Portfolios Returns and Risks
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 riskreturn profile of a security depends mostly on the security itself, the riskreturn 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 growthoriented portfolio is a collection of investments selected for their price appreciation potential, while an incomeoriented 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 riskreturn 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.
Portfolio Returns
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  ×  Return on Asset k 
n = number of assets 
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%.
Example: Calculating the Expected Return of a Portfolio of 2 Assets
Asset Weightings  

Asset A  30% 
Asset B  70% 
Expected Returns for each Asset  
E(r_{A})  13.9% 
E(r_{B})  9.7% 
The expected return of this portfolio is calculated thus:
Portfolio Expected Return = .3 × .139 + .7 × .097 = .109 = 10.9%
Portfolio Risk — Diversification and Correlation Coefficients
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. One standard deviation is equal to the average deviation of the sample.
s = Standard Deviation r_{k }= Specific Return r_{expected }= Expected Return n = Number of Returns (sample size) n – 1 = number of degrees of freedom, which, in statistics, is used for small sample sizes 
Although the diversifiable risk of a portfolio obviously depends 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. For most portfolios, diversifiable risk declines, quickly at first, then more slowly, reaching a minimum with about 20  25 securities. However, how rapidly risk declines depends on the covariance of the assets composing the portfolio.
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. The risk of a portfolio composed of these assets can be reduced to zero. 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. However, there is no reduction in risk by combining assets that are perfectly 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 Great Recession. 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 creditdefault risk, rose significantly in price, driving their yields down proportionately, with the yields of shortterm Tbills reaching almost zero.
So the corollary of this story is that correlations can and do change, and that investments always have some risk.
Calculating the Covariance and Coefficient of Correlation between 2 Assets
In this section, we will actually calculate the covariance and the coefficient of correlation between 2 assets, which is the simplest case, based on the following table:
Economic State  Probability of State  Asset A Return (%)  Asset B Return (%) 

Boom  20%  22  6 
Normal  55%  14  10 
Recession  25%  7  12 
The variance of an asset A is calculated thus:
σ^{2}  =  S ∑ s=1  P_{s}{[r_{As} – E(r_{A})]^{2} 

Risk is typically represented by the standard deviation of the expected returns of an asset, equal to the square root of its variance:
Standard Deviation = √σ^{2}
To calculate variances for the 2 assets, the probability of each state is multiplied by the return for that state minus the expected return squared. The expected returns for these 2 assets were calculated in the 1^{st} example at the top of the page:
Expected Returns  

E(r_{A})  13.9% 
E(r_{B})  9.7% 
σ^{2}_{A}  =  .2 × (22 – 13.9)^{2} + 
.55 × (14 – 13.9)^{2} +  
.25 × (7 – 13.9)^{2}  
=  25.03 
σ^{2}_{B}  =  .2 × (6 – 9.7)^{2} + 
.55 × (10 – 9.7)^{2} +  
.25 × (12 – 9.7)^{2}  
=  4.11 
Variances of Returns (%)  

σ^{2}_{A}  25.03 
σ^{2}_{B}  4.11 
Covariance is measured over time, by comparing the expected returns of each asset for each time period. The time periods are selected for the different states of the economy , comparing the expected returns of each asset during boom times, recessions, and normal times. Although returns can be selected according to other criteria, such as monthly returns, it makes sense to sample the returns based upon different states of the economy, since this is more likely to reveal their covariance.
σ_{AB}  =  S ∑ s=1  P_{s}{[r_{As} – E(r_{A})][r_{Bs} – E(r_{B})]} 

The covariance of 2 assets is equal to the probability of each economic state multiplied by the difference of the return for each asset for each economic state minus the expected return for that asset. The covariance of these 2 assets, based on the table above, is:
σ_{AB}  =  .2 × (22 – 13.9) × (6 – 9.7) + 
.55 × (14 – 13.9) × (10 – 9.7) +  
.25 × (7 – 13.9) × (12 – 9.7)  
=  9.95 
The coefficient of correlation between Asset A and Asset B, designated as σ_{AB}, which can range from 1 to +1:
σ_{AB} =  σ_{AB} σ_{A}σ_{B} 
Therefore:
σ_{AB} =  σ_{AB} σ_{A}σ_{B}  =  9.95 √25.03 √4.11  =  0.98 
As the number of assets increases, the computational complexity greatly increases, since covariance must be measured between every 2 different assets in a portfolio, which leads to (n2 – n) / 2 covariance calculations, where n = number of assets in the portfolio, in addition to the calculations of the expected returns and variances for each asset . The number of covariance calculations is divided by 2 because the covariance of Asset A to Asset B is the same as the covariance of Asset B to Asset A. To avoid this complexity, simplifying models are used. The simplest of these models is the singleindex model, which can approximate the covariance of assets in a portfolio by comparing the variance of each asset with the variance of the market.