Measuring Investment Risk
Chapter 31
Tools & Techniques of
Investment Planning
Risk Measurement
• Risk
– The uncertainty of a future outcome
• Expected Return
– The anticipated return for some future period
• Realized Return
– The actual return over some past period
• The simple fact that dominates investing is that the
realized return on an asset with any risk attached to it
may be different from what was expected.
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Measuring Investment Risk
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Risk Measurement
• Volatility
– Defined by financial economists as the range of movement (or
price fluctuation) from the expected level of return
• Increased volatility can be equated with increased risk.
– Wide price swings create more uncertainty of an eventual
outcome
• Being able to measure and determine the past volatility
of a security provides some insight into the riskiness of
that security as an investment.
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Risk Measurement
• To deal with the uncertainty of returns, investors need
to think explicitly about a security’s distribution of
probable total returns.
• With the possibility of two or more possible outcomes,
which is the norm for common stocks and virtually all
other investments, investors must consider each
possible likely outcome and assess the probability of its
occurrence.
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Measuring Investment Risk
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Risk Measurement
• The result of considering these outcomes and their
probabilities together is a probability distribution
consisting of:
– The specification of the likely returns that may occur
– The probabilities associated with these likely returns
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Measuring Investment Risk
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Probability Distributions
• Probabilities represent the likelihood of various
outcomes and are typically expressed as a decimal.
– Sometimes fractions are used
• The sum of the probabilities of all possible outcomes
must be 1.0
– They must completely describe all the perceived likely
occurrences.
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Measuring Investment Risk
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Probability Distributions
• These probabilities and associated outcomes are
largely obtained through subjective estimates by the
investor.
– Past occurrences (frequencies) are heavily relied upon but
must be adjusted for any changes expected in the future.
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Measuring Investment Risk
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Probability Distributions
• Probability distributions can be either discreet or
continuous:
– With a discreet probability distribution, a probability is assigned
to each possible outcome.
– With a continuous probability distribution, an infinite number of
possible outcomes exist.
• The most familiar continuous distribution is the normal
distribution depicted by a well-known bell-shaped curve.
– It is called a two-parameter distribution because one only
needs to know the mean and the variance to fully describe the
distribution.
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Measuring Investment Risk
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Probability Distributions
• To describe the single most likely outcome from a
particular probability distribution, it is necessary to
calculate its expected value.
– The expected value is the average of all possible return
outcomes, where each outcome is weighted by its respective
probability of occurrence.
• Investors typically use variance or standard deviation to
calculate the total risk associated with the expected
return.
– At least as a first approximation
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Measuring Investment Risk
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Variance and Standard Deviation
• The standard deviation is a statistical measure defined
as the square root of the variance of returns.
• The variance of returns is the expected value of the
average squared differences from the mean of the
distribution.
– It is a measure of how much, on average, any particular
observation of a randomly distributed variable will differ from
the average or mean value of the distribution.
• In this context, variance, volatility, and risk can be used
synonymously.
– The larger is the standard deviation, the more uncertain is the
outcome.
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Measuring Investment Risk
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Variance and Standard Deviation
• In the area of investment analysis, risk assessment
must also include differences in expected values and
the downside or loss potentials of the alternative
investments.
• Consequently, the standard deviation can best be
described as a measure of the “goodness” or
confidence one can place in a best guess estimate
(mean value) of the outcome of a random variable.
– As applied to investment returns, the expected value may be
the best guess of future returns.
– If the standard deviation of returns is large, the best guess still
may not be a very good guess.
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Measuring Investment Risk
Chapter 31
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Variance and Standard Deviation
• The standard deviation is at best only a partial or
incomplete, and sometimes misleading, measure of the
riskiness of an investment relative to another one.
• When comparing investment alternatives, investors
must use standard deviations relative to the
investments’ expected returns.
– Standard deviations become useful in conjunction with
expected returns to measure each investment’s return per unit
of risk.
• “Return bang per risk buck”
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Example
• The standard deviation of equity returns (S&P stocks
and small-capitalization stocks) is greater than the
standard deviation of fixed-income investment returns
(Corporate and government bonds and Treasury bills)
for all investment horizons.
– However, at longer investment horizons, equities dominate fixed
income investments.
– Despite greater standard deviations, equities can and should be
viewed as less risky investments than bonds and Treasury bills
for longer investment horizons.
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Variance and Standard Deviation
• If returns are normally distributed, investors can predict
that actual realized returns will fall within one standard
deviation above or below the mean about 68% of the
time.
– Within 2 standard deviations above or below the mean about
95% of the time.
• Based on history, investors can expect to earn average
annual returns of between 7.7% and 14.9% on a 20year investment in S&P stocks with about a 68%
probability, or odds of two in three.
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Measuring Investment Risk
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Variance and Standard Deviation
• Calculating a standard deviation using probability
distributions involves making subjective estimates of the
probabilities and the likely returns.
– Investors cannot avoid such estimates because future returns
are uncertain.
– The prices of securities are based on investors’ expectations
about the future.
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Variance and Standard Deviation
• The relevant standard deviation in this situation is the
ex ante standard deviation.
– Not the ex post standard deviation based on past realized
returns.
– Although ex post standard deviations may be convenient and
used as proxies for ex ante standard deviations, investors
should be careful to remember that the past cannot always be
extrapolated into the future without modifications.
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Measuring Investment Risk
Chapter 31
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Variance and Standard Deviation
• Standard deviations for well-diversified portfolios tend
to be reasonably steady over time.
– Historical calculations may be fairly reliable in projecting the
future.
– Historical calculations are much less reliable for individual
securities.
• The standard deviation is a measure of the total risk
of an asset or a portfolio.
– Includes both systematic and unsystematic risk.
– Captures the total variability in the asset’s or portfolio’s
return, whatever the sources of that variability.
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Measuring Investment Risk
Chapter 31
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Semi-Variance
• The semi-variance is a measure of downside risk: the
risk of realizing an outcome below the expected return.
– Measurement suggested by Harry Markowitz
– Preferred by investors who think only deviations below the
expected outcome return really matter.
• Standard deviation considers the possibility of
returns above the expected return as well as
below the expected return.
– Similar to variance, except that no consideration is given to
returns above the expected return when making the calculation.
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Measuring Investment Risk
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Semi-Variance
• Investors may use a measure called the downside
volatility or semi-volatility.
– They are more concerned with the chance that an investment’s
return will fall below some benchmark or target return rather
than the expected return of the investment.
• The semi-volatility is computed in the same way as the
semi-variance.
– Except that all returns above a benchmark or target return
(sometimes called the minimal acceptable rate of return) rather
than the expected return are ignored.
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Measuring Investment Risk
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Semi-Variance
• The square root of the semi-variance is sometimes
called the semi-deviation.
• Semi-deviation is to semi-variance as standard
deviation is to variance.
• The term downside deviation is sometimes used to
refer to the square root of the semi-variance.
– More frequently seems to be used to refer to the square root
of the downside volatility.
• The downside volatility is computed with respect
to a minimal acceptable rate of return different
from the expected return.
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Measuring Investment Risk
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Semi-Variance
• Most researchers and analysts use the standard
deviation as the risk measure of choice.
– Despite semi-variance being conceptually superior
– Due to some difficult math problems
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Measuring Investment Risk
Chapter 31
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Semi-Variance
• If the return distribution is symmetrical, the standard
deviation gives exactly the same answers in a portfolio
context as the semi-variance.
– With notable exceptions, stock return distributions do seem to
be reasonably symmetrical, especially for longer investment
horizons.
– For shorter-term investment horizons and certain types of
investments, such as options and other derivatives, technology
stocks, media stocks, telecom stocks, or hedge funds, the
distributions may not be normally distributed.
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Measuring Investment Risk
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Covariance and Correlation
• Covariance is a measure of the degree to which two
variables move in a systematic or predictable way,
either positively or negatively.
– If two variables move in perfect lockstep, up and down, they
exhibit perfect positive covariance.
– If two variables move in perfect lockstep, but in opposite
directions, they exhibit perfect negative covariance.
– If two variables are completely independent, showing no
systematic relationship, their covariance is zero.
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Covariance and Correlation
• Correlation is a standardized version of covariance
where values range from -1 (perfect negative
covariance) to +1 (perfect positive covariance).
• Expressed as a function of covariance, covariance is
equal to the correlation times the standard deviations of
the two variables.
• Letting 12 represent the covariance between 2
variables, 1 the standard deviation of variable 1, 2 the
standard deviation of variable 2, and 12 the correlation
of the two variables, the relation between correlation
and covariance is as follows:
12 = 1 x 2 x  12 or
 12 = 12 / (1 x 2)
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Covariance and Correlation
• Markowitz, in his seminal book, Portfolio Selection,
showed that if investors added stocks that do not
exhibit perfect covariance to their portfolio, the total risk
of the portfolio as measured by variance or standard
deviation would decline.
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Measuring Investment Risk
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Covariance and Correlation
• Assume an investor holds just one stock, S1, with a
standard deviation of 1. Thus the standard deviation of
the investor’s initial “portfolio” of one security is 1.
• If the investor sells off part of his interest in S1 and uses
the proceeds to buy another stock, S2, with a standard
deviation of 2, the variance per dollar of investment in
his portfolio, p2, is given by the following equation:
p2 = (w1 2 x 12) + (w22 x 22)+(2 x w1 x w2 x 12)
 12 is the covariance between S1 and S2, and Wi is the
proportion of the portfolio invested in stock Si.
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Measuring Investment Risk
Chapter 31
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Covariance and Correlation
• Alternatively, one can express the variance of the
portfolio in terms of correlation rather than covariance:
p2 = (w1 2 x 12) + (w22 x 22)+(2 x w1x w2 x 1 x  2 x 12)
• For each additional independent stock (covariance and
correlation are equal to 0) added to a portfolio, the
variance declines in proportion to the number of
stocks, that is:
p2 = 12/n
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Measuring Investment Risk
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Covariance and Correlation
• By splitting a portfolio into more and more securities,
the variance attributable to the nonsystematic
(uncorrelated) risks approaches zero as the number of
securities increases.
• The standard deviation of the portfolio declines by the
square root of the number of independently distributed
securities in the portfolio:
p = 1/ n
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Measuring Investment Risk
Chapter 31
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Covariance and Correlation
• If two securities are perfectly negatively correlated,
splitting one’s investment equally between the two
securities completely eliminates all variability.
• There is no risk reduction advantage to adding perfectly
positively correlated assets to one’s portfolio.
• In the real world, instances of either perfectly negatively
correlated or perfectly positively correlated securities
are extremely rare.
– Investors do not need negative correlations between securities
for them to benefit by adding securities to their portfolio.
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Measuring Investment Risk
Chapter 31
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Covariance and Correlation
• As long as a security added to a portfolio is not
perfectly positively correlated with the existing portfolio,
the addition of the security will reduce the portfolio’s risk
as measured by variance or standard deviation.
– This concept is a central principle of modern portfolio and asset
allocation theory.
• As long as there are any classes of assets whose
returns are not perfectly correlated with investors’
current portfolios, these investors can further reduce
the risk of their portfolios by adding securities from
those asset classes.
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Measuring Investment Risk
Chapter 31
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Beta
• Beta determines the volatility, or risk, of a security or
fund in relation to that of its index or benchmark.
– In contrast to standard deviation, which determines the volatility
of a security or fund according to the disparity of its returns over
a period of time.
• In the single factor Capital Asset Pricing Model, the
index or benchmark is the “market” portfolio, often
measured by the S&P 500 index.
• When beta is used to compare funds or to measure an
investment manager’s performance, the benchmark is
frequently the average of the funds in that mutual fund
category.
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Beta
• A fund with a beta very close to 1 means the fund’s
performance closely matches the index or benchmark.
– A beta greater than 1 indicates greater volatility than the overall
market.
– A beta less than 1 indicates less volatility than the benchmark.
• Investors expecting the market to be bullish may choose
funds exhibiting high betas, which increase investors’
chances of earning high returns in up markets.
– If an investor expects the market to be bearish in the near future,
the funds that have betas less than 1 are a good choice because
they would be expected to decline less in value than the index.
• Beta by itself is limited and can be skewed due to factors
other than the market risk affecting the fund’s volatility.
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R-Squared (Coefficient of Determination)
• The R-squared of a fund is a measure of what
proportion of a security’s or portfolio’s total variability is
explained by its relationship to a benchmark or index
and how much is its independent risk unrelated to the
benchmark or index.
• When used in conjunction with ratings of mutual funds
or the performance of professional managers, it advises
if the beta of a mutual fund is measured against an
appropriate benchmark.
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R-Squared (Coefficient of Determination)
• Measuring the correlation of a fund’s movements to that
of an index, R-squared describes the level of
association between the fund’s volatility and market
risk.
– More specifically, the degree to which a fund’s volatility is a
result of the day-to-day fluctuations experienced by the overall
market.
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R-Squared (Coefficient of Determination)
• R-squared values range between 0 and 100.
– Where 0 represents the least correlation and 100 represents
full correlation.
– The closer to 100, the more the beta should be trusted and
vice-versa.
• An inappropriate benchmark will skew more than just
beta.
– Alpha is calculated using beta.
• If the R-squared value of a fund is low, it is also wise not to trust
the figure given for alpha.
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Skewness
• Skewness measures the coefficient of asymmetry of a
distribution.
• While in the normal distribution both tails mirror each
other, skewed distributions have one tail of the
distribution that is longer than the other.
• A risk-averse investor does not like negative skewness.
– An investment with negative skewness has a substantial
downside tail exposing the investor to low or negative returns
below the worst potential returns on an investment with positive
skewness.
– An investment with positive skewness offer investors the
potential for upside returns far above any they could ever expect
from a negatively skewed investment.
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Measuring Investment Risk
Chapter 31
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Skewness
• In symmetric distributions, such as the normal distribution,
the mean and median are equal.
– In skewed distributions, they are different.
• The median is the point where there is a 50% probability
that realized returns will fall below (or above) that value.
– In positively skewed distributions, the median is below the mean.
• An investor has less than a 50% chance of earning the mean return
in a positively skewed investment.
– In negatively skewed distributions, the median is above the mean.
• An investor has more than a 50% chance of earning the mean return
on a negatively skewed investment.
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Measuring Investment Risk
Chapter 31
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Skewness
• All risk-averse investors will prefer the positively skewed
to the negatively skewed investment when their means
and standard deviations are identical.
• This does not mean that they would always prefer a
positively skewed investment to a symmetrically
distributed investment with the same mean and standard
deviation.
– Or other possible combinations of means, standard deviations,
and skewness.
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Measuring Investment Risk
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Kurtosis
• Kurtosis measures the degree of “fatness” in the tails of
a distribution.
• Risk averse investors will prefer a distribution with low
kurtosis (where tails are thin and returns are more likely
to fall closer to the mean), since they will always weigh
the potential downside returns heavier than the potential
upside returns.
• One of the reasons posited for the small-stock premium
being higher than it should be in theory under the meanvariance framework of the CAPM is that small cap
stocks exhibit greater excess kurtosis and negative
skewness than larger stocks.
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Measuring Investment Risk
Chapter 31
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Skewness and Kurtosis Combined
• Some distributions can exhibit both skewness and
excess kurtosis.
• As soon as skewness begins to be negative, the impact
of a high excess kurtosis is significant for a risk-averse
investor.
• When the return distribution has a negative skewness 1 (or below) and an excess kurtosis higher than 1, the
probability of having a huge negative return increases
dramatically.
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Measuring Investment Risk
Chapter 31
Tools & Techniques of
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Skewness and Kurtosis Combined
• For optimization, simulation, and investment selection,
the investor’s approach should account not only for
volatility, but also for skewness and kurtosis when the
return distribution over the relevant period is likely to
be significantly different than the normal distribution.
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