صفحه 1:
Modern portfolio:
Theory and investment
analysis
Chapter 4:
The Characteristics of the
Opportunity Set under Risk
صفحه 2:
Characteristics of the Opportunity Set under Risk
Under risk, the fundamental elements of financial decision-making remain the same as in
certainty, but the modeling process becomes more complex. Unlike the certainty case,
where each asset has a known return, under risk the return of each asset is described by a
set of possible outcomes or a probability distribution. This distribution includes all
potential results and the likelihood of each occurring.
Two key characteristics of this return distribution are:
1.Expected return (a measure of central tendency)
2.Standard deviation (a measure of risk or dispersion)
صفحه 3:
Since investors typically hold a portfolio of assets rather than a single one, analysis of return
and risk must be conducted at the portfolio level. Contrary to popular belief, the risk of a
portfolio is not the simple average of the risks of its individual assets; rather, it depends on
the relationships between asset returns. If assets are not perfectly positively correlated,
diversification can reduce the overall portfolio risk.
This chapter begins with an analysis of two-asset portfolios using algebraic and geometric
methods, examining the effect of correlation between returns. The analysis is then extended
to portfolios with multiple assets. The ultimate goal is to describe the opportunity set that
faces an investor in a world characterized by risk.
In the certainty case, financial decision-making leads to deterministic outcomes (e.g., a fixed
5% interest rate). However, under risk, outcomes are expressed probabilistically. The
frequency distribution 1015 68 عردو 4.1 Data on Three Hypothetical Events set of possible returns along
with the probability ofeach( fam ~~~ ۹
Probability 5
12
9
6
Despite the importance of the full distribution, in practice, due to its complexity and the
number of possible outcomes, it is often summarized using statistical measures such as the
mean and standard deviation. These two metrics provide the minimum necessary
information to understand the probabilistic behavior of an asset.
صفحه 4:
Determining the
Expected Value
The concept of the “mean” or expected value is a fundamental and intuitive idea
in statistics and decision-making. Just as we use it in daily life to refer to average
age or income, it also plays a central role in investment analysis. Under
conditions of risk—where multiple outcomes are possible with different
probabilities—determining the “average return” means calculating the expected
return.
fum efievicames dividedibyitheix aysaher. value is calculated using the
simple average:
For example, in Table 4.1, if the three possible returns are 12%, 9%, and 6%, the
average is 9%. 7 00 ۲ .
owever, if the outcomes have different probabilities, a weighted average is
used: =
Each return x its probability of occurrence, then summing the
Tésutsnal notation for expected return is an overline on the variable (e.g., Ri),
or alternatively, it is expressed as E(R)).
صفحه 5:
Two important properties of expected value, which
are especially useful in portfolio analysis, are:
1. Linearity of sums:
F(R (ريه+ - 5 +2
2. Multiplication by a constant:
۵ -إرية + راط
These properties are illustrated in Table 4.2. For example, if the
return of Asset 3 equals the sum of the returns of two other
assets, then its expected return will also equal the sum of their
expected returns.
Table 4.2 _Retum on Various Assets
Event Probability Asser Ane? Aner
A 14
8 0
3
Expected ع 10
صفحه 6:
~ Variance and
Risk
In addition to the average return, investment decision-making requires a measure of
how outcomes deviate from the mean. Knowing only the mean does not provide a
complete picture of risk. For example, the average depth of a river might be shallow, but
variability in depth could still lead to drowning.
The challenge with using mean deviations:
Directly comparing each outcome's deviation from the mean is not useful, as positive
and negative deviations cancel each other out. The average of these deviations is
always zero and does not convey any information about dispersion
صفحه 7:
Solutions:
Mean absolute deviation
Mean squared deviation (variance) — the standard method
Variance is defined as the mean of the squared deviations from the mean,
and the standard deviation is the square root of the variance. If all
outcomes are equally likely, variance is computed with a simple formula.
Otherwise, a weighted average is used.
Additional notes:
When using historical data, variance is sometimes adjusted by
multiplying by M/(M-1) (where M is the number of observations) to
correct for statistical bias. However, this book omits that adjustment.
Variance and standard deviation are the most common measures of
dispersion.
Downside Risk and Alternative Measures:
Sometimes investors are only concerned with deviations below the mean.
There are specific measures for this:
Semivariance: Considers only squared deviations below the mean.
Lower Partial Moments (LPMs): Can be calculated relative to a
specific threshold, such as zero return or the risk-free rate.
صفحه 8:
Value at Risk (VaR):
VaR estimates the maximum expected loss at a given confidence level (e.g.,
5%).
Despite the availability of these alternatives, using them in portfolio analysis
is often complex. Fortunately, for assets with relatively symmetric
distributions (such as many equities), variance is proportional to
semivariance. Therefore, variance or standard deviation is generally
CPAP ARASH, AWOO PARE ES a measure of dispersion in most analyses.
If two assets have the same standard deviation, the investor will prefer the
one with the higher expected return.
If two assets have the same expected return, the one with lower variance is
preferred, as it indicates more reliable outcomes.
Portfolio Variance and Risk Reduction
This section examines how combining assets into a portfolio can lead to lower
overall risk compared to holding individual assets—a core concept of portfolio
theory.
Instead of investing in a single asset, investors can combine multiple assets.
Even if the average return remains the same, the portfolio's overall risk may
decrease. Importantly, the variance (or standard deviation) of a portfolio is not
simply the average of the variances of individual assets. In fact, the portfolio
variance can be lower than the variance of any individual asset.
صفحه 9:
+
For example, a combination of assets 2 and 3 was constructed to produce a constant return of $1.10,
regardless of market conditions (good, average, or poor). In this special case, where returns are
perfectly stable across all states, risk is entirely eliminated. The key idea is that if asset returns move
in opposite directions (i.e., are negatively correlated), one can construct a portfolio with zero or very
low standard deviation.
In another example, assets 2 and 4 were examined. Their returns depended on different and
independent factors—market performance and rainfall. An equal-weighted portfolio of these assets
still led to reduced risk, though not to the same extent as in the previous example. Here, extreme
outcomes occurred less frequently, and returns became more concentrated around the mean.
In a third case, assets 2 and 5 were analyzed. Their returns were both influenced by the same factor
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By esandgetuenonthly returns and combining these stocks in pairs, it was shown that an optimal mix
could both increase return and reduce risk. For instance, a 50/50 mix of Dell and GE outperformed
GE alone in terms of both higher expected return and lower standard deviation.
Finally this section underscores that rather than analyzing the full distribution of returns, using
summary measures like mean and variance is often sufficient for portfolio analysis. In the following
parts of the chapter, the authors analytically demonstrate how these portfolio-level characteristics
depend on the properties of the individual assets that compose the portfolio.
صفحه 10:
+
2
General Characteristics of
Portfolios
The return of a portfolio is the weighted average of the returns of the assets it contains,
where each asset’s weight corresponds to its share of the total investment. Similarly, the
expected return of a portfolio is the weighted average of the expected returns of the
individual assets.
While calculating return is straightforward, computing portfolio variance is more complex.
Portfolio variance is a measure of risk and relates to how individual returns deviate from
the average return. For a two-asset portfolio, the variance formula has three components:
the variance of each asset multiplied by the square of its weight, and the covariance
between the two assets multiplied by twice the product of their weights.
Covariance measures how two assets move together. If their returns move in the same
direction, the covariance is positive; if they move in opposite directions, the covariance is
negative. Dividing the covariance by the product of the standard deviations of the two
assets yields the correlation coefficient, which ranges from -1 to +1.
صفحه 11:
When assets are negatively correlated, it is possible to
construct portfolios with lower risk—or even zero
risk. Even if the correlation is zero, the portfolio risk
will still be less than the risk of the individual assets.
However, if the assets are perfectly positively
correlated, diversification cannot reduce risk.
The formula for portfolio variance can be extended to
include more than two assets. In this case, total
portfolio variance consists of two parts: the sum of
each asset’s variance multiplied by the square of its
weight, and the sum of the covariances of all asset
pairs multiplied by twice the product of their weights.
In an equally weighted portfolio of N assets, assuming
all assets are independent (i.e., zero covariance), the
portfolio variance equals the average variance divided
by N. Therefore, as the number of independent assets
in a portfolio increases, the overall risk approaches
zero. However, in the real world, assets usually have
positive correlations, leading to positive covariances.
As a result, risk does not reach zero, though it is still
significantly lower than the risk of individual assets.
Ultimately, further analysis shows that the irreducible
risk of a diversified portfolio tends to converge toward
the average covariance between the assets. The
os
لوقه و
صفحه 12:
This chapter concludes with two practical examples that illustrate
how asset allocation decisions can be made using the concepts
introduced in the chapter.
Example 1: Allocation Between Stocks and Bonds
One of the key decisions for investors is determining the appropriate
allocation between stocks and bonds. To do this, it is essential to estimate
expected returns, standard deviations, and the correlations among assets.
These estimates are typically based on historical data.
In this example, data from Ibbotson (2011) is used for a market value-
weighted stock index and for corporate bonds. The tables and charts show
that increasing the proportion of bonds in the portfolio reduces risk, but
not linearly. In contrast, the expected return decreases linearly from 11.8%
(with full investment in stocks) to 6.4% (with full investment in bonds).
صفحه 13:
Table 1: Comparison of Portfolio Characteristics in Stock-Bond Allocation
Bond Weight (XB) Stock Weight (XS)
1.0) 100%
(°) Standard Deviation (6) Expected Return
202
2 118
This table is a hypothetical sample based on the summarized original text
and is used to illustrate the linear/nonlinear behavior of return and risk as
asset weights change.
صفحه 14:
Example 2: Allocation Between Domestic and Foreign
Stocks
This analysis also uses Ibbotson data to estimate the relevant parameters.
The results show that combining domestic and foreign stocks significantly
reduces the overall risk of the portfolio, highlighting the importance of
diversification.
Figure 1: Efficient Frontier for Portfolios Combining Domestic and
International Stocks
Each dot represents a portfolio
combining domestic and
international equities.
The curve shows that
combining the two significantly
۰ reduces risk, even when
d Return expected return remains
similar — demonstrating the
diversification benefit.
صفحه 15:
Chapter Conclusion
This chapter demonstrated that the risk of a portfolio can differ
significantly from the risk of the individual assets it contains. This
difference was evident both in portfolios composed of specific assets and
those formed from random selections. In the following chapters, we will
explore the relationship between risk and return at the level of individual
assets and then examine how to construct an investment opportunity set
based on investor preferences—namely, a preference for higher returns and
lower risk.
صفحه 16:
