Portfolios are primarily a device of risk diversification. A portfolio, which is a set of assets that is managed as a group, is an attempt to reduce the overall risk of independent securities by looking at the volatility of the group of assets as a whole rather than their volatility as individual securities.

By combining assets of different types and levels of risk, it's possible to engineer a less risky investment with higher expected return than a more-risky individual security would have provided. The portfolio spreads risk among several risk factors instead of making the investor totally reliant on one asset's performance.

### Measuring Risk

Each individual security has an associated risk (in statistical terminology, this is its standard deviation, (σ), the volatility of its value from the mean at any given moment. Unfortunately, the standard deviation of a particular security has no meaning if we wish to compare it to the performance of other assets, which will have different expected returns.

So the coefficient of variation is used instead: CV = (σ / E(R))

With CV calculated, we have a way of measuring the relative riskiness of investing in a given security. This will allow us to work with the risk of individual securities in a meaningful way - but not with a portfolio of them.

### Portfolio Risk

Portfolio risk is calculated in much the same way.

For a two-asset portfolio, just use a weighted average to determine an expected rate of return:

E(Rportfolio) = (wa * E(Ra) + wa + E(Rb), where (wa+wb = 100%)

note: you can actually do this for an arbitrary number of assets, but the math will get very complicated when calculating standard deviation, and it's usually easier to view the existing portfolio as one asset, just adding one asset at a time, unless you're trying to do complex financial analyses with multiple simultaneous changes in variables.

Two values are of major concern in calculating the risk of a portfolio: the correlation coefficient (ra,b), and the beta (β) of the portfolio.

ra,b, the correlation coefficient, measures the linearity of the relationship between the assets. An r of 1.0 indicates that the asset values move together, an r of -1.0 means they are perfect opposites, and an r of 0 means there is no linear correlation between the values. The goal is to get the portfolio as near to r =-1.0 as possible, since perfect negative correlation would mean that risk was entirely eliminated. Unfortunately, determining r precisely is very difficult.

The correlation coefficient - a concept borrowed from statistical analysis - is a measure of how closely predicted trends will match historical performance. To estimate it, we need to know the relative probabilities of all possible values for all variables! With this topic, we go far beyond the intended scope of the essay, into mathematical analysis and data modelling. Formal study of joint distibutions and variable independence is the province of calculus-based statistics, not basic financial management.

With the correlation coefficient estimated or obtained from an outside source, you can calculate the standard deviation of the two assets with a straightforward but complex equation for standard deviation with two variables:

σportfolio = √(waa + wbb + 2wawbra,bσaσb)

### Nondiversifiable Risk

Beta(β) is the nondiversifiable risk of the portfolio. It is the unremovable risk which remains after we've cancelled out as much risk as possible through combining assets. Unless you have perfect negative correlation of half the assets with the other half, a beta will exist. A portfolio's beta thus tells you the inherent risk of the asset.

To calculate β, we assume the ideal market possesses βmarket=1.0 and project our portfolio risk on its basis: β < 1.0 means our portfolio is less risky than the market, β > 1.0 indicates greater risk and less stability than the market. Industry beta values are dependent on the elasticity of the good as well as several macroeconomic factors. Obviously, the more inelastic the good, the higher the stability and the less risk.

As the portfolio grows from 1 → ∞, diversifiable risk approaches 0 and σ → β. It's easy to see that the financial technique of diversification and the creation of security portfolios is simply an application of the mathematical laws of statistics to practical finance.

sources:
Gallagher and Andrew, Financial Management: Principles and Practice (2nd Ed.), Prentice-Hall 2000
Milton and Arnold, Introduction to Probability and Statistics, 3rd Ed., McGraw Hill 1995