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)
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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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Measure of Risk: classification (2)
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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Measure of Risk: classification (3)
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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Measure of Risk: classification (4)
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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Symmetric Risk Measure:
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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Symmetric Risk Measure:
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.
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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
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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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Asymmetric Risk Measure:
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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Asymmetric Risk Measure:
Maximum Drawdown (2)
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Asymmetric Risk Measure:
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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Relative Risk Measure:
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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Relative Risk Measure:
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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Relative Risk Measure:
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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Ex Ante Risk Measure:
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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Ex Ante Risk Measure:
Value at Risk (2)
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Ex Ante Risk Measure:
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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Ex Ante Risk Measure:
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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Ex Ante Risk Measure:
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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Ex Ante Risk Measure:
Expected Shortfall (2)
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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%
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Ex Ante Risk Measure:
Expected Shortfall (3)
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.
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