Coefficient of Variation

The standard deviation can sometimes be misleading (обманчивый) in comparing the risk, or uncertainty, surrounding alternatives if they differ in size. Consider two investment opportunities A and B, whose normal probability distribution of one year return have the following characteristics

	Investment A	Investment B
Expected return Re	0.08	0.24
Standard deviation σ	0.06	0.08
Coefficient of Variation CV	0.75	0.33

Can we conclude that because the standard deviation of В is larger than that of A, it is the riskier investment? With standard deviation as our risk measure, we would have to. However, relative to the size of expected return, investment A has greater variation. This is similar to recognizing that a $10,000 standard deviation of annual income to a multimillionaire is really less significant than an $8,000 standard deviation in annual income would be to you. To adjust for the size, or scale, problem, the standard deviation can be divided by the expected return to compute the coefficient of variation (CV)

CV = σ / R_e

Thus, the coefficient of variation is a measure of relative dispersion (risk)—a measure of risk "per unit of expected return." The larger the CV, the larger the relative risk of the investment. Using the CV as our risk measure, investment A with a return distribution CV of .75 is viewed as being more risky than investment B, whose CV equals only .33.

ATTITUDES TOWARD RISK

We can, in fact, use the relationship of an individual's certainty equivalent to the expected monetary value of a risky investment (or opportunity) to define their attitude toward risk. In general, if the

• Certainty equivalent < expected value, risk aversion is present.

• Certainty equivalent = expected value, risk indifference is present.

• Certainty equivalent > expected value, risk preference is present.

For risk-averse individuals, the difference between the certainty equivalent (эквивалент надежности (гарантированная сумма, которая была бы принята вместо большей, но носящей вероятностный характер, ожидаемой суммы; рассчитывается путем корректировки дохода, носящего вероятностный характер, в соответствии с коэффициентом вероятности) and the expected value of an investment constitutes a risk premium; this is additional expected return that the risky investment must offer to the investor for this individual to accept the risky investment.

RISK AND RETURN IN A PORTFOLIO CONTEXT

So far, we have focused on the risk and return of single investments held in isolation. Investors rarely place their entire wealth into a single asset or investment. Rather, they construct a portfolio or group of investments. Therefore, we need to extend our analysis of risk a return to include portfolios.

Portfolio Return

The expected return of a portfolio is simply a weighted average of the expected returns of the securities comprising that portfolio The weights are equal to the proportion of total funds invested in each security (the weights must sum to 100%)/ The general formula for expected return of portfolio:

R_{e of p} = ΣW_jR_ej (j=[1,m]

W- is the proportion, or weight, of total funds Invested In security;

R is unexpected return for security; and

m is the total number of different securities in the portfolio.

The expected return and standard deviation of the probability distribution of possible returns for two securities are shown below

	SECURITY A	SECURITY В
Expected return, Re	14%	15%
Standard deviation σj	10.7	1.5

If equal amounts of money are invested in the two securities, the expected return of the portfolio is (0.5)14.0% + (0.5)11.5% = 12.75%.