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/ 65
Subject CM2
Revision Notes
For the 2019 exams
Stochastic models of
investment return
Booklet 2
covering
Chapter 5 Stochastic models of
investment returns
The Actuarial Education Company CONTENTS
Contents Page
Links to the Course Notes and Syllabus 2
Overview 3
Core Reading 4
Past Exam Questions 43
Solutions to Past Exam Questions 25
Factsheet 62
Copyright agreement
All of this material is copyright. The copyright belongs to Institute and
Faculty Education Ltd, a subsidiary of the Institute and Faculty of Actuaries.
The material is sold to you for your own exclusive use. You may not hire out,
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Legal action will be taken if these terms are infringed. in addition, we may
seek {o take disciplinary action through the profession or through your
employer. These conditions remain in force after you have finished using
the course.
© IFE; 2018 Examinations Page 4 LINKS TO THE COURSE NOTES AND SYLLABUS
Material covered in this booklet
Chapter 5 Stochastic models of investment returns
This chapter number refers to the 2019 edition of the ActEd course notes.
Syllabus objectives covered in this booklet
3. Stochastic investment return models (Chapter 5)
34
344
3.1.2
3413
3.4.4
3.15
Page 2
Show an understanding of simple stochastic models of
investment returns.
Describe the concept of a stochastic investment return model
and the fundamental distinétion between this and a
deterministic model.
Derive algebraically, for the model in which the annual rates of
return are independently and identically distributed and for
other simple models, expressions for the mean value and the
variance of the accumulated amount of a-single premium:
Derive algebraically, for the model jn which the annual rates of
return are independently and identically distributed, recursive
relationships which permit the evaluation of the mean value
and the variance of the accumulated amount of an annual
premium,
Derive analytically, for the model in which each year the
random variable (1 + ) has an independent log-normai
distribution, the distribution functions for the accumulated
amount of a single premium and for the present value of a
sum due al a given specified future time.
Apply the above results to the catculation of the probability
that a simple sequence of payments will accumulate to a given
amount at a specific future time
@ IPE: 2019 Examinations OVERVIEW
This booklet covers Syllabus Objective 3, which relates to stochastic models
of investment returns.
Breakdown of topics
In this chapter, we allow the interest rate to vary rather than assuming that it
is the same for the next 1 years, There are two models, namely:
+ the fixed rate model
+ the variable rate model.
We have to calculate the mean and variance of accumulated values and
also calculate probabilities based on accumulated values exceeding certain
amounts, using the lognormal model.
Exam questions
The past exam questions on the stochastic chapter can be “bookworky”,
involving proofs. but generally the questions expect you to carry out
calculations of means, variances or probabilities.
© IFE: 2019 Examinations Page 3 CORE READING _
All of the Core Reading for the topics covered in this booklet is contained in
this section.
Chapter 15— Stochastic models of investment returns
Financial contracts are often of a long-term nature. Accordingly, at the
outset of many contracts there may be considerable uncertainty about
the economic and investment conditions which will prevait over the
duration of the contract. A deterministic model is a model that
provides an output based on one set of parameter and input variables,
Howeves, deciding which set of input variables to use may be a
challenge. Thus, for example, if it is desired to determine premium
rates on the basis of one fixed rate of return, itis nearly always
necessary to adopt a conservative basis for the rate to be used in any
calculations, subject to the premium being competitive.
An alternative approach to recognising the uncertainty that in reality
exists is provided by the use of stochastic models. In such models, no
single rate is used and variations are allowed for by the application of
probability theory. Possibly one of the simplest models is that in
which each year the rate obtained is independent of the rates of return
in afl previous years and takes one of a finite set of values, each value
having a constant probability of being the actual rate for the year.
Alternatively, the rate may take any value within a specified range, the
actual value for the year being determined by some given probability
density. function.
At this stage we consider briefly an elementary example, which —
although necessarily artificial ~ provides a simple introduction to the
probabilistic ideas implicit in the use of stochastic rate models.
Suppose that an investor wishes to invest a lump sum of P into a fund
with compound investment rate growth at a constant rate for n years.
This constant investment return is not known now, but will be
determined immediately after the investment has been made.
Page 4 @IFE: 2019 Examinations 3. The accumulated value of the sum will, of course, be dependerit on the
investment rate. In assessing this value before the investment rate is
known, it could be assumed that the mean rate will apply. However,
the accumulated value using the mean rate will not equal the mean
accumulated value. In algebraic terms:
{ok "fk
Pit+ DS (ijpy) el Seton)
Lia i }
where:
i, is the jr of k possible investment rates of return
p; is the probability of the investment rates of return i;
4 In our previous example the effective annual investment rate of return
was fixed throughout the duration of the investment. A more flexible
model is provided by assuming that over each single year the annual
yield on invested funds will be one of a specified set of values or lie
within some specified range of values, the yield in any particular year
being independent of the yields in all previous years and being
determined by a given probability distribution.
5 Measure time in years. Consider the time interval [0,n} subdivided
into successive periods [0,11,14,2],....[0~1.n]. For t=1,2,....1 let i
be the yield obtainable over the tth year, ie the period [t~1,t]. Assume
that money is invested only at the beginning of each year. Let Fi
denote the accumulated amount at time-t of all money invested before
time tand tet P, be the amount of money invested at time t.
Then, for t= 1,2,3,.
Fre (1+ i Fat Pea) (4)
© (FE: 2019 Examinations Page 5 6 It follows from this equation that a. single investment of 1 at time 0 wilt
accumulate at time.n to:
Sq = (14 AM Hig) (M4 Fa) (1.2)
7 Similarly a series of annual investments, each of amount 4, at times
0,1,2,...,n~1 will accumulate at time n to:
Ag = (VEAL + ati). (+ fn)
“(V5 gL Hg) (1 fa)
lin AME iad
+(e ip) (1.3)
Note that A, and S, are random variables, each with its own
probability distribution function.
For example, if the yicid each year is 0.02, 0.04, or 0.06 and each value
is equally likely, the value of S,, will be between 1.02" and 4.08”.
Each of these extreme values will occur with probability (1/3)".
8 In general, a theoretical analysis of the distribution functions for A,
and S, is somewhat difficult. It is often more useful to use simulation
techniques in the study of practical problems. However, itis perhaps
worth noting that the moments of the random variables A, and S,
can be found relatively simply in terms of the moments of the
distribution for the yield each year. This may be seen as follows.
Page 6 © IFE: 2019 Examinations 9 From Equation (1.2) we obtain:
”
(Sp)* = T] Ai
tet
and hence:
r 1
EISh] =! hava)
if J
iets ig] (1.4)
since {by hypothesis) iy,/2.....i, ate independent, Using this last
expression and given the moments of the annual yield distribution, we
may easily find the moments of S,..
10 For example, suppose that the yield each year has mean j and
variance $?. Then, Jetting k =1 in Equation (1.4), we have:
2
EUSy] = TEU? iM]
tet
rt a
» TT ELH = 14 A (1.5)
to
since, for each value of t, Eli] =f.
41 With k =2 in Equation (1.4) we obtain:
etsty~ [Letts 24 +720
i
© [t+ 261+ B02)
tet
a(leaja fast)? (1.6)
© IFE: 2019 Examinations Page 7 since, for each value of f:
Et21 = (Eli? +vartiy] =? +8?
12 The variance of S,, is:
varlSp]= ELS] ~ (EIS)? = Ue 2jr sy? +s?) = (14 (1.7)
from Equations (1.5) and (1.6).
‘These arguments are readily extended to the derivation of the higher
moments of S, in terms of the higher moments of the distribution of
the annual investment rates of return.
43 It follows from Equation (1.3) {or from Equation (1.1}) that, for n= 2:
Ay = (tia \(t+ Ana (4.8)
The usefulness of Equation (1.8) lies in the fact that, since Ay.
depends only on the values jj,i2,.--.fy-1, the random variables i, and
Ay are independent. (By assumption the yields each year are
independent of one another.) Accordingly, Equation (1.8) permits the
development of a recurrence relation from which may be found the
moments of A,,. We illustrate this approach by obtaining the mean
and variance of A, -
14 Let:
Hn = ELAg]
and let:
my = ELAR
Page 8 @ IFE: 2019 Examinations 15
Since:
Ayedeig
it follows that:
saya ttes
and:
my = 1424 j2 +s?
where, as before, j and s* are the mean and variance of the yield each
year.
Taking expectations of Equation (1.8), we obtain (since j, and Ay.4
are independent):
by =O A+ Brad n22
This equation, combined with initial value 4, , implies that, for alt
values of n:
Bq Se atratej 1.3)
Thus the expected value of 4, is simply $7, calculated at the mean
rate of return.
Since:
AZ = (14 2iy + BMA 2g. 4 + AB 4)
by taking expectations we obtain, for 122:
Iq = (1625 +f? + 87)(44 ttn 4+ Mp td (4.10)
© IFE: 2019 Examinations Page 9 As the value of (4,4 is known (by Equation (1.9)), Equation (1.10)
provides a recurrence relation for the calculation successively of
‘Mp,MMq,M4,.... The variance of A, may be obtained as:
var{A,] = E[A2]~(ELA,))? = Ma ~ Ha (4.14)
in principle the above arguments are fairly readily extended to provide
recurrence selations for the higher moments of A, .
46 A company considers that on average it will earn interest on its funds
at the rate of 4% pa. However, the investment policy is such that in any
one year'the yield on the company’s funds is equally likely to take any
value between 2% and 6%.
For both single and annual premium accumulations with terms of 5, 10,
45, 20, and 25 years and single (or annual) investment of £1, find the
mean accumulation and the standard deviation of the accumulation at
the maturity date: (Ignore expenses.)
4
3
The annual rate of interest is uniformly distributed on the interval
[0.02,0.06]. The corresponding probability density function is
constant and equal to 25 (ie 1/(0.06-0,02)). The mean annual rate of
interest is clearly:
J+ 0.04
and the variance of the annual rate of interest is:
~ 2. 4g
qq hos 02)° = 310
We are required to find ELAn], (varLA, I, ElS,.and (var{S,]}! for
n= 5,10,15,20 and 25 .
Substituting the above values of j and s* in Equations (1.5) and (1.7),
we immediately obtain the results for the single premiums.
Page 10 @IFE: 2019 Examinations For the annual premiums we must use the recurrence relation (1.10)
(with Bp
J; at 4%) together with Equation (1.11).
The results are summarised in Table 1. It should be noted that, for
both annual and single premiums, the standard deviation of the
accumulation increases rapidly with the term.
‘Single premium €4 ‘Annual premium £1
Term [~ Mean Standard ean Standard
(years) | accum deviation accum deviation
&) a: (£) 3)
5 1.21665 0.03021 8.63298 0.09443,
10 4.48024 0.05198 1248635 | 0.28353
15 | 1.80094 0.07748 20.82453 | 0.87899
20 219112 0.10886 30.96920 | 1.00476
25 | 2.66584 0.44810 4331174 | 1.59392
‘Table 1
18 In general a theoretical analysis of the distribution functions for A,
and S$, is somewhat difficult, even in the relatively simple situation
when the yields each year are independent and identically distributed.
There is, however, one special case for which an.exact analysis of the
distribution function for S, is particularly simple.
Suppose that the random variable log(1+i;) is normally distributed
with mean yw and variance o*. In this case, the variablé (1+i,) is said
to have a lognormal distribution with parameters 4 and o*.
49 Equation (1.2) is equivalent to:
a
logS, = S,log(t+ i)
tm
@ IFE: 2019 Examinations
Page 14 20 The sum of a set of independent normal random variables is itself a
normal random variable. Hence, when the random variables
(1+f) (f 21) are independent and each has @ tognormal distribution
with parameters-and y and o®, the random variable S$, has a
lognormal distribution with parameters ny and no.
Since the distribution function of a lognormal variable is readily written
down in terms of its two: parameters, in the particular case when the
distribution function for the yield each year is lognotmal we have a
simple expression for the distribution function of S,..
21 Similarly for the present value of a sum of 1 due at the end of years:
Vp a(teiyt (iy) 4
erlogVy =--log(t+ iy) -log(t+ fn)
2
8
Since, for each value of t, log(1+-i,) is normally distributed with mean
# and variance o?, each term on the right band-side of the above
equation is normally distributed with mean —j and variance o*. Also
the terms are independently. distributed. So, log¥, is normally
distributed with mean ~ny and variance no*. Thatis, V, has
lognormal distribution with parameters ny and no*.
23 By statistically modelling V,,, it is possible to answer questions such
as;
* toagiven point in time, for a specified confidence interval, what is
the range of values for an accumulated investment?
* what is the maximum loss which will be incurred with a given level
of probability?
It can also be noted that these techniques may be extended to
calculate the risk metrics such as VaR, as introduced in a previous
chapter, of a series of investments.
Page 12 @IFE: 2019 Examinations PAST EXAM QUESTIONS
This section contains all of the past exam questions from 2008 to 2017 that
are related to the topics covered in this booklet.
Solutions are given later in this booklet. These give enough information for
you to check your answer, including working, and also show you what an
adequate examination answer should jook like. Further information may be
available in the Examiners’ Report. ASET or Course Notes.
We first provide you with a cross reference grid that indicates the main
subject areas of each exam question. You can use this, if you wish, to
select the questions that relate just to those aspects of the topic that you
may be particularly interestad in reviewing.
Alternatively, you can choose {o ignore the grid, and instead attempt each
question without having any clues as to ils content.
OIFE; 2019 Examinations Page 13 Cross reference grid
Qn see Proof way Lognormal | Attempted
1 v t
2 v
3
4 yo
5 v
6 a
2 a
8 v
9 “|
10 v
"1 v
12 v
13 v
14 v Y rn
6 v wy io
16 v yw v
7 a as)
18 v Ee
Page 14
© IFE: 2019 Examinations Subject CT1, April 2008, Question 10
‘An insurance company holds a large amount of capital and wishes to
distribute some of it to policyholders by way of two possible options.
Option A
£100 for each policyholder will be put into a fund from which the expected
annual effective rate of return from the investments. will be 5.5% and the
standard deviation of annual returns 7%. The annual effective rates of
return will be independent and (1+/,) is lognormally distributed, where i, is
the rate of return in year t. The policyholder will receive the accumulated
investment at the end of ten years.
Option B
£100 will be invested for each policyholder for five years at a rate of return of
6% per annum effective. After five years, the accumulated sum will be
invested for a further five years at the prevailing five-year spot rate. This spot
rate wil be 1% per annum effective with probability 0.2, 3% per annum
effective with probability 0.3, 6% per annum effective with probability 0.2,
and 8% per annum effective with probability 0.3. The policyholder will
receive the accumulated investment at the end of ten years.
Deriving any necessary formulae:
(i) Calculate the expected value and the standard deviation of the sum the
policyholders will receive at the end of the ten'years for each of options
Aand B. (17)
(ii) Determine the probability that the sum the policyholders will receive at
the end of ten years will be less than £116 for each of options A and B.
15)
(ii) Comment on the relative risk of the two options from the policyholders’
perspective.
[Total 24]
® IFE: 2019 Examinations Page 15 Subject CT1, September 2008, Question 6
A pension fund holds an asset with current value £4 million. The investment
return on the asset in a given year is independent of returns in all other
years, The annual investment return in the next year will be 7% with
probability 0.5 and 3% with probability 0.5. in the second and subsequent
years, annual investment retums will be 2%, 4% or 6% with probability 0.3,
0.4 and 0.3, respectively.
(@ Calculate the expected accumulated value ‘of the asset after 10 years,
showing ail steps in your calculations. (3
(i) Calculate the standard deviation of the accumulated value of the asset
after 10 years, showing all steps in you" calculations. 14}
(i) Without doing any further calculations explain how the mean and
variance of the accurnulation would be affected if the returns in years 2
to 10 were 1%, 4%, or 7%, with probability 0.3, 0.4 and 0.3 respectively.
2
a}
(Tota! 9]
Subject CT1, April 2009, Question 11
An individual wishes to receive an annuity which is payable monthly in
arrears for 15 years. The annuity is to commence in exactly 10 years at an
initial rate of £12,000 per annum. The payments increase at each
anniversary by 3% per annum. The individual would like to buy the annuily
with a single.premium 10 years from now.
(i) Calcufate the single premium required in 10 years’ time to purchase the
annuity assuming an interest rate of 6% per annum effective. 5]
The individual wishes to invest a lump sum immediately in an invesiment
product such that, over the next 10 years, it will have accumulated to the
premium calculated in ()). The annual effective returns from the investment
product are independent and (1+/,) is lognormally distributed, where i, is
the return in the f th year. The expected annual effective rate of return ts 6%
and the standard deviation of annual returns is 15%.
(i) Calculate the tump sum which the individual should invest immediately
in order to have a probability of 0.98 that the proceeds will be sufficient
to purchase the annuity in 10 years’ time. (9
(iii) Comment on your answer to (ii). (2)
TFotal 16]
Page 16 @ IFE: 2019 Examinations Subject CT1, September 2009, Question 9
A life insurance company is issuing a single premium policy which will pay
out £20,000 in twenty years’ time. The interest rate the company will earn
on the invested funds over the first ten years of the policy will be 4% per
annum with a probability of 0.3 and 6% per annum with a probability of 0.7.
Over the second ten years the interest rate eared will be 5% per annum
with probability 0.6 and 6% per annum with probability 0.5,
() Calculate the premium that the company would charge if it calculates
the premium using the expected annual rate of interest in each ten-year
period,
(i) Calculate the expected profit to the company if the premium is
calculated as in (i), The rate of interest in the second ten-year period Is
independent of that in the first ten-year period. (3)
(ii), Explain why, despite the company using the expected rate of interest to
calculate the premium, there is a positive expected profit. 2
(iv) By considering each possible outcome in (
(a) Find the range of possible profits.
(b) Calculate the standard deviation of the profit to the company. (7]
[Total 14]
Subject CT1, April 2010, Question 6
The annual returns, /, on a fund are independent and identically distributed.
Each year, the distribution of 14/ is iognormal with parameters y= 0.05
and a? = 0.004, where 7 denotes the annual return on the fund.
() Catcutate the expected accumulation in 25 years’ lime if £3,000 is
invested in the fund at the beginning of each of the next 25 years. [5]
(ii) Calculate the probability that the accumulation of a single investment of
£1 will be greater than its expected value 20 years later. 5]
[Total 10}
‘© IFE: 2019 Examinations Page 17 Subject CT1, September 2010, Question 3
‘The annual rates of retum from an asset are independently and identically
distributed. The expected accumulation after 20 years of £1 invested in this
asset is £2 and the standard deviation of the accumulation is £0.60.
(a) Calculate the expected effective rate of retum per annum from the
asset, showing all the steps in your worsing.
(b) Calculate. the variance of the effective rate of return per annum. 6]
Subject CT, April 2011, Question 10
‘The annual rates of return from a particular investment, Investment A, are
independently and identically distributed. Each year, the distribution of
(tei),where i is the tate of interest eamed in year ¢, is lognormal with
parameters wand o”.
The mean and standard deviation of i are 0.08 and 0.03 respectively
() Calculate wand o?. 15}
‘An insurance company has liabilities of £15m to meet in one year's time. it
currently has assets of £14m. Assets can either be invested in
Investment A, described above, or in Investment B, which has a guaranteed
return of 4% per annum effective.
(i) Calculate, to two decimal places, the probability that the insurance
company will be unable to meet its liabilities if
(a) all assets are invested in Investment B.
(b) 75% of assets are invested in invastment A and 25% of assets are
invested in Investment B.
(iii) Calculate the variance. of retum from each of the portfolios in (i)(a) and
3)
GO). (3)
{Total 14]
Page 18 © IFE: 2019 Examinations Subject CT1, April 2012, Question 7
The annual yields from a fund are independent and identically distributed.
Each year, the distribution of 1+/ is fog-normal with parameters jr = 0.05
and 6” = 0.004, where 7 denotes the annual yield on the fund.
(i) Calculate the expected accumulation in 20 years’ time of an annual
investment in the fund of £5,000 at the beginning of each of the next 20
years. {5)
ii) Calculate the probability that the accumulation of a single investment of
£1 made now will be greater than its expected value in 20 years’ time. [5]
TTotal 10]
Subject CT1, September 2012, Question 7
An individual wishes to make an investment that will pay out £200,000 in
twenty years’ time. The interest rate he will earn on the invested funds in the
first ten years will be either 4% per annum with probability of 0.3-or 6% per
annum with probability 0.7. The interest rate he will-earn on the invested
funds in the second ten years will also be either 4% per annum with
probability of 0.3 or 6% per annum with probability 0.7. However, the
interest rate in the second ten year period will be independent of that in the
first ten year period,
(i) Calculate the amount the individual should invest if he calculates the
investment using the expected annual interest rate in each ten year
period, el
(i) Calculate the expected value of the investment in excess of £200,000 if
the amount calculated in part (i) is invested. {3}
Gi) Calculate the range of the accumulated amount of the investment
assuming the amount calculated in part ()) is invested. (2)
Trotal 7}
© IFE: 2019 Examinations Page 19 410 Subject CT1, April 2013, Question 6
Acash sum of £10,000 is invested in a fund and heid for 15 years. The yield
‘on the investment in any year will be 5% with probability 0.2, 7% with
probability 0.6 and 9% with probability 0.2, and is independent of the yield in
any other year.
() Caloulate the mean accumulation at the end of 15 years. re
(i) Calculate the standard deviation of the accumulation at the end of 15
years,
Gi) ‘Without carrying out any further calculetions, explain how your answers
fo parts (i) and (ji) would change (if at all) if:
(a) the yields had been 6%, 7% and 8% instead of 5%, 7%, and ‘9%
per annum, respectively.
(b) the investment had been held for 13 years insiead of 15 years. [4]
otal 11}
41 Subject CT1, September 2013, Question 7
An insurance company has just written contracts that require if to make
payments to policyholders of £10 million in five years’ time. The total
premiums paid by policyholders at the outset of the contracts amounted to
£7.85 million. The insurance company is ‘o invest the premiums in assets
that have an uncertain return. The return from these assets in year t, i,
has a mean value of 5.5% per annum effective and a standard deviation of
4% per annum effective. (1+/,) is independently and lognormally
distributed,
() Calculate the mean and standard deviation of the accumulation of the
premiums over the five-year period. You should derive all necessary
formulae, [Note: You are not required to derive the formulae for the
mean and variance of a lognormal distribution } 91
A director of the insurance company is concerned about the possibility of a
considerable loss fromm the investment strategy suggested in part (i). He
therefore suggests investing in fixed-interest securities. with a guaranteed
return of 4 per cent per annum effective.
(i) Explain the arguments for and against the director's suggestion. —_ [3]
[Total 12]
Page 20 @ IFE: 2019 Examinations 12 Subject CT1, April 2014, Question 12
An investor is considering investing £18,000 for a period of 12 years. Let i,
be the effective rate of interest in the tth year, £< 12. Assume, for f<12,
that j, has mean value of 0.08 and standard deviation 0.05 and that 1+/, is
independently and fognormaily distributed.
{i) Determine the distribution of Sj where S, fs the accumulation of £1
over ¢ years. 6)
At the end of the 12 years the investor intends to use the accumulated
amount of the investment to purchase a 12-year annuity certain: paying:
£4,000 per annum monthly in advance during the first four years;
£5,000 per annum quarterly in advance during the second four years;
£6,000 per annum continuously during the final four years.
The effective tate of interest will be 7% per annum in years 13 to 18, and 9%
per annum in years 19 to 24, where the years are counted from the start of
the initial Investment.
(i) Caiculate the probability that the investor will meet the objective. [12]
{Total 47]
13 Subject CT1, September 2014, Question 2
A life insurance company is issuing a single premium policy which will pay
out £200,000 in 20 years’ time. The interest rate the company will, earn on.
the invested fund throughout the 20 years will be 4% per annum effective
with probability 0.25 or 7% per annum effective with probability 0.75. The
insurance company uses the expected annual interest rate to determine the
premium.
@ Caleulate the premium. 2
(i) Calculate the expected profit made by the insurance company at the
end of the policy
{Total 4]
© IFE: 2019 Examinations Page 21 44 Subject CT1, April 2015, Question 12
In any year, the -yield on investments with an insurance company has
mean j and standard deviation s and is independent of the yields in alt
previous years.
(i) Detive formulae for the mean and variance of the accumulated value
after a years of a single investment of 4 at time 0 with the insurance
company. fe)
Each year the value of (1+/,),where j, is the rate of interest earned in the
#” year, is lognormally distributed. The rate of interest has a mean value of
0.04 and standard deviation of 0.12 in all years.
(i) @) Calculate the parameters x2 and @” for the lognormal distribution
of (4H).
(b) Calculate the probability that an investor receives a rate of return
beiween 6% and 8% in any year. BB]
(ii) Explain whether your answer to part (ii) (b) looks reasonable. [2]
[Total 15]
15 Subject CT, April 2016, Question 8
An individual is planning to purchase £100,000 nominal of a bond on 1 June
2016 which will be redeemable at 110% on 1 June 2020. The bond will pay
coupons of 3% per annuny at the end of each year.
The individual wishes to invest the coupcn payments on deposit until the
bond is redeemed. Itis assumed that, in any year, there is a 55% probability
that the rate of interest will be 6% per annum effective and a 46% probability
that it will be 5.5% per annum éffective. Il is also assumed that the rate of
interest in any one year is independent of that in any other year.
{) Derive the necessary formula to determine the mean value of the total
accumulated investment on 1 June 2020. [4]
(i) Calculate the mean value of the total accumulated investment on 1 June
2020, 2]
{Total 6]
Page 22 © IFE: 2019 Examinations 18 Subject CT1, September 2016, Question 9
An insurance company has just written single premium contracts that require
it to make payments to policyholders of £10,000,000 in five years’ time. The
total single premiums paid by policyholders amounted to £8,000,000.
The insurance company is to invest the premiums in assets that have an
uncertain return. The return from these assets in year f, i; , is independent
of the relurns in all previous years with a mean value of 5.5% per annum
effective and a standard deviation of 4% per annum effective. (1+i,) is
lognormalty distributed.
(i) Calculate. deriving all necessary formulae, the mean and standard
deviation of the accumulation of the premiums over the five-year period.
9]
A director of thé company is concerned about the possibility of a
considerable toss from the investment in the assets suggested in part (i).
Instead, the director suggests investing in fixed interest securities with a
guaranteed return of 4% per annum effective.
(i) Set out the arguments for and against the director's position. 3)
[Total 12]
@IFE: 2019 Examinations Page 23 47 Subject CT1, Apri! 2017, Question 10
An individual aged exactly 66 intends to retire in five years’ time and receive
an annuity-certain. The annuity will be payable monthly in advance and wilt
cease afer 20 years. The annuity will increase at each anniversary of the
commencement of payment at the rate of 3% per annum.
The individual would like the initiat level of annuity to be £20,000 per annum.
The price of the annuity will be the present value of the payments on the
date it commencés usitig an interest rate of 7% per annum effective.
(Calculate the price of the annuity. (4)
In order to purchase the annuity described in part (i), the individual invests
£200,000 on his 65th birthday in a particular fund.
The investment retum on the fund in any given year is independent of
returns in all other years and the annual return is:
+ 4% with a probability of 60%
+ 7% with a probability of 40%.
(i) Calculate, showing all workings, the expected accumulation of the
investment at the time of retirement, 3]
Calculate, showing alt workings, tre standard deviation of the
investment at the time of retirement (4
(iv) Determine the probability that the individual will have sufficient funds to
purchase the annuity. (3]
{Total 14]
48 Subject CT1, September 2017, Question §
‘An individual invests £100 in an asset. The expected accumulation of this
asset after 20 years is £200 and the standard deviation of the accumulation
after 20 years is £50.
() Calculate the expected effective rate of return per annum, ay
(i) Calculate the standard deviation of the effective rate of retum per
annum. [4
{Total 5)
Page 24 ® (FE: 2019 Examinations SOLUTIONS TO PAST EXAM QUESTIONS
Subject CT1, April 2008, Question 10
(i) Expected value and standard deviation
Option A
Let j, denote the rate of interest in year f. ¢ and Si) denote the
accumulated value at time 10 of an investment of 1 at time 0. Then:
Sig = (4A) Teg) (140)
The expected value of this accumulation is:
E (Sto) EL (I+ A)(14 i) (ering) | = BA )E (ig) E (A449)
by independence. Now, for t= 12.3,..., we have:
E(t.) = 148 (ip) = 1.0855
=> E(S;9) 1.055"
The expected value of the sum each policyholder will receive is therefore:
E(1008)5} » 100E (Sjo) = 100 1.055" = £170.81
To calculate the standard deviation of the sum each policyholder will receive,
first calculate var(Sjq).
The expected value of the random variable S?, is:
E(sh)= 8] AA (i) Oriol |
7 Bl (reay Jeli) | “| (tig) | by independence.
@ IFE: 2019 Examinations Page 25 Now, for t = 1,2,3,..., we have:
el (iy? | ver(ten) eden)?
‘This follows from the variance formula var{x) =(x*)-[e(x))). Here
we have replaced X by 1+).
Since var(1+1;) = var{i;} and E(t+i,) =14EU),), ifollows thet:
elas) svarli) fe)
= 0.077 41.055"
= 4.117925
= & A17925%
Finally:
var(S,o)=€(S%)~[E (So) =1.117928"° -(1 oss?" =0.4310267
The standard deviation of the sum each policyholder will receive is therefore:
[rar (1008\0) = 100 /var (S,p) = 100V9.1310267 = £36.20
Option B
‘The table below summarises the possible accumulated values payable to
each policyholder at the end of the ten years, along with the probability
attached to each outcome.
Interest raie in final 5 |” Probability ‘Accumulated vaiue at time 10
years i
1% 0.2 100 1.065 x 1.0%
3% 03 400% 1.088 x 1,03°
6% 02 100 x 1.08° «1.06%
8% i 03 |___ 100 «1.06% x 4,08"
Page 26 @ IFE: 2019 Examinations The expected value of the sum each policyholder will receive is given by:
400 «1.065 (1.01 x0.24+1.03° x0.3 +1.08°x0.2 41.08% x03) = £169.48
The variance of the sum each policyholder will receive is:
100? «1.087 (1.0119 <0,2-+1.03"° x0,3+ 1.06" «0,24 1,08? x03}
169,487
= 29,189.88 -169,48"
= 467.538
The standard deviation of the sum each policyholder will receive is therefore:
(ii) Probability of sum received being less than £115
Option A
Now 4+ is lognormally distributed and E(j,) = 5.5% , sd(j,) = 7%. So:
E(tsi,}=t+E(i)=1.085 and var(t+i,)=var(i) = 0.07"
Using the formulae from page 14 of the Tables for the expectation and
variance of a lognormal random variable:
‘Squaring the first equation and substituting it into the second equation gives:
el! 24.085 and ete ~ 0.07%
0.077
1.055"
1.058" o* -4)+ 0.07% = aB min 13 ]-onsoos
A
Substituting this back into the first equation gives:
He = In1.055 ~ 6a? = 00513444
@ IFE: 2019 Examinations Page 27 So the distribution of Syq is:
Sio~ tog(1 op,1007) log N(0.513444,0.043928)
‘The probability we require is:
P(100Sjq < 118) = Pin Sip < In(1.1)}
_ In(i.45}—0.5134
= P{N(01
= b{-1,7829)
= 0.0373,
Option B
The accumulated amount after the first 5 years is:
100x 1.06" = £133.82
It is therefore certain that after the full 10 years each policyholder will receive
more than £115. In other words, the probability that the sum received by
each policyholder is less than £115 is zero.
(ii), Relative Risk
Option B is the less risky of the two options. This is shown by the lower
standard deviation of return and the lower probability of receiving less than
£115. Indeed, under Option B, the policyholder cannot receive less than:
100x 1.085 «1.01% = £140.65
‘The downside of Option B is that the expected sum at the end of the 10
years is lower than under Option A.
Subjéct CT1, September 2008, Question 6
(i) Expected accumulated value
Let i, be the interest rate that applies for year k. je from time k-t to
time k
Page: 28 © IFE: 2019 Examinations ‘The interest rate distribution in the first year of the investment is;
inferest Rate Probability)
[3% 05
7% 0.5
‘The expected value of the Interest rate in the first year is:
EU) = 0.03 x0.540.07 «0.5 = 0.05
The interest rate distribution in years 2 to 10 is:
The expected value of the interest rate for years 2, 3,..., 10 is:
E(j,) =0.02x0,3 + 0.04 «0.4 4.0.06 40.3 = 0.04
for k (k= 23... 10).
Letting Sj, represent the accumulated value at time 10 of an investment of
1 at time 6, and using the fact that the interest rate in any given year is
Independent of that in other years:
EiShp
© EAs WK). ig))
2 E(4+ VEC ig). EU+ iQ)
= (14 ECA) M1 + Elin)... (+ Elio)
= (4,05)(1.04)....(1.04)
=: (4.08)(1.04)?
“Therefore, the expected accumulated value of £1 million after 10 years is:
E(4,000,0008,9) ~ 4,000, 000E (Sip)
= 4,000,000(1.05)(1.04)?
~ £1,494,477
@ IFE: 2019 Examinations Page 29 (i) Standard deviation of the accumulated value
‘The expected value of the random variable Sfp is:
e(sh)= E(t+4) 5 nyt ‘0)°)
28 (CAP E( +i )..2(+Aa?)
by independence of the interest rates in different years.
Now:
E(as i?) = 1.037 0.844.077 x05
= 4.1029
For k=2,3,.... 10, we have:
e( tig?) # 1.02? «0.8 1.042 «0.4 +1.087 «0.3
= 1.08184
This gives us:
&(si,)=1.1029% 1.081849
The variance of Syq is therefore:
2
var (Sig) = E (Sfp) ~| (Sto) |
o ot
= 1.1029 *1,08184' [1,051.04 |
= 0.0052762248
Finally, the standard deviation of the accumulated value of £1 million after 10
years is:
var (1,000,0008; 9 }) = 1000,000/0.0052762248
w= £72,638
Page 30 FE: 2019 Examinations (iil) Effect of a change in the interest rate distribution for years 2 to 10
The expected value of the interest rate for the final nine years of the
investment is unchanged under the new interest rate distrubution. This is
because the new interest rate distribution, like the original one, is symmetric
around the central value of 4%. Therefore, the expected accumulated value
of the investment after 10 years will be fhe same as that calculated in (i), as
this depends solely on the expected interest rate.
The variance of the new interest rate distribution is higher than for the
original distribution, This. is because the potential interest rates are more
spread out than previously, This means that the variance of the
accumulated value of the investment would increase from the value
caiculated in (ii), as a greater range of final values is now possible.
Subject CT1, April 2009, Question 11
(i) Single premium
The single premium, SP be paid in 10 years’ time Is equal to the present
value of payments from the annuity.
sp = 12,000{a! «
Bf st.03val «...1 1.03 val) — @o%
12,0002! (141.080 4...-t.03"4v"")
The expression in bracksts is a geometric seties of 15 terms, with a=1 and
common ratio r + 1.03v . Summing this gives:
(1.03v)"°
7 1.03v
.
1
2.0004? @6%
= 12,000 x 0.969067 x 12.36365
143,774
(i) Lump sum to invest immediately
We are given that:
tei og (ua); E(i,) =0.08 ; var(i,) = 0.15?
‘© IFE: 2019 Examinations: Page 31 Using the formulae on page 14 of the Tabies, we have:
Ett) 4+ 8U,) =1.08 <0" Equation 1)
var(t-+ ip) = varliy) = 0.18% = e297 fe -1) (Equation 2}
‘Squaring Equation 1 and substituting this into Equation 2 gives:
nf O18. vs 0.01982706
\1.06*
0.18? = 4.06" {2 1} =
Substituting this into Equation 1 gives:
= In(1.06) ~ 0.5 x 9.01982706 = 0,04826538
Hence, 1+; ~ log (0.04836538,0.01982706)
Let X be the amount invested now in order to be sufficient to buy the
annuity in 10 years' time with a probabilily of 0.98. X must satisfy the
equation:
P(X Sig 2 SP) = 0.98
where Sig is a random variable representing the accumulated value at time
40 of an amount 1 invested at tirne 0, and SP is the single premium
calculated in part (i).
Since the investment returns in each year are independent, we know that:
49 ~ log N(0.4835538,0.1982708)
Our equation for X now gives us:
143,774
0.98 = °(S6 eae - ( |
= PLASye oun 924)
Page 32 @IFE: 2019 Examinations Standardising gives:
Using page 162 of the Tables, we see that:
P(Z > 2.0537)=0.02 =s P(Z<~-2,0537) =0.02
=3 P(Z > ~2.0537) «0.98
Therefore:
w(? }-0.4835538
women 8 2.0537
V0.1982706 6
ini 48.774). 480909
ae
se X = £221,219
(i), Comment
The amount the individual needs to invest now is greater than the amount of
the single premium needed in 10 years’ time.
The reason for this is the large standard deviation of the interest rate
distribution, which means that it is likely that funds will decrease in size in
some years as the interest rate would be negative.
Subject CT1, September 2009, Question 9
(i) Premium
Let i, denote the annual interest in the first 10 years and i, denote the
annual interest rate in the second 10 years. Then:
E(i,) = 4% 0.3 46% x0.7 = 5.4%
Elin) = 5% x0,5 + 6% X0.5= 5.5%
@ FE: 2019 Examinations Page 33, Let P denote the single premium. Using the expecied annual rates of
interest, we have:
P(1.084)'° (1.055)'" = 20,000 =» P= 86,919.89
(i) Expected profit
The possible outcomes are shown in the table below:
(oterest Accumulated value of premium Profit | Probability
i, =0.04 1074 omit a 03x05
6,919.89 (1.04 4.05)" ~ 16,684.98 |-3,315.02
ip = 0.08 8 (1.04) (1.06) " = 18.68 \ 20.15
i = 0.04 - / 08x 0.
‘ 6,910.80(1.04)"° (1.08)'° = 18,343.88 |~1,656.12) 7 5
i, = 0.06 =0.15
i, = 0.06 _ ~~ 7x05
, 6,919.89(1.00)"°(1.05)"° = 20,186.03 | 186.03 6 ae.
74
6,919.89(1.06)"" = 22,193,02 2.19302 | ° se
So the expected profit is:
3,315.02 x 0.15 -- 1,656.12 x 0.15 4+-186.03 x 0,36 4 2,193.02 x 0.36
= £87.00
(i). Why there is a positive expected profit
‘The expected profit is non-zero because:
e[oray” leleriy?] e[teeta) em)
The expected profit is positive because there is a fairly high chance thal the
interest rate actually earned in the first 10 years is greater than the expected
rate.
Page 34 © IFE: 2019 Examinations (iva) Range of possible profits
Using the table in part (i), the range is:
2,193.02 ~ (~3,318.02) = £5,508.04
(iv)(b) Standard deviation of profit
From part (ii), we hav
"Probability
E (profit? 3,315.02" x 0.15 + 1,656.12? 0.15
4186.03? x 0.35 + 2,193.02? * 0.35
= 3,755,194
sd(profit) » 3,755,194 87" = £4,935.88
Subject C1, April 2010, Question 6
(i) Calculate the expected accumulation
Let A, represent the accunulated value at time 1 of a series of annual
investments, each of amount 1, at the start of each of the next 7 years.
We have £3,000 invested at the start of each year for the nexi 25 years, so
the expected value of this is:
E(3,000A,5} = 3,000E (Ags
38,0008; @J%
where we use the formula for the expected vaiue of A,, in which j/
represents the mean interest rate over the period. So, j = E(i).
To find the value of j, we can use the fact that 1+/~ log(0.06,0.004) .
@ IPE: 2019 Examinations Page 35, Using the formula given on page 14 of the Tables:
E (dai) = @ 05409+0.004 .. 4 0533757
Hence:
EU ti=i+Elstef = f= 5.33767%
‘Therefore:
E(3,000Ayg) = 3, 0008.55 @5.33757%
1.05337577* =)
aa
= soo
= £158,037
(i) Calculate the probability
Let S, represent the accumulated value at time n of an investment of
amount 4 at time 0.
Since 147 ~ logM(0.08,0.004} , and the annual returns are independent, we
know that:
Sgp ~logM(20 x 0.05,20* 0,004) =? In Sgq ~ NV(1,0.08)
We need to calculate the probability P (Szq > & (S20) -
Using the formula given on page 14 of the Tables:
1.04
E (Spo) = 9088008
Page 36 © IFE: 2019 Examinations Therefore, we need to calculate:
P(Sz9 > E(Sz0}) = P [S20 > 0°)
= P{inSxo > infer }
= P(InSgq > 1.04)
Since InSz is a normal random variable, we can standardise it by
subtracting the mean and dividing by the standard deviation. Letting
Z~NO.1:
P(Spq > E (Szq)) = ef
=P(Z>0.1414)
= 1 PZ <0,1414)
= 10,5562
= 0.444
Subject CT1, September 2010, Question 3
(i) Calculate the expected rate of return pa
E(S,) BLO HAII 8). A rig)}
= E+ EM +i)... Eig) by independence
= PEGI ECE)... (1 Eq)
We are told that the interest rates are identically distributed, so they will all
have the same mean j = E(j,). Hence:
ESp)= 9
‘Therefore:
2 4 =0.035265
E(Sxq) = 14 P= 2 =>
© IFE: 2019 Exarninations Page 37 (i) Calculate the variance of the effective rate of return pa
We have:
var(Syq) = (14 J? +8? ~ 4 jy? 0.6"
= (A+) +57) -27 = 0.67
a (26457) 24.36 = of 24.36" ~2% = 0.004628
Subject CTt, April 2011, Question 10
(i) The parameters of the lognormal distribution
We are given that:
(+i) ~ fog Gino?)
E(i,) = 0.06 me E(14 i) = 1.06
var (jy) = 0.037 so var(1+i;) = 0.03"
Using the Tables:
EU+i)ne2" = 1.08 w
var(t+j,) = ete e* 2)
Substituting (1) into (2):
2
1.06? {e” -} =0.097 = (e" 208
+06
).000800676
0.0%
7087
Page 38 © IFE: 2019 Examinations Substituting into (1):
ltt $o.000800876) _ 4 9g
= 1 =In1.06-4(0,000800676) = 0.0578686
So, (1+) log N(0.0578686,0.000800676) .
(ida) ‘The probability that liabilities will not be met if all assets are in
Investment B
£14m would accumulate to 14(1.04) = £14.56m by the end of the year, so
the company will not be able to meet its liabilities of £15n7. The probability
that the company will not meet its liabilities is 1.
(ib) The probability that liabilities will not be met if 75% of assets
are in Investment A and 25% are in Investment B
The £44m will be split into £10.5m in Investment A and £3.5m in
Investment B
The £3.5n7 in Investment B will accumulate to 3.5 x 1.04 = £3.64m .
In order to accumulate to £15m in total, the assets in Investment A will need
to accumulate to 15~3.64 = £14.36m,
The accumulation factor can therefore be found as:
10.5(1+ ip) = 11.36
(144) 1.0819
We want the probability that the liabilifies are nof mel, so we want:
P[ (teh) < 1.0819]
Since (14 i,) ~ log N(a,67), then:
Indi) ~ No?) and Z omen N01) Therefore:
, ; In1,0819-0.0878686
PL (+i) <1.0819] =P] Z <=
Uarins i ( <" jo. 600800676 }
= P(Z < 9.737)
=0.768
So the probability that the liabilities will not be met is 0.77 to two decimal
places.
(il) The variance of the returns from the portfolios
The portfolio in (ii)(a) is made up entirely of Investment B. This asset has a
fixed return of 4%, so there is no variation in the return, ie the variance of the
return = 0.
The return on the portfolio in (i)(b) will be a weighted average of the returns
from Investment A and Investment B, /e:
0.75%, +0.25 x 0.04
The variance of the return will be:
var{ 0.75), +0.25 0.04} = 0.75? vat (i,)= 0.78? x 0,03" = 000050625
Subject CT1, April 2012, Question 7
(i) The expected accumulation of an annual investment
We are given that:
(+i) ~log NU = 0,05, = 0.004)
where i is the annual yield on the fund.
E(B, 000Agg) = 5,000E(Agy) = 5,00055 5,
where j=E{i).
Page 40 @ IFE: 2019 Examinations Using the Tables:
EU ei) et t8 92 PHO 8 4 9533757
=» j = E(i) =0,0839787
Therefore:
E(5,000Ayp) = 5, 00083816 a5757%
1,0533757"9 ~1
4~1.0533757"'
= §,000 x 36.0998
= £180,499
= 5,000
(i) The probability that Sjq is greater than its expected value
(Sg) = (1+ j)?° = 1.0533787"° = 2.829217
Using the information in the question:
Spo ~ log N(2041,2007)
~ log N(20 x 0.05,20 x 0.004)
~ tog (10.08)
The required probability is:
P(Sgq > 2.829217) = P(ln Sap > In 2.829217)
f, _In2.829217.~4
= Pl 27S
\ 0.08 }
104-1)
«PZ = P(Z 0.14142
( poe J PF )
=1~(0.14142)
44377 Subject CT1, September 2012, Question 7
The distribution for the interest rates in each of the ten year periods is:
4% 8%
() Amount to be invested
The mean of the annual interest rate in-each of the ten year periods is:
Efi) = 0.04 x 0.3 +0.08 «0.7 = 0.054
We need to discount for 20 years using this rate to get the amount to invest:
200,000
= £69, 858.26
1.0547°
(i) Expected vaiue in excess of £200,000
After the 20 year period the amount in part 4) will accumulate to one of three
values:
69,858.26(1.04)29, 69,858.26(1.06)%°, or 69,858.26(1.04)"(1.06)"°
We can caloulate the probabilities of each of these possibilities using the
distribution of interest rates shown earlier:
69,858.26(1.045 | 69,858.26(1.06)°° | 69,858.26(1.04)'°(1.06)"°
s
20 = 153,068.06 = 224,044.91 = 185,186.71
Probability | 0.3x 0,3 = 0.09 0.7 x0.7 = 0.49 2%0,3x0,7 «0.42
We can then calculate the expectation from first principles:
153,068.06 x 0.09 + 224,044.91 0.49 + 185,186.71 x 0.42 = 201,336.55
So the expected value in excess of £200,000 is £1,336.55.
(i), Range of accumulated values
Wé can get the range of Values straight from the solution to part (i):
224,044,91~ 153,068.06 = £70,976.85
Page 42, @ FE: 2019 Examinations 10
Subject C71, April 2013, Question 6
() Mean accumulation
The mean interest rate in-each year is:
j= EG) «0.05 «0.2 40.07 0.6 +0.09x 0.2 = 0,07
So, the mean of the accumulation of £10,000 for 15 years is:
E(10,000S,5) = 10,000E(S,5) = 10,000 x (1+ /)'> = 10,000% 1.07%
= £27,590.32
(ii) Standard deviation
First we need the variance, s” , of the annual yield.
EU) = 0.08? 0,240.07" 06 +0.097 x 0.2 = 0.00506
so 8? = 0.00506 ~ (0.077 = 0.00016
So the variance of the accumulation of £10,000 for 15 years is;
var (10, 000S;5 ) = 10,000" var(S,s)
15,
10,0007 [ou + s*) si
of, 2 15 30
= 10,000 it 07? 0.00016)» ~1.07
« 10,0007) (1.14508)"° 1.07%
1,597, 283.16
So the standard deviation of the accurnulation is:
sdi(10,0008),) » (1857, 283.76 = £1,263.84
@IFE: 2019 Examinations. Page 43 (i)(@) Different yields
If the yields had been 6%, 7% and 8%, the value of { would stil have been
7%, so the mean accumulation would be the same. However, the individual
yield values are less spread out, so the standard deviation of the
accumulation would be reduced.
(ii)(0) Shorter term
If the term had been 43 years instead of 15, the mean accumulation would
be reduced, since the investment has a shorter time in which to grow. The
standard deviation of the accumulation wou'd also have been reduced, since
the spread of possible outcomes would be narrower if the term of the
investment is. shorter.
Subject CT1, September 2013, Question 7
(i) Mean and standard deviation of the accumulation
Let i, denote the yield in year k. The accumulation after-5 years of a unit
sum of money invested al time 0 is given by:
Ss (teh )(Q+ig) (ak
The expected value of this accumulation is:
(Ss) = E[ C45 )(14 ig). 4 ig)}
= E+ JEM + ip)... EU + is) by independence
= (FEU A+ El)... + EG)
We are told that E(/,) = 0.055. Hence:
E(Ss) = 1.055°
But here there is an investment of £7.85m not a unit sum of money, so:
E(7.85S,) = 7.855 (S,) = 7.85(1.055) = 10.260, fe £10,260,000 (5 SF)
Page 44 @ FE: 2079 Examinations The variance of the accumulation afler 5 years of a unit sum of money
invested at time 0 is:
var(S;) = &(S2)-fE(S5) |"
The second moment can be written as:
£(82) | (1a Af (t4i,) (+is)']
[ta 4 aF ele vig? j-ela + ay by independence
We can simplify this using the variance:
var{t+ ig) E045 P -LEG+ iP Et+h)* =var(tei efeO ri?
So:
var (+i) +E nN of vari) [C+ ie) |
| var(i) +B} [varlis) (1+Eti "|
We are told that E(j,) = 0.055 and var(i)) = 0.047. Hence:
6
esi[o 04? + 1.056" |
= var(Ss}=[0.04" + 4.088" |
But here there is an investment of £7.85 nol a unit sum of money, so:
var(7.85Ss) = 7.85" var (S3}
37.88? {[oor 1.08857) ~(1.085)")
= 0.75875
So the standard deviation is /0.75875 = 0.87106 , ie £871,100 (4 SF).
@ IFE: 2019 Examinations Page 45 Altematively, we could use the lognormal distribution (o cafculate the mean
and standard deviation of the accumulation.
(i) Director's suggestion
If there was a guaranteed return of 4% then the accumulated value would
be:
7.85(4 04) = 9.5507 , je £9,554,000 (4 SF).
This is fess than the insurance company has to pay out (£10) $0 therefore
there is definitely going to be a loss.
However, even though the expected accumufation under the strategy in part
() is higher than the payout, that value is not guaranteed. The standard
deviation of just under £900,000 means that the accumulation could be
much higher than the mean (of £10.26m) but it could also be much lower
leading to a higher lass than investing in the fixed-interest securities.
42 Subject CTt, April 2014, Question 12
@) Distribution of S42
Sig is given by:
Sia = (Ii) (Tha)
Taking logs:
logS)z =log(t+ i.) ++ +log(t+ hy»)
Each of these factors of fog(t+i,) has a normal distribution with
parameters, say, “and a”.
Page 46 @ IPE: 2019 Examinations Since E (j;) = 0.08, we have E(1+-i,) = 1.08. Hence:
elo" 24.08 (ay
Since var(1+i,) = var (i) =0.05° we have:
enue | em” ‘| = 0.05% 2)
Squaring Equation (1), and substituting into Equation (2), we get:
1.08" [ew - 4] = 0.08"
\
&. 0.08% = 0.002141
1.08" |
to) 14
Substituting back into Equation (1), we find that:
a= ln 4.08 ~%4 0.002141 = 0.078801
fog; is the sum of twelve independent N(w.c%) random variables. So
logS,, has a N (12,2120?) distribution, and hence S$; is lognormal with
parameters 12; = 0.910686 and 120° = 0.025693.
@iFE: 2019 Examinations Page 47 (i) Probability of meeting the objective
£4, 000 par £5, 000 pa £6, 008 pai
monthly in advance quarterly in si continuo:
A arr a
Pa pe
We first need the present value at time 12 of the annuity payments made
subsequent to time 12. These are given by:
py = 4,00080 + 5,000v%s,{ Os vy | + 6,007.30, Sige
000 x 3.614346 + 6,000 x1.07°*| 1.886450 41.077 x4 856995]
+6,000 x 1.07% x 1.09"? «3.383414
= 38,825.52
So we: now want the probability that 18,0008, will exceed this value.
P (Sip > 2.15697)
= P (log N(O.910686, 0.025693) > 2.1 8697)
= P(N(0.910686,0.025693) > log2.1 5697)
= P(N(G,1) > ~0.88578)
Since the standard normal is symmetrical about zero we have:
P(Z > -0.88878) = P(Z < 0.88578) = 0.81213
So the probability of meeting the objective is about 81%.
Page 48 @ IFE: 2019 Examinations 43 Subject CT1, September 2014 Question 2
{Premium
‘The expected annual interest rate is:
E(i) = 0.04 x 0.25 + 0.07 = 0.75 = 0.0625
So we use an interest rate of 6.25% to calculate the premium.
The equation of value is:
P= 200,000 x 1.0625" = £59, 490.99
(i) Expected profit
‘The accumulated profit will take one of two values. It may be:
Profil, = 59,490.99 1.04° 200,000 = -£69,647.92
with probability 0.25.
Alternatively, the accumulated profit may be:
Profit, + $9, 490.99 x 1.07° -- 200,000 = 4£30,211.36
with probability 0.75,
So the expected value of the accumulated profit is:
E(Prof) = 0.25 x ~269, 647,92 0.75 x £30,211.36 = £5,246.54
14 Subject CT1, April 2015, Question 12
() Mean and variance
Let i, denote the yield in year k. Then E(i,)~ { and var(i,)=s*. The
accumulation after n years of a single investment of 1 at time O is given by:
Sp = [Ei] I4is lt)
GIFE: 2019 Examinations Page 49 ‘The expected value of this accumulation is:
(Sy) = E44) (ti) (1470)
SEM+A)E (+g) -E(t+i,) by independence
=[1+ Gi) }[14 Eti,)} [1+ 6G,))
a(tes(is sO +i}
=(1+i)"
‘The variance of the accumulation after n years of a single investment of 1
at time 0 is:
var(S,)=€(83)-[E(S,)}
The second moment can be writlen as:
e(s?)- elt Af (tig) 4%, F
= elf +i ia [ii [eels inf | by independence
=[var (ivi) +fedsiyf Jf var (tin) [EC Fin)
=[var( i) e e()F | ‘[vartia) ft Elif |
[sr-(tsy} - sti?
-[* aa)
a
So:
var{s,)=[3" + (te vl (te jy?
Page 50 : ® IFE: 2019 Examinations (ia) Lognormal parameters
Here we have E(i,)=0.04, and var(i,}* 0.12%. So E(t+i,)=104 and
var (1+i,)= 0.122,
Using the formulae for the mean and variance of the lognormal distribution,
we obtain:
elt? 4.04 and: e240" (e" 1-012
Squaring the first of these equations, and subsfituting into the second
equation, we get:
1.042 {6% ~ ) 0.12%
Rearranging.
fo12\}
oof 28] jro.orazzs
Substituting back to find 12, we obtain:
ystogt.04— Yea? =: 0.032608
(i(b) Probability
We now want the probability that /, lies between 0.08 and 0.08:
P(0,06 < i, $0.08} = P(1,06 $144, $1.08)
=P 1.08