rk = Specific Return
rexpected = Expected Return
n = Number of Returns (sample size).
A portfolio is the total collection of all investments held by an individual or institution, including stocks, bonds, real estate, options, futures, and alternative investments, such as gold or limited partnerships.
Most portfolios are diversified to protect against the risk of single securities or class of securities. Hence, portfolio analysis consists of analyzing the portfolio as a whole rather than relying exclusively on security analysis, which is the analysis of specific types of securities. While the risk-return profile of a security depends mostly on the security itself, the risk-return profile of a portfolio depends not only on the component securities, but also on their mixture or allocation, and on their degree of correlation.
As with securities, the objective of a portfolio may be for capital gains or for income, or a mixture of both. A growth-oriented portfolio is a collection of investments selected for their price appreciation potential, while an income-oriented portfolio consists of investments selected for their current income of dividends or interest.
The selection of investments will depend on one’s tax bracket, need for current income, and the ability to bear risk, but regardless of the risk-return objectives of the investor, it is natural to want to minimize risk for a given level of return.
The efficient portfolio consists of investments that provide the greatest return for the risk, or—alternatively stated—the least risk for a given return. To assemble an efficient portfolio, one needs to know how to calculate the returns and risks of a portfolio, and how to minimize risks through diversification.
Since the return of a portfolio is commensurate with the returns of its individual assets, the return of a portfolio is the weighted average of the returns of its component assets.
| Portfolio Return = | n ∑ k=1 | Dollar Amount of Asset k Dollar Amount of Portfolio | x | Return on Asset k |
The dollar amount of an asset divided by the dollar amount of the portfolio is the weighted average of the asset and the sum of all weighted averages must equal 100%.
Portfolio risks can be calculated, like calculating the risk of single investments, by taking the standard deviation of the variance of actual returns of the portfolio over time.
This variability of returns is commensurate with the portfolio’s risk, and this risk can be quantified by calculating the standard deviation of this variability. Standard deviation, as applied to investment returns, is a quantitative statistical measure of the variation of specific returns to the average of those returns.
| s = Standard Deviation rk = Specific Return rexpected = Expected Return n = Number of Returns (sample size). |
Although the diversifiable risk of a portfolio obviously depend on the risks of the individual assets, it is usually less than the risk of a single asset because the returns of different assets are up or down at different times. Hence, portfolio risk can be reduced by diversification—choosing individual investments that rise or fall at different times from the other investments in the portfolio. Diversifiable risk declines, quickly at first, then more slowly, reaching a minimum with about 20 - 25 securities.
The basis for diversification is that different classes of assets respond differently to different economic conditions, which causes investors to move assets from 1 class to another to reduce risk and to profit from changing conditions. For instance, when interest rates rise, stocks tend to go down as margin interest rises making it more expensive to borrow money to buy stocks, which lowers their demand, and therefore their prices, while higher interest rates also causes investors to move more money into less risky securities, such as bonds, that pay interest.
Covariance is a statistical measure of how 1 investment moves in relation to another. If 2 investments tend to be up or down during the same time periods, then they have positive covariance. If the highs and lows of 1 investment move in perfect coincidence to that of another investment, then the 2 investments have perfect positive covariance. If 1 investment tends to be up while the other is down, then they have negative covariance. If the high of 1 investment coincides with the low of the other, then the 2 investments have perfect negative covariance. If there is no discernible pattern to the up and down cycles of 1 investment compared to another, then the 2 investments have no covariance.
Because covariance numbers cover a wide range, the covariance is normalized into the correlation coefficient, which measures the degree of correlation, ranging from -1 for a perfectly negative correlation to +1 for a perfectly positive correlation. An uncorrelated investment pair would have a correlation coefficient close to zero. Note that since the correlation coefficient is a statistical measure, a perfectly uncorrelated pair of investments will rarely, if ever, have an exact correlation coefficient of zero.
The most diversified portfolio consists of securities with the greatest negative correlation. A diversified portfolio can also be achieved by investing in uncorrelated assets, but there will be times when the investments will be both up or down, and thus, a portfolio of uncorrelated assets will have a greater degree of risk, but it is still significantly less than positively correlated investments. However, even positively correlated investments will be less risky than single assets or investments that are perfectly positively correlated.
Correlations can change over time and in different economic conditions. For instance, during the late 1990’s, stock prices increased significantly, then crashed in 2000. Interest rates were lowered to boost the economy, which caused real estate prices to increase significantly from 2001 - 2006. Hence, real estate prices were increasing while stocks were either declining, or not increasing by nearly the same rate. This reflects the general negative correlation between the stock market and the real estate market. The real estate market was forming a bubble due to the extremely low interest rates at the time. The bubble finally burst in 2007, and especially 2008, leading to the 2007– 2009 credit crisis. This caused money to move into commodities during the summer of 2008, which formed another bubble, with oil prices, for instance, reaching $147 per barrel. The fast increase in prices was not due to demand, but due to the transfer of money from assets doing poorly—stocks and real estate—to commodities and future contracts. In other words, it was another bubble. However, as credit dried up, due to the prevalence of many defaults of subprime mortgages, almost every investment came crashing down in September and October of 2008: real estate, stocks, bonds, commodities. Only United States Treasuries, which are virtually free of credit-default risk, rose significantly in price, driving their yields down proportionately, with the yields of short-term T-bills reaching almost zero.
So the corollary of this story is that correlations can and do change, and that investments always have some risk.
