Portfolio Theory and Behavioral Finance PDF

Portfolio Theory and Behavioral Finance (w.IN.23HS.V) Chapter 2: Investors Choice and Utility Asset Allocation and Diversification Building Competence. Crossing Borders. Dr.-Ing. Martin Schnauss, CFA, FRM / Dr. Jan-Ale...

Portfolio Theory and Behavioral Finance (w.IN.23HS.V) Chapter 2: Investors Choice and Utility Asset Allocation and Diversification Building Competence. Crossing Borders. Dr.-Ing. Martin Schnauss, CFA, FRM / Dr. Jan-Alexander Posth [email protected], [email protected], 2020 / Vers. 1.1.3 Learning Objectives Chapter 2 After this chapter you know: How to apply concepts as: – Investors choice and utility – certainty equivalent – Risk and return objectives of investors – The asset allocation process – Leveraging of portfolios Literature: Kritzman, Mark (1992): „What Practitioners Need to Know about Utility“. Financial Analysts Journal. 48 (3), pp. 17-20. -2- Topic Investors Choice and Utility Investors Choice and Utility -3- Utility Introduction Goal of the lecture: Introduction to utility theory and its application to financial analysis. An important assumption in financial theory is that rational investors seek to maximize expected utility. What about maximizing expected wealth? Consider the famous St. Petersburg paradox (first studied by the Swiss mathematician Daniel Bernoulli in the 18th century). – Suppose the following game is proposed: A coin is flipped until head appears. You win $2𝑖𝑖 if the first head appears on the 𝑖𝑖 th toss of the coin. – How much would you be willing to pay for the game? – What is the expected value of the game? -4- Utility Introduction (cont’d) – Expected value of the game: ∞ ∞ ∞ 1 𝑖𝑖−1 1 𝑖𝑖 −1 1 E 𝑋𝑋 = 𝑖𝑖 2 = 𝑖𝑖 2 2 = 1 2 2 2 𝑖𝑖=1 𝑖𝑖=1 𝑖𝑖=1 2 3 0 1 1 1 2 1 =2 ⋅ +2 ⋅ +2 ⋅ +⋯ 2 2 2 Probability to win 1’000 is approx. 0.05%... 1 1 1 = 1 ⋅ + 2 ⋅ + 4 ⋅ + ⋯ = ∞∞ 2 4 8 – Nobody would pay an infinite amount of money for this game because significant payoffs are not very likely. – Bernoulli’s solution to the paradox was to argue that people do not maximize expected wealth but expected utility and that utility was increasing with wealth at a decreasing rate. -5- Utility as a Function of Wealth Example of a utility function 𝑢𝑢 𝑥𝑥 which is increasing in wealth and exhibits diminishing marginal utility, that is – 𝑢𝑢′ 𝑥𝑥 > 0 (marginal utility is positive, i.e. more wealth is This is the slope of u preferred over less) – 𝑢𝑢′′ 𝑥𝑥 < 0 (diminishing marginal utility, as wealth This is the curvature (or increases, the increments to utility become concavity) of u or the smaller) slope of the slope of u – Kritzman, Figure A Utility is increasing with wealth at a decreasing rate. Slope decreases from “blue” to “red”  u’’ is negative -6- Utility as a Function of Wealth (cont’d) – Kritzman, Figure B A negative second derivative implies that we would experience greater disutility from a decline in wealth than the utility we would derive from an equal increase in wealth. – People usually refuse to play fair games. They prefer certainty. -7- Log-Utility Suppose utility is given by 𝑢𝑢 𝑥𝑥 = ln(𝑥𝑥) – Verify that marginal utility is positive for every wealth level 𝑥𝑥 > 0: 𝑢𝑢′ 𝑥𝑥 > 0 ′ 1 ln 𝑥𝑥 = > 0∀𝑥𝑥 > 0 𝑥𝑥 – Verify that the function exhibits diminishing marginal utility: ′ 𝑢𝑢′′ 𝑥𝑥 < 0 1 1 = − 2 < 0∀𝑥𝑥 ≠ 0 𝑥𝑥 𝑥𝑥 – Numerical example (Kritzman, p. 17) − How much increases utility if wealth increases from 100 to 150? Ln(100) = 4.60; Ln(150) = 5.01 ln 150 − ln 100 = 0.41 Δx=50 in both cases! − If wealth increases from 150 to 200? Ln(150) = 5.01; Ln(200) = 5.30 ln 200 − ln 150 = 0.29 -8- Expected Utility An important assumption in financial theory is that rational investors seek to maximize expected utility. What is the expected utility of the game considered before if we assume a log utility function? (Also assume for convenience that initial wealth equals 0.) ∞ ∞ ∞ 1 𝑖𝑖−1 𝑖𝑖 − 1 𝐸𝐸 𝑢𝑢 𝑋𝑋 = 𝑝𝑝𝑖𝑖 𝑢𝑢 𝑥𝑥𝑖𝑖 = 𝑖𝑖 ln 2 = 𝑙𝑙𝑙𝑙(2) 𝑖𝑖 2 2 𝑖𝑖=1 𝑖𝑖=1 𝑖𝑖=1 = ln(2) ≈ 0.69 -9- Risk Aversion 𝑢𝑢′′ 𝑥𝑥 < 0 If the utility function is concave (that is, exhibits diminishing marginal utility), fair bets are avoided. Given log-utility, what is the expected utility in the following cases – $100 with certainty (initial wealth)? – $90 with probability 0.5 and $110 with probability 0.5? – $50 with probability 0.5 and $150 with probability 0.5? - 10 - Certainty Equivalent Definition: The amount of payoff (e.g. money or utility) that an agent would have to receive to be indifferent between that payoff and a given gamble is called that gamble's 'certainty equivalent'. For a risk averse agent (as most are assumed to be) the certainty equivalent is less than the expected value of the gamble because the agent prefers to reduce uncertainty. (Source: http://economics.about.com/) Given log-utility, what is the certainty equivalent of the following gambles: – $90 with probability 0.5 and $110 with probability 0.5? ln 𝐶𝐶 = 0.5 ln 90 + 0.5 ln 110 Inverse of ln() is exp() 𝐶𝐶 = 𝑒𝑒 0.5 ln 90 +0.5 ln 110 = 99.5 – $50 with probability 0.5 and $150 with probability 0.5? ln 𝐶𝐶 = 0.5 ln 50 + 0.5 ln 150 𝐶𝐶 = 𝑒𝑒 0.5 ln 50 +0.5 ln 150 = 86.6 - 11 - Graphical Representation 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 ln(150) ln(𝑥𝑥) ln(C) ln(50) 86.6 50 100 150 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 (𝑥𝑥) - 12 - Risk Premium The certainty equivalent of $86.60 means that we would be indifferent between getting $86.60 with certainty or getting $50 with a probability of 0.5 and $150 with probability 0.5. The difference in expected value that would just induce us to choose the risky prospect over the certain outcome is called the required risk premium. In our example the risk premium is 0.5 ⋅ 50 + 0.5 ⋅ 150 − 86.60 = 100 − 86.60 = $13.4 or 13.4/86.6 = 15.47% - 13 - Risk Premium (cont’d) Assume , instead, that someone is starting with initial wealth of $1,000, and ends up with $950 or $1,050, each with probability of 0.5. The certainty equivalent would then be $998.75 and the risk premium $1.25 or 0.125%. The wealthier person is less risk averse in absolute terms because the $50 gamble is just a small fraction of his wealth.  Log utility implies decreasing absolute risk aversion. If the potential gain or loss would be $500 (also 50% of initial wealth), than the certainty equivalent would be $866 and the – risk premium = 1000- 866 = $134, or – also 134/866 = 15.47%.  Log utility implies constant relative risk aversion. - 14 - Willingness to Pay for Insurance Imagine an individual that owns $100,000 and there is a probability of 1% that he loses $60,000 (say by a fire in his flat) during the next year. Further assume that his utility is logarithmic. Without insurance, the expected utility of this person will be 𝐸𝐸 𝑢𝑢(𝑋𝑋) = 0.99 ⋅ ln 100000 + 0.01 ln 40000 = 11.50376 The fair insurance premium in our example would be 0.01 ⋅ 60000 = $600 If the person gets complete insurance, his utility would be And that’s why everybody is 𝑢𝑢 100000 − 600 = ln 99400 = 11.50691 happy to overpay which is more than without insurance. for insurance… Up to how much money would this person pay for insurance? ln 100000 − 𝒙𝒙 = 11.50376 Therefore he would pay up to $912. - 15 - Measures of Risk Aversion Economists distinguish between risk aversion, risk neutrality and risk seeking. Empirical work shows that most people are risk averse. Kritzman (p. 19) states that economists distinguish between the absolute amount of one’s wealth that is exposed to risk versus the proportion of one’s wealth that is exposed to risk, and whether this amount decreases, remains constant or increases as wealth increases. - 16 - Measures of Risk Aversion (cont’d) The more concave the utility function the higher the degree of risk aversion. Important quantitative measures of risk aversion were introduced by Arrow and Pratt. Measure of absolute risk aversion: Remember slide 8! 𝑢𝑢′′ 𝑥𝑥 𝐴𝐴𝐴𝐴𝐴𝐴 𝑥𝑥 = − ′ 𝑢𝑢 𝑥𝑥 Example: If 𝑢𝑢 𝑥𝑥 = ln 𝑥𝑥 , then 𝐴𝐴𝐴𝐴𝐴𝐴 𝑥𝑥 = 1/𝑥𝑥. Therefore, log utility implies decreasing absolute risk aversion (as wealth increases). Measure of relative risk aversion: 𝑢𝑢′′ 𝑥𝑥 𝑅𝑅𝑅𝑅𝑅𝑅 𝑥𝑥 = 𝑥𝑥 ⋅ 𝐴𝐴𝐴𝐴𝐴𝐴 𝑥𝑥 = −𝑥𝑥 ′ 𝑢𝑢 𝑥𝑥 Example: If 𝑢𝑢 𝑥𝑥 = ln 𝑥𝑥 , then 𝑅𝑅𝑅𝑅𝑅𝑅 𝑥𝑥 = 1. Therefore, log utility implies constant relative risk aversion. - 17 -  Investors Choice and Utility Kritzman, p. 19f. In many applications, it is convenient to model utility as a function of expected return and risk. 𝑢𝑢 𝜇𝜇, 𝜎𝜎 = 𝜇𝜇 − 𝐴𝐴𝜎𝜎 2  𝑢𝑢 𝜇𝜇, 𝜎𝜎 : Utility dependent on expected return 𝜇𝜇 = 𝐸𝐸 𝑅𝑅 and risk 𝜎𝜎  𝐴𝐴: Measure of risk aversion. 𝐴𝐴 > 0 for risk averse individuals. The higher 𝐴𝐴, the more risk averse the individual is. 𝜕𝜕𝑢𝑢 𝜇𝜇,𝜎𝜎  > 0: Utility increases with expected return, ceteris paribus 𝜕𝜕𝜕𝜕 𝜕𝜕𝑢𝑢 𝜇𝜇,𝜎𝜎  < 0: Utility decreases with risk, ceteris paribus 𝜕𝜕𝜕𝜕 - 18 -  Investors Choice and Utility Different choices with different risk aversions. - 19 -  Investors Choice and Utility Risk in relation to utility and return 𝑢𝑢 𝜇𝜇, 𝜎𝜎 = 𝜇𝜇 − 𝐴𝐴𝜎𝜎 2 i.e.: 𝜇𝜇−𝑢𝑢 𝜇𝜇,𝜎𝜎 𝜎𝜎 = 𝐴𝐴 - 20 -  Investors Choice and Utility Risk in relation to utility and return 𝜇𝜇−𝑢𝑢 𝜇𝜇,𝜎𝜎 𝜎𝜎 = 𝐴𝐴 The graphical representation of this table gives us an indifference curve. - 21 -  Investors Choice and Utility Risk in relation to utility and return 𝜇𝜇−𝑢𝑢 𝜇𝜇,𝜎𝜎 𝜎𝜎 = 𝐴𝐴 The graphical representation of this table gives us an indifference curve. - 22 - Investors Choice and Utility Indifference Curves The investor (with the given utility function) is indifferent between different combinations of risk and expected return along a particular indifference curve. Indifference curves that are closer to the upper left yield more utility and thus are more desirable. 𝑢𝑢 = 𝜇𝜇 − 2𝜎𝜎 2 - 23 - Investors Choice and Utility Summary Interpretation: The higher the utility of an investment, the more attractive is the investment to the investor. The higher the expected return of an investment, the higher is its utility. The higher the risk (variance) of an investment, the lower is its utility. Risk averse (rational) investors like return and hate risk. - 24 - Investors Choice and Utility Summary The investor specific risk aversion is the balancing factor between risk and return. From a set of portfolios with different expected returns E(r) and investment risks σ, investors will chose the most useful portfolio, i.e. the one that maximizes their utility. Yes, we are maximizers of reward. But we are also minimizers of regret. - 25 - Investors Choice and Utility Summary Within a risk-return diagram curves of constant utility U (so called “indifference curves”) can be used to rank investments according to their utility. These curves are constructed in the following way: Given the utility U the expected return E(r) is expressed as a function of risk s: 1 𝐸𝐸 𝑟𝑟 = 𝑈𝑈 + × 𝐴𝐴 × 𝜎𝜎 2 2 Introducing ½ in the equation is just a convention; remember slide 18! - 26 - Investors Choice and Utility Summary Interpretation: In a risk-return diagram indifference curves have the shape of parabolas. All investments on the indifference curve have the same utility (for the investor). Investments that lie to the left or above the indifference curve have higher utility than investments below or to the right of the indifference curve. The higher the investor’s risk aversion, the steeper the indifference curve. - 27 - Topic Asset Allocation Asset Allocation - 28 - Asset Allocation: Complete Portfolio Definition: The asset allocation decision is an investment decision on how to distribute the available funds among broad investment classes like risk-free/money market, bonds, stocks, real-estate, …. As a result of the asset allocation process, investors should end up with a portfolio that corresponds to their risk and return objectives. - 29 - Asset Allocation: Complete Portfolio The most fundamental choice investors have to make is how to allocate their money between the following two investment options: I. Risky investment (P) II. Risk free investment (F) - 30 - Asset Allocation: Complete Portfolio Example: Investment of CHF 400,000, of which CHF 100,000 are invested risk free (money market fund) and the rest goes into two equity funds CH-BlueChips CHF 200,000, and CH-SmallCaps CHF 100,000. The allocation of the money follows two steps: 1. Split between risk-free and risky investment: a proportion y of the investors total assets is invested risky and a proportion 1-y is invested risk-free. 2. Splitting up the risky investment among all the possible risky assets: a proportion 𝑤𝑤𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 of the risky investment is invested in the BlueChip fund and a proportion 𝑤𝑤𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 in the SmallCap fund. - 31 - Asset Allocation: Complete Portfolio Example (cont.): Investment of CHF 400,000, of which CHF 100,000 are invested risk free (money market fund) the rest goes into two equity funds CH- BlueChips CHF 200,000, and CH-SmallCaps CHF 100,000. - 32 - Asset Allocation: Complete Portfolio The weight y of the risky part as a percentage of the total (“complete”) portfolio, including the risk free asset, is 𝑦𝑦 = 75% 1 − 𝑦𝑦 = 25% Question: What happens if the investor wants to reduce the risk of the portfolio, reducing the risky investment to 𝑦𝑦 = 50%? - 33 - Regrouping of investments Example cont.: (Regrouping of investments) The weight y of the risky part as a percentage 𝑦𝑦 = 50% Assumption: The relative weighting within the risky part of the investment remains constant. - 34 - Regrouping of investments As long as the relative weighting within the risky fund remains unchanged we do not have to be concerned with the individual securities or sub-funds in the risky part of the portfolio (here: CH- BlueChips and CH-SmallCaps). The probability distribution of the returns of the risky part remains unchanged. Therefore, it is sufficient to consider the risky part as one single risky investment with fixed risk and return properties. Analogy: view the risky investment as a fund of funds or strategy fund with fixed proportions. - 35 - Regrouping of investments However, the return distribution of the complete portfolio changes when assets are shifted between the risk-free and risky parts. In our further analysis in a first step we focus on complete portfolios, i.e. portfolios that consist of only one risk-free asset (F) and one risky asset (P). In a second step, we will study the risk and return properties of different combinations of two or many risky assets. - 36 - Expected Return and Risk of Portfolios Having agreed on the construction of complete portfolios we now regard the risk-return characteristics of different weightings of the risk free and risky parts. Example: E( rP ) = 15%; 𝜎𝜎𝑃𝑃 = 20% ; rf = 4% How does the risk return relationship change for different combinations of the risk-free part (1 – y) and the risky part y? - 37 - Expected Return and Risk of Portfolios Return Leverage ON! Risk - 38 - Expected Return and Risk of Portfolios Interpretation: What is the risk premium of the risky part P? 𝐸𝐸 𝑟𝑟𝑃𝑃 − 𝑟𝑟𝑓𝑓 = 15% − 4% = 11% What are the risk premium E(rC) - rf and the standard deviation σC of a complete portfolio C? Weights are 50%-50% 𝜎𝜎𝐶𝐶 = 𝑦𝑦 × 𝜎𝜎𝑃𝑃 = 0.5 × 20% = 10% 𝐸𝐸 𝑟𝑟𝐶𝐶 − 𝑟𝑟𝑓𝑓 = 𝑦𝑦 × 𝐸𝐸 𝑟𝑟𝑃𝑃 − 𝑟𝑟𝑓𝑓 = 0.5 × 11% = 5.5% Also: (50% * 15% + 50% * 4%) – 4% = (7.5% + 2.0%) – 4% = 9.5% - 4% - 39 - Expected Return and Risk of Portfolios Consider the ratio S between risk and risk premium: 100% risky portfolio 𝐸𝐸 𝑟𝑟𝑃𝑃 − 𝑟𝑟𝑓𝑓 15% − 4% S= = = 0.55 𝜎𝜎𝑃𝑃 20% This Ratio S describes the reward (return) you get for taking one (additional) unit of risk. Definition: The slope S of the Capital Allocation Line (CAL) is a parameter which describes the reward-to-variability ratio. It is also known under the name Sharpe-Ratio. - 40 - Capital Allocation Line (CAL) Comments: 1. Risk 𝜎𝜎𝐶𝐶 and risk-premium 𝐸𝐸 𝑟𝑟𝐶𝐶 − 𝑟𝑟𝑓𝑓 of a complete portfolio C are proportional to the share y of the risky investment. 2. The reward-to-variability expresses, what excess return an investor can expect for one additional unit of risk. 3. The reward-to-variability ratio remains constant for all possible complete portfolios, i.e. for all portfolios on the CAL. 4. Two equivalent formulas for the return of the complete portfolio Mathematically, this is a 𝐸𝐸 𝑟𝑟𝐶𝐶 = 𝑟𝑟𝑓𝑓 + 𝑦𝑦 × 𝐸𝐸 𝑟𝑟𝑃𝑃 − 𝑟𝑟𝑓𝑓 straight line… 𝐸𝐸 𝑟𝑟𝐶𝐶 = 𝑦𝑦 × 𝐸𝐸 𝑟𝑟𝑃𝑃 + 1 − 𝑦𝑦 × 𝑟𝑟𝑓𝑓 The first one is called the equation of the capital allocation line. - 41 - Capital Allocation Line (CAL) Can we construct complete portfolios with a higher risk-premium than the risky portfolio P? → increase the share y of the risky investment above 1 ! Leverage! If y>1 than the risk-free investment 1-y is negative, which means borrowing money (taking a loan). The resulting complete portfolio is called a leveraged portfolio. Example: Available capital CHF 100,000 Additional capital (loan) CHF 50,000 Total risky investment CHF 150,000 150,000 Share of risky investment: 𝑦𝑦 = = 1.5 100,000 Risk-free investment (loan!): 1 − 𝑦𝑦 = 1 − 1.5 = −0.5 Complete portfolio return 𝐸𝐸 𝑟𝑟𝐶𝐶 = 𝑦𝑦 × 𝐸𝐸 𝑟𝑟𝑃𝑃 + 1 − 𝑦𝑦 × 𝑟𝑟𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 - 42 - Capital Allocation Line (CAL) Example (cont.): Complete portfolio: Expected Return: 𝐸𝐸 𝑟𝑟𝐶𝐶 = 𝑦𝑦 × 𝐸𝐸 𝑟𝑟𝑃𝑃 + 1 − 𝑦𝑦 × 𝑟𝑟𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝐸𝐸 𝑟𝑟𝐶𝐶 = 1.5 × 15% + (−0.5) × 4% = 22.5% − 2% = 20.5% Risk: 𝜎𝜎𝑐𝑐 = 𝑦𝑦 × 𝜎𝜎𝑃𝑃 𝜎𝜎𝑐𝑐 = 1.5 × 20% = 30% Slope S of the Capital Allocation Line: 𝐸𝐸 𝑟𝑟𝐶𝐶 −𝑟𝑟𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 20.5%−4% S= 𝜎𝜎𝐶𝐶 = = 0.55 30% Attention: S remained at 0.55. (because 𝑟𝑟𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 = 𝑟𝑟𝑓𝑓 ) - 43 - Capital Allocation Line (CAL) Example: Excel Return Leverage ON! Risk - 44 - Capital Allocation Line (CAL) More realistic scenario: What happens if we allow for a borrowing rate that is higher than the risk-free rate? Now rf = 4% as before, but rB = 8% > rf Which of the following three numbers changes? 𝐸𝐸 𝑟𝑟𝐶𝐶 = 1.5 × 15% + (−0.5) × 8% = 22.5% − 4% = 18.5% 𝜎𝜎𝑐𝑐 = 1.5 × 20% = 30% Slope of the CAL for leveraged portfolios is now: 𝐸𝐸 𝑟𝑟𝐶𝐶 − 𝑟𝑟𝐵𝐵 18.5% − 8% Sleveraged = = = 0.35 𝜎𝜎𝐶𝐶 30% Due to higher borrowing costs, any additional unit of risk beyond the risky portfolio P is rewarded by only 0.35% additional risk premium! - 45 - Capital Allocation Line (CAL) Example: Excel Return Leverage ON! Risk - 46 - Capital Allocation Line (CAL) Graphic Representation: Slope of the CAL to the left of P («normal» portfolios): S=0.55 Slope of the CAL to the right of P («leveraged» portfolios): SL=0.35 P - 47 - Capital Allocation Line (CAL) Why is the Slope lower? Slope S - 48 - Topic Asset Combination Asset Combination and Diversification - 49 - Capital Allocation Line (CAL) Question: How do investors choose among complete portfolios on two different CALs? CAL 1 CAL 2 Remark: Every investor will only choose portfolios on the CAL with the higher Sharpe-ratio. - 50 - Capital Allocation Line (CAL) Investing in risky asset naturally comes with the risk of incurring losses. One of the most important insights in modern finance is: Combining several risky assets within a portfolio effectively stabilizes the return by lowering the portfolio risk. This follows the well-known dictum: “Don’t put all your eggs in one basket”. - 51 - Diversification and Portfolio Risk Example: Take two arbitrary assets and construct a portfolio consisting of these two assets, each with a weighting 𝑤𝑤 between 0% and 100%. Question: What is the risk-return characteristic of the resulting portfolio? Return/Risk of a combined Portfolio Expected Return Risk - 52 - Diversification Correlation vs. Cosinus theorem The Law of Cosines (also called the Cosine Rule) is very useful for solving triangles: Law of Cosines c 2 = a 2 + b 2 − 2 ⋅ a ⋅ b ⋅ cos(γ ) Note: - 53 - Diversification Correlation vs. Cosinus theorem Two positions a (Nestlé) and b (SMI-ETF) with volatilities σa, σb and correlation ρa,b We combine the two positions to form a portfolio c = a + b σ2c = σ2a + σb2 + 2 ⋅ σa ⋅ σb ⋅ ρa,b We can interpret this geometrically Equivalent to vector addition! σc cos(γ ) = −ρa,b σb γ cos(180° − γ ) = ρa,b σa - 54 - Diversification Correlation vs. Cosinus theorem σa σb γ = 180o ρ=1 Correlation tells us σc “how much of b lies σc in the direction of a” ρ = 0.5 σb ρ = 0 σb γ = 120 o γ = 90o σa σa σc σb ρ = -0.5 γ = 0o γ = 60o σc σb ρ = -1 σa - 55 - Topic Asset Combination Diversification and Portfolio Risk - 56 - Diversification Portfolio with 100 assets The total risk of the portfolio is given as σP2 = ∑ w i2σi2 + ∑ σij w i w j i i≠ j How may variance terms? How many covariance terms? What happens if we add an other instrument to the portfolio? 𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑: Variance, covariance & correlation… - 57 - Diversification Assume that we have n equally weighted assets in a portfolio For the Portfolio variance we get 1 2 1 σ = ∑ 2 σ i + ∑ 2 σ ij 2 For a portfolio with many equally p weighted assets: i n i≠ j n - the variances of the assets diversify away but  ∑ σ i2   ∑ σ ij  - the risk caused by the co- movements of the assets can 1  + n − 1  i≠ j  =  i never be diversified away n n  n  n(n − 1)        𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍: For very large n, 1/n 1 2  1 will be very small aka 0 = σ avg + 1 − σ avg n var_avg  n  covar_avg - 58 - Diversification The mathematical way… The Covariance Matrix contains the information of how assets move and interact Asset 1 Asset 2 Asset N How do How do Asset 1 How does Asset 1 Asset 2 & Asset N & 2 Asset 1 Asset 1 co- … Asset 1 σ σ1,2 …  σ1,N  move? move? co-move?  1 2  How do  σ2 ,1 σ2   … How do … How does Asset 2 Asset 2 Asset 1 & V = Asset N & σN−1,N  Asset 2 Asset 2 Asset 2 co-    move? … co-move? move? σ … σN,N−1 σN2  … …  N,1  … How do How do Asset N How does Asset N Asset 1 & Asset 2 & Asset N co- Asset N co- … Asset N move? move? move? Asset 1 Asset 2 Asset N - 59 - Diversification The mathematical way… We will only use matrix notation as an easy way to write and interpret formulas We do not have to know the whole theory behind linear objects (linear algebra) N elements per column 3 elements per column Row 1 R1C1 R1C2 R1C3 R1C4 Column 1 Column 2 Column 3 Column 4 Row 2 R2C1 R2C2 R2C3 R2C4 Row 3 R3C1 R3C2 R3C3 R3C4 N elements per row 4 elements per row If Row 2 = (a1, a2,…, aN) and Column 3 = (b1, b2,…, bN), then R2C3 = (a1b1 + a2b2 +… + aNbN) - 60 - Diversification The mathematical way… 𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑: Variance, covariance & Volatility Expected return over cash correlation… – Asset A: 20% – Asset A: 12% – Asset B: 25% – Asset B: 15% Correlation between A and B: 0.5  4% 2.5 %   1 0.5   0.4   0.12  V =  C=  w =  μ=   2.5% 6.25%   0.5 1   0.6   0.15   4% 2.5 %   0.4  σ = w ' ⋅ V ⋅ w = (0.4 0.6 ) ⋅  2 P ⋅ =  2. 5 % 6. 25 %   0. 6   0.4  = (3.1% 4.75% ) ⋅   = 4.09%  0. 6  σP = 20.22%  0.12  µP = w ' ⋅ μ = (0.4 0.6 ) ⋅   = 13.8%  0.15  - 61 - Diversification and Portfolio Risk Remark: Diversification benefits (risk reduction) exists as long as the returns of the two assets do not always behave in exactly the same way, i.e. if the correlation 𝜌𝜌 of their returns is 𝜌𝜌 < +1! Only if the two assets behave always in exactly the same way (i.e. their correlation is 𝜌𝜌 = +1), then the effect of diversification ceases to exist. - 62 - Topic Asset Combination Insertion: Splitting the risk - 63 - Diversification and Portfolio Risk What happens if we continue this diversification strategy? We construct portfolios with more and more assets: n = 10 100 1000 10000 100000 … (n is the number of assets within the portfolio) Question: How will portfolio risk or portfolio variance develop, as the number of assets in the portfolio grows? - 64 - Diversification and Portfolio Risk Systematic Risk - 65 - Diversification and Portfolio Risk Interpretation: The variance (risk) of a portfolio decreases (quite rapidly) to the level of the so called systematic risk if more and more assets are included in the portfolio. Generally, “good” diversification (i.e. a significant reduction of specific risk) can already be achieved with about 15 to 20 assets. 𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍: E.g. CTAs may have up to 200 - 66 - Splitting the Risk Systematic risk: A risk that is carried by an entire class of assets or to the entire global economy. It is impossible to reduce systematic risk for the global economy by diversification. Systematic risk is the risk that remains in fully diversified portfolio (e.g. risk of a broad stock index). But: One may mitigate some of the systematic risk by buying securities of different asset classes and/or by buying in different markets/countries. - 67 - Splitting the Risk Question: How can we understand this risk reduction and the remaining level of systematic risk? Following a top-down-approach risk emerges on several levels. For instance: Markets: GDP growth, interest rates, geopolitical events … Sector: commodity prices (e.g. oil); regulatory changes … Company: CEO resigns or new CEO; launch of new product; internal fraud … - 68 - Splitting the Risk The risk is not in the listed factors per se, but in the uncertainty of their future development and – most importantly – in the fact that we do not know the influence of this development on future returns. If shares of more and more companies and sectors are included in the portfolio, the effects of company and sector specific risk factors compensate (diversify away) and only the effect of the systematic or market risk factors remain in the portfolio returns. - 69 - Splitting the Risk The risk of a single asset or a portfolio can be split into two parts: Risk Systematic Unsystematic Non-diversiviable diversiviable Unique, company-specific or Market risk idiosyncratic risk - 70 - Splitting the Risk Systematic risk, to a large extent, results from macro-economic factors and influences all assets, while title or company-specific risk is of concern only to the company in question. Through a skillful choice of assets and markets (⇒ national and international diversification) title-specific risk can, to a large extent, be eliminated. What remains in the portfolio is systematic or market risk. - 71 - Splitting the Risk 𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍: Nowadays, more and Modern PF-Theory (MPT) states: more Alternative Risk Premiums are identified! Only systematic risk is rewarded by a sustainable risk premium. Systematic risk reflects the reaction of individual assets and portfolios to fluctuations in the market. Remark: The beta factor (𝛽𝛽 factor) describes how sensitively individual assets and/or portfolios react to fluctuations in the market and/or macro-economic factors. - 72 -