The art of professional portfolio management experienced a revolution after the concepts
in Harry Markowitz’s seminal paper explained how investors view risk and
return. In his work, he derived formulas used to calculate volatility and
expected return of a diversified portfolio with the ultimate goal of his to
find the best portfolio allocation to maximize return for a given level of
risk.
His work is
based on three assumptions: 1) Investors are generally risk-averse, 2)
investors base their portfolio decisions on risk and expected return only, and
3) investors measure risk as the variance (or standard deviation) of expected
returns. The third assumption led Markowitz to conceive the idea of a portfolio
being “mean-variance” efficient when no other portfolio offers a higher
expected return at a given level of risk, or lower level of risk for a given
expected return. Creating portfolios that have higher mean-variance efficiency
depends on certain statistical characteristics of the returns of the assets
held in a portfolio.
First, an
arithmetic mean of historical returns is used to calculate the expected value
of portfolio returns. This mean can be used to provide a reasonable
forward-looking expected return for the next time period of the same magnitude.
As Markowitz mentions, investors will seek out the highest returns. However,
based on Markowitz’s work higher expected return is accompanied by a
commensurate increase in the level of total risk.
Using
Markowitz’s framework, the total risk of a portfolio equals its volatility
(i.e. standard deviation) of past returns. Even though standard deviation
assumes a normal distribution of data which is not an accurate measurement of
financial time series, the risk quantified describes how far outcomes are
likely to be from the distribution’s mean, or expected value in this case. In
other words, the calculation of standard deviation (i.e. square root of the
variance of returns) allows for investors to see how standard deviation
measures the average deviation from the series expected value and can be used
to interpret the expectation for future volatility in return streams.
After
identifying that total risk can be measured with the standard deviation of past
returns, investors need to address the two types of portfolio volatility. The
total risk referenced above is comprised of market and firm-specific risk, also
known as undiversifiable and diversifiable risk, respectively. In other words,
firm-specific or unsystematic risk can be diversified away in a portfolio but
market or systematic risk cannot be removed entirely.
From a
statistical perspective, both macroeconomic and firm-specific risk can be
defined using two calculations. As mentioned above, standard deviation of a
component’s returns will estimate the range of returns for a security over a
specified period of time. This calculation is the first portfolio risk metric.
Second,
investors need to calculate the systematic risk that reflects how individual
stock returns in a portfolio move together and respond simultaneously to
macroeconomic news. The statistical term used for this calculation is called
covariance. By contrasting variance (i.e. the value for which the standard
deviation calculates a square root) and covariance we can better understand the
fundamental differences between these two components of portfolio risk.
For a two
stock portfolio, variance and covariance formulas differ in that when
calculating covariance, instead of multiplying the first stock’s deviation from
its mean in each period by itself (i.e. squaring the quantity) we multiply it
by the second stock’s deviation from its mean. In other words, variance measures
how one stock varies around its own mean and covariance assesses how two stocks
vary around their respective means.
Readers who
have a background in statistics may have already foreseen how based on the
calculations mentioned above diversification reduces only firm-specific risk.
That is, as the number of holdings in a portfolio increases, the individual
firm-specific variance terms will increase linearly, but the covariance terms
will increase exponentially. With more holdings in a portfolio the more
macroeconomic risk dominates the portfolio volatility.
Studies have
consistently shown that in most cases once a portfolio contains 15-30 stocks
only the effect of covariance matters. However, the optimal number of holdings
will depend on the correlations and covariance between each pair of stocks in
the portfolio. In other words, the lower the correlation and covariance across
holdings the greater the opportunity to reduce portfolio volatility even when
holding fewer positions.
Key
Takeaways:
- Basic statistics (e.g. arithmetic mean, standard
deviation, covariance, and correlation) provide the foundation for
understanding portfolio management.
- In modern finance, total risk (i.e. volatility)
is explained with the standard deviation of returns
- This total risk is comprised of macroeconomic and
firm-specific risk
- Only firm-specific risk can be reduced through
diversification
- As a portfolio increases its number of holdings,
the covariance (macro risk) effects eventually dominate the portfolio
volatility calculation.
- Creating a portfolio diversified among stocks
with low return correlations improves the portfolio’s mean-variance
efficiency.
Summary
adapted from Robert Weigand’s Applied Equity Analysis and Portfolio Management.
As always,
please feel free to contact me with any comments or questions. Thanks for
reading.
John
Comments
Post a Comment