May 22, 2013
As you are reading this article (or attempting to read!), you would most probably know that Quantitative Methods section is worth 12% of your CFA L1 exam score. And, a huge chunk of Quantitative Methods is made up of Probability distributions!
After going through a lot of recent papers/mocks, I realized that the weightage of Probability Distributions can range anywhere from 30% to 50% of your quantitative methods score. This means that Probability distribution is vital for your CFA Exam success as it can be worth almost 4-6% of your total score.
Further, probability distributions are used in topics such as equities, portfolio management, alternative investments, VAR etc. and a solid understanding of probability distributions here will help you a long way in your preparation.
Now, probability distributions may look a bit complicated at first, especially for those of you without much mathematics background. But, worry not! The questions asked are pretty straight forward and you’ll do just fine if you concentrate on a few important points.
In this article, I will take you through the 4 main types of distributions present in the CFA L1 Exam curriculum and discuss important points that shall help you crack CFA L1 Exam at one go.
Probability distribution is denoted by p(x) & is the probability that a random variable X takes on the value x, or p(x) = p(X = x). They are widely used in the field of finance.
Variable such as returns are very often modeled using probability distributions. These variables are known as random variables.
Probability distribution can be discrete or continuous. A toss of coins or a roll of dice are examples of discrete probability distribution. Whereas, if a random variable is continuous, for example, the amount of rain per day (in cm or inches), the distribution is known as continuous probability distribution.
So, a distribution is a discrete probability distribution when there are a finite number of discrete possible events. On the other hand, a distribution is a continuous probability distribution when there is an infinite set of possible events and there is an infinite number of values between any two points in the distribution.
An important point of distinction is that in case of discrete distribution, you can very well say that the probability of exactly 1 head occurring in a toss of two coins is 0.5.
But, can you say the same thing if you are asked to find the probability of getting exactly 50cm rain on a particular day? Take a moment and think about it.
As there are infinite number of possible outcomes (points) of rainfall between 0 and 100cm, the probability of receiving single point of rainfall on a particular day (say exactly 50cm) is always zero. But, the probability of the amount of rain between a range (say 40-50cm) has some positive value. Even a range of 49.999999 and 50.0000001 will have positive value of probability as this interval can be further divided into potentially infinite number of points.
When the possible number of outcomes is huge, even some discrete distributions are treated as continuous. For example, for a stock being traded on NYSE, the probability of the stock going up by exactly $1.52 is almost zero.
Therefore, probability of a single point will have a positive value in case of discrete distribution. But, in case of continuous distribution, probability of a single point will be zero and probability is always given as a range.
The figures below shows a continuous normal distribution and a discrete binomial distribution
Discrete and continuous distributions are summarized in the following table:
Discrete | Continuous |
Finite number of discrete possible events | Infinite set of possible events |
Probability of a single point has a positive value | Probability of a single point is zero and probability is always given as a range. |
Eg. Number of heads in a toss of 4 coins | Eg. Amount of rainfall (in cm) occurring on a particular day |
Now, let’s proceed to the distributions. Given below are the 4 common types of probability distributions that can be asked in CFA level 1 examination.
1. Discrete Uniform Distribution
It is the simplest type of probability distribution, where all outcomes have equal probability of occurring. P(x) = 1/k.
A roll of a die follows a uniform distribution with each outcome (i.e. 1 or 2 or 3 or 4 or 5 or 6) having probability of 1/6 each.
2. Binomial Distribution
A binary variable that takes on one of two values (usually 1 for success or 0 for failure) is known as a Bernoulli random variable. For example, a single toss of a coin is a Bernoulli random variable where we can denote H as success and T as failure or vice versa. It is represented mathematically as
p(success) = p(1) = P(X=1) = p
p(failure) = p(0) = P(X=0) = 1 – p
Now, suppose you toss 10 coins and ask me the probability of 4 heads occurring. A Binomial random variableX ~ B(n, p) is defined is the number of successes in n Bernoulli random trials where p is the probability of success on any one Bernoulli trial. Mathematically,
It is important to note that the probability (p) of success is 0.5 here, but may not always be the case. For example, I can take your favorite coin and bias it by adding weight such that the probability of getting heads becomes 0.6 and tails becomes 0.4. This will still follow a binomial distribution, just that now p = 0.6.
Expected value and variance
If you toss 10 unbiased coins, how many heads do you expect? 5, right!
Intuitively, each toss has Â½ probability of getting heads and there are 10 tosses. Therefore, you expect 5 heads on an average. This is the Expected value of a binomial trail and is given mathematically as
E(X) = np
In case of 10 unbiased coins, E(X) = 10 x Â½ = 5
Similarly, the variance is given as
Var(X) = np(1-p)
In case of 10 unbiased coins, Var(X) = 10 x Â½ x (1 – Â½) = 2.5
Continuous distributions
3. Continuous Uniform Distribution
This is another simple distribution, where the probability of occurrence all the events is constant and equal. Also, as the name suggests, it is continuous. Mathematically,
P(x_{1} â‰¤ X â‰¤x_{2}) = (x_{2} – x_{1})/(b – a),
This gives the probability of outcomes between and x_{2} and x_{1}, where a and b are lower and upper limits respectively.
A quick question for you: X is uniformly distributed between 5 and 15. What is the probability that X will be between 7 and 11?
Answer: (11 – 7)/(15 – 5) = 0.4
If you look at continuous uniform distribution carefully, you will realize that for a range that is half of the whole range, the probability of outcomes will be 50%. Similarly for a quarter of the whole range, the probability will be 25% and so on.
4. Normal Distribution
The normal or Gaussian distribution is the most commonly used distribution. In fact, it is so “normal” everywhere that people began calling it normal distribution! In finance, it is often found in stock market analysis, where the returns are assumed to follow a normal distribution. It plays an important role in portfolio theory.
It is a symmetric distribution that is completely described by two parameters: its mean, Î¼, and its variance, Ï’^{2}
Confidence interval
A 95% confidence interval is the range in which we expect the random variable to be in 95% of the time. For normal distribution, we measure this interval in terms of standard deviations.
Standard normal distribution
A normal distribution with mean = 0 and standard deviation sigma=1 is known as a standard normal distribution.
You can convert a given normal random variable into a standard normal variable by:
Standardization helps in simplifying things, makes calculations easier and also aids in comparing any two normal distribution. After standardization the value of P(Z < z) can be found by looking up the standardized table. As the normal distribution is symmetric about the y-axis F(-Z) = 1 -F(Z). Also, P(Z â‰¥ z) = 1 – P(Z < z). An example here will help in better understanding of the concept. Consider the price of a stock distributed normally with mean = $10 and standard deviation = $2. What is the probability that the stock price will be $13.70 or more? To solve this problem, we first standardize the random variable into a standard normal variable using: We have to find P(Price > 13.70) = 1 – P(Price â‰¤ 13.70) = 1 – F(1.85) = 1 – 0.9678 = 3.2% Lognormal Distribution In normal distribution, there is a possibility that asset returns become less than -100%. This would mean that the asset prices are in fact less than zero! For example, a $20 stock can go up to $50 but can’t go down to -$10! A lognormal distribution does not take values less than zero and hence avoids this problem. It is generated by the function e^{x} , where x is normally distributed. The graph below shows a lognormal distribution with mean = 0 and standard deviation = 1. We can see that the distribution is skewed to the right and is bounded from below by 0, which means that it cannot assume negative values. Conclusion We have covered the 4 types of probability distribution that are important for CFA L1 examination. Now, to reinforce these topics, it is vital that you solve some questions! Questions can be found in EduPristine’s CFA L1 question bank and quizzes. |
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