● Investing in stocks comes with uncertainty, which
can be better understood by
studying the history of security returns.
● Historical data gives insights into expected returns
and associated risks from
different securities.
● By examining past risks and returns, investors can
make more informed
predictions about potential future performance.
● Historical analysis can guide decisions on
diversification, risk management, and
investment strategies.
Market indexes are tools for summarizing the overall
performance of a range of securities, offering a
snapshot of market trends and patterns.
● The Dow Jones Industrial Average and the S&P 500
are benchmarks for tracking the American stock
market's health.
● The Dow, despite its fame, covers only 30 large
firms and can skew perceptions due to its limited
scope and equal-weight approach.
● The S&P 500 provides a more comprehensive view,
weighted by market capitalization, reflecting a
closer approximation of the average investor's
experience.
● Indexes guide investment decisions and are used
as benchmarks for portfolio performance.
performance of various investment
Comparing the growth of $1 invested Historical volatility informs about the
possible range of investment
in stocks, bonds, and Treasury bills
vehicles, like Treasury bills, bonds, and
since 1900 illustrates the impact of
outcomes, preparing investors for
stocks.
risk on investment growth.
potential market swings.
● The risk-return profile of these
● This information helps investors
investments has shown that higher
understand the potential long-term
risks are often accompanied by higher
outcomes of investing in various asset
returns over the long term.
classes.
Historical data reveals the
return foregone by investing in a
For a project with the same risk as the While historical data provides a
market, the expected return should at benchmark, the exact risk premium is
project rather than in an alternative
least match the historical market
still subject to debate, affecting cost
with
return
estimations for projects and
comparable risk.
adjusted for the current environment. investments.
● Historical averages of returns can
● The 1981 example shows the
assist in estimating the cost of capital,
necessity to adjust for contemporary
but they need to be adjusted for
interest rates when calculating the
current
expected
market conditions.
return on stocks.
● The opportunity cost of capital is the
Risk is defined in financial terms as the chance that an
outcome or investment's actual gains will differ from an
expected outcome or return. Risk includes the possibility of
losing some or all of an original investment.
Quantifiably, risk is usually assessed by considering historical
behaviors and outcomes. In finance, standard deviation is a
common metric associated with risk.
Standard deviation provides a measure of the volatility of
asset prices in comparison to their historical averages in a
given time frame.
Overall, it is possible and prudent to manage investing risks by
understanding the basics of risk and how it is measured.
Learning the risks that can apply to different scenarios and
some of the ways to manage them holistically will help all
types of investors and business managers avoid unnecessary
and costly losses.
• Risk takes on many forms but is broadly
categorized as the chance an outcome
or investment's actual gain will differ
from the expected outcome or return.
• Risk includes the possibility of losing
some or all of an investment.
• There are several types of risk and
several ways to quantify risk for
analytical assessments.
• Risk can be reduced using
diversification and hedging strategies.
Risk measures are statistical measures that are historical
predictors of investment risk and volatility, and they are
also major components in modern portfolio theory
(MPT). MPT is a standard financial and academic
methodology for assessing the performance of a stock
or a stock fund as compared to its benchmark index.
There are five principal risk measures, and each measure
provides a unique way to assess the risk present in
investments that are under consideration. The five
measures include the alpha, beta, R-squared, standard
deviation, and Sharpe ratio. Risk measures can be used
individually or together to perform a risk assessment.
When comparing two potential investments, it is wise to
compare like for like to determine which investment
holds the most risk.
Alpha measures risk relative to the market or a
selected benchmark index. For example, if the
Beta measures the volatility or systemic risk of a fund
S&P 500 has been deemed the benchmark for
in comparison to the market or the selected
a particular fund, the activity of the fund would
be compared to that experienced by the
selected index. If the fund outperforms the
benchmark, it is said to have a positive alpha. If
the fund falls below the performance of the
benchmark, it is considered to have a negative
alpha.
benchmark index. A beta of one indicates the fund is
expected to move in conjunction with the
benchmark. Betas below one are considered less
volatile than the benchmark, while those over one
are considered more volatile than the benchmark.
R-squared measures the percentage of an
investment's movement attributable to
Beta measures the volatility or systemic risk of a fund
movements in its benchmark index. An R-
in comparison to the market or the selected
squared value represents the correlation
between the examined investment and its
associated benchmark. For example, an Rsquared value of 95 would be considered to
have a high correlation, while an R-squared
value of 50 may be considered low.
benchmark index. A beta of one indicates the fund is
expected to move in conjunction with the
benchmark. Betas below one are considered less
volatile than the benchmark, while those over one
are considered more volatile than the benchmark.
Here is a very simple example showing how variance and standard deviation are
calculated.
Suppose that you are offered the opportunity to play the following game: You
start by investing $100. Then two coins are flipped. For each head that comes up, your
starting balance will be increased by 20%, and for each tail that comes up, your
starting
balance will be reduced by 10%. Clearly, there are four equally likely outcomes:
∙∙ Head + Head: You make 20 + 20 = 40%
∙∙ Head + Tail: You make 20 − 10 = 10%
∙∙ Tail + Head: You make −10 + 20 = 10%
∙∙ Tail + Tail: You make −10 − 10 = −20%
There is a chance of 1 in 4, or .25, that you will make 40%; a chance of 2 in 4, or .5,
that you will make 10%; and a chance of 1 in 4, or .25, that you will lose 20%. The
game’s expected return is, therefore, a weighted average of the possible outcomes:
Expected return
=
probability-weighted average of possible outcomes
=
( .25 × 40 ) + ( .5 × 10 ) + ( .25 × − 20 ) = +10%
When estimating the spread of possible outcomes from
investing in the stock market,
most financial analysts start by assuming that the spread of
returns in the past is a reasonable
indication of what could happen in the future. Therefore, they
calculate the
standard deviation of past returns. To illustrate, suppose that
you were presented with
the data for stock market returns shown in Table 11.3. The
average return over the
6 years from 2012 to 2017 was 15.2%. This is just the sum of the
returns over the
6 years divided by 6 (91.3/6 = 15.2%).
Column 2 in Table 11.3 shows the difference between each
year’s return and the
average return. For example, in 2013 the return of 31.7% on
common stocks was above
the 6-year average by 16.5%. In
column 3, we square these deviations
from the average.
The variance is then the average of
Because standard deviation is the
square root of the variance,
Standard deviation = square root of
variance
=√
______
102.61 = 10.13%
-The risk–return trade-off: Higher-risk assets
tend to have higher expected returns, but
also more variability in returns.
-Diversification: Combining different assets
in a portfolio can reduce the overall
variability of returns, as long as the assets
are not perfectly correlated.
-Asset versus portfolio risk: The volatility of
an individual asset is not a good measure of
its risk when it is held as part of a portfolio,
because it does not account for the
correlation with other assets.
An example of these concepts is given in Table 11.6
and Table 11.7, which show the possible returns of two
stocks, auto and gold, in different scenarios. The auto
stock has a higher expected return (5%) than the gold
stock (1%), but also a higher variance (92.7) and
standard deviation (9.6%). The gold stock has a lower
expected return (1%), but also a lower variance (53.3)
and standard deviation (7.3%). However, the gold
stock is negatively correlated with the auto stock,
meaning that it tends to do well when the auto stock
does poorly, and vice versa. This makes the gold stock
a good candidate for diversification, as it can reduce
the portfolio risk.
portfolios of two stocks, auto and gold, with different expected
returns, variances, and correlations. It shows how to calculate the
portfolio return, expected return, and standard deviation using
formulas and examples. It also explains how diversification reduces
portfolio risk and how the incremental risk of an asset depends on
its correlation with the portfolio.
-Portfolio return: The weighted average of the returns on the individual assets,
where the weights are the fractions of the portfolio invested in each asset. For a
portfolio of two assets, A and B, the formula is:
Portfolio return = (weight of A × return on A) + (weight of B × return on B)
-Expected return: The average of the possible returns, weighted by their
probabilities. For a portfolio of two assets, A and B, the formula is:
Expected return = (weight of A × expected return on A) + (weight of B × expected
return on B)
-Variance: The average of the squared deviations from the expected return,
weighted by their probabilities. For a portfolio of two assets, A and B, the
formula is:
Variance = (weight of A)^2 × (variance of A) + (weight of B)^2 × (variance of B) + 2
× (weight of A) × (weight of B) × (covariance of A and B)
-Expected return: The average of the possible returns,
weighted by their probabilities. For a portfolio of two
assets, A and B, the formula is:
Expected return = (weight of A × expected return on A)
+ (weight of B × expected return on B)
-Variance: The average of the squared deviations from
the expected return, weighted by their probabilities.
For a portfolio of two assets, A and B, the formula is:
Variance = (weight of A)^2 × (variance of A) + (weight
of B)^2 × (variance of B) + 2 × (weight of A) × (weight of
B) × (covariance of A and B)
-Standard deviation: The square root of the variance. For a
portfolio of two assets, A and B, the formula is:
Standard deviation = √(variance)
-Covariance: A measure of how the returns on two assets
move together. It is positive when the returns move in the
same direction, negative when they move in opposite
directions, and zero when they are independent. For two
assets, A and B, the formula is:
Covariance = (probability of scenario 1) × (deviation of A in
scenario 1) × (deviation of B in scenario 1) + (probability of
scenario 2) × (deviation of A in scenario 2) × (deviation of B
in scenario 2) + ... + (probability of scenario n) × (deviation
of A in scenario n) × (deviation of B in scenario n)
Table 11.8 and Table 11.9, which show the possible
returns, expected returns, variances, and standard
deviations of different portfolios formed by mixing auto
and gold stocks in varying proportions. The tables also
show the covariance of the two stocks, which is −0.18.
The example illustrates how adding a more volatile asset
(gold) to a less volatile asset (auto) can reduce the
portfolio risk, as long as the two assets are negatively
correlated. The example also shows how the incremental
risk of an asset depends on its correlation with the
portfolio. The gold stock has a negative incremental risk
when added to an all-auto portfolio, meaning that it
lowers the portfolio risk. However, the auto stock has a
positive incremental risk when added to an all -gold
portfolio, meaning that it increases the portfolio risk.
- The volatility of a portfolio is the measure of how much the portfolio's
returns vary over time. A lower volatility means a more stable and less risky
portfolio.
- The volatility of a portfolio does not depend only on the volatility of each
individual asset, but also on how the assets move together. This is measured
by the correlation between their returns, which is a number between -1 and
1.
- A correlation of 1 means that the assets move in the same direction and by
the same amount. A correlation of -1 means that the assets move in opposite
directions and by the same amount. A correlation of 0 means that the assets
move independently of each other.
- The lower the correlation between the assets, the more the diversification
benefit and the lower the portfolio volatility. This is because when some
assets lose value, others may gain value and offset the losses.
- two hypothetical stocks: one that sells umbrellas and one that sells
sunscreen. These stocks have a negative correlation because they perform
well in different weather conditions. A
portfolio that invests in both stocks would have a low volatility because the
losses from one stock would be balanced by the gains from the other.
- correlations between some major industries, calculated from historical
data. The table shows that most industries have positive correlations, but
some are higher than others. The highest correlation is between the
machinery and auto industries, which are both sensitive to the business
cycle. The lowest correlation is between the gold and tobacco industries,
which are less affected by economic fluctuations. A portfolio that invests in
different industries would have a lower volatility than a portfolio that invests
in the same industry.