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1 Class Risk and Return Historical returns in the USA. Sigma. CV. Security презентация, доклад

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Value of $1 invested in 1926

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Слайд 1Class
Risk and Return
Historical returns in the USA.
Sigma. CV. Security Characteristic

Line (SCL). Beta. Security Market Line (SML). CAPM formula.


Study materials:
KR: Ch. 11, 12.
RWJ: Ch. 9, 10
BM: Ch. 7, 8
ClassRisk and ReturnHistorical returns in the USA.Sigma. CV. Security Characteristic Line (SCL). Beta. Security Market Line (SML).

Слайд 2Value of $1 invested in 1926

Value of $1 invested in 1926

Слайд 3Risk and Return
Risk - The chance that an investment's actual

return will be different than expected. This includes the possibility of losing some

or all of the original investment. Total risk measured by the standard deviation of the historical returns.

A fundamental idea in finance is the relationship between risk and return. The greater the amount of risk that an investor is willing to take on, the greater the potential return. The reason for this is that investors need to be compensated for taking on additional risk. E.g., a U.S. Treasury bonds (bills) is considered
to be the safest investment and, when compared
to a corporate bond, provides a lower return.
The reason is that a corporation is much more
likely to go bankrupt than the U.S. government.
Risk and Return   Risk - The chance that an investment's actual return will be different than expected. This

Слайд 4Total Risk vs. Returns

Total Risk vs. Returns

Слайд 5Risk and Return
P1 – P0 (+ ∑D)
HPR, % =


P0

Return, % = P1 / P0 - 1

CV

= sigma / mean return
Risk and Return		P1 – P0 (+ ∑D) 		HPR, % = 					P0	 		Return, % = P1 / P0

Слайд 6
Measuring Returns





Historical returns:
X1=10%, X2= (-5%), X3 = 20%

Arithmetic Average:
[0.10

+ (-0.05) + 0.20] / 3 = 8.33% - usual

way

Geometric Average:
[(1.10)*(0.95)*(1.20)]1/3 - 1 = 7.84% - more correct
Measuring Returns			Historical returns: 		X1=10%,	X2= (-5%), 	X3 = 20%Arithmetic Average: 		[0.10 + (-0.05) + 0.20] / 3 =

Слайд 7Общий риск: стандартное отклонение

Общий риск: стандартное отклонение

Слайд 82. Risk and return relationship
Общий риск: стандартное отклонение

2. Risk and return relationship Общий риск: стандартное отклонение

Слайд 9Return (actual), mean return, %
CAPM-return (model return), %
Risk-free return, %
Excess

return, %
Abnormal return, %
Total risk (sigma), %
Systematic risk (beta), t
Unsystematic

risk, %
Coefficient of correlation, t
Coefficient of determination, t
Coefficient of variation (CV), t
Sharpe’s measure (S), t
Treynor’s measure (T), t
Appraisal ratio (AR), t
Z-beta, t
Z-alpha, %
Degree of volatility, t

Risk and Return measures

Return (actual), mean return, %CAPM-return (model return), %Risk-free return, %Excess return, %Abnormal return, %		Total risk (sigma), %		Systematic

Слайд 10
Measuring Risk: ERR

Expected return:
The return for an asset is

the probability weighted average return in all scenarios.


Measuring Risk: ERR	Expected return: 	The return for an asset is the probability weighted average return in all

Слайд 11 Variance: A measure of the dispersion of a set of

data points around their mean value.
Variance measures the variability

(volatility) from an average. Volatility is a measure of risk, so this statistic can help determine the risk an investor might take on when purchasing a specific security.

The variance of an asset’s expected return is the expected value of the squared deviations from the expected returns.





Measuring Risk: ERR

Variance: A measure of the dispersion of a set of data points around their mean value. 		Variance

Слайд 12

Standard deviation:
Square root of Variance. A measure of the

dispersion of a set of data from its mean. The

more spread apart the data, the higher the deviation.

In finance, it is a measure of total risk of a financial asset.


Measuring Risk: ERR

Standard deviation: 	Square root of Variance. A measure of the dispersion of a set of data from

Слайд 13
Measuring Risk: ERR



Example: Calculating ERR, Variance, and Standard deviation


Measuring Risk: ERR	Example: Calculating ERR, Variance, and Standard deviation

Слайд 14
Measuring Risk: Historical RR

Variance:
The variance of an asset’s historical

returns is the sum of the squared deviations divided by

(n-1).



Standard deviation:
Square root of variance.
Measuring Risk: Historical RR				Variance: 	The variance of an asset’s historical returns is the sum of the squared

Слайд 15
Measuring Risk: Historical RR



Example: Calculating Mean, Variance, and

Standard deviation


Measuring Risk: Historical RR  Example: Calculating Mean, Variance, and Standard deviation

Слайд 16
Risk and Return Relationship


Coefficient of Variation (CV) =
Standard deviation

/ Expected (historical) rate of return.
Measures risk-return relationship, i.e.,

sigma per 1 %
of return.
better choice

Risk and Return Relationship		Coefficient of Variation (CV) = 		Standard deviation / Expected (historical) rate of return. 		Measures

Слайд 17Risk and Return characteristics, World, 2009

Risk and Return characteristics, World, 2009

Слайд 181
2
3
4
Return, %
Sigma, %
• 2 vs. 1; return is higher

• 2

vs. 3; risk is lower

• 4 vs. 3; return is

higher

Types of investors: Risk-averse, (Risk-neutral), Risk-seeker.

A rational investor would choose the securities that locate in the left upper corner. CV MIN.

R&RR and types of investors

1234Return, %Sigma, %• 2 vs. 1; return is higher• 2 vs. 3; risk is lower• 4 vs.

Слайд 19World Stock Indices, 2009

World Stock Indices, 2009

Слайд 20Stock Behavior: Beta and Alpha

Stock Behavior: Beta and Alpha

Слайд 21SCL, Beta and Alpha

SCL, Beta and Alpha

Слайд 22
Beta
Beta: A measure of systematic risk, of a security (or a

portfolio) in comparison to the market (market index, proxy). Measure

of elasticity.

Equals the variable coefficient “b” in the linear regression equation:
Y = a + b*X + ε
Graphically, Beta = Y / X

A beta of greater than 1 indicates that the security's price is more volatile than the market, more risky. “Aggressive” security.
The Security characteristic line is more steep.
A beta of less than 1 indicates that the security's price is less volatile than the market, less risky. “Defensive” security.

Beta	Beta: A measure of systematic risk, of a security (or a portfolio) in comparison to the market (market

Слайд 23
Total Risk: Components

Systematic Risk - The risk inherent to the

entire market, or market
segment. Also known as “non-diversifiable risk"

or "market risk."
E.g., interest rates, recession and wars represent sources of systematic risk because they affect the entire market and cannot be avoided through diversification. Affects a broad range of securities. Even a portfolio of well-diversified assets cannot escape all risk.
Measured with BETA.

Unsystematic Risk – Company-specific risk that is inherent in each
investment. Can be reduced through appropriate diversification (=strategy
designed to reduce risk by spreading the portfolio across many
investments.
Also known as "specific risk", "diversifiable risk“, or "residual risk".
E.g., a sudden strike by the employees of a company is considered
to be unsystematic risk.



Total Risk: Components	Systematic Risk - The risk inherent to the entire market, or market 	segment. Also known

Слайд 24Total Risk: Components

Total Risk: Components

Слайд 25Безрисковая доходность

Безрисковая доходность

Слайд 26Безрисковая доходность

Безрисковая доходность

Слайд 27Correlations of indices with S&P500

2010 2009
North America: 0,80 – 0,95 0,88

– 0,99
South America: 0,70 – 0,76 0,83 – 0,88
Western Europe: 0,80 –

0,87 0,79 – 0,92
Australia and NZ: 0,76 – 0,80 0,47 – 0,65
Russia: 0,71 – 0,76 0,53 – 0,59
Asia: 0,45 – 0,77 0,27 – 0,76
Ukraine: 0,46 – 0,51 0,44 – 0,48



Correlations of indices with S&P500 				2010			2009North America: 	0,80 – 0,95		0,88 – 0,99South America:	0,70 – 0,76		0,83 – 0,88Western

Слайд 28
Markets: Correlations in 2010

Markets: Correlations in 2010

Слайд 29
Security Market Line (SML)





Return,%
.
rf
Risk-free rate
Market Portfolio
Beta
1.0
SML
SML equation

= rf +  ( rm - rf ) =

CAPM
Security Market Line (SML)				Return,%.rfRisk-free rate   Market PortfolioBeta1.0SMLSML equation = rf +  ( rm -

Слайд 30rm = 13%, rf = 3%

x = 1.20, then:
E(rx) = 3%

+ 1.2*(13%-3%) = 15%

y = 0.80, then:
E(ry) = 3% +

0.8*(13%-3%) = 11%

Sample Calculations for SML

rm	= 13%, 	rf = 3%x = 1.20, then:	E(rx) = 3% + 1.2*(13%-3%) = 15%y = 0.80, then:	E(ry)

Слайд 31Alpha
Alpha (abnormal return; excess return) = ract - rf
rp =

rf + p ( rm - rf ) (+/- alpha)

AlphaAlpha (abnormal return; excess return) = ract - rfrp = rf + p ( rm - rf

Слайд 32SML, 2009, DJIA

SML, 2009, DJIA

Слайд 33Beta, 2009, DJIA

Beta, 2009, DJIA

Слайд 34Markowitz’ Efficient Frontier

Markowitz’ Efficient Frontier

Слайд 35Coefficient of Determination

Coefficient of Determination

Слайд 36Stock BEHAVIOUR – from the SCL equation
Find and describe:

Beta

Alpha

Correlation

coef.

Sigma

Mean return

CV


Stock BEHAVIOUR – from the SCL equationFind and describe:BetaAlpha Correlation coef. Sigma Mean returnCV

Слайд 37Class
Risk and Return
Additional materials (Advanced level)

ClassRisk and ReturnAdditional materials (Advanced level)

Слайд 38Capital Allocation Line (CAL)

A line created in a graph of

all possible combinations of risky and risk-free assets.
The graph displays

to investors the return they can make by taking on a certain
level of risk. Also known as the "reward-to-variability ratio".





E(r)

E(rp) = 15%

rf = 7%

p = 22%

0

P

F

E(rp) -

rf = 8%


CAL

Capital Allocation Line (CAL)		A line created in a graph of all possible combinations of risky and risk-free

Слайд 39
Portfolio risk and return: Return






The rate of return on a

portfolio is a weighted average of the rates of return

of each asset comprising the portfolio, with the portfolio proportions ($) as weights.

rp = w1r1 + w2r2


w1 = Proportion of funds in Security 1, %
w2 = Proportion of funds in Security 2, %
r1 = Expected return on Security 1, %
r2 = Expected return on Security 2, %


Portfolio risk and return: Return			The rate of return on a portfolio is a weighted average of the

Слайд 40
Portfolio risk and return: Covariance


Covariance is a measure of the

linear association between the 2 variables. A measure of the

degree to which returns on two risky assets move
in tandem. A positive covariance means that asset returns
move together. A negative covariance means that returns move inversely.

Covariance between Stock 1 and Stock 2 =






OR:
Portfolio risk and return: Covariance	Covariance is a measure of the linear association between the 2 variables. A

Слайд 41
Portfolio risk and return: Covariance

Covariance between Stock 1 and Stock

2 =
= Cov 1,2 = ρ1,2 σ 1 σ

2




Portfolio risk and return: Covariance			Covariance between Stock 1 and Stock 2 = 				= Cov 1,2 = ρ1,2

Слайд 42
Portfolio risk and return: Correlation


Correlation is a measure of the

linear association between
the 2 variables. In the world of

finance, a statistical measure of
how two securities move in relation to each other.

ρ 1,2 = Cov 1,2 / σ 1 σ 2

Correlation coefficient is ALWAYS [-1; 1].

Perfect positive correlation (a correlation coefficient of +1) implies that as one security moves, either up or down, the other security will move in lockstep, in the same direction. Alternatively, perfect negative correlation means that if one security moves in either direction the security that is perfectly negatively correlated will move by an equal amount in the opposite direction. If the correlation is 0, the movements of the securities are said to have no correlation; they are completely random.

Portfolio risk and return: Correlation	Correlation is a measure of the linear association between 	the 2 variables. In

Слайд 43
Portfolio risk and return: Sigma





When two risky assets with variances

 12 and  22, respectively, are combined into a

portfolio with portfolio weights w1 and w2, respectively, the portfolio variance is given by:

p2 = w1212 + w2222 + 2w1w2 Cov(r1r2) =
= w1212 + w2222 + 2w1w2 ρ 1 2

Hence, the portfolio’s standard deviation - p - is the square root of the above formula.

Portfolio risk and return: Sigma				When two risky assets with variances  12 and  22, respectively, are

Слайд 44
Portfolio risk and return: Sigma




Rule: When a risky asset is

combined with a risk-free asset, the portfolio standard deviation equals

the risky asset’s standard deviation multiplied by the portfolio proportion invested in the risky asset.

Portfolio risk and return: Sigma			Rule: When a risky asset is combined with a risk-free asset, the portfolio

Слайд 45
DIVERSIFICATION





Coca Cola
Reebok
Standard Deviation
35% in Reebok, 65% in Coca-Cola

Expected Returns and Standard Deviations vary given different weighted combinations

of the stocks

Expected Return, %

See: file “port.xls”

DIVERSIFICATION				Coca ColaReebokStandard Deviation35% in Reebok, 65% in Coca-Cola   Expected Returns and Standard Deviations vary given

Слайд 46Efficient Frontier
Example

Correlation Coefficient = 0.4
Stocks s % of Portfolio Avg

Return
ABC Corp 28 60% 15%
Big Corp 42 40% 21%


Standard Deviation = weighted avg = 33.6
Standard Deviation = Portfolio = 28.1
Return = weighted avg = Portfolio = 17.4%

Efficient FrontierExample             Correlation Coefficient =

Слайд 47Efficient Frontier
Example

Correlation Coefficient = 0.4
Stocks s % of Portfolio Avg

Return
ABC Corp 28 60% 15%
Big Corp 42 40% 21%


Standard Deviation = weighted avg = 33.6
Standard Deviation = Portfolio = 28.1
Return = weighted avg = Portfolio = 17.4%

Let’s Add stock New Corp to the portfolio
Efficient FrontierExample             Correlation Coefficient =

Слайд 48Efficient Frontier
Example

Correlation Coefficient = 0.3
Stocks s % of Portfolio Avg

Return
Portfolio 28.1 50% 17.4%
New Corp 30 50% 19%

NEW Standard Deviation = weighted avg = 31.80
NEW Standard Deviation = Portfolio = 23.43
NEW Return = weighted avg = Portfolio = 18.20%


Efficient FrontierExample             Correlation Coefficient =

Слайд 49Efficient Frontier
Example

Correlation Coefficient = 0.3
Stocks s % of Portfolio Avg

Return
Portfolio 28.1 50% 17.4%
New Corp 30 50% 19%

NEW Standard Deviation = weighted avg = 31.80
NEW Standard Deviation = Portfolio = 23.43
NEW Return = weighted avg = Portfolio = 18.20%

NOTE: Higher return & Lower risk
How did we do that? DIVERSIFICATION
Efficient FrontierExample             Correlation Coefficient =

Слайд 50
Diversification of Unsystematic Risk





Diversification of Unsystematic Risk

Слайд 51Diversification

Diversification

Слайд 52Diversification

Diversification

Слайд 53ОНД, %
Стандартное отклонение, %
Stock B
Stock A
Corr = +1
Corr = -1
Corr

= 0
Portfolio diversification: 2 stocks

ОНД, %Стандартное отклонение, %Stock BStock ACorr = +1Corr = -1Corr = 0Portfolio diversification: 2 stocks

Слайд 54
Efficient Frontier





Standard Deviation
Each half egg shell represents the possible weighted

combinations for two stocks.
The composite of all stock sets constitutes

the efficient frontier

Expected Return, %

Efficient Frontier				Standard DeviationEach half egg shell represents the possible weighted combinations for two stocks.The composite of all

Слайд 55
Efficient Frontier





A
B
N
AB
ABN
Goal is to move up and left
Sigma
Return, %

Efficient Frontier				ABNABABNGoal is to move up and leftSigmaReturn, %

Слайд 56
Efficient Frontier


Return, %
Low Risk
High Return
High Risk
High Return
Low Risk
Low Return
High Risk
Low

Return
Sigma
Goal is to move up and left

Efficient Frontier		Return, %Low RiskHigh ReturnHigh RiskHigh ReturnLow RiskLow ReturnHigh RiskLow ReturnSigmaGoal is to move up and left

Слайд 57
Efficient (Markowitz) frontier

A line created from the risk-reward graph, comprised

of optimal portfolios.




Efficient (Markowitz) frontier	A line created from the risk-reward graph, comprised of optimal portfolios.

Слайд 58
Political Risks: Mechel Story






“В четверг (25 июля) премьер-министр Владимир Путин

встретился в Нижнем Новгороде с представителями крупнейших российских металлургических

и угольных компаний и высказался о ситуации на российском рынке стали.
Премьер-министр неожиданно сделал ряд жестких заявлений в отношении Мечела (ОАО «ЧМЗ»). По сути, он обвинил компанию в нерыночном поведении и предположил, что прокуратуре следуетобратить внимание на торговую деятельность компании.
Этих заявлений оказалось достаточно, чтобы обрушить котировки Мечела – с четверга они упали на 37,6%, заставив инвесторов гадать о будущем компании – ведущего производителя коксующегося угля в России.”
«Тройка-Диалог», 25 июля 2008 г. (www.troika.ru)

Political Risks: Mechel Story					“В четверг (25 июля) премьер-министр Владимир Путин встретился в  	Нижнем Новгороде с представителями

Слайд 59
Political Risks





Political Risks

Слайд 60
Political Risks: MICEX Index, 2008



Inauguration of president
Medvedev
Putin accuses
“Mechel” of


fishy business

Russian
intrusion into
Georgia
Putin’s speech in Sochi,
VII Economics Forum
Medvedev:

to establish a missile complex in Kaliningrad
Political Risks: MICEX Index, 2008			Inauguration of presidentMedvedevPutin accuses “Mechel” of fishy businessRussianintrusion intoGeorgiaPutin’s speech in Sochi, VII

Слайд 61
Portfolio risk and return: Beta


Systematic Risk - The risk inherent

to the entire market, or market
segment. Also known as

"un-diversifiable risk" or "market risk."

E.g., interest rates, recession and wars represent sources of systematic
risk because they affect the entire market and cannot be avoided through
diversification. Affects a broad range of securities. Even a portfolio of well-
diversified assets cannot escape all risk.

Measured with BETA.

Still, one can diversify the systematic risk – in the sense that
one can choose securities with different betas (defensive vs. aggressive stocks) and combine them.

Portfolio risk and return: Beta	Systematic Risk - The risk inherent to the entire market, or market 	segment.

Слайд 62
Portfolio risk and return: Beta






The Beta of a portfolio is

the weighted average of the rates of return of each

asset comprising the portfolio, with the portfolio proportions ($) as weights.


bp = w1 b1 + w2 b2


w1 = Proportion of funds in Security 1, %
w2 = Proportion of funds in Security 2, %
b1 = Beta of Security 1
b2 = Beta of Security 2

Portfolio risk and return: Beta			The Beta of a portfolio is the weighted average of the rates of

Слайд 63Portfolio diversification

Portfolio diversification

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