Efficient Portfolios I
We will start by studying the characteristics of
individual assets and portfolios in term of risk
and return focusing on the case of two-assets.
Combination of Two risky Assets: Short Sales
not Allowed
From last lecture, we have seen that the
expected return on a portfolio of two assets A
and B is given by:
𝑅! = 𝑥! 𝑅! + 𝑥! 𝑅!
where
𝑥! = Fraction of the portfolio held in asset A
𝑥! = Fraction of the portfolio held in asset B
𝑅! = expected return on the asset A
𝑅! = expected return on the asset B
𝑅! = expected return on the portfolio P
In addition we require the investor to fully
invest.
That is:
𝑥! + 𝑥! = 1
or
𝑥! = 1 − 𝑥!
and substituting 𝑥! = 1 − 𝑥! in the Equation of
the expected return, we get:
𝑅! = 𝑥! 𝑅! + (1 − 𝑥! )𝑅!
The expected return on the portfolio is a
weighted average of the expected returns on the
individual securities.
As discussed in the previous lecture, the
standard deviation of the return on the twoassets portfolio is:
𝜎! = 𝑥!! 𝜎!! + 𝑥!! 𝜎!! + 2𝑥! 𝑥! 𝜎!"
!/!
where:
𝜎! = standard deviation of the return on the
portfolio 𝑃
𝜎!! = variance on the return of the return on
asset 𝐴
𝜎!! = variance on the return of the return on
asset 𝐵
𝜎!" = covariance between 𝐴 and 𝐵
Substituting 𝑥! = 1 − 𝑥! in the standard
deviation and noting that the correlation
between A and B, which is defined as:
𝜎!"
𝜌!" =
𝜎! 𝜎!
can also be written as:
𝜎!" = 𝜌!" 𝜎! 𝜎!
The standard deviation of the portfolio with two
assets will become:
𝜎! =
𝑥!! 𝜎!! + 1 − 𝑥! ! 𝜎!! + 2𝑥! (1 − 𝑥! )𝜌!" 𝜎! 𝜎!
!/!
We consider several cases where the correlation
coefficient 𝜌!" is +1, 0, −1 and the intermediate
cases (that is when−1 < 𝜌 < 1).
In doing so, we consider an example based on
two stocks (C and S) that are assumed to have
the following characteristics:
Asset C
Asset S
Expected
Return (𝑅)
14%
8%
Standard
Deviation (𝜎)
6%
3%
The stock C has a higher expected return and a
higher risk than stock S.
(Note that the notation of the two assets, in the
remaining part of the lecture, will therefore
change from A and B to C and S).
Perfect Positive Correlation
(𝝆 = 𝟏)
In this case the variance will be:
𝜎!! = 𝑥!! 𝜎!! + 1 − 𝑥! ! 𝜎!! + 2𝑥! (1 − 𝑥! )𝜎! 𝜎!
-------------------------------------------------------------Since this equation has the form
𝑎! + 2𝑎𝑏 + 𝑏 ! = (𝑎 + 𝑏)!
-------------------------------------------------------------It follows that:
𝜎!! = 𝑥! 𝜎! + 1 − 𝑥! 𝜎!
!
and the standard deviation is:
𝜎! =
𝜎!! = 𝑥! 𝜎! + 1 − 𝑥! 𝜎! (1)
While the expected return on the portfolio is:
𝑅! = 𝑥! 𝑅! + (1 − 𝑥! )𝑅!
(2)
Thus, with the correlation coefficient equal to
unity, both risk and return of the portfolio are
linear combinations of the risk and return of
each security.
The form of Equations (1) and (2) means that all
combinations of two securities that are perfectly
positively correlated will lie on a straight line in
risk / return space.
To see this, first solve for 𝑥! in Equation (1):
𝜎! = 𝑥! 𝜎! + 𝜎! − 𝑥! 𝜎!
or
𝑥! 𝜎! − 𝑥! 𝜎! = 𝜎! − 𝜎!
and
𝜎! − 𝜎!
𝑥! =
𝜎! − 𝜎!
Substituting this expression in Equation (2) (the
expected return) we find that:
𝜎! − 𝜎!
𝜎! − 𝜎!
𝑅! =
𝑅! + 1 −
𝑅!
𝜎! − 𝜎!
𝜎! − 𝜎!
or
𝜎!
𝜎!
𝜎!
𝑅! =
𝑅! −
𝑅! + 𝑅! −
𝑅!
𝜎! − 𝜎!
𝜎! − 𝜎!
𝜎! − 𝜎!
𝜎!
+
𝑅!
𝜎! − 𝜎!
and, finally:
𝑅! − 𝑅!
𝑅! − 𝑅!
𝑅! = 𝑅! −
𝜎! +
𝜎!
𝜎! − 𝜎!
𝜎! − 𝜎!
which is the equation of a straight line
connecting securities C and S in the expected
return / standard deviation space.
Since this is a general result we can obtain the
straight line for our example.
In our example, we have:
𝑅! = 8
𝑅! − 𝑅! 14 − 8
=
=2
𝜎! − 𝜎!
6−3
𝜎! = 3
Substituting in the general equation we obtain:
𝑅! = 8 − 2×3 + 2𝜎!
or
𝑅! = 2 + 2𝜎!
We can see this line graphically.
(See Figure 5.1)
The two extreme cases are when we invest all in
security S or all in security C. These are the
points C and S.
The other points are portfolios where different
proportions are invested in security C and S.
• Note that since the risk of the portfolio is the
weighted average of the risk on the
individual assets, there is no reduction in
risk from buying both assets.
• This means that nothing has been gained by
diversifying rather than purchasing the
individual assets.
Perfect Negative Correlation
(𝝆 = −𝟏)
The other extreme case is when the two assets
move perfectly together, but in the opposite
direction.
In this case the variance of the portfolio is:
𝜎!! = 𝑥!! 𝜎!! + 1 − 𝑥! ! 𝜎!! − 2𝑥! (1 − 𝑥! )𝜎! 𝜎!
Which can be written as:
𝜎!! = 𝑥! 𝜎! − 1 − 𝑥! 𝜎!
!
or
𝜎!! = −𝑥! 𝜎! + 1 − 𝑥! 𝜎!
!
-------------------------------------------------------------This is because, in general, an expression like
(𝑎! − 2𝑎𝑏 + 𝑏 ! ) can be written as (𝑎 − 𝑏)! or
(−𝑎 + 𝑏)!
--------------------------------------------------------------
The standard deviation is just the square root of
the variance, that is:
𝜎! = 𝑥! 𝜎! − 1 − 𝑥! 𝜎!
or
𝜎! = −𝑥! 𝜎! + 1 − 𝑥! 𝜎!
We need to notice two things:
• Either of the above equations holds only
when its right-hand side is positive.
• The right-hand side of each question is -1
times the other.
It follows that there is a unique solution for the
return and risk of any combination of securities
C and S.
It can be shown that if two securities are
perfectly negatively correlated, it is always
possible to find a combination of these
securities that has zero risk.
This combination can be found by setting 𝜎! =
0 in one of the equations for the standard
deviation of the portfolio:
0 = −𝑥! 𝜎! + 1 − 𝑥! 𝜎!
rearranging:
𝑥! 𝜎! + 𝑥! 𝜎! = 𝜎!
and
𝜎!
𝑥! =
𝜎! + 𝜎!
Therefore, a portfolio with weights
𝑥! =
!!
!! !!!
and 𝑥! = 1 − 𝑥!
will have zero risk.
Note also, that since:
𝜎! + 𝜎! > 𝜎! > 0
0 < 𝑥! < 1
This means that the zero-risk portfolio will
always involve positive investment in both
securities.
As for the case of 𝜌 = 1, also when 𝜌 = -1 we will
have an equation for the location of all portfolio
in expected return / standard deviation space.
(See seminar exercise).
For our example (i.e. with stocks C and S), we
will find the following two equations:
2
𝑅! = 10 + 𝜎!
3
or
2
𝑅! = 10 − 𝜎!
3
(Two equations with same intercept but
opposite slopes).
Notice that since 𝜌 = -1, we shown that it will
exist a combination of these securities (C and S)
that provides a portfolio with zero risk.
Minimum risk occurs, in our example, when
𝜎!
3
1
𝑥! =
=
=
𝜎! +𝜎! 3 + 6 3
(See Figure 5.2)
We can also have a look at the plot for both
cases (that is when 𝜌 = 1 and 𝜌 = -1).
(See Figure 5.3)
It is important to notice that:
• The standard deviation reaches its lowest
value for 𝜌 = -1 (zero risk).
• The lines SC (when 𝜌 = 1) and SBC (when 𝜌
= -1), represent the limits within which all
portfolio of these securities must lie.
• This mean that an intermediate correlation
(−1 < 𝜌 < 1) should produce a curve such
as SOC.
Intermediate Risk (−𝟏 < 𝝆 < 𝟏)
In the case of an intermediate risk, one point
that deserves some attention is the point of
minimum standard deviation: that is the
portfolio that has minimum risk.
This portfolio can be obtained by considering
the general formula for the standard deviation
or risk:
𝜎! =
𝑥!! 𝜎!! + 1 − 𝑥! ! 𝜎!! + 2𝑥! (1 − 𝑥! )𝜎! 𝜎! 𝜌!"
!/!
To find the value of 𝑥! that minimize the
standard deviation, we take the first derivative
with respect to 𝑥! , set the derivative equal to
zero and solve for 𝑥! , that is:
𝑑𝜎!
=0
𝑑𝑥!
If we take the first derivative and solve for 𝑥! ,
we find the general formula:
𝜎!! − 𝜎! 𝜎! 𝜌!"
𝑥! = ! !
𝜎! +𝜎! − 2𝜎! 𝜎! 𝜌!"
or, equivalently, as:
𝜎!! − 𝜎!"
𝑥! = ! !
𝜎! +𝜎! − 2𝜎!"
(since 𝜎! 𝜎! 𝜌!" = 𝜎!" )
Finally, note that when there in no relationship
between returns, that is when for 𝜌!" = 0, the
formula becomes:
𝜎!!
𝑥! = ! !
𝜎! +𝜎!
Figure 5.5 shows the graph in the expected
return / risk space for all possible values of 𝜌.
(See Figure 5.5)
Efficient Portfolios II
So far we have found that:
• The lower (closer to -1) the correlation
coefficient between assets, the higher the
payoff for diversification.
• We have found an expression for finding the
minimum variance portfolio when two assets
are combined in a portfolio.
We now move a bit further in the study of the
curve along which all combination must lie in
the return / risk space.
The Efficient Frontier with
No Short Sales
Suppose we were able to plot all possible risky
assets and combination of risky assets in
return / risk space.
We are interested in finding a set of portfolios
that offered:
1) A Higher Return for the same Risk.
2) A Lower Risk for the same Return.
Any investor would only be interested in
holding these portfolios (also called efficient
portfolios).
If we can identify these portfolios, all other
portfolios could be ignored.
Let’s see if we can identify these portfolios by
examining the following Figure 5.8.
(See Figure 5.8)
Consider Portfolios A and C
Portfolio C would be preferred by all investors
to portfolio A because it offers less risk for the
same level of return.
Consider Portfolios A and B
Portfolio B would be preferred to portfolio A
because it offers a higher return for the same
level of risk.
Note that:
• It is impossible to find portfolios which can
dominate portfolio C or portfolio B.
• An efficient set of portfolios cannot include
interior portfolios like A.
Let’s consider only portfolios on the exterior
points.
Consider point D
Portfolio D can be eliminated in favour of
Portfolio E (which has a higher return for the
same level of risk).
This is true for every other portfolio as we move
from D to point C on the exterior curve.
Portfolio C cannot be eliminated
Note that Portfolio C is the global minimum
variance portfolio.
The global minimum variance portfolio is
defined as the portfolio that has the lowest risk
of any feasible portfolio.
Consider point F
Portfolio F is dominated by Portfolio E (which
has less risk for the same return).
This is true for every other portfolio as we move
from F to B.
Portfolio B cannot be eliminated
Note that Portfolio B is the portfolio (usually a
single security) that offers the highest expected
return of all portfolios.
To summarize:
The efficient set consist of the curve of all
portfolios that lie between the global minimum
variance portfolio and the maximum return
portfolio.
The set is called efficient frontier.
Figure 5.9 shows it graphically.
(See Figure 5.9)
The Efficient Frontier
with Short Sales
In the stock market, an investor can often sell
a security that she/he does not own.
This process is called short-selling.
Let’s refresh the concept with an example.
Example
Suppose security X cost £10 at time T and you
think it will be worth £5 at time T+1.
You can:
• At time T borrow the security from your
broker (ignoring transaction cost,
dividends, etc…) and sell it for £10: You get
£10 but you need to return security X to
your broker.
• At time T+1, if X falls by £5, you can: go to
the market and buy security X for £5. Then
return security X that you borrowed
making a profit of £5.
The main reason for short sales is that the short
seller expects the price of the share to decline
and would like to profit from the decline.
Another reason is to decrease the sensitivity of a
portfolio to market movements.
Short sales make sense of course when an
investor expected return is negative, but it can
make sense also when expected returns are
positive. This is because the cash flow received
from short selling one security with low
expected return can be used to buy a security
with a higher expected return.
When we extend the analysis to efficient
frontiers of all securities in the case of short
sales, the efficient set still starts with the
minimum variance portfolio but it has no finite
upper bound.
Figure 5.12 shows the efficient set when short
sales are allowed (curve MVBC)
(See Figure 5.12)
Note that infinite expected rate of return exists
because one can take a short position on a
security with low expected return and use the
cash flow to buy a security with higher expected
return.
Efficient Frontier with Riskless
Lending and Borrowing
So far we have only considered portfolios or
risky assets.
We will now introduce a riskless asset in our
analysis.
Note that:
• Lending at a riskless rate can be seen as
investing in an asset with a certain
outcome.
• Borrowing at a riskless rate can be seen as
selling such an asset short.
The first thing to note is that since the return of
the riskless asset is fixed (which we call 𝑅! ), the
expected return is:
𝐸[𝑅! ] = 𝑅!
(Using the property of the expectation operator)
Also, since the return is certain, the variance of
the riskless asset is zero:
𝜎!! = 𝐸[𝑅! − 𝐸[𝑅! ]]! = 𝐸[𝑅! − 𝑅! ]! = 0
Assumptions:
• Investors can lend and borrow as much as
they want.
• Investors are interested in placing part of
their funds in some portfolio A and either
lending or borrowing.
• X = fraction of funds invested in portfolio A.
• (1 - X) = fraction of funds invested in the
riskless asset.
The expected return on this combination is:
𝑅! = 1 − 𝑋 𝑅! + 𝑋𝑅!
(1)
And the standard deviation is:
𝜎! = [(1 − 𝑋)! 𝜎!! + 𝑋 ! 𝜎!!
+ 2𝑋 1 − 𝑋 𝜎! 𝜎! 𝜌!" ]!/!
However, since the value of 𝜎!! is zero, it follows
that:
𝜎! = [𝑋 ! 𝜎!! ]!/! = 𝑋𝜎!
and rearranging for X we get:
𝜎!
𝑋=
𝜎!
Substituting X in the expected return Equation
(1), we find:
𝜎!
𝜎!
𝑅! = 1 −
𝑅! + 𝑅!
𝜎!
𝜎!
or
𝑅! − 𝑅!
𝑅! = 𝑅! +
𝜎!
𝜎!
Equation of a straight line.
It follows that the combinations of riskless
lending or borrowing with portfolio A lie on a
straight line in return / risk space, where:
• The intercept is the risk-free asset
• The slope is the ratio
!! !!!
!!
This line is shown in Figure 5.13.
(See Figure 5.13)
Note that:
• From the left of A: some funds will be in a
riskless asset and some in the risky portfolio
A (lending).
• On A: all funds will be invested in the risky
portfolio A.
• From the right of A: some funds will be
borrowed. The original funds plus the
borrowed funds will be placed in portfolio A.
Portfolio A has no special property.
However, we can move a bit further and find the
best portfolio of risky assets for us to hold.
To see this point, consider the intersection of
the efficient frontier with risky assets and the
efficient frontier when a riskless asset is
present.
(See Figure 5.14)
Consider the lines 𝑅! 𝐵 and 𝑅! 𝐴.
Combinations along 𝑅! 𝐵 are superior to
combinations along 𝑅! 𝐴, since they offer greater
return for the same risk.
This is to show that we can find the set of best
combinations by extending a straight line up
from 𝑅! and swinging it up until it touches point
G.
(See Figure 5.14)
Point G is the tangency point between the
efficient frontier and a ray passing through the
point 𝑅! .
It is important to stress that:
Portfolio G is the best portfolio of risky assets
for an investor to hold, irrespective of her/his
attitude towards risk.
In fact:
1) If an investor has a higher degree of risk
aversion:
he/she would buy both the risk-free
asset and portfolio G in some combination
(ending up in the 𝑅! G line).
2) If an investor has a lower degree of risk
aversion:
he/she might want to sell the risk-free
asset and use the cash flow, in addition to
the original funds, to invest in portfolio G
(ending up in the GH line).
3) Other investors would hold risky
portfolios with the exact composition of
portfolio G.
What is important is that:
All these investors would hold risky portfolios
with the exact combination of portfolio G.
Thus for the case of riskless lending and
borrowing, we have the solution to the
portfolio problem:
We can identify the best Portfolio G without
having to know anything about the investor
attitude to risk.
The ability to be able to determine the best (or
optimum) portfolio of risky assets has a name:
It is called the Separation Theorem.
Separation Theorem: Definition
Each investor will have a utility-maximizing
portfolio that is a combination of the risk-free
asset and a portfolio of risky assets
determined by the line drawn from the riskfree rate of return tangent to the investor’s
efficient set of risky assets.