Sharpe Ratios

What is a quantile?

Dividing ordered data into essentially equal-sized data subsets in the motivation for q-quantiles 2nd quantile is called the median The 4-quantile are quartiles 5 quantile quintiles . . . 100 quantiles are called percentiles

Jarque=Bera Test: Normality test

For a normal distribution, S = 0 and K =3. So we test joint hypothesis H0 : S = 0, K = 3 Test statistic JB = n/6(S^2 + 1/4(K-3)^2) has an asymptotic chi squared distribution with two degrees of freedom X^2(2) We reject H0 at alpha % level of statistical significance if JB > X^2(2)

Historical data on RR

Historical risk-return relationships are useful indicators but there is no guarantee future will be like the past. At the same time: no reason to assume future relative relationships will differ significantly from past.

Interpretation: What does a Sharpe ratio of 1.039 mean?

If there is a 1% more risk on our security then our return will be 1.039% higher

What is easier, predicting long term or short term?

In finance, predicting long term is generally easier.

What is the Kurtosis of a normal distribution ?

K = 3

How do we measure asymmetry?

Measure the skewness: Third movement about the mean S = (1/n SUM(r - E(r))^3)/(SD of r)^3 if S > 0 Distribution is skewed to the right (positive skewed) - more observations on the right side if S < 0 Distribution is skewed to the left (negative skewed) more observations on the left side if S = 0 distribution is symmetric (normal distribution)

Other than skewness how do we measure if a distribution is not normal?

Measurement of the fatness of the tails: Kurtosis: Fourth moment about the mean It measures the thickness of the tails of the distribution K = 1/nSUM(r - E(r))^4/ SD(r)^4

Investment behavior depends on how investors perceived risk.

People are more concerned with downside risk. Many respond more to fear of extreme, rather than of typical gains and/or losses

Excess return

Rate of Return - Risk Free Rate

Sharpe Ratio (Reward to volatility)

Risk Premium(Excess return)/ Standard Deviation of excess return ( E(r) - r ) / (SD of excess return) The ratio describes how much excess return you are receiving for the extra volatility that you endure for holding a riskier asset (measures the market's " price of risk"

Sharpe Ratio only works under a normal distribution. What happens if we are not dealing with a. normal distribution

SD is no longer a complete measure of risk Sharpe Ratio is no longer a good measurement

What does Kurtosis : K > 3 mean?

The distribution of r has fatter tails than normal. Extreme outcomes (both good and bad) are more likely than predicted from a normal distribution Kurtosis of a normal distribution is 3

Risk Premium

The minimum amount of money by which the expected return on a risky asset must exceed the known return on a risk-free asset in order to induce an individual to hold the risky asset rather than the risk free asset E(r) - r

What happens when our distributions have fatter tails?

The probability of extreme values are higher. (Greater Loss or Greater Gain). Than predicted by the normal

What does it mean for r to be skewed left?

Then more extreme negative outcomes are likely, and SD(r) underestimates downside risk

Value at Risk(VaR): What we use to measure risk when our distribution is not normal

VaR answers: What is the most I can lose on a single investment? (Worst-case scenario) It is a measure of risk (potential losses) of investments; measure and quantify the level of financial risk within a firm or investment portfolio over a specific time frame Another name for Quantile

What does VaR(1%) mean?

Var(1%) = 1 % quantile *imagine a probability distribution -------------------Mean------------------------- | 1.5 million find the point where the area under the curve is equal to 1%. That point will indicate a value (dollars) If that values is say 1.5 million then... Lets say Var(1%) = 1.5 million then there is a 1 percent change that we will lose 1.5 million

What happens when our distribution is skewed left or right?

upside risk and downside risk are not equal and not measurable by a single quantity - standard deviation.