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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 MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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
EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER
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THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I 4
EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER
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
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I 5
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
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I 6
EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER
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
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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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EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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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EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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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EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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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EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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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EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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%
EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER
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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
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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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EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I 14
Expected return
Portfolio risk (σ)
Risk-free
INEFFICIENT PORTFOLIOS
EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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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EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I 16
Expected return
Portfolio risk (σ)
Risk-free
CAL (B)
CAL (C)
(B)
(A)
(C)
CAL (A)
EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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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EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I 19
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
INTERPRET THE CAPITAL MARKET LINE
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I 20
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
INTERPRET THE CAPITAL MARKET LINE
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
UNDERSTAND THE DERIVATION AND COMPONENTS OF THE CAPM DESCRIBE THE ASSUMPTIONS UNDERLYING THE CAPM
21
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
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I 22
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
UNDERSTAND THE DERIVATION AND COMPONENTS OF THE CAPM DESCRIBE THE ASSUMPTIONS UNDERLYING THE CAPM
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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
30
UNDERSTAND THE DERIVATION AND COMPONENTS OF THE CAPM DESCRIBE THE ASSUMPTIONS UNDERLYING THE CAPM
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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
24
APPLY THE CAPM IN CALCULATING THE EXPECTED RETURN ON AN ASSET
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I 25
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
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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
26
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)
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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
28
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
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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%
29
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
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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
30
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
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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
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
31
THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL FRM® PART I
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
32
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