Measuring expected Return and Risk

Probability Distribution:

When you invest in a stock you know that the return from it can take various possible values. For example, it may be – 5 percent, or 15 percent or 35 percent. Further, the likelihood of these returns can vary. Hence, you should think in terms of a probability distribution.

The probability of an event represents the likelihood of its occurrence. Suppose you say that there is a 4 to 1 chance that the market price of a stock A will rise during the next fortnight. This implies that there is an 80 percent chance that the price of stock A will increases and a 20 percent chance that it will not increase during the next fortnight. Your judgment can be represented in the form of a probability distribution as follows:

Out come Probability

Stock price will rise 0.80
Stock price will not rise 0.20

Another example may be given to illustrate the notion of probability distribution. Consider two equity stocks, Bharat Foods stock and Oriental Shipping stock. Bharat Foods stock may provide a return of 15 percent, 20 percent, or 25 percent with certain probabilities associated with them, based on the state of the economy. The second stock, Oriental Shipping stock, being more volatile, may earn a return of 20 percent, 10 percent, or 40 percent with the same probabilities, based on the state of the economy. The probability distributions of the returns on these two stocks are shown.

When you define the probability distribution of rate of return (or for that matter any other variable) remember that:

1) The possible outcomes must be mutually exclusive and collectively exhaustive.
2) The probability assigned to an outcome may vary between 0 and 1 (an impossible event is assigned a probability 0, a certain event a probability of 1, and an uncertain event a probability somewhere between 0 and 1).
3) The sum of the probabilities assigned to various possible outcomes is 1.

Probability Distribution of the Rate of Return on Bharat Foods Stocks and Oriental Shipping Stocks
Rate of Return (%)
State of the Probability of Bharat Foods Oriental Shipping
Economy Occurrence

Boom 0.30 16 40

Normal 0.50 11 10

Recession 0.20 6 — 20

Based on the probability distribution of the rate of return, you can compute two key parameters, the expected rate of return and the standard deviation of return.

Expected Rate of Return:

The expected rate of return is the weighted average of all possible returns multiplied by their respective probabilities. In symbols,
E ( R ) = Σ Ri pi ————–eq (1)
i = 1

where E ( R ) = expected return from the stock

Ri = return from stock under state i

Pi = probability that the state i occurs

n= number of possible states of the world

From eq (1), it is clear that E (R) is the weighted average of possible outcomes – each outcome is weighted by the probability associated with it. The expected a rate of return on Bharat Foods stock is:

E ( Rb) = ( 0.30) ( 16%) + (0.50) (11%) + (0.20 (6%) = 11.5%

Similarly, the expected arte of return on Oriental Shipping stock is:

E ( R o) = (0.30) ( 40%) + (0.50) (10%) + (0.20) ( — 20%) = 13.0%

Standard Deviation of Return:

Risk refers to the dispersion of a variable. It is commonly measured by the variance or the standard deviation. The variance of a probability distribution is the sum of the squares of the deviations of actual returns from the expected return, weighted by the associated probabilities. In symbols,

σ2 = Σ pi( Ri – E ( R )) 2——————-eq ( 2)


σ 2 = variance

Ri = return for the ith possible outcome

Pi = probability associated with the ith possible outcome

E (R) = expected return.

Since variance is expressed as squared returns, it is somewhat to grasp. So its square root, the standard deviation, is employed as an equivalent measure.

σ = (σ 2) ½

where σ = standard deviation

  • Sapna Dhanwani

    This article is copied from pradanna chandra’s book Investment analysis and portfolio management