**Statistical measurement of risk**

**The Mean-Variance Framework**

The Markowitz approach, while powerful and simple, boils investor choices down to two dimensions. The “good” dimension is captured in the expected return on an investment, and the “bad” dimension is the variance or volatility in that return. In effect, the approach assumes that all risk is captured in the variance of returns on an investment and that all other risk measures, including the accounting ratios and the Graham margin of safety, are redundant. We can justify the mean-variance focus in two ways: assuming that returns are normally distributed, or by assuming that investors’ utility functions push them to focus on just expected return and variance.

The deviation between actual return with expected return is risk. It is also called variability of return. Risk is measured in two ways a) Standard Deviation (σ) and b) variance (r or σ^{2}). There is also two way of calculating risk, it depends upon the nature of data such as historical data or past data & using probability.

**By using past data**

Var (_{r})=

**By using probabilities**

Var (_{r}) = ^{n}∑_{t=1 }P_{t}[r_{t} – E(r)]^{2}

To calculate standard deviation we use,

**Standard deviation of return (σ) =**

Risk measurement through expected return with coefficient of variation (C.V)

The coefficient of variation (COV) can measure the volatility of an investment. It is a ratio between the standard deviation of a expected mean. It is also known as relative standard deviation.

Coefficient of variation (C.V) = σ/E(r)

Here, E(r) = expected return

N= return at time