2 1 reading 39 portfolio risk and return part i pdf

LOS Portfolio Management • • • • • • Portfolio Management – An Overview Portfolio Risk and Return – Part I Portfolio Risk and Return – Part II Basics of Portfolio Planning and Construction Risk Management – An Introduction Fintech in Investment Management LOS LOS Calculate and interpret major return measures and describe their appropriate uses Major Return Measures Holding Period Return • The period return (HPR), refers to the change in the value of an investment over the period it is held, expressed as a percentage of the originally invested amount • It also captures any additional income that one earns from an investment 𝐄𝐧𝐝𝐢𝐧𝐠 𝐯𝐚𝐥𝐮𝐞 – 𝐁𝐞𝐠𝐢𝐧𝐧𝐢𝐧𝐠 𝐯𝐚𝐥𝐮𝐞 + 𝐀𝐬𝐬𝐞𝐭 𝐢𝐧𝐜𝐨𝐦𝐞 𝐇𝐏𝐑 = 𝐁𝐞𝐠𝐢𝐧𝐧𝐢𝐧𝐠 𝐯𝐚𝐥𝐮𝐞 𝐏𝟏 − 𝐏𝟎 + 𝐃 𝐇𝐏𝐑 = 𝐏𝟎 Example >> LOS LOS Calculate and interpret major return measures and describe their appropriate uses Example: HPR • Five years ago, a trader paid $20 per share for 500 shares in an insurance company • The current share price has grown by 50% of the purchase price • So far, he has received 12 dividend payments, each amounting to $0.50 per share • If the trader decides to sell the shares at present, what will be the holding period return? HPR = Solution: (Ending value Variable Notation Calculation Amount ($) Beginning value P0 20 × 500 10,000 Ending value P1 1.5 × 20 × 500 15,000 Asset income D 0.5 × 500 × 12 3,000 HPR = (15,000 - 10,000 + 3,000)/10,000 = 80% – Beginning value + Asset income)/Beginning value LOS LOS Calculate and interpret major return measures and describe their appropriate uses Arithmetic return • It’s a simple average of the holding period returns Example: • If a share has returned 25%, 10%, 12% and -3% over the last four years, then the arithmetic mean is given as:  Arithmetic mean = (25% + 10% + 12% + -3%) / = 11% Geometric Mean Return • The geometric mean return gives the average rate per period on an investment that is compounded over multiple periods 𝐆𝐞𝐨𝐦𝐞𝐭𝐫𝐢𝐜 𝐦𝐞𝐚𝐧 𝐫𝐞𝐭𝐮𝐫𝐧 = 𝟏 + 𝐇𝐏𝐑 𝟏 × 𝟏 + 𝐇𝐏𝐑 𝟐 … × 𝟏 + 𝐇𝐏𝐑 𝐧 𝟏 𝒏 −𝟏  From the previous example, GMR = (1.25×1.1×1.12×0.97)1/4 − 1= 10.6% LOS LOS Calculate and interpret major return measures and describe their appropriate uses Money-weighted rate of return • The money-weighted return accounts for the money invested and provides the investor with information on the return she earns on her actual investment • Its calculation is similar to that of the internal rate of return and the yield to maturity Annualized return • To annualize a return for a period less than one year, the return for the period must be compounded by the number of periods in a year • A monthly return is compounded 12 times, a weekly return 52 times, and a quarterly return times Daily returns are normally compounded 365 times LOS LOS Calculate and interpret major return measures and describe their appropriate uses Portfolio return Return of portfolio = 𝒏 𝒊=𝟏 𝒘𝒊 𝒓𝒊 Where 𝑤𝑖 is the weight of asset i, and 𝑟𝑖 is the return of asset I Others: Real return • The nominal rate of return is the amount of money generated by an investment before factoring in inflation To determine the real rate of return: 𝟏 + 𝐍𝐨𝐦𝐢𝐧𝐚𝐥 𝐫𝐚𝐭𝐞 𝐑𝐞𝐚𝐥 𝐫𝐚𝐭𝐞 𝐨𝐟 𝐫𝐞𝐭𝐮𝐫𝐧 = 𝟏 + 𝐈𝐧𝐟𝐥𝐚𝐭𝐢𝐨𝐧 𝐫𝐚𝐭𝐞 Gross and net return • Gross return is earned prior to the deduction of fees (management fees, custodial fees, etc.) A net return is the return post-deduction of fees LOS LOS Describe characteristics of the major asset classes that investors consider in forming portfolios A risk-return tradeoff refers to the relationship between risk and return • To achieve a higher return, you must accept a higher level of risk • The following historical returns (1926 – 2008) illustrate as much: Average annual Asset Class nominal return US Large Company Stocks 9.60% US Small Company Stocks 11.70% US Long-term Corporate Bonds 5.90% US Long-term Government Bonds 5.70% US Treasury Bills 4.00% World equities 8.40% World bonds 4.80% Highest return Risk (Standard deviation of return) 20.60% 33.00% 8.40% 9.40% 3.10% 17.30% 8.60% Highest risk LOS LOS Describe characteristics of the major asset classes that investors consider in forming portfolios Skewness • Skewness is present when the distribution is not symmetrical • A distribution with positive skewness has a large number of small negative values with a few large positive values – the distribution has a long right tail (why most people buy lottery tickets) • A distribution with negative skewness has a large number of small positive values with a few large negative values – the distribution has a long left tail (why most people buy insurance) • Positive skewness: Mode < Median < Mean  Investors like positive skewness because the mean is greater than the median • Negative skewness: Mean < Median < Mode LOS LOS Describe characteristics of the major asset classes that investors consider in forming portfolios Kurtosis • Kurtosis measures the degree to which a distribution is more or less peaked than a normal distribution  Positive kurtosis indicates a relatively peaked distribution  Negative kurtosis indicates a relatively flat distribution  A normal distribution has a kurtosis of • An investment characterized by high kurtosis will have “fat tails” (higher frequencies of outcomes) at the extreme negative and positive ends of the distribution curve • A distribution of returns exhibiting high kurtosis tends to overestimate the probability of achieving the mean return LOS LOS Calculate and interpret the mean, variance, and covariance (or correlation) of asset returns based on historical data Variance of asset returns: • Measures the volatility/dispersion of returns • It’s the average squared deviation from the mean (R t −μ) Example: Year Mean Deviation Square of Return (%) from mean deviation 10.5 0.4 0.16 8.5 -1.6 2.56 -4.1 16.81 12.5 2.4 5.76 13 2.9 8.41 10.1 Sum = 33.7 variance = 33.7/5 = 6.74 square root = 2.6 𝛔𝟐 = 𝐓 𝐭=𝟏 𝐑𝐭 − 𝛍 𝐓 𝟐 T = number of periods • Standard deviation (volatility) is the square root of variance LOS LOS Describe the effect on a portfolio’s risk of investing in assets that are less than perfectly correlated; Case I: When correlation = Weights Expected return std dev Corr Asset 50% 20.00% 30.00% Asset 50% 20.00% 30.00% Portfolio Return = Weighted return of assets = 0.5 × 20% + 0.5 × 20 = 20% Portfolio risk 0.52 × 0.32 ) + (0.52 × 0.32 ) + (2 × 0.5 × 0.5 × 0.3 × 0.3 × = 30% Comment: Portfolio risk is unaffected; it is simply the weighted average of the standard deviations of the two assets LOS LOS Describe the effect on a portfolio’s risk of investing in assets that are less than perfectly correlated; Case II: When correlation = Weights Expected return std dev Corr Asset 50% 20.00% 30.00% Asset 50% 20.00% 30.00% Portfolio Return = Weighted return of assets = 0.5 × 20% + 0.5 × 20 = 20% Portfolio risk = 0.52 × 0.32 ) + (0.52 × 0.32 ) + (2 × 0.5 × 0.5 × 0.3 × 0.3 × = 21% Comment: Portfolio risk is reduced with the addition of two assets that are not perfectly correlated LOS LOS Describe the effect on a portfolio’s risk of investing in assets that are less than perfectly correlated; Case III: When correlation = -1 Weights Expected return std dev Corr Asset 50% 20.00% 30.00% -1 Asset 50% 20.00% 30.00% Portfolio Return = Weighted return of assets = 0.5 × 20% + 0.5 × 20 = 20% Portfolio risk = 0.52 × 0.32 ) + (0.52 × 0.32 ) + (2 × 0.5 × 0.5 × 0.3 × 0.3 × −1 = 0% Comment: Maximum risk reduction occurs with the addition of two assets that are perfectly negatively correlated LOS LOS Explain risk aversion and its implications for portfolio selection Risk Aversion refers to individual behavior under uncertainty • What will you choose? Gamble vs guaranteed outcome?  Option 1: Guaranteed outcome of $100  Option 2: Gamble of either $200 or $0? (expected value of $100) Investor Types • • • Risk Seeker (maximizes both risk and return) Risk Neutral (maximizes return irrespective of risk) Risk Averse (very low appetite for risk; would rather accept a lower certain return than a substantial risky investment) Risk tolerance refers to the amount of risk an investor is willing to take in order to achieve their investment goals and objectives • A higher risk tolerance shows a greater willingness to take risk, implying risk tolerance and risk aversion are negatively correlated LOS LOS Describe and interpret the minimum-variance and efficient frontiers of risky assets and the global minimum-variance portfolio What’s an Efficient Frontier Although there are no securities with perfectly negative correlation, almost all assets are less than perfectly correlated • Therefore, you can reduce total risk (𝜎p) through diversification If we consider many assets at various weights, we can generate the efficient frontier • The Efficient Frontier represents all the dominant portfolios in risk/return space  A portfolio dominates all others if no other equally risky portfolio has a higher expected return, or if no portfolio with the same expected return has less risk LOS LOS Describe and interpret the minimum-variance and efficient frontiers of risky assets and the global minimum-variance portfolio Example: • Assume you have the risk/return characteristics of 20 assets • With the help of the computer, you can calculate all possible portfolio combinations • The collection of all the minimum variance portfolios would form the minimum variance frontier • Along the minimum-variance frontier, the left-most point is a portfolio with minimum variance when compared to all possible portfolios the global minimum-variance portfolio • The Efficient Frontier is the portion of the minimum-variance curve that lies above and to the right of the global minimum variance portfolio (Dominance Rule) Describe and interpret the minimum-variance and efficient frontiers of risky assets and the global minimum-variance portfolio Illustration: Efficient frontier consisting of dominant portfolios Portfolio expected return LOS LOS Dominated (inefficient) portfolios Minimum variance frontier Global minimum variance portfolio Portfolio standard deviation LOS LOS Describe and interpret the minimum-variance and efficient frontiers of risky assets and the global minimum-variance portfolio Why the emphasis is on the efficient frontier? • Portfolios lying on the efficient frontier offer the maximum expected return for their level of variance of return • Efficient portfolios use risk efficiently: investors making portfolio choices in terms of mean return and variance of return can restrict their selections to portfolios lying on the efficient frontier This simplifies the selection process • If an investor can quantify his risk tolerance in terms of variance or SD of return, the efficient portfolio for that level of variance will represent Optimal-mean variance choice LOS LOS Explain the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line Utility Theory • Relative satisfaction from consumption • Quantification of rankings of choices • Utility is different for different investors • Utility can be increased (getting higher return or lower risk), • Utility can be decreased (increase in risk) Assumptions • Investors are risk averse • Possible to rank order • Rankings are internally consistent LOS LOS Explain the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line What’s an indifferent curve? • Combination of risk-return pairs acceptable to an investor for a given level of Utility • All points on a curve have the same utility (investor is indifferent) • A curve with higher return for a given risk is superior than a curve offering lower returns for the same risk Explain the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line High utility Moderate utility b1 Expected return LOS LOS a Low utility c Standard deviation Explain the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line High risk aversion Moderate risk aversion Low risk aversion Expected return LOS LOS Risk neutral Risk lover Standard deviation Note: Most risk averse => greatest slope; Least risk averse => least slope Explain the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line Capital Allocation Line (CAL) • The combination of a risk-free asset and a risky asset (the risky asset represents multiple portfolios available to the investor) E(R p ) The CAL equation: E(R i ) Expected return LOS LOS E Rp Rf 𝜎𝑖 𝜎𝑝 E Ri − Rf = Rf + σp σi Explain the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line To get the optimal portfolio, combine the indifference curves with the CAL: Expected return LOS LOS bc v e Optimal portfolio a c d Standard deviation LOS Portfolio Management • Portfolio Management – An Overview • Portfolio Risk and Return – Part I  Portfolio Risk and Return – Part II • Basics of Portfolio Planning and Construction • Risk Management – An Introduction • Fintech in Investment Management ... portfolios Portfolio expected return LOS LOS Dominated (inefficient) portfolios Minimum variance frontier Global minimum variance portfolio Portfolio standard deviation LOS LOS Describe and interpret... (Dominance Rule) Describe and interpret the minimum-variance and efficient frontiers of risky assets and the global minimum-variance portfolio Illustration: Efficient frontier consisting of dominant... $10 0  Option 2: Gamble of either $20 0 or $0? (expected value of $10 0) Investor Types • • • Risk Seeker (maximizes both risk and return) Risk Neutral (maximizes return irrespective of risk) Risk