Question

Q4: What is the “Coefficient of Variation?

What is the inverse? (Show equations/diagrams)

What does it tell you in regards to units of risk and return?

Why is this important for a risk-averse investor?

How would you use these measurements? Show Equations/Definitions (2)!!

Coefficient of Variation:

Risk-Adjusted Rate of Return (Inverse):

Sharp Ratio:

Jensen Ratio:

Treynor Ratio:

Answer 1

Coefficient of variation: It is the amount of risk per unit of return an investor will assume. It's formula is Standard Deviation of the Stock / Mean Return. For example, suppose a stock has been giving a mean return of 11% for the past five years and it's standard deviation works out to 6%. Then it's coefficient of variation is (6/11) is 0.55. A coefficient of variation is meaningless when used in isolation, it is always used to compare a bouquet of investments. So if we also have a stock B whose mean return for the past five years is 16% and whose standard deviation is 6.25%, then CV for Stock B = 0.39. So we can say that Stock B is better in terms of risk-return trade off and has comparatively less risk per unit as compared to Stock A.

A risk averse investor will always check how much risk he is taking for one unit of return he is getting. As a rule of thumb, lesser the risk, better the stock!

Inverse Coefficient of Variation or Risk Adjusted Rate of Return: It is the reverse of Coefficient of Variation in the sense that it measures amount of return an investor is making per unit of risk assumed. So, again taking the above example of stock A and stock B, in stock A, the investor is getting 1.83% return per unit of risk assumed (calculation being: 11/6) and stock B it is (16/6.25=2.56%). Now that were just hypothetical examples for Risk Adjusted Rate of Return. In reality Risk Adjusted Rate of Return is calculated by Sharpe & Treynor ratio:

Sharpe Ratio: It's the excess return over the risk free rate per unit of standard deviation. Let's again take the above example with additional information being that risk free rate of return in the market is 4.5%.

So Sharpe Ratio for Stock A:

Sharpe Ratio for Stock B:

So, Stock B is giving far more unit of return per unit of risk assumed and is superior to Stock A.

Treynor Ratio: It's the excess return over the risk free rate per unit of beta. It is same as Sharpe Ratio with only difference being that Treynor uses 'Systematic Risk' as it's denominator whereas Sharpe Ratio uses 'Unsystematic Risk' in its denominator. Beta is a measure of system risk as it is market-driven risk, whereas Standard deviation is an unsystematic risk which generally happens due to fluctuations in company related information. Formula for Treynor is:

Jensen Ratio or Jensen's Alpha: The Jensen's measure is a risk-adjusted performance metric that depicts the average return on fund or portfolio which is above (or below) the index result with similar-characterstics stocks. It's the measure of a manager's performance in how he is able to beat the market and how much more result he can yield with respect to the risk he/she assumed. It is calculated as follows:

R(i) = realized return on the fund

R(f) = risk free rate in the market

B = Beta with respect to the market index

R(m) = Return on the index

So, let's say if the fund has given 14% return, risk free return in the market is 4.5% and the index fund has given 9%, and beta of the fund with respect to the index is 1.15 then Jensen's alpha will be:

Now given that the beta of the fund with respect to the index is 1.15 (which means the fund is more riskier than the index), it is well to be observed that it has generated some very handsome result for the risk it is taking.