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lOMoARcPSD|47172552 AC 1103 Presentations Quantitative Analysis analyst will concentrate on the quantitative facts or data https://doc...

lOMoARcPSD|47172552 AC 1103 Presentations Quantitative Analysis analyst will concentrate on the quantitative facts or data https://docs.google.com/presentation/d/1LQaPsQbaA1dN3kjDRBr associated with the problem npfsv8A2VbNTv/edit#slide=id.p2 analyst will develop mathematical expressions that describe the objectives, constraints, and other relationships that exist in the · Chapter 1 Introduction problem 1.1 Problem Solving and Decision Making analyst will use one or more quantitative methods to make a 1.2 Quantitative Analysis and Decision Making recommendation 1.3 Quantitative Analysis 1.4 Models of Cost, Revenue, and Profit 1.5- Management Science Techniques The body of knowledge involving quantitative approaches to decision making is referred to as: MOD Management Science Operations Research Decision Science It had its early roots in World War II and is flourishing in business and industry due, in part, to: numerous methodological developments (e.g.simplex method for solving linear programming problems) a virtual explosion in computing power Potential Reasons for a Quantitative Analysis Approach to Decision Making : RINC 7 Steps of Problem Solving(First 5 steps are the process of The problem is complex. decision making) PACECIE The problem is very important 1.Define the problem.. The problem is new. 2.Determine the set of alternative solutions. The problem is repetitive. 3.Determine the criteria for evaluating alternatives. 4.Evaluate the alternatives. Quantitative Analysis Process (MDMR) 5.Choose an alternative (make a decision). Model Development 6.Implement the selected alternative. Data Preparation 7.Evaluate the results. Model Solution Report Generation Quantitative Analysis and Decision Making Decision-Making Process Model Development Problems in which the objective is to find the best solution with Models are representations of real objects or situations respect to one criterion are referred to a single-criterion decision problems. Three forms of models are: Problems that involve more than one criterion are referred to as Iconic models- physical replicas (scalar representations) of real multicriteria decision problems. objects Analog models- physical in form, but do not physically resemble the object being modeled Mathematical models- represent real world problems through a system of mathematical formulas and expressions based on key assumptions, estimates, or statistical analyses Generally, experimenting with models (compared to experimenting with the real situation): RET requires less time is less expensive involves less risk Analysis Phase of Decision-Making Process The more closely the model represents the real situation, the Qualitative Analysis accurate the conclusions and predictions will be. based largely on the manager’s judgment and experience includes the manager’s intuitive Lines or arcs connecting the nodes show the direction of - influence. Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 b) Decision Making w/ Probability Expected Value Approach If probabilistic information regarding the states of nature is available, one may use the expected value (EV) approach. Here the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring. The decision yielding the best expected return is chosen. Expected value of a decision alternative Minimax Regret Approach nature 5; the probability of state of The minimax regret approach requires the construction of a regret & state of nature I payoff decision alternative table or an opportunity loss table. corresponding to This is done by calculating for each state of nature the difference between each payoff and the largest payoff for that state of nature. Then, using this regret table, the maximum regret for each possible decision is listed. The decision chosen is the one corresponding to the minimum of the maximum regrets. Decision Tree A decision tree is a chronological representation of the decision problem. Each decision tree has two types of nodes; round nodes correspond to the states of nature while square nodes correspond to the decision alternatives. - - The branches leaving each round node represent the different states of nature while the branches leaving each square node represent the different decision alternatives. At the end of each limb of a tree are the payoffs attained from the series of branches making up that limb. - Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 Risk Analysis Risk analysis helps the decision maker recognize the difference (supposed) between: (supposed) a) the expected value of a decision alternative, and b) the payoff that might actually occur The risk profile for a decision alternative shows the possible payoffs for the decision alternative along with their associated probabilities. Expected Value for each decision strong Weak Sensitivity Analysis Sensitivity analysis can be used to determine how changes to the following inputs affect the recommended decision alternative: probabilities for the states of nature values of the payoffs Expected Value of Perfect Information If a small change in the value of one of the inputs causes a change in the recommended decision alternative, extra effort and care Frequently information is available which can improve the should be taken in estimating the input value. probability estimates for the states of nature. The expected value of perfect information (EVPI) is the Decision Analysis with Sample Information increase in the expected profit that would result if one knew with certainty which state of nature would occur. Frequently, decision makers have preliminary or prior probability assessments for the states of nature that are the best The EVPI provides an upper bound on the expected value of any probability values available at that time. sample or survey information. To make the best possible decision, the decision maker may want to seek additional information about the states of nature. This new information, often obtained through sampling, can be used to revise the prior probabilities so that the final decision is based on more accurate probabilities for the states of nature. These revised probabilities are called posterior probabilities. Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 Example: Pittsburgh Development Corp. Decision Tree Let us return to the PDC problem and assume that management is considering a 6-month market research study designed to learn more about potential market acceptance of the PDC condominium project. Management anticipates that the market research study will provide one of the following two results: 1. Favorable report: A significant number of the individuals contacted express interest in purchasing a PDC condominium. 2. Unfavorable report: Very few of the individuals contacted express interest in purchasing a PDC condominium. Sample information Decision Strategy A decision strategy is a sequence of decisions and chance outcomes where the decisions chosen depend on the yet-to-be-determined outcomes of chance events. The approach used to determine the optimal decision strategy is based on a backward pass through the decision tree using the following steps: At chance nodes, compute the expected value by multiplying the payoff at the end of each branch by the corresponding branch probabilities. At decision nodes, select the decision branch that leads to the best expected value. This expected value becomes the expected value at the decision node. Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 Computing Branch Probabilities PDC’s Decision Strategy PDC's optimal decision strategy is: Conduct the market research study. If the market research report is favorable, construct the large condominium complex. If the market research report is unfavorable, construct the medium condominium complex. Branch (Posterior) Probabilities Calculation Step 1: For each state of nature, multiply the prior probability by its conditional probability for the indicator -- this gives the joint probabilities for the states and indicator. Step 2: Sum these joint probabilities over all states the marginal probability for the indicator. -- this gives Step 3: For each state, divide its joint probability by the marginal Expected Value of Sample Information probability for the indicator -- this gives the posterior probability distribution. The expected value of sample information (EVSI) is the additional expected profit possible through knowledge of the sample or survey information. Bayes' Theorem and Posterior Probabilities The expected value associated with the market research study is $15.93. The best expected value if the market research study Knowledge of sample (survey) information can be used to revise is not undertaken is $14.20. the probability estimates for the states of nature. Prior to obtaining this information, the probability estimates for the states of nature We can conclude that the difference, $15.93 - $14.20 = $1.73, is are called prior probabilities. the expected value of sample information. With knowledge of conditional probabilities for the outcomes or Conducting the market research study adds $1.73 million to indicators of the sample or survey information, these prior the PDC expected value. probabilities can be revised by employing Bayes' Theorem. Efficiency of Sample Information The outcomes of this analysis are called posterior probabilities Efficiency of sample information is the ratio of EVSI to EVPI. or branch probabilities for decision trees. Posterior Probability Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 CHAPTER 8 Risk and Rates of Return RISK: is defined as hazard: a peril exposure to loss and injury. It refers to the chance that some untavourabie event will occur. RETURN: what is earned on an investment; the sum of income and capital gains generated by an investment. TN: As a rule the more the risk. the higher the return; the lower the risk, the lower the retum. Risk and Return Valuing risky assets - a task fundamental to financial management Three-step procedure i valuing a risky asset 1. Determine the asset’s expected cash flows (future cash flow dividends 10p now and 50 p in year 5) 2. Choose discount rate that reflects asset's risk (time value of money) 3. Calculate present value (PV cash inflows - PV outflows) The three-step procedure is called discounted cash flow (DCF) analysis. Financial Return Total return: the total gain or loss experienced on an investment over a given period of time Components/ 2 types of return: 1. Income stream from the investment (form of dividends in stocks, interest in bonds) 2. Capital gain or loss due to changes in asset prices (increase and decrease in asset) Total return can be expressed either in dollar terms or in percentage terms. Dollar Returns Total dollar return= income + capital gain / loss Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 Percentage Returns Terrell’s dollar return exceeded Owen's by $100. Can we say that Terrell was better off? No, because Terrell and Owen’s initial investments were 11.7-4.1 =7.6 different: Terrell spent $2,500 in initial investment, while 11.7-5.2= 6.5 Owen spent $750. Percentage return: total dollar return divided by the initial investment Total percentage return = Total dollar return/ initial investment Probability distribution for future stock returns is unknown. We can approximate the unknown distribution by assuming a normal distribution. Equity is higher than bonds and more riskier. Equity is buying part of company. Bonds is lending money. Risks -mode of measuring risk is through probability distribution or standard deviation -more spread out the possibility the more risky. Probability distributions -A listing of all possible outcomes, and the probability of each occurrence. » Can be shown graphically. Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 Use of Standard Deviation Firm y is more risky. » 68% of time asset will have a return between expected return +/- 1 standard deviation Variability of Stock Returns » 95% of time asset will have a return -before standard deviation you have to pass variance. between expected return +/- 2 Normal distribution can be described by its mean standard deviations and its variance. » 99% of time asset will have a return between expected return +/- 3 Variance (σ2) - the expected value of squared deviations from standard deviations the mean Units of variance (%-squared) - hard to interpret, so calculate standard deviation, a measure of volatility equal to square root of σ2 Line is sml (security market line)- line all securities follow…if naa may ma sibag sa line then it might be overvalue or undervalue Comments on standard deviation as a measure of risk « Standard deviation (oi) measures total, or stand-alone, risk. « The larger oi, is, the lower the probability that actual returns will be closer to expected returns. « Larger oi, is associated with a wider probability distribution of returns. *the higher the risk the higher the sd Coefficient of Variation (CV) -useful if u comparing 2 options or investment opportunities A standardized measure of dispersion about the expected value, that shows the risk per unit of return. Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 Illustrating the CV as a measure of relative risk Diversification *the more u diversify the more the line flattens and lowers the risk -Most individual stock prices show higher volatility than the price *diversification only reduces the unsystematic risk volatility of portfolio of all common stocks. * systematic risk is market risk - How can the standard deviation for individual stocks be higher Systematic and Unsystematic Risk than the standard deviation of the portfolio? Diversification reduces portfolio volatility, but only up to a point. Diversification: investing in many different assets reduces Portfolio of all stocks still has a volatility of 21%. the volatility of the portfolio. Reduces the risk Systematic risk: the volatility of the portfolio that cannot be The ups and downs of individual stocks partially eliminated through diversification. cancel each other out. Unsystematic risk: the proportion of risk of individual assets that can be eliminated through diversification What really matters is systematic risk....how a group assets move together. In relation to the market standing or value …. Anheuser Busch stock had higher average returns than Archer-Daniels-Midland stock, with smaller volatility. American Airlines had much smaller average returns No loss no gain than Wal-Mart, with similar volatility. Standard deviation contains both systematic and unsystematic risk. Because investors can eliminate unsystematic risk through diversification, market rewards only systematic risk. Security Market Line Portfolio composed of the following two assets: *we dont like for both of these to happen - An asset that pays a risk-free (treasury bills )return Re, and - A market portfolio that contains some of every risky asset in the market. *expect return is discount rate. Composed of 2 components: risk-free asset (0 beta bec. It does not move) and the market portfolio(1 bec it morves w/ market) Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 Security market line: the line connecting the risk-free asset and the market portfolio Beta » Measures a stock’s market risk, and shows a stock’s volatility relative to the market. -measures volatility to the market « Indicates how risky a stock is if the stock is held in a well-diversified portfolio. Comments on beta « If beta = 1.0, the security is just as risky as the average stock. « If beta > 1.0, the security is riskier than average. « If beta < 1.0, the security is less risky than average. = Most stocks have betas in the range of 0.5 to 1.5. Security A is undervalued but its returns are higher because it Security Market Line and CAPM (capital asset pricing model) same level of risk. The two-asset portfolio lies on security market line Security b is overvalued. (valued at 300 but return is 100) Given two points (risk-free asset and market portfolio asset) on the security market line, the equation of the ….. line: PORTFOLIO MANAGEMENT The Required Rate of Return: The required rate of return is the nominal rate of return that an investor needs in order to make an investment worthwhile. This return varies over time and is comprised of the following: e Real risk-free rate = beta * ()risk premium = inside () market risk premium e Inflation premium The equation represents the risk and return relationship predicted by the Capital Asset Pricing Model (CAPM) e Risk premium. Real risk-free rate of return: The real risk-free rate of return (R,) is the minimum return an investor requires. This rate does not take into account expected inflation and the capital market environment. The Security Market Line Plots relationship between expected return and betas « In equilibrium, all assets lie on this line. Example: Real risk-free rate of return - If individual stock or portfolio lies above the line: Determine the real risk-free rate if the nominal risk-free rate is 8%. Expected return is too high. and the inflation rate is 3%. : Investors bid up price until expected return Answer: falls. R,= (1 +.0.08)/(1 +0.03) - 1 = 4.85% - If individual stock or portfolio lies below SML:. Expected return is too low.. Investors sell stock driving down price until Nominal risk-free rate of return (Rnominal) expected return rises. Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 This is simply the real risk-free rate of return adjusted for inflation. The line begins with the risk-free rate (with zero risk) and moves upward and to the right. As the risk of an investment increases, it is expected that the return on an investment would increase. An investor with a low risk profile would choose an investment at the beginning of the security market line. An investor Example: Nominal risk-free rate of return with a higher risk profile would thus choose an investment higher along the security market line. Determine the nominal risk-free rate of return if the risk-free rate is 3% and the rate of inflation is 3%. Security Market Line Answer: Rnominal = (1 + 0.03) x (1 + 0.03) - 1 = 6.09% In an investment setting, an investor sets his required rate of return as the base return he requires from an investment. However, given the usual uncertainty in the market, it is difficult to meet that required rate of return exactly. As such, an investor would set his return above his required rate of return to diminish the risk that his required rate of return will not be met. The excess return above the investor's required rate of return is known as the Risk-Beta risk premium. The fundamental sources of risk that contribute Given the SML reflects the return on a given investment in relation to the need of the risk premium, such as: to risk, a change in the slope of the SML could be caused by the risk premium of the investments. Recall that the risk 1. Business risk premium of an investment is the excess return required by an 2. Financial risk investor to help ensure a required rate of return is met. If the 3. Liquidity risk risk premium required by investors was to change, the slope of the 4. Exchange rate risk SML would change as well. 5. Political risk. When a shift in the SML occurs, a change that affects all These risks comprise systematic risk, and cannot be avoided investments' risk versus return profile has occurred. A shift of the through diversification since they affect the entire market. SML can occur with changes in the following: 1. Expected real growth in the economy. 1. Business Risk: Business risk is the risk that a business' cash 2. Capital market conditions. flow will not meet its needs due to uncertainty in the company's 3. Expected inflation rate. business lines. 2. Financial Risk: Financial risk is the risk to equity holders as a company increases its debt load. As debt load increases, interest expense also increases, leading to less income to be The portfolio management process is the process an investor paid out to investors. takes to aid him in meeting his investment goals. 3. Liquidity Risk: Liquidity risk is the uncertainty around the ability The procedure is as follows: to sell an investment. The more liquid an investment is the easier it is to sell. 1. Create a Policy Statement -A policy statement is the statement that contains the investor's goals and constraints as it relates to 4. Exchange-Rate Risk: Exchange-rate risk is the risk a company his investments. faces when it has businesses in other countries. When a company is in the business of producing or buying products in a country 2. Develop an Investment Strategy - This entails creating a other than its own, a company can face exchange-rate risk when strategy that combines the investor's goals and objectives with in the process when it needs to exchange currency to transact current financial market and economic conditions. business as a part of its normal business routine. 3. Implement the Plan Created -This entails putting the 5. Political Risk: Political risk is the risk of changes in the political investment strategy to work, investing in a portfolio that meets the environment of a country in which company transacts its client's goals and constraint requirements. businesses. This risk could be caused by changes in laws relating to a specific business or even more serious as a country revolution 4. Monitor and Update the Plan -Both markets and investors' that would cause disruption in a company's operations. needs change as time changes. As such, it is important to monitor for these changes as they occur and to update the plan to adjust The security market line (SML) is the line that reflects an for the changes that have occurred. investment's risk versus its return, or the return on a given investment in relation to risk. The measure of risk used for the Policy Statement security market line is beta. Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 A policy statement is the statement that contains the investor's Return objectives are important to determine. They help to focus goals and constraints as it relates to his investments. This could an investor on meeting his financial goals and objectives. be considered to be the most important of all the steps in the However, risk must be considered as well. An investor may require portfolio management process. The statement requires the a high rate of return. A high rate of return is typically accompanied investor to consider his true financial needs, both in the by a higher risk. Despite the need for a high return, an investor short run and the long run. It helps to guide the investment may be uncomfortable with the risk that is attached to that higher portfolio manager in meeting the investor's needs. When there is return portfolio. As such, it is important to consider not only return, market uncertainty or the investor's needs change, the policy but the risk of the investor in a policy statement. statement will help to guide the investor in making the necessary adjustments the portfolio in a disciplined manner. Factors Affecting Risk Tolerance An investor's risk tolerance can be affected by many factors: Expressing Investment Objectives in Terms of Risk and Return e Age- an investor may have lower risk tolerance as they get older and financial constraints are more prevalent. Return objectives are important to determine. They help to focus e Family situation - an investor may have higher income needs if an investor on meeting his financial goals and objectives. they are supporting a child in college or an elderly relative. However, risk must be considered as well. An investor may e Wealth and income - an investor may have a greater ability to require a high rate of return. A high rate of return is typically invest in a portfolio if he or she has existing wealth or high income. accompanied by a higher risk. Despite the need for a high e Psychological - an investor may simply have a lower tolerance return, an investor may be uncomfortable with the risk that is for risk based on his personality. attached to that higher return portfolio. As such, it is important to consider not only return, but the risk of the investor in a Return objectives can be divided into the following needs: policy statement. 1. Capital Preservation - Capital preservation is the need to maintain capital. To accomplish this objective, the return objective The portfolio management process is the process an investor should, at a minimum, be equal to the inflation rate. In other words, takes to aid him in meeting his investment nominal rate of return would equal the inflation rate. With this goals. objective, an investor simply wants to preserve his existing capital. The procedure is as follows: 2. Capital Appreciation -Capital appreciation is the need to grow, 1. Create a Policy Statement -A policy statement is the statement rather than simply preserve, capital. To accomplish this objective, that contains the investor's goals the return objective should be equal to a return that exceeds and constraints as it relates to his investments. the expected inflation. With this objective, an investor's intention is to grow his existing capital base. 2. Develop an Investment Strategy - This entails creating a strategy that combines the investor's 3. Current Income -Current income is the need to create income goals and objectives with current financial market and economic from the investor's capital base. With this objective, an investor conditions. needs to generate income from his investments. This is frequently seen with retired investors who no longer have income from work 3. Implement the Plan Created -This entails putting the investment and need to generate income off of their investments to meet strategy to work, investing in a living expenses and other spending needs. portfolio that meets the client's goals and constraint requirements. 4. Total Return - Total return is the need to grow the capital 4. Monitor and Update the Plan -Both markets and investors' base through both capital appreciation and reinvestment of needs change as time changes. As that appreciation. such, it is important to monitor for these changes as they occur and to update the plan to adjust Investment Constraints for the changes that have occurred. When creating a policy statement, it is important to consider an investor's constraints. There are five types of constraints that Policy Statement need to be considered when creating a policy statement. They are A policy statement is the statement that contains the investor's as follows: goals and constraints as it relates to his investments. This could be considered to be the most important of all the steps in the 1.Liquidity Constraints - Liquidity constraints identify an portfolio management process. The statement requires the investor's need for liquidity, or cash. For example, within the next investor to consider his true financial needs, both in the year, an investor needs $50,000 for the purchase of a new home. short run and the long run. It helps to guide the investment The portfolio manager in meeting the investor's needs. When there is $50,000 would be considered a liquidity constraint because it market uncertainty or the investor's needs change, the policy needs to be set aside (be liquid) for the investor. statement will help to guide the investor in making the necessary adjustments the portfolio in a disciplined manner. 2.Time Horizon - A time horizon constraint develops a timeline of an investor's various financial needs. The time horizon also affects Expressing Investment Objectives in Terms of Risk and Return an investor's ability to accept risk. If an investor has a long time horizon, the investor may have a greater ability to accept risk because he would have a longer time period to recoup any losses. Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 This is unlike an investor with a shorter time horizon whose Harry Markowitz developed the portfolio model. This model ability to accept risk may be lower because he would not have includes not only expected return, but also includes the level of the ability to recoup any losses. risk for a particular return. Markowitz assumed the following about an individual's investment behavior: 3.Tax Concerns - After-tax returns are the returns investors are focused on when creating an investment portfolio. If an investor e Given the same level of expected return, an investor will is currently in a high tax bracket as a result of his income, it choose the investment with the lowest amount of risk. may be important to focus on investments that would not e Investors measure risk in terms of an investment's variance make the investor's situation worse, like investing more heavily or standard deviation. in tax-deferred investments. e For each investment, the investor can quantify the investment's expected return and the probability of those 4.Legal and Regulatory - Legal and regulatory factors can act as returns over a specified time horizon. an investment constraint and must be considered. An example of e Investors seek to maximize their utility. this would occur in a trust. A trust could require that no more e Investors make decision based on an investment's risk and than 10% of the trust be distributed each year. Legal and return, therefore, an investor's utility curve is based on risk regulatory constraints such as this one often can't be changed and and return. must not be overlooked. The Efficient Frontier 5. Unique Circumstances - Any special needs or constraints not Markowitz’ work on an individual's investment behavior is recognized in any of the constraints listed above would fall in this important not only when looking at individual investment, but also category. An example of a unique circumstance would be the in the context of a portfolio. The risk of a portfolio takes into constraint an investor might place on investing in any account each investment's risk and return as well as the company that is not socially responsible, such as a tobacco investment's correlation with the other investments in the company. portfolio. Risk of a portfolio is affected by the risk of each investment in The Importance of Asset Allocation the portfolio relative to its return, as well as each investment's Asset Allocation is the process of dividing a portfolio among major correlation with the other investments in the portfolio. asset categories such as bonds, stocks or cash. The purpose of asset allocation is to reduce risk by diversifying the portfolio. A portfolio is considered efficient if it gives the investor a higher expected return with the same or lower level of risk as compared The ideal asset allocation differs based on the risk tolerance of the to another investment. The efficient frontier is simply a plot of investor. For example, a young executive might have an asset those efficient portfolios, as illustrated below: allocation of 80% equity, 20% fixed income, while a retiree would be more likely to have 80% in fixed income and 20% Efficient Frontier equities. Citizens in other countries around the world would have different asset allocation strategies depending on the types and risks of securities available for placement in their portfolio. For example, a retiree located in the United States would most likely have a large portion of his portfolio allocated to U.S. treasuries, since the U.S. Government is considered to have an extremely low risk of default. On the other hand, a retiree in a country with political unrest would most likely have a large portion of their portfolio allocated to foreign treasury securities, such as that of the U.S. While an efficient frontier illustrates each of the efficient portfolios Risk Aversion relative to risk and return levels, each of the efficient portfolios may Risk aversion is an investor's general desire to avoid not be appropriate for every investor. Recall that when creating an participation in "risky" behavior or, in this case, risky investment policy, return and risk were the key objectives. An investments. Investors typically wish to maximize their return investor's risk profile is illustrated with indifference curves. The with the least amount of risk possible. When faced with two optimal portfolio, then, is the point on the efficient frontier that is investment opportunities with similar returns, good investor will tangential to the investor's highest indifference curve. always choose the investment with the least risk as there is no benefit to choosing a higher level of risk unless there is also an The optimal portfolio for a risk-averse investor will not be as increased level of return. risky as the optimal portfolio of an investor who is willing to accept more risk. Insurance is a great example of investors' risk aversion. Given the potential for a car accident, an investor would rather pay for Individual Investment: The expected return for an individual insurance and minimize the risk of a huge outlay in the event of an investment is simply the sum of the probabilities of the possible accident. expected returns for the investment. Markowitz Portfolio Theory Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 Expected return Given that the standard deviation of Newco's stock is simply the square root of the variance, the standard deviation is 0.0179 or 1.79%. Covariance The covariance is the measure of how two assets relate (move) together. If the covariance of the two assets is positive, the assets move in the same direction. For example, if two assets have a covariance of 0.50, then the assets move in the same direction. If however the two assets have a negative covariance, the assets move in opposite directions. If the covariance of the two assets Portfolio is zero, they have no relationship. To determine the expected return on a portfolio, the weighted average expected return of the assets that comprise the portfolio is taken. Example: Calculate the covariance between two assets Assume the mean return on Asset A is 10% and the mean return on Asset B is 15%. Given the following returns over the past 5 Example: periods, calculate the covariance for Asset A as it relates to Asset B. Assume an investment manager has created a portfolio with the Stock A and Stock B. Stock A has an expected return of 20% and a weight of 30% in the portfolio. Stock B has an expected return of 15% and a weight of 70%. What is the expected return of the portfolio? Answer: E(R) = (0.30)(20%) + (0.70)(15%) = 6% + 10.5% = 16.5% The expected return of the portfolio is 16.5% Computing Variance and Standard Deviation for an Individual To measure the risk of an investment, both the variance and standard deviation for that investment can be calculated. Correlation The correlation coefficient is the relative measure of the relationship between two assets. It is between +1 and -1, with a +1 indicating that the two assets move completely together and a -1 indicating that the two assets move in opposite directions from each other. Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 What happens when a risk-free asset is added to a portfolio of risky assets? To begin, the risk-free asset has a standard deviation/variance equal to zero for its given level of return, hence the "risk-free" label. Example: Calculate the correlation of Asset A with Asset B. Given our covariance of 18 in the example above, what is the Expected Return - When the Risk-Free Asset is Added correlation coefficient for Asset A relative to Asset B if Asset A has Given its lower level of return and its lower level of risk, adding the a standard deviation of 4 and Asset B has a standard deviation of risk-free asset to a portfolio acts to reduce the overall return of 8. the portfolio. Answer: Example: Risk-Free Asset and Expected Return Correlation coefficient = 18/(4)(8) = 0.563 Assume an investor's portfolio consists entirely of risky assets with an expected return of 16% and a standard deviation of 0.10. The Components of the Portfolio Standard Deviation Formula investor would like to reduce the level of risk in the portfolio and Remember that when calculating the expected return of a portfolio, decides to transfer 10% of his existing portfolio into the risk-free it is simply the sum of the weighted returns of each asset in the rate with an expected return of 4%. What is the expected return of portfolio. Unfortunately, determining the standard deviation of a the new portfolio and how was the portfolio's expected return portfolio, it is not that simple. Not only are the weights of the affected given the addition of the risk-free asset? assets in the portfolio and the standard deviation for each asset in the portfolio needed, the correlation of the assets in the portfolio is Answer: also required to determine the portfolio standard deviation. The expected return of the new portfolio is: (0.9)(16%) + (0.1)(4%) = 14.4% The equation for the standard deviation for a two asset With the addition of the risk-free asset, the expected value of the portfolio is as follows: investor's portfolio was decreased to 14.4% from 16%. Standard Deviation - When the Risk-Free Asset is Added As we have seen, the addition of the risk-free asset to the portfolio of risky assets reduces an investor's expected return. Given there is no risk with a risk-free asset, the standard deviation of a portfolio is altered when a risk-free asset is added. The capital market theory builds upon the Markowitz portfolio model. The main assumptions of the capital market theory are Example: Risk-free Asset and Standard Deviation as follows: Assume an investor's portfolio consists entirely of risky assets with 1.All Investors are Efficient Investors - Investors follow an expected return of 16% and a standard deviation of 0.10. The Markowitz idea of the efficient frontier and choose to invest in investor would like to reduce the level of risk in the portfolio and portfolios along the frontier. decides to transfer 10% of his existing portfolio into the risk-free rate with an expected return of 4%. What is the standard 2.Investors Borrow/Lend Money at the Risk-Free Rate - This deviation of the new portfolio and how was the portfolio's rate remains static for any amount of money. standard deviation affected given the addition of the risk-free asset? 3.The Time Horizon is equal for All Investors - When choosing investments, investors have equal time horizons for the chosen Answer: investments. The standard deviation equation for a portfolio of two assets is rather long, however, given the standard deviation of the 4.All Assets are Infinitely Divisible - This indicates that fractional risk-free asset is zero, the equation is simplified quite nicely. shares can be purchased and the stocks can be infinitely divisible. The standard deviation of the two-asset portfolio with a risky asset is the weight of the risky assets in the portfolio multiplied by 5.No Taxes and Transaction Costs -assume that investors' the standard deviation of the portfolio. results are not affected by taxes and transaction costs. Standard deviation of the portfolio is: (0.9)(0.1) = 0.09 6. All Investors Have the Same Probability for Outcomes -When determining the expected return, assume that all investors Similar to the affect the risk-free asset had on the expected return, have the same probability for outcomes. the risk-free asset also has the affect of reducing standard deviation, risk, in the portfolio. 7. No Inflation Exists - Returns are not affected by the inflation rate in a capital market as none exists in capital market theory. As seen previously, adjusting for the risk of an asset using the risk-free rate, an investor can easily alter his risk profile. Keeping 8. There is No Mispricing Within the Capital Markets - Assume that in mind, in the context of the capital market line (CML), the the markets are efficient and that no mispricings within the markets market portfolio consists of the combination of all risky assets and exist. the risk-free asset, using market value of the assets to determine the weights. The CML line is derived by the CAPM, solving for expected return at various levels of risk. Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 Systematic and Unsystematic Risk Total risk to a stock not only is a function of the risk inherent within the stock itself, but is also a function of the risk in the overall market. Systematic risk is the risk associated with the market. When analyzing the risk of an investment, the systematic risk is the risk that cannot be diversified away. Unsystematic risk is the risk inherent to a stock. This risk is the aspect of total risk that can be diversified away when building a portfolio. When building a portfolio, a key concept is to gain the greatest return with the least amount of risk. However, it is important to note, that additional return is not guaranteed for an increased level of risk. With risk, reward can come, but losses can be magnified as well. The capital asset pricing model is a model that calculates expected return based on expected rate of return on the market, the risk-free rate and the beta coefficient of the stock. SUPPLEMENTARY: Example: CAPM model Covariance Determine the expected return on Newco's stock using the capital Covariance is a measure of the relationship between two asset pricing model. Newco's beta is 1.2. Assume the expected random variables, designed to show the degree of return on the market is 12% and the risk-free rate is 4%. co-movement between them. Covariance is calculated based on the probability-weighted average of the cross-products of each Answer: random variable's deviation from its own expected value. A E(R) = 4% + 1.2(12% - 4%) = 13.6%. positive number indicates co-movement (i.e. the variables tend to move in the same direction); a value of 0 indicates no Using the capital asset pricing model, the expected return on relationship, and a negative covariance shows that the Newco's stock is 13.6%. variables move in the opposite direction. Correlation Correlation is a concept related to covariance, as it also gives an indication of the degree to which two random variables are related, and (like covariance) the sign shows the direction of this relationship (positive (+) means that the variables move together; negative (-) means they are inversely related). Correlation of 0 means that there is no linear relationship one way or the other, and the two variables are said to be unrelated. A correlation number is much easier to interpret than covariance because a correlation value will always be between -1 and +1. Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 « -l1 indicates a perfectly inverse relationship (a unit change in one means that the other will have a unit change in the opposite direction) « +1 means a perfectly positive linear relationship (unit changes in one always bring the same unit changes in the other). Moreover, there is a uniform scale from -1 to +1 so that as correlation values move closer to 1, the two variables are more closely related. By contrast, a covariance value between two variables could be very large and indicate little actual relationship, or look very small when there is actually a strong linear correlation. Correlation and Regression: Financial variables are often analyzed for their correlation to other variables and/or market averages. The relative degree of co-movement can serve as a powerful predictor of future behavior of that variable. A sample covariance and correlation coefficient are tools used to indicate relation, while a linear regression is a technique designed both to quantify a positive relationship between random variables, and prove that one variable is dependent on another variable. When you are analyzing a security, if returns are found to be significantly dependent on a market index or some other independent source, then both return and risk can be better explained and understood. Scatter Plots A scatter plot is designed to show a relationship between two variables by graphing a series of observations on a two-dimensional graph - one variable on the X-axis, the other on the Y-axis. Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 Sirs discussion CHAPTER 8 Risk and Rates of Return RISK: is defined as hazard: a peril exposure to loss and injury. It refers to the chance that some untavourabie event will occur. RETURN: what is earned on an investment; the sum of income and capital gains generated by an investment. Average X and Y returns were found by dividing the sum by n or 5, TN: As a rule the more the risk. the higher the return; the lower the while the average of the cross-products is computed by dividing risk, the lower the retum. the sum by n - 1, or 4. The use of n - 1 for covariance is done by statisticians to ensure an unbiased estimate. Financial Return Interpreting a covariance number is difficult for those who are not Total return: the total gain or loss experienced on an investment statistical experts. The 99.64 we computed for this example has a over a given period of time sign of "returns squared" since the numbers were percentage returns, and a return squared is not an intuitive concept. The fact Components/ 2 types of return: that Cov(X,Y) of 99.64 was greater than 0 does indicate a positive 1. Income stream from the investment (form of dividends in stocks, or linear relationship between X and Y. Had the covariance been a interest in bonds) negative number, it would imply an inverse relationship, while 0 2. Capital gain or loss due to changes in asset prices (increase means no relationship. Thus 99.64 indicates that the returns have and decrease in asset) positive co-movement (when one moves higher so does the other), but doesn't offer any information on the extent of the Measures of return co-movement. Sample Correlation Coefficient By calculating a correlation coefficient, we essentially convert a raw covariance number into a standard format that can be more easily interpreted to determine the extent of the relationship between two variables. The formula for calculating a sample correlation coefficient (r) between two random variables X and Y is the following: Historical return a)holding period return -other type is future value of a stock E.g Bought 10p stock today and in year 5 it is 50p … dividend per year 5p 50-10+5(5)/10 b) alternative measures - Arithmetic - Geometric - Harmonic Ex[ected return Downloaded by rene zach ([email protected])  lOMoARcPSD|47172552 -payoff * probability and add all value Correlation and Regression Analysis Covariance Interpreting Covariance cov(X,Y)>0 X and Y are positively correlated cov(X,Y)