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CT1: CMP Upgrade 2013/14

The Actuarial Education Company © IFE: 2014 Examinations

Subject CT1

CMP Upgrade 2013/14

CMP Upgrade This CMP Upgrade lists all significant changes to the Core Reading and the ActEd material since last year so that you can manually amend your 2013 study material to make it suitable for study for the 2014 exams. It includes replacement pages and additional pages where appropriate. Alternatively, you can buy a full replacement set of up-to-date Course Notes at a significantly reduced price if you have previously bought the full price Course Notes in this subject. Please see our 2014 Student Brochure for more details.

This CMP Upgrade contains: All changes to the Syllabus objectives and Core Reading. Changes to the ActEd Course Notes, Series X Assignments and Question and

Answer Bank that will make them suitable for study for the 2014 exams.


© IFE: 2014 Examinations The Actuarial Education Company

1 Changes to the Syllabus objectives and Core Reading

1.1 Syllabus objectives No changes have been made to the syllabus objectives.

1.2 Core Reading No changes have been made to the Core Reading.



2 Changes to the ActEd Course Notes Chapter 2 Page 15 The description of the final formula on the page should refer to “compound discount” not “simple discount”. So, it should read: However, we could also use (3.3) and (3.2) to give an accumulation factor in terms of compound discount:

1 1

( )( ) (1 )n

A nv n d

Replacement pages can be found at the end of this document. Chapter 3 Page 14

In the sentence underneath the graph, should be replaced by e . It should read:

We can see that ( )

lim 1.05pi

pe e

which is the 1 i that we saw in equation (3.7).

Replacement pages can be found at the end of this document. Chapter 5 Page 22-24 Part (i) of the exam-style question has been rewritten to better reflect the current style of exam questions. Replacement pages can be found at the end of this document.



Chapter 10 Page 16 The solution to project E has been improved. Replacement pages can be found at the end of this document. Page 19, 47-48 Question 10.9 has been changed to make clear that the DPP is defined to be the first time the accumulated value becomes positive (regardless of the fact that it may become negative again afterwards). Replacement pages can be found at the end of this document.



3 Changes to the Q&A Bank Part 1 Question and solution 1.36(i) Part (i) of this question has been changed to: (i) Define the accumulation factor 1 2( , )A t t in terms of the force of interest ( )td .

The solution has also been changed. Replacement pages can be found at the end of this document.



4 Changes to the X Assignments The solutions now include a reference to the appropriate chapter. Assignment X3 Question X3.11 A new part has been added. Question X3.12 This question has been replaced. Replacement pages are included for those students who are submitting their assignments for marking as the most up-to-date version of the assignments needs to be used.



5 Other tuition services

In addition to this CMP Upgrade you might find the following services helpful with your study.

5.1 Study material We offer the following study material in Subject CT1:

Mock Exam

Additional Mock Pack

ASET (ActEd Solutions with Exam Technique) and Mini-ASET

Revision Notes

Flashcards. For further details on ActEd’s study materials, please refer to the 2014 Student Brochure, which is available from the ActEd website at www.ActEd.co.uk.

5.2 Tutorials We offer the following tutorials in Subject CT1:

a set of Regular Tutorials (usually lasting two or three full days)

a Block Tutorial (lasting two or three full days)

a Revision Day (lasting one full day)

an online classroom. For further details on ActEd’s tutorials, please refer to our latest Tuition Bulletin, which is available from the ActEd website at www.ActEd.co.uk.



5.3 Marking You can have your attempts at any of our assignments or mock exams marked by ActEd. When marking your scripts, we aim to provide specific advice to improve your chances of success in the exam and to return your scripts as quickly as possible. For further details on ActEd’s marking services, please refer to the 2014 Student Brochure, which is available from the ActEd website at www.ActEd.co.uk.



6 Feedback on the study material

ActEd is always pleased to get feedback from students about any aspect of our study programmes. Please let us know if you have any specific comments (eg about certain sections of the notes or particular questions) or general suggestions about how we can improve the study material. We will incorporate as many of your suggestions as we can when we update the course material each year. If you have any comments on this course please send them by email to [email protected] or by fax to 01235 550085.


All study material produced by ActEd is copyright and is sold for the exclusive use of the purchaser. The copyright is owned

by Institute and Faculty Education Limited, a subsidiary of the Institute and Faculty of Actuaries.

Unless prior authority is granted by ActEd, you may not hire out, lend, give out, sell, store or transmit electronically or

photocopy any part of the study material.

You must take care of your study material to ensure that it is not used or copied by anybody else.

Legal action will be taken if these terms are infringed. In addition, we may seek to take disciplinary action through the

profession or through your employer.

These conditions remain in force after you have finished using the course.

CT1-02: The time value of money Page 15


Which is:

3£10,000 0.95 £8,573.75¥ =

This is the present value of £10,000 due in 3 years and is the formula (1 )- nC d .

3.3 Discount factors

In the same way that the accumulation factor ( )A n gives the accumulation at time

n of an investment of 1 at time 0, we define ( )v n to be the present value of a

payment of 1 due at time n . Hence:

1

( )( )

v nA n

= (3.3)

So using equation (3.2) the discount factor for compound discount is:

( ) (1 )nv n d= -

However, using equation (3.3) and equation (1.2) we could also give a discount factor in terms of compound interest:

1 1

( )( ) (1 )nv n

A n i= =

+ (3.4)

However, we could also use (3.3) and (3.2) to give an accumulation factor in terms of compound discount:

1 1

( )( ) (1 )nA n

v n d= =

-

So regardless whether we’re given interest or discount rates we could calculate both accumulations and present values. Of course this does mean that we will have to read questions carefully to see what rates we are given and what we are asked to do with them!

CT1-02: The time value of money


Question 2.8

(i) Given an investment of €1,000 find the accumulation after 5 years using: (a) simple discount of 8% pa (b) compound discount of 8% pa (c) compound interest of 8% pa. (ii) Given a payment of €2,000 due in 4 years’ time, calculate the present value

using: (a) simple interest of 3% pa (b) simple discount of 3% pa (c) compound interest of 3% pa.

Note that using the definition of v from (2.2) in (3.4) we get:

1 1

( )( ) (1 )

nnv n v

A n i= = =

+

This was the discount factor that we were using in (2.3) to calculate the present value.

CT1-03: Interest rates


We assume that for each value of i there is number, d , such that:

Æ•

=( )lim p

pi d

d is the nominal rate of interest per unit time convertible continuously (or momently). This is also referred to as the rate continuously compounded. We call it the force of interest. Whilst this is all very interesting it gives us no practical way of calculating the value of d . So what we’re going to do now is derive a relationship between the force of interest and the effective rate of interest. Euler’s rule states that:

Æ•

Ê ˆ+ =Á ˜Ë ¯lim 1

nx

n

xe

n

Applying this to the right-hand-side of (3.1), which was:

( )

1 1

ppii

p

Ê ˆ+ = +Á ˜Ë ¯

gives:

1 i+ =•

Æ•

Ê ˆ+ =Á ˜

Ë ¯

( )( )lim 1

ppi

p

ie

p

But we defined ( )i • to be d . Hence:

+ =1 i ed (3.7) This gives us our connection between the effective interest rate, i , and the force of interest, d .

CT1-03: Interest rates


Taking the exponential of the ( )pi ’s that we calculated using an effective rate of interest of 5% pa in our previous graph we obtain the following:

We can see that ( )

lim 1.05pi

pe ed

Æ•= = which is the 1 i+ that we saw in equation (3.7).

The rearrangement of 1e id = + is: ln(1 )id = + (3.8)

In our first graph the limit was ln1.05 0.048790d = . Note: Since we will only ever use natural log (ie ln loge= ) in our studies, the Core

Reading, the examiners and ActEd may use loge , log and ln interchangeably.

1.0498

1.05

1.0502

1.0504

1.0506

1.0508

1.051

1.0512

1.0514

0 10 20 30 40 50

exp(i(p))

p

CT1-05: Discounting and accumulating Page 21


Question 5.14

A woman deposits £200 in a special bank account. Interest is paid to the woman every year on her birthday for five years. The capital is returned after exactly five years, along with any interest accrued since her last birthday. Interest is calculated at an effective rate of 6% pa. Calculate the present value of the interest received by the woman.

So far we have described the difference between money returned at the end of the term and the cash originally invested as “interest”. In practice, however, this quantity may be divided into interest income and capital gains, the term capital loss being used for a negative capital gain. If you invest some capital then not only might you receive income but the value of your capital may also increase (or decrease). Equities or shares are a good example of this. These were introduced earlier in the course. If you buy some shares in a company then you should receive dividends or interest from the company. However the capital value that you receive back will depend upon the market price of the shares when you decide to sell. We will consider this in more detail later in the course.

Question 5.15

A rich woman pays £2m and in return expects to receive a continuous cashflow for the next six years with a constant rate of payment. Calculate the annual payment from this cashflow and the accumulated amount of the cashflow after six years if the interest rate is 9% pa effective.

Question 5.16

True or false:

(i) 21 2

1

(0, )( , )

(0, )

A tA t t

A t=

(ii) 2 1 1 2(0, ) (0, ) ( , )A t A t A t t= +

(iii) 2 1 2 1( ) ( ) ( )v t v t v t t= -

(iv) 1 1 22

1(0, ) ( , )

( )A t A t t

v t=

CT1-05: Discounting and accumulating


4 Exam-style question

This is a typical exam-style question on this chapter. Have a go at it before turning over and having a look at the solution.

Question The force of interest at any time t (measured in years) is given by:

0.04 0 1

( ) 0.05 0.01 1 5

0.24 5

t

t t t

t

d< £Ï

Ô= - < £ÌÔ >Ó

(i) Derive and simplify as far as possible expressions for ( )A t , where ( )A t is the

total accumulated value at time t ( 0 ) of an investment of 1 at time 0. (ii) What is the present value at time 0 of a payment stream paid at a rate of

( )t t 5 1 received between t 1 and t 5 ?

CT1-05: Discounting and accumulating Page 23


Solution (i) We have to break down the times for when there is a payment or when the force

of interest changes. If we let A t( ) represent the accumulated value of the total

investments made to date at time t, then: 0 1 t :

A t e t( ) . 0 04

1 5t< £ :

2

2

0.04

1

0.04 2

1

0.04 0.025 0.01 0.025 0.01

0.025 0.01 0.025

( ) exp 0.05 0.01

exp 0.025 0.01

t

t

t t

t t

A t e s ds

e s s

e e

e

- - +

- +

È ˘Í ˙= ¥ -Í ˙Î ˚

È ˘= ¥ -Î ˚

= ¥

=

Ú

5 t< :

20.025 5 0.01 5 0.025

5

0.6 0.24 1.2

0.24 0.6

( ) exp 0.24t

t

t

A t e ds

e e

e

¥ - ¥ +

-

-

È ˘Í ˙= ¥Í ˙Î ˚

=

=

Ú

CT1-05: Discounting and accumulating


(ii) It is easiest to calculate the present value at time 1 and then discount back to time 0. The present value at time 1 is:

( )

5

1 1

52

11

52

1

52

1

52

1

0.625 0

(5 1)exp 0.05 0.01

(5 1)exp 0.025 0.01

(5 1)exp[ 0.025 0.01 0.025 0.01]

(5 1)exp[ 0.025 0.01 0.015]

100 exp 0.025 0.01 0.015

100

t

t

t s ds dt

t s s dt

t t t dt

t t t dt

t t

e- +

È ˘Í ˙- - -Í ˙Î ˚

È ˘= - - -Î ˚

= - - + + -

= - - + +

È ˘= - - + +Î ˚

= -

Ú Ú

Ú

Ú

Ú

.05 0.015 0.025 0.01 0.015

0.56100 1

42.879

e

e

+ - + +

-

È ˘-Î ˚

È ˘= - -Î ˚

=

We then need to discount this back to time 0:

1

0.04

0

42.879 exp 0.04 42.879 41.20PV dt e

CT1-10: Project appraisal Page 15


2 Different interest rates for lending and borrowing

We have assumed so far that the investor may borrow or lend money at the same rate of interest i1 . In practice, however, the investor will probably have to pay a

higher rate of interest ( j1, say) on borrowings than the rate ( j2 , say) he receives

on investments. This is because banks “make their money” by borrowing money from savers at one rate of interest and lending it out for mortgages, business loans etc at a higher rate. The difference j j1 2 between these rates of interest depends on various

factors, including the credit-worthiness of the investor and the expense of raising a loan. The concepts of net present value and yield are in general no longer meaningful in these circumstances. We must calculate the accumulation of net cashflows from first principles, the rate of interest depending on whether or not the investor’s account is in credit. In many practical problems the balance in the investor’s account (ie the accumulation of net cashflows) will be negative until a certain time t1 and positive afterwards, except perhaps when the project ends.

In some cases the investor must finance his investment or business project by means of a fixed-term loan without an early repayment option. In these circumstances the investor cannot use a positive cashflow to repay the loan gradually, but must accumulate this money at the rate of interest applicable on lending, ie j2 .

Example The cashflows for Project C are: Outlay Proceeds £100,000 (initially) £140,000 (end of year 5) The cashflows for Project E are: Outlay Income £80,000 (initially) £10,000 (end of year 1) £20,000 (start of year 2) £30,000 (end of year 2) £5,000 (start of year 3) £87,000 (end of year 3)

CT1-10: Project appraisal


A company must choose between Projects C and E, both of which would be financed by a loan, repayable only at the end of the project. The company must pay 6.25% per annum on money borrowed, but can earn only 4% on money invested in its deposit account. Calculate the accumulated profit at the end of 5 years for each project. Solution Project C Since Project C does not generate any income, the company will be relying on the loan throughout. The accumulated profit will be:

140 000 100 000 10625 5925, , . £4, Project E Since Project E does generate income, we need to consider the company’s net assets at the end of each year to see whether there are any excess funds available to invest. At the end of year 1, there will be an income payment of £10,000, but interest of 80 000 0 0625 000, . £5, is required and a further outlay of £20,000 must be made. So further borrowing of £15,000 is required (taking the total borrowing to £95,000). At the end of year 2, there will be an income payment of £30,000, but interest of 95 000 0 0625 937 5, . £5, . is required and a further outlay of £5,000 must be made. This leaves 30 000 5 937 5 5 000 062 5, , . , £19, . available for investment. At the end of year 3, the company receives an income of £87,000. It pays interest of 95 000 0 0625 937 5, . £5, . , leaving £81,062.5. This, together with the proceeds from year 2 (plus interest), will be used to pay off the loan. To obtain the accumulated profit at time 5, we take the remaining money at time 3 and accumulate it for 2 years to time 5:

( , . . , , . ) . £6,19 062 5 104 87 000 95 000 10625 104 3682



Question 10.9

The business plan for a new company that has obtained a 5 year lease for operating a local bus service is shown in the table below. Items marked with an asterisk represent continuous cashflows. Cashflow item Timing Amount (£000) Initial set up costs Immediate –250 Fees from advertising contracts 1 month +200 Purchase of vehicles 3 months –2,000 Fares from passengers* From 3 months onwards +1,000 pa Staff costs and other operating costs* From 3 months onwards – 400 pa Resale value of assets 5 years +500 Determine the discounted payback period for this project assuming that it will be financed by a flexible loan facility based on an effective annual interest rate of 10% per annum.

The discounted payback period is often employed when considering a single investment of C, say, in return for a series of payments each of R, say, payable annually in arrears for n years. The discounted payback period t1 years is

clearly the smallest integer t such that A t ( ) 0 , where:

A t C j Rstt

( ) ( ) |1 1 at rate j1

ie the smallest integer t such that: Ra Ct| at rate j1

The project is therefore viable if t n1 , in which case the accumulated profit

after n years is clearly:

P A t j Rsn tn t ( )( ) |1 21 1

1 at rate j2



Question 10.10

A speculator borrows £50,000 at an effective interest rate of 8% per annum to finance a project that is expected to generate £7,500 at the end of each year for the next 15 years. Find the discounted payback period for this investment.



Solution 10.9

Remembering that the starred cashflows are continuous, we find that the accumulated value to time t (where 3 months < t < 5 years ) is:

1/12 3/12

3/12

. ( ) 250 1.10 200 1.10 2,000 1.10

(1,000 400) 1.10

t t t

tt s

AccV t

ds

- -

-

= - ¥ + ¥ - ¥

+ - ¥Ú

Using the substitution u t s in the integral, we can write this equation in the form:

3/123/12 2/12 3/12

0

3/12|3/12

. ( ) ( 250 1.10 200 1.10 2,000) 1.10 600 1.10

2,052.83 1.10 600

tt u

tt

AccV t du

s

--

--

= - ¥ + ¥ - ¥ +

= - ¥ +

Ú

Using this formula, we find that . (3 12) 2,052.83AccV = - and . (5) 376.33AccV = . So

the accumulated value will be zero for some value of t in the range 3 months < < 5 yearst .

In fact, this will happen when:

3 123 12

2,052.83 1.10 600 0tt

s--- ¥ + =

Dividing through by 110 3 12. /t and rearranging:

3 12

2,052.833.42138

600ta - = =

ie ( 3 12)1 1.10

3.42138 3 12 4.14 4.39ln1.10

t

t t- -- = fi - = fi =

So the discounted payback period is 4.39 years.



Instead of working with accumulated values, you may find it easier to work with present values, in which case we want the time t when up to 0tNPV = . The equation will be:

1 12 3 12 3 123 12

250 200 2,000 600 0t

NPV v v v a -= - + - + =

This gives the same answer as before. Solution 10.10

The accumulated profit at the end of year t will be:

Acc.Profit( ) = , ( + )t i stt 50 000 1 7 500, |

This will be positive when:

50 000 1 7 500 0, ( + )i stt, |

Dividing through by 7 500 1, ( + )i t :

at |,

,.

50 000

7 5006 6666

Looking at the Tables (at 8%), we see that a9 6 2469| . and a10 6 7101| . .

So the discounted payback period is 10 years. Solution 10.11

If the interest rate charged on borrowing was 18%, the profit from Investment 2 would be 2 000 1 800, , £200 , which is almost the same as the profit of £199 on Investment 1. So the critical rate of interest is approximately 18%.

CT1: Q&A Bank Part 1 – Questions Page 11


Question 1.31

Calculate the present value of an annuity payable annually in advance for a term of 20 years such that the payment is £500 in year 1, £550 in year 2, £600 in year 3 etc. Assume a rate of interest of 5% pa for the first twelve years and 7% pa thereafter. [5] Question 1.32

A 90-day government bill was bought by an investor for a price of £91 per £100 nominal. After 30 days the investor sold the bill to a second investor for a price of £93.90 per £100 nominal. The second investor held the bill to maturity when it was redeemed at par. Determine which investor obtained the higher annual effective rate of return. [3] Question 1.33

A series of 10 payments is received at times 5, 6, 7, … . The first payment is $200. Each of the next five payments is 5.7692% greater than the previous one, and the remainder increase by 6.7961% each time. Find the value of these payments at time 0 using an interest rate of 10% pa effective. [5] Question 1.34

The force of interest ( ) t is:

2( ) 0.005 0.0001 t t t for all t

(i) At 8t , calculate the accumulated value of an investment of £100 made at time

0t . [3] (ii) Calculate the constant annual effective rate of interest over the eight year period. [2] [Total 5]

CT1: Q&A Bank Part 1 – Questions


Question 1.35

(i) Calculate the combined present value of an immediate annuity payable monthly in arrears such that payments are £1,000 pa for the first 6 years and £400 pa for the next 4 years, together with a lump sum of £2,000 at the end of the 10 years.

[3] (ii) Calculate the amount of the level annuity payable continuously for 10 years

having the same present value as the payments in (i). [3] (iii) Calculate the accumulated values of the first 7 years’ payments at the end of the

7th year for the payments in (i) and (ii). [3] Basis: Assume an interest rate of 12% pa convertible monthly. [Total 9] Question 1.36

(i) Define the accumulation factor 1 2( , )A t t in terms of the force of interest ( )td . [1]

(ii) The force of interest ( )td at time t (measured in years) is given by

( ) 0.01 0.04t td = + .

(a) Calculate the corresponding nominal rate of interest for the period 1t =

to 2t = .

(b) If an investment of 1 is made at time 12t = , calculate the value to which

it will have accumulated by time 6t = . [6] (iii) Calculate the accumulated value after 6 months of an investment of £100 at

time 0 at the following rates of interest: (a) a force of interest of 0.05 pa (b) a rate of interest of 5% pa convertible monthly (c) an effective rate of interest of 5% pa. [3] [Total 10]

CT1: Q&A Bank Part 1 – Solutions Page 19


Solution 1.36

(i) The formula for 1 2( , )A t t in terms of the force of interest ( )td is:

2

1

1 2( , ) exp ( )t

t

A t t t dt [1]

Note that this formula is given on page 31 of the Tables. (ii)(a) The nominal rate of interest over a one year time period is the same as the

equivalent constant effective annual rate of interest, which can be found as follows:

2 2

1 1

22

1

0.055

1 exp ( ) exp (0.01 0.04)

exp 0.005 0.04

exp[(0.02 0.08) (0.005 0.04)] 1.05654

i t dt t dt

t t

e [3]

So the annual rate of interest is 5.65%. (ii)(b) The accumulation factor from time ½ to time 6 is:

6212 1/ 2

0.39875

( ,6) exp 0.005 0.04

exp[ 0.18 0.24 (0.00125 0.02)] 1.4900

A t t

e [3]

(iii) The accumulated values are:

(a) 1

2 0.05100 100 1.025315 £102.53 e [1]

(b) 6

0.05100 1 100 1.025262 £102.53

12

[1]

(c) 1

2100 1.05 100 1.024695 £102.47 [1]

CT1: Q&A Bank Part 1 – Solutions


Solution 1.37

(i) The present value of a continuous cashflow payable at rate ( )r t from 0t to

10t is given by:

10

0

( ,0) ( ) PV A t r t dt [1]

The discount factor ( ,0)A t is:

0 0

2 2

0

( ,0) exp ( ) exp (0.2 0.01 )

exp 0.2 0.005 exp( 0.2 0.005 )

t t

t

A t s ds s ds

s s t t [2]

So the present value of the annuity is:

102

0

102 1.5

0

(200 10 )exp( 0.2 0.005 )

1,000exp( 0.2 0.005 ) 1,000( 1) £776.87

t t t dt

t t e [2]

Reasonableness check: The force of interest decreases from 0.2 at 0t to 0.1 at

10t and the payments decrease from 200 at 0t to 100 at 10t . Calculating

the present value based on averages gives @15%|10

150 £753a .

(ii) The accumulated value at time 10 can be found by accumulating the present value

at time zero for 10 years (and using the accumulation factor calculated previously):

1.5776.87776.87 (0,10) 776.87 £3,482

(10,0) A e

A [2]

Reasonableness check @15%|10

150 £3,046s .

CT1: Assignment X3 Questions Page 3


Question X3.9

The British Government wishes to issue a new fixed-interest stock with a 25-year term, which will pay half-yearly coupons (payable in arrears) and will be redeemable at par. The Bank of England has advised the government that future interest rates (expressed as nominal rates convertible half-yearly) can be assumed to be 7% pa over each of the next 5 years, 8% pa over each of the following 10 years and 8½% pa thereafter. (i) Assuming the government wishes to issue the new bond at a price of £100%,

calculate the coupon rate that should be chosen for the bond. (Quote your answer to the nearest ¼%.) [4]

(ii) After the coupon has been fixed at this level, an investor agrees a forward

contract to buy the security in four years’ time, immediately after the coupon payment then due. Calculate the forward price based on a “risk-free” rate of return equal to the interest rates advised by the Bank of England. Assume there is no arbitrage. [5]

[Total 9] Question X3.10

At time 0t an investor purchased an annuity-certain which paid her £10,000 per annum annually in arrear for three years. The purchase price paid by the investor was £25,000. The value of the retail price index at various times was as shown in the table below:

Time t (years): 0t 1t 2t 3t Retail price index: 170.7 183.3 191.0 200.9

(i) Calculate, to the nearest 0.1%, the following effective rates of return per annum

achieved by the investor from her investment in the annuity: (a) the real rate of return; and (b) the money rate of return. [7] (ii) By considering the average rate of inflation over the three-year period, explain

the relationship between your answers in (a) and (b) of (i). [2] [Total 9]

CT1: Assignment X3 Questions


Question X3.11

On 1 October 2004, the share price of Company X was £3.15. A 6-month forward contract is written on that day, for the purchase of 1,000 shares in Company X. (i) Calculate the forward price under each of the scenarios below. Assume that the

effective risk-free rate of interest is 4% pa and that there is no arbitrage. (a) There are no dividends due in the next 6 months. (b) A dividend of 10p per share is expected on 1 December 2004. (c) Dividends are payable continuously and are immediately reinvested in

the shares. The force of dividend is 3% pa. (d) Dividends of 3% per annum of the market price of the security at the

time of payment are made annually in arrear and are immediately reinvested in the shares. [7]

(ii) Calculate the value of the forward contract on 1 February 2005 under each of the

scenarios given in (a), (b) and (c) only if the share price of Company X is then £3.50. [6]

[Total 13]

CT1: Assignment X3 Questions Page 5


Question X3.12

A government issued a number of index-linked bonds on 1 June 2000 which were redeemed on 1 June 2002. Each bond had a nominal coupon rate of 3% per annum, payable half-yearly in arrears, and a nominal redemption price of 100. The actual coupon and redemption payments were indexed according to the increase in the retail price index between 6 months before the bond issue date and 6 months before the coupon or redemption payment dates. The values of the retail price index in the relevant months were:

Date Retail price index December 1999 100 June 2000 102 December 2000 107 June 2001 111 December 2001 113 June 2002 118

(i) An investor purchased £100,000 nominal at the issue date, and held it until it

was redeemed. The issue price was £94 per £100 nominal. Calculate all the investor’s cash flows from this investment, before tax. [3] (ii) The investor is subject to income tax at a rate of 25% and capital gains tax at a

rate of 35%. When calculating the amount of capital gain which is subject to tax, the price paid for the investment is indexed in line with the increase in the retail price index between the month in which the investment was purchased and the month in which it was redeemed.

(a) Calculate the investor’s capital gains tax liability in respect of this

investment. (b) Calculate the net effective yield per annum obtained by the investor on

these bonds. [6] [Total 9]

CT1: Assignment X3 Questions


Question X3.13

(i) Show that ( )

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Iaa nv

inn

n

. [4]

(ii)(a) Calculate the present value, at a rate of interest of 3% pa effective, of an annuity

where 10 is paid at the end of the first year, 12 is paid at the end of the second year, 14 is paid at the end of the third year and so on, with payments increasing by 2 pa until the payment series ends at time 20.

(ii)(b) Calculate the capital outstanding in the above annuity at the end of the 15th year,

after the payment then due has been received. (ii)(c) Determine the interest and capital components in the 16th payment. [8] [Total 12] Question X3.14

One year ago, a loan was issued bearing interest at the rate of 14% pa, payable half yearly in arrears. The loan will be redeemed at £110% in nine years’ time. (i) The loan was issued at a price such that an investor subject to income tax at

35%, but not subject to capital gains tax, would obtain a net yield of 9½% pa. Calculate the issue price for £100 nominal. [3]

Interest rates have now fallen. The investor has decided to sell and has found a potential buyer, who is subject to income tax at 25% and capital gains tax at 35%, and who is prepared to buy the stock provided that he obtains a net yield of at least 10% pa. (ii) Calculate the best price (per £100 nominal) the original investor can expect to

obtain from the potential buyer. [5] (iii) Calculate the net running yield obtained by the buyer. [2] (iv) Calculate the net yield that will be obtained by the original investor if the loan is

sold to the buyer at the price determined in (ii). [4] [Total 14]

END OF ASSIGNMENT

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