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Foundations of Risk Management: the Modern FOUNDATIONS OF RISK MANAGEMENT: THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER 푛 퐸 푋 = 휇푥 = 푃(푋 = 푋푖)( 푋푖) 푖=1 FOR TWO VARIABLES 퐸 푋 + 푌 = 푃 푋 = 푋푖, 푌 = 푌푗 푋푖 + 푌푗 = 푃푖푗 푋푖 + 푌푗 = 푖 푗 푖 푗 = 푃푖푗 푋푖 + 푃푖푗 푌푗 = 푃푖푋푖 + 푃푗푌푗 = 푬 푿 + 푬(풀) 푖 푗 푖 푗 푖 푖 푃푖푗 = 푃푖 푃푖푗 = 푃푗 푗 푖 FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 2 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER The expected return on portfolio is a weighted sum of returns on individual assets 푁 퐸 푅푝 = 휔푖 퐸 푅푖 푖=1 퐸 푅푝 – expected return on portfolio 푤푖 – weight of Asset i in portfolio 퐸 푅푖 – expected return on Asset i All weights should sum up to 100% of portfolio FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 3 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER Old (alternative approach) is the evaluate each investment opportunity on its own N σp = ωi σi i=1 Let’s consider the following example Portfolio consist of two assets $100 is invested in asset A (standard deviation σA = $4) $100 is invested in asset B standard deviation σB = $6 N 1 1 σ = ω σ = × $4 + × $6 = $5 p i i 2 2 i=1 FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 4 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER In 1950s Harry Markowitz provided a framework for measuring risk- reduction benefits of diversification; he concluded that, unless the returns of risky assets are perfectly positively correlated, risk is reduced by diversifying across assets Markowitz used standard deviation as a measure of risk of the assets; it is still used as the best proxy Diversification – process of including additional different assets in the portfolio in order to minimize market risk (i.e. include bonds and ETFs to all-stock portfolio) The modern portfolio theory (MPT) was founded in 1960s by several independent scientific studies FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 5 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER n 2 2 2 Var X = (σx) = E X − E X X − E X = E (X − E(X)) = P(Xi)(Xi − E(X)) i=1 FOR TWO VARIABLES Var X + Y = E (X + Y − E(X + Y))2 = = E (X + Y − E X − E(Y))2 = E (퐗 − 퐄 퐗 + 퐘 − 퐄(퐘))2 = 퐄 퐗 − 퐄 퐗 퐗 − 퐄 퐗 + 퐄 퐘 − 퐄 퐘 퐘 − 퐄 퐘 + 퐄 퐗 − 퐄 퐗 퐘 − 퐄 퐘 + 퐄 퐘 − 퐄 퐘 퐗 − 퐄 퐗 = 퐂퐨퐯 퐗, 퐗 + 퐂퐨퐯 퐘, 퐘 + 퐂퐨퐯 퐗, 퐘 + 퐂퐨퐯 퐘, 퐗 = 퐕퐚퐫 퐗 + 퐕퐚퐫 퐘 + ퟐ × 퐂퐨퐯 퐗, 퐘 FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 6 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER Amount Expected Expected Correlation invested return volatility ABC 20000 7% 15% 0.3 XYZ 30000 11% 22% Investor has invested cash in two companies: ABC and XYZ Weights of companies in portfolio are 20000 ◦ 푤 = = 0.4 퐴퐵퐶 20000+30000 30000 ◦ 푤 = = 0.6 푋푌푍 20000+30000 FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 7 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER Expected return 퐸 푅푝 = 푤퐴퐵퐶 ∙ 퐸 푅퐴퐵퐶 + 푤푋푌푍 ∙ 퐸 푅푋푌푍 = 0.4 ∙ 7% + 0.6 ∙ 11% = 9.4% Expected volatility 2 2 2 2 푝 = 푤1 ∙ 1 + 푤2 ∙ 2 + 2 ∙ 푤1 ∙ 푤2 ∙ 1,2 ∙ 1 ∙ 2 = = 0.42 ∙ 0.152 + 0.62 ∙ 0.222 + 2 ∙ 0.4 ∙ 0.6 ∙ 0.3 ∙ 0.15 ∙ 0.22 = 16.05% FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 8 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER ABC and XYZ example shows only one allocation of capital between two assets Expected return and risk of portfolio depends on the allocation ABC/ XYZ 100%/0% 80%/20% 60%/40% 50%/50% 40%/60% 20%/80% 0% /100% 퐸 푅푝 7.00% 7.80% 8.60% 9.00% 9.40% 10.20% 11.00% 푝 15.00% 13.97% 14.35% 15.06% 16.05% 18.72% 22.00% FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 9 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER PORTFOLIO POSSIBILITIES CURVE (RETURN) 12,00% 11,00% 11,00% 10,20% 10,00% 9,40% 9,00% 8,60% 9,00% 8,00% 7,80% 7,00% 7,00% 6,00% 10,00% 15,00% 20,00% 25,00% FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 10 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER PORTFOLIO POSSIBILITIES CURVE (RISK) 12,00% 11,00% 22,00% 10,00% 18,72% 16,05% 9,00% 14,35% 15,06% 8,00% 13,97% 7,00% 15,00% 6,00% 10,00% 15,00% 20,00% 25,00% FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 11 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER Minimum variance portfolio is a portfolio that has the minimum variance among all possible allocation of capital between assets. Allocation weights are solution of following problem 2 2 2 2 2 푝 = 푤1 ∙ 1 + 푤2 ∙ 2 + 2 ∙ 푤1 ∙ 푤2 ∙ 1,2 ∙ 1 ∙ 2 → 푚푛 푤1 + 푤2 = 1 푤1 = 75.34%, 푤2 = 24.66% MINIMUM VARIANCE PORTFOLIO 12,00% 11,00% 10,00% 9,00% E(R)=7.99%, 8,00% s.d.=13.93% 7,00% 6,00% 10,00% 15,00% 20,00% 25,00% FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 12 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER Correlation impact on PPC 12,00% 11,00% Corr = 1 10,00% Corr = 0.7 9,00% Corr = 0.3 Corr = 0 8,00% Corr = -0.5 7,00% Corr = -1 6,00% 0,00% 5,00% 10,00% 15,00% 20,00% 25,00% FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 13 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER Expected return INEFFICIENT PORTFOLIOS Risk-free Portfolio risk (σ) FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 14 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER . All portfolios on efficient frontier are made up with risky assets . Risk-free assets earns some return (at risk-free rate) and this return is expected to have zero volatility . Combination of risk-free asset and portfolio gives a new set of portfolios that form a line Expected return of combinations 퐸 푅퐶 = 푤퐹푅퐹 + 푤푃퐸 푅푃 Volatility of returns 2 2 2 2 2 퐶 = 푤퐹 퐹 + 푤푃푃 + 2푤퐹푤푃퐶표푣퐹,푃 Volatility of risk-free asset is zero, so its variance and covariance with risky portfolio are zero as well 2 2 2 퐶 = 푤푃푃 ; 푐 = 푤푃푝 FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 15 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER Expected return CAL (C) CAL (B) (B) CAL (A) (C) (A) Risk-free Portfolio risk (σ) FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 16 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER Expected return and volatility of combination of risk-free asset and risky portfolio are linear functions Relationship of expected return and volatility is also linear 퐸 푅퐶 = 푤퐹푅퐹 + 푤푃퐸 푅푃 푐 = 푤푃푝 푤퐹 + 푤푃 = 1 퐶 푤푃 = 푃 푤퐹 = 1 − 푤푃 퐶 퐶 퐸 푅퐶 = 1 − 푃 푅퐹 + 푃 ∙ 퐸 푅푃 푬 푹푷 − 푹푭 푬 푹푪 = 푹푭 + ∙ 흈푪 흈푷 FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 17 INTERPRET THE CAPITAL MARKET LINE Newly formed line that is tangent to efficient frontier is called Capital Market Line (CML) • the intercept equals a risk-free rate • the tangency point is known as a market portfolio • the slope equals a reward-to-risk ratio of risky (market) portfolio If investors have same expectations of risk and return for assets, all they will hold combination of risk-free asset and market portfolio • More risk-averse investors will buy less part of market portfolio and lend cash at risk-free rate • More risk-tolerant investors will borrow cash and buy more market portfolio FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 18 INTERPRET THE CAPITAL MARKET LINE Expected return CML 퐸 푟푚 − 푟푓 퐶푀퐿: 퐸 푟 = 푟푓 + 푀 Tangency portfolio (Market Portfolio) 퐸 푅푀 − 푅퐹 Risk-free 푇ℎ푒 푠푙표푝푒 표푓 퐶푀퐿 = 푀 Portfolio risk (σ) FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 19 INTERPRET THE CAPITAL MARKET LINE Differential Borrowing and Lending Rates. Most investors can lend unlimited amounts at the risk-free rate by buying government securities, but they must pay a premium relative to the prime rate when borrowing money. The effect of this differential is that there will be two different lines going to the Markowitz efficient frontier. Expected return New Tangency Portfolio Borrowing rate (Market Portfolio) Risk-free Portfolio risk (σ) FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 20 UNDERSTAND THE DERIVATION AND COMPONENTS OF THE CAPM DESCRIBE THE ASSUMPTIONS UNDERLYING THE CAPM The CAPM is an equilibrium model that predicts the expected return on a stock, given the expected return on the market, the stock's beta coefficient, and the risk-free rate.
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