MODERN PORTFOLIO THEORY



What we call modern portfolio theory arises from the work of Harry
Markowitz in the early 1950s. (With that date, I’m not sure how modern it
is, but we are stuck with the name.)
As we will see, the payoff from applying modern portfolio theory is
that, by combining assets in a portfolio, you can have a higher expected return
for a given level of risk; or, alternatively, you can have less risk for a
given level of expected return.
Modern portfolio theory was designed to deal with equities; so
throughout all of this first part, we are thinking about equities. We switch
to loans and other credit assets in the next part.
The Efficient Set Theorem and the Efficient Frontier
Modern portfolio theory is based on a deceptively simple theorem, called
the Efficient Set Theorem:
27
An investor will choose her/his optimal portfolio from the set of portfolios
that:
1. Offer maximum expected return for varying levels of risk.
2. Offer minimum risk for varying levels of expected return.
Exhibit 2.1 illustrates how this efficient set theorem leads to the efficient
frontier. The dots in Exhibit 2.1 are the feasible portfolios. Note that
the different portfolios have different combinations of return and risk. The
efficient frontier is the collection of portfolios that simultaneously maximize
expected return for a given level of risk and minimize risk for a given
level of expected return.
The job of a portfolio manager is to move toward the efficient frontier.
Expected Return and Risk
In Exhibit 2.1 the axes are simply “expected return” and “risk.” We need
to provide some specificity about those terms.
28 THE CREDIT PORTFOLIO MANAGEMENT PROCESS
EXHIBIT 2.1 The Efficient Set Theorem Leads to the Efficient Frontier
In modern portfolio theory, when we talk about return, we are talking
about expected returns. The expected return for equity i would be
written as
E[Ri] = μ
i
where μ
i is the mean of the return distribution for equity i.
In modern portfolio theory, risk is expressed as the standard deviation
of the returns for the security. Remember that the standard deviation for
equity i is the square root of its variance, which measures the dispersion of
the return distribution as the expected value of squared deviations about
the mean. The variance for equity i would be written as1
The Effect of Combining Assets in a
Portfolio—Diversification
Suppose that we form a portfolio of two equities—equity 1 and equity 2.
Suppose further that the percentage of the portfolio invested in equity 1 is
w1 and the percentage invested in equity 2 is w2. The expected return for
the portfolio is
E[Rp] = w1E[R1] + w2E[R2]
That is, the expected return for the portfolio is simply the weighted
sum of the expected returns for the two equities.
The variance for our two-equity portfolio is where things begin to get
interesting. The variance of the portfolio depends not only on the variances
of the individual equities but also on the covariance between the returns
for the two equities (σ1,2):
Since covariance is a term about which most of us do not have a
mental picture, we can alternatively write the variance for our two-equity
portfolio in terms of the correlation between the returns for equities 1
and 2 (ρ1,2):
σp2 wσ wσ wwρ σσ
1
2
1
2
2
2
2
2
= + +2 1 2 1,2 1 2
σp2 wσ wσ wwσ
1
2
1
2
2
2
2
2
= + +2 1 2 1,2
σi2=E[(E[Ri]−Ri )2]
Modern Portfolio Theory and Elements of the Portfolio Modeling Process 29
This boring-looking equation turns out to be very powerful and has
changed the way that investors hold equities. It says:
Unless the equities are perfectly positively correlated (i.e., unless ρ1,2 = 1)
the riskiness of the portfolio will be smaller than the weighted sum of the
riskiness of the two equities that were used to create the portfolio.
That is, in every case except the extreme case where the equities are
perfectly positively correlated, combining the equities into a portfolio will
result in a “diversification effect.”
This is probably easiest to see via an example.
Example: The Impact of Correlation
Consider two equities—Bristol-Meyers Squibb and Ford Motor Company. Using historical
data on the share prices, we found that the mean return for Bristol-Meyers Squibb was 15%
yearly and the mean return for Ford was 21% yearly. Using the same data set, we calculated
the standard deviation in Bristol-Myers Squibb’s return as 18.6% yearly and that for Ford as
28.0% yearly.
E(RBMS) = μBMS = 15% E(RF) = μF = 21%
σBMS = 18.6% σF = 28.0%
The numbers make sense: Ford has a higher return, but it is also more risky.
Now let’s use these equities to create a portfolio with 60% of the portfolio invested in
Bristol-Myers Squibb and the remaining 40% in Ford Motor Company. The expected return
for this portfolio is easy to calculate:
Expected Portfolio Return = (0.6)15 + (0.4)21 = 17.4%
The variance of the portfolio depends on the correlation of the returns on Bristol-Meyers
Squibb’s equity with that of Ford (ρBMS, F):
The riskiness of the portfolio is measured by the standard deviation of the portfolio return—
the square root of the variance.
The question we want to answer is whether the riskiness of the portfolio (the portfolio
standard deviation) is larger, equal to, or smaller than the weighted sum of the risks (the
standard deviations) of the two equities:
Weighted Sum of Risks = (0.6)18.6 + (0.4)28.0 = 22.4%
To answer this question, let’s look at three cases.
Variance of Portfolio Return= +
+
( . ) ( . ) ( . ) ( )
( )( . )( . )( , )( . )( . )
0 6 18 6 0 4 28
2 06 04 186 280
2 2 2 2
ρBMS