FOUNDATIONS OF RISK MANAGEMENT: THE MODERN PORTFOLIO … · 2020. 9. 10. · FRM® PART I THE...

FOUNDATIONS OF RISK MANAGEMENT: THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL

THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I

EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

2

𝐸 𝑋 = 𝜇𝑥 = 𝑃(𝑋 = 𝑋𝑖)(

𝑛

𝑖=1

𝑋𝑖)

FOR TWO VARIABLES

𝐸 𝑋 + 𝑌 = 𝑃 𝑋 = 𝑋𝑖 , 𝑌 = 𝑌𝑗𝑗𝑖

𝑋𝑖 + 𝑌𝑗 = 𝑃𝑖𝑗𝑗𝑖

𝑋𝑖 + 𝑌𝑗 =

= 𝑃𝑖𝑗𝑗𝑖

𝑋𝑖 + 𝑃𝑖𝑗𝑗𝑖

𝑌𝑗 = 𝑃𝑖𝑋𝑖

𝑖

+ 𝑃𝑗𝑌𝑗𝑖

= 𝑬 𝑿 + 𝑬(𝒀)

𝑃𝑖𝑗𝑗

= 𝑃𝑖 𝑃𝑖𝑗𝑖

= 𝑃𝑗


The expected return on portfolio is a weighted sum of returns on individual assets

𝐸 𝑅𝑝 = 𝜔𝑖

𝑁

𝑖=1

𝐸 𝑅𝑖

𝐸 𝑅𝑝 – expected return on portfolio

𝑤𝑖 – weight of Asset i in portfolio

𝐸 𝑅𝑖 – expected return on Asset i

All weights should sum up to 100% of portfolio


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THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I 4


Old (alternative approach) is the evaluate each investment opportunity on its own

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

σp = ωi

N

i=1

σi =1

2× $4 +

1

2× $6 = $5

σp = ωi

N

i=1

σi



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



Var X = (σx)2= E X − E X X − E X = E (X − E(X))2 = P(Xi)(Xi − E(X))2

n

i=1

FOR TWO VARIABLES

Var X + Y = E (X + Y − E(X + Y))2 = = E (X + Y − E X − E(Y))2 = E (𝐗 − 𝐄 𝐗 + 𝐘 − 𝐄(𝐘))2 = 𝐄 𝐗 − 𝐄 𝐗 𝐗 − 𝐄 𝐗 + 𝐄 𝐘 − 𝐄 𝐘 𝐘 − 𝐄 𝐘 + 𝐄 𝐗 − 𝐄 𝐗 𝐘 − 𝐄 𝐘 + 𝐄 𝐘 − 𝐄 𝐘 𝐗 − 𝐄 𝐗 = 𝐂𝐨𝐯 𝐗, 𝐗 + 𝐂𝐨𝐯 𝐘, 𝐘 + 𝐂𝐨𝐯 𝐗, 𝐘 + 𝐂𝐨𝐯 𝐘, 𝐗 =

𝐕𝐚𝐫 𝐗 + 𝐕𝐚𝐫 𝐘 + 𝟐 × 𝐂𝐨𝐯 𝐗, 𝐘


Amount invested

Expected return

Expected volatility

Correlation

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

◦ 𝑤𝐴𝐵𝐶 =20000

20000+30000= 0.4

◦ 𝑤𝑋𝑌𝑍 =30000

20000+30000= 0.6

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Expected return

𝐸 𝑅𝑝 = 𝑤𝐴𝐵𝐶 ∙ 𝐸 𝑅𝐴𝐵𝐶 + 𝑤𝑋𝑌𝑍 ∙ 𝐸 𝑅𝑋𝑌𝑍 = 0.4 ∙ 7% + 0.6 ∙ 11%

= 9.4%

Expected volatility

𝜎𝑝 = 𝑤12 ∙ 𝜎1

2 + 𝑤22 ∙ 𝜎2

2 + 2 ∙ 𝑤1 ∙ 𝑤2 ∙ 𝜌1,2 ∙ 𝜎1 ∙ 𝜎2 =

= 0.42 ∙ 0.152 + 0.62 ∙ 0.222 + 2 ∙ 0.4 ∙ 0.6 ∙ 0.3 ∙ 0.15 ∙ 0.22= 16.05%

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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%

𝐸 𝑅𝑝 7.00% 7.80% 8.60% 9.00% 9.40% 10.20% 11.00%

𝜎𝑝 15.00% 13.97% 14.35% 15.06% 16.05% 18.72% 22.00%

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7,00%

7,80%

8,60% 9,00%

9,40%

10,20% 11,00%

6,00%

7,00%

8,00%

9,00%

10,00%

11,00%

12,00%

10,00% 15,00% 20,00% 25,00%

PORTFOLIO POSSIBILITIES CURVE (RETURN)

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15,00%

13,97%

14,35% 15,06%

16,05%

18,72%

22,00%

6,00%

7,00%

8,00%

9,00%

10,00%

11,00%

12,00%

10,00% 15,00% 20,00% 25,00%

PORTFOLIO POSSIBILITIES CURVE (RISK)

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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

𝜎𝑝2 = 𝑤1

2 ∙ 𝜎12 + 𝑤2

2 ∙ 𝜎22 + 2 ∙ 𝑤1 ∙ 𝑤2 ∙ 𝜌1,2 ∙ 𝜎1 ∙ 𝜎2 → 𝑚𝑖𝑛

𝑤1 + 𝑤2 = 1

𝑤1 = 75.34%,𝑤2 = 24.66%


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E(R)=7.99%, s.d.=13.93%

6,00%

7,00%

8,00%

9,00%

10,00%

11,00%

12,00%

10,00% 15,00% 20,00% 25,00%

MINIMUM VARIANCE PORTFOLIO


6,00%

7,00%

8,00%

9,00%

10,00%

11,00%

12,00%

0,00% 5,00% 10,00% 15,00% 20,00% 25,00%

Correlation impact on PPC

Corr = 1

Corr = 0.7

Corr = 0.3

Corr = 0

Corr = -0.5

Corr = -1

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Expected return

Portfolio risk (σ)

Risk-free

INEFFICIENT PORTFOLIOS



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 𝐸 𝑅𝐶 = 𝑤𝐹𝑅𝐹 + 𝑤𝑃𝐸 𝑅𝑃

Volatility of returns

𝜎𝐶2 = 𝑤𝐹

2𝜎𝐹2 + 𝑤𝑃

2𝜎𝑃2 + 2𝑤𝐹𝑤𝑃𝐶𝑜𝑣𝐹,𝑃

Volatility of risk-free asset is zero, so its variance and covariance with risky portfolio are zero as well

𝜎𝐶2 = 𝑤𝑃

2𝜎𝑃2; 𝜎𝑐 = 𝑤𝑃𝜎𝑝

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Expected return

Portfolio risk (σ)

Risk-free

CAL (B)

CAL (C)

(B)

(A)

(C)

CAL (A)



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 𝐸 𝑅𝐶 = 𝑤𝐹𝑅𝐹 + 𝑤𝑃𝐸 𝑅𝑃

𝜎𝑐 = 𝑤𝑃𝜎𝑝

𝑤𝐹 + 𝑤𝑃 = 1

𝑤𝑃 =

𝜎𝐶𝜎𝑃

𝑤𝐹 = 1 − 𝑤𝑃

𝐸 𝑅𝐶 = 1 −𝜎𝐶

𝜎𝑃 𝑅𝐹 +𝜎𝐶

𝜎𝑃 ∙ 𝐸 𝑅𝑃

𝑬 𝑹𝑪 = 𝑹𝑭 +𝑬 𝑹𝑷 − 𝑹𝑭

𝝈𝑷∙ 𝝈𝑪

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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

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INTERPRET THE CAPITAL MARKET LINE


Expected return

Portfolio risk (σ)

Risk-free

CML

Tangency portfolio (Market Portfolio)

𝐶𝑀𝐿: 𝐸 𝑟 = 𝑟𝑓 + 𝜎𝐸 𝑟𝑚 − 𝑟𝑓

𝜎𝑀

𝑇ℎ𝑒 𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝐶𝑀𝐿 =𝐸 𝑅𝑀 − 𝑅𝐹

𝜎𝑀



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.

Portfolio risk (σ)

(Market Portfolio)

Expected return

Risk-free

Borrowing rate

New Tangency Portfolio



UNDERSTAND THE DERIVATION AND COMPONENTS OF THE CAPM DESCRIBE THE ASSUMPTIONS UNDERLYING THE CAPM

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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.

𝑅𝑖 = 𝑅𝑓 + 𝛽𝑖(𝑅𝑚 − 𝑅𝑓) 𝑅𝑖 − 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑎𝑠𝑠𝑒𝑡 𝑖 𝑅𝑓 − 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑟𝑖𝑠𝑘 − 𝑓𝑟𝑒𝑒 𝑎𝑠𝑠𝑒𝑡

𝑅𝑚 − 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑚𝑎𝑟𝑘𝑒𝑡 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝛽𝑖 − 𝑠𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑠𝑠𝑒𝑡′𝑠 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 𝑡𝑜 𝑚𝑎𝑟𝑘𝑒𝑡 𝑟𝑒𝑡𝑢𝑟𝑛𝑠

■ All investors are risk averse and

utility maximizing

■ Markets are frictionless

■ All investors have the same one-

period time horizon

■ All investors have homogeneous

expectations

■ All investments are infinitely divisible

■ All investors are price takers. Their

trades cannot affect security prices.

The assumptions of the CAPM


𝛽𝑖 =𝐶𝑜𝑣 𝑅𝑖 , 𝑅𝑚

𝑉𝑎𝑟(𝑅𝑚)

𝑅𝑖 − 𝑅𝑓 = 𝛽𝑖 𝑅𝑚 − 𝑅𝑓 + 𝑒𝑖

𝜎𝑖2 = 𝛽𝑖

2𝜎𝑚2 + 𝜎𝑒

2 + 2𝐶𝑜𝑣(𝛽𝑖𝑅𝑚, 𝑒𝑖)

Assuming 𝐶𝑜𝑣 𝛽𝑖𝑅𝑚, 𝑒𝑖 = 0 → 𝜎𝑖2 = 𝛽𝑖

2𝜎𝑚2 + 𝜎𝑒

2

SYSTEMATIC VARIANCE

NON-SYSTEMATIC VARIANCE



TOTAL RISK = SYSTEMATIC (MARKET) RISK + NON-SYSTEMATIC (SPECIFIC) RISK

*Note that non-systematic (specific) risk is not rewarded as it can be eliminated for free by

diversification

Number of securities

Portfolio risk (σ)

NON-SYSTEMATIC RISK

SYSTEMATIC RISK

cannot be diversified can be diversified

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We have following data about the company Chocolove, Inc.

Required return for Chocolove according to CAPM

𝐸 𝑅𝑖 = 𝑅𝐹 + 𝐸 𝑅𝑀 − 𝑅𝐹 ∙ 𝛽𝑖 = 1.5% + 4%− 1.5% × 1.25 = 4.625%

Expected market return 4%

Risk-free rate 1.5%

Chocolove beta 1.25

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APPLY THE CAPM IN CALCULATING THE EXPECTED RETURN ON AN ASSET


APPLY THE CAPM IN CALCULATING THE EXPECTED RETURN ON AN ASSET

SML or security market line to compare the relationship between risk and return. Unlike the

CML, which uses standard deviation as a risk measure on the X axis, the SML uses the market

beta, or the relationship between a security and the marketplace.

Beta

Expected return

SML

𝑅𝑓

𝑅𝑚

1.0

UNDERVALUED SECURITIES

OVERVALUED SECURITIES


INTERPRET BETA AND CALCULATE THE BETA OF A SINGLE ASSET OR PORTFOLIO

Weight Beta

Auto Inc 30% 1.5

Berryville 20% 0.7

Chipside Ltd 50% 1.1

Total 100% 1.14

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Beta of an investment is a measure of the risk arising from exposure to general market

movements.

Portfolio beta is a weighted sum of individual asset betas

𝛽𝑖 =𝐶𝑜𝑣 𝑅𝑖 , 𝑅𝑚

𝑉𝑎𝑟(𝑅𝑚)


CALCULATE, COMPARE, AND INTERPRET THE FOLLOWING PERFORMANCE MEASURES: THE SHARPE PERFORMANCE INDEX, THE TREYNOR PERFORMANCE INDEX, THE JENSEN PERFORMANCE INDEX, THE TRACKING ERROR, INFORMATION RATIO, AND SORTINO RATIO

27

Jensen's alpha is used to determine the abnormal return of a security or portfolio of

securities over the theoretical expected return.

Jensen′s alpha measure = Rp − Rf + βp Rm − Rf

Beta

Expected return

SML

𝑅𝑓

𝑅𝑚

1.0

UNDERVALUED SECURITIES

OVERVALUED SECURITIES

JENSEN′S ALPHA



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When we evaluate the performance of a portfolio with risk that differs from that of a

benchmark, we need to adjust the portfolio returns for the risk of the portfolio

𝑇ℎ𝑒 𝑆ℎ𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 =(𝑅𝑝 − 𝑅𝑓)

𝜎𝑝

The Sharpe ratio of a portfolio is its excess returns per unit of total portfolio risk, and higher

Sharpe ratios indicate better risk-adjusted portfolio performance

The Treynor measure is risk-adjusted returns based on systematic risk (beta) rather

than total risk

The Treynor measure =(𝑅𝑝 − 𝑅𝑓)

𝛽𝑝


Let’s calculate ratios for Chocolove, Inc

𝑆ℎ𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 =𝐸 𝑅𝑝 − 𝑅𝑓

𝜎𝑝=

4.8% − 1.5%

7%≈ 0.47

𝑇𝑟𝑒𝑦𝑛𝑜𝑟 𝑟𝑎𝑡𝑖𝑜 =𝐸 𝑅𝑝 − 𝑅𝑓

𝛽𝑝=

4.8% − 1.5%

1.25≈ 2.64

𝛼𝑝 = 𝐸 𝑅𝑝 − 𝑅𝑓 + 𝐸 𝑅𝑚 − 𝑅𝑓 ∙ 𝛽𝑖 = 4.8% − 4.625% = 0.175%

Expected Chocolove return 4.875%

Chocolove vol 7%

Risk-free rate 1.5%

Chocolove beta 1.25

Expected market return 4%

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As investor tries to earn excess return over the benchmark, the difference in returns varies over time.

𝛼 = 𝑅𝑝 − 𝑅𝐵

Tracking error is the standard deviation of difference between portfolio return and benchmark return

𝑡𝑟𝑎𝑐𝑘𝑖𝑛𝑔 𝑒𝑟𝑟𝑜𝑟 = 𝜎𝛼

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Information Ratio The ratio computes the surplus return relative to the surplus risk taken. The variability in the surplus return is a measure of the risk taken to achieve the surplus. The higher information ratio, the better performance is.

𝐼𝑅𝐴 =𝑅𝑝 − 𝑅𝐵

𝜎𝑝 −𝐵=

𝛼

𝑡𝑟𝑎𝑐𝑘𝑖𝑛𝑔 𝑒𝑟𝑟𝑜𝑟

Where:

𝑅𝑝 - average portfolio return

𝑅𝐵 - average benchmark return 𝜎𝑝 −𝐵 - standard deviation of excess return over benchmark

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Sortino ratio is similar to Sharpe ratio, but it measures not risk premium to risk, but excess return to semi-variance of returns

Down deviation is computed on observations when portfolio return falls below min acceptable return

𝑀𝑆𝐷𝑚𝑖𝑛 = (𝑅𝑃𝑡 − 𝑅𝑚𝑖𝑛)

2𝑅𝑃𝑡

<𝑅𝑚𝑖𝑛

𝑁

𝑆𝑜𝑟𝑡𝑖𝑛𝑜 𝑟𝑎𝑡𝑖𝑜 =𝐸 𝑅𝑃 − 𝑅𝑚𝑖𝑛

𝑀𝑆𝐷𝑚𝑖𝑛

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FOUNDATIONS OF RISK MANAGEMENT: THE MODERN PORTFOLIO … · 2020. 9. 10. · FRM® PART I THE...

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Transcript of FOUNDATIONS OF RISK MANAGEMENT: THE MODERN PORTFOLIO … · 2020. 9. 10. · FRM® PART I THE...