Continuous Random Variables

Understanding Continuous Random Variables | CFA Level I Quantitative Methods

Welcome back! In this lesson, we’ll dive into continuous random variables and their distributions. We’ll cover continuous uniform distributions, normal distributions, and lognormal distributions. We’ll also touch on confidence intervals, Roy’s safety-first ratio, and more. So, buckle up and let’s get started!

Continuous Random Variables vs. Discrete Random Variables

In our previous lesson, we learned about discrete random variables, where the number of possible outcomes can be counted. In contrast, a continuous random variable has an infinite number of possible outcomes, even within finite boundaries.

Unlike discrete random variables, continuous random variables have distribution functions drawn with continuous lines to reflect their continuous nature. The line represents the probability for each particular outcome.

Continuous Uniform Distribution

Let’s start by examining the continuous uniform distribution. This distribution describes equally likely outcomes within a defined range. The key property of any probability distribution function is that the sum of all probabilities must equal 1. For continuous distribution functions, this means the area under the line must be 1.

For a continuous uniform distribution within an interval from A to B, the probability is 1/(B – A). For outcomes outside of this interval, the probability is 0.

The cumulative function for this distribution is 0 for outcomes before A. Between A and B, the cumulative probability rises linearly from 0 to 1. The function for this interval is (x – A) / (B – A). After point B, the cumulative probability remains at 1.

Normal Distribution

Now let’s revisit the normal distribution, which we discussed earlier in the topic on statistical concepts. Here’s a quick recap of its properties:

  • It’s completely described by its mean and variance.
  • It has zero skewness, meaning it’s symmetrical on both sides of its center.
  • Its mean, mode, and median are equal and aligned at the center of the distribution.
  • It has a kurtosis equal to 3, with leptokurtic distributions having a kurtosis greater than 3 and platykurtic distributions having a kurtosis less than 3.
  • About 68% of observations lie within ±1 standard deviation from the mean, 95% within ±2 standard deviations, and 99% within ±3 standard deviations.

Confidence Intervals

When applied to probability distributions, these standard deviations can give us an idea of the level of confidence that the outcome will fall within specific ranges, which we call confidence intervals. There are three key confidence intervals to remember:

  1. 90% confidence interval: ±1.65 standard deviations from the expected value.
  2. 95% confidence interval: ±1.96 standard deviations from the expected value.
  3. 99% confidence interval: ±2.58 standard deviations from the expected value.

EXAMPLE

The average return of a stock index is 8.9% per year, and the variance of returns is 256. If the returns are approximately normal, what is the 90% confidence interval for the index return next year?

First, let’s find the standard deviation, which is the square root of the variance of return. We get a standard deviation, s = (256)½ = 16%

Applying the formula for a 90% confidence interval, 8.9 – 1.65×16 < X < 8.9 + 1.65×16, we get the range of -17.5% to +35.3%. This means that we are 90% sure that the returns of the stock index next year will fall within this interval.

Using the Z-Table

We’ve been discussing the standard normal distribution, where the mean is 0 and the standard deviation is 1. It’s easy to find the cumulative probability of any interval using the z-table.

For example, if we want to find the probability of X being less than 1.65, we can use the z-table. Look for 1.6 in the first column, then find 0.05 in the header row. The intersection of the row and column gives you the probability of X being less than 1.65, which is approximately 95%.

This figure means that 95% of the observations lie below 1.65, and we can expect X to be less than 1.65 with 95% confidence in the future.

The standard normal distribution is symmetrical about zero. So, any negative value of x will have cumulative probabilities equal to 1 minus the corresponding positive value. In this case, F(-1.65) = 1 – 0.9505 = 0.0495.

This shows that the cumulative probabilities within this interval are approximately 90%, which is the confidence interval. For exam purposes, you only need to remember ±1.65 standard deviations for a 90% confidence interval.

Understanding Z-Transform and Portfolio Risk

Converting Non-Standard Normal Distributions to Standard Normal

Most real-life distributions are not standard. For example, X could be the annual earnings per share (EPS) of stocks in a particular market with a mean of $2 and a standard deviation of $0.50. The distribution is still normal, but non-standard. To convert the distribution to a standard normal, we apply two transformations:

  1. Shift the distribution to align the mean to zero by subtracting the mean μ.
  2. Scale the standard deviation to 1 by dividing by the standard deviation σ.

This equation can be generalised to (x – μ) / σ, known as the z-transform.

EXAMPLE

Random variable X denotes the annual EPS of stocks in MegaLand. The mean of X is $2, and the standard deviation is $0.50. What percentage of the stocks are likely to have an EPS of less than $1?

Step 1: Find the corresponding z-value for an EPS of $1. z = (1-2)/0.5 = -2.0

Step 2: Use the z-table to find the probability. For negative z-values, use the positive z-value and subtract from 1. From the z-table, F(2.0) = 0.9772, and F(-2.0)= 1 – F(2.0) = 0.0228.

So, 2.28% of the stocks are likely to have an EPS of less than $1.

Mean-Variance Analysis and Shortfall Risk

Analysts use z-values to evaluate portfolio risk. For example, Portfolio A has an expected return of 10% and a standard deviation of 5%. An analyst views a return of less than 5% as unacceptable (threshold return). The probability of a return falling below this level is the shortfall risk.

Calculating shortfall risk using the z-table:

  1. Find the z-value: (5 – 10) / 5 = -1.0
  2. Look up the z-table: F(-1.0) = 0.1587, so the shortfall risk is 15.87%

Comparing two portfolios, B and A:

  • Portfolio B: mean of 15%, standard deviation of 7.5%, z-value of -1.33, shortfall risk of 9.18%
  • Portfolio A: mean of 10%, standard deviation of 5%, z-value of -1.0, shortfall risk of 15.87%

Despite Portfolio B having a higher standard deviation, it has a lower shortfall risk than Portfolio A. Risk evaluation is complex and requires analyzing the best method based on the investor’s objectives and risk tolerance.

Roy’s Safety First Ratio

A shortcut to evaluate the suitable portfolio is to use the Roy’s safety first ratio. The safety first ratio is the reverse of the top 2 terms of the z-formula: (Expected return – Threshold return) / Standard deviation. Due to the symmetric property of the normal distribution, the shortfall risk remains the same when using the safety first ratio. With this ratio, we select portfolios with higher values.

EXAMPLE

In summary, the z-transform and Roy’s safety first ratio are useful tools in evaluating portfolios based on their risk characteristics. Remember that risk evaluation is complex, and it’s essential to consider the investor’s objectives and risk tolerance when selecting a suitable portfolio.

Lognormal Distribution and Its Connection to Finance

Normal distribution is often a good approximation for returns, but it is unsuitable for modeling asset prices because it has a range of outcomes from negative infinity to positive infinity, while asset prices cannot fall below zero. To solve this problem, we can use the lognormal distribution.

When we take the natural log of the price relative (1 + holding period return), it directly translates to the continuously compounded return multiplied by the number of time periods. If we assume that the continuously compounded return is normally distributed, then the price relative is a lognormal distribution.

The lognormal distribution has two important properties:

  1. It is skewed towards the right with a long fat right tail.
  2. It can never take on values below zero, making it suitable for modeling asset prices.

In practice, a lognormal distribution has been found to be an accurate description of the distribution of prices for many financial assets.

Summary

In this lesson, we covered:

  • Continuous uniform distribution, where the probability is uniform for a fixed interval.
  • Normal distribution, where we can determine confidence intervals based on standard deviations from the mean.
  • Finding the cumulative probability using the z table.
  • Converting a normal distribution to a standard normal distribution.
  • Determining the most suitable portfolio using Roy’s safety first ratio.
  • Using the most appropriate distribution to model different variables.

That’s all for this lesson. In the next lesson, we will briefly discuss Monte Carlo simulation. See you again!

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