foreword for the pak study notes (ifm exam)

Foreword-PAK-IFM-Exam 2018

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Foreword for the PAK Study Notes (IFM Exam) Welcome to the PAK Study Manual group (http://www.pakstudymanual.com/). PAK is a global leader in providing study notes, video and manuals to FSA level candidates (Quantitative finance and Investment track, Corporate finance and ERM track, Individual life and annuity track, and Retirement Benefit track). For over 10 years, PAK has consistently helped many FSA students pass the very challenging FSA exams. Some students using PAK resources were able to pass their FSA exams on their first attempt with the maximum score of 10/10. Because the same rigor and hard work used in producing the FSA level PAK resources were used to produce the current PAK-IFM-study-notes, PAK is pretty confident of the contribution that the PAK-IFM-study-notes will add to the learning curve of the IFM-candidate. The field of finance can reasonably be divided into two major areas:

Corporate Finance (This area is concerned with the source of funding a corporation: debt versus capital. It looks into the theory/models for determining the cost of these sources of funding. It looks at the overall value of the firm and breaks it down into the value to various stakeholders. It seeks the optimal balance for these sources of funds) and

Derivative pricing (This area of finance is primarily concerned with the dynamics of asset prices. It formalizes these dynamics into a specific model for the asset price like the Binomial model and the lognormal model. Then it shows how derivative written on the assets can be priced using the postulated model. It is a very important branch of finance and it continues to engage other industry of the financial sector, like the insurance industry).

It is therefore no accident that the SOA syllabus for the IFM exam faithfully incorporates those two parts of Finance. In turn, The PAK Study notes for the IFM exam is also divided into two parts:

PAK-Part-I: Corporate Finance: SOA Syllabus sections 1-5 (40%-60%) and PAK-Part-II: Derivatives pricing: SOA Syllabus sections 6-10 (40%-60%)

PAK-Part-I contains at least 170 pages and PAK-Part-II contains around 450 pages. In addition:

PAK-Part-I and PAK-part-II contain clear explanations of all the concepts covered in the official source reading. Many times, they include practice examples designed to nail those concepts.

PAK-Part-II contains over 150 exam-type questions with detailed solutions. Some questions may appear more difficult than what you may realistically encounter in the SOA exam. However, they are still included in order to enhance your overall understanding of the topics.

Considering the details of explanations provided in the PAK study notes (PAK-Part-I and PAK-part-II), one will see that the notes do not require an earlier exposure to the materials in order to master the concepts and pass the IFM exam. The notes can definitely be used as a supplement or a substitute for a formal university class on the topics of derivative pricing and corporate finance (especially for those wishing to pass the IFM exam and not being able to take formal university classes that cover the entire exam syllabus).

While Part-I-Corporate-Finance is completely new to the syllabus (July 2018), PAK-part-II-Derivatives-pricing used to be included in the old MFE (Mathematics of financial economics) exam. As such, exam-type questions for PAK-Part-I will be made available at a later point in the future. For now, the student is expected to understand the materials covered in PAK-Part-I and re-work the examples provided in the study notes in order to fix the concept.

The PAK study notes are designed to help students adequately prepare for the IFM exam without 100% reference to the source manuals (including SOA approved papers). The PAK-suggested strategy is to (i) Read and master the PAK study notes, (ii) work the practice examples and the MOCK questions included in the PAK

http://www.pakstudymanual.com/�


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study notes and (iii) work the official practice questions from the SOA website, then (iv) re-work (ii) and (iii) until the IFM-candidate can answer all questions independently. The PAK study notes are organized in a user-friendly format. PAK-Part I contains 5 sections and PAK-Part II contains 5 sections. Each section contained in the study note comprises many inter-connected sub-sections. The table below summarizes the 10 PAK-sections contained in PAK-part-I and PAK-part-II:

Each section’s weight is an indication to the number of exam questions that will likely be coming from the particular section. The official source reading

comprises:

11 chapters from the book: Derivatives Markets (Third Edition), 2013, by McDonald, R.L.. 9 chapters from the book: Corporate Finance (Fourth Edition), 2017, by Berk, J. and DeMarzo, P. IFM-21-18: Measures of Investment Risk, Monte Carlo Simulation, and Empirical Evidence on the

Efficient Market Hypothesis. IFM-22-18: Actuarial Applications of Options and Other Financial Derivatives.

The feature of the PAK subsections contained in each of the 10 PAK sections is the following:

The PAK subsections synthesize the theories and concepts from the official source reading, The PAK subsections contain examples and practice to illustrate the theories and concepts from the

official source reading. Such illustrative examples, not only nail the concepts down, but also constitute example of how the concepts can be tested on a real IFM exam.

The PAK subsections are inter-connected and assembled in a way that will facilitate a holistic understanding of the official source reading and enhance the overall mastery of the official source reading,

The PAK subsections are beautifully written, they provide clear and better explanations, illustrative examples to fix the concepts.

Each of the 5 sections of PAK-Part-II contains a set of 30+ practice exam-type questions (end of section questions) and detailed solutions in addition to the illustrative examples contained in the study notes.

The PAK subsections are well-organized and user-friendly.

The SOA expects candidates to (i) Read all of the source readings and (ii) work the practice questions set for part I and part II (available on the SOA website). However, there are many practical reasons why the IFM-candidate shall consider the PAK study notes and the PAK-strategy instead:

At the strategic level, the organization of the study on a by-section-basis as done within the PAK study notes allows the candidate to take advantage of the signal send by the SOA. By communicating the weight for each of the 10 SOA sections on the official syllabus, the SOA is signaling the importance


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(in weight not necessarily concept) that they intend to place of each syllabus-section (SOA-topic). For instance, while preparing for the IFM exam, the candidate shall bear in mind that sections 1, 4, 7 and 9 can make for 60% of the entire exam.

A naïve approach to learning will be to simply cover say the Berk/DeMarzo book on corporate finance without visualizing where each chapter of the book (or portion of the chapter) falls within the SOA sections (from 1 to 10). This way, the candidate never gets to appreciate when he/she is studying items for section 1, 2,…and so on. Thus the candidate does not know whether he or she is over-learning portions that carry less weight and under-learning portions that carry more weight. PAK is not doing the naïve approach. PAK adequately selects the chapters from the official source reading that are relevant to every specific topic/section and group them together in one single section.

At times, the order of chapters in the official source reading is not representative of the way the SOA has put weights on the sections. At other times, the order of chapters in a given book does not necessarily reflect the way the SOA expects you to understand the materials being learnt. There are also situations where the SOA has selected a paper (IFM-21-18/ IFM-22-18 for instance) that comes to complement a specific topic covered in the book. All these situations were factored in when designing the PAK-sections and sub-sections.

For instance, when McDonald discusses derivatives, the book is looking from the perspective of derivatives as an asset (or a hedge instrument, or a speculative bet). In contrast, SOA paper IFM-21-18 discusses derivatives from the perspective of the insurance industry: The derivative is a liability that sits on the balance sheet of an insurance company. However, the SOA expects candidates to understand both aspects of derivatives.

After all, one may simply recognize that the two books (McDonald in Derivatives and Berk in Corporate Finance) included in the official syllabus were not written exclusively for IFM exam takers. As such, the IFM papers (IFM-22-18) and (IFM-21-18) were added to supplement them and make the subject of financial economics more relevant to actuaries-to-be. PAK strongly believes that blending various parts of the official source readings within the PAK-sections is more of an effective learning strategy and is in line with the way students need to prepare for the IFM exam.

Another instance is the Black-Scholes-Merton (BSM) model: McDonald derives the BSM model in chapter 12 by eliminating risk in the hedging portfolio (by using Ito Lemma). However, all of Ito Calculus/stochastic processes have been eliminated from the IFM syllabus (to now be part of the QFI FSA exams). So, to arrive at the BSM model, PAK wants you to learn the ways of the Binomial model and risk-neutral valuation, the lognormal model and the conditional expectation of the risk neutral lognormal model. PAK believes that this is what the SOA wants from IFM-candidates, since there is no more Ito Calculus/stochastic processes in the syllabus and we still have to deal with BSM, lognormal and Binomial. The problem here is that in McDonald, the lognormal model is discussed in chapter 18, while the BSM model is derived at in chapter 12. So, in order to make the derivation of the BSM (without using Ito Lemma) comprehensive to the IFM exam-taker, the PAK study notes discusses McDonald chapter 18 before McDonald chapter 12. This will not have come easy to someone following the naïve approach.

McDonald discusses models of the asset price (Binomial and lognormal) while (IFM-22-18: Actuarial Applications of Options and Other Financial Derivatives) presents various insurance-related derivative (variable annuities for instance). PAK effectively shows examples of how these instruments can be valued using the models discussed in McDonald (including the Black Scholes model). PAK does not believe that the SOA exam is about retrieving information from IFM-22-18. Rather, PAK believes the IFM-exam will be about using the models presented in


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McDonald in order to solve the insurance’s business problems mentioned in (IFM-22-18). As such, many MOCK-questions and some illustrative examples are included in the PAK-study-notes to enhance your understanding of how models in McDonald (Binomial, BSM) can be implemented to value the simple guarantees (GMDB and GMMB), and the Equity Index Annuities (Though the EIA was not covered in IFM-22-18, it can be found in the SOA practice questions (the difficult questions) available on the SOA website).

At the end of the day, the organization of the PAK sections and subsection allows a more efficient learning process. PAK is very confident of the contribution its study notes brings to the learning process of the IFM-exam-candidate, or any person interested in application of derivatives in insurance in general.

Best of luck! Bell F. Ouelega FSA CERA MAAA CQF PAK Instructor for IFM Exam, QFI Exams and RP-RB-IRM Exam


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About the Author of the PAK Study Notes for the IFM Exam:

After completing a Masters degree in Statistical science from the George Washington University (Washington D.C), Bell started his career in Actuarial science as an Actuarial Analyst. Bell completed his ASA (associate of the Society of Actuaries), CERA (Certified Enterprise Risk Analyst) and his fellowship (FSA) in roughly 7 years. Immediately after his FSA, Bell completed his CQF (Certificate in Quantitative Finance) while he was the Director in Quantative Research in the asset and liability management are of life insurance in Des Moines Iowa. While still working with insurance companies, he joined the PAK Study Group as a contributor roughly 5 years ago. He started to write the PAK study notes for the financial economics and the risk management FSA exams. Ever since, Bell has continued to consistently help FSA students pass their exams, especially those of the Quantitative finance/investment track, and those of taking the investment risk management for retirement benefit track. Every session, Bell receives a lot of strong feedback from FSA students. Bell can be reached at his email at ([email protected]).

PAK testimonials (More on our website) The QFIC PAK course was an excellent supplement to the core reading materials. The practice questions and helpful summaries are essential for understanding how very conceptual and abstract concepts from mathematical finance are applied on the exam. Bell was also a terrific resource and provided thoughtful responses to challenging questions N.M. (QFIC Exam). Thank you very much for putting the PAK study package for the RP-IRM exam- I thought the study manual was very helpful and incredibly easy to read. I passed. Thanks again. P.R. (RP IRM Exam).

Bell, with the help of PAK study manual and your personal explanations, I was able to pass the QFIC exam. The 90 MOCK questions you designed and the solutions you provided saved me. After going through them once, I was able to fly through the past SOA exams easily before the actual sitting. The morning quantitative section for spring 2015 exam was a real blood bath, I found it significantly more difficult than any of the past QFIC exams. Calvin Y (QFI-IRM; QFIC and QFIA). Bell, I found the Study Manual and the other PAK materials quite useful for passing the exam. Thanks for the continued motivation, email exchanges and answering all the syllabus related questions. Don.C (QFIC Exam).

The PAK Study materials, the proposed study schedule and the grouping of topics, and especially the PAK MOCK problems were very useful. Thanks so much for a great PACKAGE. Jackson (QFIA Exam).

mailto:[email protected]�

Sample-PAK-Part-I-IFM-Exam 2018

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Section-1-1) Common Measures of Risk and Return

Suppose that you invest an amount (𝑆0) to purchase one share of an asset currently trading at (𝑆0) per share.

1) If the asset is a risk-free asset, then you already know what return you will get from this investment. For instance, you may invest in a government security promising a fixed rate of return for a holding period.

2) If the asset is a risky asset, then you do not know the actual return you will make on this investment.

For a one-year holding period, you return is calculated as:

𝑅 =𝑆1 − 𝑆0𝑆0

Where: 𝑆1 = 𝑇ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝑎𝑠𝑠𝑒𝑡 𝑜𝑛𝑒 𝑦𝑒𝑎𝑟 𝑓𝑟𝑜𝑚 𝑛𝑜𝑤

At the time of your initial investment (t=0), the terminal value of this asset (𝑆1) is unknown to you. That makes the return (R) a random variable. There are common measures for the random return:

1) We can specify its probability distribution function, 2) We can calculate its mean or 3) We can calculate its variance (or standard deviation/also called volatility).

For instance, assume that at time 1, the asset can take a value among a set of ‘n’ possible values (𝑆𝑖) where (𝑖 = 1, … ,𝑛) with probability (𝑝𝑖). Then:

(1) The probability distribution function of returns is a discrete random variable of ‘n’ possible values (𝑅𝑖) and associated probabilities (𝑝𝑖), where:

𝑅𝑖 =𝑆𝑖 − 𝑆0𝑆0

(2) The expected return is calculated as:

𝐸(𝑅) = �𝑅𝑖𝑝𝑖

𝑛

𝑖=1

(3) And the variance and standard deviation (SD) of returns are:

𝑉𝑎𝑟(𝑅) = �𝑝𝑖 × �𝑅𝑖 − 𝐸(𝑅)�2𝑛

𝑖=1

= �𝑝𝑖 × 𝑅𝑖2 − �𝐸(𝑅)�2𝑛

𝑖=1

While: 𝑆𝐷(𝑅) = �𝑉𝑎𝑟(𝑅)


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To illustrate the concept, assume that we currently invested $100.00 to acquire an asset. If it is known that one year from now the asset price (𝑆1) can take one of three possible values (140; 110; 80) with probabilities (0.25; 0.5; 0.25), then we specify (i) the probability distribution function for the returns, we then calculate (ii) the expected return and (iii) the standard deviation of returns as follows:

For a risk-free investment with a certain return, the variance of returns is zero and the distribution….

…. If there is no perfect correlation between two assets (X) and (Y), then, the opportunity set is a curve

:

If there is perfect positive correlation between two assets (X) and (Y), then, the opportunity set is a straight line

:

…


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….

Practice

Assume we are given the returns distribution for asset X and asset Y in six different states of the world as follows:

Calculate the mean returns, variance returns and covariance of returns for these two assets. We have the simple calculation below: …

…. Let us work through a practice problem to fix ideas:

Practice 1: Finding the slope of the CML

A portfolio is made up of two risky assets. Those assets and the Treasury rates are the only assets in the economy. The table below summarizes their expected returns and standard deviations:

You are asked to:

(a) (1 point) Find the mathematical expression for the portfolio’s Sharpe ratio. (b) (1 point) Find the optimal portfolio (in terms of the weights a and b) that optimizes the Sharpe ratio measure of performance.

(c) (1 point) Find the slope of the Capital Market Line (CML) passing through the risk-free asset and the optimal portfolio found in question (b) Solution


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a)

Sharpe Ratio of a two assets portfolio

….

Instead of holding this well-diversified portfolio (the market portfolio), the evidence suggests that:

Individual investors systematically fail to diversify their portfolios adequately, The median number of stocks held by investors in 2001 was four (4), 90% of investors held less than 10 different stocks, The holdings are concentrated in companies that are in the same industry or geographical

zone, This under-diversification is worldwide and not unique to USA investors only.

1.

Behavioral factors that explain the under-diversification of individual investors

Familiarity bias2.

: Investors prefer to invest in companies they are familiar with. Relative wealth concern: Investors care about the performance of their portfolios relative to that of their peers. As a result, they choose undiversified portfolios that match those of their peers.

…

Practice

Assume that we have gathered the values of the equity index for 5 years, where at the beginning of the first year, the index was at $100.000, then at the end of the first year, it was at $95.00 and so on. We are interested in calculating relevant statistics (variance, standard deviation, semi variance and semi standard deviation). We have:

Note the following about these two popular risk measures:


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….

Tail-Value at Risk (VaR)

The VaR estimate fails to convey information about how worse things are. Though the estimate returns a threshold portfolio value, it does not adequately capture and convey the magnitude of the portfolio values below that threshold. In other words, ‘we do not know how bad things can be’. The ….

The 5% (=α) expected shortfall for this risk is found as: …

VaR and Tail VaR for insurance losses

For insurance claim distributions, the adverse event is not to the left of the claim distribution but it is to the right of the claim distribution. In other words, while for an investment portfolio we are mainly concern about losses (or negative returns), for an insurance portfolio we are mainly… ….

The expected NPV upon deciding to produce films I and II is 125 (0.5 × 375 − 0.5 × 125), the same as the DTA without the information. The tree is:


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… What if managers of US have the flexibility to delay the production of film II until after film I is released and the information about it being a blockbuster or a moderate hit is available?

If the movies are produced in sequel, the total cost of production rises and is above 525. By waiting to produce film I after production of film I, the firm has better information about

film I and the book. Such information also affects the cost of production of film II. By production film I first, the cost of production for film I is 300.00.

…

Section-5-6) MM2 and the cost of capital of a levered firm As seen in PAK-section-2-5, the cost of capital of the levered firm is the weighted average cost of capital of its financing components (this is in a world without taxes):

𝒓𝑼 =𝑬

𝑬 + 𝑫× 𝒓𝑬 +

𝑫𝑬 + 𝑫

× 𝒓𝑫 Note1

: 𝒓𝑼 = 𝑻𝒉𝒆 𝒖𝒏𝒍𝒆𝒗𝒆𝒓𝒆𝒅 𝒄𝒐𝒔𝒕 𝒐𝒇 𝒄𝒂𝒑𝒊𝒕𝒂𝒍 𝒐𝒇 𝒂 𝒍𝒆𝒗𝒆𝒓𝒆𝒅 𝒇𝒊𝒓𝒎

𝑬 = 𝑻𝒉𝒆 𝑴𝒂𝒓𝒌𝒆𝒕 𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝒆𝒒𝒖𝒊𝒕𝒚 𝑫 = 𝑻𝒉𝒆 𝑴𝒂𝒓𝒌𝒆𝒕 𝒗𝒂𝒍𝒖𝒆 𝒐𝒇 𝒅𝒆𝒃𝒕 𝒓𝑬 = 𝑻𝒉𝒆 𝒆𝒒𝒖𝒊𝒕𝒚 𝒄𝒐𝒔𝒕 𝒐𝒇 𝒄𝒂𝒑𝒊𝒕𝒂𝒍 𝒓𝑫 = 𝑻𝒉𝒆 𝒆𝒒𝒖𝒊𝒕𝒚 𝒄𝒐𝒔𝒕 𝒐𝒇 𝒅𝒆𝒃𝒕

Notice that there is no corporate tax in this equation as of yet. The cost of equity (𝑟𝐸) is solved as2

:

1 This is equation 12.8 from the official reading. 2 This is equation 14.5 from the official reading.


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𝒓𝑬 =𝑬 + 𝑫𝑬

𝒓𝑼 −𝑫𝑬𝒓𝑫 = 𝒓𝑼 +

𝑫𝑬

(𝒓𝑼 − 𝒓𝑫) Thus:

The cost of equity for the levered firm increases with leverage. This is precisely the second proposition of Modigliani and Miller. In other words, with

perfect capital markets, a firm’s WACC (𝒓𝑼) is independent of its capital structure, or put another way,

Leverage has no effect on the overall cost of capital for the firm. In a perfect capital market

, as leverage increases (the ratio of debt to equity), the WACC (weighted average cost of capital) of the firm does not change, but only its cost of equity (𝒓𝑬) increases. The situation is depicted below:

Note that the cost of equity of a levered firm (𝒓𝑬) is now made up of two sources of risk:

Risk of the firm without leverage (𝒓𝑼) and The additional risk of the firm due to leverage (𝑫

𝑬(𝒓𝑼 − 𝒓𝑫)).

Let us work through some practice questions to illustrate the concept of MM-II.

Practice

Determine whether the following statement is true/false about MM-II proposition3

in a world without taxes:

…

… We conclude the following:

In a world with taxes/or without taxes, the cost of equity for a levered firm (𝑟𝐸) increases with the debt ratio.

….

3 The cost of equity-capital of a levered firm increases with the firm’s market value debt-equity ratio.


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The cost of debt (𝑟𝐷) and the cost of equity (𝑟𝐸) increase with leverage. As leverage increases, the weight put on the cost of debt increases and then is reduced by the tax rate.

The net effect is a reduction in the cost of capital.

Practice

Assume the firm in the prior practice question (The firm has a current debt cost of capital of 6% and its equity cost of capital is 13.85%). Its current debt-equity ratio is 3 and the corporate tax rate is 35%. What is the unlevered cost of capital of this firm? Because the… Let us work another practice problem to illustrate MM1 and MM2.

Practice

Assume that two firms operate in a world with zero corporate taxes. They have the following cash flow and capital structures:

You are also told that the unlevered cost of capital is 10%. The earnings and debt cash flows are assumed to be perpetuities.

What is the value of the levered firm?

What is the MV of debt and the MV of equity for the levered firm?

The MV of the debt is calculated as follows:

Calculate the cost of equity to the levered firm

Approach 1:


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The MV of equity is the PV of the cash flows to equityholders using the cost of equity as the

Calculate the overall WACC of the levered firm

Approach 1

However, in the world without corporate taxes, leverage increases the risk of the firm and thus increases the cost of equity of firm. …

Section-5-7) Computing the WACC using multiple securities If the firm capital structure is more complex, then the cost of capital of the firm (𝑟𝑊𝐴𝐶𝐶) is.. problems to fix ideas:

Practice

Assume the following information about two firms (a levered firm) and (an unlevered firm):

Confirm that given the beta of the unlevered firm of 1, the beta of the equity of the levered firm is indeed equal to 1.65 We have: …

Sample-PAK-Part-II-IFM-Exam 2018

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… Let us summarize the payoffs for the four ways to own an asset below:

Notice how (𝑭 = 𝑭𝑽�𝑭𝑷�). Let us work through some basic examples to fix the concept: …

Continuous dividend yield

If instead of paying a discrete dividend stream as above, the asset pays a continuous dividend with yield of (𝑞), then the prepaid-forward price becomes1

𝑭𝑷 = 𝑺(𝟎) × 𝒆−𝒒𝑻

:

And the no-arbitrage forward price is:

𝑭 = 𝑭𝑽�𝑭𝑷� = �𝑺(𝟎) × 𝒆−𝒒𝑻� × �𝒆𝒓𝑻� = 𝑺(𝟎) × 𝒆(𝒓−𝒒)𝑻 … Assuming that at maturity the underlying asset S(T) trades at different values other than 1200.00, we can now calculate the payoffs for the short and long call option and the profits for the long and short call options below:

1 Under this formula, the dividend amount (𝑞 × 𝑆(0)) is continuously paid throughout and reinvested at ……..


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Notice the following:

The short call option (the writer of the call option) has a liability only when the underlying asset (S(T)) exceeds the strike price (K=1000).

When the underlying asset is below the strike level (S(T)<K), the short position total profit is positive, but it never exceeds the FV(Premium) for the premium that was received for the position initially.

Likewise, when S(T) decreases to be below the strike (K), the long call is at a loss, but the loss canner decrease below the FV(premium).

Contrast the payoff/profit for the long call option to that of the long forward contract: For the long forward, when the underlying asset trades below the forward price (F), the payoff/profit of the long position can actually decrease to even reach the low value of (-F) at (S(T)=0). For a long call option, even if S(T)=0, the profit can never be below –FV(Premium).

…

Section-6-19) Life insurance application: Equity Index Based Insurance products

Variable annuities

A variable annuity is an investment-type product that is offered exclusively by a life insurance company in order to give protection to the policyholder. The insured/policyholder has invested …. But, the investor is aware that at the start of the retirement (aged 65-67), there is the possibility that he/she will not be able to purchase the desired annuity in the open market. Thus, the investor runs the risk of having an annual retirement income that is insufficient to him/her. As a result, the investor needs protection against the risk of not being able to use the proceeds of his/her separate …. The life insurance will then draft a contract called a variable annuity. The most important clause of the contract is that at the maturity of the variable annuity (or commencement of the retirement of the policyholder), the insurance company is responsible for making sure that the … The insurance company is the writer of a put option (has a short position in a put option) written on the value of the separate account:

The maturity of this option is the commencement of the retirement for the policyholder. The strike of this option, though unknown to both parties when signing the contract, is the

market value of the desired annuity at the maturity of the contract. ….

Let us briefly consider how the GMMB works: The policyholder is the owner of a separate account fund (𝐹𝑢𝑛𝑑(𝑇)). The policyholder is exposed to … Let us further illustrate the concept GMIB with a simple example. Let us assume that at contract inception we have gathered the following information from the investor:


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So, this investor, currently aged (x), has deposited $10.00 into an aggressive investment fund. After purchasing the GAO from the life insurer, the investor is guaranteed that when he/she retires (in n years) at age (x+n), he/she will be able to use the proceeds of the fund to purchase an annuity granting him/her an annual benefit of $1.20 for life. At this point, even if the proceeds of the fund is not sufficient to pay for the annuity, the investor does not care. With the GAO contract, the insurer is liable for financing any deficit. Therefore, the investor has successfully transferred the mortality/longevity risk and interest rate risk to the insurance company. Let us assume that at time t=n, the prevailing prices of annuities is as follows:

So, technically, the life insurance company is on the hook if the balance in the SA is below the MV of …

MOCK You have gathered the following data for a short sale (with margin requirements):

The short sale transaction is on 100 shares of XYZ stock. What is the cash amount that is available for withdrawal at the settlement (your cash profit)?

a) 192 b) 193 c) 194 d) 195


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MOCK Suppose the futures price on the S&P 500 index is $1000.00. The notional amount for one futures is $250,000. You long 10 futures contracts on the S&P 500 index. The initial margin is 10% of the position notional amount and you mark-to-market over a 10 week-period. The margin account credits continuous interest at 6% per annum. You are told that the Futures price on the index over the 10 weeks horizon is tabulated below:

The margin balance after 5 weeks is calculated as:

a) 310,000 b) 315,000 c) 320,000 d) 325,000 e) 330,000

…. At which point, we will expect early exercise of the put to be optimal (The premium is much smaller than the payoff). In fact, it can be very optimal to exercise American put option before the expiration of the option. We summarize the findings in the table: …

We illustrate with the same numerical example as follows:


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Note:

As the stock price increases, the payoff of the bull spread and the box spread are capped at the strike differential (𝐾2 − 𝐾1) while losses of the 2:1 ratio spread and the collar continue to increase.

Losses on the collar are somehow mitigated by the net premium (cash in-flow) whereas those on the 2:1 ratio are not mitigated.

The collar is beneficial when you expect the terminal asset price will not increase.

… Suppose that the put and the call options underlying the chooser contract have the same maturity (𝑇2) and the same strike price (K). Graphically…. You can further exploit the put-call parity between the instruments as of valuation time (𝑇1) to get the formula2

𝐦𝐚𝐱(𝒄,𝒑) = 𝒄 + 𝒆−𝒒(𝑻𝟐−𝑻𝟏)𝐦𝐚𝐱 (𝟎,𝑲𝒆−(𝒓−𝒒)(𝑻𝟐−𝑻𝟏) − 𝑺𝟏)

:

Practice Question 1

Suppose that the put and the call options underlying the chooser contract, and the chooser contract have the same maturity (𝑇1): Thus, at maturity (𝑇1), the payoff of the chooser is3

𝑀𝑎𝑥�𝑀𝑎𝑥�𝑆𝑇1 − 𝐾, 0�;𝑀𝑎𝑥�𝐾 − 𝑆𝑇1 , 0��

:

To complete this sub-section, we will illustrate the covered put insurance strategy with the numerical example. Example

:

We have gathered the following information about a stock price and options written on it:

2 This solves problem 14.20 (b) as required by the official syllabus. 3 This solves problem 14.20 (a) as required by the official syllabus.


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We want to calculate the profit of the covered put portfolio. We have:

….. MOCK The risk management area of your organization is looking to use linear interpolation to obtain the values of the American call and put options. Apparently the financial engineering software that was used could not produce the grid of option prices at all in-between strike levels. In order to identify the intern suited to carry on the work, you have asked them to sketch the slope of the linear interpolation that the/she intends to use for American call and put for three strikes level (𝐾1 < 𝐾2 < 𝐾3). In return, you obtained the following:


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Which of the following statement is accurate?

a) Only (a) and (c) are true b) Only (a) and (d) are true c) None of them is true d) Only (b) and (c) are true e) Only (b) and (d) are true

MOCK Based on a no-arbitrage pricing model, you have gathered the following data about barrier down-an-out options:


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Based on the above data which of the following is accurate?

MOCK You have observed the following call prices for three different strikes (out of a pricing model in your company). The model was produced by your manager and you have been asked to share your feedback on these findings:

Which of the following statements is accurate regarding arbitrage opportunities?

a) Apparently, there is no arbitrage opportunity to be made in this case, b) There is arbitrage opportunity and one could set up a butterfly spread with a minimum

payoff of -15 c) There is arbitrage opportunity and one could set up a butterfly spread with a minimum

payoff of -10 d) There is arbitrage opportunity and one could set up a butterfly spread with a minimum

payoff of -5 e) There is arbitrage opportunity and one could set up a butterfly spread with a minimum

payoff of 0

Pricing options on a Futures contract

Suppose that the current value of the Futures contract is $31.00. There is an American put option expiring in 9 months written on this Futures contract. The risk-free rate of interest is 5%. The binomial step is 3 months. The option is struck at $30.00, and the up level for the Futures is 1.1772 while the down level is 0.871.


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First, we calculate the RN up-probability of the futures price as follows:

𝒑∗ =𝒆(𝟎−𝟎)∆𝒕 − 𝒅

𝒖 − 𝒅=𝟏 − 𝒅𝒖 − 𝒅

= 𝟎.𝟒𝟐𝟏𝟑𝟐 To value this contract, you built the tree for the underlying asset (the futures contract with three time steps). In the graphics below, we built the tree and calculated the option value while checking for early exercise of the American put.

The American put option on this Futures contract is worth $2.73.

Section-8-13) Applying binomial models to price a gap option

To illustrate how binomial trees can be used to value gap option, let us work through one example:

Gap call option

In PAK-section-7-27, we defined gap options. Assume we want to price the following European gap put option written on an underlying asset:


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…

MOCK You have built a two-steps binomial tree to price a European call option on an underlying asset as follows:

In this tree, you have conveniently calculated the bond component and the stock component of your replicating portfolio at the end of the first time step: On the up, you have B=-93.469 and Delta*stock=109.763, while on the low, B=-22.204 and delta*stock=24.7717. The underlying asset is currently priced at $100.00. The dividend yield on this stock is 2.75%. The interest rate is 6.5% per annum. The option expires in six months, and the time step is a quarter of a year. All you know is that the strike price exceeds or equal 90. Which of the following is accurate about this situation?


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Section-9-7) The conditional expected price under the lognormal model We now calculate the conditional expectations under the lognormal model:

𝐸(𝑆𝑇|𝑆𝑇 > 𝐾) In the statistical literature, the partial expectation of the random variable (𝑆𝑇) with threshold (𝐾) is defined as:

𝒈(𝑲) = 𝑬(𝑺𝑻|𝑺𝑻 > 𝐾) × 𝑷𝒓𝒐𝒃(𝑺𝑻 > 𝐾) For the lognormal distribution of parameters (𝜇,𝜎2), the partial expectation is well-known. It is given by:

𝒈(𝑲) = 𝒆𝝁+𝟏𝟐𝝈

𝟐× 𝑵�

𝝁 + 𝝈𝟐 − 𝒍𝒏(𝑲)𝝈

�

For the distribution of (𝑆𝑇), we adjust as follows:

𝐸(𝑆𝑇|𝑆𝑇 > 𝐾) × 𝑃𝑟𝑜𝑏(𝑆𝑇 > 𝐾)

= (𝑆0) × 𝑒�𝛼−𝛿−12𝜎

2�𝑇+12𝑇𝜎2

× 𝑁�𝑙𝑛(𝑆0) − 𝑙𝑛(𝐾) + �𝛼 − 𝛿 − 1

2𝜎2� 𝑇 + 𝜎2𝑇

𝜎√𝑇�

Thus:

𝐸(𝑆𝑇|𝑆𝑇 > 𝐾) = �𝑆0

𝑃𝑟𝑜𝑏(𝑆𝑇 > 𝐾)� × �𝑒(𝛼−𝛿)𝑇 × 𝑁�𝑙𝑛(𝑆0) − 𝑙𝑛(𝐾) + �𝛼 − 𝛿 + 1

2𝜎2� 𝑇

𝜎√𝑇��

Recall that:

𝑃𝑟𝑜𝑏(𝑆𝑇 > 𝐾) = 1 − 𝑁�−�̂�2� = 𝑁��̂�2� Where:

�̂�2 =𝑙𝑛 �𝑆0𝐾� + �𝛼 − 𝛿 − 1

2𝜎2� × 𝑇

𝜎√𝑇

Also, Let:

�̂�1 =𝑙𝑛 �𝑆0𝐾� + �𝛼 − 𝛿 + 1

2𝜎2� × 𝑇

𝜎√𝑇

We now have:

𝐸(𝑆𝑇|𝑆𝑇 > 𝐾) = 𝑆0 × 𝑒(𝛼−𝛿)𝑇 ×𝑁��̂�1�𝑁��̂�2�

With the same reasoning, we have:


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𝐸(𝑆𝑇|𝑆𝑇 < 𝐾) = 𝑆0 × 𝑒(𝛼−𝛿)𝑇 ×𝑁�−�̂�1�𝑁�−�̂�2�

Therefore: ….

… Assume that we have an underlying exchange rate (𝑥0) as the units of domestic currency per unit of foreign currency. We want to write plain vanilla options on this exchange rate. We have the strike exchange rate (K) as the strike units of domestic currency per unit of foreign currency. Assume that the volatility of this exchange rate is (σ) per annum. The options (put and call) expire in T years. The domestic risk-free rate of interest is (𝑟𝐷) and the foreign risk-free rate of interest is (𝑟𝐹). As explained in (PAK-section-7-6), to value these options, it is convenient to treat the exchange rate (𝑥0) as a domestic dividend-paying asset of dividend yield equal to the foreign risk-free rate (𝑟𝐹). Thus, the Black Scholes option prices are:

𝒄 = �𝒙𝟎𝒆−𝑟𝐹𝑻�𝑵(𝒅𝟏) −𝑲𝒆−𝑟𝐹×𝑻𝑵(𝒅𝟐) While:

𝒑 = 𝑲𝒆−𝑟𝐹×𝑻𝑵(−𝒅𝟐) − �𝒙𝟎𝒆−𝑟𝐹𝑻�𝑵(−𝒅𝟏) Where:

𝑑1 =𝑙𝑛 �𝑥0𝐾 � + �𝑟𝐷 − 𝑟𝐹 + 1

2𝜎2� × 𝑇

𝜎√𝑇

And 𝑑2 = 𝑑1 − 𝜎√𝑇

In PAK-section-7-6, we showed how put call parity works for currencies options. Let us now show how to values these options: We have: ….

Section-9-16) Black Scholes model for equity-based Life insurance liabilities

One of the most important life insurance application if Black Scholes is in the area of modeling the equity embedded derivatives in variable annuities and equity index annuities. In PAK-section-8-12, we describe how the options embedded in the European put option of a variable annuity with a guaranteed maturity benefit and the European call option of an equity index annuity can be valued via binomial trees. MOCK-9 and MOCK-8 of PAK-section-8 provide examples of binomial trees implementation. Though an advanced treatment of the Black-Scholes application to life insurance is way beyond the scope of this exam, it is however important to be able to use the model and price basic contracts.

Point-to-point equity-index annuity

The insurance company offers a Single Premium Deferred Annuity (SPDA). The policyholder pays the fixed premium to the insurance company (Premium). The company guarantees that 95% of the


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….. We clearly see that the insurer’s initial liability is made up of two components:

o A fixed portion that evolves like a bond and o A derivative component that is in fact an equity index call option

Therefore, to fulfill its future obligations, the life insurer might want to purchase these instruments. That is why we are interested in finding the Black Scholes value of the derivative portion of the insurer’s liability. In this sub-section, we show how we can use the Black Scholes call option to value the derivative component of the life insurer’s liability: …. What about the other guarantees (PAK-section-7)? First let us look at the mathematics underlying each of the following guarantees: …. We have:

𝐆𝐌𝐌𝐁 𝐯𝐚𝐥𝐮𝐞 = 𝑷𝒓𝒐𝒃(𝑻∗(𝒙) > 𝑇) × �(𝟏 −𝒎)𝑻 × 𝑭𝟎

𝑺𝟎× 𝑩𝑺𝑷𝟎 �𝑻,

𝑮 × 𝑺𝟎(𝟏 −𝒎)𝑻 × 𝑭𝟎

, 𝑺��

… Using the law of total probability and ignoring lapsation, the value of the GMMD is calculated as follows:

𝐆𝐌𝐃𝐁 𝐯𝐚𝐥𝐮𝐞 = � 𝑩𝑺𝑷𝟎(𝒕,𝑮, 𝑺) × 𝒇𝑻(𝒙)(𝒕)𝒅𝒕∞

𝟎

Where: 𝒇𝑻(𝒙)

= 𝑻𝒉𝒆 𝒑𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚 𝒅𝒆𝒏𝒔𝒊𝒕𝒚 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝒐𝒇 𝒕𝒉𝒆 𝒇𝒖𝒕𝒖𝒓𝒆 𝒍𝒊𝒇𝒆𝒕𝒊𝒎𝒆 𝒐𝒇 (𝒙) 𝒂𝒕 𝒊𝒏𝒄𝒆𝒑𝒕𝒊𝒐𝒏 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒐𝒏𝒕𝒓𝒂𝒄𝒕 …. Let us work through a practice problem to fix the concept.

Practice

A life aged (x) is interested in purchasing a 4-year Variable annuity with two types of guarantees:

The GMMB (that pays the initial fund value at maturity of the contract upon survival of (x)) and

The GMDB that pays a death benefit equal to the initial fund value at the end of the year o death shall death occurs during the 5 years term of the contract.

The insurer intends for the guarantees to be funded at the inception of the contract. The initial fund value of the separate account is $100.00. The volatility of the separate account is 30%. The annual rate of interest is 4%.


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Based on information about the policyholder, the insurer has estimated the survival probability to time T and the value of the Black Scholes Put option written on his/her separate account maturating at future time T below:

Calculate the total cost of the guarantees that must be paid by the policyholder at the inception of the contract. ….. MOCK You want to compare the ability and usage of the Binomial Tree model (BT) and the Black Scholes Merton model (BSM) for valuing options. You are given the following grid:

Where you will interpret each statement and think through as to whether the statement is false or accurate. For instance, statement 1 is accurate, it says that the BSM and the BT are able to price plain vanilla European Options. ….

… The Vega of a long position in a European or an American option is always positive. To visualize it, we carry on with the same example and plot: Let us work through an illustrative example where we will Gamma and Vega hedge a portfolio using ….

Section-10-11) Extension of the Greeks: Dividend paying assets, currency options, forward

and futures How does the formula for the Greeks change when we consider a dividend paying stock of yield q? The table below summarizes the answer:


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….

…. Day 1

:

The maturity of the option is reduced by 1 day to be 90/365 and the European call option is now worth $3.0621. Ignoring the calculation for the new delta of the option at this point, we have at the end of this day:

𝐺𝑎𝑖𝑛 𝑜𝑛 𝑡ℎ𝑒 100 𝐸𝑢𝑟𝑜𝑝𝑒𝑎𝑛 𝑐𝑎𝑙𝑙 𝑜𝑝𝑡𝑖𝑜𝑛𝑠 = $278.04 − 306.21 = −28.17 𝐺𝑎𝑖𝑛 𝑜𝑛 𝑡ℎ𝑒 58.24 𝑠ℎ𝑎𝑟𝑒𝑠 𝑜𝑓 𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝑎𝑠𝑠𝑒𝑡 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒𝑑 𝑜𝑛 𝑑𝑎𝑦 0 = 58.24 × (40.50 − 40) = 29.12

𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑖𝑛𝑐𝑢𝑟𝑟𝑒𝑑 𝑓𝑟𝑜𝑚 𝑑𝑎𝑦 0 𝑡𝑜 𝑑𝑎𝑦 1 = −0.449 Finally, the overnight profit4

29.12 − 28.17 − 0.449 = $𝟎.𝟓𝟎

at end of day 1 is calculated as:

What if we consider rebalancing on day 1? The new delta of the option is 0.6142. ….

Section-10-19) Delta hedging for several days

Using the formula from the prior section, it is pretty straightforward to obtain the following table:

The official reading calculates the capital gain (Change in MV-Additional Investment in Stock) and works out the daily profit as (Capital Gain – Interest Cost). So, the official reading uses the approach: 4 This is the amount of cash that we can pocket if there is a profit at end of day 1, or the amount of cash that we must pay if there is a loss.


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𝑇ℎ𝑒 𝑚𝑎𝑟𝑘 𝑡𝑜 𝑚𝑎𝑟𝑘𝑒𝑡 𝑝𝑟𝑜𝑓𝑖𝑡 𝑓𝑜𝑟 𝑑𝑎𝑦 (𝑖) = [(𝑀𝑉𝑖 − 𝑀𝑉𝑖−1) − (𝑆𝑖) × (∆𝑖 − ∆𝑖−1)] − (𝑟 × 𝑀𝑉𝑖−1)

Where:

[(𝑴𝑽𝒊 −𝑴𝑽𝒊−𝟏) − (𝑺𝒊) × (∆𝒊 − ∆𝒊−𝟏)] = 𝑻𝒉𝒆 𝒄𝒂𝒑𝒊𝒕𝒂𝒍 𝒈𝒂𝒊𝒏 Or essentially:

𝑇ℎ𝑒 𝑚𝑎𝑟𝑘 𝑡𝑜 𝑚𝑎𝑟𝑘𝑒𝑡 𝑝𝑟𝑜𝑓𝑖𝑡 𝑓𝑜𝑟 𝑑𝑎𝑦 (𝑖) = 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝑔𝑎𝑖𝑛(𝑖) − 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑐𝑜𝑠𝑡(𝑖) While we use:

𝑇ℎ𝑒 𝑚𝑎𝑟𝑘 𝑡𝑜 𝑚𝑎𝑟𝑘𝑒𝑡 𝑝𝑟𝑜𝑓𝑖𝑡 𝑓𝑜𝑟 𝑑𝑎𝑦 (𝑖) = ∆𝑖−1 × (𝑆𝑖 − 𝑆𝑖−1) − (𝐶𝑖 − 𝐶𝑖−1) − (𝑟 × 𝑀𝑉𝑖−1) The Delta-Gamma approximation is pretty good and we not calculate delta for the other level of the stock price. Gamma and Delta are calculated for S=40. We formalize the Delta-Gamma approximation into something that looks like the Taylor expansion as follows:

Delta-Gamma approximation Let us assume that over a small time interval of length (h), the stock price moves by an amount (ε) from (𝑆𝑡) to (𝑆𝑡+ℎ = 𝑆𝑡 + 𝜀). Let:

𝐶(𝑆𝑡) = 𝑇ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝐸𝑢𝑟𝑜𝑝𝑒𝑎𝑛 𝑐𝑎𝑙𝑙 𝑜𝑝𝑡𝑖𝑜𝑛 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡 𝑢𝑠𝑖𝑛𝑔 (𝑆𝑡)

∆(𝑆𝑡) = 𝑇ℎ𝑒 𝑑𝑒𝑙𝑡𝑎 𝑜𝑓 𝑡ℎ𝑒 𝐸𝑢𝑟𝑜𝑝𝑒𝑎𝑛 𝑐𝑎𝑙𝑙 𝑜𝑝𝑡𝑖𝑜𝑛 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡 𝑢𝑠𝑖𝑛𝑔 (𝑆𝑡)

𝛤(𝑆𝑡) = 𝑇ℎ𝑒 𝐺𝑎𝑚𝑚𝑎 𝑜𝑓 𝑡ℎ𝑒 𝐸𝑢𝑟𝑜𝑝𝑒𝑎𝑛 𝑐𝑎𝑙𝑙 𝑜𝑝𝑡𝑖𝑜𝑛 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡 =∆(𝑆𝑡+ℎ) − ∆(𝑆𝑡)

𝜀

The approximated value of the option is:

𝑪(𝑺𝒕+𝒉) = 𝑪(𝑺𝒕) + 𝜺 × ∆(𝑺𝒕) +𝟏𝟐

× 𝝐𝟐 × 𝜞(𝑺𝒕) Notice: …. Let us work through a practice problem in order to fix ideas:

Practice

Consider the following information about a levered firm and the market:


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Assume that there is no corporate tax in this economic system. Find the MV of the firm’s equity and the MV of the firm’s debt

MOCK You are in the Black Scholes economy. There is only one stock on which two derivatives instruments are considered. Based on a numerical method (Monte Carlo for instance), you have estimated the following Greeks for the two European options:

You are also given the following information:

The values for option 1 and option 2 are:

foreword for the pak study notes (ifm exam)

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