2016 Exam MFE Study Guide

Transcription

1 Exam MFE Study Guide 2016 Exam MFE Study Guide Howard C. Mahler, FCAS, MAAA In 2013 and prior years the SOA and CAS jointly administered Exam MFE/3F. Starting in 2014 the SOA administered Exam MFE without the CAS. How much detail is needed and how many problems need to be done varies by person and topic. In order to help you to concentrate your efforts: 1. About 1/6 of the many problems are labeled highly recommended, while another 1/6 are labeled recommended. 2. Important Sections are listed in bold in the table of contents. Extremely important Sections are listed in larger type and in bold. 3. Important ideas and formulas are in bold. 4. A section of Important Ideas and Formulas. 5. A chart of past exam questions by Section. My Study Guide is a thick stack of paper. 1 However, many students find they do not need to look at the textbook. For those who have trouble getting through the material, concentrate on sections in bold. Sections and material in italics should be skipped on the first time through. Highly Recommended problems (about 1/6 of the total) are double underlined. Recommended problems (about 1/6 of the total) are underlined. Do at least the Highly Recommended problems your first time through. It is important that you do problems when learning a subject and then some more problems a few weeks later. Be sure to do all the recent exam questions at some point. 2 I have written some easy and tougher problems. 3 The former exam questions are arranged in chronological order. The more recent exam questions are on average more similar to what you will be asked on your exam, than are less recent exam questions. In the electronic version use the bookmarks / table of contents in the Navigation Panel in order to help you find what you want. You may find it helpful to print out selected portions, such as the Table of Contents and the Important Ideas Section. My Practice Exams are sold separately. My Seminar Slides are sold separately. 1 The number of pages is not as important as how long it takes you to understand the material. One page in a textbook might take someone as long to understand as ten pages in my Study Guides. 2 Unfortunately, there are only a few released exams, plus the sample exam questions. 3 Points in my study guides are based on 100 points = a 4 hour exam. Questions on your exam are worth the equivalent of 2.5 points.

2 Exam MFE Study Guide The MFE CBT exam will provide a formula document as well as a normal distribution calculator that will be available during the test by clicking buttons on the item screen. Details are available on the Prometric Web Site. To try it out: Similar to other exam reference buttons, the normal distribution calculator button will be available throughout the exam in the top right corner of every item screen. Click the button to call up the calculator and calculate cumulative normal distribution and inverse cumulative normal distribution values. Use these values to answer the question as needed. When using the normal distribution calculator, values should be entered with five decimal places. Use all five decimal places from the result in subsequent calculations. The normal distribution calculator button replaces the Normal Table. The previous rule on rounding no longer applies. 4 You can try the normal distribution calculator button at the Prometric Web Site. You will benefit from using it at least part of the time when you are studying. The formula sheet contains the same information about the Normal and LogNormal distributions as was provided in the past. Besides many past exam questions from the CAS and SOA, my study guides include some past questions from exams given by the Institute of Actuaries and Faculty of Actuaries in Great Britain. These questions are copyright by the Institute of Actuaries and Faculty of Actuaries, and are reproduced here solely to aid students studying for actuarial exams. These IOA questions are somewhat different in format than those on your exam, but should provide some additional perspective on the syllabus material. I suggest you buy and try the TI-30XS Multiview calculator. You will save time doing repeated calculations using the same formula. The BA II Plus Professional calculator is useful for calculations involving interest. Many people find it helpful to have both calculators during the exam. Download from the SOA website, a copy of the tables to be attached to your exam. 5 Read the Hints on Study and Exam Techniques in the CAS Syllabus. 6 Read Tips for Taking Exams. 7 4 Unfortunately, most of my solutions were written up using the prior rule: On Joint Exam 3F/MFE, when using the normal distribution, choose the nearest z-value to find the probability, or if the probability is given, choose the nearest z-value. No interpolation should be used. For example, if the given z-value is 0.759, and you need to find Pr(Z < 0.759) from the normal distribution table, then chose the probability for z-value = 0.76: Pr(Z < 0.76) = This should not make a significant difference. 5 You will be supplied with information on the Normal and LogNormal Distributions

3 Exam MFE Study Guide Starting in Spring 2011, MFE/3F is 3 hours and given via Computer Based Testing (CBT). While studying, you should do as many problems as possible. Going back and forth between reading and doing problems is the only way to pass this exam. The only way to learn to solve problems is to solve lots of problems. You should not feel satisfied with your study of a subject until you can solve a reasonable number of the problems. There are two manners in which you should be doing problems. First you can do problems in order to learn the material. Take as long on each problem as you need to fully understand the concepts and the solution. Reread the relevant syllabus material. Carefully go over the solution to see if you really know what to do. Think about what would happen if one or more aspects of the question were revised. 8 This manner of doing problems should be gradually replaced by the following manner as you get closer to the exam. The second manner is to do a series of problems under exam conditions, with the items you will have when you take the exam. Take in advance a number of points to try based on the time available. For example, if you have an uninterrupted hour, then one might try either 60/2.5 = 24 points or 60/3 = 20 points of problems. Do problems as you would on an exam in any order, skipping some and coming back to some, until you run out of time. I suggest you leave time to double check your work. Expose yourself somewhat to everything on the syllabus. Concentrate on sections and items in bold. Do not read sections or material in italics your first time through the material. 9 My chart of where the past exam questions have been may also help you to direct your efforts. 10 Try not to get bogged down on a single topic. On hard subjects, try to learn at least the simplest important idea. The first time through do enough problems in each section, but leave some problems in each section to do closer to the exam. Make a schedule and stick to it. Spend a minimum of one hour every day. I recommend at least two study sessions every day, each of at least 1/2 hour. 8 Some may also find it useful to read about a dozen questions on an important subject, thinking about how to set up the solution to each one, but only working out in detail any questions they do not quickly see how to solve. 9 Material in italics is provided for those who want to know more about a particular subject and/or to be prepared for more challenging exam questions; it could be directly needed to answer perhaps one question on an exam. 10 While this may indicate what ideas questions on your exam are likely to cover, every exam contains a few questions on ideas that have yet to be asked. Your exam will have its own mix of questions.

4 Exam MFE Study Guide Throughout do Exam Problems and Practice Problems in my study guides. At least 50% of your time should be spent doing problems. As you get closer to the Exam, the portion of time spent doing problems should increase. Review the important formulas and ideas sections, at the end of each study guide. During the last several weeks do my practice exams, sold separately. Here is a schedule that may help some people go through my study guide. 11 A 13 week Study Schedule for Exam MFE: 1. Sections Sections Sections Sections Sections Sections Sections Sections Sections Sections Sections Sections Sections Throughout, go back and review the important ideas and do some more problems in sections you have gone through previously. 11 This is just an example of one possible schedule. Adjust it to suit your needs or make one up yourself.

5 Exam MFE Study Guide Past students helpful suggestions and questions have greatly improved this study guide. I thank them! Feel free to send me any questions or suggestions: Howard Mahler, hmahler@mac.com Please do not copy the Study Guide, except for your own personal use. Giving it to others is unfair to yourself, your fellow students who have paid for them, and myself. 12 If you found them useful, tell a friend to buy his own. Please send me any suspected errors by prior to the exam. (Please specify as carefully as possible the page, Study Guide, and Exam.) Author Biography: Howard C. Mahler is a Fellow of the Casualty Actuarial Society, and a Member of the American Academy of Actuaries. He has taught actuarial exam seminars and published study guides since He spent over 20 years in the insurance industry, the last 15 as Vice President and Actuary at the Workers' Compensation Rating and Inspection Bureau of Massachusetts. He has published many major research papers and won the 1987 CAS Dorweiler prize. He served 12 years on the CAS Examination Committee including three years as head of the whole committee ( ). Mr. Mahler has taught live seminars and/or classes for Exam C, Exam MFE, CAS Exam ST, CAS Exam 5, and CAS Exam 8. He has written study guides for all of the above. hmahler@mac.com 12 This study guide represents thousands of hours of work.

6 Exam MFE Study Guide Pass Marks and Passing Percentages for Past Exams: 13 Pass Mark as % of Raw Effective Available Number of Number Number Passing Passing Exam Points Candidates Passing Ineffective Percent Percent MFE S07 N.A % 52.5% MFE F07 60% % 55.5% MFE/3F S08 55% % 53.2% MFE/3F F08 63% % 49.2% MFE/3F S09 60% % 43.1% MFE/3F F09 63% % 39.2% MFE/3F S10 59% % 42.2% MFE/3F F10 60% % 39.4% MFE/3F S11 71% % 50.9% MFE/3F F11 72% % 47.5% MFE/3F 4/12 76% % 53.4% MFE/3F 8/12 72% % 53.1% MFE/3F 11/12 72% % 48.5% MFE/3F 3/13 72% % 52.6% MFE/3F 7/13 72% % 54.9% MFE/3F 11/13 72% % 47.3% MFE 3/14 72% % 55.3% MFE 7/14 72% % 51.7% MFE 11/14 72% % 54.2% MFE 3/15 72% % 55.4% MFE 7/15 72% % 52.4% 13 Information taken from the SOA webpage. Check the webpage for updated information. 14 Starting in May 2011, Exam 3F/MFE is administered using computer-based testing (CBT). Under CBT, it is not possible to schedule everyone to take the examination at the same time. As a result, each administration consists of multiple versions of the examination given over a period of several days. The examinations are constructed and scored using Item Response Theory (IRT). Under IRT, each operational item that appears on an examination has been calibrated for difficulty and other test statistics and the pass mark for each examination is determined before the examination is given. All versions of the examination are constructed to be of comparable difficulty to one another. For the May 2011 administration of Examination MFE/3F, an average of 71% correct was needed to pass the exam.

7 Mahlerʼs Guide to Financial Economics Exam MFE prepared by Howard C. Mahler, FCAS Copyright 2016 by Howard C. Mahler. Study Aid 2016-MFE Howard Mahler

8 2016-MFE, Financial Economics, HCM 11/28/15, Page 1 Mahlerʼs Guide to Financial Economics Copyright 2016 by Howard C. Mahler. Concepts in Derivatives Markets by Robert L. McDonald are demonstrated. 1 Information in bold or sections whose title is in bold are more important for passing the exam. Larger bold type indicates it is extremely important. Information presented in italics (and sections whose titles are in italics) should not be needed to directly answer exam questions and should be skipped on first reading. It is provided to aid the readerʼs overall understanding of the subject, and to be useful in practical applications. Highly Recommended problems are double underlined. Recommended problems are underlined. 2 Solutions to the problems in each section are at the end of that section. Section # Pages Section Name Introduction European Options Properties of Premiums of European Options Put-Call Parity Bounds on Premiums of European Options Options on Currency Exchange Options Futures Contracts Synthetic Positions American Options Replicating Portfolios Risk Neutral Probabilities Utility Theory and Risk Neutral Pricing Binomial Trees, Risk Neutral Probabilities Binomial Trees, Valuing Options on Other Assets Other Binomial Trees Binomial Trees, Actual Probabilities Jensen's Inequality Normal Distribution LogNormal Distribution The Table of Contents is continued on the next page. 1 All references are to the third edition. 2 Note that problems include both some written by me and some from past exams. The latter are copyright by the Society of Actuaries and the Casualty Actuarial Society and are reproduced here solely to aid students in studying for exams. The solutions and comments are solely the responsibility of the author; the SOA and CAS bear no responsibility for their accuracy. While some of the comments may seem critical of certain questions, this is intended solely to aid you in studying and in no way is intended as a criticism of the many volunteers who work extremely long and hard to produce quality exams.

9 2016-MFE, Financial Economics, HCM 11/28/15, Page 2 Section # Pages Section Name Limited Expected Value A LogNormal Model of Stock Prices Black-Scholes Formula Black-Scholes, Options on Currency Black-Scholes, Options on Futures Contracts Black-Scholes, Stocks Paying Discrete Dividends Using Historical Data to Estimate Parameters of the Stock Price Model Implied Volatility Histograms Normal Probability Plots Option Greeks Delta-Gamma Approximation Option Greeks in the Binomial Model Profit on Options Prior to Expiration Elasticity Volatility of an Option Risk Premium of an Option Sharpe Ratio of an Option Market Makers Delta Hedging Gamma Hedging Relationship to Insurance Exotic Options Asian Options Barrier Options Compound Options Gap Options Valuing European Exchange Options Forward Start Options Chooser Options Options on the Best of Two Assets Cash-or-Nothing Options Asset-or-Nothing Options Random Walks Standard Brownian Motion Arithmetic Brownian Motion Geometric Brownian Motion Geometric Brownian Motion Model of Stock Prices Ito Processes Ito's Lemma The Table of Contents is continued on the next page.

10 2016-MFE, Financial Economics, HCM 11/28/15, Page 3 Section # Pages Section Name Valuing a Claim on S^a Black-Scholes Equation Simulation Simulating Normal and LogNormal Distributions Simulating LogNormal Stock Prices Valuing Asian Options via Simulation Improving Efficiency of Simulation Bonds and Interest Rates The Rendleman-Bartter Model The Vasicek Model The Cox-Ingersoll-Ross Model The Black Model Interest Rate Caps Binomial Trees of Interest Rates The Black-Derman-Toy Model Important Formulas and Ideas My practice exams and my seminar slides are each sold separately.

11 2016-MFE, Financial Economics, HCM 11/28/15, Page 4 Throughout I make many references to Derivatives Markets by McDonald; these are to the third edition. One does not need the textbook in order to use my study guide; the references are to help those who are also using the textbook. Chapter of Third Edition Derivatives Markets Sections of Study Guide , 12, 14, 15, , 13, 16-17, , 28, 31, , , 27, , 28, Appendix B.1 1 Appendix C 18 I have included in my early sections, the 9 questions from the 2007 FM Sample Exam for Derivatives Markets, based on earlier chapters of the textbook. Unless otherwise stated chapter appendices are not included in the required readings from this text. 3 Excluding Options on Commodities on pages 315 and Including Appendices 11.A and 11.B. 5 Including Appendix 12.A. 6 Including Appendix 13.B. 7 Sections (up to but excluding Modeling Correlated Asset Prices on pages ), 20.4 (excluding Multivariate Itôʼs Lemma on pages ), (up to but excluding Valuing a Claim on S a Q b on pages ). 8 Sections (excluding What If the Underlying Asset Is Not an Investment Asset on pages ) and 21.3 (excluding The Backward Equation on pages , and excluding the last two paragraphs of the section on page 639). 9 But with only those definitions in Tables 23.1 and 23.2 that are relevant to Section Up to the second paragraph on page 721, but including footnote 4 on page 721 and the top panel in Figure 24.3 on page Sections (up to the first paragraph on page 773), 25.5 (excluding LIBOR Market Model on pages ), Appendix 25.A (this appendix contains only a reference to the following site for download, )

12 2016-MFE, Financial Economics, HCM 11/28/15, Page 5 For those who have the second edition of the textbook by McDonald: 12 Chapter of Second Edition Derivatives Markets Sections of Study Guide , 12, 14, , 13, 16-17, 27, , 28, 31, , , 27, , 28, Appendix B.1 1 Appendix C 18 I have included in my early sections, the 9 questions from the 2007 FM Sample Exam for Derivatives Markets, based on earlier chapters of the textbook. Unless otherwise stated chapter appendices are not included in the required readings from this text. 12 The current syllabus refers only to the third edition. 13 Excluding Options on Commodities on page Including Appendices 11.A and 11.B. 15 Including Appendix 12.A. 16 Including Appendix 13.B. 17 Sections (up to but excluding Multivariate Itôʼs Lemma on pages ) and 20.7 (up to but excluding Valuing a Claim on S a Q b on pages and excluding Finding the lease rate on top one-half of page 669). 18 Sections (excluding What If the Underlying Asset Is Not and Investment Asset on pages ) and 21.3 (excluding The Backward Equation on pages , and excluding the paragraph on page 692 that begins If a probability and through the end of the section). 19 But with only those definitions in Tables 22.1 and 22.2 that are relevant to Section Up to but excluding Exponentially Weighted Moving Average on page 746 and through the end of the section. 21 Up to but excluding Forward rate agreements on pages

13 2016-MFE, Financial Economics, HCM 11/28/15, Page 6 This study guide covers all of the material on SOA MFE. 22 The syllabus consists of various sections of the 3rd edition of Derivatives Markets by Robert L. McDonald, plus a short study note Some Remarks On Derivatives Markets by Elias S. W. Shiu. 23 Unless stated otherwise in a question assume: The market is frictionless. There are no taxes, transaction costs, bid/ask spreads, or restrictions on short sales. All securities are perfectly divisible. Trading does not affect prices. Information is available to all investors simultaneously. Every investor acts rationally (i.e., there is no arbitrage.) The risk-free rate is constant The notation is the same as used in Derivatives Markets by Robert L. McDonald. The MFE CBT exam will provide a formula document as well as a normal distribution calculator that will be available during the test by clicking buttons on the item screen. Details are available on the Prometric Web Site. Similar to other exam reference buttons, the normal distribution calculator button will be available throughout the exam in the top right corner of every item screen. Click the button to call up the calculator and calculate cumulative normal distribution and inverse cumulative normal distribution values. Use these values to answer the question as needed. When using the normal distribution calculator, values should be entered with five decimal places. Use all five decimal places from the result in subsequent calculations. The normal distribution calculator button replaces the Normal Table. The previous rule on rounding no longer applies. 24 You can try the normal distribution calculator button at the Prometric Web Site. You will benefit from using it at least part of the time when you are studying. The formula sheet contains the same information about the Normal and LogNormal distributions as was provided in the past, as reproduced on the next page. 22 In 2007 the CAS and SOA gave separate exams. Starting in 2008 they gave a joint exam. Starting in 2014 the SOA administers MFE alone. 23 The study note is available on the SOA webpage. 24 Unfortunately, most of my solutions were written up using the prior rule: On Joint Exam 3F/MFE, when using the normal distribution, choose the nearest z-value to find the probability, or if the probability is given, choose the nearest z-value. No interpolation should be used. For example, if the given z-value is 0.759, and you need to find Pr(Z < 0.759) from the normal distribution table, then chose the probability for z-value = 0.76: Pr(Z < 0.76) = This should not make a significant difference.

14 2016-MFE, Financial Economics, HCM 11/28/15, Page 7

15 2016-MFE, Financial Economics, HCM 11/28/15, Page 8 Changes to the Syllabus from 2007: 25 Section 12.6 of Derivatives Markets, Perpetual American Options, is no longer on the syllabus. 26 The final portion of Section 20.6, Multivariate Itoʼs Lemma, is no longer on the syllabus. 27 The final portion of Section 24.5, Forward Rate Agreements, is no longer on the syllabus. Added, the first portion of Section 20.7, Valuing a Claim on S a, up to but excluding the last subsection on Valuing a Claim on S a Q b. Changes to the Syllabus for Spring 2009: Chapter 10 of Derivatives Markets, exclude Options on Commodities on page 334. Exclude Section 11.5 of Derivatives Markets, on Binomial Trees, Discrete Dividends. Add Appendices 11.A and 11.B of Derivatives Markets. Add Appendix 13.B of Derivatives Markets. Chapter 20 of Derivatives Markets: exclude Finding the lease rate on top one-half of page 669. Add parts of Chapter 21: Sections (excluding What If the Underlying Asset Is Not and Investment Asset on pages ) and 21.3 (excluding The Backward Equation on pages , and excluding the paragraph on page 692 that begins If a probability and through the end of the section). Add parts of Chapter 22: Section 22.1 (but with only those definitions in Tables 22.1 and 22.2 that are relevant to Section 22.1.) Add parts of Chapter 23: Sections (up to but excluding Exponentially Weighted Moving Average on page 746 and through the end of the section.) Add Appendix B.1. Add Appendix C. Changes to the Syllabus for Fall 2009: 28 Add Chapter 18 of Derivatives Markets, about the LogNormal Stock Price Model. Add Chapter of Derivatives Markets, about Monte Carlo Valuation, in other words simulation. Changes for 2011: Computer based testing. 3 hours and approximately 30 questions. 25 Starting in 2008 there was a joint exam, 3F/MFE. 26 In 2007 Section 12.6 was on SOA MFE, but not CAS In 2007 this final portion of Section 20.6 was on CAS 3, but not SOA MFE. 28 Material was moved from Exam 4/C onto Exam 3F/MFE. Exam 3F/MFE was extended from 2 hours to 2.5 hours and will consist of approximately 25 multiple-choice questions.

16 2016-MFE, Financial Economics, HCM 11/28/15, Page 9 Exam Questions by Section of This Study Guide: MFE CAS 3 MFE CAS 3 MFE/3F Section Section Name Sample 5/07 5/07 11/07 5/09 1 Introduction 25 2 European Options 3 Properties of Premiums of Euro. Options 2 4 Put-Call Parity 1 3, 4 1, 4 14, Bounds on Premiums of Euro. Options 6 Options on Currency Exchange Options 8 Futures Contracts 9 Synthetic Positions American Options Replicating Portfolios Risk Neutral Probabilities Utility Theory and Risk Neutral Pricing 14 Bin. Trees, Risk Neutral Probs. 4, 49 15, , 19, Binomial Trees, Options on Other Assets 5, Other Binomial Trees Binomial Trees, Actual Probabilities Jensen's Inequality 19 Normal Distribution 20 LogNormal Distribution 21 Limited Expected Value 22 A LogNormal Model of Stock Prices Black-Scholes Formula 6 20, 21 3, Black-Scholes, Options on Currency Black-Scholes, Options on Futures Black-Scholes, Discrete Dividends Historical Data to Estimate Parameters 17, Implied Volatility 29 Histograms 30 Normal Probability Plots The SOA did not release its 11/07 exam MFE. The CAS/SOA did not release the 5/08, 11/08,11/09, and subsequent exams 3F/MFE. Continued on the next page

17 2016-MFE, Financial Economics, HCM 11/28/15, Page 10 MFE CAS 3 MFE CAS 3 MFE/3F Section Section Name Sample 5/07 5/07 11/07 5/09 31 Option Greeks 8, Delta-Gamma Approximation Option Greeks in the Binomial Model 44, 45, Profit on Options Prior to Expiration Elasticity 20, Volatility of an Option 5 37 Risk Premium of an Option 38 Sharpe Ratio of an Option Market Makers 40 Delta Hedging 9, 47, Gamma Hedging Relationship to Insurance 43 Exotic Options 44 Asian Options Barrier Options Compound Options Gap Options Valuing European Exchange Options 49 Forward Start Options 19, Chooser Options Options on the Best of Two Assets Cash-or-Nothing Options 28, Asset-or-Nothing Options 4 54 Random Walks 55 Standard Brownian Motion Arithmetic Brownian Motion 57 Geometric Brownian Motion 58 Geo. Brown. Mot. Model Stock Pr. 10, 11, 32, , Ito Processes 12, 23, 48, , 18 63, 66, 67, Ito's Lemma 13, 24, 35, , 68, 73 Continued on the next page

18 2016-MFE, Financial Economics, HCM 11/28/15, Page 11 MFE CAS 3 MFE CAS 3 MFE/3F Section Section Name Sample 5/07 5/07 11/07 5/09 61 Valuing a Claim on S^a 16, 62, 71, Black-Scholes Equation Simulation 64 Simulating Normals & LogNormals 65 Simulating LogNormal Stocks Valuing Asian Options via Sim. 67 Improving Efficiency of Simulation 57, 58, 59, Bonds and Interest Rates The Rendleman-Bartter Model 70 The Vasicek Model 14, The Cox-Ingersoll-Ross Model 21, 38, The Black Model 7 73 Interest Rate Caps 3 74 Binomial Trees of Interest Rates 5 75 The Black-Derman-Toy Model 15, 29, 30, Questions no longer on the syllabus: MFE, 5/07, Q. 16. In August 2010, the SOA/CAS updated the file of MFE Sample Exam questions. There are now a total of 76 sample questions. Check the SOA webpage to see if any additional Sample Exam questions have been added. Valuing a Claim on S a was added to the syllabus in Spring Some material on simulation was moved here from Exam 4/C in Fall 2009.

19 2016-MFE, Financial Economics 1 Introduction, HCM 11/28/15, Page 12 Section 1, Introduction Earlier Chapters of Derivatives Markets by McDonald are on Exam 2/FM. 29 Some of the ideas covered in those chapters are used in the chapters on your exam. Derivatives: 30 A derivative is an agreement between two people that has a value determined by the price of something else. For example, Alan gives Bob the right to buy from Alan a share of IBM stock one year from now at a price of $120. This is an example of a stock option. The value of this option depends on the price of IBM stock one year from now. Options: A call is an option to buy. For example, Bob purchased a call option on IBM stock from Alan. A put is an option to sell. For example, if Debra purchased a put option on IBM stock from Carol, then Debra will have the option in the future to sell a share of IBM stock to Carol at a specified price. Continuously Compounded Risk Free Rate: 31 If r is the continuously compounded annual risk free rate, then the present value of $1 T years in the future is: e -rt. r as used by McDonald is what an actuary would call the force of interest. Effective Annual Rate: 32 If r is the effective annual risk free rate, then the present value of $1 T years in the future is: 1/(1+r) T. An effective annual rate is what an actuary would call the rate of interest. Effective annual rate will be used in Interest Rate Caps and the Black-Derman-Toy Model, to be discussed in subsequent sections. Otherwise, we will use continuously compounded rates. 29 3rd edition, Chapters 1 3, Chapter 4 ( ), Chapter 5 ( and Appendix 5.B), Chapter 8 ( ). 30 Warren E. Buffett has said, Derivatives are financial weapons of mass destruction, carrying dangers that, while now latent, are potentially lethal. 31 See Appendix B.1 of Derivatives Markets by McDonald. 32 See Appendix B.1 of Derivatives Markets by McDonald.

20 2016-MFE, Financial Economics 1 Introduction, HCM 11/28/15, Page 13 Selling Short: If we sell a stock short, then we borrow a share of stock and sell it for the current market price. We will give this person a share of stock at the designated time in the future. We also must pay this person any stock dividends they would have gotten on the stock, when they would have gotten them. Forward Contracts: A forward contract is an agreement that sets the terms today, but the buying or selling of the asset takes place in the future. For example, Ed will be moving in a month, and his friend Fred agrees to buy Edʼs TV one month from now for $200. The purchaser of an option has bought the right to do something in the future, but has no obligation to do anything. In contrast, in a forward contract both parties are obligated to fulfill their parts of the contract. Value of a Forward Contract: F 0,T = forward price at time T in the future. For example, if Joe buys a forward contract to buy one share of ABC stock in two years at $120, then F 0,2 = $120. At time 2 years, Joe pays $120 and gets one share of stock. 33 PV[F 0,T ] is the present value at time 0 of a forward contract to be executed at time T. PV[F 0,T ] = F 0,T e -rt. Let us assume the current price of XYZ stock is S 0. Assume XYZ stock pays no dividends. Charlie can buy a forward contract to buy one share of XYZ stock in exchange for paying F 0,T at time T. If Charlie invests F 0,T e -rt at the risk free rate, then at time T he will have F 0,T. 34 He uses that amount to fulfill his forward contract and at time T Charlie has one share of XYZ Stock. Lucy can instead buy one share of XYZ stock now, for the current market price of S 0, and hold onto the share of stock until at least time T. 33 This differs from the prepaid price. Joe might instead be able to pay $110 now and get a share of stock 2 years from now. This is an example of a prepaid futures contract. 34 Charlie could invest in a Treasury Bond.

21 2016-MFE, Financial Economics 1 Introduction, HCM 11/28/15, Page 14 Both Lucy and Charlie end up in the same situation, with one share of XYZ stock at time T. Therefore, their investments must have equal present value. S 0 = F 0,T e -rt. F 0,T = S 0 e rt, in the absence of dividends. If instead XYZ stock pays dividends, then Lucy would have collected any dividends paid from time 0 to T, while she owned the stock. Charlie would not. Thus Lucyʼs position is equal to Charlieʼs position plus a receipt of dividends. Therefore, S 0 - PV[Div] = F 0,T e -rt. F 0,T = S 0 e rt - PV[Div] e rt. If the dividends are paid at discrete points in time, with amount D t i paid at time t i, then F 0,T = S 0 e rt - e r(t - t i) Dt i. Exercise: The current price of a stock is $100. It will to pay a dividend 3 months from now, a dividend 6 months from now, a dividend 9 months from now, and a dividend 12 months from now. Each dividend is of size $2. r = 6%. Determine the the forward price for a share of stock one year from now [Solution: F 0,1 = (100) e (2)(e e e e 0 ) = $ Comment: Both sides of the equation are valued one year from now.] Futures Contracts: A futures contract is similar to a forward contract except: A futures contract is typically traded on an exchange. A futures contract is marked to market periodically. 35 The buyer and the seller post margin Marked-to-market means the item is revalued to reflect current market prices. 36 A deposit which compensates the other party to a futures contract in case one of the parties does not fulfill its obligation.

22 2016-MFE, Financial Economics 1 Introduction, HCM 11/28/15, Page 15 Forward Contracts Versus Futures Contracts: 37 Forward Contract Futures Contract Type of Market Dealer or Broker (Commodities) Exchange Liquidity Low High Contract Form Customized Standard Performance Guarantee Creditworthiness Mark-to-Market Transaction Costs Bid-ask spread Fees or Commissions Continuous Dividends: We often assume that dividends are paid at a continuous rate δ. 38 Over a short period of time dt, stock dividends of: δ S(t) dt are paid, where S(t) is the stock price at time t. So that if one buy a share of stock at time 0, and reinvests the dividends in the stock, at time T one would have e Tδ shares of the stock. 39 Exercise: One buys 1 million shares of a stock that pays dividends at the continuous annual rate of 2%. The dividends are reinvested in that stock. After 3 years how many shares of the stock does one own? [Solution: (1 million)e (3)(0.02) = 1,061,837 shares.] If XYZ stock pays continuous dividends at a rate δ, and Lucy buys e -Tδ shares of XYZ stock now, then Lucy would have one share of the stock at time T. Charlie buys a future contract for one share of the stock. If Charlie invests F 0,T e -rt at the risk free rate, then at time T he will have F 0,T. 40 At time T Charlieʼs position equals Lucyʼs; they both own a share of stock. Therefore, F 0,T e -rt = S 0 e -Tδ. Therefore, in the case of dividends paid continuously: F 0,T = S 0 e T(r - δ). Prepaid Forward Price: The forward price is the price we would pay in the future for a forward contract. In contrast, the prepaid forward price, F P 0,T, is the price we would pay today for a forward contract. F P 0,T = F0,T e -rt. 37 Taken from Table 2.2 Financial Economics, Harry H. Panjer, editor. 38 This is a good approximation for a stock index fund. 39 δ acts similarly to a force of interest. 40 Charlie could invest in a Treasury Bond.

23 2016-MFE, Financial Economics 1 Introduction, HCM 11/28/15, Page 16 For example, let us assume we are paying today in order to own a share stock at time 3. Then the prepaid forward price is F P 0,3 (S). We would pay this price at time 0 in exchange for receiving the stock at time 3. However, we would not receive any dividends the stock would pay between time 0 and 3. Therefore, in the case of discrete dividends, F P 0,T (S) = S0 - PV[Div]. Exercise: The current price of a stock is 120. The stock will pay a dividend of 3 in 2 months. What is the 5 month prepaid forward price of the stock? r = 6%. [Solution: S 0 - PV[Div] = 120-3e -(2/12)(6%) = ] In the case of continuous dividends, F P 0,T (S) = S0 e -δt. Exercise: The current price of a stock is 80. The stock pays dividends at a continuous rate of 1%. What is the 5 month prepaid forward price of the stock? [Solution: S 0 e -δt = 80e -(5/12)(1%) = ] If we pay S 0 e -δt in order to buy e -δt shares of stock today and reinvest the dividends we would have one share of stock at time T. Thus S 0 e -δt is the price we would pay today to own one share of stock at time T. More generally, F P t, T (S) = St e -δ(t-t). Continuously Compounded Returns: Let S t and S t+h be the stock prices at times t and t+h. Then the continuously compounded return on the stock between time t and t+h is: ln[s t+h / S t ]. On an annual basis, this return is: ln[s t+h / S t ] / h. For example, if the stock price is $80 at time 0 and $90 at time 2 years, then the continuously compounded return from time 0 to 2 is: ln[90/80] = 11.78%. On an annual basis, this return is: 11.78% / 2 = 5.89%. Exercise: The stock price is $90 at time 2 years and $85 at time 2.5 years, what is the annual continuously compounded return? [Solution: ln[85/90] / 0.5 = -11.4%.]

24 2016-MFE, Financial Economics 1 Introduction, HCM 11/28/15, Page 17 One can get the future stock price from the current stock price and the continuously compounded return. For example, if the current stock price is $100, and the continuously compounded return over the next three years is 8% per year, then the stock three years from now is: 100 e 0.24 = $ Exercise: The current price of a stock price is $60. Over the next four years the annual continuously compounded return are: 17%, 33%, -140%, and 6%. What is the stock price in four years? [Solution: 60 exp[0.17] exp[0.33] exp[-1.40] exp[0.06] = 60 exp[-0.84] = $ Comment: The continuously compounded returns add; the return over the whole four years is: 17% + 33% - 140% + 6% = -84%. When the stock price declines by a very large amount, one can have a continuously compounded return of less than -100%.] Volatility: The volatility of a stock is the standard deviation of its continuously compounded returns. 41 Actuarial Present Values: 42 Let us assume that one year from now an insurer will pay either $50 with probability 70% or $100 with probability 30%. Then the expected payment in one year is: (0.7)(50) + (0.3)(100) = $65. Assume that the continuously compound annual rate of interest is now 5%. Then the actuarial present value of the insurerʼs payment is: 65 e = $ In general in order to calculate an actuarial present value, one takes a sum of the expected payments at each point in time each multiplied by the appropriate discount factor. The discount factor adjusts for the difference between the time value of money at the present and at the time when the payment is made. Exercise: In addition to the payments one year from now, the insurer will pay two years from now either $50 with probability 50%, $100 with probability 40%, or $200 with probability 10%. Assume that one year from now the continuously compound annual rate of interest will be 6%. Determine the actuarial present value of the insurerʼs total payments, including those made one year from now and two years from now. [Solution: The expected payment in two years is: (0.5)(50) + (0.4)(100) + (0.1)(200) = $85. Discounting back to the present: 85 exp[ ] = $ Adding in the actuarial present value of the payments made in one year, the actuarial present value of the insurerʼs total payments is: $ $76.15 = $ ] 41 Volatility will be discussed in subsequent sections and is usually stated on an annual basis. 42 Covered extensively on CAS Exam LC and SOA Exam MLC. 43 If instead the 5% were an effective annual rate, then the actuarial present value would be: 65/1.05 = $61.90.

25 2016-MFE, Financial Economics 1 Introduction, HCM 11/28/15, Page 18 Named Positions: 44 One can buy various combinations of options and stock. The more common such positions have been given names. Bear Spread: The sale of an option together with the purchase of an otherwise identical option with a higher strike price. Can construct a bear spread using either puts or calls. The owner of the Bear Spread hopes that the stock price moves down. Box Spread: Buy a call and sell a put at one strike price, plus at another (higher) strike price sell a call and buy a put. 45 Bull Spread: The purchase of an option together with the sale of an otherwise identical option with a higher strike price. Can construct a bull spread using either puts or calls. The owner of the Bull Spread hopes that the stock price moves up. Butterfly Spread: Buying a K strike option, selling two K + ΔK strike options, and buying a K + 2ΔK strike option. Collar: Purchase a put and sell a call with a higher strike price. Ratio Spread: Buying m of an option and selling n of an otherwise identical option at a different strike. Straddle: Purchase a call and the otherwise identical put. Strangle: The purchase of a put and a higher strike call with the same time until expiration. For example, Gene Green buys a Straddle with K = 80. He buys an 80-strike call and a similar 80-strike put. His payoff at expiration is: Max[0, S T - 80] + + Max[0, 80 - S T ] = S T The further the stock price at expiration is from 80, the larger Geneʼs payoff. Gene is hoping there is a large movement in the stock price See Chapter 3 of Derivatives Markets by McDonald, on the syllabus of Exam FM. 45 For European options, the box spread is equivalent ot a zero-coupon bond. 46 In other words, Gene is betting that the stockʼs volatility is high. In contrast, the seller of a straddle is betting that the stockʼs volatility is low.

26 2016-MFE, Financial Economics 1 Introduction, HCM 11/28/15, Page 19 For example, Vanessa buys a 90 strike call and sells an otherwise identical 100 strike call. This is an example of a Call Bull Spread. Vanessa hopes the stock price increases. If Vanessa bought her 90 strike call from Nathan and sold her 100 strike call to Nathan, than Nathan owns a Call Bear Spread. Nathan hopes the stock price declines. Long and Short Positions: Entering into a long position is buying. Entering into a short position is selling or writing. For example if you long one call option and long the similar put option, then you bought the call and put, and you have purchased a straddle. If instead you short a call option and the similar put option, then you have written (sold) a straddle. If you short a 60-strike 3-month call and long a 80-strike 3-month call, then you have purchased a Bear Spread.

27 2016-MFE, Financial Economics 1 Introduction, HCM 11/28/15, Page 20 Problems: 1.1 (1 point) A stock price is 160. Assume r = 0.08 and there are no dividends. What is the 4-year forward price? A. less than 200 B. at least 200 but less than 210 C. at least 210 but less than 220 D. at least 220 but less than 230 E. at least (1 point) A stock has a current price of 120. The stock pays dividends at a continuously compounded rate of 1.5%. r = What is the 4-year prepaid forward price? A. 113 B. 115 C. 117 D. 119 E (1 point) A stock has a two-year forward price of The stock pays dividends at a continuously compounded rate of 3%. r = 7%. What is the current price of this stock? A. 90 B. 92 C. 94 D. 96 E (1 point) A stock has a current price of 90. The stock pays dividends at a continuously compounded rate of 2%. r = What is the 5-year forward price? A. 100 B. 105 C. 110 D. 115 E (1 point) A stock has a current price of $100. In 3 months the stock will pay a dividend of $2. r = What is the 4-month prepaid forward price? 1.6 (1 point) A stock has a four-year forward price of The stock pays dividends at a continuously compounded rate of 0.8%. r = 5.2%. What is the current price of this stock? A. 65 B. 70 C. 75 D. 80 E (1 point) Options are extremely risky investments. The variance of returns is great, yet most people are assumed to be risk-averse. Moreover, brokerage commissions on options are high. So why are options and other derivative securities such popular financial instruments?

28 2016-MFE, Financial Economics 1 Introduction, HCM 11/28/15, Page (CAS5B, 11/93, Q. 31) (2 points). a. (1.5 points) List and briefly describe three examples of derivative instruments. b. (0.5 points) Why do firms use them? 1.9 (CAS5B, 11/91, Q. 60) (1 point) Which of the following statements concerning corporate securities is FALSE? A. One reason firms utilize derivative instruments is to protect themselves against the effects of adverse changes in various external factors. B. Firms do not issue derivative securities to raise money. C. A futures contract is an order than you place in advance to buy or sell an asset or commodity. D. In a futures contract, the price is fixed when you place the order and paid at the time of the order. E. A forward contract is a tailor-made product that is not traded on an organized exchange (CAS5B, 5/94, Q. 11) (1 point) Which of the following are true? 1. A future is an order that you place in advance to buy or sell an asset or commodity at a price that is agreed upon when the order is placed. 2. A forward contract is traded on an organized exchange. 3. Swaps are agreements that grant bond owners the right to exchange the bond for a predetermined number of common shares by the exercise date. A. 1 B. 2 C. 1, 2 D. 2, 3 E 1, 2, (CAS5B, 11/95, Q. 31) (2 points) a. (1/2 point) Briefly describe warrants and convertible bonds. b. (3/4 points) For each, describe what the rational holder probably will do on the expiration date if the price of stock rises significantly from the date of issuance. c. (3/4 points) For each, describe what the holder probably will do if the price of the company's stock falls significantly (CAS5B, 11/98, Q.10) (1 point) Which of the following are true regarding financial derivatives? 1. Firms typically issue derivatives to raise money on short notice. 2. A forward contract may be traded on an organized exchange. 3. A warrant is a derivative. A. 1 B. 3 C. 1, 3 D. 2, 3 E. 1, 2, 3

29 2016-MFE, Financial Economics 1 Introduction, HCM 11/28/15, Page (FM Sample Exam, Q.4) Zero-coupon risk-free bonds are available with the following maturities and yield rates (effective, annual): Maturity (years) Yield You need to buy corn for producing ethanol. You want to purchase 10,000 bushels one year from now, 15,000 bushels two years from now, and 20,000 bushels three years from now. The current forward prices, per bushel, are 3.89, 4.11, and 4.16 for one, two, and three years respectively. You want to enter into a commodity swap to lock in these prices. Which of the following sequences of payments at times one, two, and three will NOT be acceptable to you and to the corn supplier? A. 38,900, 61,650, 83,200 B. 39,083, 61,650, 82,039 C. 40,777, 61,166, 81,554 D. 41,892, 62,340, 78,997 E. 60,184, 60,184, 60, (FM Sample Exam, Q.6) The current price of one share of XYZ stock is 100. The forward price for delivery of one share of XYZ stock in one year is 105. Which of the following statements about the expected price of one share of XYZ stock in one year is TRUE? A. It will be less than 100 B. It will be equal to 100 C. It will be strictly between 100 and 105 D. It will be equal to 105 E. It will be greater than (FM Sample Exam, Q.7) A non-dividend paying stock currently sells for 100. One year from now the stock sells for 110. The risk-free rate, compounded continuously, is 6%. The stock is purchased in the following manner: You pay 100 today You take possession of the security in one year Which of the following describes this arrangement? A. Outright purchase B. Fully leveraged purchase C. Prepaid forward contract D. Forward contract E. This arrangement is not possible due to arbitrage opportunities

30 2016-MFE, Financial Economics 1 Introduction, HCM 11/28/15, Page (FM Sample Exam, Q.8) You believe that the volatility of a stock is higher than indicated by market prices for options on that stock. You want to speculate on that belief by buying or selling at-the-money options. What should you do? A. Buy a strangle B. Buy a straddle C. Sell a straddle D. Buy a butterfly spread E. Sell a butterfly spread 1.17 (CAS3, 11/07, Q.25) (2.5 points) On January 1, 2007, the Florida Property Company purchases a one-year property insurance policy with a deductible of $50,000. In the event of a hurricane, the insurance company will pay the Florida Property Company for losses in excess of the deductible. Payment occurs on December 31, For the last three months of 2007, there is a 20% chance that a single hurricane occurs and an 80% chance that no hurricane occurs. If a hurricane occurs, then the Florida Property Company will experience $1,000,000 in losses. The continuously compounded risk-free rate is 5%. On October 1, 2007, what is the risk-neutral expected value of the insurance policy to the Florida Property Company? A. Less than $185,000 B. At least $185,000, but less than $190,000 C. At least $190,000, but less than $195,000 D. At least $195,000, but less than $200,000 E. At least $200, (IOA CT8, 9/08, Q.6) (6 points) Consider an asset S paying a dividend at a constant instantaneous rate of δ, a forward contract with maturity T written on S and a constant, instantaneous (continuously compounded) risk-free rate of r. Derive the price at time t of the forward contract, using the no-arbitrage principle.