MEMO
TO: Dr. Richard Burgess
FROM: Casper Shoghi
SUBJECT: Project 2
DATE: 21 January 2013
OVERVIEW
This project examined 30 random stocks’ monthly returns from the period of January
2010 to October 2012. By using summary statistics and graphing the efficiency frontier
for visual representation, four portfolios were found and examined for their standard
deviations and returns per month. These portfolios are the model portfolio, the modified
model portfolio, the minimum variance portfolio, and the equally allocated portfolio. In
addition to this, the capital market line (CML) was also found and points along the CML
were examined and related to the model portfolio.
What was found was that the model portfolio was concentrated in only 8 stocks with
much of the weight in an individual stock. After introducing constraints and finding the
modified model portfolio, the concentration went up to ten stocks, and the portfolio was
not weighted as largely in individual stocks. Unlike the model and minimum variance
portfolios, the modified model portfolio is off of the efficiency frontier due to
constraining the weights. The equally allocated portfolio is also off of the efficiency
frontier because the weights are set at a certain amount.
WHAT WAS LEARNED
In addition to learning how to find summary statistics for portfolios and graphing them on
an efficiency frontier, one can learn how to find the capital market line and find points
along it using the Solver tool in Microsoft Excel. Additionally, one will find out that the
modified portfolio lies on the tangency point to the CML on the efficiency frontier. By
constraining the weights of the model portfolio, the ability to further diversify the
portfolio and create the modified model portfolio can also be learned. The sum of the
weights for each portfolio provides clues to just how diversified each of the four
portfolios actually is.
MOVING ALONG THE CAPITAL MARKET LINE
The Capital Market Line (CML) was found by plotting the risk free rate upon the y axis
(at standard deviation = 0) and by plotting the point along the efficiency frontier with the
maximum slope. Using these two points, the slope of the CML was found and it was
drawn into Figure 1. The risk free rate was also added into the portfolio weights and
returns per month. The Solver tool was then used to find points along the CML both
above and below the model portfolio by inserting targeted monthly returns and finding
the standard deviation.
The points along the CML closest to the risk free rate had the most allocation to the risk
free. For example, at a target monthly return of .5%, 81% of the portfolio was allocated to
the risk free. This decreased towards zero allocation in the risk free as the portfolio
returns got closer to the model portfolio. The levered portfolios above the model portfolio
had negative allocations to the risk free rate that progressively increased as they got
further away from the model portfolio. For example, at a return of 3.5%, the largest return
examined, the negative allocation to the risk free was the most at -59%.
The CML greatly simplifies portfolio optimization and the portfolio management process
because it increases the understanding of risk free allocation and allows for the model
portfolio to be found. One can narrow down the results from an infinite number of
portfolios along the CML, to just one possibility (the model portfolio). This model
portfolio appears optimal and would be the portfolio recommended by managers. This
reduction from infinite possibilities to one portfolio greatly simplifies the portfolio
management process.
MODEL PORTFOLIO
First, the equation was created to find the slope by taking the mean return in % per month
of the portfolio less the risk free rate of 9% and dividing that by the standard deviation.
The Solver tool was used to find the point with the maximum slope along the efficiency
frontier. The point with the maximum slope that lies upon the efficiency frontier is the
model portfolio.
The model portfolio resulted in a portfolio in which only 8 of the 30 potential stocks were
invested in with 43.7% allocated to Watson Pharmaceuticals Inc. 100% of the allocation
was in stocks and none of it was in the risk free. This result is synonymous with the
results in Moving Along the CML in that as the tangency point on the efficiency frontier
(the model portfolio) was approached from either direction, the amount allocated to risk
free decreased towards zero.
With only 8 out of the 30 stocks being invested in, one may wish to add upper constraints
upon how much weight can be in an individual security. In this case, 43.7% of the model
portfolio is in Watson Pharmaceuticals. If the manager believed that this was too risky to
have so much of the portfolio weighted in one stock, a maximum percentage allocation
could be added to the portfolio to spread out the weights and increase diversification.
Since the portfolio returns used in this portfolio are historical data, there is no guarantee
on these expected returns, and so it might be beneficial to add these constraints.
MODIFIED MODEL PORTFOLIO
To find the Modified Model Portfolio, upper level constraints were added to how much
weight could be in a particular stock. A constraint capping the amount allocated into
individual stocks at 15% was added to the Solver tool, and once again, the maximum
slope was found.
By limiting how much weight could be in one stock, the portfolio went from being made
up of 8 stocks in the model portfolio to being made up of 10 stocks in the modified, and
the allocation in each of those 10 stocks is less concentrated. Due to this, the standard
deviation has reduced from 3.52 to 3.1955 as can also be seen in Figure 1. This
demonstrates that the modified portfolio is less risky, but it also has less return decreasing
from 2.227% to 1.8372%. On Figure 1, it can be seen that the modified model portfolio
has moved off of the efficiency frontier and is down and to the left of the model portfolio
supporting the change in data above.
EQUALLY ALLOCATED PORTFOLIO AND MINIMUM VARIANCE
PORTFOLIO
The Minimum Variance Portfolio was found by minimizing the variance in the Solver
tool by changing the weights held in each security. For the Equally Allocated Portfolio,
the weight of 3.3% was inserted for each of the 30 stocks.
The minimum variance portfolio had a standard deviation of 2.4164 which is the lowest
possible standard deviation for the portfolio meaning that it lies at the point on the
efficiency frontier with the lowest standard deviation as shown in Figure 1. The minimum
variance portfolio also has a lower return than the model and modified model portfolios
at .7901%.
The equally weighted portfolio can be found off of the efficiency frontier to the right.
This means that the same return could be realized with lower risk. The same return
could be achieved keeping the same return of 1.1492% except with a standard deviation
of around 2.5 instead of the equally weighted portfolio’s standard deviation of 4.8075.
SUM OF WEIGHTS ANALYSIS
The weights in individual securities vary greatly across the four portfolios. In the model
portfolio, 44% was in one stock and 91% was in five. Once modified, the amount in one
stock went down to 15% and the amount in five to 69%. The minimum variance portfolio
more closely resembles the modified portfolio with 40% in one stock, and 88% in five.
The Equally Weighted Portfolio was the only portfolio to not have 100% of the allocation
in a maximum of ten stocks with only 33% in ten stocks.
RESULTS AND ANALYST COMPARISON
One of the stocks that was heavily weighted in the model and modified portfolio was VF
Corporation. According to CNNFN.com, VF Corp. was upgraded on February 19th to a
buy stock with a rating of A-. This is consistent with the results of this project.
Interestingly enough, the most heavily weighted stock in the model portfolio, Watson
Pharmaceuticals, changed its name to Actavis Inc. CNNFN.com also reports as recently
as February 14th that this also a “buy” stock. In looking more at the minimum variance
portfolio, Campbell’s Soup Company is the most heavily weighted, but according to
MSN.com, analysts’ rate Campbell’s as a “moderate sell.”
Disagreement between the portfolios and analysts’ predictions, such as the one with
Campbell’s above, are completely possible because the portfolios take only historical
returns into account and not any future outlooks. Selling the portfolio would undoubtedly
be more difficult if analysts’ predictions disagreed with the portfolio. It is not wise to
invest solely on historical values and so potential investors are more likely to do their
research before investing in a portfolio. If a potential investor found that most analysts
were disagreeing with the heavily weighted stocks in a portfolio, they would be much less
likely to invest in that portfolio.
FROST AND SAVARINO SUMMARY
In the article “For Better Performance: Constrain Portfolio Weights” authors Peter Frost
and James Savarino discuss the topic of weight constraints on portfolios, and how they
can be beneficial to the portfolio manager. The article argues that by constraining the
maximum amount that can be invested in individual securities, a portfolio manager can
decrease the estimation bias and improve performance of a portfolio. Assuming that the
optimal portfolio expected returns and variances are found using historical samples, the
article claims that completely trusting this information causes the expected return
exaggerated.
In addition to this, Frost and Savarino also discuss that disallowing short-selling. Frost
and Savarino found that by disallowing short-selling the estimation bias is reduced
significantly. With this information, the article concludes that constraining the proportion
invested in individual securities in a portfolio can be justified. This justification is due to
the reduction in estimation bias and the improvement in performance.
KRITZMAN SUMMARY
The article “Are Portfolio Optimizers Error Maximizers?” by Mark Kritzman examines
the effect portfolio optimizers can have on portfolios and explores whether or not small
errors are maximized in the returns. Kritzman argues that even though errors in estimates
can greatly effect allocations to the optimal portfolio, the distributions of the returns will
overall be relatively close. One reason for this, Kritzman says, is that assets may be
substitutes for each other. This being the case, misallocations do not have large impact on
the loss. After providing a few hypothetical situations and other exhibits for evidence,
Kritzman concludes that input errors are no reason for alarm in estimating potential
exposure to loss and answers the title of the article by saying that the idea of optimizers
being error maximizers is, in the end, just “hype.”