Choosing an Investment Portfolio
P.V. Viswanath
Based on Bodie and Merton
Returns and Return Uncertainty
 If an investor buys an asset, a, at time 0 for $P0, receives a
cashflow at time 1 of $D1 and sells it at time 1, for $P1, the
return on the asset is defined as Ra,1 = (P1 + D1 – P0)/P0
 At time 0, the actual value of Ra,1 is unknown.
 Suppose there are three different possible outcomes,
corresponding to three different states: states 1, 2 and 3.
 Thus Ra,1 could be equal to 3%, 7% or 10% with
probabilities of 0.3, 0.2 and 0.5 for the three different states.
 The mean or expected value of Ra,1 is computed as 0.3(3%)
+ 0.2(7%) + 0.5(10%) = 7.3%
 The expected value is a kind of average outcome, where the
different possible outcomes are weighted by the
probabilities.
 E(Ra) is the way we represent the expected return on asset a.
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Return Volatility
 Consider another asset, b, which has returns of 1%,
7% and 11.2% in the three different states.
 The expected return of asset b is 0.3(1%) + 0.2(7%)
+ 0.5(11.20%), which is also 7.3%.
 However, asset b’s returns are said to be more
volatile than those of asset a, meaning roughly that
the range of possible outcomes is greater, or that
more extreme outcomes – both positive and
negative – are more likely.
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Variance
 The variance of returns is computed as the mean squared
deviation from the expected return.
 The squared deviation is a measure of how extreme a return
is.
 Taking the mean computes the average value of these
deviations.
 The square root of the variance is called the standard
deviation or volatility.
 Variance is measured in percent-squared.
 Standard deviation is measured in the same units as returns,
viz. percent.
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Return volatilities of Assets a & b
Probabilities
0.3
0.2
0.5
Probabilities
Return (Asset a)
Deviation from
Mean Return
3
7
10
-4.3
-0.3
2.7
Return (Asset b)
Deviation from
Mean Return
Squared
Deviation
18.49
0.09
7.29
Squared
Deviation
0.3
1
-6.3
39.69
0.2
7
-0.3
0.09
0.5
11.2
3.9
19.53
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Variance and Standard Deviations
 We see that the variance of returns for asset a is 9.21,
computed as 0.3(18.49) + 0.2(0.09) + 0.5(7.29) and the
standard deviation is 3.035%
 The variance of returns for asset b can be similarly
computed and is 19.53; the standard deviation is 4.419%.
 sa is used to represent the standard deviation of returns on
asset a, while Var(Ra) or sa2 is used to represent the variance
of returns on asset a.
 We see that, as expected asset b, which has more extreme
outcomes, also has a higher volatility.
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Portfolios
 A portfolio is defined as a set of assets and the proportions
of money invested in those assets.
 Thus, if we agree on a set of assets, i = 1,…,n, then a set of
weights wi, i=1,..,n represents a portfolio.
 If the assets are all stocks – SBUX, VZ and F, then (0.1,
0.45, 0.45) is a portfolio representing an investment of $100
in SBUX, $450 in VZ and $450 in F; or equivalently, $50 in
SBUX, $225 in VZ and $225 in F.
 Even though the dollar returns will vary for the two
portfolios, the percentage returns will always be the same;
hence the mean return and variance of returns will be the
same.
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Portfolio Return
 Suppose we construct a portfolio, consisting of $30 in asset
a and $70 in asset b; i.e. wa = 0.3 and wb = 0.7.
 Then the return on the portfolio in any given state will be
waRa + wbRb.
 We can then compute the expected return for this portfolio,
as well as its variance of returns.
 A riskfree asset always has the same return independent of
the state.
 Hence, its standard deviation of returns is zero.
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Combining risky and riskless assets
 Suppose we combine a risky portfolio, P with a risk-free
asset, f.
 Let C denote the complete portfolio, and y the proportion
invested in the risky portfolio. Then,
 rC = y rP + (1-y) rf
 E(rC) = y E(rP) + (1-y) rf, and
 sC = y sP
 Suppose E(rP) = 15%, rf = 7%, and sP = 22%, then we can
compute the trade-off between expected return and volatility
obtained by putting different proportions in the risky
portfolio and the risk-free asset.
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The Risk-Reward Trade-Off Line
Capital Allocation Line
P
E(rp)=15%
E(rp) - rf = 8%
rf = 7%
Risk-reward ratio = 8/22
f
sp= 22%
s
Figure: Capital Allocation Line
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Covariance and Correlation
 If we have two assets a and b, the covariance
between the returns on the two assets is defined as
the weighted average of the product of deviations of
each return from its mean return.
 The correlation is the ratio of the covariance to the
product of the two standard deviations. The
correlation can only take values between -1 and +1.
 We can use the deviations of the returns on assets a
and b in our earlier example to compute the
covariance coefficient and the correlation
coefficient.
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Covariance
Probabilities
Ra – E(Ra)
Rb – E(Rb)
Product of
deviations
0.3
-4.3
-6.3
27.09
0.2
-0.3
-0.3
0.09
0.5
2.7
3.9
10.53
 The covariance can be computed as 0.3(17.49) +
0.2(0.49) + 0.5(19.24) = 13.41
 The correlation coefficient is
13.41/(3.035 x 4.419) = 0.99988
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Covariance
 The covariance (and correlation coefficient) is a measure of how
closely two variables move together.
 In our case, both assets have high returns in state 1 and low returns
in state 2; hence they have a high correlation. If there were another
asset, c, with low returns in state 1 and high returns in state 2, it
would have a negative correlation with asset a.
 An asset, d, with a return of 1%, 7% and 45% in the three states can
be shown to have a correlation coefficient of only 0.93 with asset a.
 This correlation is less than the correlation between assets a and b
because asset d behaves quite differently from asset a in state 3,
even though state 3 is a good state for both assets.
 If we have many different states, we can construct assets with
different kinds of correlation coefficients between their returns,
corresponding to reality.
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Combinations of risky assets
 Suppose we invest positive amounts in two risky assets that
are imperfectly positively correlated.
 Then, because of the potential for diversification or offset
between the returns of different assets in a portfolio, the
standard deviation of returns on a portfolio is less than the
weighted average of the standard deviation of returns on the
assets making up the portfolio.
 The smaller the correlation coefficient, the greater the
possibility of diversification and vice-versa.
 Thus, if we combine assets a and b in a portfolio, there will
not be much diversification; however, if we combine assets a
and d, there will be more diversification, while there will be
even more diversification and potential for variance
reduction if we combine assets a and c.
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Combinations of risky assets
 Suppose we combine two risky assets, D and E,
with portfolio weights wD and wE.
 Then the expected return and the variance of returns
on the portfolio are given by
E ( rp )  wD E ( rD )  wE E ( rE )
s  w s  w s  2wD wE Cov(rD rE )
2
p
2
2
D
D
2
E
2
E
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Combinations of risky assets
The reward-to-variability ratio at
portfolio p is equal to the slope of the
tangent to the combination line at p.
E(r)
E
*
p
*
*
D
Minimum Variance Portfolio
Hypothetical combination line if
standard deviations combined linearly
s(r)
Figure: Combination Line of Portfolios
constructed from two risky assets
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Optimal Combinations of risky assets and
the risk-free asset
 If we have assets D and E, as above, but we have, in
addition, a risk-free asset as well.
 Then, we can think of the optimal combination of the three
assets as being achieved in two steps.
 First compute the optimal combination, P, of assets D and E.
This portfolio is also called the tangency portfolio.
 Next combine this portfolio with the risk-free asset.
 In practice, the optimal risky portfolio will depend upon the
risk-free rate, as well, as can be seen from the graph in the
next slide.
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Optimal Combinations of risky assets and
the risk-free asset
Capital Allocation Line
E
*
P
*
*
*
D
rf
Optimal Portfolio
Figure: Location of the optimal portfolio
when there is a risk-free asset
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The Tangency Portfolio
 The proportions of assets D and E in the tangency portfolio
are given by the formulas below:
wD 
[ E (rD )  r f ]s E2  [ E (rE )  r f ]Cov(rD , rE )
[ E (rD )  r f ]s E2  [ E (rE )  r f ]s D2  [ E (rD )  r f  E (rE )  r f ]Cov(rD , rE )
wE  1  wD
 The particular proportions that the investor would then invest
in the tangency portfolio versus the risk-free asset depends on
his/her risk aversion. The greater the risk aversion, the more
will be invested in the risk-free asset.
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The minimum variance portfolio
 The proportions of assets D and E in the minimum
variance portfolio are given by the formulas below:
s 2E  Cov( rD , rE )
wmin ( D)  2
; wmin ( E )  1  wmin ( D)
2
s D  s E  2Cov( rD , rE )
 Of course, no rational investor would actually hold
this portfolio, unless s/he were extremely riskaverse.
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Efficient Frontier with Two Assets
 The efficient frontier is the set of portfolios of risky assets
offering the highest possible expected rate of return for any
given standard deviation.
 With two risky assets and one risk-free asset, the efficient
frontier is precisely the line starting from the risk-free asset
that is tangential to the combination line consisting of
combinations of the two risky assets.
 For any standard deviation of returns the highest return is
given by some portfolio on the above straight line.
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Efficient Frontier with many assets
 Suppose there are many risky assets, but no risk-free asset.
 Then conceptually, we can imagine taking pairs of assets
and drawing their combination lines. Any point on one of
these combination line is a portfolio of those two particular
assets. We could then imagine combination lines where the
two risky investments, themselves, are from the
“combination” portfolios generated above.
 There would be a very large, potentially infinite number of
these combination lines.
 If we consider all of the points on these infinite combination
lines, we would have a dense area, bounded on the left by a
parabola.
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Efficient Frontier with Many Risky Assets
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Efficient Frontier with Many Risky Assets
and a Risk-free Asset
 In this case, the optimal combination of risky assets
is once again found as the point of tangency
between a straight line from the point representing
the riskless asset and the efficient frontier of risky
assets.
 The straight line connecting the riskless asset and
the tangency point representing the optimal
combination of risky assets is the best risk-reward
trade-off line available – that is, the efficient
frontier.
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Efficient Frontier with Many Risky Assets
and a Risk-free Asset
Optimal Combination of the
risk-free asset and Portfolio P.
Efficient Frontier

E(r)

Efficient Portfolio, P, of Risky Assets:
Forms the risky part of all investors' optimal portfolios.
rf
s(r)
Figure: Optimal Portfolio Selection with Many Risky
Assets and a Risk-free Asset
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