Measurement of Risk

Measurement of Risk

To measure risk, an investor should first understand the fact that risk cannot be measured accurately because it is surrounded with complex environment factors and social, economic and political forces. The uncertainties make the measurement of risk an approximation or a fairly accurate estimation. The analyst must be very cautious while making predictions because much depends on his accuracy in predicting risks.

The quantification of risk ensures comparison as well as uniformity in measurement, analysis and interpretation. To eliminate guesses and haunches in measurement is possible by finding out the difference between actual return and estimated return that is the dispersion around the expected return.Discussion as to how probability distributions are framed was made. These distributions are calculated through ‘standard deviations’ and ‘variances’. They are used for quantifying risk. Fischer and Jordan describe risk in the following manner.

“The variability of return around the expected average is thus a quantitative description of risk”.

‘Standard Deviation and Variance:

The most useful method for calculating the variability is the standard deviation and variance. Risk arises out of variability. If we compare the stocks of Company-A and Company-B in the following example (Table 7.1) we find that the expected returns for both the companies are same but the spread is not the same. Company-A is riskier than Company-B because returns at any particular time are uncertain with respect to its stock.

The average stock for Company-A and B is 12 but appears riskier than B as future outcomes are to be considered. Another example may be cited here (Table 7.2) of probability distributions to specify expected return as well as risk.

The expected return is the weighted average of returns. When each return is multiplies by the associated probability and added together the result is termed as the weighted average return or in other words expected average returns, for example:

Stocks of Company -A and Company-B have identical expected average returns, but the spread is different. The range in Company-A is from 8 to 12 and for Company-B it ranges between 9 and 11 only. The range does not imply greater risk.

The spread or dispersion can be measured by standard deviation. The following example (Table 7.2) calculates return and risk through probability distribution and standard deviation and variance method of two companies

or Variance = Ө 2 = 0.000800

Comparison of return and risk for stocks of Company-A and Company-B with standard deviation 4.9% of Company- A and 2.8% of Company-B.

The standard deviations and probability distributions show that stock of Company-A has a higher expected return, and a higher level of risk as measured by standard deviation.

In figure 7.2 are plotted probability distributions, expected returns and standard deviations of returns. Company-B’s stocks are symmetrical about its expected return, Company A’s is not. This diagram also highlights important properties of standard deviation and variance.

The deviations are squared and added on both sides of expected return. Many investors are contented when deviations in return are higher than expected.

Standard deviation and variance is the best method for calculations in upward returns but when deviations are below the expected return, then instead of standard deviation, the investors or the security analysts should use semi- variance as a measurement of risk as it reflects only downside variations in return.

tandard deviation measures risk for both individual assets and for portfolios. It measures the total variation return about expected return. Another example of risk through standard deviation measurement is given through mean. Given below in Table 7.4 (a) are the stocks of two companies ABC and XYZ. Show the relationship between risk and return. Which of the two stocks show higher risk?

The relationship of risk and return is clearly established in this example. Company-ABC has a higher return and a higher standard deviation than Company-XYZ and the return related to risk (standard deviation) is higher than the XYZ which shows that the stock in, ABC has performed better than XYZ but is somewhat riskier.

Risk associated with individual stocks is as discussed earlier of two types, systematic or non-diversifiable and unsystematic or diversifiable. Systematic risk is often referred to as market risk and unsystematic part as financial risk.

Return of all stocks consists of an element of both types of risks. While systematic risk is correlated with the variability of overall stock market, the unsystematic risk is the remaining portion of the total variability of stock prices.

The risk of stocks in terms of systematic and unsystematic compounds is tested through the ‘market model’. According to the market model, the return on any stock is related to the return on the market index in a linear manner.

This widely accepted market model is based on ‘Empirical Testing’. This measure of quantifying risk is also referred to as Beta analysis or ‘volatility’. The application of the Beta concept or market model is done through the use of statistical measurement through a regression equation.

According to Amling, Investment Management through this model:

“returns of stocks are regressed against the return of the market index”.

The basic equation for calculating risk can then be formulated as:

Regression

Regression Equation:

Y = a + βX + E…. Equation 7.1

Y = Return from the security in a given period.

α = Alpha or the intercept (where the line crossed vertical axis.)

β = Beta or slope of the regression formula.

Σ = Epsilon or Error involved in estimating the value of the stock.

i. Beta:

The most important part of the equation is β or beta. It is used to describe the relationship between the stock’s return and market index’s returns. If the regression line is at an exact 45° angle, beta will be equal to +1.0. A 1% change in the market index (horizontal axis) shows that it is on an average accompanied by a 1% change in the stock on the vertical axis.

The percentage changes in the price of the stock are regressed against the percentage changes in the price of a market index.

S&P 500 Price Index Beta may be positive or negative. Usually, betas are found to be positive. We rarely find a negative beta which reflects a movement contrary to the market. A .5 beta indicates that the market index change of 1% was reflected by a .5% price change in stocks. Similarly, a 1.5 beta would reflect that whenever the market index rose or fell by 1%, the stocks would rise and fall by 1.5%.

Beta is referred to as systematic risk to the market and a + E the unsystematic risk. Beta is a useful piece of information both for individual stock as well as portfolios, but as a measure of risk it is better used in the analysis of portfolios.

Also, beta measures risk satisfactorily for diversified efficient portfolios but not inefficient portfolios. The concept of efficient and inefficient portfolios will be clarified later in the book. For the present, it may be said that beta is a satisfactory measure for portfolios because risk other than that reflected by beta is diversified.

Beta has certain limitations within which it must be considered. While calculating past betas, the length of time will affect beta size. When estimating future betas, the markets expected return should also be estimated.

If high beta is accurately predicted and the market also goes up predicted, the relationship will work. On the contrary, high beta estimation and low market or downward, market will show that the beta will drop much faster than the market.

Finally, its shortcoming as a measure for individual stock as already explained should be realized while calculating stock. For the total portfolio beta is effective. Figure 7.3 shows the beta along with alpha, Rho and Epsilon and Figure 7.4 establish the alpha and beta relationship between the stock and the market.

The stock has a beta or systematic risk to the market of 99. This shows that the stock does have as much risk as the market but it has a slightly higher unsystematic risk. Based on it, the stock in the past has provided a return and risk comparable to the market.

ii. Alpha:

The distance between the inter-section and the horizontal axis is called (a) alpha. The size of the alpha exhibits the stock’s unsystematic return and its average return independent of the market’s return. If alpha gives a positive value it is a healthy sign but alphas expected value is zero. The belief of many of the investors is that they can find stocks with positive alpha’s and have a profitable return.

It must be recalled, however, that in an efficient market positive alphas cannot be predicted in advance. The portfolio theory also maintains that the alphas of stocks will average out to 0 in a properly diversified portfolio. The third factor besides alpha and beta is Rho.

iii. Rho (p):

Rho (p) is the correlation coefficient which describes the dispersion of the observations around the regression line. The correlation coefficient expresses correlation between two stocks, for example i and j.

The correlation coefficient would be + 1.0 if an upward movement in one security is accompanied by an upward movement of another security. Conversely, downward movement of one security is followed in the same direction, i.e., downward by another security.

If the movement of two stocks is not in the same direction, the correlation coefficient will be negative and would show -1.0. If there was no relationship between the movements of the two stocks, the correlation coefficient would be 0.

The correlation coefficient can be calculated in the following manner:

Correlation Coefficient:

This formula is normally calculated through the computer. The relationship or degree of correlation among securities indicate that if there is perfect correlation diversification will not reduce portfolio risk below the lower of the two individual security risks.

If the securities are negatively correlated portfolio risk can be greatly reduced. If the relationship is to be drawn between two security returns, portfolio risk can be eliminated. When actual common stocks are analysed, it will be found that usually these stocks are highly correlated but not perfectly correlated.

Correlation coefficients also help in determining the extent to which a portfolio has eliminated unsystematic risk. For example, if a correlation with the market index is+.95 on squaring the correlation, the result is .9025 which means that 90% of the portfolio, risk is now systematic and 10% of unsystematic risk remains. Figure 7.3 shows correlation relationships in terms of scatter diagrams and regression lines

Co-Variance:

While standard deviation is an excellent measure for calculation of risk of individual stocks, it has its limitations as a measure of a total portfolio. With the correlation the covariance approach should also be considered when there are 2 or 3 stocks on the portfolio. Covariance can be used to achieve the highest portfolio expected return for a predetermined portfolio variance level or the lowest portfolio return.

An individual security’s expected return and variance express return and risk for portfolios of stocks, the expected return is the weighted average of the expected return on the individual securities. This is weighted according to each securities’ rupee proportion in the portfolio.

Since stocks tend to cover or move together, portfolio risk cannot be expressed for an individual stock. The formula for calculation of covariance of two stocks i and j, and the covariance of stocks with beta coefficients is shown in Figure 7.6.

Covariance Equation:

Cov.ij = P ij θ i θj (Equatin7.3).

P ij = joint probability that ij will move simultaneously.

θi = standard deviation of i.

θj = standard deviation of j.

Investor’s Attitude towards Return and Risk:

Before concluding the discussion on risk and its measurement, let us turn back to the investor’s attitude towards risk and return. Understanding and measuring return and risk is fundamental to the investment process and increases an awareness of the investment problem.

Most investors are ‘risk averse’. They must be aware of the risks in different investments whether they are confronted with high, moderate or low risk and the kinds of risks investment are exposed to before making their investments.

Table 7.6 gives a ready reference to the kinds of risks an investor is exposed to. To have a higher return, the investor should be able to accept the fact that he has to be faced with greater risks. In commercial bank and life insurance savings, most of the risks are low but purchasing power risk is high.

Figure 7.7 graphically establishes the attitudes of two investors about the combination of return and risks they would be able to accept. The investor has to decide for himself whether he would like to choose a group of securities which will give him 15% return with 10% risk or a return of 25% with 20% risk

The curvesi12 3 and j1,2,3 show the attitudes of two different investors about the return and risk combination they are willing to accept. The curves show that investors i and j are quite happy with combination of risk and return at point A(i) and B(j).

The first investor can with all his efforts only obtain that combination of securities at point A, that is the best the investor can do within his limited ability, the market will not offer any more. The second investor j is willing to shoulder greater risk to earn a higher return. This investor prefers to take a higher risk for a higher return.

Risk of Single Asset:

The concept of risk is more difficult to quantify. Statistically we can express risk in terms of standard deviation of return. For example, in case of gilt edged security or government bonds, the risk is nil since the return does not vary – it is fixed.

But strictly speaking if we consider inflation and calculate real rate of return (inflation adjusted) we find that even government bonds have some amount of risk since the rate of inflation may vary.

Return from unsecured fixed deposits appear to have zero variability and hence zero risk.

But there is a risk of default of interest as well as the principal.

In such case the rate of return can be negative.

Hence, this investment has high risk though apparently it carries zero risk. For other investments like shares, business etc., where the rate of return is not fixed, there may be a schedule of return with associated probability for each rate of return.

The mean of the probable returns gives the expected rate of return and the standard deviation or variance which is square of standard deviation measures risk.

Higher the range of the probable return, higher the standard deviation and hence higher the risk.

A risk averse investor will look for return where the range is low. Hence, low standard deviation means low risk.

The problem is to minimize the standard deviation without sacrificing expected rate of return. This is possible by diversification. Risk is measured in terms of variability of returns.

If Investment ‘A’ and Investment ‘B’ whose mean rate of return is same ,the returns of Investment ‘A’ show more variability than Investment ‘B’. In view of the variability of returns, Investment ‘A’ is more risky, even though both the investments are having the same mean returns. The following illustrations explain the quantification of risk in terms of standard deviation.

Risk of Portfolio of Two Assets:

The risk of a security is measured in terms of variance or standard deviation of its returns. The portfolio risk is not simply a measure of its weighted average risk. The securities consisting in a portfolio are associated with each other. The portfolio risk also considers the covariance between the returns of the investment, covariance of two securities is a measure of their co-movement, it expresses the degree to which the securities vary together.

The standard deviation of two share portfolio is calculated by applying the formula given below:

Where:

σp = Standard deviation of portfolio consisting securities A and B

WA, WB = Proportion of funds invested in Security A and Security B

σA , σB = Standard deviation of returns of Security A and Security B

ρAB = Correlation coefficient between returns of Security A and Security B

The correlation coefficient (ρAB) can be calculated as follows:

ρAB = CovAB/ σA σB

The covariance of Security A and Security B (CovAB) can be presented as follows:

CovAB = σA σB ρAB

The diversification of unsystematic risk, using two security portfolio, depends upon the correlation that exists between the returns of those two securities. The quantification of correlation is done through calculation of correlation coefficient of two securities (ρAB).

The value of correlation ranges between -1 to 1, it can be interpreted as follows:

If ρAB = 1 No unsystematic risk can be diversified

If ρAB = -1 All unsystematic risk can be diversified

If ρAB = 0 No correlation exists between the returns of Security A and Security B.