risk-return measures 2012

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RISK-RETURN MEASURES

Asset Pricing and Portfolio ChoiceUniversit degli Studi di Torino April 2012

Giulio CasuccioHead of Quantitative Strategies and [email protected]


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Risk-Return Trade Off

Risk aversion is a common factor among all

different types of in investors, so higher uncertainty

and volatility should be rewarded with higher

expected return.Risk taking is the main driver of return so any

financial performanceshould be always evaluated

taking into accountrisk and return jointly.

Correctly measuring risk is not obvious and doing

it in the proper way is crucial in evaluating and

choosing investments.

Asset Pricing and Portfolio Choice April 2012Risk-Return Measures 2


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Coherent Measure of Risk (1)

A coherent measure of risk R is defined by

satisfying four main axioms:

Monotonicity the larger the loss, the larger

the the risk.

If X < Y then R(X) < R(Y)

Positive HomogeneityIf the loss is multipliedby a positive factor, risk should increase by the

same factor.

If n > 0 then R(nX) > R(X)

Risk-Return Measures 3Asset Pricing and Portfolio Choice April 2012


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Coherent Measure of Risk (2)

Translational Invariance Risk free position

decrease the risk.

R(X + a) = R(X) a where a is a risk free position

Sub-Additivity The risk of aggregated is less

than or equal to sum of the individual risk

(diversification benefit).

R(X + Y) < R(X) + R(Y)



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Measure of Risk: classification (1)

Measures of risk should be classified across

different dimensions, depending on their

characteristics and objectives:

Scale Independent They do not consider the

risk aversion degree of the investor.

Or

Utility Based Risk is corrected by theinvestorsdegree of risk aversion, which should

be specified, but it is not unique for all

investors.Risk-Return Measures 5Asset Pricing and Portfolio Choice April 2012


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Absolute Risk Measure Risk is calculated

without considering any market index or

benchmark as reference.

Or

Relative Risk Measure Risk is defined with

respect to a specific market index or

benchmark, consistent with the characteristics

of the investment, or a cash equivalent risk free

position.



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Symmetric Risk Measure Risk is defined as

the volatility with respect to a certain value,

calculated or expected, not distinguishing

between higher or lower results.

Or

Asymmetric Risk MeasureExclusively returns

below a certain value, calculated or expected,

are taking into account: only negative events

are considered as risk.



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Ex Post Risk Measure They calculate the

realized risk and return over a specific period,

which represents simply one of the different

possible scenarios.

Or

Ex Ante Risk MeasureThey aim to shape the

whole distribution of expected returns,

empirical or theoretical, and to quantify the

probability of specific well defined events.



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Symmetric Risk Measure:

Standard Deviation

Thestandard deviation(volatility) is a measure of

the dispersionof a collection on returns defined as

the root-mean-square (RMS) on the values from

their mean.

It is expressed in the same unit of the data and it

implies normal (symmetrical) distribution ofreturns (central limit theorem and 68-95-99.7 rule).

It is the most common measure of risk but not the

most appropiate: it violates monotonocity axiom.



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Sharpe Ratio (1)

The Sharpe ratio (reward to volatility ratio) is a

measure of the excess return with respect to the

risk free rate per unit risk.

It is always calculated ex-postover a specific time

period and assuming a constant risk free rate.It implies any investor to select the investment

instrument with the higher Sharpe ratio,

independently from his risk aversion.Risk-Return Measures 10Asset Pricing and Portfolio Choice April 2012


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Sharpe Ratio (2)

Sharpe ratio assumes zero investment strategy:

each level of volatility is efficiently obtained

through the combination of risk free asset with the

investment which offers the highest Sharpe ratio.

Risk-Return Measures 11

Exp. Return Volatility Sharpe Ratio

Risk free 3,00% 0,00%

Investment 1 5,00% 10,00% 0,20

Investment 2 8,00% 20,00% 0,25

Investor Risk-Aversion: Vol. < 10%

Exp. Return Volatility Sharpe Ratio

Strategy 1

100% Investment 1: 5,00% 10,00% 0,20

Startegy 2

50% Investment 2 + 50% Risk Free 5,50% 10,00% 0,25

Asset Pricing and Portfolio Choice April 2012


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Asymmetric Risk Measure:

Downside Volatility

The downside volatlity is a measure of the

dispersion of the negative realizations of a

collection on returns.

It is defined as the root-mean-square (RMS) of the

negative values from their mean, assuming the

positive ones equal zero.

Only the probability of a negative return is

considered as risk.Risk-Return Measures 12Asset Pricing and Portfolio Choice April 2012


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Maximum Drawdown (1)

The Maximum Drawdown is calculated starting

from the cumulative returns series.

Drawdown is defined as the difference between

any local maximum and its relative minimum and

the Maximum Drawdown is the largest drawdown

found over the considered period.



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Maximum Drawdown (2)



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Stirling Ratio

MDD is commonly used as measure of risk for

commodity investments and hedge funds through

the widespread utilization of different indices.

Stirling Ratio measures the profit divided by the

maximum drawdown over a specified period.

It is calculated as the ratio between the rate of

return (or the excess return) and the MDD.It can be considered as a modification of the

Sharpe ratio where the denominator is replaced by

the MDD.



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Relative Risk Measure:

Tracking Error (1)

Tracking Error is a measure of how closely a

portfolio follows the market or its benchmark.

It is defined as the standard deviation of the

difference between the portfolio returns and the

market index or benchmark returns.



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Tracking Error (2)

Tracking Error is a useful measure to distinguish

between passive, enhanced passive and active

portfolios.

The former would have a TE close to zero, ideally

lower than 2%, while the latter have a higher TE,

usually within 5%.

Too large tracking errors can be due to amisspecified benchmark or to an incoherent

management style.



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Information Ratio

Information Ratiomeasures the active return of a

portfolio divided by the amount of risk the

manager takes relative to a benchmark.

It is defined as the ratiobetween the excess return

with respect to a defined benchmark and the

realized tracking error.

The ratio shows the risk-adjusted active returnmeasuring the excess return obtained for each unit

of active risk taken so it directly evaluates the

managersability.



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Jensens Alpha

Jensens Alpha is used to determine the excess

return of a portfolio over its theoretical expected

return.

The theoretical return is predicted by a market

model, most commonly the CAPM, measuring the

sensitivity of the portfolio to the market by its

market beta.

Jensen's alpha = Portfolio Ret. - (Risk Free Rate + Portfolio Beta *

(Market Ret. - Risk Free Rate))



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Ex Ante Risk Measure:

Shortfall Probability

Given the expected return distribution of a

portofolio, Shortfall Probability measures the

weight of the negative estimated values.

So it indicates the ex ante probablity of realized

returns lower than zero or lower than a reference

index return.

Obviously how to calculate Shortfall Probabilityand the accuracy of the result depends on the

quality of the estimation of the expected returns

and so it is not unique.



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Value at Risk (1)

Given the expected returns distribution of a

portfolio, Value at Riskis calculated with respect to

a specified probability and time horizon.

VaR measures the maximum expected losswithin

the confidence level defined by the specified

probability and time horizon.

For example, a 5% VaR at 1 month equal to 2%means that the expected loss in one month is lower

than 2% with a probability of 95%.



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Value at Risk (2)



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Value at Risk (3)

Given some confidence level (0,1) the VaR of

the portfolio at the confidence level is given by

the smallest number l such that the probability

that loss L exceedes l is not larger than (1 ).

VaR definition assumes no trades during thespecified time horizon and normal distribution of

returns.



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Value at Risk (4)

Common parametersfor VaR are 1%, 5% and 10%

and it is usually calculated on different time

horizons, from 1 day to 1 year, even if the longer

the time horizon the less meaningful the result.The definition of VaR is nonconstructive, it specifies

a property it must have but not how to compute

the VaR: it depends on the chosen probability

distributions of returns.

It is not a complete coherent measures of risk

because it violates the sub-additivity axiom.



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Expected Shortfall (1)

Expected Shortfall (ES) is an alternative to VaR and

it measures the expected return of a portfolio at

each quantile (q) of its returns distribution.

ES is computed taking into account the wholedistribution(probability density function) up to the

specified quantile and not only a single event

probability.



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Risk-Return Measures 26

Probability Profit/Loss

10% -100

30% -20

40% 0

20% 50

q ES q

5% -100 [5%*(-100)]/5%

10% -100 [10%*(-100)]/10%

20% -60 [10%*(-100)+10%*(-20)]/20%

40% -40 [10%*(-100)+30%*(-20)]/40%

100% -6 [10%*(-100)+30%*(-20)+40%*(0)+20%*(50)]/100%

Asset Pricing and Portfolio Choice April 2012


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ES evaluates risk in a conservativeway by focusing

on the less profitable outcomes: for high values of

q it ignores the most profitable but unlikely

possibilities, for small q it focuses on worst losses.ESq increases as q increases and the 100% ES

equals the expected value of the portfolio.

For a given portfolio ESq is worse (or equal) thanthe VaR(q) at the same q level.

Expected Shortfall is a coherent risk measure.


risk-return measures 2012

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