Risk refers to the difference between the expected return and actual return on an investment
. Risk plays a very important role in deciding the amount of return that one gets from his/her investments. Higher the risk, higher the return and vice versa. There are various methods of measuring risks, and one should be aware of them in order to invest in an informed manner.
Measuring risk involves the use of mathematical tools and techniques. So following are the various methods of measuring risks:
The easiest way to know the risk of an investment is to calculate the standard deviation. Standard deviation measures how much return on an investment is deviating from the expected normal or average returns. Higher the standard deviation, the more variable the return and higher will be the risk.
E[R] = the expected return on the stock,
p = the probability
R = the return on the stock
The following example shows the probability and return of a stock, and on the basis of these figures, we can calculate the standard deviation.
The Expected Return can be calculated as follows:
E[R] = (0.2)(5%) + (0.35)(-14%) + (0.1)(18%) + (0.35)(29%) = 8.05%
The Expected Return = 8.05%
The Variance can be calculated as follows:
Variance = (0.2)(0.05 – 0.08)2 + (0.35)(-0.14 – 0.08)2 + (0.1)(0.18 – 0.08)2 + (0.35)(0.29 – 0.08)2 = 0.0336
The Variance = 0.0336
The Standard Deviation equals the square root of the variance:
The Standard Deviation = 0.1832 = 18.32%
Beta measures an individual investment’s volatility in relation to the market in general. It represents the systematic risk.
β >1 : aggressive security (β =1.2, 10% change in market will have 12% change in stock)
β <1 : defensive security (β =0.8, 10% change in market will have only 8% changes in stock)
β =0 : Risk-free security (No change)
β =1 : Security is neutral (Portfolio replicating market index) (Same Change)
Beta is calculated with the help of the following formula:
CoVAM = Co-variance between the security and the market
VarM = Variance of the market
The co-variance between the security and the market can be calculated with the help of the following formula:
- When probabilities of the return are not given
Co-variance = sum of product of deviation of individual security and market/(n-1)
- When probabilities of the return are given
Co-variance = sum of (product deviation of individual security and market and the probabilities)
Alpha measures an investment’s beta against its actual performance. A positive or high alpha implies that the investment has outperformed the market, whereas a negative or low alpha implies the opposite.
- Sharpe ratio
The Sharpe ratio, developed by William F. Sharpe, is the ratio of a portfolio’s total return minus the risk-free rate divided by the standard deviation of the portfolio, which is a measure of its risk.
Risk Premium = Total Portfolio Return – Risk-free Rate
Sharpe Ratio = Risk Premium / Standard Deviation of Portfolio Return
= Rp – Rf/Standard deviation of portfolio
The Sharpe ratio measures the performance of the portfolio compared to the risk taken. The higher the Sharpe ratio, the better the performance and the greater the profits for taking on additional risk.
Treynor’s ratio, popularized by Jack L. Treynor, compares the portfolio risk premium to the systematic risk of the portfolio as measured by its beta.
Treynor Ratio Formula
Treynor Ratio = Total Portfolio Return – Risk-Free Rate/ Portfolio Beta
= Rp – Rf/ β
Note that since the beta of the general market is defined to be 1, the Treynor Ratio of the market equals its return minus the risk-free rate. The Treynor ratio measures the return per unit risk: it is higher with either higher portfolio returns or lower portfolio betas.
- Jenson’s Index
Alpha is a coefficient that is proportional to the excess return of a portfolio over its required return, or its expected return, for its expected risk as measured by its beta. Jensen’s index, developed by Michael C. Jensen, uses the capital asset pricing model (CAPM) to determine the amount of the return that is firm-specific over that which is due to market volatility as measured by the firm’s beta in relation to the market beta. It attempts to measure the constant return that the portfolio manager earned above, or below the return of an unmanaged portfolio with the same market risk or the fund manager’s expertise.
Jensen’s Alpha = Total Portfolio Return – (Risk-Free Rate – [Portfolio Beta × (Market Return – Risk-Free Rate)])
Jensen’s alpha can be positive, negative, or zero. Note that, by definition, Jensen’s alpha of the market is zero. If the alpha is negative, then the portfolio is underperforming the market; thus, higher alphas are more desirable.