<p>This course introduces some of the discrete and continuous probability distributions&nbsp;most commonly used to describe the behavior of random variables.</p>

Study Notes for Common Probability Distributions

Common Probability Distributions

<p><a title="Monte Carlo Simulation – Example" href="https://financetrain.com/monte-carlo-simulation-example/">Monte Carlo Simulation</a> can be used to price various financial instruments such as derivatives.</p>
<p>In this article, we will learn how to calculate the price of an option using the Monte Carlo Simulation.</p>
<p>Even though the option value can be easily calculated using the <a title="Black-Scholes Option Pricing Model" href="https://financetrain.com/black-scholes-option-pricing-model/">Black-Scholes Option pricing</a> formula, we can make use of the Monte Carlo Simulation technique to achieve the same results.</p>
<p>Let us calculate the price of a call option. Assume that the underlying stock price (S) is 195, the exercise price(X) is 200, risk free rate (rf) is 5%, volatility (s) is 30%, and the time to expiry (t) is 0.25.</p>
<h3>Step 1</h3>
<p>The role of Monte Carlo simulation is to generate several future value of the stock based on which we can calculate the future value of the call option. The changes in the stock prices can be calculated using the following formula:</p>
<p>[latex]\Delta S=Sr_{f}\Delta t +S\sigma \varepsilon \sqrt{t}[/latex]</p>
<p>In this equation, ε represents the random number generated from a standard normal probability distribution. In our example, we will calculate this number using the Rand() function in excel. A standard normal variable can be approximated using the Excel formula “= Rand() + Rand() + Rand() + Rand() + Rand() + Rand() + Rand() + Rand() + Rand() + Rand() + Rand() + Rand() - 6.0”.</p>
<p>For the purpose of this example, we will generate 1000 random paths. In reality, the higher the number of trials, the more accurate will be the result.</p>
<p>If the random number is 0.576548, ΔS will be 19.29978.</p>
<h3>Step 2</h3>
<p>Once we have ΔS, we can calculate the future value of the stock price (S + ΔS). We need to do this for each path.</p>
<p>The stock value at expiry will be 195 + 19.29978 = 214.29978</p>
<h3>Step 3</h3>
<p>The option value at expiry will be given by the formula =MAX (0,S-X)</p>
<p>= MAX(214.29978 – 200) = 14.29978</p>
<p>The above three steps will be repeated 1000 times to get 1000 option values.</p>
<h3>Step 4</h3>
<p>We will take the average of these 1000 option values. In our example, this value comes to approx. 9.671. Please note that this value will change every time the spreadsheet is recalculated, so you may never get the same answer.</p>
<h3>Step 5</h3>
<p>The option value will be discounted to the present value by multiplying it with exp(-r*t)</p>
<p>In our example, this value comes to 9.95.</p>
<p>The result can be compared to the results from a Black-Scholes calculator. You will notice that the results will get closer as you increase your number of trials.</p>

Option Pricing Using Monte Carlo Simulation

We know that a <a title="What is a Random Variable?" href="https://financetrain.com/what-is-a-random-variable/">random variable</a> is an uncertain quantity or a number. Its value is determined by chance. For example, the outcome of rolling a die is random. We could get any number from 1 to 6. In case of a die, the probability of getting any number is 1/6. Each outcome has the same probability. However, the probability of each outcome could be different.

A probability distribution is a graph or a table that describes the probabilities of each outcome of a random variable.

In a probability distribution, each value or outcome of the random variable is represented as x. The probability of getting x is represented as P(x). So, if X is the random variable, we are saying that the probability of random variable X being equal to x is P(X=x) or P(x). This is called the probability function.

The probability distribution of rolling a die is shown below:
<table>
<tbody>
<tr>
<td width="102"><strong>xi</strong></td>
<td width="123"><strong>P(xi)</strong></td>
</tr>
<tr>
<td width="102"><strong>1</strong></td>
<td width="123">1/6</td>
</tr>
<tr>
<td width="102"><strong>2</strong></td>
<td width="123">1/6</td>
</tr>
<tr>
<td width="102"><strong>3</strong></td>
<td width="123">1/6</td>
</tr>
<tr>
<td width="102"><strong>4</strong></td>
<td width="123">1/6</td>
</tr>
<tr>
<td width="102"><strong>5</strong></td>
<td width="123">1/6</td>
</tr>
<tr>
<td width="102"><strong>6</strong></td>
<td width="123">1/6</td>
</tr>
</tbody>
</table>
Note that the sum of all probabilities should be equal to 1 and the probability of each outcome, P(x) is between 0 and 1.

What is a Probability Distribution

<h3>Discrete Random Variable</h3>
A <a title="What is a Random Variable?" href="https://financetrain.com/what-is-a-random-variable/">random variable</a> is said to be discrete if the total number of values it can take can be counted. Alternatively, we can say that a discrete random variable can take only a discrete countable value such as 1, 2, 3, 4, etc. For example, in case of the roll of a die, there could be only 6 outcomes. This is an example of a discrete random variable. Similarly, if we have to pick a stock from S&amp;P 500 index, it’s a discrete random variable. Each outcome should also have a positive probability.

Let’s say a variable X can take values 1, 2, 3, 4. The probabilities of each of these outcomes are given below:
<table>
<tbody>
<tr>
<td width="121"><strong>xi</strong></td>
<td width="132"><strong>P(xi)</strong></td>
</tr>
<tr>
<td width="121"><strong>1</strong></td>
<td width="132">0.2</td>
</tr>
<tr>
<td width="121"><strong>2</strong></td>
<td width="132">0.3</td>
</tr>
<tr>
<td width="121"><strong>3</strong></td>
<td width="132">0.4</td>
</tr>
<tr>
<td width="121"><strong>4</strong></td>
<td width="132">0.1</td>
</tr>
</tbody>
</table>
We can draw this distribution in the form of a histogram.

<img class="aligncenter size-full wp-image-19263" src="https://financetrain.com/wp-content/uploads/pd1.gif" alt="pd1" width="411" height="297">

<strong>Note:</strong> What would be the probability of the random variable X being equal to 5?

P(5) = 0 because as per our definition the random variable X can only take values, 1, 2, 3 and 4.
<h3>Continuous Random Variable</h3>
In contrast to discrete random variable, a random variable will be called continuous if it can take an infinite number of values between the possible values for the random variable. Examples include measuring the height of a person, or the amount of rain fall that a city receives. The number of possible outcomes is infinite. In that case, what is the probability that the random variable X will get a certain value x?

P(x) will be 0 because we are talking about the possibility of one outcome from an infinite number of outcomes.

In finance, some variables such as price change of a stock, or the returns earned by an investor are considered to be continuous, even though they are actually discrete, because the number of possible outcomes is large, and the probability of each outcome is very small. For example, the probability of an investor earning a return of exactly 8.25% is almost zero.

The probability distribution of a continuous random variable is called probability density function.

Discrete Vs. Continuous Random Variable

We can also construct a cumulative distribution function for a <a title="What is a Random Variable?" href="https://financetrain.com/what-is-a-random-variable/">random variable</a>. A cumulative distribution function gives the probability that the random variable X is less than or equal to x, for every value x. In case of discrete random variables, the cumulative distribution function is the sum of the probabilities of all outcomes unto and including the specific outcome x.

The cumulative distribution function is expressed as &nbsp;.

We will build upon our earlier probability distribution example.
<table>
<tbody>
<tr>
<td width="121">xi</td>
<td width="132">P(xi)</td>
<td width="132">F(xi)</td>
</tr>
<tr>
<td width="121"><strong>1</strong></td>
<td width="132">0.2</td>
<td width="132">0.2</td>
</tr>
<tr>
<td width="121"><strong>2</strong></td>
<td width="132">0.3</td>
<td width="132">0.5</td>
</tr>
<tr>
<td width="121"><strong>3</strong></td>
<td width="132">0.4</td>
<td width="132">0.9</td>
</tr>
<tr>
<td width="121"><strong>4</strong></td>
<td width="132">0.1</td>
<td width="132">1</td>
</tr>
</tbody>
</table>
Probability that X =1 is 0.2

Probability that X = 1 or 2 = 0.2 + 0.3 = 0.5

Probability that X = 1 or 2 or 3 = 0.2 + 0.3 +0.4 = 0.9

Probability that X = 1 or 2 or 3 or 4= 0.2 + 0.3 +0.4 +0.1= 1.0

The histogram for cumulative distribution will look as follows:

<img class="aligncenter size-full wp-image-19267" src="https://financetrain.com/wp-content/uploads/cdp.gif" alt="cdp" width="411" height="297">

The above cumulative distribution was for a discrete random variable. Even a continuous random variable will have a cumulative distribution function.

Cumulative Distribution Function

A discrete uniform random variable is a <a title="Discrete Vs. Continuous Random Variable" href="https://financetrain.com/discrete-vs-continuous-random-variable/">discrete random variable</a> for which the probability of each outcome is the same.

<strong>Example</strong>

The roll of a die is a discrete uniform random variable and has a discrete uniform probability distribution.

The random variable X can take the values X = {1, 2, 3, 4, 5, 6}

Each outcome has a probability of 1/6.

The <a title="What is a Probability Distribution" href="https://financetrain.com/probability-distribution/">probability distribution</a> and <a title="Cumulative Distribution Function" href="https://financetrain.com/cumulative-distribution-function/">cumulative distribution functions</a> are shown below:
<table>
<tbody>
<tr>
<td width="102"><strong>xi</strong></td>
<td width="151"><strong>Probability distribution, P(xi)</strong></td>
<td width="151"><strong>Cumulative Distribution, F(xi)</strong></td>
</tr>
<tr>
<td width="102"><strong>1</strong></td>
<td width="151">1/6</td>
<td width="151">1/6 = 0.1667</td>
</tr>
<tr>
<td width="102"><strong>2</strong></td>
<td width="151">1/6</td>
<td width="151">2/6 = 0.333</td>
</tr>
<tr>
<td width="102"><strong>3</strong></td>
<td width="151">1/6</td>
<td width="151">3/6 = 0.5</td>
</tr>
<tr>
<td width="102"><strong>4</strong></td>
<td width="151">1/6</td>
<td width="151">4/6 = 0.667</td>
</tr>
<tr>
<td width="102"><strong>5</strong></td>
<td width="151">1/6</td>
<td width="151">5/6 = 0.833</td>
</tr>
<tr>
<td width="102"><strong>6</strong></td>
<td width="151">1/6</td>
<td width="151">6/6 = 1</td>
</tr>
</tbody>
</table>
<img class="aligncenter size-full wp-image-19270" src="https://financetrain.com/wp-content/uploads/dpd.gif" alt="dpd" width="402" height="298">

Let’s observe a few values from the above table.

P(3) = 1/6 or 0.1667

F(3) = 0.5

P(2&lt;=X&lt;=5) = 4*1/6 = 0.667

We can generalize this as follows:

P(x) = 1/6 or 0.1667

Cumulative distribution function for any outcome i is F(xi) = i*P(x)

Probability function for a range with k outcomes = k*P(x)

Discrete Uniform Random Variable

A Bernoulli random variable is a <a title="What is a Random Variable?" href="https://financetrain.com/what-is-a-random-variable/">random variable</a> that takes a value of 1 in case of a success and a value of 0 in case of a failure. We can also say that this random variable has a Bernoulli distribution. A classic example is a <strong>single</strong> toss of coin. When we toss a coin, the outcome can be heads (success) with a probability p or tails (failure) with a probability of (1 – p). The important point here is a single toss of coin.

Now suppose we perform n number of trials. Each trial is independent and will result in a success with a probability p or a failure with a probability (1-p). From the n trials, suppose X represents the number of successes. Then X is a binomial random variable with parameters (n, p). Note that Bernoulli random variable is a special case of binomial random variable with parameters (1, p). The variable X will have a binomial distribution.

The binomial distribution has the following characteristics:
<ul>
 	<li>For each trial there are only two possible outcomes, success or failure.</li>
 	<li>Probability of success, p, of each trial is fixed.</li>
 	<li>There are n trials.</li>
 	<li>Each trial is independent</li>
 	<li>The binomial probability function defines the probability of x successes from n trials.</li>
</ul>
The binomial probability function is given using the following formula.

<img class="aligncenter size-full wp-image-19275" src="https://financetrain.com/wp-content/uploads/bd12.gif" alt="bd1" width="525" height="58">

Let’s take an example to understand how this can be applied.

You have a pool of stocks having returns either <em>above 5%</em> or <em>below 5%</em>. The probability of selecting a stock with above 5% returns is 0.70. You are going to pick up 5 stocks. Assuming binomial distribution, what is the probability of picking 2 stocks with above 5% returns?

Let’s define our problem.

Success = Pick stock with above 5% returns

p = 0.70

n = 5

x = 2

<img class="aligncenter size-full wp-image-19274" src="https://financetrain.com/wp-content/uploads/bd22.gif" alt="bd2" width="518" height="56">

<strong>Expected Value and Variance of a Binomial Distribution</strong>

For a binomial distribution, the <a title="Expected Value of Investments" href="https://financetrain.com/expected-value-investments/">expected value</a> and variance are given as below:

<img class="aligncenter size-full wp-image-19273" src="https://financetrain.com/wp-content/uploads/bd3.gif" alt="bd3" width="554" height="196">

Bernoulli and Binomial Distribution

The future price movement of a stock can be approximated using a <a title="Constructing Binomial Tree to Describe Stock Price Movement" href="https://financetrain.com/constructing-binomial-tree-to-describe-stock-price-movement/">binomial tree</a>.

Let’s say the current stock price is S. The price of the stock can either move up or move down. We will refer to the up movement as u and the down movement as d. A down movement d will then be given by d where d is assumed to be equal to 1/u. We also need the probability of an up move (p) and the probability of a down move (1 - p).

The following diagram shows the price movement in a small time period Dt.

<img class="aligncenter size-full wp-image-19480" src="https://financetrain.com/wp-content/uploads/bt13.gif" alt="bt1" width="229" height="133">

From each node at Su and Sd, the prices can again take two possible paths.
<ul>
 	<li>Su can move to Suu or Sud. Note that Sud will be the same as S (S*u*1/u).</li>
 	<li>Sd can move to Sdu (or just S) or Sdd.</li>
</ul>
This way a binomial tree can be built for multiple periods. The following diagram shows the binomial tree for 4 periods.

<img class="aligncenter size-full wp-image-19481" src="https://financetrain.com/wp-content/uploads/bt21.gif" alt="bt2" width="455" height="366">

Let’s take an example to understand these values.

Initial stock price, S = $20

u = 1.02

d = 1/1.02

Probability of up move, p = 0.60

Probability of down move, (1-p) = 0.40

The possible stock values for period 1:
<ul>
 	<li>Su = 20*1.02 = 20.40 with a probability of 0.60</li>
 	<li>Sd = 19.60 with a probability of 0.40</li>
</ul>
The possible stock values for period 2:
<ul>
 	<li>Suu = 20.81 with a probability of 0.60*0.60 = 0.36</li>
 	<li>S = $20 with a probability of 2*0.60*0.40 = 0.48</li>
 	<li>Sdd = 19.22 with a probability 0.40*0.40 = 0.16</li>
</ul>
Similarly we can calculate the values for period 3 and period 4. Binomial trees have their application in pricing options and other financial concepts.

The following diagram shows this binomial tree upto two periods.

<img class="aligncenter size-full wp-image-19482" src="https://financetrain.com/wp-content/uploads/bt3.gif" alt="bt3" width="449" height="294">

Stock Price Movement Using a Binomial Tree

Tracking error is a measure of how closely a portfolio follows its benchmark. A tracking error of zero means that the portfolio exactly follows its benchmark. The benchmark could be an index such as S&amp;P 500 index. Let’s say the S&amp;P 500 index provides a return of 6% and your portfolio tracking the S&amp;P 500 index earns 4% returns. The tracking error is calculated as follows:

<img class="aligncenter size-full wp-image-19485" src="https://financetrain.com/wp-content/uploads/tr1.gif" alt="tr1" width="547" height="149">

In our example, the tracking error will be:

<strong>Tracking error = 4% - 6% = -2%</strong>

Morningstar defines tracking error as trailing returns.

<strong>Tracking Risk:</strong> Tracking error sometimes also refers to tracking risk, which is the standard deviation of returns of the portfolio to benchmark returns over a period of time. This is a more commonly used method of calculating tracking error.

<img class="aligncenter size-full wp-image-19486" src="https://financetrain.com/wp-content/uploads/tr2.gif" alt="tr2" width="558" height="118">

Tracking error is an important measure for investors to know how well the portfolio is replicating the index. A low tracking error indicates the portfolio is closely following the benchmark. A high tracking error means the portfolio is moving away from the benchmark index. Investors desire a low tracking error.

Tracking Error and Tracking Risk

We know that a discrete uniform random variable is a discrete random variable for which the probability of each outcome is the same. We also know that a random variable is continuous if it can take an infinite number of values between the possible values for the random variable.

Keeping these two characteristics in mind, we can say that a continuous random variable will have a uniform distribution if all the possible values in the given range have a uniform or same probability.

A continuous uniform distribution has the following characteristics.
<ul>
 	<li>It has an infinite number of possible values between two possible values a and b.</li>
 	<li>The probability for each outcome, P(X=x) or p(x) = 0</li>
 	<li>The probability of an outcome outside the range is 0.</li>
 	<li>The probability of an outcome being between two values n1 and n2 is simply the ratio of two ranges.</li>
</ul>
<strong>P(n1 &lt;= X &lt;= n2) = (x2-x1)/(b-a)</strong>

<strong>Example</strong>

If X is uniformly distributed between 0 and 100, it can be represented as follows:

<img class="aligncenter  wp-image-19519" src="https://financetrain.com/wp-content/uploads/cud.png" alt="cud" width="408" height="246">

Let’s say you want to calculate the probability of the value falling between 20 and 40. This will be calculated as:

<strong>P(20 &lt;=X &lt;= 40) = (40 – 20)/(100 – 0) = 0.20</strong>

This is shown below:

<img class="aligncenter  wp-image-19520" src="https://financetrain.com/wp-content/uploads/cud2.png" alt="cud2" width="419" height="253">

Since each possibility has an equal probability of occurring, the cumulative uniform distribution function will be a straight line, as shown below:

<img class="aligncenter  wp-image-19521" src="https://financetrain.com/wp-content/uploads/cud3.png" alt="cud3" width="314" height="276">

Continuous Uniform Distribution

The <a title="Understanding Normal Distribution" href="https://financetrain.com/understanding-normal-distribution/">normal distribution</a> is the well-known bell-shaped curve depicted below. The bell-shaped curve comes from a statistical tendency for outcomes to cluster symmetrically around the mean (or average).

Deviations from the mean are described in terms of standard deviations. In all normal distributions, 68% of outcomes will fall within 1 standard deviation to either side of the mean.

Let’s illustrate the concept of mean and standard deviation with a simple example. My New York subway commute every day is 30 minutes on average, with a standard deviation of 5 minutes.

Assuming a normal distribution for the time it takes me to get to work, this would imply that:
<ul>
 	<li>68% of the time, I can expect my daily commute to be between 25 minutes and 35 minutes (i.e., the mean of 30 minutes plus or minus 1 standard deviation, or 5 minutes).</li>
</ul>
<ul>
 	<li>16% of the time, my commute is less than 25 minutes (because the normal distribution is symmetrical around the mean, I expect this event to occur 16% of the time, or (100%-68%)/2).</li>
</ul>
<ul>
 	<li>16% of the time, my commute is greater than 35 minutes (again, because the normal distribution is symmetrical). Or, in other words, my 84% confidence level worst-case commute is 35 minutes (Only 16% of the time I would expect longer commute).</li>
</ul>
From this example, it makes sense that the more standard deviations we move from the mean, the lower the probability is of such an event occurring. For example, a delay of 10 minutes or more (2 standard deviations) only has a 2.5% chance of occurring, compared to a 16% probability of a delay of 5 minutes or more (1 standard deviation).

The table below relates standard deviations to lower tail probabilities (lower tail probabilities quantify the chance of an event of that magnitude or greater occurring):
<table width="565">
<tbody>
<tr>
<td width="177"><strong>Standard Deviations</strong></td>
<td width="180"><strong>Lower Tail Probability</strong></td>
<td width="208"><strong>Commuting Example</strong></td>
</tr>
<tr>
<td width="177"><strong>1</strong></td>
<td width="180">16%</td>
<td width="208">Delay of 5 minutes or more</td>
</tr>
<tr>
<td width="177"><strong>1.28</strong></td>
<td width="180">10</td>
<td width="208">Delay of 6.4 minutes or more</td>
</tr>
<tr>
<td width="177"><strong>1.65</strong></td>
<td width="180">5</td>
<td width="208">Delay of 8.25 minutes or more</td>
</tr>
<tr>
<td width="177"><strong>2</strong></td>
<td width="180">2.5</td>
<td width="208">Delay of 10 minutes or more</td>
</tr>
<tr>
<td width="177"><strong>2.33</strong></td>
<td width="180">1</td>
<td width="208">Delay of 11.65 minutes or more</td>
</tr>
</tbody>
</table>
<strong>Characteristics of Normal Distribution</strong>
<ul>
 	<li>The distribution is characterized by bell curve which has more weight in the center and tapers off on either side which means it has tails on either side. It takes only two moments i.e. the mean µ and the variance σ<sup>2</sup>&nbsp;to describe this function and is therefore a parametric function.</li>
 	<li>The mean gives more information about the location and the variance gives an idea of how dispersed the values are.</li>
 	<li>A normal distribution has a skewness=0 and kurtosis=3. The excess kurtosis is 0.</li>
 	<li>A linear combination of 2 or more normally distributed random variables is also normally distributed. This means if X and Y are normally distributed then X + Y is also normally distributed.</li>
</ul>
The density function for a normal distribution is given using the following formula:

<img class="aligncenter size-full wp-image-19525" src="https://financetrain.com/wp-content/uploads/nd21.gif" alt="nd2" width="515" height="123">

Normal Distribution

A <strong>univariate distribution</strong> refers to the distribution of a single <a title="What is a Random Variable?" href="https://financetrain.com/what-is-a-random-variable/">random variable</a>. Note that the above characteristics we saw of a normal distribution are for the distribution of one normal random variable, representing a univariate distribution.

On the other hand, a <strong>multivariate distribution</strong> refers to the probability distribution of a group of random variables. For example, a multivariate <a title="Normal Distribution" href="https://financetrain.com/normal-distribution/">normal distribution</a> is used to specify the probabilities of returns of a group of n stocks. This has relevance because the returns of different stocks in the group influence each other’s behaviour, that is, the behaviour of one random variable in the group is influenced by the behaviour of another variable.

<strong>How to Construct Multivariate Distribution?</strong>

For discrete random variables, joint probabilities are used to describe the multivariate distribution

For continuous random variables, if each random variable follows a normal distribution, a multivariate normal distribution is created. Remember that a linear combination of 2 or more normally distributed random variables is also normally distributed.

If we want to describe the multivariate normal distribution of the returns of a group of stocks, we need the following three parameters:
<ul>
 	<li>List of means returns of each stock</li>
 	<li>List of variances of returns of each stock</li>
 	<li>List of correlations between each pair of stocks.</li>
</ul>
A univariate normal distribution is described using just the two variables namely mean and variance. For a multivariate distribution we need a third variable, i.e., the correlation between each pair of random variables. This is what distinguishes a multivariate distribution from a univariate distribution. If there are n random variables in the group, we will have n*(n-1)/2 pairs of correlations.

Univariate Vs. Multivariate Distribution

A confidence interval is an interval in which we expect the actual outcome to fall with a given probability (confidence). Consider the following statement:

<strong><em>In a normal distribution, 68% of the values fall within 1 standard deviation of the mean. </em></strong>

So, if X is a normal random variable, the 68% confidence interval for X is -1s &lt;= X &lt;= 1s. This is demonstrated in the following diagram.

<img class="aligncenter size-full wp-image-19530" src="https://financetrain.com/wp-content/uploads/ci1.gif" alt="ci1" width="568" height="334">

The four commonly used confidence intervals for a normal distribution are:
<ul>
 	<li>68% of values fall within 1 standard deviation of the mean (-1s &lt;= X &lt;= 1s)</li>
 	<li>90% of values fall within 1.65 standard deviations of the mean (-1.65s &lt;= X &lt;= 1.65s)</li>
 	<li>95% of values fall within 1.96 standard deviations of the mean (-1.96s &lt;= X &lt;= 1.96s)</li>
 	<li>99% of values fall within 2.58 standard deviations of the mean (-2.58s &lt;= X &lt;= 2.58s)</li>
</ul>
The confidence interval is generally represented as , where n is the number of standard deviations. For example, n=1.65 for 90% confidence interval.

<strong>Example</strong>

A stock portfolio has mean returns of 10% per year and the returns have a standard deviation of 20%. The returns are normally distribution. Calculate the 99% confidence interval.

<strong>95% confidence interval = 10% +/- 2.58*20%.</strong>

The confidence interval is -41.6% to 61.6%. This means with 99% confidence, the returns will range from -41.6% to 61.6%.

Confidence Intervals for a Normal Distribution

A <a title="Normal Distribution" href="https://financetrain.com/normal-distribution/">normal distribution</a> can be described using just two parameters, namely (μ), mean and variance (σ<sup>2</sup>).

In a normal distribution, these two variables could take any value. For example, for a normally distributed stock portfolio, the mean could be 10% and the standard deviation could be 20%.

A standard normal distribution is a standardized form of normal distribution with a mean μ = 0 and standard deviation σ = 1.

We can standardize any normal random variable, by computing a z-score for it. z-scores make it easier to compare data values measured on different scales. A z-score reflects how many standard deviations above or below the mean a raw score is. The z-score is positive if the data value lies above the mean and negative if the data value lies below the mean. Z-score is represented using the following formula:

<img class="aligncenter size-full wp-image-19565" src="https://financetrain.com/wp-content/uploads/snd1.gif" alt="snd1" width="555" height="113">

Where x represents the observation, m is the population mean, and s is the standard deviation.

Suppose the dividends paid by a company every year are normally distributed with a mean of $10 and a standard deviation of $2. If the company pays a dividend of $14 this year, what will be its z-score?

<img class="aligncenter size-full wp-image-19564" src="https://financetrain.com/wp-content/uploads/snd2.gif" alt="snd2" width="541" height="58">

A z-score of 2 indicates that the current dividends are 2 standard deviations above the mean.

Standard Normal Distribution

Once we have the z-scores, we can use the standard normal table to calculate the probabilities. The standard normal table shows the area (as a proportion, which can be translated into a percentage) under the standard normal curve corresponding to any Z-score or its fraction, i.e., the probability of observing a z-value that is less than the given value.

The following z-table shows the probabilities for z values ranging between – 3 and 3.

<img class="aligncenter  wp-image-19569" src="https://financetrain.com/wp-content/uploads/cp1.png" alt="cp1" width="634" height="605">

In the above table, F(Z) is the probability that a variable from a standard normal distribution will be less than or equal to Z.

<strong>Example 1: Finding Probability</strong>

Let’s continue with our earlier example. The dividends paid by a company every year are normally distributed with a mean of $10 and a standard deviation of $2. If the company pays a dividend of $14 this year, its z-score is 2 indicating that the current dividends are 2 standard deviations above the mean.

We can look up the z-score of 2 in the above table. Look for a z-score in the first column, and note the corresponding F(Z) value. For a z-score of 2, F(Z) = 0.9772. This means that 97.72% of observations lie below a z-score of 2 (2 standard deviations). We can also interpret this as – there is a 97.72% probability that the dividends paid by the company will be below $14.

<img class="aligncenter  wp-image-19570" src="https://financetrain.com/wp-content/uploads/cp2.png" alt="cp2" width="471" height="202">

<strong>Example 2: Finding Probability</strong>

Compute p (-1.2 &lt; Z &lt; 0.80)

We are required to calculate the probability of z being between -1.2 and 0.80. This is demonstrated below:

<img class="aligncenter size-full wp-image-19571" src="https://financetrain.com/wp-content/uploads/cp3.png" alt="cp3" width="488" height="204">

From the table, we can observe that p(z=0.80) = 0.7881 and p(z=-1.2) = 0.1151

<img class="aligncenter size-full wp-image-19572" src="https://financetrain.com/wp-content/uploads/cp4.gif" alt="cp4" width="526" height="36">

Calculating Probabilities Using Standard Normal Distribution

Shortfall risk refers to the probability that a portfolio will not exceed the minimum return level (target return; benchmark return). Shortfall-risk is more consistent with the investors’ intuitive perception of risk in that it focusses more on the real economic risk of an investor. On the other hand, the standard deviation is rather a measure of the volatility of financial assets. Shortfall risk is also known as downside risk.

The following diagram illustrates shortfall risk. As we can see, it is the risk that the portfolio value will fall below some minimum acceptable level, RL.

<img class="aligncenter size-full wp-image-19576" src="https://financetrain.com/wp-content/uploads/sfr.gif" alt="sfr" width="327" height="188">

Shortfall Risk

Roy’s Safety first criterion states that the optimal portfolio minimizes the probability that portfolio return, RP, falls below RL.

According to this criteria, we select the portfolio with the highest safety first ratio (SFRatio).

<img class="aligncenter size-full wp-image-19579" src="https://financetrain.com/wp-content/uploads/safety.gif" alt="safety" width="525" height="60">

<strong>Example</strong>

An investor has a minimum threshold of 2%. He has to make a choice between two portfolios A and B.
<table>
<tbody>
<tr>
<td width="92"><strong>&nbsp;</strong></td>
<td width="142"><strong>Portfolio A</strong></td>
<td width="151"><strong>Portfolio B</strong></td>
</tr>
<tr>
<td width="92"><strong>E(R)</strong></td>
<td width="142">10%</td>
<td width="151">15%</td>
</tr>
<tr>
<td width="92"><strong>s</strong></td>
<td width="142">20%</td>
<td width="151">25%</td>
</tr>
</tbody>
</table>
&nbsp;

According to Roy’s Safety-first criteria, which portfolio should he choose?

SFRatio of Portfolio A = (10-2)/20 = 0.40

SFRatio of Portfolio B = (15-2)/25 = 0.52

According to safety-first criteria, we should select portfolio B. Minimum threshold is 0.40 standard deviations away for portfolio A while it is 0.52 standard deviations away for portfolio B.

Safety-first Ratio

The concept of lognormal distribution is very closely related to the concept of normal distribution.

Let’s say we have a random variable Y. This variable Y will have a lognormal distribution if the natural log of Y (ln Y) is normally distributed. So, we check if the natural logarithm of a <a title="What is a Random Variable?" href="https://financetrain.com/what-is-a-random-variable/">random variable</a> is <a title="Normal Distribution" href="https://financetrain.com/normal-distribution/">normally distributed</a> or not. If it is, then the random variable itself will have a lognormal distribution.

A <a title="Properties of Log-Normal Distribution" href="https://financetrain.com/properties-of-log-normal-distribution/">lognormal distribution</a> has two important characteristics:
<ul>
 	<li>It has a lower bound of zero.</li>
 	<li>The distribution is skewed to the right, i.e., it has a long right tail.</li>
</ul>
Note that this is in contrast with a normal distribution which has zero skew and can take both negative and positive values. Just like a normal distribution, a lognormal distribution is also described by just two parameters, namely, m and s.

A lognormal distribution is commonly used to describe distributions of financial assets such as share prices. A lognormal distribution is more suitable for this purpose because asset prices cannot be negative. An important point to note is that when the continuously compounded returns of a stock follow normal distribution, then the stock prices follow a lognormal distribution. Even in cases where returns do not follow a normal distribution, stock prices are better described by a lognormal distribution.

Consider the expression Y = exp(X).

Exp(X) or e<sup>x </sup>is the opposite of taking logs. If we take log on both side, we will have ln y = X

So, if we assume that X has normal distribution, then Y has lognormal distribution (because ln Y is normally distributed).

We can compare this with how stock prices move. Let’s say that the initial stock price is S<sub>0</sub> and the stock price after period t is S<sub>t</sub>. If the rate of return r is continuously compounded then the future stock price can be expressed as:

S<sub>t</sub> = S<sub>0</sub>*EXP(r)

S<sub>0</sub> is a known quantity and is a constant. This expression is the same as Y = exp(X).

Therefore, if r is normally distributed, the stock price will be lognormally distributed.

Lognormal Distribution and Stock Prices

A discretely compounded rate of return is simply a compounded rate of return with a discrete compounding frequency such as daily, monthly, quarterly, or semi-annually.

As the frequency of compounding increases, the annual effective yield also increases because the interest or income earned is compounded more frequently.

<strong>Example</strong>

Suppose an investment grows at an annual rate of 10% compounded quarterly. At the end of one year, the investment will grow to:

<img class="aligncenter size-full wp-image-19585" src="https://financetrain.com/wp-content/uploads/dr1.gif" alt="dr1" width="519" height="33">

The effective annual yield is given as:

<img class="aligncenter size-full wp-image-19584" src="https://financetrain.com/wp-content/uploads/dr2.gif" alt="dr2" width="503" height="52">

Discretely Compounded Rate of Return

In contrast to <a title="Discretely Compounded Rate of Return" href="https://financetrain.com/discretely-compounded-rate-return/">discrete compounding</a>, continuous compounding means that the returns are compounded continuously. The frequency of compounding is so large that it reaches infinity. These are also called log returns.

Suppose the rate of return is 10% per annum. The effective annual rate on a continuously compounded basis will be:

<strong>Effective Annual Rate = e<sup>r</sup> – 1</strong>

<strong>=e^0.10 – 1</strong>

<strong>=10.517%</strong>

This means that if 10% was continuously compounded, the effective annual rate will be 10.517%.

We can also perform the reverse calculations. If a portfolio earned 10.517% in one year, then what would be the equivalent continuously compounded rate?

It will be ln(1+r) = ln (1.10517) = 10%

Let’s take another example. An investment of $100 grows to $130 in one year. What is the continuously compounded rate of return?

We can calculate it in two ways. First calculate the return earned, i.e., (130 -100)/100 = 30%. 30% is the holding period return.

The continuously compounded rate of return = ln(1.30) = 26.23%

Alternatively you can simply calculate ln(130/100) = 26.23%

An important property of continuously compounding rates is that they are additive.

Continuously Compounded Rate of Return

<p>The fundamental assumption of the Historical Simulations methodology is that you base your results on the past performance of your portfolio and make the assumption that the past is a good indicator of the near-future. The following is a comparison of historical simulation with Monte Carlo simulation on various factors.</p>



<figure class="wp-block-table"><table><tbody><tr><td>&nbsp;</td><td>Historical Simulation</td><td>Monte Carlo Simulation</td></tr><tr><td><strong>General</strong></td><td>Estimates prices by reliving history; we take actual historical rates and revalue a the asset each change in the market</td><td>Estimates prices by simulating random scenarios.</td></tr><tr><td><strong>Use</strong></td><td>Appropriate for all types of instruments, linear or non-linear</td><td>Appropriate for all types of instruments, linear or nonlinear</td></tr><tr><td><strong>Distribution of risk factors</strong></td><td>The historical simulation method replicates the actual distribution of risk factors.</td><td>Monte-Carlo simulation is general in nature. &nbsp;</td></tr><tr><td><strong>Distribution Assumptions</strong></td><td>No need to make distributional assumptions</td><td>You can use various distributional assumptions (normal, T-distribution, and so on)</td></tr><tr><td><strong>Possibility of extreme events happening</strong></td><td>In the case of historical simulation the possibility of extreme events happening is only more relevant if it happened in recent history.</td><td>Monte-Carlo method due to its complete random nature accounts for these events completely.</td></tr><tr><td><strong>Disadvantage</strong></td><td>You need a significant amount of daily rate history (at least a year, preferably much more) You need significant computational power for revaluing the portfolio under each scenario.</td><td>Takes a lot of computational power (and hence a longer time to estimate results)</td></tr></tbody></table></figure>

Historical Simulation Vs Monte Carlo Simulation

This quiz is a part of the CFA Level 1 reading 'Common Probablity Distributions'. 

Common Probablity Distributions

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