Introduction

In this chapter, we will delve into the calculation of realized returns and historical volatility, crucial components in understanding the behavior of stock prices. These metrics provide valuable insights into a stock’s past performance, helping investors and analysts assess risk and make informed decisions. We will explore the step-by-step process for computing these metrics, including the relevant formulas and practical examples.

Realized Returns

Realized returns, often referred to as historical returns, are a measure of how much an investment has earned or lost over a specified period. They are calculated as the percentage change in the price of the stock from the beginning to the end of the period, plus any dividends received during that time. The formula for realized return is:

$\text{Realized Return (\%)}=\frac{(\text{Ending Stock Price}-\text{Beginning Stock Price}+\text{Dividends})}{\text{Beginning Stock Price}}\times 100$

For example, suppose you invested in a stock priced at USD 100, and after one year, the stock price has increased to USD 110, and you received USD 5 in dividends during the year. The realized return would be:

$\text{Realized Return (\%)}=\frac{(110-100+5)}{100}\times 100=15\mathrm{\%}$

This means that your investment earned a 15% return over the one-year period.

Historical Volatility

Historical volatility measures the degree of variation in a stock’s price over a specified historical period. It quantifies the stock’s past price fluctuations, providing an indication of how volatile or stable the stock has been. To calculate historical volatility, follow these steps:

1) Calculate Daily Returns: For a given historical period, record the daily closing prices of the stock. Calculate the daily returns by taking the natural logarithm of the ratio of the closing price for each day to the previous day’s closing price:

$$r_t=\ln \left(\frac{P_t}{P_{t-1}}\right)$$

Where:

• $r_t$ = Daily return on day $t$
• $P_t$ = Closing price on day $t$
• $P_{t-1}$ = Closing price on the previous day $(t-1)$

2) Calculate Variance: Compute the variance of the daily returns. The variance measures how much the returns deviate from their mean and is a proxy for volatility. The formula for variance is:

$${\sigma }^2=\frac{{\mathrm{\Sigma }}_i(r_t-\mu {)}^2}{(n-1)}$$

Where:

• ${\sigma }^2$ = Variance of daily returns
• $\mu$ = Mean of daily returns
• $n$ = Number of daily returns

3) Calculate Volatility: Finally, calculate the historical volatility as the square root of the variance:

$$\sigma =\sqrt{{\sigma }^2}$$

Where:

• $\sigma$ = Historical Volatility