Order Details;

06 Jan 15 Futures-handbook-ug-2015_16 45

Assignment – VaR, interest rates, swaps, etc

Due: Tuesday 21 April 2015 E-submission using turnitinUK for the written report

and email for the spreadsheet.

Value: 50% of the overall unit mark

Groups: You may submit work individually or in groups of up to 4. You do not have to use

the same groups as for the first assignment.

Each group should submit only one copy of the work required.

The assignment mark will be given to all group members unless the group members

request it to be split differently.

Groups of more than 1 MUST submit additional information on how the group worked

together. Details of this are given below (page Error! Bookmark not defined.).

Note: If your group plans to work over the Easter period you should arrange details of the work

and communication before Easter.

Required: A word processed report not longer than 2,000 words excluding appendices.

Your report must be submitted to turnitinUK – details provided separately.

AND The completed spreadsheet listed below sent by attachment to an email to

james.hicks@port.ac.uk

Details of the report can be found on the spreadsheet:

Futures-term_2-question-ug

The spreadsheet will be used to generate material used in the report.

The ‘completed’ spreadsheet must be submitted by email.

DO NOT add any menus or macros to the spreadsheet.

Marking: See the grading criteria which will be used on the next page.

Marks are awarded for evidence of thinking and learning (not reading & copying).

Description gets low marks; eg simply describing what you did – although part of

this assignment specifically requests ‘documentation’ which may be descriptive.

Analysis and evaluation are high value BUT these may be superficial, hence low

value, or ‘in depth’ and thus high value. The latter is often called ‘critical analysis’;

eg answering why you did it & what happened.

Please see links on my web page (www.hicksj.myweb.port.ac.uk) to:

Reports How to write a report.

Referencing My referencing page & Referencing@Portsmouth.

Plagiarism Definitions, examples and consequences of plagiarism.

Further links to material related to the assignment are available from the unit Moodle site.

U21130ug Futures & Options Markets U21130

46 Futures-handbook-ug-2015_16 06 Jan 15

Assignment – Grading criteria (Undergraduate level 6)

80+ As below plus:

Likely to add new insights to the topic & approaches the quality of published material.

Outstanding work – contains accurate, relevant material and demonstrates understanding

of complex subject matter & is able to view it in a wider context.

Shows originality & confidence in analysing and criticising assumptions.

Is aware of the limits of knowledge.

Evidence of extensive research, uses & presents references effectively.

Outstanding presentation in terms of organisation, structure, use & flow of language,

grammar, spelling, format, diagrams, tables, etc.

70-79 As below plus:

Outstanding work – contains accurate, relevant material and demonstrates understanding

of complex subject matter & is able to view it in a wider context.

Shows originality & confidence in analysing and criticising assumptions.

Is aware of the limits of knowledge.

Evidence of extensive research, uses & presents references effectively.

Excellent presentation in terms of organisation, structure, use & flow of language,

grammar, spelling, format, diagrams, tables, etc.

60-69 As below plus:

Very good work – contains most of the information required, is accurate & relevant &

demonstrates understanding of the subject matter & attempts to view it in a wider

context.

Shows some originality of thought with good critique & analysis assumptions.

Is aware of the limits of knowledge.

Well researched, good use & presentation of references.

Very good presentation in terms of organisation, structure, presentation, use & flow of

language, grammar, spelling, format, diagrams, tables, etc.

50–59 As below plus:

Work that attempts to address the topic with some understanding & analysis, key aspects

of the subject matter covered.

Research extends to primary sources. Appropriately cited and presented references.

Satisfactory presentation in terms of organisation, structure, use & flow of language,

grammar, spelling, format, diagrams, tables, etc.

40-49 Adequate work which attempts to address the topic with limited understanding &

analysis.

Some research using texts, Internet & key reference sources with reference citation and

presentation according to convention.

An attempt to follow directions regarding organisation, structure, use & flow of

language, grammar, spelling, format, diagrams, tables, etc.

30-39 FAIL

Anything which is inadequate in most or all of the following: length, content, structure,

analysis, expression, argument, relevance, research and presentation.

Work in this range attempts to address the question/problem but is substantially

incomplete and deficient.

Serious problems with a number of aspects of language use are often found in work in

this range.

< 29 No serious attempt to address the question or problem, and/or manifests a serious

misunderstanding of the requirements of the assignment. Acutely deficient in all aspects.

Assignment – VaR, interest rates, swaps, etc

06 Jan 15 Futures-handbook-ug-2015_16 47

Working in groups

Groups of more than 1 MUST submit additional information on how the group worked together.

Individuals not working in a group may also complete this information if they wish to practice

the skills involved.

This information has been developed along the lines of proposed ‘graduate and employability

skills’ for the University of Portsmouth.

Management of tasks and of others

Group members should share sufficient contact information to enable them to communicate with

each other in the timescales required. They should also ensure that every group member has a

copy of, or access to, material produced in relation to the assignment. This can be done by

exchanging copies of material at group meetings (see below), by emailing copies of material to

other group members or by using a shared workspace such as Google Docs.

Groups should arrange to meet on a regular basis. At each meeting, one member of the group

should be appointed to record:

1 Date, time and location of meeting.

2 Student reference (hemis numbers) of those attending and apologies for non-attendance.

3 Brief details of the topics discussed.

4 Review of the work done to date, or since the last meeting, including:

Whether or not the work is progressing according to schedule (or plan).

How the work relates to other work done.

What further work needs to be done, and by whom.

Any feedback and / or constructive criticism given.

5 Details of any decisions taken, including:

Plans for future work, especially work to be done before the next meeting.

How that work is to be split amongst group members.

How that work is to be reviewed and / or evaluated by other members of the group.

Any assistance and / or support needed or offered.

A different group member should record the above information at different meetings so that all

group members have the opportunity to keep these records for at least one meeting.

Meetings may take place formally to cover the requirements above and may be in person or via

electronic means such as email, text, chat, etc.

Often meetings will be less formal and typically will be used to work on the assignment together.

The information listed above should still be recorded. In this case, item 5 should also cover work

that was carried out while the group worked together. (It may also be the case that item 5 relating

to work to be carried out before the next meeting will not apply. This occurs when group

members do not carry out work between meetings but do all the work together.)

Requirement:

You must submit a summary of not more than one page covering:

Who was in the group? How did the group maintain contact and arrange meetings?

How did the group share out work? What was done by whom? What was done jointly?

How did the group review, evaluate and act on the work done?

How did the group provide feedback, constructive criticism, assistance and support?

U21130ug Futures & Options Markets U21130

48 Futures-handbook-ug-2015_16 06 Jan 15

Record of group work

Date, time, location:

Date, time, location:

Brief details of the work done and / or topics discussed:

(for example, the parts of the workbook you worked on during a lab session)

Review of the work done to date, or since the last meeting:

Details of any plans made or decisions taken:

Note: This is provided as a template for keeping the records required for groups.

Term 2 assignment question Futures & Options: undergraduate jrh

You are in charge of managing a portfolio for Reckless Bank plc, a major international bank based in the UK.

The portfolio consists of two companies from the FTSE 100 (FTSE100) index: 6796.63 at 22 January 2015

number of shares price in £ on total value

shareholding in millions code 22 Jan 2015 in millions weighting Sector

Tesco 10 900803 2.3585 23.59 33.07% Food Retailers & Wholesalers

Vodafone Group 20 953133 2.3870 47.74 66.93% Telecommunication services

total value 71.33 100.00%

The bank wishes to introduce Value at Risk (VaR) as one of its risk management tools and has asked you to

investigate this using your portfolio as a test case with a view to introducing it across all bank investments which,

in addition to share holdings, include hedges, options, swaps, forwards, futures and other derivatives.

The bank wants to comply with existing and expected future regulations on bank capital in the UK.

A member of your team has gathered price data for each of the shares in your portfolio for the past two years

together with the price of two FTSE indexes for the same period. These are tabulated on the DATA page.

You have decided to try three different methods for calculating VaR:

1 Historical simulation based on the data for the past two years.

2 Monte Carlo simulation for the portfolio as a whole.

3 Exponentially weighted moving average (EWMA) using the linear model.

For the second method you know that you will need to calculate the volatility of the daily returns.

For the third method you will also need to calculate related correlations and/or covariances.

Each of these methods has its own advantages and disadvantages in terms of accuracy, complexity of calculation,

amount of data required and suitability for not only your portfolio but also the bank’s other portfolios and total portfolio.

You are unsure as to what period should be used to calculate volatilities from the base price data and how often

these need to be recalculated, and thus how long the estimates will be used before recalculation.

For the test, you have decided to calculate your portfolio’s VaR for 22 January 2015

(the next trading day after 21 January 2015) using the three different methods.

1 For the historical method you decided to use the last 500 trading days up to, and including, 21 January 2015.

2 For the Monte Carlo method, you decided to use the volatilities from the actual data up to, and including,

21 January 2015 for [calendar] periods of 1 quarter (= 3 months) and one year.

3 For the EWMA method, you decided to use (as the starting point) the volatilities and related correlations, covariances, etc

from the actual data up to, and including, 21 January 2015 for a [calendar] period of 1 month,

and to update these using the information from 22 January 2015.

In addition,

the bank wishes to gain a better understanding of its risk position with respect to specific instruments in interest rates.

They have asked you to provide answers with respect to the following scenarios:

A The term structure is flat at 3% per annum with continuous compounding. Some time ago the bank entered into a 6-year

swap with a principal of $150 million in which every year it pays 12-month LIBOR and receives 4.6%.

The swap now has two years six months to run. Four months ago 12-month LIBOR was 1.5% (with annual compounding).

What is the value of the swap today?

What is the financial institution’s credit exposure on the swap?

B The bank has an interest rate swap with a principal of $200 million involving the exchange of 2% per annum (semiannually

compounded) for 6-month LIBOR. The remaining life is 15 months. Interest is exchanged every six months. The 3 month,

9 month and 15 month rates are 0.5%, 0.7%, and 0.8% with continuous compounding.

Six-month LIBOR was 0.6% four months ago. What is the value of the swap?

What is the nature of the credit risks to the bank in these swap agreements?

C The bank’s assets and liabilities both have a duration of 6 years. Is the bank hedged against interest rate movements?

(Explain carefully any limitations of the hedging scheme it has chosen.)

D Explain what is meant by basis risk in the situation where a company knows it will be purchasing

a certain asset in two months and uses a three-month futures contract to hedge its risk.

Required:

Prepare a report for the directors of Reckless Bank plc evaluating the results of your tests, assessing the introduction of

VaR across the bank as a whole, and responding to the additional questions with regard to interest rates.

The report should deal with the issues raised above and should make clear any assumptions relevant to the bank’s use of

particular methods of VaR. (It is expected that about 75% of the report will be on VaR and the remainder on interest rates.)

In order to facilitate implementation of VaR throughout the bank, the report should also contain appendices documenting

the data and calculations required. These should be sufficient to allow the development of a business system but

not particularly detailed with respect to any particular model, such as a spreadsheet.

As with any report like this, numbers should be shown in tables or graphs and not buried in words.

Save the words for interpretation and analysis of the numbers.

Futures-term_2-question-ug / revised: 29 Jan 2015 / © Dr J R Hicks page 1 (of 2)

Term 2 assignment question Futures & Options: undergraduate jrh

NOTES:

Unless indicated otherwise, the following refer to daily changes. Non-trading days are not counted.

Many of the formulae, etc, are used to estimate the required value and may not be strictly mathematically correct.

Accuracy and rounding Given that we are dealing with inherently uncertain future values, the full accuracy of modern

computers is not required in all cases. However, it should not necessarily be discarded.

prices Prices should normally be rounded (not just displayed) to the nearest penny, £, £000’s, etc

as is appropriate in the particular context. In some cases it may be appropriate to retain

values in intermediate calculations to a few significant places more than the final rounding.

For example, sample prices used to calculate sample portfolio values might be rounded to the nearest 100th of a penny.

other values Normally, 3 to 5 significant places is appropriate for final values

but, as for prices, more significant places may be used in intermediate values.

For example, percentages do not normally require more than 4 significant places, ie 2 decimal places.

The daily [continuously compounded] return on day n is =ln (price on day n / price on day n-1)

usually expressed as a percentage.

The population of [continuously compounded] daily returns over a number of days forms a distribution

which will have a standard deviation (stdev). This standard deviation is the daily volatility of the asset.

The standard deviation of a holding is the value of the holding times the volatility of the holding.

The square of the standard deviation of a distribution is called the variance of the distribution.

The N day volatility is equal to the one day volatility times the square root of N.

Alternatively, the N day VaR is equal to the one day VaR times the square root of N.

The variance of [daily returns of] a portfolio (in a linear model) is equal to

the sum of: the variance times the square of the weighting for each constituent asset

plus 2 times the sum of the weighting of asset A times the weighting of asset B times the covariance of A and B

for all distinct pairings of the constituent assets

The covariance of A and B is the stdev of A times the stdev of B times the correlation between A and B

The exponentially weighted moving average uses a constant, lambda, between 0 and 1

the variance on day n is lambda times the variance on day n-1

plus (1 – lambda) times the square of (the daily [continuously compounded] return on day n-1)

the covariance on day n is lambda times the covariance on day n-1

plus (1 – lambda) times the daily returns of A and B on day n-1

Monte Carlo simulation uses a random number for each of the portfolio’s constituent assets

(and the characteristics of the related distribution of daily returns on the share price)

to calculate a new portfolio value and the daily change in value, or ‘daily VaR’.

This is done many (thousands) of times and the population of ‘daily VaR’s forms a distribution from which the daily

VaR at the 99% level can be read.

Most spreadsheet contain statistical functions which simplify the above calculations.

‘range’ simply refers to a rectangular block in a spreadsheet ‘cell’ to a single cell in a spreadsheet

The most useful ones for this exercise are listed below. Precede them by by = in Excel.

Name Use

AVERAGE(range) calculates the average (mean) of all numeric values in the range

STDEVP(range) calculates the population standard deviation of all values in the range

STDEV(range) calculates the sample standard deviation of all values in the range

VARP(range) calculates the population variance of all values in the range

VAR(range) calculates the sample variance of all values in the range

SMALL(range,k) Returns the k-th smallest value in the range

LARGE(range,k) Returns the k-th largest value in the range

PERCENTILE(range,p) Returns the value from the range at a specified percentile [ 0 <= p <= 1 ]

WEIGHTAVG(range1,range2,type) Returns a weighted average of the values in range1 using the weights in range2

COVAR(range1,range2) calculates the covariance of two data sets in range1 and range2

CORREL(range1,range2) calculates the correlation coefficient of two data sets in range1 and range2

NORMINV(Prob,Mean,SDev) Computes the inverse of the cumulative normal distribution function

NORMSINV(Prob) Returns the inverse of the standard normal cumulative distribution

LN(cell) calculates the [natural] logarithm of the value in ‘cell’

EXP(cell) calculates the exponential function of the value in ‘cell’

Futures-term_2-question-ug / revised: 29 Jan 2015 / © Dr J R Hicks page 2 (of 2)

Share prices

Daily closing prices for a selection of companies and the FTSE 100 index Other daily closing prices for a selection of companies

Start 22-Jan-13 End 21-Jan-15 and the FTSE 350 index

(ORD $0.50)

FTSE 100 TESCO Vodafone Daily continuously compounded return FTSE 350 BP HSBC BP HSBC Vodafone TESCO

price index PLC group FTSE TESCO Vodafone price index Holdings Holdings group

FTSE100 900803(p) 953133(p) 100 PLC group FTSE350 900995(p) 507534(p) 900995(PI)507534(PI)953133(PI) 900803(PI)

day date £ pence pence % % % £ pence pence index index index index

22-Jan-13 6179.17 349.55 166.44 3307.33 455.45 693.00 46616.58 1778.32 2904.93 117217.80

23-Jan-13 6197.64 352.80 167.52 3315.02 463.25 695.10 47414.93 1783.71 2923.71 118307.60

24-Jan-13 6264.91 352.50 172.85 3350.71 467.70 704.90 47870.40 1808.86 3016.73 118207.10

25-Jan-13 6284.45 355.65 174.49 3362.18 465.30 710.30 47624.76 1822.72 3045.35 119263.40

28-Jan-13 6294.41 355.95 175.36 3366.65 468.45 717.10 47947.17 1840.17 3060.55 119363.90

29-Jan-13 6339.19 361.25 177.21 3386.44 475.65 720.10 48684.11 1847.86 3092.75 121141.30

30-Jan-13 6323.11 357.35 177.82 3376.88 475.90 724.10 48709.70 1858.13 3103.48 119833.40

31-Jan-13 6276.88 356.30 176.39 3354.98 466.75 716.70 47773.17 1839.14 3078.44 119481.30

01-Feb-13 6347.24 360.60 177.77 3396.07 472.50 719.60 48361.70 1846.58 3102.59 120923.30

04-Feb-13 6246.84 357.50 174.75 3346.27 462.05 706.10 47292.11 1811.94 3049.82 119883.80

05-Feb-13 6282.76 359.20 175.51 3365.16 468.70 709.40 47972.75 1820.41 3063.23 120453.80

06-Feb-13 6295.34 363.20 174.59 3374.44 465.75 713.10 47670.81 1829.90 3047.13 121795.20

07-Feb-13 6228.42 360.35 176.13 3342.12 456.80 700.40 46754.76 1797.31 3073.96 120839.40

08-Feb-13 6263.93 362.60 178.23 3361.13 455.60 716.70 46631.93 1839.14 3110.63 121593.90

11-Feb-13 6277.06 367.90 177.97 3367.13 454.75 718.30 46544.93 1843.25 3106.16 123371.30

12-Feb-13 6338.38 371.95 177.82 3399.03 460.05 728.90 47087.40 1870.45 3103.48 124729.40

13-Feb-13 6359.11 374.25 175.92 3413.06 453.60 730.90 47041.47 1875.58 3070.39 125500.60

14-Feb-13 6327.36 370.55 171.72 3394.80 450.05 724.20 46673.32 1858.39 2997.05 124259.90

15-Feb-13 6328.26 364.75 171.98 3397.26 448.10 726.40 46471.09 1864.03 3001.52 122314.90

18-Feb-13 6318.19 366.95 170.95 3390.52 445.00 726.00 46149.60 1863.01 2983.63 123052.70

19-Feb-13 6379.07 369.10 167.57 3424.10 446.35 728.80 46289.59 1870.19 2924.60 123773.70

20-Feb-13 6395.37 374.40 166.55 3434.61 447.75 732.90 46434.79 1880.71 2906.72 125550.90

21-Feb-13 6291.54 373.65 163.98 3380.18 443.00 716.40 45942.18 1838.37 2862.00 125299.40

22-Feb-13 6335.70 372.25 167.11 3404.58 444.05 721.40 46051.07 1851.20 2916.56 124830.00

25-Feb-13 6355.37 373.95 168.19 3414.25 451.15 729.90 46787.39 1873.01 2935.34 125400.10

12-Dec-14 6300.63 165.75 215.30 3447.88 385.65 603.40 43648.07 1696.20 4175.52 60271.64

15-Dec-14 6182.72 164.80 213.20 3388.52 373.25 592.20 42244.64 1664.72 4134.80 59926.18

16-Dec-14 6331.83 167.45 219.45 3464.01 383.95 603.50 43455.68 1696.48 4256.01 60889.82

17-Dec-14 6336.48 168.15 219.65 3466.76 397.55 592.90 44994.93 1666.68 4259.89 61144.36

18-Dec-14 6466.00 175.80 225.60 3536.66 402.55 596.80 45560.83 1677.65 4375.28 63926.13

19-Dec-14 6545.27 185.40 225.00 3578.71 413.00 607.50 46743.57 1707.73 4363.64 67416.94

22-Dec-14 6576.74 181.00 225.20 3596.85 413.65 612.00 46817.14 1720.38 4367.52 65817.00

23-Dec-14 6598.18 184.60 224.70 3610.37 415.85 612.60 47066.13 1722.06 4357.83 67126.00

24-Dec-14 6609.93 186.00 225.00 3616.16 416.85 614.20 47179.31 1726.56 4363.64 67635.13

29-Dec-14 6633.51 188.45 224.80 3625.09 418.00 619.90 47309.47 1742.58 4359.77 68526.00

30-Dec-14 6547.00 186.90 221.05 3583.11 409.30 609.20 46324.80 1712.50 4287.04 67962.38

31-Dec-14 6566.09 189.00 222.65 3595.28 411.00 608.60 46517.20 1710.82 4318.07 68726.00

02-Jan-15 6547.80 188.00 221.90 3585.67 410.45 612.00 46454.95 1720.38 4303.52 68362.38

05-Jan-15 6417.16 181.60 216.45 3520.88 389.70 603.90 44106.46 1697.61 4197.83 66035.13

06-Jan-15 6366.51 178.80 215.35 3492.47 391.05 593.70 44259.25 1668.93 4176.49 65017.01

07-Jan-15 6419.83 182.00 216.00 3519.62 393.75 601.00 44564.84 1689.45 4189.10 66180.56

08-Jan-15 6569.96 209.25 225.75 3597.06 405.00 606.20 45838.12 1704.07 4378.19 76089.50

09-Jan-15 6501.14 204.10 224.90 3562.73 398.65 602.00 45119.42 1692.26 4361.71 74216.81

12-Jan-15 6501.42 204.55 226.80 3560.46 396.55 602.70 44881.74 1694.23 4398.55 74380.44

13-Jan-15 6542.20 212.00 229.00 3584.26 396.30 604.20 44853.45 1698.45 4441.22 77089.50

14-Jan-15 6388.46 214.00 225.25 3505.90 382.15 590.10 43251.94 1658.81 4368.49 77816.75

15-Jan-15 6498.78 219.25 227.90 3558.24 392.60 596.90 44434.68 1677.93 4419.89 79725.81

16-Jan-15 6550.27 219.05 227.80 3582.21 413.35 594.00 46783.18 1669.78 4417.95 79653.06

19-Jan-15 6585.53 221.40 229.45 3603.59 408.90 596.30 46279.52 1676.24 4449.95 80507.56

20-Jan-15 6620.10 224.90 230.00 3621.61 412.20 595.70 46653.02 1674.55 4460.62 81780.31

21-Jan-15 6728.04 229.00 238.15 3675.83 426.00 605.60 48214.91 1702.38 4618.68 83271.19

22-Jan-15 3711.88 428.40 620.30 48486.55 1743.71 4629.34 85762.06

total return index

Futures-term_2-question-ug data / 29 Jan 2015 / © Dr J R Hicks page 1 (of 1)

Standard Normal Distribution table

Number of standard deviations from the mean of a

Standard Normal (Cumulative) Distribution

Probability (%) 0.0 0.1 0.2 0.3 0.4

0.00 1.282 0.842 0.524 0.253

0.01 2.326 1.227 0.806 0.496 0.228

0.02 2.054 1.175 0.772 0.468 0.202

0.03 1.881 1.126 0.739 0.440 0.176

0.04 1.751 1.080 0.706 0.412 0.151

0.05 1.645 1.036 0.674 0.385 0.126

0.06 1.555 0.994 0.643 0.358 0.100

0.07 1.476 0.954 0.613 0.332 0.075

0.08 1.405 0.915 0.583 0.305 0.050

0.09 1.341 0.878 0.553 0.279 0.025

For probabilities p > 0.5 use the fact that number of standard deviations

of p = number of standard deviations of 1 – p.

For example, probabilities of 15% and 85% correspond to 1.036 standard

deviations (below and above the mean respectively).

Areas of the standard normal distribution

Probability that a standard normal random variable is less than x

Standard Normal (Cumulative) Distribution

x 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.00 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.10 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753

0.20 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141

0.30 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517

0.40 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

0.50 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224

0.60 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549

0.70 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852

0.80 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133

0.90 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

1.00 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.10 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830

1.20 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015

1.30 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177

1.40 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319

1.50 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441

1.60 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545

1.70 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633

1.80 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706

1.90 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767

2.00 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817

2.10 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857

2.20 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890

2.30 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916

2.40 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936

2.50 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952

2.60 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964

2.70 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974

2.80 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981

2.90 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986

3.00 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

3.10 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993

3.20 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995

3.30 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997

3.40 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998

3.50 0.9998

4.00 0.99997 For intermediate values use interpolation. For example:

4.50 0.999997 N(0.6278) = N(0.62) + 0.78 x ( N(0.63) – N(0.62) )

5.00 0.9999997 = 0.7324 + 0.78 x (0.7357 – 0.7324) = 0.7351

For example, the probability that a standard normal random variable is less than 1.65, denoted N(1.65), is 0.9505.

For negative x use the fact that N(-x) = 1 – N(x).

Thus the probability that a standard normal random variable is less than -1.65, denoted N(-1.65), is 1 – 0.9505 = 0.0495.

Futures-term_2-question-ug / revised: 6 March 2006 / © Dr J R Hicks page 1 (of 1)

Exponential and Logarithmic functions

are inverse functions; ie ln(exp(x)) = x exp(ln(x)) = x

Exponential function = e raised to the power x

exp (x) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.000 1.0000 1.0101 1.0202 1.0305 1.0408 1.0513 1.0618 1.0725 1.0833 1.0942

0.001 1.0010 1.0111 1.0212 1.0315 1.0419 1.0523 1.0629 1.0736 1.0844 1.0953

0.002 1.0020 1.0121 1.0222 1.0325 1.0429 1.0534 1.0640 1.0747 1.0855 1.0964

0.003 1.0030 1.0131 1.0233 1.0336 1.0439 1.0544 1.0650 1.0757 1.0865 1.0975

0.004 1.0040 1.0141 1.0243 1.0346 1.0450 1.0555 1.0661 1.0768 1.0876 1.0986

0.005 1.0050 1.0151 1.0253 1.0356 1.0460 1.0565 1.0672 1.0779 1.0887 1.0997

0.006 1.0060 1.0161 1.0263 1.0367 1.0471 1.0576 1.0682 1.0790 1.0898 1.1008

0.007 1.0070 1.0171 1.0274 1.0377 1.0481 1.0587 1.0693 1.0800 1.0909 1.1019

0.008 1.0080 1.0182 1.0284 1.0387 1.0492 1.0597 1.0704 1.0811 1.0920 1.1030

0.009 1.0090 1.0192 1.0294 1.0398 1.0502 1.0608 1.0714 1.0822 1.0931 1.1041

For negative x use the fact that exp(-x) is the reciprocal of exp(x), ie 1 / exp(x).

For intermediate values use interpolation.

For example, exp(0.0333) = exp(0.033) + .3 ( exp(0.034) – exp(0.033) ) = 1.0336 + .3 ( 1.0346 – 1.0336 ) = 1.0339

Logarithmic function = ln(x)

ln (x) 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

0.00 0.0000 0.0953 0.1823 0.2624 0.3365 0.4055 0.4700 0.5306 0.5878 0.6419

0.01 0.0100 0.1044 0.1906 0.2700 0.3436 0.4121 0.4762 0.5365 0.5933 0.6471

0.02 0.0198 0.1133 0.1989 0.2776 0.3507 0.4187 0.4824 0.5423 0.5988 0.6523

0.03 0.0296 0.1222 0.2070 0.2852 0.3577 0.4253 0.4886 0.5481 0.6043 0.6575

0.04 0.0392 0.1310 0.2151 0.2927 0.3646 0.4318 0.4947 0.5539 0.6098 0.6627

0.05 0.0488 0.1398 0.2231 0.3001 0.3716 0.4383 0.5008 0.5596 0.6152 0.6678

0.06 0.0583 0.1484 0.2311 0.3075 0.3784 0.4447 0.5068 0.5653 0.6206 0.6729

0.07 0.0677 0.1570 0.2390 0.3148 0.3853 0.4511 0.5128 0.5710 0.6259 0.6780

0.08 0.0770 0.1655 0.2469 0.3221 0.3920 0.4574 0.5188 0.5766 0.6313 0.6831

0.09 0.0862 0.1740 0.2546 0.3293 0.3988 0.4637 0.5247 0.5822 0.6366 0.6881

For 0 < x < 1 use the fact that ln(x) = -ln(1/x). Ie, ln(0.8) = -ln(1.25).

For intermediate values use interpolation.

For example, ln(1.0345) = ln(1.03) + .45 ( ln(1.04) – ln(1.03) ) = 0.0296 + .45 ( 0.0392 – 0.0296 ) = 0.0339.

Futures-term_2-question-ug / revised: 6 March 2006 / © Dr J R Hicks page 1 (of 1)

Bibliography

06 Jan 15 Futures-handbook-ug-2015_16 49

Bibliography

Ansell, J, & Wharton, F (1992). Risk: analysis, assessment and management. Chichester: Wiley.

Arnold, G (2013). Corporate Financial Management (5th ed.). Harlow: Financial Times Prentice

Hall.

Bischoff, V (2012, Jun 29). Q&A: what is Libor and what did Barclays do to it? Retrieved from

Citywire: http://citywire.co.uk/money/qanda-what-is-libor-and-what-did-barclays-do-toit/

a600479

Chisholm, A (2010). Derivatives demystified: a step-by-step guide to forwards, futures, swaps

and options. Chichester ; Hoboken, NJ: John Wiley & Sons. Retrieved from

http://site.ebrary.com/lib/portsmouth/docDetail.action?docID=10419235

Durbin, M (2010). All about derivatives (2nd ed.). New York: McGraw-Hill. Retrieved from

http://site.ebrary.com/lib/portsmouth/docDetail.action?docID=10433701

Fabozzi, F J (2002). Fixed income securities (2nd ed.). Chichester: Wiley-Academy.

Fabozzi, F J, & Choudhry, M (2004). The handbook of European fixed income securities.

Hoboken, NJ: J Wiley. Retrieved from

http://site.ebrary.com/lib/portsmouth/docDetail.action?docID=10114004

Fabozzi, F J, & Mann, S V (2012). The handbook of fixed income securities (8th ed.). New York:

McGraw-Hill. Retrieved from http://lib.myilibrary.com/Open.aspx?id=342557

Hull, J C (2008). Fundamentals of Futures and Options Markets (6th ed.). Upper Saddle River,

NJ: Pearson Prentice Hall.

Hull, J C (2011). Fundamentals of Futures and Options Markets (7th ed.). Boston, Mass. ;

London: Pearson. 602 p.

Hull, J C (2012). Options, Futures and other Derivatives (8th ed.). Upper Saddle River, NJ:

Pearson Prentice Hall. 847 p

Hull, J C (2014). Fundamentals of Futures and Options Markets (8th ed.). Boston, Mass. ;

London: Pearson. 579 pages [eBook] http://lib.myilibrary.com/?id=543613

McDonald, R L (2009). Fundamentals of derivatives markets (International ed.). Boston, Mass ;

London: Pearson Addison-Wesley.

McLaney, E J (2006). Business finance: theory and practice (7th ed.). Harlow: Financial Times

Prentice Hall. Retrieved from

http://www.dawsonera.com/depp/reader/protected/external/AbstractView/S9781405898188

McLaney, E J (2009). Business finance: theory and practice (8th ed.). Harlow, England ; New

York: Prentice Hall/Financial Times. Retrieved from

http://lib.myilibrary.com/Open.aspx?id=231724

McLaney, E J (2011). Business finance: theory and practice (9th ed.). Harlow: Financial Times

Prentice Hall. Retrieved from http://lib.myilibrary.com/Open.aspx?id=327562

Samuels, J M, Brayshaw, R, & Wilkes, F M (1996). Management of company finance (6th ed.).

London: International Thomson Business Press.

Samuels, J M, Brayshaw, R, & Wilkes, F M (1996). Management of company finance: solutions

manual (6th ed.). London: International Thompson Business.

U21130ug Futures & Options Markets U21130

50 Futures-handbook-ug-2015_16 06 Jan 15

Web links relating to the unit

GloriaMundi http://gloria-mundi.com/ claims to be All About Value at Risk

requires a free Sign up

Basel

Committee

www.bis.org/bcbs/ on Banking Supervision & the Basel

Accord

Financial

Risk

Management

http://glynholton.com/ trading, financial engineering and risk

management

(formerly Contingency Analysis)

Risk Archive http://riskarchive.com/ Financial Risk Management Forum: 1996

– 2009 (up to Winter 2009)

MSCI

(RiskMetrics)

www.msci.com financial analytics and wealth

management solutions

Simtools http://home.uchicago.edu/~rmyerson

/addins.htm

for doing Monte Carlo simulation and risk

analysis in spreadsheets

Other useful web links

my web page www.hicksj.myweb.port.ac.uk, especially the links to:

Reports How to write a report.

Referencing My referencing page & Referencing@Portsmouth.

Plagiarism Definitions, examples and consequences of plagiarism.

Google Scholar http://scholar.google.co.uk/ and the

Library Discovery Service http://www.port.ac.uk/library/infores/discovery/

Term 2 assignment question | Futures & Options: undergraduate | jrh | ||||||||

You are in charge of managing a portfolio for Reckless Bank plc, a major international bank based in the UK. | ||||||||||

The portfolio consists of two companies from the FTSE 100 (FTSE100) index: | 6796.63 | at 22 January 2015 | ||||||||

number of shares | price in £ on | total value | ||||||||

shareholding | in millions | code | 22 Jan 2015 | in millions | weighting | Sector | ||||

Tesco | 10 | 900803 | 2.3585 | 23.59 | 33.07% | Food Retailers & Wholesalers | ||||

Vodafone Group | 20 | 953133 | 2.3870 | 47.74 | 66.93% | Telecommunication services | ||||

total value | 71.33 | 100.00% | ||||||||

The bank wishes to introduce Value at Risk (VaR) as one of its risk management tools and has asked you to | ||||||||||

investigate this using your portfolio as a test case with a view to introducing it across all bank investments which, | ||||||||||

in addition to share holdings, include hedges, options, swaps, forwards, futures and other derivatives. | ||||||||||

The bank wants to comply with existing and expected future regulations on bank capital in the UK. | ||||||||||

A member of your team has gathered price data for each of the shares in your portfolio for the past two years | ||||||||||

together with the price of two FTSE indexes for the same period. These are tabulated on the DATA page. | ||||||||||

You have decided to try three different methods for calculating VaR: | ||||||||||

1 | Historical simulation based on the data for the past two years. | |||||||||

2 | Monte Carlo simulation for the portfolio as a whole. | |||||||||

3 | Exponentially weighted moving average (EWMA) using the linear model. | |||||||||

For the second method you know that you will need to calculate the volatility of the daily returns. | ||||||||||

For the third method you will also need to calculate related correlations and/or covariances. | ||||||||||

Each of these methods has its own advantages and disadvantages in terms of accuracy, complexity of calculation, | ||||||||||

amount of data required and suitability for not only your portfolio but also the bank’s other portfolios and total portfolio. | ||||||||||

You are unsure as to what period should be used to calculate volatilities from the base price data and how often | ||||||||||

these need to be recalculated, and thus how long the estimates will be used before recalculation. | ||||||||||

For the test, you have decided to calculate your portfolio’s VaR for | 22 January 2015 | |||||||||

(the next trading day after 21 January 2015) using the three different methods. | ||||||||||

1 | For the historical method you decided to use the last 500 trading days up to, and including, 21 January 2015. | |||||||||

2 | For the Monte Carlo method, you decided to use the volatilities from the actual data up to, and including, | |||||||||

21 January 2015 for [calendar] periods of 1 quarter (= 3 months) and one year. | ||||||||||

3 | For the EWMA method, you decided to use (as the starting point) the volatilities and related correlations, covariances, etc | |||||||||

from the actual data up to, and including, 21 January 2015 for a [calendar] period of 1 month, | ||||||||||

and to update these using the information from 22 January 2015. | ||||||||||

In addition, | ||||||||||

the bank wishes to gain a better understanding of its risk position with respect to specific instruments in interest rates. | ||||||||||

They have asked you to provide answers with respect to the following scenarios: | ||||||||||

A | The term structure is flat at 3% per annum with continuous compounding. Some time ago the bank entered into a 6-year | |||||||||

swap with a principal of $150 million in which every year it pays 12-month LIBOR and receives 4.6%. | ||||||||||

The swap now has two years six months to run. Four months ago 12-month LIBOR was 1.5% (with annual compounding). | ||||||||||

What is the value of the swap today? | ||||||||||

What is the financial institution’s credit exposure on the swap? | ||||||||||

B | The bank has an interest rate swap with a principal of $200 million involving the exchange of 2% per annum (semiannually | |||||||||

compounded) for 6-month LIBOR. The remaining life is 15 months. Interest is exchanged every six months. The 3 month, | ||||||||||

9 month and 15 month rates are 0.5%, 0.7%, and 0.8% with continuous compounding. | ||||||||||

Six-month LIBOR was 0.6% four months ago. What is the value of the swap? | ||||||||||

What is the nature of the credit risks to the bank in these swap agreements? | ||||||||||

C | The bank’s assets and liabilities both have a duration of 6 years. Is the bank hedged against interest rate movements? | |||||||||

(Explain carefully any limitations of the hedging scheme it has chosen.) | ||||||||||

D | Explain what is meant by basis risk in the situation where a company knows it will be purchasing | |||||||||

a certain asset in two months and uses a three-month futures contract to hedge its risk. | ||||||||||

Required: | ||||||||||

Prepare a report for the directors of Reckless Bank plc evaluating the results of your tests, assessing the introduction of | ||||||||||

VaR across the bank as a whole, and responding to the additional questions with regard to interest rates. | ||||||||||

The report should deal with the issues raised above and should make clear any assumptions relevant to the bank’s use of | ||||||||||

particular methods of VaR. (It is expected that about 75% of the report will be on VaR and the remainder on interest rates.) | ||||||||||

In order to facilitate implementation of VaR throughout the bank, the report should also contain appendices documenting | ||||||||||

the data and calculations required. These should be sufficient to allow the development of a business system but | ||||||||||

not particularly detailed with respect to any particular model, such as a spreadsheet. | ||||||||||

As with any report like this, numbers should be shown in tables or graphs and not buried in words. | ||||||||||

Save the words for interpretation and analysis of the numbers. | ||||||||||

NOTES: | ||||||||||

Unless indicated otherwise, the following refer to daily changes. Non-trading days are not counted. | ||||||||||

Many of the formulae, etc, are used to estimate the required value and may not be strictly mathematically correct. | ||||||||||

Accuracy and rounding | Given that we are dealing with inherently uncertain future values, the full accuracy of modern | |||||||||

computers is not required in all cases. However, it should not necessarily be discarded. | ||||||||||

prices | Prices should normally be rounded (not just displayed) to the nearest penny, £, £000’s, etc | |||||||||

as is appropriate in the particular context. In some cases it may be appropriate to retain | ||||||||||

values in intermediate calculations to a few significant places more than the final rounding. | ||||||||||

For example, sample prices used to calculate sample portfolio values might be rounded to the nearest 100th of a penny. | ||||||||||

other values | Normally, 3 to 5 significant places is appropriate for final values | |||||||||

but, as for prices, more significant places may be used in intermediate values. | ||||||||||

For example, percentages do not normally require more than 4 significant places, ie 2 decimal places. | ||||||||||

The | daily [continuously compounded] return on day n is | =ln (price on day n / price on day n-1) | ||||||||

usually expressed as a percentage. | ||||||||||

The population of [continuously compounded] daily returns over a number of days forms a distribution | ||||||||||

which will have a standard deviation (stdev). | This standard deviation is the daily volatility of the asset. | |||||||||

The standard deviation of a holding is the value of the holding times the volatility of the holding. | ||||||||||

The square of the standard deviation of a distribution is called the variance of the distribution. | ||||||||||

The N day volatility is equal to the one day volatility times the square root of N. | ||||||||||

Alternatively, the N day VaR is equal to the one day VaR times the square root of N. | ||||||||||

The variance of [daily returns of] a portfolio (in a linear model) is equal to | ||||||||||

the sum of: the variance times the square of the weighting for each constituent asset | ||||||||||

plus | 2 times the sum of | the weighting of asset A times the weighting of asset B times the covariance of A and B | ||||||||

for all distinct pairings of the constituent assets | ||||||||||

The | covariance of A and B is the | stdev of A times the stdev of B times the correlation between A and B | ||||||||

The | exponentially weighted moving average uses a constant, lambda, between 0 and 1 | |||||||||

the variance on day n is | lambda times the variance on day n-1 | |||||||||

plus | (1 – lambda) times the square of (the daily [continuously compounded] return on day n-1) | |||||||||

the covariance on day n is | lambda times the covariance on day n-1 | |||||||||

plus | (1 – lambda) times the daily returns of A and B on day n-1 | |||||||||

Monte Carlo simulation uses a random number for each of the portfolio’s constituent assets | ||||||||||

(and the characteristics of the related distribution of daily returns on the share price) | ||||||||||

to calculate a new portfolio value and the daily change in value, or ‘daily VaR’. | ||||||||||

This is done many (thousands) of times and the population of ‘daily VaR’s forms a distribution from which the daily | ||||||||||

VaR at the 99% level can be read. | ||||||||||

Most spreadsheet contain statistical functions which simplify the above calculations. | ||||||||||

‘range’ simply refers to a rectangular block in a spreadsheet | ‘cell’ to a single cell in a spreadsheet | |||||||||

The most useful ones for this exercise are listed below. Precede them by by = in Excel. | ||||||||||

Name | Use | |||||||||

AVERAGE(range) | calculates the average (mean) of all numeric values in the range | |||||||||

STDEVP(range) | calculates the population standard deviation of all values in the range | |||||||||

STDEV(range) | calculates the sample standard deviation of all values in the range | |||||||||

VARP(range) | calculates the population variance of all values in the range | |||||||||

VAR(range) | calculates the sample variance of all values in the range | |||||||||

SMALL(range,k) | Returns the k-th smallest value in the range | |||||||||

LARGE(range,k) | Returns the k-th largest value in the range | |||||||||

PERCENTILE(range,p) | Returns the value from the range at a specified percentile [ 0 <= p <= 1 ] | |||||||||

WEIGHTAVG(range1,range2,type) | Returns a weighted average of the values in range1 using the weights in range2 | |||||||||

COVAR(range1,range2) | calculates the covariance of two data sets in range1 and range2 | |||||||||

CORREL(range1,range2) | calculates the correlation coefficient of two data sets in range1 and range2 | |||||||||

NORMINV(Prob,Mean,SDev) | Computes the inverse of the cumulative normal distribution function | |||||||||

NORMSINV(Prob) | Returns the inverse of the standard normal cumulative distribution | |||||||||

LN(cell) | calculates the [natural] logarithm of the value in ‘cell’ | |||||||||

EXP(cell) | calculates the exponential function of the value in ‘cell’ |

CHAPTER 1

Introduction

Practice Questions

Problem 1.8.

Suppose you own 5,000 shares that are worth $25 each. How can put options be used to

provide you with insurance against a decline in the value of your holding over the next four

months?

You should buy 50 put option contracts (each on 100 shares) with a strike price of $25 and an

expiration date in four months. If at the end of four months the stock price proves to be less

than $25, you can exercise the options and sell the shares for $25 each.

Problem 1.9.

A stock when it is first issued provides funds for a company. Is the same true of an exchangetraded

stock option? Discuss.

An exchange-traded stock option provides no funds for the company. It is a security sold by

one investor to another. The company is not involved. By contrast, a stock when it is first

issued is sold by the company to investors and does provide funds for the company.

Problem 1.10.

Explain why a futures contract can be used for either speculation or hedging.

If an investor has an exposure to the price of an asset, he or she can hedge with futures

contracts. If the investor will gain when the price decreases and lose when the price increases,

a long futures position will hedge the risk. If the investor will lose when the price decreases

and gain when the price increases, a short futures position will hedge the risk. Thus either a

long or a short futures position can be entered into for hedging purposes.

If the investor has no exposure to the price of the underlying asset, entering into a futures

contract is speculation. If the investor takes a long position, he or she gains when the asset’s

price increases and loses when it decreases. If the investor takes a short position, he or she

loses when the asset’s price increases and gains when it decreases.

Problem 1.11.

A cattle farmer expects to have 120,000 pounds of live cattle to sell in three months. The livecattle

futures contract on the Chicago Mercantile Exchange is for the delivery of 40,000

pounds of cattle. How can the farmer use the contract for hedging? From the farmer’s

viewpoint, what are the pros and cons of hedging?

The farmer can short 3 contracts that have 3 months to maturity. If the price of cattle falls, the

gain on the futures contract will offset the loss on the sale of the cattle. If the price of cattle

rises, the gain on the sale of the cattle will be offset by the loss on the futures contract. Using

futures contracts to hedge has the advantage that it can at no cost reduce risk to almost zero.

Its disadvantage is that the farmer no longer gains from favorable movements in cattle prices.

Problem 1.12.

It is July 2013. A mining company has just discovered a small deposit of gold. It will take six

months to construct the mine. The gold will then be extracted on a more or less continuous

basis for one year. Futures contracts on gold are available on the New York Mercantile

Exchange. There are delivery months every two months from August 2013 to December 2014.

Each contract is for the delivery of 100 ounces. Discuss how the mining company might use

futures markets for hedging.

The mining company can estimate its production on a month by month basis. It can then short

futures contracts to lock in the price received for the gold. For example, if a total of 3,000

ounces are expected to be produced in September 2014 and October 2014, the price received

for this production can be hedged by shorting a total of 30 October 2014 contracts.

Problem 1.13.

Suppose that a March call option on a stock with a strike price of $50 costs $2.50 and is held

until March. Under what circumstances will the holder of the option make a gain? Under

what circumstances will the option be exercised? Draw a diagram showing how the profit on

a long position in the option depends on the stock price at the maturity of the option.

The holder of the option will gain if the price of the stock is above $52.50 in March. (This

ignores the time value of money.) The option will be exercised if the price of the stock is

above $50.00 in March. The profit as a function of the stock price is shown in Figure S1.1.

‐5

0

5

10

15

20

20 30 40 50 60 70

Profit

Stock Price

Figure S1.1 Profit from long position in Problem 1.13

Problem 1.14.

Suppose that a June put option on a stock with a strike price of $60 costs $4 and is held until

June. Under what circumstances will the holder of the option make a gain? Under what

circumstances will the option be exercised? Draw a diagram showing how the profit on a

short position in the option depends on the stock price at the maturity of the option.

The seller of the option will lose if the price of the stock is below $56.00 in June. (This

ignores the time value of money.) The option will be exercised if the price of the stock is

below $60.00 in June. The profit as a function of the stock price is shown in Figure S1.2.

‐10

0

10

20

30

40

50

60

0 20 40 60 80 100 120

Profit

Stock Price

Figure S1.2 Profit from short position in Problem 1.14

Problem 1.15.

It is May and a trader writes a September call option with a strike price of $20. The stock

price is $18, and the option price is $2. Describe the investor’s cash flows if the option is

held until September and the stock price is $25 at this time.

The trader has an inflow of $2 in May and an outflow of $5 in September. The $2 is the cash

received from the sale of the option. The $5 is the result of the option being exercised. The

investor has to buy the stock for $25 in September and sell it to the purchaser of the option

for $20.

Problem 1.16.

An investor writes a December put option with a strike price of $30. The price of the option is

$4. Under what circumstances does the investor make a gain?

The investor makes a gain if the price of the stock is above $26 at the time of exercise. (This

ignores the time value of money.)

Problem 1.17.

The CME Group offers a futures contract on long-term Treasury bonds. Characterize the

investors likely to use this contract.

Most investors will use the contract because they want to do one of the following:

a) Hedge an exposure to long-term interest rates.

b) Speculate on the future direction of long-term interest rates.

c) Arbitrage between the spot and futures markets for Treasury bonds.

Problem 1.18.

An airline executive has argued: “There is no point in our using oil futures. There is just as

much chance that the price of oil in the future will be less than the futures price as there is

that it will be greater than this price.” Discuss the executive’s viewpoint.

It may well be true that there is just as much chance that the price of oil in the future will be

above the futures price as that it will be below the futures price. This means that the use of a

futures contract for speculation would be like betting on whether a coin comes up heads or

tails. But it might make sense for the airline to use futures for hedging rather than

speculation. The futures contract then has the effect of reducing risks. It can be argued that an

airline should not expose its shareholders to risks associated with the future price of oil when

there are contracts available to hedge the risks.

Problem 1.19.

“Options and futures are zero-sum games.” What do you think is meant by this statement?

The statement means that the gain (loss) to the party with the short position is equal to the

loss (gain) to the party with the long position. In total, the gain to all parties is zero.

Problem 1.20.

A trader enters into a short forward contract on 100 million yen. The forward exchange rate

is $0.0080 per yen. How much does the trader gain or lose if the exchange rate at the end of

the contract is (a) $0.0074 per yen; (b) $0.0091 per yen?

a) The trader sells 100 million yen for $0.0080 per yen when the exchange rate is $0.0074

per yen. The gain is 10000006 millions of dollars or $60,000.

b) The trader sells 100 million yen for $0.0080 per yen when the exchange rate is $0.0091

per yen. The loss is 10000011 millions of dollars or $110,000.

Problem 1.21.

A trader enters into a short cotton futures contract when the futures price is 50 cents per

pound. The contract is for the delivery of 50,000 pounds. How much does the trader gain or

lose if the cotton price at the end of the contract is (a) 48.20 cents per pound; (b) 51.30 cents

per pound?

a) The trader sells for 50 cents per pound something that is worth 48.20 cents per pound.

Gain = ($0.5000 − $0.4820) × 50,000 = $900.

b) The trader sells for 50 cents per pound something that is worth 51.30 cents per pound.

Loss = ($0.5130 − $0.5000) × 50,000 = $650.

Problem 1.22.

A company knows that it is due to receive a certain amount of a foreign currency in four

months. What type of option contract is appropriate for hedging?

A long position in a four-month put option can provide insurance against the exchange rate

falling below the strike price. It ensures that the foreign currency can be sold for at least the

strike price.

Problem 1.23.

A United States company expects to have to pay 1 million Canadian dollars in six months.

Explain how the exchange rate risk can be hedged using (a) a forward contract; (b) an

option.

The company could enter into a long forward contract to buy 1 million Canadian dollars in

six months. This would have the effect of locking in an exchange rate equal to the current

forward exchange rate. Alternatively the company could buy a call option giving it the right

(but not the obligation) to purchase 1 million Canadian dollars at a certain exchange rate in

six months. This would provide insurance against a strong Canadian dollar in six months

while still allowing the company to benefit from a weak Canadian dollar at that time.

Problem 1.24

A trader buys a call option with a strike price of $30 for $3. Does the trader ever exercise the

option and lose money on the trade. Explain.

Yes, the trader will exercise if the asset price is greater than $30, but will cover the cost of the

call option only if the price is greater than $33. The trader exercises and loses money if the

price is between $30 and $33.

Problem 1.25

A trader sells a put option with a strike price of $40 for $5. What is the trader’s maximum

gain and maximum loss? How does your answer change if it is a call option?

The trader’s maximum gain from the put option is $5. The maximum loss is $35,

corresponding to the situation where the option is exercised and the asset price is zero. If the

option were a call, the trader’s maximum gain would still be $5, but there would be no bound

to the loss as there is in theory no limit to how high the asset price could rise.

Problem 1.26

‘‘Buying a stock and a put option on the stock is a form of insurance.’’ Explain this

statement.

If the stock price declines below the strike price of the put option, the stock can be sold for

the strike price.

Further Questions

Problem 1.27 (Excel file)

Trader A enters into a forward contract to buy an asset for $1000 an ounce in one year.

Trader B buys a call option to buy the asset for $1000 in one year. The cost of the option is

$100. What is the difference between the positions of the traders? Show the profit as a

function of the price of the asset in one year for the two traders.

Trader A makes a profit of ST ̶ 1000 and Trader B makes a profit of max(ST ̶ 1000, 0) –100

where ST is the price of the asset in one year. Trader A does better if ST is above $900 as

indicated in the diagram below

Problem 1.28

On June 25, 2012, as indicated in Table 1.2, the spot offer price of Google stock is $561.51

and the offer price of a call option with a strike price of $560 and a maturity date of

September is $30.70. A trader is considering two alternatives: buy 100 shares of the stock

and buy 100 September call options. For each alternative, what is (a) the upfront cost, (b)

the total gain if the stock price in September is $620, and (c) the total loss if the stock

price in September is $500. Assume that the option is not exercised before September and

if stock is purchased it is sold in September.

a) The upfront cost for the share alternative is $56,151. The upfront cost for the option

alternative is $3,070.

b) The gain from the stock alternative is $62,000−$56,151=$5,849. The total gain from

the option alternative is ($620-$560)×100−$3,070=$2,930.

c) The loss from the stock alternative is $56,151−$50,000=$6,151. The loss from the

option alternative is $3,070.

Problem 1.29

What is arbitrage? Explain the arbitrage opportunity when the price of a dually listed

mining company stock is $50 on the New York Stock Exchange and $52 CAD on the Toronto

Stock Exchange. Assume that the exchange rate is such that 1 USD equals 1.01 CAD.

Explain what is likely to happen to prices as traders take advantage of this opportunity.

Arbitrage involves carrying out two or more different trades to lock in a profit. In this case,

traders can buy shares on the NYSE and sell them on the TSX to lock in a USD profit of

52/1.01−50=1.485 per share. As they do this the NYSE price will rise and the TSX price will

fall so that the arbitrage opportunity disappears

Problem 1.30

In March, a US investor instructs a broker to sell one July put option contract on a stock. The

stock price is $42 and the strike price is $40. The option price is $3. Explain what the

investor has agreed to. Under what circumstances will the trade prove to be profitable? What

are the risks?

‐ 300

‐ 200

‐ 100

0

100

200

300

700 900 1100 1300

Profit

Asset Price

Trader A

Trader B

The investor has agreed to buy 100 shares of the stock for $40 in July (or earlier) if the party

on the other side of the transaction chooses to sell. The trade will prove profitable if the

option is not exercised or if the stock price is above $37 at the time of exercise. The risk to

the investor is that the stock price plunges to a low level. For example, if the stock price

drops to $1 by July (unlikely but possible), the investor loses $3,600. This is because the put

options are exercised and $40 is paid for 100 shares when the value per share is $1. This

leads to a loss of $3,900 which is offset by the premium of $300 received for the options.

Problem 1.31

A US company knows it will have to pay 3 million euros in three months. The current

exchange rate is 1.4500 dollars per euro. Discuss how forward and options contracts can be

used by the company to hedge its exposure.

The company could enter into a forward contract obligating it to buy 3 million euros in three

months for a fixed price (the forward price). The forward price will be close to but not

exactly the same as the current spot price of 1.4500. An alternative would be to buy a call

option giving the company the right but not the obligation to buy 3 million euros for a a

particular exchange rate (the strike price) in three months. The use of a forward contract locks

in, at no cost, the exchange rate that will apply in three months. The use of a call option

provides, at a cost, insurance against the exchange rate being higher than the strike price.

Problem 1.32 (Excel file)

A stock price is $29. An investor buys one call option contract on the stock with a strike price

of $30 and sells a call option contract on the stock with a strike price of $32.50. The market

prices of the options are $2.75 and $1.50, respectively. The options have the same maturity

date. Describe the investor’s position.

This is known as a bull spread and will be discussed in Chapter 11. The profit is shown in the

diagram below

Problem 1.33

The price of gold is currently $1,800 per ounce. Forward contracts are available to buy or

sell gold at $2,000 per ounce for delivery in one year. An arbitrageur can borrow money at

5% per annum. What should the arbitrageur do? Assume that the cost of storing gold is zero

and that gold provides no income.

The arbitrageur should borrow money to buy a certain number of ounces of gold today and

short forward contracts on the same number of ounces of gold for delivery in one year. This

means that gold is purchased for $1800 per ounce and sold for $2000 per ounce. The cost of

the borrowing is $90 per ounce. A riskless profit of $110 per ounce is generated.

Problem 1.34.

Discuss how foreign currency options can be used for hedging in the situation described in

Example 1.1 so that (a) ImportCo is guaranteed that its exchange rate will be less than

1.5800, and (b) ExportCo is guaranteed that its exchange rate will be at least 1.5400.

ImportCo can buy call options on $10,000,000 with a strike price of 1.5800. This will ensure

that it never pays more than $15,800,000 for the sterling it requires. ExportCo can buy put

options on $30,000,000 with a strike price of 1.5400. This will ensure that the price received

for the sterling will be above1.5430,000,000 $46,200,00.

Problem 1.35.

The current price of a stock is $94, and three-month call options with a strike price of $95

currently sell for $4.70. An investor who feels that the price of the stock will increase is

trying to decide between buying 100 shares and buying 2,000 call options (20 contracts).

Both strategies involve an investment of $9,400. What advice would you give? How high does

the stock price have to rise for the option strategy to be more profitable?

The investment in call options entails higher risks but can lead to higher returns. If the stock

price stays at $94, an investor who buys call options loses $9,400 whereas an investor who

buys shares neither gains nor loses anything. If the stock price rises to $120, the investor who

buys call options gains

2000(120 95) 9400 $40600

An investor who buys shares gains

100(120 94) $2600

The strategies are equally profitable if the stock price rises to a level, S , where

100(S 94) 2000(S 95) 9400

or

S 100

The option strategy is therefore more profitable if the stock price rises above $100.

Problem 1.36.

On June 25, 2012, an investor owns 100 Google shares. As indicated in Table 1.3, the bid

share price is $561.32 and a December put option with a strike price $520 costs $26.10. The

investor is comparing two alternatives to limit downside risk. The first involves buying one

Decmber put option contract with a strike price of $520. The second involves instructing a

broker to sell the 100 shares as soon as Google’s price reaches $520. Discuss the advantages

and disadvantages of the two strategies.

The second alternative involves what is known as a stop or stop-loss order. It costs nothing

and ensures that $52,000, or close to $52,000, is realized for the holding in the event the

stock price ever falls to $520. The put option costs $2,610 and guarantees that the holding can

be sold for $52,000 any time up to December. If the stock price falls marginally below $520

and then rises the option will not be exercised, but the stop-loss order will lead to the holding

being liquidated. There are some circumstances where the put option alternative leads to a

better outcome and some circumstances where the stop-loss order leads to a better outcome.

If the stock price ends up below $520, the stop-loss order alternative leads to a better

outcome because the cost of the option is avoided. If the stock price falls to $480 in

November and then rises to $580 by December, the put option alternative leads to a better

outcome. The investor is paying $2,610 for the chance to benefit from this second type of

outcome.

Problem 1.37.

A trader buys a European call option and sells a European put option. The options have the

same underlying asset, strike price and maturity. Describe the trader’s position. Under what

circumstances does the price of the call equal the price of the put?

The trader has a long European call option with strike price K and a short European put

option with strike price K . Suppose the price of the underlying asset at the maturity of the

option is ST . If ST K , the call option is exercised by the investor and the put option expires

worthless. The payoff from the portfolio is TS K . If TS K , the call option expires

worthless and the put option is exercised against the investor. The cost to the investor is

T K S . Alternatively we can say that the payoff to the investor is TS K (a negative

amount). In all cases, the payoff is TS K , the same as the payoff from the forward contract.

The trader’s position is equivalent to a forward contract with delivery price K .

Suppose that F is the forward price. If K F , the forward contract that is created has zero

value. Because the forward contract is equivalent to a long call and a short put, this shows

that the price of a call equals the price of a put when the strike price is F .

CHAPTER 2

Mechanics of Futures Markets

Practice Questions

Problem 2.8.

The party with a short position in a futures contract sometimes has options as to the precise

asset that will be delivered, where delivery will take place, when delivery will take place, and

so on. Do these options increase or decrease the futures price? Explain your reasoning.

These options make the contract less attractive to the party with the long position and more

attractive to the party with the short position. They therefore tend to reduce the futures price.

Problem 2.9.

What are the most important aspects of the design of a new futures contract?

The most important aspects of the design of a new futures contract are the specification of the

underlying asset, the size of the contract, the delivery arrangements, and the delivery months.

Problem 2.10.

Explain how margin protect investors against the possibility of default.

Margin is money deposited by an investor with his or her broker. It acts as a guarantee that

the investor can cover any losses on the futures contract. The balance in the margin account is

adjusted daily to reflect gains and losses on the futures contract. If losses are above a certain

level, the investor is required to deposit further margin. This system makes it unlikely that the

investor will default. A similar system of margin accounts makes it unlikely that the

investor’s broker will default on the contract it has with the clearing house member and

unlikely that the clearing house member will default with the clearing house.

Problem 2.11.

A trader buys two July futures contracts on frozen orange juice. Each contract is for the

delivery of 15,000 pounds. The current futures price is 160 cents per pound, the initial

margin is $6,000 per contract, and the maintenance margin is $4,500 per contract. What

price change would lead to a margin call? Under what circumstances could $2,000 be

withdrawn from the margin account?

There is a margin call if more than $1,500 is lost on one contract. This happens if the futures

price of frozen orange juice falls by more than 10 cents to below 150 cents per lb. $2,000 can

be withdrawn from the margin account if there is a gain on one contract of $1,000. This will

happen if the futures price rises by 6.67 cents to 166.67 cents per lb.

Problem 2.12.

Show that, if the futures price of a commodity is greater than the spot price during the

delivery period, then there is an arbitrage opportunity. Does an arbitrage opportunity exist if

the futures price is less than the spot price? Explain your answer.

If the futures price is greater than the spot price during the delivery period, an arbitrageur

buys the asset, shorts a futures contract, and makes delivery for an immediate profit. If the

futures price is less than the spot price during the delivery period, there is no similar perfect

arbitrage strategy. An arbitrageur can take a long futures position but cannot force immediate

delivery of the asset. The decision on when delivery will be made is made by the party with

the short position. Nevertheless companies interested in acquiring the asset will find it

attractive to enter into a long futures contract and wait for delivery to be made.

Problem 2.13.

Explain the difference between a market-if-touched order and a stop order.

A market-if-touched order is executed at the best available price after a trade occurs at a

specified price or at a price more favorable than the specified price. A stop order is executed

at the best available price after there is a bid or offer at the specified price or at a price less

favorable than the specified price.

Problem 2.14.

Explain what a stop-limit order to sell at 20.30 with a limit of 20.10 means.

A stop-limit order to sell at 20.30 with a limit of 20.10 means that as soon as there is a bid at

20.30 the contract should be sold providing this can be done at 20.10 or a higher price.

Problem 2.15.

At the end of one day a clearing house member is long 100 contracts, and the settlement price

is $50,000 per contract. The original margin is $2,000 per contract. On the following day the

member becomes responsible for clearing an additional 20 long contracts, entered into at a

price of $51,000 per contract. The settlement price at the end of this day is $50,200. How

much does the member have to add to its margin account with the exchange clearing house?

The clearing house member is required to provide 20×$2,000 = $40,000 as initial margin for

the new contracts. There is a gain of (50,200 50,000) 100 $20,000 on the existing

contracts. There is also a loss of (51,000 – 50,200) × 20 = $16,000 on the new contracts. The

member must therefore add

40,000 – 20,000 + 16,000 = $36,000

to the margin account.

Problem 2.16.

On July 1, 2013, a Japanese company enters into a forward contract to buy $1 million with

yen on January 1, 2014. On September 1, 2013, it enters into a forward contract to sell $1

million on January 1, 2014. Describe the profit or loss the company will make in dollars as a

function of the forward exchange rates on July 1, 2013 and September 1, 2013.

Suppose 1 F and 2 F are the forward exchange rates for the contracts entered into July 1, 2013

and September 1, 2013, respectively. Suppose further that S is the spot rate on January 1,

2014. (All exchange rates are measured as yen per dollar). The payoff from the first contract

is (S F1) million yen and the payoff from the second contract is 2 (F S) million yen. The

total payoff is therefore 1 2 2 1 (S F ) (F S) (F F ) million yen.

Problem 2.17.

The forward price on the Swiss franc for delivery in 45 days is quoted as 1.1000. The futures

price for a contract that will be delivered in 45 days is 0.9000. Explain these two quotes.

Which is more favorable for an investor wanting to sell Swiss francs?

The 1.1000 forward quote is the number of Swiss francs per dollar. The 0.9000 futures quote

is the number of dollars per Swiss franc. When quoted in the same way as the futures price

the forward price is111000 09091. The Swiss franc is therefore more valuable in the

forward market than in the futures market. The forward market is therefore more attractive

for an investor wanting to sell Swiss francs.

Problem 2.18.

Suppose you call your broker and issue instructions to sell one July hogs contract. Describe

what happens.

Hog futures are traded by the CME Group. The broker will request some initial margin. The

order will be relayed by telephone to your broker’s trading desk on the floor of the exchange

(or to the trading desk of another broker). It will be sent by messenger to a commission

broker who will execute the trade according to your instructions. Confirmation of the trade

eventually reaches you. If there are adverse movements in the futures price your broker may

contact you to request additional margin.

Problem 2.19.

“Speculation in futures markets is pure gambling. It is not in the public interest to allow

speculators to trade on a futures exchange.” Discuss this viewpoint.

Speculators are important market participants because they add liquidity to the market.

However, contracts must be useful for hedging as well as speculation. This is because

regulators generally only approve contracts when they are likely to be of interest to hedgers

as well as speculators.

Problem 2.20.

Explain the difference between bilateral and central clearing for OTC derivatives.

In bilateral clearing, two market participants enter into an agreement with each other covering

all outstanding derivative transactions between the two parties. Typically the agreement

covers collateral arrangements, events of default, the circumstances under which one side can

terminate the transactions, etc. In central clearing a CCP (central clearing party) stands

between the two sides of an OTC derivative transaction in much the same way that the

exchange clearing house does for exchange-traded contracts. It absorbs the credit risk but

requires initial and variation margin from each side.

Problem 2.21.

What do you think would happen if an exchange started trading a contract in which the

quality of the underlying asset was incompletely specified?

The contract would not be a success. Parties with short positions would hold their contracts

until delivery and then deliver the cheapest form of the asset. This might well be viewed by

the party with the long position as garbage! Once news of the quality problem became widely

known no one would be prepared to buy the contract. This shows that futures contracts are

feasible only when there are rigorous standards within an industry for defining the quality of

the asset. Many futures contracts have in practice failed because of the problem of defining

quality.

Problem 2.22.

“When a futures contract is traded on the floor of the exchange, it may be the case that the

open interest increases by one, stays the same, or decreases by one.” Explain this statement.

If both sides of the transaction are entering into a new contract, the open interest increases by

one. If both sides of the transaction are closing out existing positions, the open interest

decreases by one. If one party is entering into a new contract while the other party is closing

out an existing position, the open interest stays the same.

Problem 2.23.

Suppose that on October 24, 2013, a company sells one April 2014 live-cattle futures

contract. It closes out its position on January 21, 2014. The futures price (per pound) is 91.20

cents when it enters into the contract, 88.30 cents when it closes out the position and 88.80

cents at the end of December 2013. One contract is for the delivery of 40,000 pounds of

cattle. What is the profit? How is it taxed if the company is (a) a hedger and (b) a speculator?

Assume that the company has a December 31 year end.

The total profit is

40,000 × (0.9120 – 0.8830) = $1,160

If you are a hedger this is all taxed in 2014. If you are a speculator

40,000 × (0.9120 – 0.8880) = $960

is taxed in 2013 and

40,000 × (0.8880 – 0.8830) = $200

is taxed in 2014.

Problem 2.24

Explain how CCPs work. What are the advantages to the financial system of requiring all

standardized derivatives transactions to be cleared through CCPs?

A CCP stands between the two parties in an OTC derivative transaction in much the same

way that a clearing house does for exchange-traded contracts. It absorbs the credit risk but

requires initial and variation margin from each side. In addition, CCP members are required

to contribute to a default fund. The advantage to the financial system is that there is a lot

more collateral (i.e., margin) available and it is therefore much less likely that a default by

one major participant in the derivatives market will lead to losses by other market

participants. There is also more transparency in that the trades of different financial

institutions are more readily known. The disadvantage is that CCPs are replacing banks as the

too-big-to-fail entities in the financial system. There clearly needs to be careful oversight of

the management of CCPs.

Further Questions

Problem 2.25

Trader A enters into futures contracts to buy 1 million euros for 1.4 million dollars in three

months. Trader B enters in a forward contract to do the same thing. The exchange (dollars

per euro) declines sharply during the first two months and then increases for the third month

to close at 1.4300. Ignoring daily settlement, what is the total profit of each trader? When the

impact of daily settlement is taken into account, which trader does better?

The total profit of each trader in dollars is 0.03×1,000,000 = 30,000. Trader B’s profit is

realized at the end of the three months. Trader A’s profit is realized day-by-day during the

three months. Substantial losses are made during the first two months and profits are made

during the final month. It is likely that Trader B has done better because Trader A had to

finance its losses during the first two months.

Problem 2.26

Explain what is meant by open interest. Why does the open interest usually decline during

the month preceding the delivery month? On a particular day, there were 2,000 trades in a

particular futures contract. This means that there were 2,000 buyers (going long) and

2,000 sellers (going short). Of the 2,000 buyers, 1,400 were closing out positions and 600

were entering into new positions. Of the 2,000 sellers, 1,200 were closing out positions and

800 were entering into new positions. What is the impact of the day’s trading on open

interest?

Open interest is the number of contract outstanding. Many traders close out their positions

just before the delivery month is reached. This is why the open interest declines during the

month preceding the delivery month. The open interest went down by 600. We can see this in

two ways. First, 1,400 shorts closed out and there were 800 new shorts. Second, 1,200 longs

closed out and there were 600 new longs.

Problem 2.27

One orange juice future contract is on 15,000 pounds of frozen concentrate. Suppose that in

September 2013 a company sells a March 2015 orange juice futures contract for 120 cents

per pound. In December 2013, the futures price is 140 cents. In December 2014, the futures

price is 110 cents. In February 2015, the futures price is 125 cents. The company has a

December year end. What is the company’s profit or loss on the contract? How is it realized?

What is the accounting and tax treatment of the transaction is the company is classified as a)

a hedger and b) a speculator?

The price goes up during the time the company holds the contract from 120 to 125 cents per

pound. Overall the company therefore takes a loss of 15,000×0.05 = $750. If the company is

classified as a hedger this loss is realized in 2015, If it is classified as a speculator it realizes a

loss of 15,000×0.20 = $3000 in 2013, a gain of 15,000×0.30 = $4,500 in 20104 and a loss of

15,000×0.15 = $2,250 in 2015.

Problem 2.28.

A company enters into a short futures contract to sell 5,000 bushels of wheat for 250 cents

per bushel. The initial margin is $3,000 and the maintenance margin is $2,000. What price

change would lead to a margin call? Under what circumstances could $1,500 be withdrawn

from the margin account?

There is a margin call if $1000 is lost on the contract. This will happen if the price of wheat

futures rises by 20 cents from 250 cents to 270 cents per bushel. $1500 can be withdrawn if

the futures price falls by 30 cents to 220 cents per bushel.

Problem 2.29.

Suppose that there are no storage costs for crude oil and the interest rate for borrowing or

lending is 5% per annum. How could you make money if the June and December futures

contracts for a particular year trade at $80 and $86, respectively.

You could go long one June oil contract and short one December contract. In June you take

delivery of the oil borrowing $80 per barrel at 5% to meet cash outflows. The interest

accumulated in six months is about 80×0.05×0.5 or $2. In December the oil is sold for $86

per barrel and $82 is repaid on the loan. The strategy therefore leads to a profit of $4. Note

that this profit is independent of the actual price of oil in June 2010 or December 2009. It will

be slightly affected by the daily settlement procedures.

Problem 2.30.

What position is equivalent to a long forward contract to buy an asset at K on a certain date

and a put option to sell it for K on that date?

The equivalent position is a long position in a call with strike price K .

Problem 2.31

A company has derivatives transactions with Banks A, B, and C which are worth +$20

million, −$15 million, and −$25 million, respectively to the company. How much margin or

collateral does the company have to provide in each of the following two situations?

a) The transactions are cleared bilaterally and are subject to one-way collateral agreements

where the company posts variation margin, but no initial margin. The banks do not have to

post collateral.

b) The transactions are cleared centrally through the same CCP and the CCP requires a

total initial margin of $10 million.

If the transactions are cleared bilaterally, the company has to provide collateral to Banks A,

B, and C of (in millions of dollars) 0, 15, and 25, respectively. The total collateral required is

$40 million. If the transactions are cleared centrally they are netted against each other and the

company’s total variation margin (in millions of dollars) is –20 + 15 + 25 or $20 million in

total. The total margin required (including the initial margin) is therefore $30 million.

Problem 2.32

A bank’s derivatives transactions with a counterparty are worth +$10 million to the bank

and are cleared bilaterally. The counterparty has posted $10 million of cash collateral.

What credit exposure does the bank have?

The counterparty may stop posting collateral and some time will then elapse before the bank

is able to close out the transactions. During that time the transactions may move in the bank’s

favor, increasing its exposure. Note that the bank is likely to have hedged the transactions and

will incur a loss on the hedge if the transactions move in the bank’s favor. For example, if the

transactions change in value from $10 to $13 million after the counterparty stops posting

collateral, the bank loses $3 million on the hedge and will not necessarily realize an offsetting

gain on the transactions.

Problem 2.33. (Excel file)

The author’s Web page (www.rotman.utoronto.ca/~hull/data) contains daily closing prices

for crude oil futures contract and gold futures contract. You are required to download the

data and answer the following:

a) How high do the maintenance margin levels for oil and gold have to be set so that

there is a 1% chance that an investor with a balance slightly above the maintenance

margin level on a particular day has a negative balance two days later (i.e. one day

after a margin call). How high do they have to be for a 0.1% chance. Assume daily

price changes are normally distributed with mean zero. Explain why the exchange

might be interested in this calculation.

b) Imagine an investor who starts with a long position in the oil contract at the

beginning of the period covered by the data and keeps the contract for the whole of

the period of time covered by the data. Margin balances in excess of the initial margin

are withdrawn. Use the maintenance margin you calculated in part (a) for a 1% risk

level and assume that the maintenance margin is 75% of the initial margin. Calculate

the number of margin calls and the number of times the investor has a negative

margin balance and therefore an incentive to walk away. Assume that all margin calls

are met in your calculations. Repeat the calculations for an investor who starts with a

short position in the gold contract.

The data for this problem in the 7th edition is different from that in the 6th edition.

a) For gold the standard deviation of daily changes is $15.184 per ounce or $1518.4 per

contract. For a 1% risk this means that the maintenance margin should be set at

1518.4 2 2.3263 or 4996 when rounded. For a 0.1% risk the maintenance

margin should be set at 1518.4 2 3.0902 or 6636 when rounded.

For crude oil the standard deviation of daily changes is $1.5777 per barrel or $1577.7

per contract. For a 1% risk, this means that the maintenance margin should be set at

1577.7 2 2.3263 or 5191 when rounded. For a 0.1% chance the maintenance

margin should be set at 1577.7 2 3.0902 or 6895 when rounded. NYMEX

might be interested in these calculations because they indicate the chance of a trader

who is just above the maintenance margin level at the beginning of the period having

a negative margin level before funds have to be submitted to the broker.

b) For a 1% risk the initial margin is set at 6,921 for on crude oil. (This is the

maintenance margin of 5,191 divided by 0.75.) As the spreadsheet shows, for a long

investor in oil there are 157 margin calls and 9 times (out of 1039 days) where the

investor is tempted to walk away. For a 1% risk the initial margin is set at 6,661 for

gold. (This is 4,996 divided by 0.75.) As the spreadsheet shows, for a short investor in

gold there are 81 margin calls and 4 times (out of 459 days) when the investor is

tempted to walk away. When the 0.1% risk level is used there is 1 time when the oil

investor might walk away and 2 times when the gold investor might do so.

CHAPTER 3

Hedging Strategies Using Futures

Practice Questions

Problem 3.8.

In the Chicago Board of Trade’s corn futures contract, the following delivery months are

available: March, May, July, September, and December. State the contract that should be

used for hedging when the expiration of the hedge is in

a) June

b) July

c) January

A good rule of thumb is to choose a futures contract that has a delivery month as close as

possible to, but later than, the month containing the expiration of the hedge. The contracts

that should be used are therefore

(a) July

(b) September

(c) March

Problem 3.9.

Does a perfect hedge always succeed in locking in the current spot price of an asset for a

future transaction? Explain your answer.

No. Consider, for example, the use of a forward contract to hedge a known cash inflow in a

foreign currency. The forward contract locks in the forward exchange rate, which is in

general different from the spot exchange rate.

Problem 3.10.

Explain why a short hedger’s position improves when the basis strengthens unexpectedly and

worsens when the basis weakens unexpectedly.

The basis is the amount by which the spot price exceeds the futures price. A short hedger is

long the asset and short futures contracts. The value of his or her position therefore improves

as the basis increases. Similarly it worsens as the basis decreases.

Problem 3.11.

Imagine you are the treasurer of a Japanese company exporting electronic equipment to the

United States. Discuss how you would design a foreign exchange hedging strategy and the

arguments you would use to sell the strategy to your fellow executives.

The simple answer to this question is that the treasurer should

1. Estimate the company’s future cash flows in Japanese yen and U.S. dollars

2. Enter into forward and futures contracts to lock in the exchange rate for the U.S.

dollar cash flows.

However, this is not the whole story. As the gold jewelry example in Table 3.1 shows, the

company should examine whether the magnitudes of the foreign cash flows depend on the

exchange rate. For example, will the company be able to raise the price of its product in U.S.

dollars if the yen appreciates? If the company can do so, its foreign exchange exposure may

be quite low. The key estimates required are those showing the overall effect on the

company’s profitability of changes in the exchange rate at various times in the future. Once

these estimates have been produced the company can choose between using futures and

options to hedge its risk. The results of the analysis should be presented carefully to other

executives. It should be explained that a hedge does not ensure that profits will be higher. It

means that profit will be more certain. When futures/forwards are used both the downside

and upside are eliminated. With options a premium is paid to eliminate only the downside.

Problem 3.12.

Suppose that in Example 3.4 the company decides to use a hedge ratio of 0.8. How does the

decision affect the way in which the hedge is implemented and the result?

If the hedge ratio is 0.8, the company takes a long position in 16 December oil futures

contracts on June 8 when the futures price is $8. It closes out its position on November 10.

The spot price and futures price at this time are $95 and $92. The gain on the futures position

is

(92 − 88)×16,000 = $64,000

The effective cost of the oil is therefore

20,000×95 − 64,000 = $1,836,000

or $91.80 per barrel. (This compares with $91.00 per barrel when the company is fully

hedged.)

Problem 3.13.

“If the minimum-variance hedge ratio is calculated as 1.0, the hedge must be perfect.” Is this

statement true? Explain your answer.

The statement is not true. The minimum variance hedge ratio is

S

F

It is 1.0 when 05 and 2 S F . Since 10 the hedge is clearly not perfect.

Problem 3.14.

“If there is no basis risk, the minimum variance hedge ratio is always 1.0.” Is this statement

true? Explain your answer.

The statement is true. Using the notation in the text, if the hedge ratio is 1.0, the hedger locks

in a price of 1 2 F b . Since both 1 F and 2 b are known this has a variance of zero and must be

the best hedge.

Problem 3.15

“For an asset where futures prices are usually less than spot prices, long hedges are likely to

be particularly attractive.” Explain this statement.

A company that knows it will purchase a commodity in the future is able to lock in a price

close to the futures price. This is likely to be particularly attractive when the futures price is

less than the spot price. An illustration is provided by Example 3.2.

Problem 3.16.

The standard deviation of monthly changes in the spot price of live cattle is (in cents per

pound) 1.2. The standard deviation of monthly changes in the futures price of live cattle for

the closest contract is 1.4. The correlation between the futures price changes and the spot

price changes is 0.7. It is now October 15. A beef producer is committed to purchasing

200,000 pounds of live cattle on November 15. The producer wants to use the December livecattle

futures contracts to hedge its risk. Each contract is for the delivery of 40,000 pounds of

cattle. What strategy should the beef producer follow?

The optimal hedge ratio is

0 7 1 2 0 6

1 4

The beef producer requires a long position in 200000 06 120000 lbs of cattle. The beef

producer should therefore take a long position in 3 December contracts closing out the

position on November 15.

Problem 3.17.

A corn farmer argues “I do not use futures contracts for hedging. My real risk is not the

price of corn. It is that my whole crop gets wiped out by the weather.”Discuss this viewpoint.

Should the farmer estimate his or her expected production of corn and hedge to try to lock in

a price for expected production?

If weather creates a significant uncertainty about the volume of corn that will be harvested,

the farmer should not enter into short forward contracts to hedge the price risk on his or her

expected production. The reason is as follows. Suppose that the weather is bad and the

farmer’s production is lower than expected. Other farmers are likely to have been affected

similarly. Corn production overall will be low and as a consequence the price of corn will be

relatively high. The farmer’s problems arising from the bad harvest will be made worse by

losses on the short futures position. This problem emphasizes the importance of looking at

the big picture when hedging. The farmer is correct to question whether hedging price risk

while ignoring other risks is a good strategy.

Problem 3.18.

On July 1, an investor holds 50,000 shares of a certain stock. The market price is $30 per

share. The investor is interested in hedging against movements in the market over the next

month and decides to use the September Mini S&P 500 futures contract. The index is

currently 1,500 and one contract is for delivery of $50 times the index. The beta of the stock

is 1.3. What strategy should the investor follow? Under what circumstances will it be

profitable?

A short position in

1 3 50 000 30 26

50 1 500

contracts is required. It will be profitable if the stock outperforms the market in the sense that

its return is greater than that predicted by the capital asset pricing model.

Problem 3.19.

Suppose that in Table 3.5 the company decides to use a hedge ratio of 1.5. How does the

decision affect the way the hedge is implemented and the result?

If the company uses a hedge ratio of 1.5 in Table 3.5 it would at each stage short 150

contracts. The gain from the futures contracts would be

1.501.70 $2.55

per barrel and the company would be $0.85 per barrel better off.

Problem 3.20.

A futures contract is used for hedging. Explain why the daily settlement of the contract can

give rise to cash flow problems.

Suppose that you enter into a short futures contract to hedge the sale of an asset in six

months. If the price of the asset rises sharply during the six months, the futures price will also

rise and you may get margin calls. The margin calls will lead to cash outflows. Eventually the

cash outflows will be offset by the extra amount you get when you sell the asset, but there is a

mismatch in the timing of the cash outflows and inflows. Your cash outflows occur earlier

than your cash inflows. A similar situation could arise if you used a long position in a futures

contract to hedge the purchase of an asset and the asset’s price fell sharply. An extreme

example of what we are talking about here is provided by Metallgesellschaft (see Business

Snapshot 3.2).

Problem 3.21.

The expected return on the S&P 500 is 12% and the risk-free rate is 5%. What is the expected

return on the investment with a beta of (a) 0.2, (b) 0.5, and (c) 1.4?

a) 005 02 (012 005) 0064 or 6.4%

b) 005 05 (012 005) 0085 or 8.5%

c) 005 14 (012 005) 0148 or 14.8%

Further Questions

Problem 3.22

It is now June. A company knows that it will sell 5,000 barrels of crude oil in September.

It uses the October CME Group futures contract to hedge the price it will receive. Each

contract is on 1,000 barrels of ‘‘light sweet crude.’’ What position should it take? What price

risks is it still exposed to after taking the position?

It should short five contracts. It has basis risk. It is exposed to the difference between the

October futures price and the spot price of light sweet crude at the time it closes out its

position in September. It is also possibly exposed to the difference between the spot price of

light sweet crude and the spot price of the type of oil it is selling.

Problem 3.23

Sixty futures contracts are used to hedge an exposure to the price of silver. Each futures

contract is on 5,000 ounces of silver. At the time the hedge is closed out, the basis is $0.20

per ounce. What is the effect of the basis on the hedger’s financial position if (a) the trader

is hedging the purchase of silver and (b) the trader is hedging the sale of silver?

The excess of the spot over the futures at the time the hedge is closed out is $0.20 per ounce.

If the trader is hedging the purchase of silver, the price paid is the futures price plus the basis.

The trader therefore loses 60×5,000×$0.20=$60,000. If the trader is hedging the sales of

silver, the price received is the futures price plus the basis. The trader therefore gains

$60,000.

Problem 3.24

A trader owns 55,000 units of a particular asset and decides to hedge the value of her

position with futures contracts on another related asset. Each futures contract is on 5,000

units. The spot price of the asset that is owned is $28 and the standard deviation of the

change in this price over the life of the hedge is estimated to be $0.43. The futures price of

the related asset is $27 and the standard deviation of the change in this over the life of the

hedge is $0.40. The coefficient of correlation between the spot price change and futures

price change is 0.95.

(a) What is the minimum variance hedge ratio?

(b) Should the hedger take a long or short futures position?

(c) What is the optimal number of futures contracts with no tailing of the hedge?

(d) What is the optimal number of futures contracts with tailing of the hedge?

(a) The minimum variance hedge ratio is 0.95×0.43/0.40=1.02125.

(b) The hedger should take a short position.

(c) The optimal number of contracts with no tailing is 1.02125×55,000/5,000=11.23 (or

11 when rounded to the nearest whole number)

(d) The optimal number of contracts with tailing is

1.012125×(55,000×28)/(5,000×27)=11.65 (or 12 when rounded to the nearest whole

number).

Problem 3.25

A company wishes to hedge its exposure to a new fuel whose price changes have a 0.6

correlation with gasoline futures price changes. The company will lose $1 million for each 1

cent increase in the price per gallon of the new fuel over the next three months. The new

fuel’s price change has a standard deviation that is 50% greater than price changes in

gasoline futures prices. If gasoline futures are used to hedge the exposure what should the

hedge ratio be? What is the company’s exposure measured in gallons of the new fuel? What

position measured in gallons should the company take in gasoline futures? How many

gasoline futures contracts should be traded?

The hedge ratio should be 0.6 × 1.5 = 0.9. The company has an exposure to the price of 100

million gallons of the new fuel. It should therefore take a position of 90 million gallons in

gasoline futures. Each futures contract is on 42,000 gallons. The number of contracts required

is therefore

2142.9

42,000

90,000,000

or, rounding to the nearest whole number, 2143.

Problem 3.26

A portfolio manager has maintained an actively managed portfolio with a beta of 0.2. During

the last year the risk-free rate was 5% and equities performed very badly providing a return

of −30%. The portfolio manage produced a return of −10% and claims that in the

circumstances it was good. Discuss this claim.

When the expected return on the market is −30% the expected return on a portfolio with a

beta of 0.2 is

0.05 + 0.2 × (−0.30 − 0.05) = −0.02

or –2%. The actual return of –10% is worse than the expected return. The portfolio manager

has achieved an alpha of –8%!

Problem 3.27.

It is July 16. A company has a portfolio of stocks worth $100 million. The beta of the

portfolio is 1.2. The company would like to use the CME December futures contract on the

S&P 500 to change the beta of the portfolio to 0.5 during the period July 16 to November 16.

The index is currently 1,000, and each contract is on $250 times the index.

a) What position should the company take?

b) Suppose that the company changes its mind and decides to increase the beta of the

portfolio from 1.2 to 1.5. What position in futures contracts should it take?

a) The company should short

(1 2 0 5) 100 000 000

1000 250

or 280 contracts.

b) The company should take a long position in

(1 5 1 2) 100 000 000

1000 250

or 120 contracts.

Problem 3.28. (Excel file)

The following table gives data on monthly changes in the spot price and the futures price for

a certain commodity. Use the data to calculate a minimum variance hedge ratio.

Spot Price Change 050 061 022 035 079

Futures Price Change 056 063 012 044 060

Spot Price Change 004 015 070 051 041

Futures Price Change 006 001 080 056 046

Denote i x and i y by the i -th observation on the change in the futures price and the change in

the spot price respectively.

0 96 1 30 i i x y

2 2 4474 2 2 3594 i i x y

2 352 i i x y

An estimate of F is

2 4474 0 962 0 5116

9 10 9

An estimate of S is

2 3594 1 302 0 4933

9 10 9

An estimate of is

2 2

10 2 352 0 96 1 30 0 981

(10 2 4474 0 96 )(10 2 3594 1 30 )

The minimum variance hedge ratio is

0 981 0 4933 0 946

0 5116

S

F

Problem 3.29.

It is now October 2013. A company anticipates that it will purchase 1 million pounds of

copper in each of February 2014, August 2014, February 2015, and August 2015. The

company has decided to use the futures contracts traded in the COMEX division of the CME

Group to hedge its risk. One contract is for the delivery of 25,000 pounds of copper. The

initial margin is $2,000 per contract and the maintenance margin is $1,500 per contract. The

company’s policy is to hedge 80% of its exposure. Contracts with maturities up to 13 months

into the future are considered to have sufficient liquidity to meet the company’s needs. Devise

a hedging strategy for the company.

Assume the market prices (in cents per pound) today and at future dates are as follows. What

is the impact of the strategy you propose on the price the company pays for copper? What is

the initial margin requirement in October 2013? Is the company subject to any margin calls?

Date Oct 2013 Feb 2014 Aug 2014 Feb 2015 Aug 2015

Spot Price 372.00 369.00 365.00 377.00 388.00

Mar 2014 futures price 372.30 369.10

Sept 2014 futures price 372.80 370.20 364.80

Mar 2015 futures price 370.70 364.30 376.70

Sept 2015 futures price 364.20 376.50 388.20

To hedge the February 2014 purchase the company should take a long position in March

2014 contracts for the delivery of 800,000 pounds of copper. The total number of contracts

required is 800000 25000 32 . Similarly a long position in 32 September 2014 contracts

is required to hedge the August 2014 purchase. For the February 2015 purchase the company

could take a long position in 32 September 2014 contracts and roll them into March 2015

contracts during August 2014. (As an alternative, the company could hedge the February

2015 purchase by taking a long position in 32 March 2014 contracts and rolling them into

March 2015 contracts.) For the August 2015 purchase the company could take a long position

in 32 September 2014 and roll them into September 2015 contracts during August 2014.

The strategy is therefore as follows

Oct 2013: Enter into long position in 96 Sept. 2014 contracts

Enter into a long position in 32 Mar. 2014 contracts

Feb 2014: Close out 32 Mar. 2014 contracts

Aug 2014: Close out 96 Sept. 2014 contracts

Enter into long position in 32 Mar. 2015 contracts

Enter into long position in 32 Sept. 2015 contracts

Feb 2015: Close out 32 Mar. 2015 contracts

Aug 2015: Close out 32 Sept. 2015 contracts

With the market prices shown the company pays

36900 08 (37230 36910) 37156

for copper in February, 2014. It pays

36500 08 (37280 36480) 37140

for copper in August 2014. As far as the February 2015 purchase is concerned, it loses

37280 36480 800 on the September 2014 futures and gains 37670 36430 1240 on

the February 2015 futures. The net price paid is therefore

37700 08800 081240 37348

As far as the August 2015 purchase is concerned, it loses 37280 36480 800 on the

September 2014 futures and gains 38820 36420 2400 on the September 2015 futures.

The net price paid is therefore

38800 08800 082400 37520

The hedging strategy succeeds in keeping the price paid in the range 371.40 to 375.20.

In October 2013 the initial margin requirement on the 128 contracts is 128$2000 or

$256,000. There is a margin call when the futures price drops by more than 2 cents. This

happens to the March 2014 contract between October 2013 and February 2014, to the

September 2014 contract between October 2013 and February 2014, and to the September

2014 contract between February 2014 and August 2014. (Under the plan above the March

2015 contract is not held between February 2014 and August 2014, but if it were there would

be a margin call during this period.)

Problem 3.30. (Excel file)

A fund manager has a portfolio worth $50 million with a beta of 0.87. The manager is

concerned about the performance of the market over the next two months and plans to use

three-month futures contracts on the S&P 500 to hedge the risk. The current level of the

index is 1250, one contract is on 250 times the index, the risk-free rate is 6% per annum, and

the dividend yield on the index is 3% per annum. The current 3 month futures price is 1259.

a) What position should the fund manager take to eliminate all exposure to the market

over the next two months?

b) Calculate the effect of your strategy on the fund manager’s returns if the level of the

market in two months is 1,000, 1,100, 1,200, 1,300, and 1,400. Assume that the onemonth

futures price is 0.25% higher than the index level at this time.

a) The number of contracts the fund manager should short is

0 87 50 000 000 138 20

1259 250

Rounding to the nearest whole number, 138 contracts should be shorted.

b) The following table shows that the impact of the strategy. To illustrate the

calculations in the table consider the first column. If the index in two months is 1,000,

the futures price is 1000×1.0025. The gain on the short futures position is therefore

(1259 100250) 250138 $8849 250

The return on the index is 32 12 =0.5% in the form of dividend and

250 1250 20% in the form of capital gains. The total return on the index is

therefore 195%. The risk-free rate is 1% per two months. The return is therefore

205% in excess of the risk-free rate. From the capital asset pricing model we

expect the return on the portfolio to be 087205% 17835% in excess of the

risk-free rate. The portfolio return is therefore 16835%. The loss on the portfolio is

01683550000000 or $8,417,500. When this is combined with the gain on the

futures the total gain is $431,750.

Index now 1250 1250 1250 1250 1250

Index Level in Two Months 1000 1100 1200 1300 1400

Return on Index in Two Months ‐0.20 ‐0.12 ‐0.04 0.04 0.12

Return on Index incl divs ‐0.195 ‐0.115 ‐0.035 0.045 0.125

Excess Return on Index ‐0.205 ‐0.125 ‐0.045 0.035 0.115

Excess Return on Portfolio ‐0.178 ‐0.109 ‐0.039 0.030 0.100

Return on Portfolio ‐0.168 ‐0.099 ‐0.029 0.040 0.110

Portfolio Gain ‐8,417,500 ‐4,937,500 ‐1,457,500 2,022,500 5,502,500

Futures Now 1259 1259 1259 1259 1259

Futures in Two Months 1002.50 1102.75 1203.00 1303.25 1403.50

Gain on Futures 8,849,250 5,390,625 1,932,000 ‐1,526,625 ‐4,985,250

Net Gain on Portfolio 431,750 453,125 474,500 495,875 517,250

CHAPTER 4

Interest Rates

Practice Questions

Problem 4.8.

The cash prices of six-month and one-year Treasury bills are 94.0 and 89.0. A 1.5-year bond

that will pay coupons of $4 every six months currently sells for $94.84. A two-year bond that

will pay coupons of $5 every six months currently sells for $97.12. Calculate the six-month,

one-year, 1.5-year, and two-year zero rates.

The 6-month Treasury bill provides a return of 6 94 6383% in six months. This is

26383 12766% per annum with semiannual compounding or 2 ln(106383) 1238%

per annum with continuous compounding. The 12-month rate is 11 89 12360% with

annual compounding or ln(11236) 1165% with continuous compounding.

For the 1 1

2 year bond we must have

4e0123805 4e011651 104e15R 9484

where R is the 1 1

2 year zero rate. It follows that

1 5

1 5

3 76 3 56 104 94 84

0 8415

0 115

R

R

e

e

R

or 11.5%. For the 2-year bond we must have

5e0123805 5e011651 5e011515 105e2R 9712

where R is the 2-year zero rate. It follows that

2 0 7977

0 113

e R

R

or 11.3%.

Problem 4.9.

What rate of interest with continuous compounding is equivalent to 15% per annum with

monthly compounding?

The rate of interest is R where:

12 1 0 15

12

eR

i.e.,

12ln 1 0 15

12

R

01491

The rate of interest is therefore 14.91% per annum.

Problem 4.10.

A deposit account pays 12% per annum with continuous compounding, but interest is actually

paid quarterly. How much interest will be paid each quarter on a $10,000 deposit?

The equivalent rate of interest with quarterly compounding is R where

4

0 12 1

4

e R

or

R 4(e003 1) 01218

The amount of interest paid each quarter is therefore:

10 000 0 1218 304 55

4

or $304.55.

Problem 4.11.

Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are 4%,

4.2%, 4.4%, 4.6%, and 4.8% per annum with continuous compounding respectively. Estimate

the cash price of a bond with a face value of 100 that will mature in 30 months and pays a

coupon of 4% per annum semiannually.

The bond pays $2 in 6, 12, 18, and 24 months, and $102 in 30 months. The cash price is

2e00405 2e004210 2e004415 2e00462 102e004825 9804

Problem 4.12.

A three-year bond provides a coupon of 8% semiannually and has a cash price of 104. What

is the bond’s yield?

The bond pays $4 in 6, 12, 18, 24, and 30 months, and $104 in 36 months. The bond yield is

the value of y that solves

4e05 y 4e10 y 4e15 y 4e20 y 4e25 y 104e30 y 104

Using the Goal Seek or Solver tool in Excel y 006407 or 6.407%.

Problem 4.13.

Suppose that the 6-month, 12-month, 18-month, and 24-month zero rates are 5%, 6%, 6.5%,

and 7% respectively. What is the two-year par yield?

Using the notation in the text, m 2 , d e0072 08694 . Also

A e00505 e00610 e006515 e00720 36935

The formula in the text gives the par yield as

(100 100 0 8694) 2 7 072

3 6935

To verify that this is correct we calculate the value of a bond that pays a coupon of 7.072%

per year (that is 3.5365 every six months). The value is

3536e00505 35365e00610 3536e006515 103536e00720 100

verifying that 7.072% is the par yield.

Problem 4.14.

Suppose that zero interest rates with continuous compounding are as follows:

Maturity( years) Rate (% per annum)

1 2.0

2 3.0

3 3.7

4 4.2

5 4.5

Calculate forward interest rates for the second, third, fourth, and fifth years.

The forward rates with continuous compounding are as follows: to

Year 2: 4.0%

Year 3: 5.1%

Year 4: 5.7%

Year 5: 5.7%

Problem 4.15.

Use the rates in Problem 4.14 to value an FRA where you will pay 5% for the third year on

$1 million.

The forward rate is 5.1% with continuous compounding or e00511 1 5232% with annual

compounding. The 3-year interest rate is 3.7% with continuous compounding. From equation

(4.10), the value of the FRA is therefore

[1000000(005232 005)1]e00373 207885

or $2,078.85.

Problem 4.16.

A 10-year, 8% coupon bond currently sells for $90. A 10-year, 4% coupon bond currently

sells for $80. What is the 10-year zero rate? (Hint: Consider taking a long position in two of

the 4% coupon bonds and a short position in one of the 8% coupon bonds.)

Taking a long position in two of the 4% coupon bonds and a short position in one of the 8%

coupon bonds leads to the following cash flows

Year 0: 90 − 2×80 = −70

Year 10: 200 – 100 = 100

because the coupons cancel out. $100 in 10 years time is equivalent to $70 today. The 10-

year rate, R, (continuously compounded) is therefore given by

100 70e10R

The rate is

1 ln 100 0 0357

10 70

or 3.57% per annum.

Problem 4.17.

Explain carefully why liquidity preference theory is consistent with the observation that the

term structure of interest rates tends to be upward sloping more often than it is downward

sloping.

If long-term rates were simply a reflection of expected future short-term rates, we would

expect the term structure to be downward sloping as often as it is upward sloping. (This is

based on the assumption that half of the time investors expect rates to increase and half of the

time investors expect rates to decrease). Liquidity preference theory argues that long term

rates are high relative to expected future short-term rates. This means that the term structure

should be upward sloping more often than it is downward sloping.

Problem 4.18.

“When the zero curve is upward sloping, the zero rate for a particular maturity is greater

than the par yield for that maturity. When the zero curve is downward sloping the reverse is

true.” Explain why this is so.

The par yield is the yield on a coupon-bearing bond. The zero rate is the yield on a zerocoupon

bond. When the yield curve is upward sloping, the yield on an N-year coupon-bearing

bond is less than the yield on an N-year zero-coupon bond. This is because the coupons are

discounted at a lower rate than the N-year rate and drag the yield down below this rate.

Similarly, when the yield curve is downward sloping, the yield on an N-year coupon bearing

bond is higher than the yield on an N-year zero-coupon bond.

Problem 4.19.

Why are U.S. Treasury rates significantly lower than other rates that are close to risk free?

There are three reasons (see Business Snapshot 4.1).

1. Treasury bills and Treasury bonds must be purchased by financial institutions to fulfill a

variety of regulatory requirements. This increases demand for these Treasury instruments

driving the price up and the yield down.

2. The amount of capital a bank is required to hold to support an investment in Treasury

bills and bonds is substantially smaller than the capital required to support a similar

investment in other very-low-risk instruments.

3. In the United States, Treasury instruments are given a favorable tax treatment compared

with most other fixed-income investments because they are not taxed at the state level.

Problem 4.20.

Why does a loan in the repo market involve very little credit risk?

A repo is a contract where an investment dealer who owns securities agrees to sell them to

another company now and buy them back later at a slightly higher price. The other company

is providing a loan to the investment dealer. This loan involves very little credit risk. If the

borrower does not honor the agreement, the lending company simply keeps the securities. If

the lending company does not keep to its side of the agreement, the original owner of the

securities keeps the cash.

Problem 4.21.

Explain why an FRA is equivalent to the exchange of a floating rate of interest for a fixed

rate of interest?

A FRA is an agreement that a certain specified interest rate, K R , will apply to a certain

principal, L, for a certain specified future time period. Suppose that the rate observed in the

market for the future time period at the beginning of the time period proves to be M R . If the

FRA is an agreement that K R will apply when the principal is invested, the holder of the FRA

can borrow the principal at M R and then invest it at K R . The net cash flow at the end of the

period is then an inflow of K R L and an outflow of M R L . If the FRA is an agreement

that K R will apply when the principal is borrowed, the holder of the FRA can invest the

borrowed principal at M R . The net cash flow at the end of the period is then an inflow

of M R L and an outflow of K R L . In either case, we see that the FRA involves the exchange of a

fixed rate of interest on the principal of L for a floating rate of interest on the principal.

Problem 4.22.

Explain how a repo agreement works and why it involves very little risk for the lender.

The borrower transfers to the lender ownership of securities which have a value

approximately equal to the amount borrowed and agrees to buy them back for the amount

borrowed plus accrued interest at the end of the life of the loan. If the borrower defaults, the

lender keeps the securities. Note that the securities should not have a value significantly more

than the amount borrowed. Otherwise the borrower is subject to the risk that the lender will

not honor its obligations.

Further Questions

Problem 4.23

When compounded annually an interest rate is 11%. What is the rate when expressed with (a) semiannual

compounding, (b) quarterly compounding, (c) monthly compounding, (d) weekly compounding, and (e) daily

compounding.

We must solve 1.11=(1+R/n)n where R is the required rate and the number of times per year the rate is

compounded. The answers are a) 10.71%, b) 10.57%, c) 10.48%, d) 10.45%, e) 10.44%

Problem 4.24

The following table gives Treasury zero rates and cash flows on a Treasury bond:

Maturity (years Zero rate Coupon payment Principal

0.5 2.0% $20

1.0 2.3% $20

1.5 2.7% $20

2.0 3.2% $20 $1000

Zero rates are continuously compounded

(a) What is the bond’s theoretical price?

(b) What is the bond’s yield?

The bond’s theoretical price is

20×e-0.02×0.5+20×e-0.023×1+20×e-0.027×1.5+1020×e-0.032×2 = 1015.32

The bond’s yield assuming that it sells for its theoretical price is obtained by solving

20×e-y×0.5+20×e-y×1+20×e-y×1.5+1020×e-y×2 = 1015.32

It is 3.18%.

Problem 4.25 (Excel file)

A five-year bond provides a coupon of 5% per annum payable semiannually. Its price is 104.

What is the bond’s yield? You may find Excel’s Solver useful.

The answer (with continuous compounding) is 4.07%

Problem 4.26 (Excel file)

Suppose that LIBOR rates for maturities of one month, two months, three months, four

months, five months and six months are 2.6%, 2.9%, 3.1%, 3.2%, 3.25%, and 3.3% with

continuous compounding. What are the forward rates for future one month periods?

The forward rates for the second, third, fourth, fifth and sixth months are (see spreadsheet)

3.2%, 3.5%, 3.5%, 3.45%, 3.55%, respectively with continuous compounding.

Problem 4.27

A bank can borrow or lend at LIBOR. The two-month LIBOR rate is 0.28% per annum with

continuous compounding. Assuming that interest rates cannot be negative, what is the

arbitrage opportunity if the three-month LIBOR rate is 0.1% per year with continuous

compounding. How low can the three-month LIBOR rate become without an arbitrage

opportunity being created?

The forward rate for the third month is 0.001×3 − 0.0028×2 = − 0.0026 or