CM2 Summary Exam 2023 PDF

CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE CM2 LOSS RESERVING AND FINANCIAL ENGINEERING SUMMARY EXAM FOR 2023 CA PRAVEEN PATWARI 0 JAI SHREE RAM CM2 SUMMARY...

CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE CM2 LOSS RESERVING AND FINANCIAL ENGINEERING SUMMARY EXAM FOR 2023 CA PRAVEEN PATWARI 0 JAI SHREE RAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE CM2-SUMMARY INDEX QUESTIONS CHAPTER NO DETAILS PAGE NO 1. THE EFFICIENT MARKETS HYPOTHESIS 2-6 2. UTILITY THEORY 7-17 3. STOCHASTIC DOMINANCE AND BEHAVIOURAL FINANCE 18-27 4. MEASURES OF INVESTMENT RISK 28-34 5. STOCHASTIC MODELS OF INVESTMENT RETURNS 35-40 6. PORTFOLIO THEORY 41-47 7. MODELS OF ASSET RETURNS 48-53 8. ASSET PRICING MODELS 54-59 9. BROWNIAN MOTION AND MARTINGALES 60-69 10. STOCHASTIC CALCULUS AND ITO PROCESSES 70-79 11. STOCHASTIC MODELS OF SECURITY PRICES 80-82 12. CHARACTERISTICS OF DERIVATIVE SECURITIES 83-93 13. THE GREEKS 94-98 14. THE BINOMIAL MODEL 99-109 15. THE BLACK-SCHOLES OPTION PRICING FORMULA 110-121 16. THE 5-STEP METHOD IN DISCRETE TIME 122-130 17. THE 5-STEP METHOD IN CONTINUOUS TIME 131-145 18. THE TERM STRUCTURE OF INTEREST RATES 146-152 19. CREDIT RISK 153-162 20. RUIN THEORY 163-176 21. RUN-OFF TRIANGLES 177-183 CA PRAVEEN PATWARI 1 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE CHAPTER THE EFFICIENT MARKETS 1 HYPOTHESIS Q1. State what is meant by an efficient market. SOLUTION Efficient market An efficient market is one in which security prices quickly and accurately reflect all available and re- levant information. Q2. Describe the three main forms of the Efficient Markets Hypothesis (EMH). SOLUTION Three main forms of the EMH 1. Strong form EMH: market prices incorporate all information, both publicly available and also that available only to insiders. 2. Semi-strong form EMH: market prices incorporate all publicly available information. 3. Weak form EMH: the market price of an investment incorporates all information contained in the price history of that investment. Q3. State the key implication of the EMH for investment returns. SOLUTION Key implication of the EMH for investment returns The EMH suggests that it is impossible for investors to systematically generate excess risk-adjusted investment returns. However, it may be possible to systematically generate higher absolute investment returns by tak- ing additional risk. CA PRAVEEN PATWARI 2 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q4. What is the implication for investment management if markets are efficient? SOLUTION Implication for investment management of market efficiency If markets are efficient, then all securities will be correctly priced and there will be no mispricings. So, active investment management, which aims to identify and exploit any mispricings, will be of no use Instead, investment management should be passive and aim to diversify across a whole market, eg tracking an index. Q5. State the implications for investment analysis of each of the three forms of the EMH. SOLUTION Implications for investment analysis of the three forms of the EMH 1. The strong form EMH implies that private or insider information is of no use in generating excess risk-adjusted returns. So, insider trading cannot add value. 2. The semi-strong form EMH implies that fundamental analysis based on publicly available ac- counting and economic data cannot add value. 3. The weak form EMH implies that technical analysis (or chartism) based on historical market data cannot add value. Q6. State a particular difficulty of testing: (i) the strong form of the EMH (ii) the semi-strong form of the EMH (iii) the weak form of the EMH. SOLUTION Difficulties with testing each form of the EMH A particular difficulty with testing: (i) the strong form of the EMH is that inside information is not generally available. CA PRAVEEN PATWARI 3 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE (ii) the semi-strong form of the EMH is that it is not always clear whether information is public or private, eg professional fund managers are able to gain access to the senior management of companies, but individual investors are not. (iii) the weak form of the EMH is that it is impossible to test all the different trading strategies sug- gested by technical analysis, in order to see if it can add value. Q7. State five other difficulties with testing the EMH. SOLUTION Five other difficulties with testing the EMH 1. The EMH does not tell us how the market should react to new information. 2. It is not always clear when new information arrives. 3. Allowance must be made for risk. 4. Allowance must be made for taxes, transaction costs and research costs. 5. The tests used rely on (possibly invalid) statistical assumptions. Q8. Summarise the results of studies of: (i) the strong form of the EMH (ii) the weak form of the EMH. SOLUTION Studies of the EMH (i) Strong form Studies of directors’ share dealings suggest that even with inside information it is difficult to out-perform. (ii) Weak form Studies have failed to identify a difference between the returns on stocks using technical anal- ysis and those from purely random stock selection after allowing for transaction costs. Credible challenge has emerged to the weak form EMH from econometric research suggesting that stocks tend to exhibit short-term momentum effects (trending in the same direction) and medium-term mean-reversion (trending in the opposite direction). CA PRAVEEN PATWARI 4 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q9. State the two main categories of tests of the semi-strong form of the EMH. SOLUTION Categories of tests of semi-strong form EMH Tests of informational efficiency - whether information is incorporated efficiently into prices, which therefore react quickly and accurately to new information. Tests of excessive volatility - whether prices are excessively volatile, ie the change in the market value of stocks cannot be justified by the arrival of new information. Q10. Give three examples of over-reaction to events that appear to contradict informa- tional efficiency. SOLUTION Over-reactions that contradict informational efficiency 1. The market appears to over-react to past performance. 2. Certain accounting ratios appear to have predictive powers. 3. New share issues have poor subsequent long-term performance. Q11. Give three examples of under-reaction to events that appear to contradict informa- tional efficiency. SOLUTION Under-reactions that contradict informational efficiency 1. Stock prices continue to respond to earnings announcements up to a year after their an- nouncement. 2. Abnormal excess returns follow de-mergers. 3. Abnormal negative returns follow mergers. Q12. What is an ‘excessively volatile’ market? Describe how Shiller tested whether the equity market is excessively volatile. SOLUTION Excessively volatile market and Shiller’s test for excessive volatility An excessively volatile market is one in which the change in the market value of stocks cannot be justified by the arrival of new information. CA PRAVEEN PATWARI 5 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Shiller used a discounted cashflow model of equities based on the actual dividends paid and a ter- minal value for the stock to calculate the perfect foresight price, the ‘correct’ equity price if market participants had been able to predict future dividends correctly. If markets are efficient, broad movements in the perfect foresight price should be correlated with moves in the actual price, as both react to the same news. He found strong evidence that the observed level of volatility contradicted the EMH, ie that actual prices were much more volatile than perfect foresight prices. However, subsequent studies using different formulations found that the violation of the EMH only had borderline statistical signific- ance. Q13. State four criticisms of Shiller’s test of excessive volatility. SOLUTION Criticisms of Shiller’s test of excessive volatility Criticisms of Shiller’s methodology include: 1. the choice of terminal value for the stock price 2. the use of a constant discount rate 3. bias in the estimates of the variances because of autocorrelation 4. possible non-stationarity of the series, ie the series may have stochastic trends that invalidate the measurements obtained for the variance of the stock price. CA PRAVEEN PATWARI 6 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE CHAPTER UTILITY THEORY 2 Q1. Define ‘utility’. SOLUTION Definition of utility Utility is the satisfaction that an individual obtains from a particular course of action, eg consuming a good. Q2. State the expected utility theorem. SOLUTION The expected utility theorem The expected utility theorem states that a function, U(w), can be constructed representing an inves- tor’s utility of wealth, w, at some future date. Decisions are made on the basis of maximising the expected value of utility under the investor’s particular beliefs about the probability of different outcomes. ___________________ The expected utility yielded by investment in a risky asset is given by: N E  U    pi U  w i  i 1 where a set of N possible outcomes  w1 ,....w N  , has associated probabilities of occurring of  p1,....p N . CA PRAVEEN PATWARI 7 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q3. State the names of four axioms from which the expected utility theorem can formally be derived. SOLUTION Derivation of expected utility theorem 1. Comparability 2. Transitivity 3. Independence 4. Certainty equivalence The four axioms listed above are not the only possible set, but they are the most commonly used. Q4. State the definition of: (i) comparability (ii) transitivity SOLUTION Definitions (i) Comparability An investor can state a preference between all available certain outcomes. (ii) Transitivity Suppose that A, B and C are three available certain outcomes. If A is preferred to B and B is preferred to C, then A is preferred to C. Q5. State the definition of: (i) independence (ii) certainty equivalence. CA PRAVEEN PATWARI 8 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE SOLUTION Definitions (i) Independence Suppose that an investor is indifferent between two outcomes, A and B. The investor is then also indifferent between the following two gambles:  A with probability p and C with probability (1 – p), and  B with probability p and C with probability (1 – p). (ii) Certainty equivalence Suppose that A is preferred to B and B is preferred to C. Then there is a unique probability, p, such that the investor is indifferent between B and a gamble giving A with probability p and C with probability (1 – p). B is known as the certainty equivalent of the above gamble Q6. Define non-satiation and express it in terms of an investor's utility function, U(w). SOLUTION Non-satiation The principle of non-satiation is the assumption that people prefer more wealth to less. It can be expressed as: U’(w) > 0 Q7. Define what it means for an investor to be: (i) risk-averse (ii) risk-seeking (iii) risk-neutral and express each of these as a condition on the utility function, U(w). CA PRAVEEN PATWARI 9 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE SOLUTION Risk aversion A risk-averse investor values an incremental increase in wealth less highly than an incremental de- crease and will reject a fair gamble. The utility function condition is: U’’(w) < 0 A risk-seeking investor values an incremental increase in wealth more highly than an incremental decrease and will seek a fair gamble. In this case: U’’(w) > 0 A risk-neutral investor is indifferent between a fair gamble and the status quo. In this case: U’’(w) = 0 Q8. What can we say about the certainty equivalent for a fair gamble for a risk-averse in- vestor? SOLUTION Risk aversion and the certainty equivalent For a risk-averse investor, the certainty equivalent of a fair gamble is negative, ie the investor would have to be paid to accept the gamble. Q9. Describe how the certainty equivalent of a gamble is related to absolute and relative risk aversion. SOLUTION The certainty equivalent of a gamble and absolute risk aversion If the absolute value of the certainty equivalent decreases with increasing wealth, the investor is said to exhibit declining absolute risk aversion. If the absolute value of the certainty equivalent in- creases with increasing wealth, the investor exhibits increasing absolute risk aversion. The certainty equivalent of a gamble and relative risk aversion If the absolute value of the certainty equivalent decreases (increases) as a proportion of total wealth as wealth increases, the investor is said to exhibit declining (increasing) relative risk aver- sion. CA PRAVEEN PATWARI 10 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q10. Describe how the amount of wealth invested in risky assets is related to absolute and relative risk aversion. SOLUTION The amount invested in risky assets and absolute risk aversion Investors who hold an increasing absolute amount of wealth in risky assets as they get wealthier exhibit declining absolute risk aversion. Investors who hold a decreasing absolute amount of wealth in risky assets as they get wealthier ex- hibit increasing absolute risk aversion. The amount invested in risky assets and relative risk aversion Investors who hold an increasing proportion of their wealth in risky assets as they get wealthier exhibit declining relative risk aversion. Investors who hold a decreasing proportion of their wealth in risky assets as they get wealthier ex- hibit increasing relative risk aversion. Q11. Define absolute and relative risk aversion in terms of the utility function U(w). SOLUTION Absolute risk aversion U"  w  Aw   U ' w  Relative risk aversion U"  w  R  w   wA  w    w U ' w  These formulae appear on page 43 of the Tables. Q12a. For the quadratic utility function: (i) State its simplified form. (ii) State, with reasons, conditions on the parameter value and on wealth. (iii) State the type of absolute and relative risk aversion exhibited. CA PRAVEEN PATWARI 11 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE SOLUTION Quadratic utility function U  w   w  dw 2 (i) (ii) For risk-aversion to hold, we require d < 0. 1 For non-satiation to hold, we require   w   2d (iii) The quadratic utility function exhibits both increasing absolute and relative risk aversion (be- cause the first derivatives of A(w) and R(w) are positive). Q12b. Demonstrate that the quadratic utility function exhibits both increasing absolute and relative risk aversion. SOLUTION Quadratic utility function – absolute risk aversion  U ''  w  2d Aw   U ' w  1  2dw 2 4d A ' w    0 , ie increasing 1  2dw  2 Quadratic utility function – relative risk aversion  U ''  w  2dw R w  w  U ' w  1  2dw 2 2d 4d w 2d R ' w      0 , ie increasing 1  2dw 1  2dw  2 1  2dw  2 Q13a. For the log utility function: (i) State its form. (ii) State the type of absolute and relative risk aversion exhibited. CA PRAVEEN PATWARI 12 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE SOLUTION Log utility function (i) U  w   In  w  , w 0 (ii) The log utility function exhibits decreasing absolute risk aversion (ie A'(w) < 0) and constant relative risk aversion (ie R‘(w) = 0). Q13b. Demonstrate that the log utility function exhibits decreasing absolute risk aversion and constant relative risk aversion. SOLUTION Log utility function – absolute risk aversion  U ''  w  1 Aw   U ' w  w 1 A ' w    2  0 , ie decreasing w Log utility function – relative risk aversion  U ''  w  R w  w 1 U ' w  R '  w   0 , ie constant Q14a. For the power utility function: (i) State its form. (ii) State, with reasons, a condition on the parameter value γ (iii) State the type of absolute and relative risk aversion exhibited. SOLUTION Power utility function  w 1 (i) Uw  , w 0  CA PRAVEEN PATWARI 13 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE (ii) For risk-aversion to hold, we require   1. (iii) The power utility function exhibits decreasing absolute risk aversion (ie A'(w) < 0) and con- stant relative risk aversion (ie R'(w) = 0). Q14b. Demonstrate that the power utility function exhibits decreasing absolute risk aver- sion and constant relative risk aversion. SOLUTION Power utility function – absolute risk aversion  U ''  w     1 Aw   U ' w  w    1 A ' w   2  0 , ie decreasing w Power utility function – relative risk aversion  U ''  w  R w  w      1 U ' w  R '  w   0 , ie constant Q15. (i) What is the defining property of an iso-elastic utility function? (ii) Why are such functions useful? (iii) Give two examples of iso-elastic utility functions. SOLUTION Iso-elastic utility functions (i) An iso-elastic utility function is one that exhibits constant relative risk aversion, ie R'(w) = 0. (ii) The use of iso-elastic utility functions simplifies the determination of an optimal strategy for a multi-period investment decision because it is possible to make a series of so-called ‘myopic’ decisions. That is, the decision at the start of each period only has to consider the possible out- comes at the end of that period and can ignore subsequent periods. (iii) The log and power utility functions are iso-elastic. CA PRAVEEN PATWARI 14 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q16. When would a state-dependent utility function be used? Give an example. SOLUTION State-dependent utility functions State-dependent utility functions can be used to model the situation where there is a discontinuous change in the state of the investor at a certain level of wealth. Such a situation arises for an insurance company that will become insolvent if the value of its assets falls below a certain level. At asset levels just above the insolvency position, the company will be highly risk-averse and this can be modelled by a utility function that has a discontinuity at the in- solvency point. However, the consequence of applying the same utility function when the company has just become insolvent would be that the company would be prepared to accept a high probability of losing its remaining assets for a chance of regaining solvency. This is unlikely to reflect reality and so a different form of utility function would be required to model the company’s behaviour in this state. Q17. Describe one approach for constructing utility functions SOLUTION Construction of utility functions One approach that has been proposed is to devise a series of questions that allow the shape of an individual’s utility function to be roughly determined. A utility curve of a predetermined functional form can then be fitted by a least squares method to the points determined by the answers to the questions. The curve fitting is constrained by the requirement that the function has the desired economic properties (non-satiation, risk aversion and, perhaps, declining absolute risk aversion). CA PRAVEEN PATWARI 15 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q18. State the formula for finding the maximum premium P that a person with initial wealth a will be prepared to pay to insure against a random loss X. Explain what each side of the formula represents. SOLUTION Maximum premium the insured will be prepared to pay The maximum premium is given by the solution to the equation: E[U(a – X)] = U(a – P) LHS = expected utility of wealth without insurance RHS = certain utility of wealth with insurance Q19. State the formula for finding the minimum premium Q that an insurer with initial wealth b will be prepared to charge to insure against a risk with random loss X. Explain what each side of the formula represents. SOLUTION Minimum premium the insurer will be prepared to charge The minimum premium is given by the solution to the equation: E[U(b + Q – X)] = U(b) LHS = expected utility of wealth with insurance contract RHS = certain utility of wealth without taking on the insurance contract Q20. Describe the situation in which an insurance contract is feasible SOLUTION Feasible insurance contract An insurance contract is feasible if the minimum premium the insurer is prepared to charge is less than the maximum premium the policyholder is prepared to pay. CA PRAVEEN PATWARI 16 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q21. State three limitations of utility theory. SOLUTION Limitations of utility theory 1. To calculate expected utility, we need to know the precise form and shape of the individual's utility function. Typically, we do not have such information. 2. The expected utility theorem cannot be applied separately to each of several sets of risky choices facing an individual. 3. For corporate risk management, it may not be possible to consider a utility function for the firm as though the firm was an individual. This is because the stakeholders in the firm will have different stakes (both in the firm and in other assets) and different preferences, all of which will be unknown to the firm. CA PRAVEEN PATWARI 17 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE CHAPTER STOCHASTIC DOMINANCE 3 AND BEHAVIOURAL FINANCE Q1. Define absolute dominance SOLUTION Absolute dominance Absolute dominance exists when one investment portfolio provides a higher return than another in all possible circumstances. Q2. Consider two investment portfolios, A and B, with cumulative probability distribu- tion functions of returns FA and FB respectively. State the first-order stochastic dominance theorem and give an example. SOLUTION First-order stochastic dominance The first-order stochastic dominance theorem states that, assuming an investor prefers more to less, A will dominate B (ie the investor will prefer Portfolio A to Portfolio B) if: FA  x   FB  x  , for all x, and FA  x   FB  x  , for some value of x. For example, if two normal distributions have the same variance but different means, the one with the higher mean displays first-order stochastic dominance over the other. Q3. Consider two investment portfolios, A and B, with cumulative probability distribu- tion functions of returns FA and FB respectively. State the second-order stochastic dominance theorem and give an example. CA PRAVEEN PATWARI 18 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE SOLUTION Second-order stochastic dominance The second-order stochastic dominance theorem applies when the investor is risk-averse, as well as preferring more to less. In this case, the condition for A to dominate B is that: a FA  y  dy  a FB  y  dy, for all x, x x with strict inequality holding for some value of x. Here a is the lowest return that the portfolios can possibly provide. For example, if two normal distributions have the same mean but different variances, the one with the lower variance displays second-order stochastic dominance over the other. Q4. Describe what is meant by the term ‘behavioural finance’. SOLUTION Behavioural finance definition The field of behavioural finance looks at how a variety of mental biases and decision-making errors affect financial decisions. It relates to the psychology that underlies and drives financial decision- making behavior. Q5. State the two phases of decision-making as described in prospect theory. SOLUTION Prospect theory – two phases of decision-making 1. Editing/framing phase - where outcomes of a decision are initially appraised and ordered 2. Evaluation phase - choosing among the appraised options. CA PRAVEEN PATWARI 19 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q6. Describe the editing/framing phase of decision-making in prospect theory. SOLUTION Prospect theory – editing/framing phase Editing leads to a representation of the acts, outcomes and contingencies associated with a particu- lar choice problem. It involves a number of basic operations that simplify and provide context for the choice, including:  Acceptance - people are unlikely to alter the formulation of the choices presented  Segregation -people tend to focus on the most ‘relevant’ factors of a decision problem. Within this process, framing refers to the way in which a choice can be affected by the order or manner in which it is presented. Standard economic theory implies that framing should have no substantive effect on decisions, although some evidence from research suggests otherwise. Q7. List eight behavioural patterns that are observed in the evaluation phase of decision- making in prospect theory. SOLUTION Prospect theory – evaluation phase – eight behavioural patterns 1. Reference dependence 2. Loss aversion 3. Endowment effects 4. Changing risk attitudes 5. Diminishing sensitivity 6. Probability weighting 7. Certainty effect 8. Isolation effect Q8. Describe what is meant by ‘reference dependence' in the context of prospect theory. SOLUTION Prospect theory – evaluation phase – reference dependence Reference dependence relates to the idea that people derive utility from gains and losses measured relative to some reference point, rather than from absolute levels of wealth. This emerges from the CA PRAVEEN PATWARI 20 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE idea that people are more attuned to changes in attributes rather than absolute magnitudes. This generates utility curves with a point of inflexion at the chosen reference point. Q9. Describe what is meant by ‘loss aversion’ in the context of prospect theory. SOLUTION Prospect theory – evaluation phase – loss aversion Loss aversion relates to the idea that people are much more sensitive to losses (even small ones) than to gains of the same magnitude. In experiments, the pain from a loss is estimated to be twice as strong as the pleasure from an equivalent gain. Thus, utility curves are steeper in the domain of losses than they are in the domain of gains. Q10. Describe what is meant by ‘endowment effects’ in the context of prospect theory. SOLUTION Prospect theory – evaluation phase – endowment effects An endowment effect occurs when a person’s preferences depend upon what they already possess. This implies that a person’s preferences depend upon a certain reference point, perhaps deter- mined by the person’s possessions. Ownership itself creates satisfaction. Q11. Describe what is meant by ‘changing risk attitudes’ in the context of prospect theory. SOLUTION Prospect theory – evaluation phase – changing risk attitudes Changing risk attitudes relates to the idea that individuals tend to be risk-averse in the domain of gains and risk-seeking in the domain of losses. Q12. Describe what is meant by ‘diminishing sensitivity’ in the context of prospect theory. SOLUTION Prospect theory – evaluation phase – diminishing sensitivity Diminishing sensitivity relates to the idea that, as an individual gains (loses) more, the marginal im- pact on utility of additional gains (losses) falls. Thus, a utility function is concave in the region of gains but convex in the region of losses. CA PRAVEEN PATWARI 21 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q13. Describe what is meant by ‘probability weighting’ in the context of prospect theory. SOLUTION Prospect theory – evaluation phase – probability weighting Probability weighting relates to the idea that people do not weight outcomes by their objective probabilities, but instead by transformed probabilities or decision weights. In general, weights are computed using a weighting function such that low probabilities are overweighted, while high probabilities are underweighted. Q14. Describe what is meant by the ‘certainty effect’ in the context of prospect theory. SOLUTION Prospect theory – evaluation phase – certainty effect The certainty effect relates to the idea that when an outcome was certain and becomes less proba- ble, the impact on an individual’s utility is greater than a similar reduction in probability for an out- come that was previously probable. For example, the impact on an individual's utility of reducing the probability of earning $100 from 100% to 99% is greater than the impact of reducing the prob- ability of earning $100 from 50% to 49%. Q15. Describe what is meant by the ‘isolation effect’ in the context of prospect theory. SOLUTION Prospect theory – evaluation phase – isolation effect The isolation effect relates to the idea that when choosing between alternatives, people often disre- gard components that the alternatives share and instead focus on what sets them apart, in order to simplify the decision. Since different choices can be decomposed in different ways, this invariably leads to inconsistent choices and preferences. Q16. Describe what is meant by ‘System 1’ and ‘System 2’ in the context of decision- making. SOLUTION System 1 and System 2 System 1 is the instinctive part of the brain that makes quick and effortless decisions, while System 2 is slow, deliberate and calculating, akin to the ‘rational’ thinker assumed by Expected Utility Theory. CA PRAVEEN PATWARI 22 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Which part of the brain is engaged depends largely on the decision context, including the environ- ment and prevailing emotional state. Nonetheless, according to psychologists, System 1 is crucial to daily decisions. People essentially re- ly on System 1 unless System 2 regulates it. Q17. Define ‘heuristic’. List four examples of heuristics. SOLUTION Heuristics A heuristic is a shortcut used by the brain to deliver a quick decision. System 1 depends on the use of heuristics, such as:  anchoring and adjustment  representativeness  availability  familiarity Q18. Explain what is meant by ‘anchoring and adjustment’. SOLUTION Heuristics – anchoring and adjustment Anchoring and adjustment is a term used to explain how people produce estimates. People start with an initial idea of the answer (‘the anchor') and then adjust away from this initial anchor to ar- rive at their final judgement. People may use experience or ‘expert’ opinion as the anchor, or the anchor may be more or less ar- bitrary. The anchor will be amended to allow for evident differences to the current conditions. Experimental evidence suggests that the adjustments to the anchor tend to be insufficient, even if people are aware that the anchor is ridiculous. CA PRAVEEN PATWARI 23 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q19. Explain what is meant by ‘representativeness’. SOLUTION Heuristics - representativeness The idea behind representativeness is that decision-makers often use similarity as a proxy for probabilistic thinking. This is because it is easier and quicker for our brain to compare a situation to a similar one (System 1) than to assess it probabilistically on its own merits (System 2). Representativeness is one of the most commonly used heuristics and can, at times, work reasonably well. Nonetheless, similarity does not always adequately predict true probability, leading to irra- tional outcomes. This is also related to the so-called ‘law of small numbers’, where people assess the probability of something occurring based on its occurrence in a small, statistically unrepresentative sample due to a desire to make sense of the uncertain situation. Q20. Explain what is meant by ‘availability’. SOLUTION Heuristics – availability Availability applies when people assess the probability of an event occurring by the ease with which instances of its occurrence can be brought to mind. Vivid outcomes are more easily recalled than other (perhaps more sensible) options, and this can lead to biased judgements. For example, individuals living in areas that are prone to extreme weather events may only be com- pelled to take out home insurance after they have been directly affected by such events rather than beforehand, and may even cease their coverage after some time as the memory of the event sub- sides. Q21. Explain what is meant by ‘familiarity’ and ‘home-country bias’. SOLUTION Heuristics – familiarity and home–country bias Familiarity is closely related to availability, and describes the process by which people favour situa- tions or options that are familiar over others that are new. This may lead to an undiversified portfo- CA PRAVEEN PATWARI 24 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE lio of investments if people simply put their money in industries or companies that they are familiar with rather than others in alternative markets or sectors. Home-country bias is an example of familiarity and refers to people’s tendency to disproportionally invest in stocks from their home country, rather than forming an internationally diversified portfolio. Q22. List four behavioural biases that arise from the interaction between System 1 and System 2. SOLUTION Behavioural biases 1. Overconfidence 2. Self-serving bias 3. Status quo bias 4. Herd behaviour Q23. Describe the behavioural bias ‘overconfidence’ SOLUTION Behavioural biases - overconfidence Overconfidence occurs when people systematically overestimate their own knowledge, judgement and abilities. Studies show that the discrepancy between accuracy and overconfidence increases with knowledge. This may result from:  Hindsight bias -events that happen/don’t happen will be thought of as having been predictable/ unlikely prior to the event.  Confirmation bias -people tend to look for evidence that confirms their point of view (and tend to dismiss evidence that does not justify it). CA PRAVEEN PATWARI 25 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q24. Describe the behavioural biases ‘self-serving bias’, ‘status quo bias’ and ‘herd behaviour’. SOLUTION Behavioural biases – self-serving bias, status quo bias, herd behaviour Closely related to overconfidence, self-serving bias occurs when people credit favourable or posi- tive outcomes to their own capabilities or skills, while blaming external forces or others for any negative outcomes. This may be done in order to maintain a positive self-image. Status quo bias is the inherent tendency of people to stick with their current situation, even in the presence of more favourable alternatives and even when no transaction costs are involved. Herd behaviour describes the tendency of people to follow the actions and decisions taken by oth- ers, as a mechanism to deal with uncertain situations. The underlying rationale may be that others must know better (ie safety in numbers), learning or conformity preferences. However, such deci- sions may lead to mass hysteria or delusion if the initial actions are themselves biased. Q25. Describe the ‘equity premium puzzle’. SOLUTION The equity premium puzzle The equity premium puzzle is related to the observed gap in returns from risk-bearing stocks when compared to the returns from lower-risk government securities. The standard explanation for this excess return (or premium) from investing in equities is that it re- flects the difference in the level of risk associated with stocks relative to government securities, with higher-risk investments requiring higher returns to attract investors. However, the equity premium levels observed are far too large to be explained by risk aversion alone using the models typically used in mainstream financial economics. CA PRAVEEN PATWARI 26 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q26. Describe the term ‘myopic loss aversion' and how this can be used to explain the eq- uity premium puzzle. SOLUTION Myopic loss aversion and the equity premium puzzle Myopic loss aversion suggests that investors are less risk-averse when faced with a multi-period se- ries of ‘gambles’. They are generally much more concerned by losses than equivalent gains, leading them to focus on very short-term returns and volatility rather than long-run earnings. Since stock prices are typically more volatile in the short run, this may dissuade myopic investors from buying stocks unless the returns premium on stocks is sufficiently high to compensate for this loss aversion. Thus, the equity premia observed across the world are so high since they take into ac- count both risk aversion as well as this aversion to short-term losses. The past equity premium is consistent with loss aversion, and an assumption that people evaluate their portfolios based on the last 12 months of returns (which is myopic, since investment decisions should rationally be made over longer timescales). CA PRAVEEN PATWARI 27 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE CHAPTER MEASURES OF 4 INVESTMENT RISK Q1. Define the variance of return for: (i) a continuous distribution, f(x) (ii) a discrete distribution, P(X = x). SOLUTION Variance of return (i) For a continuous distribution, f(x):  V    x  f  x  dx 2  (ii) For a discrete distribution, P(X = x): V     x  P  X  x  2 x where  is the mean return at the end of the chosen period. Q2. State the main advantage of using variance as a measure of investment risk. State the main disadvantage. SOLUTION Main advantage and disadvantage of variance The main advantage of using variance as a measure of investment risk is that it is mathematically tractable. In particular, it leads to elegant solutions for the optimal portfolio in mean-variance port- folio theory. The main disadvantage of using variance is that it treats upside and downside risk equally, whereas investors are typically concerned much more with downside risk. CA PRAVEEN PATWARI 28 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q3. Define the downside semi-variance of return for: (i) a continuous distribution, f(x) (ii) a discrete distribution, P(X = x). SOLUTION Downside semi-variance of return (i) For a continuous distribution, f(x):  DSV      x  f  x  dx 2  (ii) For a discrete distribution, P(X = x):    x  P  X  x  2 DSV  x  where  is the mean return at the end of the chosen period. Q4. State the main advantage and the three main disadvantages of using downside semi- variance as a measure of investment risk. SOLUTION Downside semi-variance compared to variance The main advantage of using downside semi-variance as a measure of investment risk is that it fo- cuses on downside risk. The main disadvantages are that it: 1. completely ignores upside risk above the mean 2. can be mathematically less tractable than the variance 3. measures downside risk relative to the mean, rather than a benchmark chosen by the investor. Q5. Define the shortfall probability below L for: (i) a continuous distribution, f(x) (ii) a discrete distribution, P(X = x). CA PRAVEEN PATWARI 29 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE SOLUTION Shortfall probability (i) For a continuous distribution, f(x): f  x  dx L SP    (ii) For a discrete distribution, P(X = x): SP   P X  x xL where L is the chosen benchmark level. It measures the probability of returns falling below a certain level, L. Q6. State the three main advantages and two main disadvantages of the shortfall proba- bility. SOLUTION Shortfall probability Advantages 1. It is easy to understand 2. It is tractable. 3. The investor chooses the benchmark level. Disadvantages 1. It doesn’t tell us anything about the extent of any shortfall. 2. It ignores upside risk above the benchmark level. Q7. State what is meant by Value at Risk (VaR). SOLUTION Value at Risk (VaR) VaR represents the maximum potential loss on a portfolio over a given future time period with a given degree of confidence (normally expressed as 1 – p). CA PRAVEEN PATWARI 30 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE For a continuous distribution: VaR(X) = – t, where P(X < t) = p For a discrete random variable: VaR(X) = – t, where t  max x : P  X  x   p So, with this notation, there’s only a probability of p (eg 1%) that the outcome is worse than losing t. Q8. State the assumption that is often made when calculating the Value at Risk. Does this assumption hold in practice? SOLUTION Assumption underlying Value at Risk When calculating the Value at Risk, it is often assumed that investment returns are normally distri- buted. However, in practice investment returns may be skewed and/or be fat-tailed. This is particularly the case for portfolios that are exposed to credit risk or that include derivatives. Q9. Define the expected shortfall below L for: (i) a continuous distribution, f(x) (ii) a discrete distribution, P(X = x). SOLUTION Expected shortfall (i) For a continuous distribution, f(x), the expected shortfall below a chosen benchmark level L is: ES  E  max  L  X,0      L  x  f  x  dx L  (ii) For a discrete distribution, P(X = x), the expected shortfall is: ES  E  max  L  X,0    L  x  P X  x  x L CA PRAVEEN PATWARI 31 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q10. State two ways in which the TailVaR may be defined in terms of the expected short- fall measure. SOLUTION TailVaR The expected shortfall is given by: ES =E[max(L – X, 0)] where L is a chosen benchmark level. If L is chosen to be a particular percentile point on the distribution, eg the 95% confidence level, then the risk measure is known as the TailVaR. However, TailVaR can also be expressed as the expected shortfall conditional on there being a shortfall. To calculate this, we would take the expected shortfall formula and divide by the shortfall probability. Q11. If investment decisions are based only on expected return and variance of return, this is equivalent to the investor having what type of utility function? SOLUTION Utility function corresponding to variance If investment decisions are based only on expected return and variance of return, this is equivalent to the investor having a quadratic utility function. Q12. If investment decisions are based only on expected return and downside semi- variance of return, this is equivalent to the investor having what type of utility func- tion? SOLUTION Utility function corresponding to downside semi-variance If investment decisions are based only on expected return and downside semi-variance, this is equivalent to a utility function that is quadratic below the expected return and linear above. CA PRAVEEN PATWARI 32 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q13. To what form of utility function does the use of a shortfall risk measure in invest- ment decision making correspond? SOLUTION Utility function corresponding to shortfall risk measure The use of a shortfall risk measure corresponds to a utility function that has a discontinuity at the minimum required return. Q14. Explain how an individual or corporation is likely to view each of the following types of unexpected event in a risk management context: (i) low-frequency-low-severity (ii) high-frequency-low-severity (iii) high-frequency-high-severity (iv) low-frequency-high-severity. SOLUTION Risk management of unexpected events (i) low-frequency-low-severity: such an event does not warrant any worry to an individual or corporation. (ii) high-frequency-low-severity: the overall cost of such events may be damaging, due to the high frequency. These types of event may need to be assessed to determine how they can be controlled. Low severity events are not generally insurable, as the cost of management per claim is too expen- sive (iii) high-frequency-high-severity: such events are to be avoided. (iv) low-frequency-high-severity: such events tend to be insured. CA PRAVEEN PATWARI 33 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q15. Define ‘adverse selection’ in the context of insurance. Describe how an insurer will try to reduce the problems of adverse selection. SOLUTION Adverse selection Adverse selection describes the fact that those who are bad risks are more inclined to take out in- surance than those who are good risks. This is sometimes known as self-selection or anti-selection. To try to reduce the problems of adverse selection, insurance companies find out relevant informa- tion about the risk characteristics of prospective policyholders, and use this information to split them into small, reasonably homogeneous pools in order to charge appropriate premiums. Q16. Define ‘moral hazard’ in the context of insurance. State the main implication of moral hazard for insurance companies. SOLUTION Moral hazard Moral hazard describes the fact that policyholders may, because they have insurance, behave in a way that makes the insured event more likely. This makes insurance more expensive, and may even push the price of insurance above the maxi- mum premium that a person is willing to pay. CA PRAVEEN PATWARI 34 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE CHAPTER STOCHASTIC MODELS OF 5 INVESTMENT RETURNS Q1. Outline the difference between a deterministic model and a stochastic model. SOLUTION Deterministic and stochastic models A deterministic model is a model that provides an output based on one set of inputs. However, de- ciding which set of inputs to use may be a challenge. A stochastic model recognises the uncertainty that exists in reality, by allowing the inputs to vary according to an assumed probability distribution. As such, the output will vary along with the in- puts, and the model produces a distribution of results. Q2. Describe the fixed rate model of investment returns. Give an expression for:  S n , the accumulation at time n of a single payment of 1 made at time 0  An , the accumulation at time n of a series of payments of 1 made at the start of each year for n years. SOLUTION Fixed rate model of investment returns The fixed rate model assumes that the unknown future investment return, i, does not vary from year to year, ie it is fixed over the whole period in question. Sn  1  i  n A n  s n CA PRAVEEN PATWARI 35 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q3. For the fixed rate model, give the formulae for the mean and variance of S n , the ac- cumulated value at time n of a single payment of 1 made at time 0. SOLUTION Fixed rate model: mean and variance of S n We calculate the accumulation under each possible investment return and then find the mean and variance of them: E  Sn   E 1  i     1  i  P  I  i  n n   i   Var  Sn   E Sn   E  Sn   2 2 where: E Sn   E 1  i     1  i  P  I  i  2 2n 2n     i Q4. For the fixed rate model, give the formulae for the mean and variance of An , the ac- cumulated value at time n of a series of payments of 1 made at the start of each year for n years. SOLUTION Fixed rate model: mean and variance of An We calculate the accumulation under each possible investment return and then find the mean and variance of them: s n    E  A n   E  s n @ i% P  I  i  i Var  A n   E  A n    E  A n   2 2   where:   E  A n   E   2   2    s n @ i% P  I  i  2 s    n  i CA PRAVEEN PATWARI 36 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q5. Describe the varying rate model of investment returns. Give an expression for:  S n , the accumulation at time n of a single payment of 1 made at time 0  An , the accumulation at time n of a series of payments of 1 made at the start of each year for n years. SOLUTION Varying rate model The varying rate model assumes that the investment return, i t , changes from year to year and the returns in each year are independent. Sn  1  i1 1  i 2 ... 1  i n  A n  1  i1 1  i 2 ... 1  i n   1  i 2 ... 1  i n  ...  1  i n  Note that An   An 1  1  1  i n . Q6. Write down formulae for the mean and variance of S n , the accumulated value at time n years of a single payment of 1 made at time 0, using the varying rate model. Assume 2 that the investment return, i t , in each year has mean j and variance s , and returns in each year are independent. SOLUTION Varying rate model: mean and variance of S n Assuming that E i t   j, Var  i t   s and the returns in each year are independent, then: 2 E Sn   1  j n   n Var  Sn   1  j  s  1  j 2 2 2n CA PRAVEEN PATWARI 37 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q7. In any year, the rate of investment return per annum has mean value j and standard deviation s and is independent of the rates in all previous years. If S n is the accumulated amount at time n years of a single investment of 1 made at time 0, show that: E S n  = 1 + j n SOLUTION Varying rate model: E Sn  proof  n  E Sn   E  1  i t    t 1  n   E 1  i t   by independence t 1 n   1  E i t   t 1  1  j n Q8. In any year, the rate of investment return per annum has mean value j and standard deviation s and is independent of the rates in all previous years. If S n is the accumulated amount at time n years of a single investment of 1 made at time 0, show that:   n Var Sn  = 1 + j  s  1  j 2 2 2n SOLUTION Varying rate model: Var Sn  proof  n   2  n 2 E Sn   E   1  i t     E   1  i t   2    t 1  t 1       CA PRAVEEN PATWARI 38 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE   n   E  1  2i t  i t  2 by independence    t 1   n   1  2E i t   E i t  2   t 1 E  X   Var  X    E  X   : 2 2 Using       n 2 n E Sn   1  2 j  j  s  1  j  s 2 2 2 2     n So: Var  Sn   E Sn    E  Sn    1  j  s  1  j 2 2 2 2 2n   Q9. How do you calculate the mean and variance of An , the accumulated value at time n years of regular payments of 1 at times 0,1…., n – 1, using the varying rate model? SOLUTION Varying rate model: mean and variance of An Assuming that E i t   j, Var  i t   s and the returns in each year are independent, then: 2 E  A n    sn evaluated at j Var  A n   E  A n    E  A n   2 2   where E  A 0   0 and: 2     2  E  A k   1  j  s 2 2  1  2E A k 1   E A k2 1   2 Q10. Each year, the distribution of 1 + i is lognormal with parameters μ and σ. The an- nual rates of investment return are independent and identically distributed. S n is the accumulated amount at time n years of a single investment of 1 made at time 0. CA PRAVEEN PATWARI 39 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE  What is the distribution of S n ?  Write down a formula for P  S n  s . SOLUTION Varying rate model: lognormal distribution of S n 2 The distribution of 1 + i is lognormal with parameters μ and σ.The annual rates of investment re- turn are independent and identically distributed. Then:  Sn ~ log N n, n 2   InS nn ~ N  0,1 n  In s  n  and: P  Sn  s        n   Q11. If  1 + i t  ~ logN μ , σ 2  and the annual rates investment return are independent, show  . n that Sn =  1 + i t  ~ logN nμ,nσ 2 t=1 SOLUTION Varying rate model: proof of the lognormal distribution of S n n Let Sn   1  i t . Then: t 1 ln Sn  ln 1  i1   ln 1  i 2  ...  ln 1  i n  Each of the ln 1  i t  has a N  ,   2  distribution, and the sum of independent normal random va- riables is a normal random variable (with mean and variance equal to the sum of the individual means and variances respectively), so:  lnSn ~ N n , n 2   So finally Sn ~ log N n , n  2 . CA PRAVEEN PATWARI 40 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE CHAPTER PORTFOLIO THEORY 6 Q1. What does mean-variance portfolio theory enable the investor to do? SOLUTION Aim of portfolio theory Mean-variance portfolio theory enables the investor to construct a portfolio that offers the highest expected return for a given risk (variance), or the lowest risk (variance) for a given expected return. If the investor’s utility function is known, then mean-variance portfolio theory allows the investor to choose the portfolio that has the optimal balance between expected return and risk (variance) in order to maximise expected utility. Q2. State the seven main assumptions underlying mean-variance portfolio theory. SOLUTION Assumptions underlying portfolio theory 1. All expected returns, variances and covariances of pairs of assets are known. 2. Investors make their decisions purely on the basis of expected return and variance of returns. 3. Investors consider a single time period only. 4. Investors are non-satiated. 5. Investors are risk-averse. 6. Taxes and transaction costs are ignored. 7. Assets may be held in any amounts, ie short-selling is possible, we can have infinitely divisible holdings and there are no maximum investment limits CA PRAVEEN PATWARI 41 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q3. Define the following terms: (i) opportunity set (ii) efficient portfolio (iii) efficient frontier. SOLUTION Definitions (i) The opportunity set is the set of all possible combinations of variance/standard deviation and expected return that are attainable by the investor based on the available combinations of se- curities. (ii) An efficient portfolio is one such that no other portfolio gives a higher expected return for the same or lower level of variance of return or a lower variance for the same or higher expected return. (iii) The efficient frontier is the set of all efficient portfolios. It is the upper half of the curve around the edge of the opportunity set. Q4. Define the following terms: (i) indifference curve (ii) optimal portfolio. SOLUTION Definitions (i) An indifference curve represents the combinations of expected return and variance that yield the same expected utility to the investor. The investor is therefore indifferent between all such combinations of mean and variance. (ii) The optimal portfolio is the combination of security holdings that maximises the investor’s ex- pected utility, as a function of variance and expected return. It is the portfolio given by the point on the efficient frontier at which it is tangential to an indifference curve. CA PRAVEEN PATWARI 42 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q5a. State a formula for each of the following in the case where there are only two securi- ties, A and B: (i) the portfolio expected return, E (ii) the portfolio variance of return, V. SOLUTION Formulae with two securities Let x i be the proportion of the investor’s wealth invested in security i. (i) Portfolio expected return E  x A EA  x BEB (ii) Portfolio variance of return 2 2 V  x A VA  x B VB  2x A x BCAB where C AB   AB  A  B. Q5b. Derive the formula for the portfolio proportion invested in Security A in the mini- mum-variance portfolio, in the case where there are only two securities, A and B. SOLUTION Proportion invested in Security A in the minimum-variance portfolio The portfolio variance is given by: 2 2 V  x A VA  x B VB  2x A x BCAB  x A VA  1  x A  VB  2x A 1  x A  C AB 2 2 Differentiating this with respect to x A gives: V  2x A VA  2 1  x A  VB   2  4x A  C AB  0 x A From which: VB  C AB xA  VA  VB  2C AB CA PRAVEEN PATWARI 43 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q6. State the number of parameter values needed in order to calculate the portfolio mean and variance in the N-security case. SOLUTION Parameter values with N securities In order to calculate the portfolio mean and variance in the N-security case we need:  N means, one for each security  N variances, one for each security  N(N – 1)/2 covariances. So, a total of N(N + 3)/2 parameter values are required. Q7. State the formula for: (i) the portfolio mean (ii) the portfolio variance in the general N-security case. SOLUTION Formulae in N-security case Let x i be the proportion of the investor’s wealth invested in security i. (i) Portfolio mean N E  E  R P    x i Ei i 1 (ii) Portfolio variance N N V  Var  R P     x i x jC ij i 1 j1 Where Cij iji  j. CA PRAVEEN PATWARI 44 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q8. State the formula for the Lagrangian function in the general N-security case. SOLUTION Formula for Lagrangian function The Lagrangian function is: N N  N   N  L   x i x jCij     Ei x i  E P      x i  1 i 1 j1  i 1   i 1  Q9. State the N + 2 conditions that need to be solved to find the efficient frontier in the N- security case. SOLUTION Lagrangian conditions in N-security case L N   2  x jC ij –  E i    0 (one equation for each of N securities) x i j1 L N    x i Ei – E P  0  i 1 L N    xi – 1  0  i 1 Q10. What type of function is the equation for the minimum portfolio variance in terms of the portfolio mean? SOLUTION Portfolio variance in terms of portfolio mean  The optimal security weightings, the x i 's , are all linear in the portfolio mean E,  The portfolio variance V is a quadratic function of the x i 's.  So, the portfolio variance V must be a quadratic function of the portfolio mean E, ie a parabola. CA PRAVEEN PATWARI 45 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q11. State the form of the efficient frontier in expected return-standard deviation space, if a risk-free asset is available. SOLUTION Efficient frontier with a risk-free asset If a risk-free asset is available, the efficient frontier is a straight line in expected return-standard deviation space. Q12. Indifference curves can be drawn in E – V space if expected utility can be expressed as a function of mean and variance only. Give two instances when this is the case. SOLUTION Assumptions to draw indifference curves If: 1. investment returns are normally distributed, so that they depend only on the mean and va- riance, and/or 2. the investor has a quadratic utility function then expected utility can be expressed as a function of mean and variance only. Q13. Consider a portfolio consisting of N securities, in which an equal amount is invested in each security Derive an expression for the portfolio variance in terms of the average of the indi- vidual security variances and covariances. SOLUTION Portfolio variance in terms of individual variances and covariances With equal investment, the proportion invested in any one asset x i is 1/N and the formula for the variance of the portfolio becomes: N N N V   1 / N  Vi   1 N 1 N  C ij 2 i 1 i 1 j1 j i CA PRAVEEN PATWARI 46 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE N  N  1 N N Cij ie V  1 N   1 N  Vi   N  N  1 i 1 N i 1 j1 j i Replacing the variances and covariances in the summations with their averages V and C , we have: V N 1 V  C N N Q14. Use the expression from the previous question to explain what happens as the num- ber of securities N becomes very large. SOLUTION As N becomes very large Recall that: V N 1 V  C N N where V and C are the average variance and covariance. V N 1 As N  , 0 and C C, so V C N N ie the portfolio variance tends to the average of the individual covariances. This illustrates that the individual risk of securities can be diversified away, but the contribution to the total risk caused by the covariance terms cannot be. CA PRAVEEN PATWARI 47 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE CHAPTER MODELS OF ASSET RETURNS 7 Q1. State the formula and assumptions for the single-index model, defining the notation used. SOLUTION Single-index model The single-index model express the return on a security as: R i   i  i R M   i where:  Ri is the return on security i   i , i are constants  RM is the return on the market  i is a random variable representing the component of R i that is uncorrelated with the market  i is uncorrelated with R M and is independent of  j for all i  j. Q2. State equations for the expected return and variance of return on security i under the single-index model. SOLUTION Expected return and variance under the single-index model (i) Expected return E i   i  i E M CA PRAVEEN PATWARI 48 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE (ii) Variance 2 Vi  i VM  Vi where Vi is the variance of  i. Q3. Interpret the equation for the variance. SOLUTION Equation for the variance 2 Vi  i VM  Vi This equation models the variance of the return on security i as the sum of two terms. The first term captures the variability of return that is related to the variance of the return on the market. This is called the systematic risk. The second term captures the variability of return that is specific to security i. This is called the spe- cific risk. Q4. Distinguish between systematic risk and specific risk. SOLUTION Systematic risk and specific risk Systematic risk is the uncertain element of the security’s return related to the market as a whole. It cannot be diversified away. Specific risk is the uncertain element of the security's return that depends on factors peculiar to the individual security. It can be diversified away. Q5. State and interpret the equation for the covariance of the returns on securities i and j under the single-index model. SOLUTION Covariance of returns under the single-index model The covariance is equal to: Cij  i jVM CA PRAVEEN PATWARI 49 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE The equation shows that in the single-index model any correlation between the returns on two se- curities comes only from their joint correlation with the market as a whole. So, the reason that se- curities move together is a common response to market movements - there are no other possible common factors. Q6. How does the single-index model reduce the number of data items required com- pared to mean-variance portfolio theory? SOLUTION Reduction in number of data items Mean-variance portfolio theory requires estimates of N means, N variances and N(N –1)/2 cova- riances. The single-index model requires estimates of N  i 's, N i 's and N Vi 's , plus the mean and va- riance of market returns. So, compared to mean-variance portfolio theory, the number of data items required has been re- duced from N(N + 3)/2 to 3N+ 2. Q7. Derive an expression for the variance of returns for a portfolio of N securities under the single-index model. SOLUTION Portfolio variance under single-index model N N N N R p   x i R i   x i  i  R M  x i i   x i  i i 1 i 1 i 1 i 1 So, recalling that  i is uncorrelated with R M and independent of  j for all i  j : 2  N  N   var R p  Vp  0  VM   x ii    x i Vi 2  i 1  i 1 This term represents the systematic risk, whilst this term represents the specific risk. CA PRAVEEN PATWARI 50 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q8. Use the expression derived in the previous question to explain what happens as the number of securities N in a portfolio becomes very large. SOLUTION What happens as the number of securities becomes very large? 2  N  N Vp  VM   x ii    x i Vi 2  i 1  i 1 Let x i =1/N for all i, Then: 2  N   1 N V 1 Vp  VM   i    i  VM   V 2  i 1 N  N i 1 N N where  and V are the average beta and specific risk respectively. 1 2 So, as N , V 0 , ie the specific risk disappears, and Vp VM . N Q9. State the formula for the multifactor model, defining all notation used. SOLUTION Multifactor model The investment return is given by: R i  a i  bi , 1l1  bi , 2 l2 ....  bi , L l L  ci where:  Ri is the return on security i  a i , ci are the constant and random parts respectively of the component of return unique to security i  11 ,....,1L are the changes in a set of L factors which explain the variation of R i about the expected return a i.  bi , k is the sensitivity of security i to factor k. CA PRAVEEN PATWARI 51 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE Q10. Describe the three main categories of multifactor models. SOLUTION Three main categories of multifactor models 1. Macroeconomic factor models use observable economic time series as the factors, eg annual rates of inflation and economic growth, short-term interest rates, yields on long-term govern- ment bonds, and the yield margin on corporate bonds over government bonds. 2. Fundamental factor models use company-specific variables (such as the level of gearing, the industry sector, the price/earnings ratio, the level of research and development spending) in- stead of, or in addition to, macroeconomic variables. 3. Statistical factor models do not rely on specifying the factors independently of the historical returns data. Instead a technique called principal components analysis can be used to deter- mine a set of indices based on the historical returns data that explain as much as possible of the observed variance. Q11. Suppose we have two correlated indices I1 and I 2. I1 could be a market index and I 2 could be an industry sector index. Explain how two new, uncorrelated factors, * * I1 and I 2 , can be derived. SOLUTION Derivation of uncorrelated factors * First, let I1  I1. We then carry out a linear regression analysis to determine the parameters and 1 and  2 in the equation: I2  1   2 I1*  d 2 This is possible because I1 and I 2 are correlated. We then set: *  * I 2  d 2  I 2  1   2 I1  * * By construction I 2 (which equals the ‘residual’ d 2 ) is uncorrelated with I1. CA PRAVEEN PATWARI 52 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE * Q12. Interpret the meaning of the index I 2 in the previous question. SOLUTION * The meaning of the index I 2 * Changes in I 2 can be interpreted as the changes in the observed value of I 2 that cannot be ex- plained by the observed changes in I1 , ie the changes in I 2 that are independent of changes in I1. Q13. Discuss whether multifactor models are better at predicting future returns than the single-index model. SOLUTION Are multifactor models better at predicting future returns than the single-index model? Although many studies have found that incorporating more factors into the model (for example in- dustry indices) leads to a better explanation of the historical data, correlation with the market is the largest factor in explaining security price variation. Furthermore, there is little evidence that multifactor models are significantly better at forecasting the future correlation structure. CA PRAVEEN PATWARI 53 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE CHAPTER ASSET PRICING MODELS 8 Q1. State the five additional assumptions of the Capital Asset Pricing Model (CAPM) as compared to mean-variance portfolio theory. SOLUTION Additional assumptions of CAPM compared to mean-variance portfolio theory 1. All investors have the same one-period horizon. 2. All investors can borrow or lend unlimited amounts at the same risk-free rate, r. 3. The markets for risky assets are perfect. Information is freely and instantly available to all in- vestors and no investor believes that they can affect the price of a security by their own ac- tions. 4. Investors have the same estimates of the expected returns, standard deviations and cova- riances of securities over the one-period horizon. 5. All investors measure returns in the same ‘currency’, eg pounds or dollars or in ‘real' or ‘mon- ey’ terms. Q2. State two consequences of the additional assumptions for the efficient frontier under CAPM. SOLUTION Efficient frontier under CAPM 1. As investors have homogeneous expectations (regarding means, variances and covariances), they all face the same efficient frontier of risky assets. CA PRAVEEN PATWARI 54 JAI SHREE SHYAM CM2 SUMMARY ACTUATORS EDUCATIONAL INSTITUTE 2. In addition, as they are all subject to the same risk-free rate of interest r, the efficient frontier collapses to the same straight line in E   space, that passes through the risk-free rate of re- turn on the E-axis and is tangential to the efficient frontier for risky securities. Q3. Explain what is meant by the market portfolio. SOLUTION Market portfolio The market capitalisation of a company is the total value of its shares, ie the number of shares is- sued multiplied by the share price  NiPi . The market portfolio consists of all risky assets held in proportion to their market capitalisation, ie the portfolio weighting of Share i is: N iPi xi   i NiPi In expected return-standard deviation space, the market portfolio is situated at the point where a straight line through the risk-free return is tangential to the efficient frontier of risky assets only. Q4. State the prediction of CAPM concerning the investment choice of all rational inves- tors. SOLUTION