What Is Standard Deviation - Essay Example

Aftab Ahmed Aftab 01 Karachi, Pakistan 19-July-2008 Every rational investor wants to invest money in an investment which will give him more money. But the question arises how can one make choice between a set of investments There are number of theories regarding how an investor view risk and prefers one investment over the other ones. Standard deviation is one of several procedures to predict profit / loss on an investment. What is standard deviation Standard deviation can defined as "a measure of the dispersion of outcomes around the mean (or expected value), used to measure total risk. It is the square root of the variance."(1) Standard deviation, represented by the lowercase form of the Greek letter sigma, is a statistic that tells you how tightly the data points are clustered around the mean for a given process, which in turn tells you how much variation exists. When data points are tightly clustered around the mean and the bell-shaped curve is steep, the standard deviation -- and hence the variation -- is small. When the data points are spread apart and the bell-shaped curve is flat, the standard deviation -- and the variation -- is great. Formula for Standard Deviation. Aftab 02 For instance, if the random variable X takes on N values (which are real numbers) with equal probability, then its standard deviation can be calculated as follows: Where is the mean of X. Standard Deviation and finance Standard deviation with regard to finance can be defined as "Statistical measure that shows the likelihood of an investment to yield above- or below-average returns over a period of time. For example, if hypothetical XYZ Fund has an average annual total return of 11% and a standard deviation of 6.00, which means XYZ Fund's performance is likely to vary from a low of 5% to a high of 17%. Calculated by the fund, standard deviation is only relative to the asset class being measured." (2) Limitation to the Standard Deviation. The standard deviation of investment returns is widely accepted as the best, and perhaps only commonly used indicator of portfolio risk in the investment management business. However, its usefulness is actually quite limited. In fact, relying on it can often produce misleading and inaccurate Aftab 03 conclusions. Although standard deviation does provide some insight, and in many circumstances is in fact meaningful but there are a number of flaws associated with relying on the standard deviation of returns as a risk measure. The bigger flaw with standard deviation is that it isn't intuitive. Sure, a standard deviation of seven is obviously higher than a standard deviation of five, But are those high or low figures Because a fund's standard deviation is not a relative measure-which means it's not compared with other funds or with a benchmark-it is not very useful to you without some context. Another limitation to standard deviation lies with the underlying data. Most investors will recall normal distribution from their introduction to statistics course. This bell curve underlies all of the assumptions about standard deviation. If the underlying data is not normally distributed, then the standard deviation is likely to give misleading results. It's worth noting that a number of studies show that investment returns are not normally distributed. There are some drawbacks to using standard deviation as a measure of risk, however. It interprets any difference from the average, above or below, as bad. This runs contrary to the way most investors feel about returns. Few investors fret about their portfolios doubling; most only worry about the downside--their returns being below average. There is another theory which is called Utility Theory. "This theory gives us a way to measure investor's preferences for wealth and the amount of risk they Aftab 04 are willing to undertake in the hope of attaining greater wealth. This makes it possible to develop a theory of portfolio optimization. Thus utility theory lies at the heart of modern portfolio theory." (3) Definition of utility "In economics, utility means the real or fancied ability of a good or service to satisfy a human want."(4) Utility Theory and finance Financially utility function is defined as "A mathematical expression that assigns a value to all possible choices. In portfolio theory, the utility function expresses the preferences of economic entities with respect to perceived risk and expected return." (5) In other words a utility function measures an investor's relative preference for different level of total wealth. Its non-satiation property states that utility increases with wealth i.e. that more wealth is preferred is to less wealth, and the investor is never satiated. He never has so much wealth that getting more would not be al least a little bit desirable. Its risk aversion property states that the utility function is concave or, in other words, that the marginal utility of wealth decreases as wealth increases. Aftab 05 Generally it can be stated as that the increase in utility caused by the acquisition of an additional euro decreases as wealth increases. Different investors can and will have different utility functions, but we assume that any such utility function satisfies the two critical properties of non-satiation and risk aversion. The comparison of utility theory with standard deviation with regard to risk will be explained with a given data later in this article. We will explain the above in the light of the following data. Read the following data carefully. The following chart shows the yearly return in percentage of two assets A and B. Assets 1 2 3 4 5 6 7 8 9 10 11 12 A 10 12 5 19 4 18 11 1 20 4 13 7 B 12 11 12 10 12 11 3 -12 13 17 12 13 Assets 13 14 15 16 17 18 19 20 21 22 23 24 A 4 10 11 9 15 6 5 20 12 8 3 12 B 13 12 8 10 12 7 13 5 13 17 12 12 Aftab 06 Using formulae give above, we can calculate standard deviation of both assets A and B. The Standard deviation for asset "A" and "B" is 5.44 and 5.57 respectively. It means asset "A" is less volatile or less risky than asset "B". It should be further noted that the expected profit or return from asset "A" is less than expected profit of asset "B" Now the question arises what will be the choice of a rational investor "X". Whether he will go to asset "A" or "B". Here standard deviation fails to answer. Because it only shows data variance from the mean (average data). But how this variance affect investor's choice is another matter. But Utility function explains the investor's behavior. The principle of expected utility maximization states that a rational decision maker or investor, when faced with a choice among a set of competing feasible investment alternatives, acts to select an investment which maximizes his expected utility of wealth. What is meant by rational investor In developing the logic of choice, economists assume that people are rational, which means that people have well-defined goals and that they purposefully and logically act to attain those goals as best they can, given their circumstances. You might note that economics does not judge the goals themselves as rational or irrational. Economists almost always take goals as Aftab 07 given. People are irrational if, given their goals, they act in ways that do not lead to the accomplishment of their goals.(6) Calculation of Expected utility. If there is a p% chance of winning $X, the expected value is equal to pX. If there is a p% chance of X and a q% chance of Y, then EV=pX+qY.(7) If we examine the data described above, we can infer the following results. Asset "A" has a mean of 9.96 and standard deviation of 5.44. It means that that investor will expect maximum return of 15.4 and a minimum return of 4.52. The possibility of return between these two values are equal so the expected utility will be same to mean. In case of asset "A" expected utility is 9.96 and similarly we can calculate Expected utility of asset "B" which will be 10.33. Hence If the investor "X" follows expected utility function he will maximize expected utility and he will choose asset "B". What is downside risk It can defined as "A risk metric that distinguishes between "good" and "bad" returns by assigning risk only to those returns below a return specified by an investor".(8) Downside risk, as the name implies, measures risk below a certain point. Aftab 08 For example, if an investor is only worried about losing money, that point would be zero and the possibility of negative returns would be viewed as risky. If an investor needs to earn a 7% annual return in order to meet their goals, any return under 7% would be considered risky. This investment return floor, which serves as the dividing line between good and bad outcomes, is called the minimum acceptable re-turn (MAR). The amount of risk contained in a set of returns changes considerably when the MAR changes. For example, if Investor A needs an investment which returns 10% annually, any amount less than this will result in the underfunding of A's pension plan. Investor B wants a good return but doesn't want to incur losses. By raising the MAR from zero to 10%, a larger amount of the return distribution violates the MAR. This additional area--the amount between zero and 10%--is considered risk for Investor A but not for Investor B. Difference between standard deviation and downside risk measurement. Although the standard deviation identifies risk associated with the volatility of returns, it does not separate upside from downside changes in returns. But the distinction of upside and downside is important in practice. Risk averse investors will naturally be averse to downside volatility, but would welcome any upside changes. Unlike standard deviation, downside risk accommodates different views of risk. Institutional investors often view investment risk as the possibility of Aftab 09 underperforming the benchmark, whereas retail investors tend to regard risk in absolute terms as the possibility of loss. What will be the choice under downside measure If we look back to our chart, we consider that our investor "X" is a risk averse investor and is not interested in losses. Surely he will invest in asset "A" because asset "B" has resulted in loss of 12% in 8th year. Calculation of Downside risk. There are a number of ways to calculate downside risk and they are likely to yield different results. It is essential that individuals interpreting downside risk statistics understand the calculation methodology because downside risk statistics calculated using different assumptions are not comparable. One of the method for calculation of downside risk is called the sortino ratio. The Sortino ratio measures the risk-adjusted return of an investment asset, portfolio or strategy. It is a modification of the Sharpe ratio but penalizes only those returns falling below a user-specified target, or required rate of return, while the Sharpe ratio penalizes both upside and downside volatility equally. It is thus a measure of risk-adjusted returns that some people find to be more relevant than the Sharpe. The ratio is calculated as: , Aftab 10 where R is the asset or portfolio realized return; T is the target or required rate of return for the investment strategy under consideration, (T was originally known as the minimum acceptable return, or MAR); DR is the downside risk. The downside risk is the target semi-deviation = square root of the target semi-variance (TSV). TSV is the return distribution's lower-partial moment of degree 2 (LPM2). where T is often taken to be the risk free interest rate and f() is the pdf of the returns. DR can also be thought of, and calculated from a sequence of historical returns x as, the root mean square underperformance U, where U = x T if x T < 0, otherwise U = 0. Thus, the ratio is the actual rate of return in excess of the investor's target rate of return, per unit of downside risk. Aftab 11 WORKS CITED (1) "Glossary of Investment Terms" 15-july-2005 (2) Wachovia Personal Finance Glossary. 15-july-2008 (3) John Norstad's Finance Page "homepage.mac.com/j.norstad/index.html" 16-july-2008 (4) Principia Cybernetica Web "Utility" 16-july-2008 (5) The Free Dictionary By Farlex "Financial definition of Utility Function" 17-july-08 (6) "Utility Function" 16-july-08 Aftab 12 (7) Expected Value, Expected utility " www.calculemus.org" 16-july-08 (8) UTIMCO 2006 AR "Frequently Used Terms": 18-July-08 Read More