Two Methods of Measuring Risk
Two Methods of Measuring Risk ( Measures of Risk ) – Risk arises from the expected volatility in asset’s return over time caused by one or more of the following sources of returns on investment:
▪ fluctuations in expected income
▪ fluctuations in the expected future price of the asset
▪ fluctuations in the amount an investor can reinvest and fluctuations in returns earned from reinvestment
In this section we examine the measures of risk arising from an investment.
There are two methods of measuring risk viz.,
i) absolute measure and
ii) relative measure of risk.
The absolute measures of risk for an investment are:
▪ Variance of rates of return (σ^{2}_{i})
▪ Standard deviation of rates of return (σ_{i})
▪ Covariance (σ_{im})
These measures of risk can be influenced by the magnitude of original numbers. Hence, to compare series with greatly different values, we need a relative measure of dispersion.
The relative/standardize measures of risk for investments are:
▪ Coefficient of variation (CV) of rates of return which is calculated as:
CV = σ_{i}/E(r_{i})
▪ Correlation coefficient (ρ_{ij}), being a statistical measure of the extent to which two variables are associated which is calculate as:
ρ_{ij} = σ_{ij}/σ_{i} ×σ_{j}
▪ Covariance of returns with the market portfolio called beta (b_{i}) which is calculated as:
b_{i} = σ_{i m}/ σ^{2}_{m}
Statistical measures allow an investor to compare the return and risk measures for alternative investments directly. Variance and standard deviation of the estimated distribution of expected returns are, however, two possible measures of uncertainty. Calculation of the expected return of the probability distribution is essential to calculate the variance and standard deviation.
Standard deviation
Standard deviation is a measure of the dispersion of forecast returns when such returns approximate a normal probability distribution. It is a statistical concept and is widely used to measure risk from holding a single asset. The standard deviation is derived so that a high standard deviation represents a large dispersion of return and is a high risk; a low deviation is a small dispersion and represents a low risk. Standard deviation of the rates of return is simply the square root of the variance of the rates of return which tells us about the potential for deviation of the return from its expected values. Variance is directly related to the standard deviation and is of considerable importance in finance. However, the variance is simply the standard deviation squared (or the standard deviation is the root of the variance). The following is the formula for calculating the variance of historical returns (using ex post data):
_{N}
s^{2}_{r} = 1/(N -1) ∑ (r_{it }– `r_{i})^{2}
^{ t = 1}
Dividing by N-1 gives us an unbiased estimate for the variance when the sample is relatively a small size. In the case of the example of historical data the computation of the variance would be as follows:
(-.09 -.04)^{2 }= (-.13)^{2} = .0169
(.12 -.04)^{2 }= (.08)^{2 }= .0064
(-.03 -.04)^{2 }= (-.07)^{2} = .0049
(.04 -.04)^{2 }= (00)^{2 }= 0000
(.14 -.04)^{2 }= (.10)^{2 }= .0100
(-.02 -.04)^{2 }= (-.06)^{2 }= .0036
(.10 -.04)^{2 }= (.06)^{2 }= .0036
(.15 -.04)^{2 }= (.11)^{2 }= .0121
(.04 -.04)^{2 }= (00)^{2 }= 0000
(-.05 -.04)^{2 }= (-.09)^{2 }= .0081
=====================
Total = .0656
s^{2}_{r} = 1/9 (.0656)
= .0073
Standard deviation is the positive square root of the variance.
s_{r} = Ös^{2}_{r} = Ö .0073 = .085 = 8.5 %
The variance of expected rates of return by using ax ante data could be found as follows:
_{N}
s^{2}(r) = ∑ h_{t}[r_{i}_{t }– E(r_{i})]^{2 }
^{ t}^{ = 1}
In case of our example of probability distribution of possible return, the variance of the population by using ex ante data is computed as follows:
.10(-.09 -.04)^{2 }= .10(-.13)^{2} = .00169
.10(.12 -.04)^{2 }= .10(.08)^{2 }= .00064
.10(-.03 -.04)^{2 }= .10(-.07)^{2} = .00049
.10(.04 -.04)^{2 }= .10(00)^{2 }= 00000
.10(.14 -.04)^{2 }= .10(.10)^{2 }= .00100
.10(-.02 -.04)^{2 }= .10(-.06)^{2 }= .00036
.10(.10 -.04)^{2 }= .10(.06)^{2 }= .00036
.10(.15 -.04)^{2 }= .10(.11)^{2 }= .00121
.10(.04 -.04)^{2 }= .10(00)^{2 }= 00000
.10(-.05 -.04)^{2 }= .10(-.09)^{2 }= .00081
=====================
Total = .00656
s^{2}_{r} = .00656
Also, s = Ö.00656 = .081 = 8.1%
Thus, the ex ante standard deviation is considered as a weighted average of the potential deviations from the expected returns and a reasonable measure of risk.
Covariance
The expected rates of return and the variance provide the information about the natures of the probability distribution of a single stock or a portfolio of stocks indicating nothing about the returns on securities interrelated. Certainly a stock may generate a rate of return above its expected value. If it is known in advance, will it impact on the expected rate of return on other stock? If one stock generates a rate of return above its expected value, will other stock produce the same? A statistical measure giving answer to these questions is the variance between two stocks. Covariance refers to the measure of the degree of association between the returns of a pair of securities. It is defined as the extent to which two random variables covary over time.
However, the properties of covariance are given below:
▪ A positive covariance:
The returns on two securities tend to move in the same direction at the same time indicating that if one increases (decreases), the other does the same.
▪ A negative covariance:
The returns on two securities tend to move inversely at the same time indicating that if one increases (decreases), the other decreases (increases).
▪ Zero covariance:
The returns on two securities are independent having no tendency to move in the same or opposite directions together.
To understand the concept of covariance,
let us assume that we have two stocks called stock A and stock B. In a period of six months the stocks produce the following rates of return:
Month | Jan | Feb | Mar | Apl | May | Jun | Mean |
Stock-A | .12 | .14 | -.10 | .08 | -.04 | .04 | .04 |
Stock-B | .09 | .12 | .06 | .10 | -.09 | .08 | .06 |
We like to estimate the covariance from the sample of the six monthly returns. As the data given above are historical returns,
The sample covariance may be computed by using the following formula:
_{ N}
s_{A},_{B} = 1/ (N-1) ∑ [(r_{A,t} – `r_{A})( r_{B,t} –`r_{B})]
^{ t}^{ = 1}
Using the data in our example, we can estimate the covariance of the returns from stock A and stock B as follows:
(.12 – .04)(.09 -.06) = (.08)(.03) = .0024
(.14 – .04)(.12 -.06) = (.10)(.06) = .0060
(-.10 – .04)(.06 -.06) = (-.14)(00) = 0000
(.08 – .04)(.10 -.06) = (.04)(.04) = .0016
(-.04 -.04)( -.09 -.06) = (-.08)(-.15) = .0120
(.04 – .04)(.08 -.06) = (00)(.02) = 0000
=============================
Total = .0220
σ _{AB} = 1/5 (.022)
=.0044
The above equation is used to calculate the covariance from a sample of paired returns. Now the question arises when the actual probabilities of getting various pairs of returns at the same time. The probabilities of getting various pairs of returns on two investments at the same time could be presented in the joint probability distribution. If we have the actual probability distribution rather than a sample estimate, the population covariance of underlying joint distribution may be estimated by applying the following formula:
_{ N}
σ _{AB} = ∑ h_{t} [r_{At} – E(r_{A})][ r_{Bt} – E(r_{B})]
^{ t = 1}
Suppose we have the following observation about the expected rates of return from two stocks:
Probability | .20 | .25 | .30 | .15 | .10 | E (r) |
Stock-A | .02 | .07 | .10 | .11 | .21 | .09 |
Stock-B | .24 | .18 | .10 | -.01 | -.12 | .10 |
Covariance describes the relationship between the returns on stock A and B. We compute the population covariance as follows:
.20(.02 -.09)(.24 -.10) = .20 (-.07)(.14) = -.00196
.25 (.07 -.09)(.18 -.10) = .25(-.02)(.08) = -.0004
.30(.10 -.09)(.10 -.10) = .30(.01)(00) = 0000
.15(.13 -.09)(-.01 -.10) = .15(.04)( -.11) = -.00066
.10(.21 -.09)( -.12 -.10) = .10(.12)(-.22) = -.00264
=====================================
Total = -.00566
σ _{AB} = -.00566
The calculation of covariance is very important since it is a critical input in determining the variance of a portfolio of stocks. However, it doesn’t accurately describe the nature of the relationship between two investments. Furthermore, we can standardize the covariance obtaining a better descriptor called the correlation coefficient.
Correlation coefficient
Correlation coefficient: Correlation coefficient is a statistical measure of the relative comovements between returns on securities. It measures the extent to which the returns on any two securities are related. As the covariance number is unbounded, we can bound it by dividing it by the product of the standard deviations for the two investments as follows:
r_{A,B} = σ _{AB} / s(r_{A})s(r_{B})
The resulting number of the above equation is called the coefficient correlation falling within the range-1 to + 1. With perfect positive correlation, the returns on securities have a perfect direct linear relationship indicating that the return on one security allows an investor to forecast perfectly what the other security will result. With perfect negative correlation, the returns on securities have a perfect inverse linear relationship to each other indicating that the return on one security provides knowledge about the return on the other security i.e. when return on one security is high, the other is low. With zero correlation, there exists no relationship between the returns on two securities indicating that the return on one security provides no value in predicting the return on the other security. However, we can rewrite the covariance as the product of the correlation coefficient and the standard deviations of the two stocks as:
σ _{AB} = r_{A,B} [s(r_{A})s(r_{B})]
Coefficient of determination (CD)
If we square the correlation coefficient we obtain a number called the coefficient of determination. Coefficient of determination (CD) tells the fraction of the variability in the returns on the one investment that can be associated with the variability in the returns on the other.
CD (R^{2}) = (r_{ij})^{2}
Table: Different Aspects of Statistical Measures of Returns on Securities.
Ex post (historical) | Ex ante (expected) |
Mean return:_{ N}
`r_{i} = 1/ n ∑ r_{it} ^{ t}^{ = 1} |
Expected return:_{ N}
E(r) = ∑ h_{i}r_{i} ^{ i}^{ = 1} |
Variance: _{ N}
s^{2} = 1/(n -1) ∑ (r_{it }– `r_{i})^{2} ^{ t}^{ = 1} |
Variance: _{N}
s^{2}(r) = ∑ h_{i }[r_{i }– E(r)]^{2} ^{ i}^{ = 1} |
Standard deviation: _{ }
_{ N} s = Ö1/(n -1) ∑ (r_{it }– `r_{i})^{2} ^{ t=}^{1 } |
Standard deviation:
_{ N} s(r) = Ö ∑ h_{i }[r_{i }– E(r)]^{2} ^{ i}^{ = 1} |
Covariance(sample): _{ }_{ N}
s _{AB} =1/(N-1) ∑[(r_{At} -`r_{A})( r_{Bt} -`r_{B})] ^{ t}^{ = 1} |
Covariance (population):
_{ N} s _{AB }=∑hi [r_{Ai} – E(r_{A})][ r_{Bi} – E(r_{B})] ^{ i }^{= 1} |
Calculating a standard deviation using probability distributions involves making subjective estimates of the probabilities and the likely returns. We cannot avoid such estimates as future returns are uncertain. The prices of the securities are based on investors’ expectations about the future. The relevant standard deviation in this situation is the ex ante standard deviation rather than ex post based on the realized returns. Although standard deviations based on realized returns could be used as proxies for ex ante standard deviations, it should be remembered that past may not always be extrapolated into the future without modifications. Though ex post standard deviations are convenient, they are subject to error.