Study Notes for Portfolio Risk and Return: Part I

Portfolio Risk and Return: Part I

Study Notes for Portfolio Risk and Return: Part II

Portfolio Risk and Return: Part II

For the purpose of portfolio construction, the financial assets are primarily looked at from the perspective of risk and returns. Based on the analysis of risk and returns we analyse thousands of securities and portfolio combinations before making the right selection for our own portfolio. Therefore, it is important that we have a deeper understanding of the risk and return and how these are calculated.

When we talk about the returns from a financial asset, we can broadly classify them into two types. First, financial assets may provide some income in the form of dividends or interest payments. Second, some financial assets may also offer capital appreciation as the prices of these securities change themself. For example, stocks provide both these types of returns, that is dividend income and capital appreciation. There are some stocks which don’t pay dividends, ad only provide price appreciation, such as Google, and Apple. Other assets such as bonds and pension plans only provide income such as monthly income payments in case of pension funds.

While making investment decisions, we should be aware of what kind of returns we are actually talking about. Some of the information you directly observe from the stock market may be misleading. For example, the stock indices will do a good job of reporting the price appreciation but may not truly reflect the adjustment for dividend payment. The return from the dividend payment and capital gain will put together will be the total return but may not be directly observable. Similarly the period for which you are observing the returns also becomes important; whether you are using annualized returns or the returns for your selected holding needs to be considered.

Apart from the use of correct data, how we process the return information is also important. Are we going to calculate arithmetic mean returns or geometric mean returns needs to be understood. In the next few articles we will look at the different type of return measures and how they can be applied in different situations.

Major Types of Return Measures

For investments, the Holding Period Return (HPR) refers to the total return earned from an investment or an investment portfolio over the holding period, that is, the period for which the asset or portfolio was held by the investor. The holding period can be anything such as 1 day, 1 month, 6 months, 1 year, 5 years and so on.

If you buy an asset now at $100 and sell it at $120 after 2 years, the holding period return will be (120 – 100)/100 = 20%. The time when the asset was bough can be labelled t and the current time when the asset is sold can be labelled t+1. If the asset pays any income such as dividend income on maturity, then that should also be added to the total returns. If P represents the price of the asset, then the holding period return formula can be presented as follows:

<img class="aligncenter size-full wp-image-8707" title="hpr1" src="https://financetrain.com/wp-content/uploads/hpr1.gif" alt="" width="254" height="76" />

Let's take a simple example to understand the HPR calculation.

Let's say that we purchased one share of a stock for $100 at the beginning of the year. After three months, the stock price has gone up to 102 and it also pays a dividend of $2. The holding period return will be:

HPR = (102 - 100 + 2)/100 = 4%

The holding period returns can be annualized from either longer periods or shorter periods.

If the original HPR is calculated over multiple years, then the annualized returns can be calculated as follows:

<img class="aligncenter size-full wp-image-8708" title="hpr2" src="https://financetrain.com/wp-content/uploads/hpr2.gif" alt="" width="435" height="82" />

If the original HPR that we have are quarterly, then we can annualize them using the following formula:

<img class="aligncenter size-full wp-image-8709" title="hpr3" src="https://financetrain.com/wp-content/uploads/hpr3.gif" alt="" width="493" height="51" />

The same above formula can also be used if we had the annual returns and wanted to calculate the holding period return for the multiple period.

For example, let’s say that our investment had a price appreciation of 10%, 8%, and -6% over the three year period.

The HPR can be calculated as follows:

HPR = [(1+ 0.10)(1+0.08)(1-0.06)] – 1 = 11.67%

How to Calculate the Holding Period Returns

This video by Arif Irfanullah provides a very clear conceptual understanding of the portfolio risk and return concepts as a part of the CFA Level 1 syllabus. This first video talks about

<ul>
	<li>Major return measures</li>
	<li>Major asset classes for investments</li>
	<li>Evaluating investments</li>
	<li>Covariance and correlation</li>
	<li>Risk and return of a two asset portfolio</li>
	<li>Efficient Frontier</li>
	<li>Effect of Correlation on Diversification Benefit</li>
</ul>

<iframe width="600" height="338" src="http://www.youtube.com/embed/glpYC8e9zs4?list=PLE066FFBD4766125B&amp;hl=en_US" frameborder="0" allowfullscreen></iframe>

Portfolio Risk & Return - Part 1A - Video

This video by Arif Irfanullah provides a very clear conceptual understanding of the portfolio risk and return concepts as a part of the CFA Level 1 syllabus. This video talks about
<ul>
	<li>Markowitz Efficient Frontier</li>
	<li>Capital Allocation Line</li>
	<li>Risk Aversion</li>
	<li>Risk-Return Indifference Curve</li>
	<li>Optimal Investor Portfolio</li>
</ul>
<iframe src="http://www.youtube.com/embed/J11dJ5zjULM?list=PLE066FFBD4766125B&amp;hl=en_US" frameborder="0" width="600" height="338"></iframe>

Portfolio Risk & Return - Part 1B - Video

This video by Arif Irfanullah provides a very clear conceptual understanding of the portfolio risk and return concepts as a part of the CFA Level 1 syllabus. This video talks about:
<ul>
	<li>CAL vs. CML</li>
	<li>Risk and Return on the CML</li>
	<li>CML with Different Lending and Borrowing Rates</li>
	<li>Types of Risk</li>
	<li>Return Generating Models, such as Market model and Multifactor models.</li>
</ul>
<iframe src="http://www.youtube.com/embed/J11dJ5zjULM" frameborder="0" width="600" height="338"></iframe>

Portfolio Risk and Return - Part 2A - Video

This video by Arif Irfanullah provides a very clear conceptual understanding of the portfolio risk and return concepts as a part of the CFA Level 1 syllabus. This video talks about:
<ul>
	<li>Beta - Calculation and Estimation</li>
	<li>Capital Asset Pricing Model (CAPM) Assumptions</li>
	<li>Security Market Line</li>
	<li>Applications of CAPM, including estimating expected returns, and evaluating portfolio performance</li>
</ul>
<iframe src="http://www.youtube.com/embed/J11dJ5zjULM" frameborder="0" width="600" height="338"></iframe>

Portfolio Risk and Return - Part 2B - Video

While calculating the returns on financial assets, we will often look at the returns from multiple holding periods. For example, one may hold an asset for five years, and the asset may have earned total 150% returns over this period of 7 years. However, it is difficult to interpret these returns as we cannot compare them with returns on other assets. For comparison purpose, we will have to aggregate these returns for the same period such as daily returns, monthly returns, or yearly returns. In the example above, it will be more suitable to calculate average annual returns than to know the returns earned over 7 years.

While calculating the aggregate returns, our return measure will vary depending on what method we use to calculate the aggregate returns. Two common methods are arithmetic returns and geometric returns.

Let’s take an example to understand both these methods. Let’s say that our portfolio generated the following returns in 5 years.

<img class="aligncenter size-full wp-image-8785" title="mean" src="https://financetrain.com/wp-content/uploads/mean.gif" alt="" width="138" height="141" />

Arithmetic Returns

To calculate the arithmetic average, we take the simple average of the 5 yearly returns as follows:

Arithmetic Returns = (100%+(-50%)+35%+(-20%)+50%)/5

= 23%

Geometric Returns

One problem with arithmetic mean is that it assumes the returns on the investment made at the beginning of each period. So, for each period the beginning investment amount is assumed to be the same. It ignores the compounding effect of investment returns made in the previous years. Using arithmetic returns, our measure can be majorly flawed. Consider an investment of $100 at the beginning. Say in first year the investment value rises to $200. The returns are 100%. In year 2, the investment falls back to $100, which will be a return of -50% in the year 2. If we take the average of two year returns, i.e., 100% in year 1 and -50% in year 2, it shows an average annual return of 25% on this investment, even though our investment value is back to $100 (from where we started). This problem can be solved by calculating geometric returns which incorporates the compounding effect.

In our example the geometric returns can be calculated as follows:

Geometric returns = [(1+100%)*(1-50%)*(1+35%)*(1-20%)*(1+50%)]^(1/5) - 1

= 10%

As you can see, geometric return is lower than the arithmetic return, and is a better method for aggregating returns over multiple holding periods.

Arithmetic Returns Vs. Geometric Returns

In the previous article, we learned about <a href="https://financetrain.com/arithmetic-returns-vs-geometric-returns/" title="Arithmetic Returns Vs. Geometric Returns">arithmetic returns and geometric returns</a>. However, the problem with these measures is that they do not consider the amount of investment made in each period. For example, in the first year, we may have an investment of USD 5,000 while in the second year, the investment may only be $2,000. So, the returns when looked at along with how much money was invested will make a huge difference to our actual return on investment. This will be called money-weighted return on internal rate of return.

Let’s say we had the following investments and returns in the past 3 years:

In the first year, we made an investment of $1000, and we had a 100% return in the first year. By the end of the year, our investment has grown to $2000. Then at the beginning of the second year we invested $2000 more making a total investment of $4000. The returns in the second year were -50%, and our investment value reduced to $2000. Then assume we withdrew $500 from the investment fund, leaving only $1500 invested. In the third year there was no new investment, and our returns were 35%, making our investment grow to $2025. The cash flows are shown in the table below.

<img class="aligncenter size-full wp-image-8790" title="money-weighted1" src="https://financetrain.com/wp-content/uploads/money-weighted1.gif" alt="" width="433" height="201" />

The money-weighted returns can be calculated using the same formula as that of the <a href="https://financetrain.com/net-present-value-npv-internal-rate-of-return-irr-excel/" title="Calculating Net Present Value (NPV) and Internal Rate of Return (IRR) in Excel">Internal rate of Return (IRR)</a>.

<img class="aligncenter size-full wp-image-8791" title="money-weighted2" src="https://financetrain.com/wp-content/uploads/money-weighted2.gif" alt="" width="203" height="98" />

Our cash flows are as follows:

CF0 = -$1,000

CF1 = =$2,000

CF2 = +$500

CF4 = $2,025

Applying the above formula and solving for IRR we get:

IRR or money-weighted returns = -8%

This tells the investor about what she actually earned on the money invested for the entire three year period. Note that this return is negative because a significantly large amount of money was invested in the year of negative returns compared to other years.

How to Calculate Money-weighted Returns

When we make investments, we invest our money in different assets and earn returns for different periods of time. For example, an investment in a short-term <a title="US Treasury Bills, Notes and Bonds" href="https://financetrain.com/us-treasury-bills-notes-and-bonds/">Treasury bill</a> will be for 3 months. We may invest in a stock and exit after a week for a few days. For the purpose of making the returns on these different investments comparable, we need to annualize the returns. So, all daily, weekly, monthly, or quarterly returns will be converted to annualized returns. The process for annualizing the returns is as follows:

The basic idea is to compound the returns to an annual period. So, if we have monthly returns, we know that there are 12 months in the year, similarly there are 52 weeks, 4 quarters, and 365 days. We compound our returns by the number of periods in the whole year.

<img class="aligncenter size-full wp-image-8803" title="annual-return" src="https://financetrain.com/wp-content/uploads/annual-return.gif" alt="" width="394" height="54" />

Let’s take a few examples to understand this.

Example 1: Quarterly Returns

Let’s say we have 5% quarterly returns. Since there are four quarters in a year, the annual returns will be:

Annual returns = (1+0.05)^4 – 1 = 21.55%

Example 2: Monthly Returns

Let’s say we have 2% monthly returns. Since there are 12 months in a year, the annual returns will be:

Annual returns = (1+0.02)^12 – 1 = 26.8%

Example 3: Weekly Returns

Let’s say we have 0.5% weekly returns. Since there are 52 weeks in a year, the annual returns will be:

Annual returns = (1+0.005)^52 – 1 = 29.6%

Example 4: Daily Returns

Let’s say we have 0.1% daily returns. Since there are 365 days in a year, the annual returns will be:

Annual returns = (1+0.001)^365 – 1 = 44.02%

Example 5: 100 Days Returns

We can actually have returns for any number of days and convert them to annualized returns. Let’s say we have 6% returns over 100 days. The annual returns will be:

Annual returns = (1+0.06)^(365/100) – 1 = 23.69%

Annualized returns however have one limitation – they assume that we will be able to reinvest the money at the same rate. However this may not always be possible. If we earned 5% in a quarter there is no guarantee that we will be able to replicate these returns over the next three quarters in the year.

How to Calculate Annualized Returns

We have learned about how to<a title="Major Types of Return Measures" href="https://financetrain.com/major-types-of-return-measures/"> calculate the returns</a> on single assets. However, portfolio managers will have many assets in their portfolios in different proportions. The portfolio manager will have to therefore calculate the returns on the entire portfolio of assets. The returns on the portfolio are calculated as the weighted average of the returns on all the assets held in the portfolio.

The formula for portfolio returns is presented below:

<img class="aligncenter size-full wp-image-8806" title="pr-1" src="https://financetrain.com/wp-content/uploads/pr-1.gif" alt="" width="374" height="58" />

w represents the weights of each asset, and r represents the returns on the assets. For example, if an asset constitutes 25% of the portfolio, its weight will be 0.25. Note that sum of all the asset weights will be equal to 1, as it will represent 100% of the investment. The returns here are single period returns with same periods for each asset’s returns.

Let’s take an example of a two asset portfolio to understand how portfolio returns are calculated. Let’s say that our portfolio comprises of two assets A and B and has the following details.

<img class="aligncenter size-full wp-image-8807" title="pr-2" src="https://financetrain.com/wp-content/uploads/pr-2.gif" alt="" width="258" height="85" />

The table presents the amount invested in each asset and the returns from each asset. The total amount invested is $100,000. We can calculate the weights for each asset as follows:

wA = 25000/100000 = 0.25

wB = 75000/100000 = 0.75

We can now calculate the portfolio returns as follows:

<img class="aligncenter size-full wp-image-8808" title="pr-3" src="https://financetrain.com/wp-content/uploads/pr-3.gif" alt="" width="374" height="52" />

The same calculation can be extended for multiple assets.

In the later articles we will also learn how to calculate the risk of the portfolio.

How to Calculate Portfolio Returns

In an investment management company such as a mutual fund, you will come across two types of returns being calculated, namely, returns on a gross, or net of fee basis. While there are some industry practices for calculating these returns, all fund managers don’t strictly follow one method.

<img class="size-full wp-image-8813" title="Gross and Net Returns" src="https://financetrain.com/wp-content/uploads/gross-net.gif" alt="" width="279" height="202" /> Gross and Net Returns

In simple words, the gross returns refer to the returns calculated before deducting any fee, while the net returns refer to the returns calculated after deducting the fee. The fee may include managerial and administrative expenses such as management fees, custodial fees, taxes, and any other administrative expenses. The fee however does not include commissions, and any other expenses directly related to trading.

Since the gross returns are unaffected by the management and administrative expenses, they are a preferred measure for selecting and evaluating the relative performance of the portfolio managers. It is also preferred for attribution, risk, and other value-added reporting.

If we take two identical portfolios, they will have same gross returns; however, it’s not necessary that their net of fee returns will also be the same. This is because as the size of the assets grows, the magnitude of fee paid will reduce. So, on a risk-adjusted basis, a larger portfolio will seem superior compared to an identical smaller portfolio. This is the reason why gross returns are more suitable for evaluating performance of portfolio managers.

The net of fee returns are the returns the fund has actually earned for the investors. It’s calculated after deducting all the managerial, custodial and administrative expenses. These returns are what the investor is interested in knowing. The net returns can also be used for measuring the actual performance of the manager and the fund level.

Again if we take two identical funds, the fund having less assets will seem to have lower net returns compared to a fund with larger asset base because of the fee benefit for larger fund. For this reason, many mutual funds that don’t have large assets under management will waive off some fees to make their returns lucrative to investors.

Gross and Net Returns Calculations

We have looked at a <a title="Major Types of Return Measures" href="https://financetrain.com/major-types-of-return-measures/">variety of return measures</a>. However, till now we assumed that the investment is made by the investor’s own money. However, in reality, the investor will not use only his money for making investments. The position will be leveraged. For example, while trading in <a title="Characteristics of Futures Contracts" href="https://financetrain.com/characteristics-of-futures-contracts/">futures contracts</a>, the investor may have to put about 10% of the notional value as <a title="Role of Margin Money in Futures Trading" href="https://financetrain.com/role-of-margin-money-in-futures-trading/">margin money</a>. Even while buying stocks, the investor may invest part his money and part borrowed money.

When the investor invests 10% of the total required investment, then his profits and losses are amplified by 10 times. Similarly when he invests 50% of his money and 50% borrowed money, then the returns will double. In case of borrowed money, however, the investor may want to adjust his returns for the interest paid on the borrowed money.

In general, leverage increases the rate of return. The reason is mainly because a leveraged position is riskier compared to an unleveraged one. This is especially true while talking about the expected rate of return from an investment.

Let’s take an example. Let’s say an investment grows in value from $1000 to $1200.

If the entire $1000 was the investor’s money, then it’s an unleveraged position, and the investor’s returns would be:

R = (1200-1000)/1000 = 20%

However, if the investor had invested $500 of his money and the remaining $500 was borrowed money, then it’s a leveraged position. Assuming no interest cost, the return on the leveraged position would be:

R = (1200-1000)/500 = 40%

If there was an interest paid on the borrowed money, that would be deducted from the numerator while calculating the leveraged returns.

How to Calculate Leveraged Returns

While calculating the returns on an investment, what we directly observe is the nominal returns. These are the returns which have not been adjusted for the inflation. The nominal returns can also be looked at as pre-tax nominal returns and post-tax nominal returns. Pre-tax nominal return, as the name suggests, is the return without adjusting for the tax one has to pay on the returns earned.

In financial investments, the investors are usually concerned about the after-tax returns that they will get as the tax liability can vary substantially. The returns that an investor gets can be classified into two types: capital gains arising from the price appreciation, and the income arising from the interest or dividend received. Both capital gains and income get different treatment from each other and also the tax rates vary in different countries. The income in the form of interest or dividends is usually treated as ordinary income and clubbed with other income for the purpose of calculating tax burden. The capital gains can be long-term capital gains or short-term capital gains. In many countries long-term capital gains get preferential tax treatment.

An investment manager may reduce the overall tax liability by choosing suitable securities for their portfolios.

The real returns refer to what the investor has actually earned after adjusting for the inflation. The relation between the real rate and nominal rate can be expressed as follows:

Rnominal = (1 + rreal) * (1 + inflation rate)

Real returns are useful while comparing returns over different time periods because of the differences in inflation rates. Also, they are useful for comparing investments in different countries because the investments will be in different local currencies and will have different inflation rates.

Nominal Returns and Real Returns in Investments

After discussing the <a title="Major Types of Return Measures" href="https://financetrain.com/major-types-of-return-measures/">calculation of returns</a> on investments, let’s now learn about how to measure the risks associated with these returns.



In general, the risk of an asset or a portfolio is measured in the form of the standard deviation of the returns, where standard deviation is the square root of variance. Let’s look at how standard deviation and variance is calculated.



The variance is calculated as follows. Once you have the price data, the first step is to calculate the returns. What kind of returns we have depends on the periodicity of the data. For example, if we have daily prices, then we calculate the daily returns, which is calculated as (P(t1) – P(t0))/p(t0).



Using the returns data, we calculate the mean/average returns. The variance of the asset returns will then be the average of square of the difference between the returns and the mean.



<div class="wp-block-image"><figure class="aligncenter size-large"><img src="https://financetrain.com/wp-content/uploads/asset-variance.gif" alt="" class="wp-image-26655"/></figure></div>



Where n is the number of observations in the data, Rt represents the returns for each time period and μ represents the mean returns.



The standard deviation will simply be the square root of the variance.



The following is a simple example that illustrates the calculation:



<div class="wp-block-image"><figure class="aligncenter"><img src="https://financetrain.com/wp-content/uploads/risk-sd.gif" alt="" class="wp-image-8838"/></figure></div>

Calculate Variance and Standard Deviation of an Asset

We learned about how to calculate the standard deviation of a single asset. Let’s now look at how to calculate the standard deviation of a portfolio with two or more assets.

The returns of the portfolio were simply the weighted average of returns of all assets in the portfolio. However, the calculation of the risk/standard deviation is not the same. While calculating the variance, we also need to consider the covariance between the assets in the portfolio. If the assets are perfectly correlated, then the simple weighted average of variances will work. However, when we have to account for the covariance, the equation will change.

Covariance reflects the degree to which two securities vary or change together, and is represented as Cov (Ri,Rj). The problem with covariance is that it has no units, and is difficult to compare across assets. Using covariance, we can calculate the correlation between the assets using the following formula:

[caption id="attachment_8857" align="aligncenter" width="290"]<img class="size-full wp-image-8857" title="pr1" src="https://financetrain.com/wp-content/uploads/pr1.gif" alt="" width="290" height="90" /> Correlation[/caption]

After incorporating covariance, the standard deviation of a two-asset portfolio can be calculated as follows:

[caption id="attachment_8858" align="aligncenter" width="458"]<img class="size-full wp-image-8858" title="pr2" src="https://financetrain.com/wp-content/uploads/pr2.gif" alt="" width="458" height="167" /> Standard Deviation of a two Asset Portfolio[/caption]

In general as the correlation reduces, the risk of the portfolio reduces due to the diversification benefits. Two assets a perfectly negatively correlated provide the maximum diversification benefit and hence minimize the risk.

Let’s take an example to understand the calculation. Assume we have a portfolio with the following details:

[caption id="attachment_8859" align="aligncenter" width="498"]<img class="size-full wp-image-8859" title="pr3" src="https://financetrain.com/wp-content/uploads/pr3.gif" alt="" width="498" height="142" /> Example[/caption]

The standard deviation can be calculated as follows:

<img class="aligncenter size-full wp-image-8860" title="pr4" src="https://financetrain.com/wp-content/uploads/pr4.gif" alt="" width="470" height="79" />

This portfolio has an expected return of 16.80% and portfolio risk of 11.09%.

Standard Deviation and Variance of a Portfolio

We learned that the calculation of risk for a portfolio of two assets is not straight forward as we also have to account for the covariance between the assets in the portfolio.

Depending on the correlation between the assets, the risk-return profile of the portfolio changes. Note that we can combine the two assets in varying proportions in the portfolio two arrive at an infinite number of portfolios. Say the two assets are A and B. You can start with a portfolio that has 100% money invested in Stock A, then create many portfolios with different proportions of A and B, and end with a portfolio that has 100% money invested in stock B.

When you plot the risk-return profile of all these portfolios, what you see is the feasible set of portfolios.

Let’s take an example to understand this. Let’s say we have two securities A and B with the following expected returns, standard deviation, and correlation. Below you will also see 6 different portfolios with varying proportions of A and B.

[caption id="attachment_8885" align="aligncenter" width="469"]<img class="size-full wp-image-8885" title="ef-1" src="https://financetrain.com/wp-content/uploads/ef-1.gif" alt="" width="469" height="288" /> Portfolio of Two Assets[/caption]

When we plot the risk-return profile of these portfolios, it looks as follows:

[caption id="attachment_8886" align="aligncenter" width="505"]<img class="size-full wp-image-8886" title="ef-2" src="https://financetrain.com/wp-content/uploads/ef-2.gif" alt="" width="505" height="331" /> Efficient Frontier[/caption]

&nbsp;

Notice that there is a minimum variance portfolio with lowest risk on the efficient frontier.

We can make the following observations from this:
<ul>
	<li>There are benefits of diversification, as the risk reduces when we combine assets in the portfolio.</li>
	<li>There is a Minimum Variance Portfolio (Portfolio 3) that has the minimum risk.</li>
	<li>The feasible set or opportunity set is represented by the entire curved line.</li>
	<li>The curve bends backwards.</li>
	<li>Investors invest above the Minimum Variance Portfolio (MVP), as any portfolio below it does not have an optimal risk-return profile.</li>
</ul>

Efficient Frontier for a Portfolio of Two Assets

In this article, we will look at how <a title="Correlation and Covariance" href="https://financetrain.com/correlation-and-covariance/">correlation</a> affects the diversification benefits of a portfolio. Let’s take a portfolio with two assets. The correlation between the two assets can range from -1.0 to 1.0 and depending on the correlation figure the shape of the efficient frontier will change.

The following graph shows how the <a title="Efficient Frontier for a Portfolio of Two Assets" href="https://financetrain.com/efficient-frontier-for-a-portfolio-of-two-assets/">efficient frontier</a> will look based on different values of correlation. When the correlation between the assets is 1, it means that any change in one asset is equally reflected in the change in the other asset. So, if one asset goes down 10%, the other asset will also go down 10%. There is no diversification benefit. In such a case any portfolio derived from a combination of A and B will fall on the straight line, reflecting the risk simply weighted with both assets.

<img class="aligncenter size-full wp-image-8889" title="corr-eff" src="https://financetrain.com/wp-content/uploads/corr-eff.gif" alt="" width="451" height="254" />

However, as the correlation starts reducing, the portfolio starts getting diversification benefit and the efficient frontier starts bending towards the left. The lesser the correlation between the two assets, the more the diversification benefit and more it will bend towards the left. By this logic, when the two assets are perfectly negatively correlated, that is, the correlation between them is -1, the diversification should yield maximum risk reduction. So, there could be a combination of A and B where the risk reduces to 0. That point is represented as O in the above chart. It is not however really possible to find assets that are perfectly negatively correlated. For this reason the efficient frontier will usually be a curved line bending towards the left, signifying some levels of diversification.

Effect of Correlation on Diversification

We have seen that different asset classes such as bonds, stocks, and commodities provide different levels of risk and return to investors. However, we also know that these investment options are not equally preferred by all investors. An equity stock providing high returns may be suitable for one investor but another investor may want to avoid such an investment. This happens because of the different attitudes of investors towards risk. A portfolio manager will consider the risk profile of his client investors and try to match his portfolio investment in such financial instruments that have the similar risk-return profile.

This concept is called risk aversion. In general, risk aversion refers to the behaviour of investor to prefer less risk to more risk. A risk averse investor will:
<ul>
	<li>Prefer lower to higher risk for a given level of expected return</li>
	<li>Accept high risk investment only if expected returns are greater</li>
</ul>
Let’s take a simple example to understand this. Assume that an investor has been offered two choices. In the first choice he will receive $100 for sure. In another choice, there is a 50% chance that he will receive $200 and another 50% chance that he will receive nothing. We can have some understanding of what kind of an investor he is depending on the choice he makes. The following figure shows the risk-return profiles of different types of investors.

[caption id="attachment_9104" align="aligncenter" width="533"]<img class="size-full wp-image-9104 " title="risk-averse" src="https://financetrain.com/wp-content/uploads/risk-averse.gif" alt="" width="533" height="295" /> Risk Aversion[/caption]

Risk Neutral Investor

A risk neutral investor is someone who is only concerned about the return but does not worry about the risk posed by the investment. As long as an investment provides high returns, this type of investor will go for it. In our example, a risk-neutral investor will be indifferent between the two choices. He will look at both choices 1) $100 for sure, and 2) 50% nothing and 50% $200, as the same. In general, investors are not risk neutral. An investor may be risk neutral if the investment is not so significant. For example, a very wealthy investor will be indifferent to whether he receives $100 guaranteed or goes for the gamble.

Risk Averse Investor

As we mentioned earlier, most investors are risk-averse, that is, they want to reduce the amount of risk they take for a given level of return. If the returns provided are higher, they will be willing to take proportionately higher risk. In our example, such an investor will go for guaranteed $100. In fact, depending on his risk aversion, he may even be willing to accept slightly lower returns (say $90 instead of $100) for the certainty he gets. Different investors will exhibit different degrees of risk aversion. In the figure above, the two investors IP and IQ have different levels of risk aversion. Investor IP is more risk averse than investor IQ as investor IP demands more return for every additional unit of risk in the investment.

Risk Lover/Risk Seeker

The third category of investor is a risk lover. In our example, such an investor will choose to gamble because it’s not about the returns. Rather he loves taking risk, and gets additional pleasure by taking the risk. A risk lover definitely wants to increase his returns but his utility (level of happiness) increases from the extra risk that he takes. For risk lovers, the risk-return curve slopes downwards, i.e. the return they receive reduces and the risk increases. This is the typical profile of a gambler.

Risk Aversion of Investors and Portfolio Selection

In the previous article we learned that different investors exhibit different levels of risk aversion. For each investor the degree of risk aversion translates into certain utility (read satisfaction) that he gets from an investment. In our example of $100 for sure vs. a gamble where you get $200 or nothing, when a risk averse chooses to go with $100 for sure, it means that the $100 with certainty provides him more utility compared to the gamble. For another person, even $90 may be good enough for him to find more utility compared to going for a gamble. However, if the certain amount is reduced more, say to just $20, then he might as well try the gamble option.

The risk-return utility function of an investor can be exhibited using the following formula:

U = E(R) – ½ A * Variance

Where:
<ul>
	<li>U is a given level of happiness/utility</li>
	<li>A is the level of risk averseness</li>
</ul>
We can make some interesting observations from this function:
<ol>
	<li>Utility increases as expected returns increase</li>
	<li>Utility decreases when variance or risk increases</li>
	<li>Utility reduces as risk-aversion (A) increases</li>
</ol>
As the risk aversion increases, an investor demands more return for every unit of increase in risk. When the risk increases, the investor demands more return based on his utility function, thereby keeping the level of utility the same. This concept can be explained with the help of indifference curve. An indifference curve presents the risk-return requirements of an investor at a certain level of utility. The following graph shows three indifference curves for the same investor.

[caption id="attachment_9117" align="aligncenter" width="574"]<img class=" wp-image-9117 " title="indifference" src="https://financetrain.com/wp-content/uploads/indifference.jpg" alt="" width="574" height="430" /> Indifference Curves for an Investor[/caption]

&nbsp;

Note that each of the curves (C1, C2, and C3) represents different levels of utility for the investor. However, on any one indifference curve, the utility is the same anywhere on the curve. For example, on indifference curve C1, the investor gets same level of utility (satisfaction) at both points S1 and S2. Similarly, the utility is the same at S3 and S4.

Now let’s compare points S4 on C2 and S6 on C3. On point S4, the investor faces a certain amount of risk for certain return. On point S6, he gets the same return as on S4 but at lower risk. So, given a choice, he can increase is utility by moving to the indifference curve C3.

In general, the utility of risk-averse investors increases as we move leftwards in the graph. Given a choice between C1, C2, and C3, the investor would want to be on C3 to maximize this utility.

The indifference curve will have different shape for different types of investors as shown in the following graph.

[caption id="attachment_9104" align="aligncenter" width="533"]<img class="size-full wp-image-9104" title="risk-averse" src="https://financetrain.com/wp-content/uploads/risk-averse.gif" alt="" width="533" height="295" /> Risk Aversion[/caption]

Utility Indifference Curves for Risk-averse Investors

We know that an investor can combine many risky assets to create a portfolio with lower risks. By varying the weights of different assets in the portfolio many portfolios with different risk-return profiles can be created. If we plot the risk-return profiles of these different portfolios, what we get is an efficient frontier. The efficient frontier represents the set of portfolios that will give the highest return at each level of risk or the lowest risk for each level of return.

Let’s assume that investor chooses to invest in a risky portfolio that lies on this efficient frontier. Let’s call it Ri.

The investor can combine the risky portfolio, Ri, with a risk-free asset Rf, to create a new portfolio, RP.

The risk-return profile of this combined portfolio of risk assets and risk-free asset will be as follows:

<img class="aligncenter size-full wp-image-9122" title="caleq1" src="https://financetrain.com/wp-content/uploads/caleq1.gif" alt="" width="369" height="89" />

In the above equation, the risk (standard deviation) for the risk-free asset is 0. So, the equation will be reduced to the following:

<img class="aligncenter size-full wp-image-9123" title="caleq2" src="https://financetrain.com/wp-content/uploads/caleq2.gif" alt="" width="127" height="48" />

We can create different portfolios with different weights for the risk-free asset and the risky portfolio. The following graph shows the risk-return of the portfolio by varying the weight from 0 to 100%. The line is called the Capital Allocation Line.

<img class="aligncenter size-full wp-image-9124" title="CAL" src="https://financetrain.com/wp-content/uploads/CAL.gif" alt="" width="455" height="267" />If 100% of the money is invested in the risk-free asset, then the return will be Rf at zero risk. If 100% money is invested in the risky portfolio, the investor will get expected returns of E(Ri) with a risk of σi.

Any portfolio can be created on this line by varying the proportion of funds invested in risk-free asset and risky asset.

The Capital Allocation Line (CAL) can be explained using the standard regression equation Y = MX + C, as follows:

<img class="aligncenter size-full wp-image-9125" title="caleq3" src="https://financetrain.com/wp-content/uploads/caleq3.gif" alt="" width="243" height="70" />

Note that when you invest in a risk-free asset, you are actually lending money to the government. For this reason, any portfolio created between Rf and Ri is called a lending portfolio. An investor can earn returns higher than the risky portfolio by borrowing money at the risk-free rate and investing this money in the portfolio of risky assets. This is represented in the following chart:

<img class="aligncenter size-full wp-image-9126" title="lend-borrow" src="https://financetrain.com/wp-content/uploads/lend-borrow.gif" alt="" width="477" height="350" />Let’s say you borrow 25% at the risk-free rate. In that case w = -0.25 and 1-w = 1.25. The returns on the portfolio will be:

<img class="aligncenter size-full wp-image-9127" title="caleq4" src="https://financetrain.com/wp-content/uploads/caleq4.gif" alt="" width="394" height="44" />

Capital Allocation Line with Two Assets

In the previous articles, we learned that an investor can invest in a combination of risk-free asset and risky assets anywhere on the <a href="https://financetrain.com/capital-allocation-line-with-two-assets/">capital allocation line</a>. A rational investor is also risk-averse and has a <a href="https://financetrain.com/utility-indifference-curves-for-risk-averse-investors/">utility indifference curve</a> that characterizes his risk-return expectations.

However, the problem is that on the capital allocation line, the investor can create an infinite number of portfolios with different risk-return combinations. So, how does one go about selecting the right portfolio that is suitable for an investor? This question can be answered by combining the capital allocation line or the efficient frontier with the indifference curves of the investor, as shown in the graph below:

[caption id="attachment_9154" align="aligncenter" width="395"]<img class="size-full wp-image-9154" title="optimal" src="https://financetrain.com/wp-content/uploads/optimal.jpg" alt="" width="395" height="300" /> Optimal Portfolio[/caption]

The graph above shows the efficient frontier, the capital allocation line, and the three indifference curves for the investor.

Let’s first take the case where the investor wants to put all his money in only risky assets. In that case the feasible set for the investor is depicted on the curve ABNME. We also have the indifference curves for the investor, namely, I1, I2, and I3. The efficient frontier is what the investor can invest in. Any portfolio below it is not preferred and any portfolio above it is not attainable. Therefore, there is no portfolio that is attainable on indifference curves I2 and I3. There is only one portfolio that meets his utility indifference curve, I1, and that is portfolio N. In this situation, portfolio N will be the optimal portfolio for the investor.

Now, let’s assume that the investor wants to invest in both risky and risk-free assets. The capital allocation line (Rf-PMKZ) will depict the feasible set of portfolios. Again, any portfolio on I3 is unattainable. We also know that as we move leftwards, the utility of the investor increases. When we look at I1, there is a portfolio where CAL and I1 intersect. However, we have a better portfolio P where I2 intersects with CAL. In this scenario, portfolio P will be the optimal portfolio for the investor.

We can also note that between Portfolio N and Portfolio P, portfolio P is a better portfolio as the investor earns almost the same returns, but with lesser risk.

Selecting Optimal Portfolio for an Investor

In this article, we will learn how to compute the risk and return of a portfolio of assets. Let’s start with a two asset portfolio.

Portfolio Return

Let’s say the returns from the two assets in the portfolio are R1 and R2. Also, assume the weights of the two assets in the portfolio are w1 and w2. Note that the sum of the weights of the assets in the portfolio should be 1. The returns from the portfolio will simply be the weighted average of the returns from the two assets, as shown below:

RP = w1R1 + w2R2

Let’s take a simple example. You invested $60,000 in asset 1 that produced 20% returns and $40,000 in asset 2 that produced 12% returns. The weights of the two assets are 60% and 40% respectively.

The portfolio returns will be:

RP = 0.60*20% + 0.40*12% = 16.8%

Portfolio Risk

Let’s now look at how to calculate the risk of the portfolio. The risk of a portfolio is measured using the standard deviation of the portfolio. However, the standard deviation of the portfolio will not be simply the weighted average of the standard deviation of the two assets. We also need to consider the covariance/correlation between the assets. The covariance reflects the co-movement of the returns of the two assets. Unless the two assets are perfectly correlated, the covariance will have the impact of reduction in the overall risk of the portfolio.

The portfolio standard deviation can be calculated as follows:

<img class="aligncenter size-medium wp-image-9192" title="covar" src="https://financetrain.com/wp-content/uploads/covar-300x105.png" alt="" width="300" height="105" />

In the above example, let’s say the standard deviation of the two assets are 10 and 16, and the correlation between the two assets is -1. The standard deviation of the portfolio will be calculated as follows:

σP = Sqrt(0.6^2*10^2 + 0.4^2*16^2 + 2*(-1)*0.6*0.4*10*16) = 0.4

Correlation and Covariance

As you can see, the standard deviation can be calculated using either covariance or correlation. The relationship between the two is depicted below:

<img class="size-full wp-image-9193 aligncenter" title="risk2" src="https://financetrain.com/wp-content/uploads/risk2.gif" alt="" width="208" height="71" />

In this example, we had taken a simple portfolio of two assets however as the number of assets in the portfolio increase, the complexity will increase as we will have to consider the covariance between each pair of the assets in the portfolio. For a three asset portfolio, the risk and return will be calculated as follows:

<img class="size-full wp-image-9194 aligncenter" title="risk3" src="https://financetrain.com/wp-content/uploads/risk3.gif" alt="" width="551" height="138" />

The calculation can be simplified by representing the formula in the form of an equation. For a two asset portfolio the formula can be represented as:
<img class="size-full wp-image-9195" title="risk4" src="https://financetrain.com/wp-content/uploads/risk4.gif" alt="" width="246" height="71" />
Note that there are there matrices in the calculation. The first one is a one-dimensional matrix of the asset weights in the portfolio. The second matrix is called the variance covariance matrix with the variances of the assets as the diagonal elements and the covariance as the other elements. The third matrix is the transpose of the first matrix. Multiplying the three matrices gives us the variance of the portfolio. In Microsoft Excel, you can use the MMULT function to perform matrix multiplication.

This form of variance calculation can easily be expanded to as many assets as we want.

Let’s take the case of a three asset portfolio with asset weights as 0.3, 0.4, and 0.3. The variance covariance matrix is given below:

<img class="size-full wp-image-9196 aligncenter" title="risk5" src="https://financetrain.com/wp-content/uploads/risk5.gif" alt="" width="223" height="94" />

The variance of the portfolio can be calculated as follows:

<img class="size-full wp-image-9197 aligncenter" title="risk6" src="https://financetrain.com/wp-content/uploads/risk6.gif" alt="" width="508" height="101" />

How to Calculate Portfolio Risk and Return

Portfolio Risk and Return

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Portfolio Risk and Return

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