## Wednesday, December 2, 2009

### A statistical measure of the dispersion of data points in a data series around the mean

A statistical measure of the dispersion of data points in a data series around the mean. It is calculated as follows:

The coefficient of variation represents the ratio of the standard deviation to the mean, and it is a useful statistic for comparing the degree of variation from one data series to another, even if the means are drastically different from each other.

In the investing world, the coefficient of variation allows you to determine how much volatility (risk) you are assuming in comparison to the amount of return you can expect from your investment. In simple language, the lower the ratio of standard deviation to mean return, the better your risk-return tradeoff.

Note that if the expected return in the denominator of the calculation is negative or zero, the ratio will not make sense.

Expected return

The average of a probability distribution of possible returns, calculated by using the following formula:

How do you calculate the average of a probability distribution? As denoted by the above formula, simply take the probability of each possible return outcome and multiply it by the return outcome itself. For example, if you knew a given investment had a 50% chance of earning a 10% return, a 25% chance of earning 20% and a 25% chance of earning -10%, the expected return would be equal to 7.5%:

= (0.5) (0.1) + (0.25) (0.2) + (0.25) (-0.1) = 0.075 = 7.5%
Although this is what you expect the return to be, there is no guarantee that it will be the actual return.

Standard Deviation
1. A measure of the dispersion of a set of data from its mean. The more spread apart the data is, the higher the deviation.
2. In finance, standard deviation is applied to the annual rate of return of an investment to measure the investment's volatility (risk).

A volatile stock would have a high standard deviation. In mutual funds, the standard deviation tells us how much the return on the fund is deviating from the expected normal returns.

Standard deviation can also be calculated as the square root of the variance.

Variance

A measure of the dispersion of a set of data points around their mean value. Variance is a mathematical expectation of the average squared deviations from the mean.

Variance measures the variability (volatility) from an average. Volatility is a measure of risk, so this statistic can help determine the risk an investor might take on when purchasing a specific security.

Calculation of the coefficient of variation

Let’s consider the two asset portfolio with 60% of the asset A and the 40% of the asset B. Each security’s future return is considered as a random variable (RA and RB). Our portfolio is a weighed combination of assets. The return of a portfolio is also a random variable and we can calculate portfolio expected return and the variance of the portfolio.

First, let’s calculate the expected return and the variance for asset A. Assume, that the asset A has the following estimated rate of return distribution:

Rate of Return Probability

(-5%) 20%

10% 50%

20% 30%

The expected rate of return for RA is:

rA = (-5%)*0.2 + 10%*0.5 + 20%*0.3 = 10%

The formula for variance RA is:

= (-5-10)(-5-10)*0.2 + (10-10)(10-10)*0.5 + (20-10)(20-10)*0.3 = 75

Standard deviation is simply the square root of the variance:

= 8.66%

Let us assume that the asset B has the expected value of return rB = 15%, and the standard deviation of RB is 12%.

The standard deviation reflects the degree of risk for each individual assets A and B. When we want to compare the risk of different assets with the different expected return, we need to use the relative measure - the coefficient of variation:

Hence,asset's A coefficient of variation is: 8.66/10 = 0.866
asset's B coefficient of variation is: 12/15 = 0.8

So, we can say that asset A is a riskier investment than asset B.

In the next post, we'll calculate the expected return of our portfolio, define the covariance and coefficient of correlation for two assets, and how these values affect the variance and the standard deviation of the portfolio.

How to calculate standard deviation:

Suppose that an investor has \$600 to invest and is considering investing all of it in the shares of one firm, currently trading at \$30. The investor assesses a 0.75 probability that the shares will increase in market value to \$33 over the coming period and a 0.25 probability that the share will decrease in its market value to \$26. Assume that the firm will pay \$1 dividend per share at the end of the year.

The payoffs from the proposed investment are as follows:-

If shares increase: \$33 x 20 shares + \$20dividend = \$680

If shares decrease: \$26 x 20 shares + \$20dividend = \$540

PAYOFF

(\$) RATE OF

RETURN PROBABILITY EXPECTED

RATE OF

RETURN VARIANCE

(1) (2) (3) (4) = (2) x (3) (5)

680 (680 - 600)/600 = 0.13 0.75 0.0975 (0.13 - 0.0725)^2 x 0.75 = 0.0025

540 (540 - 600)/600 = - 0.10 0.25 -0.025 (-0.1 - 0.0725)^2 x 0.25 = 0.0074

sum (x) 0.0725 Add the above two results to get σ² = 0.0099

The standard deviation is the square root of the variance. In the above example, the standard deviation is square root of 0.0099 i.e. 0.0995 or 9.95%

How to calculate the standard deviation using an ordinary calculator?

Key in 0.0099 and then press the √ key to get 0.0095 or 9.95%