Financial Mathematics
Understanding Portfolio Math part I
Mean, Variance, Standard Deviation and Correlation
Posted on June 9, 2010 by Manish
While making an investment decision, it is important to assess the risk/return profile of any investment. The relation between risk and return raises three basic questions：
◎ How do I estimate the percentage return that I will receive on an investment?
◎ How much risk does an asset add to a portfolio?
◎ What can I do to eliminate some of that risk?
To answer these questions, we need to understand the key statistical concepts that are applied to financial assets. They are: expected returns, variance and standard deviation, and correlation. Let us look at each of them in detail:
Expected Return
The expected return is the market’s, or an investor’s, best guess as to the return on an asset. Any technique can be used to arrive at the guess. This section will review two common techniques. One uses a simple average of historical returns. Another technique uses the returns from possible outcomes and the probabilities of those outcomes to arrive at an expected return.
Variance and Standard Deviation
Risk is the possibility that actual returns might differ, or vary, from expected returns. In fact, actual returns will likely differ from expected returns. It is important for decision-makers to estimate the magnitude and likelihood of the difference between actual and estimated returns. After all, there is a big difference if your predictions result in an error of only $100 versus an error of $1 million.
By using the concepts of variance and standard deviation, investors can judge not only how wrong their estimates might be, but also estimate the likelihood, or probability, of favorable or unfavorable outcomes. With the tools of expected return and standard deviation, financial decision-makers are better able to evaluate alternative investments based on risk-return tradeoffs, and their own risk preferences.
Diversification
The Diversification topic answers the third question regarding what one can do to minimize risk of a group, or portfolio, of investments. By selecting investments that perform differently under the same market conditions, one can create a portfolio that has less risk for the same level of expected return. The concepts of covariance and correlation are used to measure how the returns on assets relate to each other and the market in general and how they can be used to reduce the overall risk to the investor.
Challenges
The following challenges present you the problem situations that require applications of the risk and return concepts discussed above.
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