Portfolio Selection
It is not easy to get the optimal portfolio. In fact, it is tedious to calculate the returns and risks for all possible combinations of even two assets/instruments. At least we should calculate 100 times of the return and risk of the portfolios which combines A and B assets at 100%:0%, 99%:1%, 98%:2%...... 3%:97%, 2%, 98%, 1% 99%, and 0%:99%.
However, there is a software program called ‘Shazam’ which does all these by one command(‘Portfolio’), get the best (combination) portfolio and the minimum variance portfolio, and even to plot the efficient frontier, such as:
The PORTFOLIO command provides features for calculating efficient portfolios.
Example
Berndt [1991, Chapter 2] provides a data set of monthly returns for a number of companies for the period January 1978 to December 1987. From this data set, a file (filename: P.txt) was prepared with returns for Mobil, IBM, Weyerhauser and Citicorp as well as the return on 30-day Treasury Bills (a measure of the risk-free return) and a value-weighted composite monthly market return based on all stocks listed at the New York and American Stock Exchanges.
The SHAZAM commands below solve a portfolio selection problem.
**The following two lines are standard SHAZAM protocol to read the data into the program.SAMPLE 1 120
READ (P.txt) DATE MOBIL IBM WEYER CITCRP MARKET RKFREE / SKIPLINES=1
** we do the following 4 lines of command in order to Convert to percentages
GENR MOBIL=100*MOBIL
GENR IBM=100*IBM
GENR WEYER=100*WEYER
GENR CITCRP=100*CITCRP
** Set a risk-free rate of return
GEN1 RF=100*(RKFREE:120)
** Here comes the magic command:
PORTFOLIO MOBIL IBM WEYER CITCRP / INRATES RISKFREE=RF EQUALW PFRONT &
GRAPHDATA GRAPHLINE
STOP
Below is SHAZAM output generated by the PORTFOLIO command.
|_PORTFOLIO MOBIL IBM WEYER CITCRP / INRATES RISKFREE=RF EQUALW PFRONT &| GRAPHDATA GRAPHLINE
PORTFOLIO ANALYSIS - RATES OF RETURN 4 ASSETS 120 OBSERVATIONS
MEAN RISKFREE RATE OF RETURN = 0.27700
VARIABLE MEAN ST.DEV SHARPE
MOBIL 1.6192 8.0308 0.16713
IBM 0.96167 5.9024 0.11600
WEYER 0.96333 8.5066 0.80682E-01
CITCRP 1.1858 8.0972 0.11224
COVARIANCE MATRIX
MOBIL 64.493
IBM 15.225 34.838
WEYER 26.403 24.694 72.363
CITCRP 20.227 20.250 37.195 65.564
MOBIL IBM WEYER CITCRP
EFFICIENT PORTFOLIOS
MINIMUMVARIANCE RISKFREE=ZERO RETURN=ZERO ACTUAL
MEAN 1.1521 1.3300 0.44409E-15 1.1825
VARIANCE 27.751 32.037 207.44 32.828
STDEV 5.2680 5.6602 14.403 5.7296
SHARPE 0.16612 0.18604 -0.19233E-01 0.15804
PORTFOLIO WEIGHTS
MOBIL 0.23486 0.48422 -1.3797 0.25000
IBM 0.59096 0.39925 1.8322 0.25000
WEYER 0.13607E-01 -0.10732 0.79655 0.25000
CITCRP 0.16057 0.22385 -0.24910 0.25000
The figure below shows the minimum-variance risk-return frontier. The x-axis measures the standard deviation of return.
The portfolios marked M and Z are the MINIMUMVARIANCE and RISKFREE=ZERO portfolios respectively. The curve that extends upward from point M gives the efficient frontier. That is, all portfolios on the efficient frontier have greater expected return than the portfolio with the global minimum variance.
The portfolio A is the equal-weighted portfolio calculated with the EQUALWEIGHT option and reported on the SHAZAM output in the ACTUAL column. Clearly, this is not an efficient portfolio since it is possible to find an alternative portfolio with the same risk but a higher expected return.
The straight line with an intercept at the risk-free rate of return r that is tangential to the efficient frontier at point Q has a slope that is the maximimum Sharpe ratio of all possible portfolios.
So by using the Shazam’s portfolio command, you can combine a few assets/instruments and come up with the best portfolio, and thus can be a ‘Portfolio Manager’. You may be able to start your own mutual fund company as well. The next hand-out goes into a little more details and shows how to actually use the Shazam and to get the graphs and the best portfolio for your given assets.