gram cost each year?

Problems from AITAS2E:Chapter 3: 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18

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Problem Set 7

1. Using the following values from Australian Life Tables 1985-87 (Males),estimate the parameters in Makeham�s formula and then estimate �65:

�30 = 0:00129; �40 = 0:00174; �50 = 0:00477:

2. Consider the survival function

s(x) = 1� 0:01x

for 0 � x � 100. Find an expression for ex.

3. The table below gives age speci�c fertility rates for a certain country,together with survival probabilities for females from birth.

Age speci�c SurvivalAge group fertility rate probability15-19 0.030 0.966120-24 0.092 0.960725-29 0.126 0.955630-34 0.088 0.950435-39 0.029 0.943540-44 0.005 0.933345-49 0.001 0.9165

Each survival probability is the probability of surviving from birth tothe mid-point of the age group. Assuming a sex ratio at birth of 105,estimate the gross reproduction rate and the net reproduction rate forthis population.

4. The population of a certain country was 5.3 million at 30 June 2000and 5.7 million at 30 June 2005. Estimate the country�s population at30 June 2002 assuming

(a) a linear model of population growth, and

(b) a geometric model of population growth.

5. Consider the linear model of population growthPt = P0(1 + rt), andthe geometric model ~Pt = P0(1 + r)t. Is ~Pt always greater than Pt?

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6. Census data for a certain state reveal a population size in 1993 of 4million, in 1998 of 4.34 million and in 2003 of 4.52 million. Fit alogistic curve to these data and hence estimate the state�s populationin mid-2013. Assume that each census took place mid-year.

7. You have been asked to project the male population of Victoria overage 60 as at September 2020.

(a) What data would you require as at September 2010?

(b) What assumptions would you have to make in order to carry outthe projection?

Exam Questions

1. The table below shows data for a city over the last calendar year.

Age Number of Number of Femalegroup male births female births population15-24 5,500 5,200 60,00025-34 5,800 5,400 65,00035-44 4,000 3,800 63,00045-49 500 480 32,000

Calculate the following:

(a) the sex ratio at birth,

(b) age speci�c fertility rates for each age group,

(c) the gross reproduction rate,

(d) the net reproduction rate assuming that l0 = 100; 000 and thatfor 15 � x � 50;

lx = 99; 000� 100x:

2. Brie�y describe the trends in age speci�c fertility rates in Australiaover the last 10 years.

3. State with reasons which you would expect to be the largest, and whichthe smallest of

(a) gross reproduction rate,

(b) sex ratio at birth, and

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(c) net reproduction rate

in Australia in 2006. (You may assume each is quoted as a rate per1,000 women.)

Problems from AITAS2E:Chapter 3: 19, 20, 21, 22, 23, 24 and 25

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Problem Set 8

1. The probability that a life aged 20 survives for t years is 0:998t for0 � t � 20: Using an interest rate of 6% per annum e¤ective, calculate

(a) the EPV of a payment of $100,000 if (20) survives for 20 years,

(b) the EPV of a payment of $100,000 at the end of the year of (20)�sdeath should death occur before age 40,

(c) the EPV of payments of $10,000 annually in arrear for 20 yearsprovided (20) is alive, and

(d) the EPV of payments of $10,000t at times t years from the presentprovided (20) is alive where t = 1; 2; : : : ; 20:

Verify your answers by setting up spreadsheet calculations.

2. A 20 year old male is subject to a constant force of mortality of 0.012over the next 10 years. An insurance contract provides him with$100,000 in 10 years time if he is alive then. Assuming an e¤ectiverate of interest of 6% per annum e¤ective for �ve years, and 5.5% perannum e¤ective thereafter, what is the EPV of this payment to thenearest dollar?

3. Consider a �xed coupon bond paying interest at 8% per annum half-yearly and redeemable at par in six years�time. Using an interest rateof 6% per annum convertible half-yearly, and assuming the probabilityof a payment due time t (years) hence is 0:98t, calculate the expectedpresent value of the payments under the bond per $100 nominal (face)value. Show your answer correct to three decimal places.

Verify your answer using a spreadsheet calculation.

Exam question

1. A local government bond has a face value of $100 and a coupon of6% per annum, payable half yearly. An investor assumes that theprobability that a payment due t years from the issue date of the bondis 0:995t for t = 0:5; 1; 1:5; :::; 10.

(a) Show that under an e¤ective rate of interest of 5% per annum, theinvestor calculates the expected present value of payments fromthis bond as $104.17.

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(b) Suppose that the investor pays $103.50 and that all payments dueunder the bond are made. Find the e¤ective rate of interest perannum the investor obtains on his purchase. Give your answer tofour decimal places.

Problems from AITAS2E:Chapter 5: 1, 2, 9, 10, 16 and 17

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Problem Set 9

1. By general reasoning, deduce which single premium required to providethe bene�t described in each of the following pairs will be higher. Ineach case, the sum insured is identical.

(a) A 20 year endowment insurance for a 30 year old and a 20 yearendowment insurance for a 50 year old.

(b) An endowment insurance for a 20 year old for a term of 20 yearsan endowment insurance for a 20 year old for a term of 40 years.

(c) An endowment insurance for a person aged x for 30 years and awhole life policy for a person aged x.

(d) An endowment insurance with term 30 years for a 30 year old anda term insurance with term 30 years for a 30 year old.

(e) A whole life policy for a 25 year old that is participating and awhole life policy for a 25 year old that is non-participating.

(f) A ten year term insurance for a male aged 20 and a ten year terminsurance for a female aged 20.

(g) A 10 year term insurance for a male aged 20 and a 10 year terminsurance for a male aged 30.

2. You have been asked to write, in no more than 120 words, a responseto the following question for a brochure that will be distributed to thepublic by the life insurance industry body: �What would happen if alife insurance company asked you no questions when you applied forinsurance?�What major points would you cover in your reply?

3. A trainee actuary in a life insurance company has suggested that aperson�s postcode be used as part of the underwriting process. Do youthink this is reasonable?

4. (a) Use Google to �nd a de�nition of moral hazard.

(b) Do you think that suicide is an example of a moral hazard in thecontext of a term insurance policy? If so, how might an insurancecompany protect itself against this risk?

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Exam Questions

1. What features do an endowment insurance policy and a whole life in-surance policy have in common, and what features di¤er?

2. Bill and Ben are both aged 50. Bill has never smoked, but Ben hasbeen a regular smoker since the age of 20.

(a) Bill and Ben decide to buy whole life insurance with sum insured$100,000 by annual premiums from the same life insurance com-pany. Who pays the higher annual premium? Justify your answer.

(b) Bill and Ben decide to buy deferred annuities commencing at age60 by single premium from the same life insurance company. Theannual amount of the annuity will be $40,000, payable in advance.Who pays the higher single premium? Justify your answer.

Problems from AITAS2E:Chapter 4: 1, 2, 3 and 4

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Problem Set 10

1. Calculate �a30, the expected present value of a payment of 1 per annumat the beginning of every year while a life now aged 30 is alive using aforce of interest of 6% per annum and assuming a force of mortality of0.005 per annum for all ages.

2. Let l30+t = 100�t for t = 0; 1; 2; :::10. Using an e¤ective rate of interestof 5% per annum, set up a spreadsheet to calculate the following:

(a) �a30:10 ;

(b) A30:10 , and

(c) the annual premium for a 10 year endowment insurance issued toa life aged 30 with sum assured $200,000, with the death bene�tbeing payable at the end of the year of death.

3. Use the recursive relationship �ax = 1 + vpx �ax+1 to calculate all valuesof �ax for x = 25; 26 and 27, using an e¤ective interest rate of 5%per annum given �a28 = 18:464, q25 = 0:000695, q26 = 0:000672 andq27 = 0:000656:

4. Calculate A30, the expected present value of 1 payable at the end ofthe year of death of a life now aged 30 assuming a rate of interest of6% per annum e¤ective and a force of mortality of 0.002 per annum forages up to 50 and 0.03 for ages above 50. Show your answer correct to�ve decimal places.

Exam Questions

1. A life insurance company issues 10 year endowment insurance policiesto lives aged 30 with sum insured S. The insurance company assumesthat the lives are subject to a constant force of mortality � betweenages 30 and 40, and assumes it can earn interest at rate i per annum.

(a) Find expressions for tp30 and q30+t.

(b) Show that

P = S

�v � v e

�� � v10 e�10�1� v10 e�10�

�:

Derive all results that you use.

(c) Use general reasoning to explain why P is an increasing functionof �.

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2. Suppose that tp50 = expf�t=60g for t � 0 and that the e¤ective rateof interest will be 5% per annum for the next ten years, and 4.5% perannum thereafter.

(a) Calculate the expected present value of a payment of 1 to (50) ifhe is alive �ve years from now. Give your answer to four decimalplaces.

(b) Calculate the expected present value of a payment of 1 to (50)if he is alive �fteen years from now. Give your answer to fourdecimal places.

(c) Calculate the expected present value of an annuity of $50,000 perannum payable annually in advance to (50) (so that the �rst pay-ment is now) as long as (50) is alive. Give your answer to thenearest dollar.

Problems from AITAS2E:Chapter 5: 3, 4, 5, 6, 7, 8, 12, 13, 14, 15, 18, 19, 20, 21, 22 and 23.

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Problem Set 11

1. (a) What is the di¤erence between �Comprehensive�insurance coverand �Third party, �re and theft cover�for motor vehicles? (Hint:visit the CGU website.)

(b) Would you expect an insurance company to use the same approachto calculating an outstanding claims reserve under each type ofcover in part (a)? (Hint: what, if any, are the possible reasons fordi¤erences in claims settlement patterns?)

(c) What is the purpose of rating factors in general insurance? Whatdo you think are the main advantages and disadvantages of a largenumber of rating factors and a small number of rating factors?

2. Under proportional reinsurance, the reinsurer pays an agreed propor-tion of claims that occur in a year. If the agreed proportion is 30%,

(a) how much does the insurer pay if the total amount of claims inthe year is $3,000,000?

(b) how much does the reinsurer pay if the total amount of claims inthe year is $5,000,000?

3. Under a stop loss reinsurance arrangement, there is a retention level. Ifthe amount of claims is below the retention level, the insurer pays theclaims in full. If the amount is above the retention level, the insurer�sclaims payment is limited to the retention level and the reinsurer paysthe di¤erence between the amount of claims and the retention level.Suppose the retention level is $1 million.

(a) Find the share of claims for the insurer and the reinsurer if theamount of claims is $800,000.

(b) Find the share of claims for the insurer and the reinsurer if theamount of claims is $1.2 million.

(c) Find the share of claims for the insurer and the reinsurer if theamount of claims is $10 million.

4. Based on your answers to questions 3 and 4,

(a) do you think that in general proportional reinsurance would bemore attractive to a reinsurer than stop loss reinsurance?

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(b) do you think that in general proportional reinsurance would be amore expensive form of reinsurance than stop loss reinsurance?

5. Visit the UniSuper website, and look at the investment performanceof the Accumulation Funds over the last 5 �nancial years. (Financialyears start on 1 July in Australia.) Consider an academic whose salarywas as follows in each of these �nancial years:

Year Salary1 72,4562 76,3283 80,7884 84,5605 85,000

Suppose that this person�s employer contributed 14% of salary at thevery start of each of these �ve years to UniSuper. What would be theaccumulated amount of these contributions at the end of year 5 if

(a) the person�s Investment Option was Balanced.

(b) the person�s Investment Option was Growth.

Are you surprised by these answers?

http://www.unisuper.com.au/investments/investment-performance

Exam question

1. (a) What is the purpose of rating factors in general insurance?

(b) Give three examples of rating factors for motor vehicle insuranceother than age and gender, saying why each is appropriate as arating factor.

(c) A woman has owned the same car for ten years. Give three reasonswhy her premium might reduce over the ten year period.

Problems from AITAS2E:Chapter 4: 6, 7, 8 and 9

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Problem Set 12

1. As a contributing member of a superannuation fund, what do you seeas the principal positive and negative features of a de�ned bene�t fundas compared with a de�ned contribution fund? How is your answera¤ected if you are a contributing employer rather than a member?

2. Give three examples of di¤erences between the operation of a socialinsurance scheme and a life insurance fund.

3. When the government provides a �xed rate age pension, what do youthink are the pros and cons of having

(a) a non-contributory means-tested system? (i.e. a system underwhich there are no contributions and entitlement to bene�t de-pends on income)

(b) a contributory �at rate system? (i.e. a scheme under which con-tributions are at a �xed level)

Exam Questions

1. (a) Explain how a disability income insurance policy operates.

(b) What features of these policies are intended to reduce claims?

(c) Apart from disability, what other policyholder related risk doesan insurance company issuing such policies face? Can this risk bemanaged?

2. State whether each of the following statements is true or false (no jus-ti�cation is required):

(a) under de�ned bene�t superannuation, scheme members bear theinvestment risk,

(b) underwriting takes place before a person is admitted to a socialinsurance scheme,

(c) under de�ned contribution superannuation, a scheme member�s re-tirement bene�t is a multiple of the member�s salary at retirement,and the multiple depends on the length of scheme membership.

Problems from AITAS2E:Chapter 4: 5 and 10

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