CT1-PU-10 cmp upgrade cover sheet 2010 - Actuarial … Upgrades/CT… · · 2009-09-14Subject CT1...

CT1: CMP Upgrade 2009/10 Page 1

The Actuarial Education Company © IFE: 2010 Examinations

Subject CT1

CMP Upgrade 2009/2010

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 2009 study material to make it suitable for study for the 2010 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 2010 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 2010 exams.

Page 2 CT1: CMP Upgrade 2009/10

© IFE: 2010 Examinations The Actuarial Education Company

1 Changes to the Syllabus objectives and Core Reading

1.1 Syllabus objectives There have been no changes to the Syllabus objectives.

1.2 Core Reading There have been no changes to the Core Reading.



2 Changes to the ActEd Course Notes Chapter 1 Page 13 The following paragraph has been added after the first paragraph on this page: Some of these questions are taken from past exam papers; some are written in the style of exam questions (“exam-style”). Where the question is taken from an exam, we have included a reference to the past paper in which the question appeared. Chapter 5 Pages 24 and 25 The inequalities on the time periods in part (i) of the solution to the exam-style question have been changed. Replacement pages have been included with this upgrade note. Chapter 12 An exam question has been added to the end of this chapter. Replacement pages have been included with this upgrade note. Chapter 13 Pages 21 to 23 The exam question included in the chapter has been changed. Replacement pages have been included in this upgrade note. Chapter 14 Pages 33 to 35 Exam-style question 1 has been replaced with a past exam question. Replacement pages have been included in this upgrade note.



Chapter 15 Page 14 Underneath the formula:

2 2 443

1 (0.06 0.02) 1012

s −= − = ×

the following sentence has been added: This formula for the variance of a uniform random variable is given on page 13 of the Tables.



3 Changes to the Q&A Bank Each of the first four parts of the Q&A Bank has been split into two sections:

• Section 1 – Development Questions. The aim of these questions is to build on your understanding, test key Core Reading and bring your knowledge and skills up to the level required to tackle exam-style questions.

• Section 2 – Exam-style Questions. These questions are of the level of difficulty

you are likely to face in the examination. It is very important that you focus on these questions as preparation for the exam.

The split of questions into development and exam-style is given for each part of the Q&A Bank below. Q&A Bank Part 1 Development Questions – 1 to 13, 16 to 27, 29, 33. Exam-style Questions – 14, 15, 28, 30 to 32, 34 to 39. Question 1.38 The solution to this question has been corrected. The minus sign at the start of the first expression for the present value after the sentence: “We can integrate this by inspection (or by substitution):” should be removed. Additional Question Question X4.6 from Assignment X4 has been added to this Q&A bank, in the exam-style section.



Q&A Bank Part 2 Development Questions – 1 to 3, 5, 8, 9, 11.

Exam-style Questions – 4, 6, 7, 10, 12 to 21. Additional Question Question X2.10 from Assignment X2 has been added to this Q&A bank, in the exam-style section, with the following additional explanation added to the solution to (i): The formula for ( )Ia ∞ is derived as follows:

( ) lim ( ) limn

nnn n

a nvIa Ia

i∞ →∞ →∞

−= =

As n →∞ , 0nnv → and 1na

d→ , so:

( )1 1( ) dIa

i id∞ = =

Q&A Bank Part 3 Development Questions – 2 to 5, 9, 10, 12 to 16, 19.

Exam-style Questions – 1, 6 to 8, 11, 17, 18, 20 to 27. Question 3.2 In the fourth sentence of the solution, “DM” has been replaced with “Euro”. Question 3.23 The total number of marks for this question has been corrected to 13 (previously 14).



Question 3.27 The solution to the first part of the question has been updated to give additional explanation: (i) First we need to find out if there is a capital gain:

1(1 0.4) 8(1 ) 4.57%

105g t − ×

− = = [1]

(4) 0.25@ 6% 4(1.06 1) 5.8695%i = − = [1] Since (4)

1(1 )i g t> − , the investor makes a capital gain. We need to calculate the price to give the investor a yield of at least 6% pa, so we must consider the worst case scenario for the investor. As the investor makes a capital gain, the worst case is that this gain is received at the latest possible date, ie in 20 years’ time. [1] This leads to the following equation for the price per £100 nominal (with all functions calculated at 6%): (4) 20

|200.6 8 [105 0.3(105 )]P a P v= × + − − [2]

0.6 8 11.7248 [105 0.3(105 )] 0.311805P= × × + − − × [1] 56.279 22.918 0.09354P= + + [1] Rearranging to find P gives:

79.197 / 0.90646 £87.370P = = [1] So for a nominal holding of £100,000 the price to ensure a yield of at least 6% would be not greater than £87,370. [1]

Additional Question Question X3.10(ii) from Assignment X3 has been added to this Q&A bank, in the development section.



Q&A Bank Part 4 Development Questions – 1 to 3, 5, 6, 9, 10, 13 to 15, 20, 21.

Exam-style Questions – 4, 7, 11, 12, 16 to 19, 22 to 28. Question 4.8

This question has been removed from the Q&A bank and is now Question X4.9 on Assignment X4.

Additional Question

Question X4.5 from Assignment X4 has been added to this Q&A bank, in the development section. Q&A Bank Part 5 Question 5.15 The words “redeemable at par” have been added to the description of Stock A and Stock B.



4 Changes to the X Assignments Assignment X1 Question 9 The question has been altered slightly to state that time is measured in years, and that in (i) a force of interest per annum is required. Replacement pages are included. Question 10 The question has been reworded slightly, by adding “effective” after each of the percentage increase figures, and stating that “each payment is higher than the previous one”. Replacement pages are included. Question 11 The mark allocation for (i) has been changed to 3 (previously 4) and the mark allocation for (ii) has been changed to 4 (previously 3). The marking schedule has been amended accordingly. Replacement pages are included. Question 13 Part (ii) of the question has been changed to ask for the value at time 0 of £100 due at time 9. The solution has been updated accordingly. Replacement pages are included.



Assignment X2 Question 4 The following alternative expression for the net present value has been added to the solutions: 3

3 3300 90 30( ) 80NPV a Ia v= − + + +

Question 5 The following expression has been added as the first line of the present value calculation in the solutions: 2 3 4 15 16100 100 120 120 240 240= + + + + + +PV v v v v v v An additional alternative expression for the present value has been added to the list at the end of the solutions: 8 8 8 880 20( ) 1.05(80 20( ) )a Ia a Ia+ + + at a rate of interest of 21.05 1−

Question 6 A second alternative approach has been added to the solutions to part (iv). Replacement pages are included. Question 8 The mark allocation for (ii) has been changed to 2 (previously 4) and the total mark allocation for this question has been changed to 9 (previously 11). The marking schedule has been amended accordingly. The first paragraph of the solution to (iii) has been simplified. Replacement pages are included. Question 9 An alternative solution to part (iii) has been added. Replacement pages are included. Question 10 This question has been replaced with a new question. Replacement pages are included.



Assignment X3 Question 2 This question has been replaced with Question 3.10(i) from the 2009 assignments, with the mark allowance altered to 3 (previously 4). Replacement pages are included. Question 5 This question has been replaced. Replacement pages are included. Question 6 The question has been reworded as follows: Outline how a debenture differs from a government bond. Replacement pages are included. Question 7 In the second sentence of this question “(running from 1st April to 31st March)” has been added after “preceding year”. Question 10 This question has been replaced. Replacement pages are included. Question 12 A note has been added to the question to explain that a risk discount rate is an annual effective interest rate. An alternative solution for (ii) has been added in terms of accumulated values, as follows: “Alternatively, working in terms of accumulated values: At time 4, 10.3905AVc = and 9.5587AVr = , so 9.5587 10.3905 0.832AV = − = − . At time 5, 12.2223AVc = and 12.7057AVr = , so 12.7057 12.2223 0.483AV = − = .”



Assignment X4 Two questions (X4.5 and X4.6) have been removed from this assignment, and replaced with a new question, which is labelled X4.9. Consequently, the question numbering has altered throughout this assignment. To avoid confusion, a replacement X4 assignment document is included here. Question 2 The comment in italics at the end of the solution has been removed. Questions 5 and 6 These questions have been removed. Question 9 This is a new question. The solution is attached. Question 11 This question is labelled X4.10 in the 2010 version of the assignment. In the solution to (ii)(a), the numerical value of 6.245 years has been added for the DMT, along with the calculation of the volatility. An alternative approach to this solution has also been added where the calculations are in terms of the nominal amounts of the bonds purchased. Replacement pages of the solutions are provided. Question 13 This question is labelled X4.12 in the 2010 version of the assignment. Part (ii)(a) of the question has been changed to: Show that the parameters μ and 2σ of the lognormal distribution take the values 0.05384 and 0.008861 respectively. The mark allowance for (ii) has been changed to 10 (previously 11), and the total mark allowance for the question has been changed to 16 (from 17). Replacement pages of the solutions are provided.



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:

• Series Y Assignments

• Mock Exam 2009 and Mock Exam 2010

• ASET (ActEd Solutions with Exam Technique) and mini-ASET

• Revision Notes

• Flashcards For further details on ActEd’s study materials, please refer to the 2010 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 (lasting 4 half days, 2 full days or 3 full days)

• a Block Tutorial (lasting two or three full days)

• a Revision Day (lasting one full day). 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 2010 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.

CT1-05: Discounting and accumulating Page 23


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 50.24 5

tt t t

tδ

< ≤⎧⎪= − < ≤⎨⎪ >⎩

(i) What is the total accumulated value at any time t (> 0 ) of investments of 1 at

times 0, 4 and 6? (ii) What is the present value at time 0 of a payment stream paid at a rate of

ρ( )t t= −5 1 received between t = 1 and t = 5 ?

Page 24 CT1-05: Discounting and accumulating


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 4t< < :

2

2

0.04

1

0.04 21

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

− − +

− +

⎡ ⎤⎢ ⎥= × −⎢ ⎥⎣ ⎦

⎡ ⎤= × −⎣ ⎦

= ×

=

∫

4 5t≤ ≤ :

2

0.025 16 0.01 4 0.025

4 4

0.385 0.025 0.01 0.36

( ) exp 0.05 0.01 exp 0.05 0.01

( 1)

t t

t t

A t e s ds s ds

e e

× − × +

− −

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= − + −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= +

∫ ∫

5 6t< < :

0.385 0.625 0.05 0.36

5

0.385 0.215 0.24 1.2

0.385 0.24 0.985

( ) ( 1) exp 0.24

( 1)

( 1)

t

t

t

A t e e ds

e e e

e e

− −

−

−

⎡ ⎤⎢ ⎥= +⎢ ⎥⎣ ⎦

= +

= +

∫

CT1-05: Discounting and accumulating Page 25


6t ≥

0.385 0.455

6 6

0.84 0.455 0.24 1.44

( ) ( 1) exp 0.24 exp 0.24

( 1)

t t

t

A t e e ds ds

e e e −

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= + +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= + +

∫ ∫

(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

521

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− +

⎡ ⎤⎢ ⎥− − −⎢ ⎥⎣ ⎦

⎡ ⎤= − − −⎣ ⎦

= − − + + −

= − − + +

⎡ ⎤= − − + +⎣ ⎦

= −

∫ ∫

∫

∫

∫

.05 0.015 0.025 0.01 0.015

0.56100 1

42.879

e

e

+ − + +

−

⎡ ⎤−⎣ ⎦

⎡ ⎤= − −⎣ ⎦

=

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−⎡ ⎤

= × − = =⎢ ⎥⎢ ⎥⎣ ⎦∫

Page 26 CT1-05: Discounting and accumulating


This page has been left blank so that you can keep the chapter summaries

together for revision purposes.

CT1-12: Elementary compound interest problems Page 41


6 Exam question

Have a go at the following past exam question on some of the material introduced in this chapter, before turning the page and checking the solution.

Question (Subject 102, April 2002, Question 6) An investor purchases a bond 3 months after issue. The bond will be redeemed at par ten years after issue and pays coupons of 6% per annum annually in arrears. The investor pays tax of 25% on both income and capital gains (with no relief for indexation). (i) Calculate the purchase price of the bond per £100 nominal to provide the

investor with a rate of return of 8% per annum effective. [6] (ii) The real rate of return expected by the investor from the bond is 3% per annum

effective. Calculate the annual rate of inflation expected by the investor. [2] [Total 8]

Page 42 CT1-12: Elementary compound interest problems


Solution (i) Purchase price of bond The capital gains test is: ( )

1(1 ) capital gainpi g t> − ⇒ So in this case:

0.08

6(1 0.25) 0.045100

i =

− =

Since 0.08 0.045> , there is a capital gain. A timeline for the cashflows is:

. . . 100 1 2¼ time

cashflows (before tax)6 6 6100

time bondpurchased

The value of these payments at time 0 is: 10 10

100.75 6 100 0.25(100 )a v P v× + − −

But the investor has purchased the bond at time ¼t = , so to calculate the value of the bond at ¼t = we use: ¼(value at ¼) (value at 0) (1 )t t i= = = × + So:

( )¼ 10 10

10

¼ ¼ 10 ¼ 1010

1.08 0.75 6 100 0.25(100 )

1.08 4.5 1.08 75 1.08 0.25

P a v P v

a v v P

= × + − −

= × + × + ×

CT1-12: Elementary compound interest problems Page 43


Rearranging gives:

¼ 10 ¼ ¼ 1010(1 1.08 0.25 ) 1.08 4.5 1.08 75

66.196 £75.06 per £100 nominal0.88195

P v a v

P

− = × + ×

⇒ = =

(ii) Annual rate of inflation To calculate the real rate of return, *i , given a money rate of return, i , and constant inflation, e , we use:

*1i ei

e−

=+

Therefore:

0.080.031

0.03 0.03 0.081.03 0.05

4.85%

ee

e ee

e

−=

+⇒ + = −⇒ =⇒ =

Page 44 CT1-12: Elementary compound interest problems


This page has been left blank so that you can keep the chapter summaries

together for revision purposes.

CT1-13: Arbitrage and forward contracts Page 21


3 Exam question

Have a go at the following past exam question before turning the page and checking the solution.

Question (Subject 102, April 2004, Question 6) (i) Explain what is meant by the “no arbitrage” assumption in financial

mathematics. [2] (ii) A three-year forward contract is to be issued on a particular company share. The

current market value of the share is £4.50 and a dividend of £0.20 per share has just been paid. The parties to the contract assume that the future quarterly dividends will increase by 1% per quarter-year compound for the first two years and by 1½% per quarter-year compound for the final year.

Assuming a risk-free force of interest of 5% per annum, and no arbitrage,

calculate the forward price. [7] [Total 9]

Page 22 CT1-13: Arbitrage and forward contracts


Solution (i) Arbitrage Arbitrage is a risk-free trading profit. An arbitrage opportunity exists if:

● an investor can make an immediate profit with no risk of future loss, or

● an investor can have zero initial cost, no risk of future loss and a non-zero probability of a future profit.

No arbitrage means that arbitrage opportunities do not exist. (ii) Forward price The forward price is given by: 3 0.15

0 0( ) ( )S I e S I eδ− = − where I is the present value of the dividends during the three year period and 0S is the current price. The first eight quarterly dividends are: 2 3 81.01 0.2, 1.01 0.2, 1.01 0.2, , 1.01 0.2× × × ×… After that the increase changes to 1.5%, so the next four dividends are: 8 8 2 8 41.01 0.2 1.015, 1.01 0.2 1.015 , , 1.01 0.2 1.015× × × × × ×… Working in quarters:

2 2 8 8

8 9 8 4 12

1.01 0.2 1.01 0.2 1.01 0.2

1.01 0.2 1.015 1.01 0.2 1.015

I v v v

v v

= × + × + + ×

+ × × + + × ×

which needs to be evaluated using a quarterly effective interest rate, j . The annual rate of interest is 0.05 1 5.1271%e − = , so: 0.251.051271 1 1.258%j = − =

CT1-13: Arbitrage and forward contracts Page 23


We now need to evaluate the expression for I . The first eight terms are a geometric progression. Summing, we get:

( )81.01 0.2 1 (1.01 )

1.58181 1.01

v v

v

× −=

−

The next four terms are also a geometric progression. Summing, we get:

( )8 9 41.01 0.2 1.015 1 (1.015 )

0.788541 1.015

v v

v

× × −=

−

Summing these two geometric progressions, we get: 1.5818 0.78854 2.3703I = + = So the forward price is: 0.15(4.5 2.3703) 2.47e− = The forward price is £2.47.

Page 24 CT1-13: Arbitrage and forward contracts


4 End of Part 3

You have now completed Part 3 of the Subject CT1 Notes. Review Before looking at the Question and Answer Bank we recommend that you briefly review the key areas of Part 3, or maybe re-read the summaries at the end of Chapters 11 to 13. Question and Answer Bank You should now be able to answer the questions in Part 3 of the Question and Answer Bank. We recommend that you work through several of these questions now and save the remainder for use as part of your revision. Assignments On completing this part, you should be able to attempt the questions in Assignment X3. Reminder If you have not booked a tutorial, then maybe now is the time to do so.

CT1-14: Term structure of interest rates Page 33


5 Practice questions

It’s quite common to get a long (and tricky) question in the exam on duration, volatility, convexity and immunisation. The other topics covered in this chapter are also examined frequently. Have a go at the following two (short) questions yourself before looking at the solution. Question 1 is a past paper question; question 2 is exam-style.

Question 1 (Subject 102, April 2004, Question 4) In a particular bond market, the two-year par yield at time 0t = is 4.15% and the issue price at time 0t = of a two-year fixed interest stock, paying coupons of 8% annually in arrears and redeemed at 98, is £105.40 per £100 nominal. Calculate: (a) the one-year spot rate (b) the two-year spot rate. [6]

Question 2 At 1 July 2004, an investor has a liability of £20,000 to be paid on 1 January 2008 and a liability of £18,000 to be paid on 1 July 2010. The investor currently holds assets with a present value equal to the present value of the liabilities. The investor wishes to immunise its position by investing in two zero coupon bonds with outstanding terms of four years and seven years. Determine whether or not this is possible assuming an effective interest rate of 10% per annum. [6]

Page 34 CT1-14: Term structure of interest rates


Solution 1 Let 1i be the one-year spot rate and 2i be the two-year spot rate. Using the par yield, assuming that we have £100 nominal we get:

21 2

4.15 104.151001 (1 )i i

= ++ +

The timeline for the bond is:

0 1 2

105.40 8 8+98

This gives us a second equation:

21 2

8 106105.401 (1 )i i

= ++ +

Rearranging both equations to get 22

1(1 )i+

gives:

21 12

1 4.15 1 8 1100 105.401 104.15 1 106(1 ) i ii

⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟+ ++ ⎝ ⎠ ⎝ ⎠

This can be solved as follows:

1 1

1

1

439.90 833.2010,600 10,977.411 1

393.3 377.411

4.21%

i i

ii

− = −+ +

⇒ =+

⇒ =

CT1-14: Term structure of interest rates Page 35


Substituting this into the equation for the par yield:

22

22

2

4.15 104.151001.0421 (1 )

(1 ) 1.0846964.15%

i

ii

= ++

⇒ + =⇒ =

So we have: (a) The one-year spot rate is 4.21%. (b) The two-year spot rate is 4.15%.

Solution 2 Let X be the present value of the investment in the zero-coupon bond with a term of 4 years. Let Y be the present value of the investment in the zero-coupon bond with a term of 7 years. The present values of the assets and liabilities are equal. Therefore:

X Y v v+ = + =20 000 18 000 4883 5 6, , £24,. For immunisation the discounted mean terms must also be equal or since the present values are equal this is equivalent to:

4 7 35 20 000 6 18 000 1083 5 6X Y v v+ = × + × =. , , £111,. Solving the above two equations simultaneously gives:

X = £20,103 and Y = £4,385

We must now check whether the convexity of the assets is greater than the convexity of the liabilities.

Page 36 CT1-14: Term structure of interest rates


Since the convexity of a series of cashflows of ktC at time kt is:

( )2 2

1 1

1 1

( 1) ( 1)k kk k

k kk k

n nt t

t k k k k tk k

n nt t

t tk k

C t t v t t C v v

C v C v

+

= =

= =

+ +=

∑ ∑

∑ ∑

and the denominator is just the present value calculated above, we have:

Conv Xv YvA =

× × + × ×=

4 5 7 824 488

21862 2

,.

Alternatively, first write X Av and Y Bv= =4 7 where A and B are the nominal amounts of the stocks, differentiate twice and substitute back to get X and Y.

Conv v vL =

× × + × ×=

35 4 5 20 000 6 7 18 00024 488

22 025 5 8. . , ,,

..

Since the convexity of the liabilities is greater than the convexity of the assets, immunisation using the stocks given is not possible.

CT1: Assignment X1 Questions Page 1


Question X1.1

Calculate the present value of a perpetuity paying £50 pa in arrears at an annual effective rate of interest of 6%. [1] Question X1.2

Calculate the length of time it will take £800 to accumulate to £1,000 at a simple rate of interest of 4% pa. [2] Question X1.3

A 91-day government bill is discounted at a simple rate of discount of 10% pa. Calculate the annual effective rate of interest earned on this investment. [3] Question X1.4

An annuity of $300 pa is paid annually in advance for seven years, followed by $100 pa paid quarterly in arrears for a further five years. The rate of interest is 6% pa convertible half-yearly. Calculate the accumulated amount at the end of twelve years. [4] Question X1.5

An investor, who has a sum of £10,000 to invest, wishes to purchase an annuity certain with a term of 10 years. Calculate the amount of the payments that can be provided if the annuity takes each of the following forms (assuming interest of 8% pa effective): (i) a level annuity payable monthly in arrears [2] (ii) a level annuity due payable half-yearly, commencing in 2 years’ time. [2] [Total 4] Question X1.6

Explain when you would use real and money rates of interest. Give an example of when each rate of interest would be used. [4]

Page 2 CT1: Assignment X1 Questions


Question X1.7

(i) A nominal rate of interest of 10% pa convertible quarterly is equivalent to what rate of discount convertible quarterly? [2]

(ii) A single investment of £500 is accumulated at a nominal rate of discount of

6% pa convertible half-yearly for 1 year, followed by a nominal rate of interest of 6% pa convertible every 4 months for 1 year. Calculate the accumulated amount of this investment after 2 years. [3]

[Total 5] Question X1.8

For each of the following calculate the equivalent effective annual rate of interest: (i) an effective rate of interest of 12.7% paid every 2 years [1] (ii) an effective rate of discount of 5.75% pa [1] (iii) a force of interest of ½% per month [1] (iv) a nominal rate of discount of 6% pa convertible quarterly [1] (v) a rate of interest of 14% pa convertible every 2 years. [1] [Total 5] Question X1.9

The force of interest, ( )tδ , is a function of time and at time t , measured in years, is given by: 2( ) 0.03 0.005 0.001 0 10t t t tδ = − + ≤ ≤ (i) Calculate the equivalent constant force of interest per annum for the period 0t =

to 10t = . [3] (ii) Calculate the accumulated value at time 7t = of an investment of £250 at time

0t = plus a further investment of £150 at time 5t = . [4] [Total 7]



Question X1.10

A man receives payments half-yearly in arrears for 20 years. The first payment is £250, and each payment is higher than the previous one. The payments increase by 5% pa effective for 7 years (ie the last increase of this amount is at time 7.5 years). The remaining payments increase by 4.5% pa effective. The interest rate is 6% pa convertible quarterly for the first 10 years and 5.5% pa effective for the last 10 years. Calculate the present value of the payments. [9] Question X1.11

(i) An annuity payable annually in arrears has a first payment of £300, with subsequent payments decreasing by £10 each year to £110 in the final year.

Find an expression for the present value of this annuity. Hence, calculate the

present value of the annuity payable at an effective rate of interest of 6% pa. [3] (ii) A 15-year annuity-due provides annual payments starting at £50 in year 1, £70

in year 2, £90 in year 3, and so on, until the payments have increased to £150. Payments then continue at £150 pa until the 15th payment has been made.

Calculate the present value of this annuity at an effective rate of interest of

5.2% pa. [4] (iii) An annuity certain provides payments annually in arrear for 8 years. The first

payment is £500, with subsequent payments increasing by 5% pa compound. Calculate the present value of this annuity at an effective rate of interest of

8% pa. [3] [Total 10]



Question X1.12

(i) Prove from first principles that:

|

|( )n

nn

a nvIa

δ

−= [5]

(ii) An annuity certain payable continuously for a term of 10 years provides

payments at the rate of:

• 1 pa for the first 3 months • 2 pa for the next 3 months • 3 pa for the next 3 months, and so on.

To what amount will the annuity payments accumulate by the end of the term if

interest is 10% pa convertible half yearly? [6] [Total 11] Question X1.13

The force of interest ( )tδ is a function of time, and at any time t , measured in years, is given by the formula:

0.04 0.005 0 6

( ) 0.16 0.015 6 80.04 8

+ ≤ <⎧⎪= − ≤ <⎨⎪ ≤⎩

t tt t t

tδ

(i) Derive expressions in terms of t for the accumulated amount at time t of an

investment of 1 at time 0. [6] (ii) Calculate the value at time 0 of £100 due at time 9. [2] (iii) Calculate the accumulated value at time 10 of a payment stream, paid

continuously from 6t = to 8t = , under which the rate of payment at time t is ( ) 16 1.5t tρ = − . [7]

[Total 15]

CT1: Assignment X1 Solutions Page 11


Summing this as a geometric progression:

207 3 0.5

10 0.5 0.5

0.5

0.5

250 1.05 1.045 1.0451(1 ) (1 ) (1 )

4,119.961.0451(1 )

⎛ ⎞⎛ ⎞× × ⎜ ⎟− ⎜ ⎟⎜ ⎟⎜ ⎟+ + +⎝ ⎠⎝ ⎠ =−

+

i f f

f

The total present value is: 3,506.34 1,099.41 4,119.96 8,725.71+ + = Solution X1.11

(i) Present value The cashflow diagram is:

0 1 2 20

300 290 110

. . .

. . .

years

cashflow

Note that there are 19 decreases of £10, so there are 20 payments altogether. The present value is given by: ( )20 20310 10a Ia− [1]

Evaluating the present value:

( )| |20 20310 10

310 11.4699 10 98.7004

PV a Ia= −

= × − × [1] 3,555.68 987.00 £2,569= − = [1]

Page 12 CT1: Assignment X1 Solutions


(ii) Present value The cashflow diagram is:

0 1 2 3 4 5 6 14. . .

50 70 90 110 130 150 150 . . . 150

years

cashflow

Note that the 15th payment is at time 14 (since the 1st payment is at time 0). There are many expressions for the present value. One of them is: ( ) 4

5 10530 20 150PV a Ia v a= + + [1]

30 4.529538 20 13.12989 150 6.2437= × + × + × [2] 135.89 262.60 936.56 £1,335.04= + + = [1] Alternative expressions include:

( ) ( )

( ) ( )

5 66 9 6 96 6

4 55 10 5 104 4

30 20 150 30 20 150

50 20 150 50 20 150

a Ia v a a Ia v a

a Ia v a a Ia v a

+ + + +

+ + + +

(iii) Present value The cashflow diagram is:

0 1 2 3 8. . .

500 . . .

years

cashflow500 × 1.05 500 × 1.052 500 × 1.057

The present value for these payments is: 2 7 8500 500 1.05 500 1.05PV v v v= + × + + ×



So: 0.44 0.04( 8) 0.12 0.04(0, ) t tA t e e e− += = (8 t≤ ) [1] In summary:

2

2

0.04 0.0025

0.36 0.16 0.0075

0.12 0.04

0 6

(0, ) 6 8

8

t t

t t

t

e t

A t e t

e t

+

− + −

+

⎧ ≤ <⎪⎪= ≤ <⎨⎪

≤⎪⎩

(ii) Present value The present value here is:

0.12 0.04 9 0.48100 100 100 £61.88(0,9)A e e+ ×= = = [2]

Alternatively, we could calculate this from first principles:

[ ]

6 8 9

0 6 8

6 82 2 90 6 8

0.04 0.005 0.16 0.015 0.04

0.04 0.0025 0.16 0.0075 0.04

0.33 (0.8 0.69) (0.36 0.32)

0.48

100

100

100

100 £61.88

t dt t dt dt

t t t t t

PV e e e

e e e

e e e

e

− + − − −

⎡ ⎤ ⎡ ⎤− + − − −⎣ ⎦ ⎣ ⎦

− − − − −

−

∫ ∫ ∫=

=

=

= =

(iii) Accumulated value The accumulated value at time 8 will be:

882

2

8 80.16 0.015 0.16 0.0075

6 6

80.8 0.16 0.0075

6

(16 1.5 ) (16 1.5 )

(16 1.5 )

t ts ds s s

t t

t e dt t e dt

t e dt

− ⎡ ⎤−⎣ ⎦

− +

∫− = −

= −

∫ ∫

∫ [2]



But 2 2

8 80.8 0.16 0.0075 0.8 0.16 0.0075

66

( 0.16 0.015 ) t t t tt e dt e− + − +⎡ ⎤− + = ⎢ ⎥⎣ ⎦∫ , so the accumulated

value is:

2 2

2

8 80.8 0.16 0.0075 0.8 0.16 0.0075

6 6

80.8 0.16 0.0075

6

0 0.11

(16 1.5 ) 100 ( 0.16 0.015 )

100

100( ) 11.6278

t t t t

t t

t e dt t e dt

e

e e

− + − +

− +

− = − − +

⎡ ⎤= − ⎢ ⎥⎣ ⎦

= − − =

∫ ∫

[3] Thus the accumulated value at time 10 is:

10

80.04

0.0811.6278 11.6278 12.596dt

e e∫

= = [2] Alternatively, students could find the expression for ( ,10)A t and calculate the accumulated amount as:

8

6

( ) ( ,10)t A t dtρ∫



Question X2.8

Two projects A and B have the following expected cashflows: Project A Project B Initial outlay: £170,000

£200,000

Other expenses: £20,000 at the end of year 1 £10,000 at the end of year 2

– –

Income: £20,000 at the end of year 1 £20,000 at the end of year 2 £200,000 at the end of year 3

£14,000 pa at the end of each of the first 6 years £200,000 at the end of year 6

(i) Calculate the internal rate of return (correct to 1 decimal place) for each project. [4] (ii) Calculate the net present value of each project using a risk discount rate of

6% pa. [2] (iii) If funds for the projects can be raised by borrowing from a bank, determine the

interest rate charged by the bank above which each project becomes unprofitable. Mention any other factors that should be taken into account when deciding between the projects. [3]


Mr Smith is struggling to repay his loan of £20,000 with payments of £427.90 made monthly in arrears for 5 years. (i) Calculate the APR of Mr Smith’s loan. [5] After exactly one year, a loan company offers to ‘help’ Mr Smith by restructuring his loan with new monthly payments of £274.49 made in arrears. (ii) Assuming the company charges the same APR as Mr Smith’s original loan,

calculate the term of the new loan. [4] (iii) How much more interest in total will Mr Smith pay on his restructured loan than

on his original loan? [2] [Total 11]



Question X2.10

(i) Explain what is meant by the following terms: (a) equation of value (b) discounted payback period from an investment project. [4] (ii) An insurance company is considering setting up a branch in a country in which it

has not previously operated. The company is aware that access to capital may become difficult in twelve years’ time. It therefore has two decision criteria. The cashflows from the project must provide an internal rate of return greater than 9% per annum effective and the discounted payback period at a rate of interest of 7% per annum effective must be less than twelve years.

The following cashflows are generated in the development and operation of the

branch. Cash Outflows Between the present time and the opening of the branch in three years’ time the

insurance company will spend £1.5m per annum on research, development and the marketing of products. This outlay is assumed to be a constant continuous payment stream. The rent on the branch building will be £0.3m per annum paid quarterly in advance for twelve years starting in three years’ time. Staff costs are assumed to be £1m in the first year, £1.05m in the second year, rising by 5% per annum each year thereafter. Staff costs are assumed to be incurred at the beginning of each year starting in three years’ time and assumed to be incurred for 12 years.

Cash Inflows The company expects the sale of products to produce a net income at a rate

of £1m per annum for the first three years after the branch opens rising to £1.9m per annum in the next three years and to £2.5m for the following six years. This net income is assumed to be received continuously throughout each year. The company expects to be able to sell the branch operation 15 years from the present time for £8m .

Determine which, if any, of the decision criteria the project fulfils. [17] [Total 21]



(iv) Amount outstanding Due to the nature of a repayment loan, we would expect half of the loan to be repaid some time after half the term of the loan. The capital outstanding after the t th quarterly instalment is: (4)

5 ¼4 170.07t t

L a−

= ×

So we need to find the smallest value of t for which: (4) (4)

5 ¼ 5 ¼680.28 1,000 1.470

t ta a

− −< ⇒ < [1]

Solving the equation (4)

5 ¼1.470

ta

−= , we find:

14

14

(5 )

(4)

5

1 1.05 1.47

1.05 0.928

1 ln 0.9285 1.535 13.94 ln1.05

t

t

i

t t

− −

−

−=

⇒ =

⇒ − = = − ⇒ = [1]

For the amount outstanding to be below 1,000, we require 14t = , ie the capital outstanding will fall below £1,000 immediately after the 14th payment. [1] Alternatively, working in quarters we would solve the equation: 20170.07 1,000ta − =

using an interest rate of 1.2272%. Another alternative is to work years and let n be the remaining term of the loan. This gives the equation: (4)4 170.07 1,000

na× =

Solving this gives 1.535n = years, so the time elapsed since the loan was taken out is 3.465 years, leading to the same conclusion as above.



Solution X2.7

(i)(a) Money-weighted rate of return The money-weighted rate of return is found from the equation:

1

234.2(1 ) 0.8(1 ) 5.5i i+ + + = [1] Using a binomial approximation for the first guess gives: 4.2(1 3 ) 0.8(1 ½ ) 5.5 3.85%i i+ + + ≈ ⇒ Trying out values gives:

3% 5.4014

4% 5.5403

i

i

= ⇒

= ⇒ [1]

Using interpolation gives:

5.5 5.40143 (4 3) 3.7%5.5403 5.4014

i pa−≈ + × − =

− [1]

(i)(b) Time-weighted rate of return The time-weighted rate of return is found from the equation:

3 5.1 0.8 5.5(1 ) 1.10414.2 5.1

i −+ = × = [1]

3.4%i pa⇒ = [1] (i)(c) Linked annual rate of return There were no cashflows during 2002 or 2003. So the values of 1 i+ for these years were just:

4.6 5.11.0952 and 1.10874.2 4.6

= = [1]

The effective rate of return for 2004 solves the equation of value:

1

25.1(1 ) 0.8(1 ) 5.5i i+ + + =



Using the quadratic formula gives:

1

2(1 ) 0.96300 1 0.92737 7.263%i i i+ = ⇒ + = ⇒ = − [1] So the linked annual rate of return is found from the equation:

3(1 ) 1.0952 1.1087 0.92737 1.126

4.0%

i

i pa

⇒ + = × × =

⇒ = [1] (ii) Relative performance of the fund managers The MWRR is lower for the second fund, whereas the TWRR is higher. The TWRR is the better measure of fund performance as it is not affected by the timing and size of cashflows (which are not under the control of the fund manager). [1] Therefore on this basis we would say the second fund manager has performed better. [1] Solution X2.8

(i) Internal rate of return The IRR for Project A is the rate of interest that satisfies the equation of value:

2 2 3170,000 20,000 10,000 20,000 20,000 200,000 0ν− − − + + + =v v v v Simplifying this and expressing it in £000s: 2 310 200 170+ =v v [1] By trial and error, we find that:

2 3

2 3

7% 10 200 171.99

8% 10 200 167.34

= ⇒ + =

= ⇒ + =

i v v

i v v [1]

Using interpolation:

170 171.997 (8 7) 7.4%167.34 171.99

i pa−≈ + × − =

− [1]



For Project B, the income (paid at the end of each of the first 6 years) represents 7% of the initial capital, which is returned at the end of the 6 years. So we can see immediately that the IRR for Project B is exactly 7%. [1] Alternatively, we can solve: 6

614 200 200a v+ =

(ii) Net present value The net present values are found by discounting the payments at 6%. Project A: 2 3NPV 170,000 10,000 200,000 £6,823.82v v= − + + = [1]

Project B: 6|6NPV 200,000 14,000 200,000 £9,834.65a v= − + + = [1]

(iii) Interest rate at which it is unprofitable Each project will be profitable if the rate of interest at which funds can be borrowed is less than the internal rate of return. So, to be profitable, Project A requires a borrowing rate less than 7.4% and Project B requires a rate lower than 7%. [1] Other factors to be considered include: • Project A does not provide any net income during the first year. • The lower internal rate of return for Project B applies for a longer period. • The rates of interest available in years 4 to 6 (ie after Project A has finished) will

affect the comparison between the accumulated profits at the end of year 6 (ie when Project B finishes).

• The risk associated with the receipt of income. If Project A involves greater uncertainty or risk, it may not be accepted even though it has a higher IRR.

[1 mark for each point, up to maximum of 2]



Alternatively, we could consider the interest paid over the whole term of the loan. The original loan period was five years: total interest 5 12 £427.90 £20,000 £5,674= × × − = The total interest paid over the whole term, allowing for restructuring, is: total interest 1 12 £427.90 7¼ 12 £274.49 £20,000 £9,015.43= × × + × × − = Therefore Mr Smith will pay £9,015.43 £5,674 £3,341.43− = more interest. Solution X2.10

This question is taken from Subject CT1, September 2005, Question 11. (i)(a) Equation of value The equation of value equates the value of the payments made to the value of the payments received at an appropriate rate of interest. [1] The values can be determined at any point in time, but it is common to equate present values, ie the values at time 0. [1] (i)(b) Discounted payback period An investment project usually requires expenditure (or investment) in the early years before returns (or profits) emerge in the later years. So during the early years the ‘project account’ will be overdrawn and interest will be incurred at the borrowing rate. [1] The discounted payback period is the time from the beginning of the project until the ‘project account’ balance first becomes positive. [1] Markers: As an alternative to this final point, students should be awarded one mark if they state that the discounted payback period ends at the first point in time such that the net present value (or accumulated value) of project cashflows up to that time is positive.



(ii) Checking the investment criteria Is the internal rate of return > 9% pa effective? If the net present value at 9% is positive then the internal rate of return will be greater than 9%, and similarly if the net present value at 9% is negative then the internal rate of return will be less than 9%. Cash outflows The present value of the research and associated costs is: 3 9%1.5 £3.96535a m= [1]

The present value of the rent is:

( )439% 12 9%

0.3 £1.75112v a m= [1]

The present value of the staff costs is:

{ }( )

3 2 2 3 3 11 119% 9% 9% 9% 9%

121.051.093

9% 1.051.09

1 1.05 1.05 1.05 1.05

1£7.60691

1

v v v v v

v m

+ + + + +

−= =

− [2]

Alternatively, the present value of the staff costs may be calculated using an annuity:

39% 12 @ %Iv a where 1.09 1 3.8095%

1.05I = − =

Total present value of cash outflow is £13.32338m.



Cash inflows The present value of the net sales income is: 3 6 9

9% 9% 9%3 9% 3 9% 6 9%1.9 2.5 £10.42887v a v a v a m+ + = [2]

The present value of the proceeds from the sale of branch operation is: 15

9%8 £2.19630v m= [1] Total present value of cash inflow is £12.62517m. So the net present value of the project at 9% pa is: 12.62517 13.32338 £0.698m− = − [1] Since the net present value is negative at 9% interest, the 9%IRR < . So the criterion is not satisfied. [1] Is the discounted payback period less than 12 years? If the net present value of the cashflows up to time 12 (excluding the payments then due) is negative the discounted payback period is 12 years≥ , and similarly if this is positive the discounted payback period is 12 years< . Present value of cash outflows up to time 12 Research etc 3 7%1.5 £4.07270a m= [1]

Rent

( )437% 9 7%

0.3 £1.66472v a m= [1]



Staff costs

{ }( )

3 2 2 3 3 8 87% 7% 7% 7% 7%

91.051.073

7% 1.051.07

1 1.05 1.05 1.05 1.05

1£6.82069

1

v v v v v

v m

+ + + + +

−= =

− [2]

Alternatively, the present value of the staff costs may be calculated using an annuity:

37% 9 @ %Iv a where 1.07 1 1.9048%

1.05I = − =

Total present value of cash outflow up to time 12 is £12.55811m. Present value of cash inflows up to time 12 Net sales income 3 6 9

7% 7% 7%3 7% 3 7% 3 7%1.9 2.5 £9.34598v a v a v a m+ + = [2]

So the net present value of the cashflows up to time 12 is: 9.34598 12.55811 £3.212m− = − . [1] Since this is negative, the 12 yearsDPP ≥ . The criterion is not satisfied. [1] Note that if you include the cash outflows at time 12, the net present value is –£3.934m, and the conclusion is the same. Alternatively, if we work in terms of accumulated values:

( ) 9.17248( ) 3.74928( ) 15.36153( ) 21.04883

AV researchAV rentAV staffAV sales

====

The accumulated value of cashflows just before time 12 is £7.234m− , giving the same conclusion.



Question X3.1

An investor obtains a nominal rate of return on a 10-year bond of 6.4% pa effective. Calculate the annual effective real rate of return if inflation is assumed to be 3% pa throughout the 10 years. [2] Question X3.2

Define an interest rate swap and describe briefly the two kinds of risk facing each counterparty to the swap. [3] Question X3.3

Describe the role of the following in the markets for derivatives: (i) margin [2] (ii) the clearing house. [2] [Total 4] Question X3.4

An equity that pays annual dividends is purchased immediately after a dividend payment has been made. The next dividend is expected to be d per £1 invested. If dividends are expected to grow at a compound rate g and price inflation is expected to operate at rate e , show that the real rate of return obtained on this equity will be

1d g ei

e+ −

=+

. Assume that the equity is held indefinitely. [4]

Question X3.5

An investor, who is liable to income tax at 20% but is not liable to capital gains tax, wishes to earn a net effective rate of return of 5% per annum. A bond bearing coupons payable half-yearly in arrear at a rate 6.25% per annum is available. The bond will be redeemed at par on a coupon date between 10 and 15 years after the date of issue, inclusive. The date of redemption is at the option of the borrower. Calculate the maximum price that the investor is willing to pay for the bond. [5]



Question X3.6

Outline how a debenture differs from a government bond. [5] Question X3.7

An investor purchases a newly issued bond with six-monthly coupons of 9% pa on 1 January 2004. He is liable to 40% tax on income, which he pays on 1st April each year for the income earned in the preceding year (running from 1st April to 31st March). He is also liable for capital gains tax of 35%, payable when the gain is made. The bond is redeemable at 110% in 5 years’ time. What price should the investor pay in order to achieve a yield of 8.5% pa? [5] Question X3.8

An investor purchases a 5-month forward contract on 1 January 2004 to buy 1,000 shares at the end of the contract. The price of a share on 1 January 2004 is £50. Dividends are received continuously and the dividend yield is 6% pa. The risk-free rate of interest is 4% pa effective and there is no arbitrage. (i) Calculate the forward price. [3] (ii) The investor decides to sell the contract on the 1 April 2004, when the price of

the stock is £49.50 per share. Calculate the value of the contract at this time. [3] [Total 6]



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) If the government wishes to issue the new bond at a price of £100%, what

coupon rate should be chosen for this 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]


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]



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. [5] (ii) Calculate the value of the forward contract on 1 February 2005 under each of the

scenarios listed above if the share price of Company X is then £3.50. [6] [Total 11] Question X3.12

Company XYZ is considering investing in a capital project. The costs for the project will be £5 million at the beginning of the first year, and

£ ( )20.5 1.04k−× million payable at the beginning of year k , 2, 3, ..., 10k = .

The returns will be £2 million at the end of each of the first 5 years, increasing by £250,000 at the end of each of years 6 to 10. Company XYZ decides to use a risk discount rate of 12% pa to assess this project. (Note that a risk discount rate is an annual effective interest rate.) (i) Calculate the net present value of the project. [7] (ii) Calculate the discounted payback period for the project. [4] [Total 11]



Assignment X3 Solutions Markers: This document sets out one approach to solving each of the questions. Please give credit for other valid approaches. Solution X3.1

Using the formula:

1−

=′+

i eie

where e is the rate of inflation and i is the nominal rate of return and ′i is the real rate of return, gives: [1]

0.064 0.03 3.30%1.03

i −= =′ [1]

Solution X3.2

In an interest rate swap, one party agrees to pay to the other a regular series of fixed amounts for a certain term. In exchange, the second party agrees to pay a series of variable amounts based on the level of a short-term interest rate. Both sets of payments are in the same currency. [1] Each counterparty faces market risk and credit risk:

● Market risk is the risk that market conditions will change so that the present value of the net outgo under the agreement increases. [1]

● Credit risk is the risk that the counterparty will default on its payments. This will only occur if the swap has a negative value to the defaulting party. [1]



Solution X3.3

(i) Margin ● This is the sum of money that each party to a futures contract must deposit with

the clearing house. [1]

● For an option contract, only the writer of the contract has to deposit margin. [½]

● It acts as a cushion against potential losses which the parties may suffer as a result of adverse market movements. [½]

● When the contract is first struck, initial margin is deposited with the clearing house. This is changed on a daily basis through additional payments of variation margin. This variation margin ensures that the clearing house’s exposure to credit risk is controlled. [½]

● The amount of margin will be small compared with the value of the underlying asset. [½]

● Margin deposits earn interest. [½]

[Maximum 2]

(ii) The clearing house ● Acts as a party to every trade, standing between the buyer and seller so that the

link between the two is broken. [1]

● It is informed when a contract in futures or options has been agreed. [½]

● It checks that the buy and sell orders match each other. [½]

● It guarantees that the contract will be honoured by removing the credit risk of individual investors. The credit risk would otherwise be caused by the chance that one of the counterparties would renege. [½]

● This makes each contract readily tradable: each contract is indistinguishable from all the others. [½]

[Maximum 2]



Solution X3.4

If 1 unit is invested in the equity, it will generate expected dividends as follows:

cashflows

years0 1 32

d d(1 + g) d(1 + g)2

So, the (real) equation of value, allowing for inflation between when the equity is purchased and the receipt of each dividend, is:

2

2 32 3

(1 ) (1 )1(1 ) (1 ) (1 )

+ += + + +

+ + +

d d g d gv v ve e e

[1]

Summing the RHS as an infinite geometric series with 1

da ve

=+

and 11

gr ve

+=

+ using

1aS

r∞ =−

gives:

1

1(1 )1

1 (1 )(1 )d e v

g e v

−

−+

=− + +

[1]

Multiplying through by the factor (1 )(1 )e i+ + :

1(1 )(1 ) (1 )

de i g

=+ + − +

Rearranging and simplifying to isolate the real rate of interest i :

1(1 )(1 ) (1 ) 1 1 1

d g d g ee i d g i ie e

+ + + −+ + = + + ⇒ + = ⇒ =

+ + [2]



Solution X3.5

This question is taken from Subject CT1, April 2006, Question 4. Carrying out the capital gains test:

12(2)

1

@5% 2 (1 ) 1 4.939%

6.25(1 ) (1 0.2) 5%100

i i

g t

⎛ ⎞= + − =⎜ ⎟⎝ ⎠

− = − = [1]

Since (2)

1(1 )i g t< − , there is a loss for the investor and therefore a gain for the borrower. Therefore the borrower will want to redeem as early as possible, ie after 10 years. [1] We will work with £100 nominal. The equation of value for the price P is: (2) 10

106.25 0.8 100P a v= × + [1]

Calculating the necessary annuity: (2)

107.817079a = [1]

So the price per £100 nominal is: 39.085 61.391 £100.48P = + = [1]



Solution X3.6

How debentures differ from government bonds Security

In most developed countries, there is no risk of default on either income or capital payments on government bonds. [1] A debenture is less secure, in spite of being secured on assets of the company. The level of security depends on the company that has issued it, and the term. [1] Marketability

Government bonds are usually are more marketable than debentures. This is because they tend to be issued in large volumes. [1] Fewer investors hold debentures and trade is much less active. Marketability is poorest on small unlisted issues. [1] Yield

The yields on government bonds are lower than on equivalent debentures, mainly due to the greater security and higher marketability of government bonds. [1] Other differences

Government bonds are issued in the form of index-linked bonds in large volumes. [½]

Government bonds have lower dealing costs. [½] [Maximum 5]



Solution X3.7

We need to test for a capital gain:

(2)1(1 )Di t

R> − for a capital gain

where 1t is the rate of income tax, D is the coupon rate and R is the redemption rate. Here we have:

(2)1

9(1 ) (1 0.4) 0.0491 0.0833110

D t iR

− = − = =

so we do have a capital gain. [1] Let the price paid be P, then: (2) 0.25 5 5

|| 559 3.6 110 (110 ) 0.35P a a v v P v= − + − − × [1]

where all functions are calculated at 8.5%. Working with this:

36.2041 13.8999 73.1550 (110 ) 0.232766

69.8549 91.050.767

P P

P

= − + − − ×

⇒ = = [2]

So the investor pays £91.05 per £100 nominal. [1] The capital gains test above implicitly assumes that the income tax is payable immediately that income is received. Note that since the payment of income tax is deferred in this question, the capital gains test is only approximate. The safest way to proceed in this type of question is to carry out the standard capital gains test, do the calculations, then check that the final answer agrees with the outcome of the test ie does the price calculated give the capital gain the test predicted. If not, rework the calculations. Here (as in the majority of such cases) the capital gains test gives the correct answer and marks should be awarded for carrying it out.



Solution X3.10

This question is taken from Subject CT1, April 2005, Question 6. (i)(a) Real rate of return The diagram showing the cashflows is:

-25 10 1010 Payment

Time

RPI

0 1 2 3

170.7 183.3 191.0 200.9

The equation of value for the real cashflows is:

2 3

2 3

170.7 170.7 170.725 10 10 10183.3 191 200.9

9.312602291 8.937172775 8.496764559

= × + × + ×

= + +

v v v

v v v [2] Using trial and improvement: At 3%, the RHS is 25.241 At 3.5%, the RHS is 25.004 [1] So the real rate of return is approximately 3.5%. [1] Markers: in the trial and improvement section, award half a mark for each of up to two correct values on the way to the true value. (i)(b) Money rate of return The equation of value for the actual cashflows is: 2 325 10 10 10v v v= + + [1] Using trial and improvement: At 9.5%, the RHS is 25.089 At 10%, the RHS is 24.869 [1]



Interpolating between these values, we get:

9.5 25 25.089 9.710 9.5 24.869 25.089i i− −

= ⇒− −

So the money rate of return is approximately 9.7%. [1] Markers: in the trial and improvement section, award half a mark for each of up to two correct values on the way to the true value. (ii) Relationship The average rate of inflation is:

3 200.9 1 5.6%170.7

− = [1]

We have that 1i ji

j−′ =+

, where i′ is the real rate of interest, j is the rate of inflation and

i is the money rate of interest. Substituting our values into the RHS of this equation, we get:

0.097 0.056 3.9%1.056−

This is very approximately the real rate of interest that we obtained. [1]



Question X4.1

At time 0t = , the 2-year spot rate is 4% pa effective, the 3-year spot rate is 5% pa effective and the 4-year spot rate is 6% pa effective. Calculate the 2-year continuous-time forward rate from time 2t = . [2] Question X4.2

The n-year forward rate for transactions beginning at time t and maturing at time t n+ is denoted as ft n, . You are given: f0 1, = 6.0% per annum f0 2, = 6.5% per annum f1 2, = 6.6% per annum Determine the 3-year par yield. [3] Question X4.3

A life insurance company is modelling its investment performance over the next 10 years by assuming that the yield obtained during the first 5 years will have a constant value of 4%, 5% or 6% pa, each with equal probability, and that the yield during the second 5 years will be 1% pa lower than the yield during the first 5 years. Find the expected accumulated value at the end of the 10 years of a single investment of £100,000 at time 0. [3]



Question X4.4

The 1, 2, 3, 4 and 7-year spot rates are 4%, 3.75%, 3.6%, 3.5% and 3.9% pa respectively. The 3-year forward rate from time 3 is 3.8% pa and the 3-year forward rate from time 2 is 3.7% pa. Calculate: (i) the 2-year forward rate from time 5 [2] (ii) the present value of payments of 5 at the end of each of the 7 years [2] (iii) the accumulated value at time 6 of a payment of £100 at time 1. [1] [Total 5] Question X4.5

An investor has purchased a 99-year property lease, which will pay £100 at the end of each of the first 33 years, £200 at the end of each of the next 33 years and £300 at the end of each of the last 33 years. Calculate using 8% pa interest: (i) the present value of the income from the lease [2] (ii) the discounted mean term of the income from the lease. [4] [Total 6] Question X4.6

(i) Calculate, using an effective annual interest rate of 10%, the present value, volatility and convexity of the portfolio consisting of the following set of payments:

£1,000 payable in 5 years’ time £5,000 payable in 10 years’ time and £25,000 payable in 25 years’ time. [6] (ii) Calculate the present value and volatility in 2 years’ time assuming no change in

the interest rate. [3] [Total 9]



Question X4.7

The annual rates of interest from a particular investment, in which part of an insurance company’s funds is invested, are independently and identically distributed. Each year, the distribution of ( )1+ it , where it is the rate of interest earned in year t, is log-normal

with parameters μ and σ 2 . it has mean value 0.07 and standard deviation 0.02, the parameter 0.06748μ = and

2 0.0003493σ = . (i) The insurance company has liabilities of £1m to meet in one year from now. It

currently has assets of £950,000. Assets can either be invested in the risky investment described above or in an investment which has a guaranteed return of 5% per annum effective. Find, to two decimal places, the probability that the insurance company will be unable to meet its liabilities if:

(a) All assets are invested in the investment with the guaranteed return. (b) 85% of assets are invested in the investment which does not have the

guaranteed return and 15% of assets are invested in the asset with the guaranteed return. [7]

(ii) Determine the variance of return from the portfolios in (i)(a) and (i)(b) above. [3] [Total 10] Question X4.8

The liabilities of a fund consist of two lump sum payments due at known times in the future. The second lump sum is due for payment 5 years after the first and is twice the amount of the first. (i) If the total present value and the discounted mean term of the liabilities (both

calculated using a market interest rate of 6% pa effective) are £75,000 and 8 years, respectively, determine the timing and amounts of the payments. [6]

(ii) If the assets of the fund consist of a single zero coupon bond that will mature 8

years from now with a redemption payment of £119,540, what can you say about this portfolio on the basis of Redington’s theory of immunisation? [4]

[Total 10]



Question X4.9

£1,000 is invested for 10 years. In any year the yield on the investment will be 4% with probability 0.4, 6% with probability 0.2 and 8% with probability 0.4 and is independent of the yield in any other year. (i) Calculate the mean accumulation at the end of 10 years. [2] (ii) Calculate the standard deviation of the accumulation at the end of 10 years. [5] (iii) Without carrying out any further calculations, explain how your answers to (i)

and (ii) would change (if at all) if: (a) the yields had been 5%, 6% and 7% instead of 4%, 6% and 8% per

annum, respectively; or (b) the investment had been made for 12 years instead of 10 years. [4] [Total 11]



Question X4.10

(i) A company has to pay £ ( )2,000 10 t− at the end of year t , for ,5, 6, 7, 8 9t = . It values these liabilities assuming that there will be a constant effective annual rate of interest of 6% pa.

(a) Express the present value of the liabilities in terms of level and

increasing annuities. (b) Hence calculate the present value of the liabilities. [3] (ii) The company wants to immunise its exposure to the liabilities by investing in 2

bonds: Bond A pays coupons of 5% pa annually in arrears and is redeemable at

par in 15 years’ time. Bond B is a zero-coupon bond that is redeemable at par in 5 years’ time. The gross redemption yield on both stocks is the same as the interest rate used to

value the liabilities. (a) Calculate the amount that the company should invest in each of the two

bonds to ensure that the present value and volatility of the assets are equal to those of the liabilities.

(b) What other condition is required for immunisation? [9] [Total 12]



Question X4.11

An economist’s model of interest rates indicates that the n year spot rate of interest is

( ) 10.20.1 1 ne−−+ .

(i) Sketch a yield curve based on this formula, indicating clearly the values of the

immediate spot rate and the limiting yield on long-dated stocks. [3] (ii)(a) Explain what is meant by the term structure of interest rates. (ii)(b) Explain briefly the shape of the yield curve by reference to the liquidity

preference theory. [4] (iii) Assuming that the economist’s model is correct, calculate: (a) the price of a bond, purchased now, paying coupons of 6% annually in

arrears and redeemable at par in 3 years’ time (b) the par yield for the bond in (a). [6] [Total 13]



Question X4.12

In any year, the yield on funds invested with a given insurance company has mean j and standard deviation s , and is independent of the yields in all previous years. Let ti be the rate of interest earned in the t th year (i) Derive formulae for the mean and variance of the accumulated value after n

years of a single investment of 1 at time 0. [6] (ii) Each year the value of (1 )ti+ is lognormally distributed. The rate of interest has

a mean value of 0.06j = and a standard deviation of 0.1s = in all years. (a) Show that the parameters μ and 2σ of the lognormal distribution take

the values 0.05384 and 0.008861 respectively. (b) Derive the distribution of 10S , where 10S denotes the accumulation of

one unit of money for 10 years. (c) Find the single amount that should be invested to give an accumulation

of at least £50,000 in 10 years’ time with probability 0.95. [10] [Total 16]


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Alternatively:

5

5

5 5

5 5

2 ( 5) 82

2( 5) 8 68 5.0044 years1 2 1 2

t t

t tXtv X t vDMT

Xv Xv

t t v vtv v

+

++ +

= =+

+ + +⇒ = ⇒ = =

+ +

Substituting back into the present value equation:

5 5.004475,000 £40, 245

(1 2 )X

v v= =

+ [1]

ie the payments are £40,245 after 5 years and £80,491 after 10 years. [1] (ii) Redington’s conditions The present value of the assets is: 8119,540 75,001v = ie the same as the present value of the liabilities (near enough). [1] We know that the durations of the assets and the liabilities are the same (both 8 years). [1] However, the convexity of the assets will be less than the convexity of the liabilities because the asset cashflow falls between the liability cashflows. [1] Redington’s theory requires the opposite to be true for immunisation. So this portfolio is “reverse immunised”, ie a small change in interest rates in either direction will lead to a deficit! [1]



Solution X4.9

This question is taken from Subject 102, April 2003, Question 9. (i) Mean Let:

( )10

101

1 tt

S i=

= +∏

where ( 1,2, ,10)ti t = … are independent and identically distributed random variables. The mean of the rate of return in each year, j , is: [ ] 0.04 0.4 0.06 0.2 0.08 0.4 0.06tE i j= = × + × + × = [1] So:

( ) ( )10 1010[1,000 ] 1,000 1 1,000 1.06 £1,790.85E S j= + = = [1]

(ii) Standard deviation The variance of the rate of return in each year, 2s , is:

[ ] 2 2 2 2 2var 0.04 0.4 0.06 0.2 0.08 0.4 0.06

0.00032ti s= = × + × + × −

= [2] The variance of the accumulation is:

( )

( ) ( )( )( ) ( )

210 10

210 102 2 2

10 202 2

2 2

var 1,000 1,000 var[ ]

1,000 (1 ) 1

1,000 1.06 0.00032 1.06

1,000 0.095633

=

⎡ ⎤= + + − +⎢ ⎥

⎣ ⎦⎡ ⎤= + −⎢ ⎥⎣ ⎦

= ×

S S

j s j

[2] So the standard deviation is £95.63. [1]



(iii)(a) Change of yield The average yield in any year, j , will remain unchanged because the random variable is still symmetrically distributed about 6%. The values of the yield are more closely packed around the mean, so 2s will be smaller. [1] The mean accumulated value will be unchanged, but the standard deviation of this value will decrease. [1] (iii)(b) Change of term The values of j and 2s are unchanged. The accumulation is over a longer period so both the mean and the standard deviation of the accumulated value will increase. [2] Solution X4.10

(i)(a) Expressing present value of liabilities in terms of level and increasing annuities

The liabilities are illustrated on the timeline below:

0 5 6 7 8 9 time

10 8 6 4 2 payment, £1,000s

The present value of the liabilities is:

( ) ( ){ }

5 6 7 8 9

4 2 3 4 5 45 5

10,000 8,000 6,000 4,000 2,000

2,000 5 4 3 2 2,000 6

LPV v v v v v

v v v v v v v a Ia

= + + + +

= + + + + = − [2]



(i)(b) Calculating present value of liabilities If 0.06i = , then: 42,000 1.06 (25.2742 12.1469) £20,796LPV −= × × − = [1] (ii)(a) Amount to be invested in each bond Let A denote the present value of the amount invested in Bond A and B denote the amount invested in Bond B. The present value of the assets is then: APV A B= + Setting this equal to the present value of the liabilities, we have: (1) … 20,796A B+ = [1] We have that equating volatilities is the same as equating discounted mean terms. Here it is easier to use the discounted mean term. The discounted mean term of the liabilities is:

5 6 7 8 950,000 48,000 42,000 32,000 18,000

129,865 6.245 years

LL

L

v v v v vDMTPV

PV

+ + + +=

= = [1]

Alternatively, calculating the volatility of the liabilities gives:

6 7 8 9 105 10,000 6 8,000 7 6,000 8 4,000 9 2,000

20,796122,514 5.89120,796

L

L

PVPV

v v v v v

ν′

= −

− × − × − × − × − ×= −

= =



To calculate the discounted mean term of the assets, we first need to calculate the price of Bond A per £100 nominal. This is:

( )1515 15

1005 100 5 9.7122 90.2881.06

P a v= + = × + = [1]

So an investment of A in Bond A buys 90.288

A lots of £100 nominal.

The discounted mean term of the assets is then:

( )

( )( )

( )15

2 15 1590.288

1590.288 15

1,50090.288 1.06

5 10 75 1,500 5

5 1,500 5

5 67.2668 5

10.657 5

+ + + + +=

+ +=

× + +=

+=

A

AA

A

A

A

A

A

v v v v BDMT

PV

Ia v B

PV

B

PV

A BPV

[2]

Setting the discounted mean term of the assets equal to the discounted mean term of the liabilities we obtain: (2) … 10.657 5 129,865A B+ = [1] Now multiplying (1) by 5, we get: (3) … 5 5 103,980A B+ = and subtracting (3) from (2): 5.657 25,885 £4,576A A= ⇒ = [1] Substituting this back into (1), we find that: 20,796 4,576 £16,220B = − = [1]



Alternatively, defining A to be the nominal amount purchased of Bond A , and B to be the nominal amount purchased of Bond B , the present value of the assets is:

( )15 5150.05 0.90288 0.74726APV A a v Bv A B= + + = +

The DMT of the assets is:

( )

( )( )

2 15 15 5

15 515

0.05 2 0.05 15 0.05 15 5

0.05 15 5

9.6223 3.7363

AA

A

A

A v v v v BvDMT

PV

A Ia v Bv

PV

A BPV

+ × + + × + +=

+ +=

+=

These give us the simultaneous equations:

0.90288 0.74726 20,7969.6223 3.7363 129,865

A BA B

+ =+ =

Solving these gives 5,068A = and 21,707B = . Converting these values into the amounts invested in the bonds (ie the present values) asked for in the question, gives the same answers as above. (ii)(b) Other condition for immunisation We also require that the convexity of the assets is greater than the convexity of the liabilities at the current rate of interest, ie:

2 2

2 2A L

A Ld PV d PVPV PV

di di> [1]



(ii)(a) Parameter values Equating the formulae for the mean and variance of a lognormal distribution to the values we want:

2½ (1 ) 1 1.06e E i jμ σ+ = + = + = [1]

2 22 2 2( 1) var(1 ) 0.1e e i sμ σ σ+ − = + = = [1]

Solving these simultaneous equations:

22 2 21.06 ( 1) 0.1 0.008861eσ σ− = ⇒ = [½ ]

ln1.06 ½ 0.008861 0.05384μ μ= − × ⇒ = [½ ] (ii)(b) Derivation Now since ( ) ( ) ( )10 1 2 101 1 1S i i i…= + + + :

( ) ( ) ( )

( ) ( ) ( )10 1 2 10

1 2 10

log log 1 1 1

log 1 log 1 log 1

S i i i

i i i

…⎡ ⎤= + + +⎣ ⎦

= + + + + + + [1] Since ( )ln 1 ~ (0.05384,0.008861)ti N+ we can use the additive property of independent normal distributions: 10ln ~ (10 0.05384,10 0.008861)S N × × [1] So 10 ~ log (0.5384,0.08861)S N . [1]



(ii)(c) Investment required Let X be the amount invested at time 0. We require: 10( 50,000) 0.95> =P X S [1]

10

10

50,000 0.95

50,000ln ln 0.95

⎛ ⎞⇒ > =⎜ ⎟⎝ ⎠

⎛ ⎞⇒ > =⎜ ⎟⎝ ⎠

P SX

P SX

[1]

Since we require 95% of the population to be more than our value of z, we must have a negative z value:

0z

95%

Standardising:

50,000ln 0.53841.6449

0.08861Xz

−= = − [1]

Rearranging:

50,000ln 0.5384 1.6449 0.08861 0.0487

£47,621

X

X

= − =

⇒ = [1]

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