The normal distribution 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.

<img class="aligncenter size-full wp-image-1134" title="normald" src="https://financetrain.com/wp-content/uploads/2010/10/normald.png" alt="" width="567" height="326" />

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 (for example, 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 border="1" cellspacing="0" cellpadding="0">
<tbody>
<tr>
<td width="197" valign="top"><strong>Standard Deviations</strong></td>
<td width="197" valign="top"><strong>Lower Tail   Probability</strong></td>
<td width="197" valign="top"><strong>Commuting Example</strong></td>
</tr>
<tr>
<td width="197" valign="top">1</td>
<td width="197" valign="top">16%</td>
<td width="197" valign="top">Delay of 5 minutes   or more</td>
</tr>
<tr>
<td width="197" valign="top">1.28</td>
<td width="197" valign="top">10</td>
<td width="197" valign="top">Delay of 6.4   minutes or more</td>
</tr>
<tr>
<td width="197" valign="top">1.65</td>
<td width="197" valign="top">5</td>
<td width="197" valign="top">Delay of 8.25   minutes or more</td>
</tr>
<tr>
<td width="197" valign="top">2</td>
<td width="197" valign="top">2.5</td>
<td width="197" valign="top">Delay of 10 minutes or more</td>
</tr>
<tr>
<td width="197" valign="top">2.33</td>
<td width="197" valign="top">1</td>
<td width="197" valign="top">Delay of 11.65   minutes or more</td>
</tr>
</tbody>
</table>

Understanding Normal Distribution

Many financial calculations and estimations require a statistical analysis of the variability of past market returns. Because we have strong evidence that market returns are approximately normally distributed, we can estimate potential market movements with a given probability using two simple parameters: mean (or average return) and standard deviation (or volatility).

<strong>Mean</strong>

Mean describes where the distribution is centered, and standard deviation describes dispersion from the mean, or uncertainty (for example, the width of the distribution). Conveniently, we only need to estimate the standard deviation for short-term risk prediction, because mean returns tend to be close to 0 (zero).

<strong>Standard Deviation</strong>

From a risk perspective, the most important variable that defines the normal distribution is the standard deviation, because it measures uncertainty. The larger the standard deviation, the larger the uncertainty, or risk.

For example, we could say that an asset with a standard deviation of 10% is twice as risky as an asset with a standard deviation of 5%.

Multiples of standard deviation can be used to estimate lower tail probabilities of loss. Lower tail probabilities of loss refer to the chance of losses exceeding a specified amount. For instance, there is a 16% chance of losses exceeding 1 standard deviation from the mean (we know this because 1 standard deviation included 68% of returns to either side of the mean of a normal distribution, leaving 32%/2 = 16% probability in each tail). Because returns tend to cluster around the mean, larger standard deviation moves have a lower probability of occurring.

We use confidence level multipliers to arrive at the tail probability of loss levels and implied VaR confidence levels (Confidence levels are 1- lower tail probability of loss)
<table border="1" cellspacing="0" cellpadding="0">
<tbody>
<tr>
<td width="197" valign="top"><strong>Confidence Level   Multiplier</strong></td>
<td width="197" valign="top"><strong>Lower Tail   Probability of Loss</strong></td>
<td width="197" valign="top"><strong>Implied VaR   Confidence Level</strong></td>
</tr>
<tr>
<td width="197" valign="top">1</td>
<td width="197" valign="top">16%</td>
<td width="197" valign="top">84%</td>
</tr>
<tr>
<td width="197" valign="top">1.28</td>
<td width="197" valign="top">10</td>
<td width="197" valign="top">90</td>
</tr>
<tr>
<td width="197" valign="top">1.65</td>
<td width="197" valign="top">5</td>
<td width="197" valign="top">95</td>
</tr>
<tr>
<td width="197" valign="top">2</td>
<td width="197" valign="top">2.5</td>
<td width="197" valign="top">97.5</td>
</tr>
<tr>
<td width="197" valign="top">2.33</td>
<td width="197" valign="top">1</td>
<td width="197" valign="top">99</td>
</tr>
</tbody>
</table>

Statistical Foundations: Mean and Standard Deviation

Whereas standard deviation shows how risky individual assets are, correlations show how asset risks are interrelated. Correlations are calculated by observing historical comovement in returns and range between –1 and 1.

A correlation of 1 means that returns move together perfectly, whereas a correlation of -1 implies perfect opposite movement. A 0 (zero) correlation implies independence. We generally observe positive correlations within an asset class (for example, equities) and between the major assets classes (for example, stocks and bonds); however, FX prices and commodity prices tend to have lower correlations to other asset classes.

<img class="aligncenter size-full wp-image-1144" title="correlation" src="https://financetrain.com/wp-content/uploads/2010/10/correlation.gif" alt="" width="402" height="502" />

<strong>Positive correlation (75%):</strong> These stocks generally move in the same direction

<strong>Negative Correlation (75%):</strong> These stocks generally move opposite each other

<strong>Independent:</strong> There is no apparent relationship between these stocks r.
<h3>Correlations Affect Diversification</h3>
Portfolio managers look for assets with low (or even negative) correlations to achieve better diversification.

Correlations are dynamic and often change during volatile market conditions, which may significantly affect portfolio risk.
<h3>Correlation vs. Causality</h3>
It is important to distinguish between correlation and causality. Causality is result of one event precipitating another event (for example, Thai baht devaluation precipitating an Asian market meltdown), whereas correlation is a statistical measure of observed comovement over a period of time.
<h3>Correlations are Dynamic</h3>
Correlations can easily change from positive to negative. For example, while we tend to see positive correlations between stocks and bonds (because rising interest rates generally have a negative effect on stock prices and vice versa), you can see that this relationship is not stable. Note in particular how correlation turned negative in October 1997, when there was a high stock market volatility (may be due to flight to safety--people exiting stocks in favor of bonds).

Statistical Foundations: Understanding Correlations

Following are the major steps we take to estimate volatility, using the S&amp;P 500 Index:
<ul>
	<li>Let's look at historical market rates. We start by converting daily prices into log changes. (Daily log changes are conceptually similar to percentage returns, except they are continuously compounded.)</li>
</ul>
<ul>
	<li>Our goal is to use past returns to predict the volatility of future returns. We can plot changes in market rates onto a histogram and fit a normal distribution. You can see here that the normal distribution is a reasonable but not perfect fit for stock returns. We expect this distribution of returns to stay reasonably stable over time. However, every day we re-estimate the standard deviation of the distribution to predict tomorrow's volatility (hence our predicted distribution of returns changes each day).</li>
</ul>
<ul>
	<li>By re-estimating volatilities every day, we      can get a dynamic estimate of risk. Observe the dynamic nature of      risk--several periods of high and low volatility. We expect approximately      5% excessions on both the upside and the downside. Actual daily excessions      over this 5-year period were approximately 6% on the upside and 4% on the      downside.</li>
</ul>

Statistical Foundations: Predicting Volatility

We will now learn how to calculate the VaR of one position and two positions by applying the concept of volatilities and correlations.

The following examples of how to calculate the risk of one and two positions illustrate the basic concept of parametric (delta) VaR estimation for linear instruments.

The general steps for calculating VaR are:

<strong>Step 1: </strong>Set VaR parameters: probability of loss and confidence level, time horizon, and base currency.

<strong>Step 2: </strong>Determine market value of each position, in base currency.

<strong>Step 3: </strong>Calculate VaR of individual positions, given market volatilities.

<strong>Step 4: </strong>Calculate portfolio VaR, given correlations between all variables.

VaR can be estimated as follows:

<strong>VaR = Market value X Price volatility</strong>

We will generally use the term volatility to express a multiple of standard deviation, depending on our chosen confidence level for VaR (i.e., confidence level multiple X standard deviation). Therefore, if we are using a VaR confidence level of 95%, volatility will refer to 1.65 X standard deviation.

Parametric VaR Estimation

<strong>BetaSquare are a USD-based firm with one asset: JPY 14 billion in cash. What is the 95% worst-case loss over a 1-day period?</strong>

You have the following information:
<ul>
	<li>The daily price volatility of the JPY/USD exchange rate is 1.78%, using a 95% confidence level. (Note: This implies that 1 standard deviation equals 1.78%/1.65 = 1.08%.)</li>
</ul>
<ul>
	<li>The JPY/USD exchange rate is 140.</li>
</ul>
<strong>Solution:</strong>
<table border="1" cellspacing="0" cellpadding="0">
<tbody>
<tr>
<td width="197" valign="top"><strong>Step</strong></td>
<td width="197" valign="top"><strong>Calculation</strong></td>
<td width="197" valign="top"><strong>Comment</strong></td>
</tr>
<tr>
<td width="197" valign="top">1. Choose a probability of   loss</td>
<td width="197" valign="top">5%</td>
<td width="197" valign="top">=95% confidence worst-case loss</td>
</tr>
<tr>
<td width="197" valign="top">2. Measure value in USD</td>
<td width="197" valign="top">$100 million</td>
<td width="197" valign="top">Assuming 140 JPY/USD</td>
</tr>
<tr>
<td width="197" valign="top">3. 1-day volatility JPY/USD   (or 1.65 standard deviations)</td>
<td width="197" valign="top">1.78%</td>
<td width="197" valign="top">Data Set</td>
</tr>
<tr>
<td width="197" valign="top">4. Calculate Risk</td>
<td width="197" valign="top">$100 million * 1.78 =

$1.78</td>
<td width="197" valign="top">Market value volatility</td>
</tr>
</tbody>
</table>
This means that your 95% worst-case loss due to adverse movements in the JPY IUSD over 1 day would be $1.78 million (or, you have a 5% chance of losing $1.78 million or more overnight). Now let's view an example of two cash positions. To calculate the total risk of two or more positions, however, we need to include correlations.

Risk of a Single Cash Position

<strong>You now have two assets: JPY 1 billion + THB 4 billion. What is the risk over a 1-day period?</strong>

<strong>Solution</strong>
<table border="1" cellspacing="0" cellpadding="0">
<tbody>
<tr>
<td width="197" valign="top"><strong>Step</strong></td>
<td width="197" valign="top"><strong>Calculation</strong></td>
<td width="197" valign="top"><strong>Comment</strong></td>
</tr>
<tr>
<td width="197" valign="top">1. Measure value in USD</td>
<td width="197" valign="top">$200 million</td>
<td width="197" valign="top">140 JPY/USD and 40 THB/USD</td>
</tr>
<tr>
<td width="197" valign="top">2. 1-month volatility</td>
<td width="197" valign="top">JPY/USD 1.78%

THB/USD 1.96%</td>
<td width="197" valign="top"></td>
</tr>
<tr>
<td width="197" valign="top">3. What are risks?/$</td>
<td width="197" valign="top">On JPY $ 1.78 million

On THB $ 1.96 million</td>
<td width="197" valign="top">Data Set</td>
</tr>
<tr>
<td width="197" valign="top">4. What is total risk?</td>
<td width="197" valign="top">$ 1.78 + $ 1.96 m = $ 3.74   million</td>
<td width="197" valign="top">This is incorrect you cannot   just add the risks!</td>
</tr>
</tbody>
</table>
Simply adding risks together is incorrect: This would assume that both JPY and THB would move together in perfect lockstep. To calculate the total risk of two or more positions, we need to know correlations between all variables.

We will now show you how it is actually calculated.

<strong> </strong>

<strong>Risk of Two Cash Position (With Correlation)</strong>

The data set shows a correlation of 55% between JPY/USD and THB/USD.

Because the correlation is less than 1, we expect the total risk of our JPY and THB positions to be less than $3.74 million.
<table border="1" cellspacing="0" cellpadding="0">
<tbody>
<tr>
<td width="295" valign="top">Undiversified risk</td>
<td width="295" valign="top">$3.74. million</td>
</tr>
<tr>
<td width="295" valign="top">Diversification benefit</td>
<td width="295" valign="top">-$0.45 million</td>
</tr>
<tr>
<td width="295" valign="top">Net risk</td>
<td width="295" valign="top">$3.29 million</td>
</tr>
</tbody>
</table>
<img class="aligncenter size-full wp-image-1160" title="risk-two-assets" src="https://financetrain.com/wp-content/uploads/2010/10/risk-two-assets.png" alt="" width="412" height="147" />The diversification shows the difference between net portfolio risk and gross risk assuming perfect correlation (i.e., net portfolio risk minus gross risk). We can calculate the risk of two linear positions using the following formula:

Risk of Two Cash Positions

Discover how to scale standard deviation to different time horizons. We know that risk increases with time: The longer we hold a position, the greater the potential loss. Following is a simple approximation to help you scale volatility estimates to a longer (or shorter) -time horizon. Note, however, that this is just an approximation.

Volatility (or standard deviation) may be roughly approximated by scaling by the square root of time, assuming independent price moves. Note that we use the number of trading days (5 for 1 week, 21 for 1 month), as opposed to actual days to scale volatility.

<img class="aligncenter size-full wp-image-1164" title="time-scaling1" src="https://financetrain.com/wp-content/uploads/2010/10/time-scaling1.png" alt="" width="325" height="75" />

<strong>Square Root of Time Scaling Rule</strong>

Intuitively, we can picture the square root of time scaling rule as follows: Imagine Zeus flipping coins every day--if heads come up, the stock market goes up, if tails come up, the stock market falls. Assuming 1 % daily volatility (or standard deviation) in the markets, what would you expect weekly volatility to be?

We could just guess and multiply 1 % by 5 to get 5%. However, it is unlikely (1/32 chance) that we'll have 5 bad days in a row, because every day's coin flip is independent--it has no memory of the past. An accurate prediction of a 5-day risk would 1 % * square root of 5 = 2.24% (but only if Zeus is not biased!).

<strong>Assumptions</strong>

Square root of time scaling assumes independent price moves and constant volatility:
<ul>
	<li>If there is significant mean reversion, time      scaling will overestimate volatility (mean reversion is a statistical tendency      to revert to a long-term average).</li>
	<li>If there is significant trending, time      scaling will underestimate volatility (trending is a statistical tendency      to keep moving in one direction).</li>
	<li>If volatility changes over time, or there      are jumps, time scaling will be inaccurate.</li>
</ul>

Time Scaling of Volatility

Statistical Foundations of VaR

This frontmatter description will appear below the title

Statistical Foundations of VaR

Lessons

Take the Next Step in Your Data Career