Mean Variance Portfolio Theory - Financial Engineering

07/06/2024, 13:03 2 Mean variance portfolio theory | Financial Engineering
2 Mean variance portfolio theory
2.1 Learning objectives
Describe and discuss the assumptions of mean-variance portfolio theory.

Discuss the conditions under which application of mean-variance portfolio theory leads to
the selection of an optimum portfolio.
Calculate the expected return and risk of a portfolio of many risky assets, given the
expected return, variance and covariance of returns of the individual assets, using mean-
variance portfolio theory.
Explain the benefits of diversification using mean-variance portfolio theory.
2.2 Background
First, it’s worth pausing to understand the name of this theory. “Mean variance portfolio theory”
sounds quite technical. In practical terms, and building on the concepts of the preceding
Chapter, think of it this way:
Mean - expected return, Variance - risk related to the investment.
Then we can see that this theory is really just about the two basic return measures we’ve been
talking about. This theory looks at how investments can be selected to maximise an investor’s
utility where that utility if defined just as a trade off between risk (as measured by variance)
and return.
Almost all investors will choose to split their investments across a number of different types of
asset (e.g. equities, bonds, cash, property, commodities, etc.) and different securities within
each asset type. Together these multiple assets make up the investment portfolio. A decision
therefore needs to be made as to what proportions. Typically, the decision on the split between
types of assets (known as asset allocation) is made first and then the portion of the portfolio
within each asset type (or equivalently asset class) is allocated to specific funds or securities.
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This is known as a ‘top down’ approach. An alternative approach is to adopt a completely

‘bottom up’ approach and focus on selecting individual securities and the split between asset
types is then simply a function of the different securities chosen.
What is critical is that it is the joint behaviour of the assets that becomes important when
considering a portfolio. In particular, what is the combined risk and return of the assets
together and how does this change under different conditions? In order to assess this, a
definition of risk needs to be chosen. As we saw in the previous chapter, there are a number of
different possible risk measures that an investor could use. Mainly for simplicity, we will
assume that the risk measure is the variance of returns. We also need to specify how
investors make choices. In this chapter we will be assuming that investors only consider the
expected return and the variance of return over a single time period.
Check your understanding
Can you think of why a “top down” approach is typically used in preference to “bottom up”
in the areas of asset management actuaries are involved in?
Answer
Why is the joint behaviour of assets in a portfolio critical to allow for? What mathematical
quantity could we use to represent the joint behaviour?
Answer
In order to estimate these two key quantities for a portfolio of assets, we will need to know the
expected returns and variances of returns for all of the assets individually, as well as the
covariance of returns for each possible pair of assets. The collection of all possible portfolios
(i.e. combinations of assets) is called the opportunity set.
Suppose that there are \(N\) securities that the investor can invest in and that a proportion \
(x_i\) is invested in each, with return \(R_i\) on security \(i\).
Then the portfolio return \(R_P\) is defined by \(R_P=\sum_{i}x_iR_i\). In other words it is just
the weighted average return, where the weights are the proportions invested in each security.
The expected return is \(E_P=\sum_{i}x_iE_i\), where \(E_i\) is the expected return on security
\(i\).
The variance of the portfolio return is \[V_P=Var[R_P]=\sum_{i}\sum_{j}x_ix_jC_{ij},\]
where \(C_{ij}\) is the covariance of the returns on security \(i\) and security \(j\). It is common
to write \(C_{ii}\) as \(V_i\), the variance on security \(i\).
In the simplest case of only two assets, these formulae are:
\[\begin{align*} E_P &= x_1E_1+x_2E_2 \\ V_P &= x_1^2V_1+x_2^2V_2+2x_1x_2C_{12}

\end{align*}\]
Note that we are not considering any liabilities that the investor may have, but these could be
added to the analysis. In that case changes in the gap between the value of the assets and
liabilities is likely to be the focus, rather than just the expected return/expected change of
either assets or liabilities considered separately.
2.3 Efficient portfolios
We make two further assumptions regarding investor preferences:
Investors are never satiated: they always prefer a higher return than a lower return, for a
given level of risk
Investors are risk averse: for a given level of return they will always choose a portfolio with
a lower risk over one with a higher risk
The portfolios which have the highest return for a given level of risk (variance of return), or the
lowest level of risk for a given level of return are called efficient portfolios. Investors will only
invest in efficient portfolios and therefore all other portfolios can be ignored.
Why might investors choose portfolios that are not efficient as defined above, even if they
have all the knowledge and information to do so?
Answer
It is also worth noting that we are implicitly making assumptions that the market is frictionless:
There are no taxes or transaction costs

Securities are infinitely divisible and can be traded in any (fractional) amount
Short selling is permitted in any amount
These assumptions can all be relaxed, but that would significantly complicate the analysis and
so obscure the key results. This does not mean that real life limitations should be ignored, but
rather for the purposes of developing the theory, they will be ignored. You should always be
aware of what assumptions you are making, especially when they are implicit assumptions.
This will help you decide how plausible the results are.
What does short selling mean?
Answer
2.4 Minimum variance (i.e. minimum risk)

portfolio
It is always possible to choose a portfolio that has the least risk of all possible portfolios for a
given expected return (bearing in mind that this portfolio might contain only one asset). To see
how the joint behaviour of the assets influences this, consider the chart of possible portfolios
made up of two assets (A and B) with the following characteristics. We have plotted these
portfolios in expected return-risk space, with the different levels of risk and return achieved by
varying the weights of the two assets in the portfolio.
Here is a static version of the image.
\(E(R_A)=8\%; \quad E(R_B)=4\%; \quad \sigma_A=10\%; \quad \sigma_B=1%\)
What do the red dot and the blue dots represent?
Answer
Consider the shape of the efficient frontier graph. Why is it upward sloping, and why does
its gradient flatten out with increasing risk taken?
Answer
Check that you can calculate the position of the red dot manually, for a particular split of
assets.
Answer
Try changing the correlation to see how the possible portfolios change. In particular, notice
how the minimum risk (i.e. portfolio at the turning point) changes.
We have specified the joint behaviour via the correlation, which determines the covariance but
is perhaps easier to specify. Notice that the risk is in terms of standard deviation, not variance.
If we plotted return against variance the shape of the curve would change, but the overall
behaviour would be the same. Notice also that the upper half of the curve (above the minimum
risk portfolio) represents the efficient frontier, so no portfolios from the lower half would ever be
selected. They are shown for completeness only.
The key fact that you should have determined is that the lower the positive correlation, or the
larger the negative correlation, the lower the risk of the minimum risk portfolio. This is because
the behaviour of the assets is more offsetting under those lower correlation scenarios. Notice
also that the risk/return combinations are “better” for assets with correlations less than +1, in
the sense that you can achieve higher levels of return for the same risk levels and vice versa.
The minimum variance portfolio can be found using calculus: for the simple case of only 2
assets, the derivation is as follows, remembering that the weights have to sum to 1.
\[\begin{align} \begin{split} V_P &= x_1^2V_1+(1-x_1)^2V_2+2x_1(1-x_1)C_{12}\\ \Rightarrow

V_P &= x_1^2V_1+(1-2x_1+x_1^2)V_2+2x_1C_{12}-2x_1^2C_{12} \end{split} \tag{2.1}
\end{align}\]
Taking the derivative with respect to the weight \(x_1\) and setting this to zero:
\[\begin{align*} \frac{dV}{dx_1} &= 2x_1V_1-2V_2+2x_1V_2+2C_{12}-4x_1C_{12}=0\\

\Rightarrow x_1 &= \frac{V_2-C_{12}}{V_1+V_2-2C_{12}} \end{align*}\]
This result can be generalised to the case for \(N\) securities using linear algebra. It can be
shown that when there are \(N\) securities, the weights in the minimum variance portfolio are:
\[\frac{\mathbf{V^{-1}}}{\mathbf{1^T}.\mathbf{V^{-1}}.\mathbf{1}}.\mathbf{1}\]
where \(\mathbf{1}\) is the \(N\)-dimensional vector containing \(1\) in every entry and \
(\mathbf{V}\) is the variance-covariance matrix.
2.5 Benefits of diversification
The result that you have observed for two assets can be generalised to N assets. It is one of
the fundamental results of portfolio theory and a very important concept in asset management.
The key point is that risk can be effectively controlled through portfolio selection.
Using the notation in (2.1), we can write the portfolio variance as:
\[V_P=\sum_ix_i^2V_i+\sum_i\sum_jx_ix_jC_{ij}\]
If we assume that equal amounts are invested in each of \(N\) securities, then the expression
for the variance becomes:
\[\begin{align*} V_P &= \sum_i(1/N)^2V_i+\sum_i\sum_j(1/N)(1/N)C_{ij}\\ \Rightarrow V_P &=

(1/N)\sum_i(1/N)V_i+(N-1)/N)\sum_i\sum_j(1/(N(N-1))C_{ij}\\ &= (1/N)\times\bar{V}+(N-
1)/N\times\bar{C} \end{align*}\]
where \(\bar{V}\) and \(\bar{C}\) are the average variance and covariance, respectively.
It can be seen that when N becomes large, the variance (or risk) of the individual securities will
tend to zero and the portfolio return approaches the average covariance between the
securities. In other words, it is the joint movement (and therefore covariance) of the assets that
determines the risk in a larger portfolio, not the variability of each asset in isolation. The risk
arising from the covariance is driven by factors that affect all securities. The ‘specific risk’ or
variance of individual securities due to factors affecting them only is ‘diversified away’.
In the special case when the returns on the securities are independent and the covariances
are therefore zero, the portfolio variance will tend to zero as the number of securities in the
portfolio increases. This is highly unlikely to be encountered in practice.
This demonstrates mathematically the reason why it is better to spread your assets across a
number of different securities and indeed asset classes as well. The concept of specific risk
and risk that affects all assets (referred to as ‘systematic’ risk) is important as well and we will
return to this in the next chapter when we focus on the capital asset pricing model.
2.6 Finding the efficient frontier with more than

two assets
To generalise the above results to portfolios containing \(N\) assets, we need to determine how
to find the weights of each of these in the portfolios. To do so, we need to either pick a given
level of portfolio return and minimise the portfolio variance, or pick a level of portfolio risk and
maximise the level of return. The minimisation or maximisation is always subject to the
constraint that the weights of all of the assets in the portfolio have to add up to 100%. Since
we are assuming short sales are possible, negative weights in some assets are allowed.
We assume that we wish to minimise \(V_P\) for a given expected return \(E_P=E\), subject to
the constraint that \(\sum_{i}x_i=1\), where the \(x_i\) are the proportions invested in security \
(i\). The \(x_i\) are the variables that we are trying to find the values for that will meet our
conditions. To solve this constrained optimisation problem, we use Lagrange multipliers. The
Lagrangian function is then:
\[W=V_P-\lambda(E-E_P)-\mu(\sum_ix_i-1),\] where \(\lambda\) and \(\mu\) are Lagrange

multipliers.
To find the required minimum, we take partial derivatives of \(W\) with respect to the \(x_i\) and
\(\lambda\) and \(\mu\) and set these equal to zero. This leads to a system of linear equations
that can then be solved. These partial derivatives are:
\[\begin{align*} \frac{\partial W}{\partial x_i} &= 2\sum_{j}C_{ij}x_j-\lambda E_i-\mu\\

\frac{\partial W}{\partial \lambda} &= -\left(\sum_{i}E_ix_i-E\right)\\ \frac{\partial W}{\partial \mu}
&= -\left( \sum_{i}x_i-1 \right) \end{align*}\]
Setting each of these equal to zero results in:
\[\begin{align*} 2\sum_{j}x_jC_{ij}-\lambda E_i-\mu &= 0\\ \sum_{i}E_ix_i &= E\\ \sum_{i}x_i &=
1 \end{align*}\]
which are \(N+2\) equations, since the first equation must hold for each of the \(N\) assets.
These can be represented using matrix notation as:
\[\mathbf{A.y}=\mathbf{b},\] where
\[\mathbf{A}= \begin{pmatrix} \mathbf{2C} & \mathbf{-E} & \mathbf{-1} \\ \mathbf{E^T} &

\mathbf{0} & \mathbf{0} \\ \mathbf{1^T} & \mathbf{0} & \mathbf{0} \end{pmatrix},\]
and
\[\begin{align*} \mathbf{y^T} &= (\mathbf{x^T},\lambda,\mu) \\ \mathbf{b^T} &=

(0,0,\dots,0,E_P,1) \quad \text{(n zeros)} \\ \mathbf{x^T} &= (x_1,x_2,\dots,x_n) \\ \mathbf{E^T}
&= (E_1,E_2,\dots,E_n) \\ \mathbf{1^T} &= (1,1,\dots,1) \quad \text{(n 1s)} \end{align*}\]
The solution is then \(\mathbf{y}=\mathbf{A^{-1}.b}\). This shows that the \(x_i\) are linear in \
(E_P=E\). Substituting for \(\mathbf{x}\) in the definition of the variance-covariance matrix \
(\mathbf{x^TCx}\), shows that \(V_P\) is quadratic in \(E_P\).These solutions can be generated
for the full range of \(V\) and \(E\) values to generate the efficient frontier.
Note that the portfolios generated by this method can all still be plotted in 2 dimensions as for
the case with two assets. This is because we are assuming that investors are only interested
in the expected returns and variance of returns.
2.7 Choosing the efficient portfolio
An individual investor will select the efficient portfolio that they wish to hold based on their
preferences. These can be expressed using indifference curves. These can be plotted against
the curve of efficient portfolios in expected return-risk space, as in the figure below. Note that
the indifference curves are upward sloping due to the risk aversion of the investor.
Explain the last sentence.
Answer
Explain what an indifference curve is, in the above context.
Answer
Figure 2.1: Efficient frontier together with indifference curves

The investor will select the portfolio where the efficient frontier is tangent to the indifference
curve, as this maximises utility. In this case, it is the point where the efficient frontier touches
the 11% utility curve. For those investors with quadratic utility functions, the mean-variance
approach will always lead to optimal portfolios. For those who consider that a more
sophisticated approach to defining risk is required, higher moments can be included, for
example to capture skewness and kurtosis. The approach would then need to be extended to
more dimensions than the two that have been considered here. As in the case of risk
measures, it is important to consider whether the disadvantages associated with the additional
mathematical complexity, degree of effort required and difficulties in interpreting the results of
such an alternative approach outweigh the benefits of using this more complicated approach
or not.
Explain why, for those investors with quadratic utility functions, the mean-variance
approach will always lead to optimal portfolios.
Answer
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Practitioner’s corner
The assumptions of mean-variance portfolio theory are very strong and are unlikely to be
met in practice. However, the fundamental point about the benefits of diversification is
really important. This is one of the few theoretical concepts. It is worth highlighting that
the degree of diversification depends heavily on the correlations (or equivalently
covariances) between the assets. These vary over time and, in particular, will increase
towards +1 under times of financial stress. This is because of investor behaviour,
e.g. everyone selling during a market crash.
It is therefore important to remember that diversification works under normal market

conditions: it will not necessarily help you in an extreme event. As noted in the chapter on
measures of investment risk, it is important to understand how your portfolio is likely to
behave under those extreme conditions. The method of choice for testing this possible
behaviour is stress and scenario testing. The choice of scenarios is critical in order for
this analysis to be an effective tool. It is not a simple task to decide what an extreme, but
plausible, combination of events should look like. You also need to capture all of the
particularly negative outcomes to avoid missing key gaps that need attention. It is also
not possible to capture “unknown unknowns”: the (in)famous ‘black swans’ that can have
the most significant impact on any portfolio.
Another point about diversification is that there can be practical limitations to how many
assets you can include in your portfolio. These include increased costs, increased
complexity, potential liquidity issues if you include very illiquid assets, etc. Fortunately,
you can achieve reasonably good diversification with a relatively small number of well
chosen assets (perhaps 20-30).
Finally, remember that risk and return always go together. If you reduce your risk then you
also reduce your maximum possible returns. The same is true of a well-diversified
portfolio. It will protect you against losses on individual assets, but dilute big gains on
specific securities that perform really well. The big gains that you sometimes see hedge
funds achieving are often as a result of taking highly concentrated bets: the complete
opposite of diversification.
1. An investor has a quadratic utility function. What assumptions of mean-variance

portfolio theory can be relaxed for this investor?
Answer
2. A portfolio is made up of 2 securities, one issued by a manufacturing company and

one by a services company. A new study suggests that the services company will
make more sales to the manufacturing company, when the manufacturer is making
strong profits. What effect will this have on the portfolio? How will the minimum
variance portfolio change?
Answer
3. State three key factors that are likely to lead to correlations between different
securities and therefore causing each to have an element of systematic risk.
Answer
4. How does the shape of the curve showing the opportunity set in \(E(R)/\sigma\)-space
change when there are no short sales allowed, compared to the case when short
sales are allowed?
Answer
2.8 Past IFoA exam questions.
Past IFoA exam questions from CM2 and CT8 papers are a very valuable resource to help you
develop your understanding and exam technique. Question papers and Examiners’ Reports
can be downloaded from the IFoA website free of charge.
At the end of each chapter we have suggested two or three past questions and you should
work through these and compare your answers with the Examiners’ Report as part of your
weekly study.
CM2A April 2019 Question 5 This is a fairly straightforward question involving some
definitions and a calculation.
Note that the examiners were not impressed by candidates who tried to define an inefficient
portfolio as one that isn’t efficient!
CM2B September 2020 Question 2 Parts (i) to (iii) This is an Excel-based question, which
might take you a little while, but you will have a good understanding of constructing portfolios
when you’ve finished.
We’ll come back to Part (iv) after you’ve studied CAPM.
2.9 Homework questions
We recommend you do the homework questions each week and submit them for feedback.
You’ll find them in the Assessment & Feedback section of the Blackboard site.

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07/06/2024, 13:03 2 Mean variance portfolio theory | Financial Engineering

2 Mean variance portfolio theory

2.1 Learning objectives

Describe and discuss the assumptions of mean-variance portfolio theory.

Mean - expected return, Variance - risk related to the investment.

This is known as a ‘top down’ approach. An alternative approach is to adopt a completely

Check your understanding

The variance of the portfolio return is \[V_P=Var[R_P]=\sum_{i}\sum_{j}x_ix_jC_{ij},\]

In the simplest case of only two assets, these formulae are:

\[\begin{align*} E_P &= x_1E_1+x_2E_2 \\ V_P &= x_1^2V_1+x_2^2V_2+2x_1x_2C_{12}

2.3 Efficient portfolios

We make two further assumptions regarding investor preferences:

Check your understanding

There are no taxes or transaction costs

Check your understanding

What does short selling mean?

2.4 Minimum variance (i.e. minimum risk)

Here is a static version of the image.

\(E(R_A)=8\%; \quad E(R_B)=4\%; \quad \sigma_A=10\%; \quad \sigma_B=1%\)

Check your understanding

What do the red dot and the blue dots represent?

\[\begin{align} \begin{split} V_P &= x_1^2V_1+(1-x_1)^2V_2+2x_1(1-x_1)C_{12}\\ \Rightarrow

\[\begin{align*} \frac{dV}{dx_1} &= 2x_1V_1-2V_2+2x_1V_2+2C_{12}-4x_1C_{12}=0\\

2.5 Benefits of diversification

\[\begin{align*} V_P &= \sum_i(1/N)^2V_i+\sum_i\sum_j(1/N)(1/N)C_{ij}\\ \Rightarrow V_P &=

2.6 Finding the efficient frontier with more than

\[W=V_P-\lambda(E-E_P)-\mu(\sum_ix_i-1),\] where \(\lambda\) and \(\mu\) are Lagrange

\[\begin{align*} \frac{\partial W}{\partial x_i} &= 2\sum_{j}C_{ij}x_j-\lambda E_i-\mu\\

Setting each of these equal to zero results in:

\[\mathbf{A}= \begin{pmatrix} \mathbf{2C} & \mathbf{-E} & \mathbf{-1} \\ \mathbf{E^T} &

\[\begin{align*} \mathbf{y^T} &= (\mathbf{x^T},\lambda,\mu) \\ \mathbf{b^T} &=

2.7 Choosing the efficient portfolio

Check your understanding

Explain the last sentence.

Explain what an indifference curve is, in the above context.

Figure 2.1: Efficient frontier together with indifference curves

Check your understanding

It is therefore important to remember that diversification works under normal market

Check your understanding

1. An investor has a quadratic utility function. What assumptions of mean-variance

2. A portfolio is made up of 2 securities, one issued by a manufacturing company and

2.8 Past IFoA exam questions.

We’ll come back to Part (iv) after you’ve studied CAPM.

2.9 Homework questions

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