Simple Probability Distributions and Powerful Use Cases in Risk Management
In this article, I have penned down my learning on probability distributions based on my working knowledge and not based on academical knowledge. The emphasis is on how anyone even with minimal working knowledge of statistics can use these distributions to their advantage at work.
Probability Distributions - It’s just a plot/ distribution of different values a variable can take and its corresponding probabilities/ frequencies.
Why I use probability distribution at my work? Based on empirical/academic evidence or expert suggestion, if I know a variable follows a particular probability distribution, then I can answer the below questions
· What is the probability for the variable to take a particular range of values?
· What would be the value of the variable at different probability or confidence levels#?
#Confidence level is a concept in statistics which means what is the probability that the estimated value based on sample would represent the true parameter value of the population. In simple terms, how confidently can you say that your estimate is correct? For example, if one estimates at 90% confidence level, the maximum loss would be Rs. 100, it means the person is 90% confident on his/ her estimate that the actual loss would be less than Rs. 100
Let us take 3 cases below to understand it better.
I. 3+2 = 5. This is what mathematics says! So, we are 100% sure (sure event- 100% Probability) on the outcome. Hence there will be no probability distribution in this case.
II. Now let us assume in an imaginary world; where the function of exact addition has not been discovered yet, 3+2 ranges from 4.5 to 5.5 and possible values are 4.6, 4.7, 4.8, 4.9, 5.0, 5.1, 5.2, 5.3, 5.4. 5.5.
Here in this case, we have 10 discrete outcomes possible. And for simplicity, let us assume all outcomes are equally possible i.e. probability of arriving at each value is same (1/10). So, the probability distribution will look as below. Since we can see probability, which is assigned to 10 different discrete values, this distribution is known as discrete probability distribution
III. 3+2 ranges from 4.5 to 5.5 and can take any value in between- continuous probability distribution. In continuous probability distribution, since there are infinite integers possible (in this case- there are infinite integers possible between 4.5 to 5.5), technically the probability of a single value is very low i.e., close to 0. So, in a continuous probability distribution, the practitioners would always be interested in finding the probability of a range of values.
In this case, for instance we could be interested in finding the probability of values > 4.7 or values < 5.2 or probability for values in the range 4.7 to 5.2; rather than a discrete value.
In this article, I have discussed on two most commonly used continuous probability distributions in finance.
Normal Distribution usage in Market Risk
Let us assume that I have a portfolio of stocks and I want to get some sense on how much market risk I am taking, assuming the return of my portfolio follows a normal distribution (it is a symmetrical bell- shaped curve- the shape of the curve is dependent on the mean and standard deviation).
Based on historical performance,
· Mean of the annual portfolio returns = 9.46%
· Standard Deviation of the annual portfolio returns = 11.75%
Below are the inferences that can be made after assuming normal distribution,
· I can be 78.96% confident that I would make a positive annual return.
· 90% VaR* (Market Risk) of my portfolio is 5.60%
· 95% VaR* (Market Risk) of my portfolio is 9.86%
· 99% VaR* (Market Risk) of my portfolio is 17.87%
*VaR- Value at Risk in simple terms is a concept in Risk Management which helps us to estimate what would be the maximum loss at a particular confidence level. For example, if 90% VaR of a portfolio is Rs. 100, it means the maximum loss the portfolio can face at 90% confidence is Rs. 100
Below graphical representation is to show the returns across different probability/ confidence intervals. I have also plotted the mean return to understand how deviated the returns can be as we move towards the end of the spectrum.
Log Normal distribution usage in Credit Risk
Again, based on empirical studies and analyzing the historical bankruptcy and bond default data, we have understood that credit losses follow a log normal distribution (it is a distribution bounded by zero and have a long-right tail- the shape of the curve here is again dependent on mean and standard deviation)
Suppose I have estimated the expected loss (EL) of a credit instrument to be 5% and I am interested in estimating the Unexpected Loss (UL) at different confidence intervals. In simple terms, unexpected losses are the worst-case losses that has very low probability of occurrences.
The below graphical representation shows the losses at different confidence intervals. It can be easily observed from the chart that the losses increase exponentially at high confidence intervals.
By this I have explained why probability distributions are used in general, what inferences we can make on a variable by assuming to have a particular probability distribution, general use cases of how two most common distributions are used in finance.
I have done all my workings in excel. If you are interested to review the excel or want to have a discussion on this, please feel free to drop an email at sampathkumar0592@gmail.com
