<p>While making an investment decision, it is important to assess the risk/return profile of any investment. The relation between risk and return raises three basic questions:</p>



<ul><li>How do I estimate the percentage return that I will receive on an investment?</li><li>How much risk does an asset add to a portfolio?</li><li>What can I do to eliminate some of that risk?</li></ul>



<p>To answer these questions, we need to understand the key statistical concepts that are applied to financial assets. They are: expected returns, variance and standard deviation, and correlation. Let us look at each of them in detail:</p>



<h4>Expected Return</h4>



<p>The expected return is the market's, or an investor's, best guess as to the return on an asset. Any technique can be used to arrive at the guess. This section will review two common techniques. One uses a simple average of historical returns. Another technique uses the returns from possible outcomes and the probabilities of those outcomes to arrive at an expected return.</p>



<h4>Variance and Standard Deviation</h4>



<p>Risk is the possibility that actual returns might differ, or vary, from expected returns. In fact, actual returns will likely differ from expected returns. It is important for decision-makers to estimate the magnitude and likelihood of the difference between actual and estimated returns. After all, there is a big difference if your predictions result in an error of only $100 versus an error of $1 million.</p>



<p>By using the concepts of variance and standard deviation, investors can judge not only how wrong their estimates might be, but also estimate the likelihood, or probability, of favorable or unfavorable outcomes. With the tools of expected return and standard deviation, financial decision-makers are better able to evaluate alternative investments based on risk-return tradeoffs, and their own risk preferences.</p>



<h4>Diversification</h4>



<p>The Diversification topic answers the third question regarding what one can do to minimize risk of a group, or portfolio, of investments. By selecting investments that perform differently under the same market conditions, one can create a portfolio that has less risk for the same level of expected return. The concepts of covariance and correlation are used to measure how the returns on assets relate to each other and the market in general and how they can be used to reduce the overall risk to the investor.</p>

Mean, Variance, Standard Deviation and Correlation

The concept of Efficient Frontier was first introduced by Harry Markowitz in his paper on Portfolio Selection (1952 Journal of Finance).

The portfolio theory considers a universe of risky investments and explores these possible investments in order to find the optimum portfolio. So, for a given amount of risk, MTP explains how select a portfolio with maximum returns, and with a given amount of return, MTP explains how to select a portfolio with minimum risk.

Suppose you have all the required data (expected returns, volatility, and correlations) for all the investments you are considering. Using this data, you can create various portfolios with different portfolio risk and return profiles.

Among all these portfolios, choose the optimal portfolios in either of the following way:
<ol>
	<li>Identify all the portfolios that have the same risk (volatility). From this sub-set of portfolios, choose the one that has the highest return.</li>
	<li>Identify all the portfolios that have the same returns. From this sub-set of portfolios, choose the one that has the lowest risk.</li>
</ol>
Both the methods will product a set of optimal portfolios. This set of optimal portfolios is called the efficient frontier. (Later in this post, we will learn how to construct an efficient frontier in Excel)

If you plot all the portfolios that you could make using the universe of risky securities that you have, the graph will look something like the one below:

<a href="https://financetrain.com/wp-content/uploads/2010/06/Efficient-Frontier.gif"><img class="aligncenter size-full wp-image-195" title="Efficient Frontier" src="https://financetrain.com/wp-content/uploads/2010/06/Efficient-Frontier.gif" alt="" width="491" height="376" /></a>

Each red dot represents the mean and standard deviation of a portfolio. The blue line is the efficient frontier. The efficient frontier has all the optimal portfolios we selected above. Portfolios on the efficient frontier have maximum return for a given level of risk or, alternatively, minimum risk for a given level of return. Clearly, a rational investor will select a portfolio on the efficient frontier.
<h2>Dominated Portfolio</h2>
In the above graph, you see that for the same level of risk there are multiple portfolios providing different levels of return. A portfolio is said to be <strong>dominated</strong>, if there is atleast one portfolio that has higher return and lower standard deviation. In other words, a portfolio is dominated if there is any other portfolio to its north-west. So, in the above graph, all portfolios other than those on the efficient frontier are dominated.

Typically, the portfolios that make the efficient frontier are the most diversified ones. Less diversified portfolios tend to be closer to the middle of the achievable region.
<h2>Constructing an Efficient Frontier in Excel</h2>
The attached PDF file shows you how you can construct an efficient frontier using excel.
<h5><a href="https://financetrain.com/wp-content/uploads/2010/06/EfficientFrontier-Excel.pdf">Constructing Efficient Frontier in Excel</a></h5>

Constructing an Efficient Frontier

One problem with using futures contracts to hedge a portfolio of spot assets, is that a perfect futures contracts may not exist, that is, a perfect hedge cannot be achieved. For example, if an airline wishes to hedge its exposure to variation in jet fuel prices, it will find that there is no jet fuel futures market. A variation on the theme might go as follow. Although there exists a futures market for an underlying asset, that futures market is so illiquid that it is functionally useless.

Thus, we need to find way to use sub-optimal contracts, contracts that are highly correlated with the underlying asset and who have a similar variance. This is achieved using the <strong>minimum variance hedge ratio.</strong>

The minimum variance hedge ratio (or optimal hedge ratio) is the ratio of futures position relative to the spot position that minimizes the variance of the position.

<Ad />

The minimum variance hedge ratio is given as follows:

<a href="https://financetrain.com/wp-content/uploads/2010/06/mvhr_207.gif"><img class="size-full wp-image-210 alignnone" title="mvhr_207" src="https://financetrain.com/wp-content/uploads/2010/06/mvhr_207.gif" alt="" width="94" height="61" /></a>

Where,
[latex]\rho[/latex] is the correlation and [latex]\sigma[/latex] is the standard deviation

Let us take an example to understand this.

An airlines company wishes to hedge their annual 2,000,000 gallons jet fuel requirement. In short they fear that the price of jet fuel will rise.

Unfortunately, there exists no jet fuel futures contract.

However, a futures contract for heating oil trades at the NYMEX (New York Mercantile Exchange) and, it is known that jet fuel is a derivative of heating oil..

The contract size of the NYMEX heating oil contract is 42,000 gallons. Thus, if heating oil were a perfect hedge, the Airline would purchase 2,000,000/42,000 contracts = 47.61 contract (either 46 or 47 contracts)

Since heating oil is not a perfect hedge, they would use the minimum-variance hedge ratio to calculate the optimal number of contracts to purchase. i.e.,

<a href="https://financetrain.com/wp-content/uploads/2010/06/example_207.gif"><img class="alignnone size-full wp-image-230" title="example_207" src="https://financetrain.com/wp-content/uploads/2010/06/example_207.gif" alt="" width="511" height="79" /></a>

The airlines company collects 15 month’s worth of data on spot jet fuel (S) and heating oil futures (F) prices.
<table border="1" cellspacing="0" cellpadding="0">
<tbody>
<tr>
<td width="197" valign="top"><strong>Month</strong></td>
<td width="197" valign="top"><strong>Δ in Future Price (ΔF)</strong></td>
<td width="197" valign="top"><strong>Δ in Spot Jet Fuel Price (ΔS)</strong></td>
</tr>
<tr>
<td width="197" valign="top">1</td>
<td width="197" valign="top">0.021</td>
<td width="197" valign="top">0.029</td>
</tr>
<tr>
<td width="197" valign="top">2</td>
<td width="197" valign="top">0.035</td>
<td width="197" valign="top">0.020</td>
</tr>
<tr>
<td width="197" valign="top">3</td>
<td width="197" valign="top">-0.046</td>
<td width="197" valign="top">-0.044</td>
</tr>
<tr>
<td width="197" valign="top">4</td>
<td width="197" valign="top">0.001</td>
<td width="197" valign="top">0.008</td>
</tr>
<tr>
<td width="197" valign="top">5</td>
<td width="197" valign="top">0.044</td>
<td width="197" valign="top">0.026</td>
</tr>
<tr>
<td width="197" valign="top">6</td>
<td width="197" valign="top">- 0029</td>
<td width="197" valign="top">-0.019</td>
</tr>
<tr>
<td width="197" valign="top">7</td>
<td width="197" valign="top">-0.026</td>
<td width="197" valign="top">-0.010</td>
</tr>
<tr>
<td width="197" valign="top">8</td>
<td width="197" valign="top">-0.029</td>
<td width="197" valign="top">-0.007</td>
</tr>
<tr>
<td width="197" valign="top">9</td>
<td width="197" valign="top">0.048</td>
<td width="197" valign="top">0.043</td>
</tr>
<tr>
<td width="197" valign="top">10</td>
<td width="197" valign="top">-0.006</td>
<td width="197" valign="top">0.011</td>
</tr>
<tr>
<td width="197" valign="top">11</td>
<td width="197" valign="top">-0.036</td>
<td width="197" valign="top">-0.036</td>
</tr>
<tr>
<td width="197" valign="top">12</td>
<td width="197" valign="top">-0.011</td>
<td width="197" valign="top">-0.018</td>
</tr>
<tr>
<td width="197" valign="top">13</td>
<td width="197" valign="top">0.019</td>
<td width="197" valign="top">0.009</td>
</tr>
<tr>
<td width="197" valign="top">14</td>
<td width="197" valign="top">-0.027</td>
<td width="197" valign="top">-0.032</td>
</tr>
<tr>
<td width="197" valign="top">15</td>
<td width="197" valign="top">0.029</td>
<td width="197" valign="top">0.023</td>
</tr>
</tbody>
</table>
We calculate the following:

σF = 0.0313

σS = 0.0263

ρ = 0.928

So the Optimal number of contract is:     = 37.14

Rounding, Airlines company would buy: 37 heating oil futures contracts

If indeed jet fuel prices rose over the course of the year. The losses Airline would incur due to the higher spot jet fuel prices, would be offset by the gains they made from buying 37 contracts of heating oil futures.

Minimum Variance Hedge Ratio

Correlation is a familiar concept used to describe the strength of the relationship between variables. Serial correlation (also known as autocorrelation) is the term used to describe the relationship between observations on the same variable over independent periods of time. If the serial correlation of observations is zero, observations are said to be independent. However, if observations are serially correlated, it means they don’t evolve in a random process, but rather they are related to their prior values.

Serial correlation is calculated as a function of the mean and variance.

If the observations are independent, the serial correlation is zero. However, if serial correlation has a statistically significant difference from zero, the observations are not likely to be independent. In this case, the observations may exhibit positive or negative serial correlation, and the proportional relationship between standard deviation and the square root of time does not hold.
<ul>
	<li>If observations are positively serially correlated, they are said to exhibit mean aversion. This means observations are prone to trends and returns measured over longer periods will have higher standard deviation than if the subperiod returns were independent.</li>
</ul>
<ul>
	<li>If observations are negatively serially correlated they are said to be mean reverting. Mean reversion indicates that observations tend towards the average value over time and returns measured over longer periods will have a lower standard deviation than if the subperiod returns were independent.</li>
</ul>
This can be used to relate the variability in returns in a single asset over different time period. For example, we can answer the following question:

How is the standard deviation of annual returns of a single asset related to the standard deviation of monthly returns for the same asset?
<ol>
	<li>Standard deviation of annual returns is Sqrt(12) times larger than standard deviations of monthly returns</li>
	<li>Standard deviation of annual returns is Sqrt(12) times larger than standard deviations of daily returns (assuming 250 business days in a year.)</li>
</ol>
As we can see, the standard deviation of annual returns is only Sqrt(12) times larger than monthly returns (and not 12 times larger). This is similar to diversification effect but within the asset, that is, losses in some months are offset by gains in other months.

This "square root of time" formula is also used to convert value-at-risk from one time horizon to another.

What is Serial Correlation (Autocorrelation)?

<!-- wp:embed {"url":"https://www.youtube.com/watc...

Diversification and Portfolio Risk

Value-at- Risk (VaR) is a general measure of risk developed to equate risk across products and to aggregate risk on a portfolio basis. VaR is defined as the predicted worst-case loss with a specific confidence level (for example, 95%) over a period of time (for example, 1 day). For example, every afternoon, J.P. Morgan takes a snapshot of its global trading positions to estimate its DEaR (Daily-Earnings-at-Risk), which is a VaR measure defined as the 95% confidence worst-case loss over the next 24 hours due to adverse price moves.

This definition of VaR uses a 5% risk level: You would anticipate exceeding your VaR amount only 5% of the time (or, 95% of the time you expect to lose less than your VaR amount) over a 1-day horizon.

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.
<div class="special-area">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.</div>
VaR can be estimated as follows:
<p class="formula">VaR = Market value X Price volatility</p>
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.
<h2>Risk of a Single Position</h2>
<div class="special-area">

<b>AlphaSquare 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? </b>

<br><br>You have the following information: 
<ul class="list2">
<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>
<li>The JPY/USD exchange rate is 140.</li>
</ul>
</div>
<h3>Solution</h3>
<br>
<table class="mytable" cellspacing="0" style="margin-left:10px;">

  <tbody><tr>

    
    <th scope="col">Step</th>
	<th scope="col">Calculation</th>
	<th scope="col">Comment</th>
	
  </tr>
  <tr>
    <td>1. Choose a probability of loss</td>
    <td>5%</td>

	<td>=95% confidence worst-case loss</td>
	
  </tr>
  <tr>
    <td class="alt">2. Measure value in USD</td>
    <td class="alt">$100 million</td>
	<td class="alt">Assuming 140 JPY/USD</td>
	
  </tr>

      <tr>
    <td>3. 1-day volatility JPY/USD (or 1.65 standard deviations)</td>
    <td>1.78%</td>
	<td>Data Set</td>
	
  </tr>
  <tr>
    <td class="alt">4. Calculate Risk</td>

    <td class="alt">$100 million * 1.78 = $1.78</td>
	<td class="alt">Market value volatility</td>
	
  </tr>
</tbody></table>

<p>
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. 
</p>

<h2>Risk of Two Cash Positions</h2>
<div class="special-area">

<b>You now have two assets: JPY 1 billion + THB 4 billion. What is the risk over a 1-day period? </b>
</div>
<h3>Solution</h3>
<br>
<table class="mytable" cellspacing="0" style="margin-left:10px;">

  <tbody><tr>
    
    <th scope="col">Step</th>
	<th scope="col">Calculation</th>
	<th scope="col">Comment</th>

	
  </tr>
  <tr>
    <td>1. Measure value in USD</td>
    <td>$200 million</td>
	<td>140 JPY/USD and 40 THB/USD</td>
	
  </tr>
  <tr>

    <td class="alt">2.	1-month volatility</td>
    <td class="alt">JPY/USD 1.78%<br>THB/USD 1.96%
</td>
	<td class="alt">Dataset</td>
	
  </tr>
      <tr>
    <td>3.	What are risks?/$</td>
    <td>On JPY $ 1.78 million<br>On THB $ 1.96 million

</td>
	<td>Data Set</td>
	
  </tr>
  <tr>
    <td class="alt">4.	What is total risk?</td>
    <td class="alt">$ 1.78  + $ 1.96 m = $ 3.74 million</td>
	<td class="alt">This is incorrect you cannot just add the risks!</td>

	
  </tr>
</tbody></table>

<p>
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. 
</p>
<p>The data sets shows a correlation of 55% between JPY/USD and THB/USD. </p>

<p>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. </p>
<table class="mytable" cellspacing="0" style="margin-left:10px;">

  <tbody>  
  <tr style="border-top:1px solid #C1DAD7;">
    <td class="alt first-row" >Undiversified risk</td>

	<td class="alt first-row">$3.74. million</td>
	
  </tr>
  <tr>

    <td>Diversification benefit</td>
    <td>-$0.45 million</td>

	
  </tr>
  <tr>
    <td class="alt">Net risk</td>

	<td class="alt">$3.29 million</td>

	
  </tr>


</tbody></table>
<p>
The diversification shows the difference between net portfolio risk and gross risk assuming perfect 
correlation (i.e., net portfolio risk minus gross risk). <br><br>We can calculate the risk of two linear positions using the following formula:
</p>
<p style="text-align:center;" class="formula"><a href="https://financetrain.com/wp-content/uploads/2010/06/var_equation.jpg"><img src="https://financetrain.com/wp-content/uploads/2010/06/var_equation.jpg" alt="" title="var_equation" width="521" height="61" class="alignnone size-full wp-image-259" /></a></p>

<p>In our example, the calculation will be as follows:</p>
<p style="text-align:center;" class="formula"><a href="https://financetrain.com/wp-content/uploads/2010/06/var_calculation.jpg"><img src="https://financetrain.com/wp-content/uploads/2010/06/var_calculation.jpg" alt="" title="var_calculation" width="461" height="86" class="alignnone size-full wp-image-260" /></a></p>

Value at Risk (VaR) of a Portfolio

Let us consider two assets A and B with the following details:
<table border="1" cellspacing="0" cellpadding="0">
<tbody>
<tr>
<td width="148" valign="top"></td>
<td width="148" valign="top">Mean</td>
<td width="148" valign="top">Standard Deviation</td>
<td width="148" valign="top">Correlation</td>
</tr>
<tr>
<td width="148" valign="top">A</td>
<td width="148" valign="top">[latex] \mu_{A}=10\%[/latex]</td>
<td width="148" valign="top">$latex \sigma_{A}=20\%$</td>
<td rowspan="2" width="148">$latex \rho_{AB}=30\%$</td>
</tr>
<tr>
<td width="148" valign="top">B</td>
<td width="148" valign="top">$latex \mu_{B}=12\%$</td>
<td width="148" valign="top">$latex \sigma_{B}=26\%$</td>
</tr>
</tbody>
</table>
We have a total of $10 million to invest. Our objective is to reach a target return of $5 million. Let us look at the following three options and find out the probability of reaching our target in each case:
<ol>
	<li>Entire      $10 million in Asset A</li>
	<li>Entire      $10 million in Asset B</li>
	<li>$5      million in A and $5 million in B</li>
</ol>

Assuming r as the return from each portfolio, our objective can be expressed as follows:
<div class="formula">

P (10million*r &gt; 5million)

Or

P (r &gt; 0.5)

</div>
We know that if the returns of an asset are normally distributed, it can be expressed as a function of standard normal distribution. We can associate the return distribution to a standard normal distribution, which has a zero mean and a standard deviation of one. 

Therefore, our returns can be expressed as follows:

$latex r = \mu_{r} + Z\sigma_{r}$

or<sub> </sub>$latex Z = (r - \mu_{r})/ \sigma_{r}$

Our probability can be expressed as follows:

<a href="https://financetrain.com/wp-content/uploads/2010/06/formula_266.gif"><img class="alignnone size-full wp-image-271" title="formula_266" src="https://financetrain.com/wp-content/uploads/2010/06/formula_266.gif" alt="" width="267" height="121" /></a>

Let us now evaluate each of the three portfolios.

<strong>1. Entire $10 million in Asset A</strong>

$latex \mu_{A}=10\%$

$latex \sigma_{A}=20\%$

<a href="https://financetrain.com/wp-content/uploads/2010/06/A_266.gif"><img class="alignnone size-full wp-image-274" title="A_266" src="https://financetrain.com/wp-content/uploads/2010/06/A_266.gif" alt="" width="261" height="55" /></a>

That is, P(Z &gt; 2) = 2.28%

<strong>2. Entire $10 million in Asset B</strong>

$latex \mu_{B}=12\%$

$latex \sigma_{B}=26\%$

<a href="https://financetrain.com/wp-content/uploads/2010/06/B_2661.gif"><img class="alignnone size-full wp-image-276" title="B_266" src="https://financetrain.com/wp-content/uploads/2010/06/B_2661.gif" alt="" width="273" height="58" /></a>

That is, P(Z &gt; 1.46) = 7.19%

<strong>3. $5 million in A and $5 million in B</strong>

First we need to calculate the expected return and standard deviation of the portfolio.

<a href="https://financetrain.com/wp-content/uploads/2010/06/c_266.gif"><img class="alignnone size-full wp-image-277" title="c_266" src="https://financetrain.com/wp-content/uploads/2010/06/c_266.gif" alt="" width="336" height="192" /></a>

That is, P(Z &gt; 2.09) = 1.8%
<h3>Conclusion</h3>
You can observe that the portfolio with the highest risk has the greatest chance of reaching the target, while the diversified portfolio has the least chance of reaching the target.

Probability of One Portfolio Outperforming Another Portfolio

Earlier we looked at calculating the probability of beating a fixed target. Now we will look at calculating the probability of beating a benchmark which is itself stochastic.

Let us consider two assets A and B with the following details:
<table border="1" cellspacing="0" cellpadding="0">
<tbody>
<tr>
<td width="148" valign="top"></td>
<td width="148" valign="top">Mean</td>
<td width="148" valign="top">Standard Deviation</td>
<td width="148" valign="top">Correlation</td>
</tr>
<tr>
<td width="148" valign="top">A</td>
<td width="148" valign="top">[latex] \mu_{A}=10\%[/latex]</td>
<td width="148" valign="top">$latex \sigma_{A}=20\%$</td>
<td rowspan="2" width="148">$latex \rho_{AB}=30\%$</td>
</tr>
<tr>
<td width="148" valign="top">B</td>
<td width="148" valign="top">$latex \mu_{B}=12\%$</td>
<td width="148" valign="top">$latex \sigma_{B}=26\%$</td>
</tr>
</tbody>
</table>
We have a total of $10 million to invest. Our objective is to beat a benchmark. 

Let us take the 50-50 portfolio, which has the following returns:

$latex r_{1} = 0.5A + 0.5B$

Suppose the benchmark has the following returns:

$latex r_{2} = 0.4A + 0.6B$

We need to find that probability that our portfolio will beat the benchmark index, i.e., $latex P(r_{1} > r_{2})$

This can be expressed as:

$latex P(r_{1} - r_{2} > 0)$

We can write this as:

$latex P(0.5A + 0.5B - 0.4A - 0.6B > 0)$

or

$latex P(0.1A - 0.1B) > 0 $

0.1A - 0.1B is normally distributed.

Therefore, it's mean and standard deviation will be given as follows:

Mean, $latex E(0.1A-0.1B) = 0.1\mu_{A} - 0.1\mu_{B} = 0.1(10\%)-0.1(12\%) = -0.2\% $

Standard Deviation, $latex \sigma(0.1A-0.1B) = \sqrt{0.1^{2}\sigma_{A}^{2}+0.1^{2}\sigma_{B}^{2}-2(0.1)(0.1)\sigma_{A}\sigma_{B}\rho}$

$latex = 2.76\% $

We can write our probability as follows:

$latex P(0.1A-0.1B>0) = P\left ( \frac{0.1A-0.1B+0.002}{0.0276} > \frac{0 + 0.002}{0.0276} \right ) $

$latex = P(Z>0.0725) $

where Z is the standard normal variable.

$latex P(Z>0.0725) = 47.1\%$, using 1-NORMSDIST(0.0725) in excel.

Therefore, the 50-50 portfolio has a 47.1% chance of beating the benchmark portfolio of 40-60.

This probability of beating the benchmar depends on the correlation between the assets. With high correlation, the probability will decrease and vice verse. 

Probability of Attaining a Return Goal

Understanding Portfolio Math

This frontmatter description will appear below the title

COURSE

Understanding Portfolio Math

Lessons

R Programming Bundle: 25% OFF