In this post we see how to compute historical volatility in python, and the different measures of risk-adjusted return based on it. We have also provided the python codes for these measures which might be of help to the readers.

**Introduction**

Volatility measures the dispersion of returns for a given security. Volatility can be measured by the standard deviation of returns for a security over a chosen period of time. Historic volatility is derived from time series of past price data, whereas, an implied volatility is derived using the market price of a traded derivative instrument like an options contract.

#### Example: Computing historic volatility for NIFTY

First, we use the log function from numpy to compute the logarithmic returns using NIFTY closing price, and then use the rolling_std function from pandas plus the numpy square root function to compute the annualized volatility. The rolling function uses a window of 252 trading days. Each of the days in the selected lookback period is assigned an equal weight. The user can choose a longer or a shorter period as per his need.

## Computing Volatility # Load the required modules and packages import numpy as np import pandas as pd import pandas.io.data as web # Pull NIFTY data from Yahoo finance NIFTY = web.DataReader('^NSEI',data_source='yahoo',start='6/1/2012', end='6/1/2016') # Compute the logarithmic returns using the Closing price NIFTY['Log_Ret'] = np.log(NIFTY['Close'] / NIFTY['Close'].shift(1)) # Compute Volatility using the pandas rolling standard deviation function NIFTY['Volatility'] = pd.rolling_std(NIFTY['Log_Ret'], window=252) * np.sqrt(252) print(NIFTY.tail(15)) # Plot the NIFTY Price series and the Volatility NIFTY[['Close', 'Volatility']].plot(subplots=True, color='blue',figsize=(8, 6))

**Measures of risk-adjusted return based on volatility**

**Sharpe ratio**

The Sharpe ratio which was introduced in 1966 by Nobel laureate William F. Sharpe is a measure for calculating risk-adjusted return. The Sharpe ratio is the average return earned in excess of the risk-free rate per unit of volatility.

Sharpe ratio = (Mean return − Risk-free rate) / Standard deviation of return

Following is the code to compute the Sharpe ratio in python. The inputs required are the returns from the investment, and the risk-free rate (rf).

# Sharpe Ratio import numpy as np def sharpe(returns, rf, days=252): volatility = returns.std() * np.sqrt(days) sharpe_ratio = (returns.mean() - rf) / volatility return sharpe_ratio

**Information ratio (IR)**

The information ratio is an extension of the Sharpe ratio which replaces the risk-free rate of return with the returns of a benchmark portfolio. It measures a trader’s ability to generate excess returns relative to a benchmark.

Following is the code to compute the Information ratio in python. The inputs required are the returns from the investment, and the benchmark returns.

import numpy as np def information_ratio(returns, benchmark_returns, days=252): return_difference = returns - benchmark_returns volatility = return_difference.std() * np.sqrt(days) information_ratio = return_difference.mean() / volatility return information_ratio

**Modigliani ratio (M2 ratio)**

The Modigliani ratio measures the returns of the portfolio, adjusted for the risk of the portfolio relative to that of some benchmark. To calculate the M2 ratio, we first calculate the Sharpe ratio and then multiply it with the annualized standard deviation of a chosen benchmark. We then add the risk-free rate to the derived value to give M2 ratio.

Following is the code to compute the Modigliani ratio in python. The inputs required are the returns from the investment, benchmark returns, and the risk-free rate.

# Modigliani Ratio import numpy as np def modigliani_ratio(returns, benchmark_returns, rf, days=252): volatility = returns.std() * np.sqrt(days) sharpe_ratio = (returns.mean() - rf) / volatility benchmark_volatility = benchmark_returns.std() * np.sqrt(days) m2_ratio = (sharpe_ratio * benchmark_volatility) + rf return m2_ratio

#### Next Steps

In our coming posts we will cover other risk measures, and the measures of risk-adjusted returns. You can also catch our other blogs available under the Risk and portfolio management category.

To understand Risk Management in Financial Institutions, have a look at the blogpost. The key requirement in successful options trading involves understanding and implementing options pricing models. In this post, a brief understanding about Greeks is given, which will help in creating and understanding the pricing models.