There are many mathematical formulas and tools that can be used for risk management in securities trading. Some of the most common include the Sharpe ratio, the Kelly criterion, and the value at risk (VaR) metric. Even if you might not use them, it is good to be familiar with the terminology.

The Sharpe ratio

The Sharpe ratio is a measure of risk-adjusted return that compares the return of an investment to that of a risk-free asset, such as a government bond. To calculate the Sharpe ratio, you first need to determine the excess return of the investment, which is the difference between the investment’s return and the return of the risk-free asset. This excess return is then divided by the standard deviation of the investment’s returns, which is a measure of its volatility.

Here is an example of a Sharpe ratio calculation:

Suppose you have an investment that has a return of 10% over a certain period of time. The risk-free rate of return during that same period was 3%. To calculate the excess return of the investment, you would subtract the risk-free rate from the investment’s return:

Excess return = 10% – 3% = 7%

Next, you would need to calculate the standard deviation of the investment’s returns. This is a measure of the volatility of the investment, and it can be calculated using historical data on the investment’s returns. For the purposes of this example, let’s assume that the standard deviation of the investment’s returns is 5%.

To calculate the Sharpe ratio, you would divide the excess return by the standard deviation:

Sharpe ratio = 7% / 5% = 1.4

This Sharpe ratio indicates that the investment has a higher return relative to its volatility, which means that it is a good risk-adjusted investment. Generally, a Sharpe ratio of 1 or higher is considered good, while a Sharpe ratio of less than 1 is considered less attractive.

The Kelly criterion

The Kelly criterion is a mathematical formula used to determine the optimal size of a series of bets in order to maximize the growth rate of a portfolio. It is based on the idea that the optimal betting strategy is to bet a fixed fraction of your current portfolio on each bet, in order to maximize your long-term growth rate.

Here is an example of how to calculate the Kelly criterion for a series of bets:

Suppose you have a portfolio with a current value of $100,000, and you are considering making a series of bets with the following characteristics:

• The probability of winning each bet is 50%
• The payout for each winning bet is 1.5x the amount bet
• The probability of losing each bet is 50%
• The loss for each losing bet is the amount bet

In this case, the Kelly criterion formula would be as follows:

Kelly criterion = (probability of winning * payout for winning) – (probability of losing * loss for losing)

= (0.5 * 1.5) – (0.5 * 1)

= 0.25

This means that the optimal betting strategy is to bet 25% of your current portfolio on each bet. In this case, that would mean betting $25,000 on each bet. By following this strategy, you can maximize the growth rate of your portfolio over the long term.

Value at Risk

Value at Risk (VaR) is a measure of the risk of loss on a portfolio of investments. It estimates the maximum loss that a portfolio might incur over a given time period, with a given level of confidence. In other words, it is a measure of the worst possible loss that could be incurred on a portfolio, given a certain level of confidence.

Here is an example of how to calculate VaR for a portfolio:

Suppose you have a portfolio with a current value of $100,000, and you want to calculate the VaR for a one-day holding period, with a 95% confidence level.

First, you need to determine the standard deviation of the portfolio’s daily returns. Let’s say that the standard deviation of the portfolio’s daily returns is 3%. You can use your historical performance for this.

Next, you need to use the inverse normal distribution function to determine the value that corresponds to the 95% confidence level. This value is typically called the “z-score.” For a 95% confidence level, the z-score is 1.65.

Now, you can use the following formula to calculate the VaR for the portfolio:

VaR = portfolio value * standard deviation of returns * z-score

= $100,000 * 0.03 * 1.65

= $495

This means that there is a 95% probability that the portfolio will not lose more than $495 over a one-day holding period. In other words, there is a 5% probability that the portfolio will lose more than $495 over a one-day holding period.

These are just a few examples of the many mathematical formulas and tools that can be used for risk management in securities trading. The specific formulas and techniques used will depend on the trader’s goals, risk tolerance, and trading strategy. Even if some of the data is “soft”, the formulas can help you compare the risk between assets.