 # Measurement of risk in the financial markets

## Introduction

Risk is defined as uncertainty in respect of whether the realized return will equal expected return. Risk is seen as either:

• The extent to which return varies from the average return, i.e. variability of return. This is measured by the standard deviation of expected rates of return. This measure measures total risk.

• The volatility of return relative to the market of which the security is a part. This risk is measured by the so-called beta coefficient. It measures systematic risk (the area below the horizontal line in Figure 4).

Both historical data and expected rates of return may be used in the computation of risk. The former employs existing "hard" data and the latter employs expected rates of return (ER) and their associated probabilities (P).

Thus we have two measures of risk: standard deviation and beta. It must be added that some analysts also regard variance as a measure of risk; it is, but it is a blood relative of standard deviation (it is the square of the standard deviation as we shall see).

## Standard deviation (one asset)

We start with four possible outcomes of an investment (share A) and associated probabilities of outcome (see Table 5).

We showed earlier that the expected rate of return on share A (ERA) is computed as follows:

ERA = P1R1 + P2R2 + P3R3 + -

Using the numbers in Table 5:

ERA = (0.15 x 25%) + (0.40 x 15%) + (0.25 x 0%) + (0.20 x -5%) = 3.75% + 6.0% + 0% + -1% = 8.75%.

 Outcome: Number Expected rate of return (R) (share A) Probability (P) 1 25% 0.15 2 15% 0.40 3 0% 0.25 4 -5% 0.20 1.00

Table 5: Expected rates of return and their associated probabilities (on one share)

We are now able to compute the variance of expected rate of return (a2), and the standard deviation, which if the square root of the a2, i.e. a. The variance is the weighted sum of squared deviations from the expected return. The reason the deviations are squared is that positive and negative deviations from the ER contribute in the same way to the measure of variability.

We can now calculate the a2 on share A:

a2A = [Pt x (Rt - ERA)2] + [P2 x (R2 - ERA)2} +

[P3 x (R3 - ERA)2] + [P4 x (R4 - ERA)2] = [0.15 x (25.0 - 8.75)2] + [0.40 x (15.0 - 8.75)2] +

[0.25 x (0.0 - 8.75)2] + [0.20 x (-5.0 - 8.75)2] = (0.15 x 264.06) + (0.4 x 39.06) + (0.25 x 76.56) + (0.2 x 189.06) = 39.61 + 15.62 + 19.14 + 37.81 = 112.18%.

The a, as noted above, is the square root of the a2, and it is equal to 10.59%. Again, this is a measure of the dispersion around the mean (i.e. the average return). The higher the standard deviation, the higher the risk is.

It may be useful to provide an example of computation of the variance in the case of the use of historical data (see Table 6).

a2A = Z (R - RA)2 / n - 1

= [(25.0 - 8.75)2 + (15.0 - 8.75)2 + (0.0 - 8.75)2 + (-5.0 - 8.75)2] / 3 = (264.06 + 39.06 + 76.56 + 189.06) / 3

= 568.74 / 3 = 189.58

aA = 13.76%.

 Year Annual return (share A) 1999 25% 2000 15% 2001 0% 2002 -5% Mean 8.75%

Table 6: Annual return on share A

## Standard deviation (a portfolio of shares)

The standard deviation of a portfolio may also be computed by including more than one asset. A portfolio has an average return and dispersion around the average return.

The standard deviation of a portfolio is the square root of the sum of (in the case of a two share portfolio):

• The squared standard deviation of the return of the first asset times its squared weight in the portfolio, plus

• The squared standard deviation of the return of the second asset times its squared weight in the portfolio, plus

• Two times the weight of the first asset times the weight of the second asset times the covariance of the two assets.16

This will not be discussed further here; it is the matter of portfolio theory.

## Beta

As seen above, the second risk measure is the volatility of return relative to the market of which the security is a part. This risk is measured by the so-called beta coefficient. It measures the tendency of a share's return to fluctuate relative to fluctuations in the market (in practice a market index).

If a share has a beta of 2, this means that the share has a tendency to rise / fall twice as much as the market over the chosen period of time, i.e. when the chosen index rises by z percent over a period, the share has a tendency to rise by 2 x z percent.

The beta is an important input in the required rate of return (rrr), which is the "rate" mostly used in the valuation of equity (the CGDDM).

Essentially, the rrr has to be determined in some way to take account of the risk inherent in equity, i.e. there must be a risk premium for equity. Much research has been done on determining the correct rrr, and the one most followed is the Capital Asset Pricing Model (CAPM) estimate.

According to the CAPM the rrr is equal to the risk-free rate of interest plus a multiple of the market risk premium as represented by the share's beta coefficient:

rrr = rfr + ((3 x (mr - rfr))

where

rfr = risk-free rate

( = beta

mr = market rate of return, i.e. the return observed over the period chosen

mr - rfr = the premium over the risk-free rate.