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Lecture 5 - Optimal Portfolios
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FINS2624Portfolio Management
Lecture 5: Optimal Portfolios
P
R
O
B
A
B
I
L
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T
YPOSSIBLE OUTCOMES
Returns Under Uncertainty
Broader Asset Space & Uncertainty
On most asset types our cash flows are not known
U
Decision Making Framework based on Risk & Return
The satisfaction/utility of
investors captured via a trade
off between risk and return
They serve as the basis for asset and portfolio selection
Harry M Markowitz - Portfolio Selection:
Efficient Diversification of Investments
(1959)
E(r)
ฯ
๐ผ = ๐ฌ ๐ โ ๐ ๐๐จ๐๐ as general representation
P
R
O
B
A
B
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L
I
T
YPOSSIBLE OUTCOMES
Returns Under Uncertainty
Conditions of UncertaintyMeasuring Return and Risk
Expected return โ ๐ธ ๐๐ = ๐ ๐๐ ๐๐๐
as a measure of central tendency
Easily transferred to portfolios
๐ ๐๐ =
๐=๐
๐ต
๐๐๐ฌ ๐๐
P
R
O
B
A
B
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L
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YPOSSIBLE OUTCOMES
Returns Under Uncertainty
Variance โ ๐๐2 = ๐ธ ๐๐ โ ๐ธ ๐๐
2
as a measure of dispersion
Not so easily transferred to portfolios
๐๐ท๐ =
๐=๐
๐ต
๐๐๐๐๐๐ +
๐=๐
๐ต
๐=๐
๐ต
๐๐๐๐๐ช๐๐ ๐๐, ๐๐
iโ j
โจฏ
Simplifying Variance/CovarianceThe Covariance Matrix
Multiplying out the components as product of 2 vectors and summing the parts.
Method can be applied not just to variance but to any portfolio combination covariance
๐ค๐ด๐๐ด โ ๐ธ ๐ค๐ด๐๐ด ๐ค๐ต๐๐ต โ ๐ธ ๐ค๐ต๐๐ต โฆ ๐ค๐๐๐ โ ๐ธ ๐ค๐๐๐
๐ค๐ด๐๐ด โ ๐ธ ๐ค๐ด๐๐ด๐ค๐ต๐๐ต โ ๐ธ ๐ค๐ต๐๐ต
โฎ๐ค๐๐๐ โ ๐ธ ๐ค๐๐๐
๐ค๐ด2๐๐๐ ๐๐ด ๐ค๐ด๐ค๐ต๐ถ๐๐ฃ ๐๐ด, ๐๐ต โฆ ๐ค๐ด๐ค๐๐ถ๐๐ฃ ๐๐ด, ๐๐
๐ค๐ต๐ค๐ด๐ถ๐๐ฃ ๐๐ต , ๐๐ด ๐ค๐ต2๐๐๐ ๐๐ต โฆ ๐ค๐ต๐ค๐๐ถ๐๐ฃ ๐๐ต, ๐๐
โฎ๐ค๐๐ค๐ด๐ถ๐๐ฃ ๐๐ , ๐๐ด
โฎ๐ค๐๐ค๐ต๐ถ๐๐ฃ ๐๐, ๐๐ต
โฎ๐ค๐2๐๐๐ ๐๐
๐ค๐ด๐๐ด๐ค๐ต๐๐ตโฎ๐ค๐๐๐
๐ค๐ด๐๐ด ๐ค๐ต๐๐ต โฆ ๐ค๐๐๐
๐ค๐ด2๐๐๐ ๐๐ด ๐ค๐ด๐ค๐ต๐ถ๐๐ฃ ๐๐ด, ๐๐ต โฆ ๐ค๐ด๐ค๐๐ถ๐๐ฃ ๐๐ด, ๐๐
๐ค๐ต๐ค๐ด๐ถ๐๐ฃ ๐๐ต , ๐๐ด ๐ค๐ต2๐๐๐ ๐๐ต โฆ ๐ค๐ต๐ค๐๐ถ๐๐ฃ ๐๐ต, ๐๐
โฎ๐ค๐๐ค๐ด๐ถ๐๐ฃ ๐๐ , ๐๐ด
โฎ๐ค๐๐ค๐ต๐ถ๐๐ฃ ๐๐, ๐๐ต
โฎ๐ค๐2๐๐๐ ๐๐
OR
Investment Decision & Way Forward
E(r)
ฯ
Dominated (Excluded)
Portfolios
Eligible Portfolios
The Efficient FrontierOptimal
Portfolio
We choose the portfolio
which maximises utility
amongst all the possible
combinations mapped out
Conceptually this is given by
the intersection of the
Efficient Frontier (which
includes only the best
risk/return combinations)
with the highest possible
utility indifference curve
The task from here is to translate this idea into an actual portfolio of choice
Introducing a Risk Free Asset
Consider an asset which is risk free, defined by:
๐ธ ๐๐ = ๐๐ and ๐๐๐ ๐๐ = 0
This could be thought of as a short term US Government obligation
(Treasury Bill) or the like โ basically a security with a certain cash flow
If we combine a holding in a risky portfolio (proportion y) with the risk free
asset we can express the return on the Complete Portfolio (C) as:
๐๐ = 1 โ ๐ฆ ๐๐ + ๐ฆ๐๐. From that we can deduce:
๐ธ ๐๐ถ = 1 โ ๐ฆ ๐๐ + ๐ฆ๐ธ(๐๐) or:
๐ธ ๐๐ถ = ๐๐ + ๐ฆ[๐ธ ๐๐ โ ๐๐]
๐๐๐(๐๐ถ) = 1 โ ๐ฆ2๐๐๐ ๐๐ + ๐ฆ
2๐๐๐ ๐๐+2 1 โ ๐ฆ ๐ฆ๐ถ๐๐ฃ(๐๐, ๐๐ )
= ๐ฆ2๐๐๐ , ๐๐ = ๐ฆ2๐๐2 (since ๐๐ is fixed)
Hence ๐๐ถ = ๐ฆ๐๐
Both the expected return and the standard deviation are linear functions increasing
in y with โ๐ธ(๐๐ถ)
โ๐ฆ= [๐ธ ๐๐ โ ๐๐] and
โ๐๐
โ๐ฆ= ๐๐
The Risk Free Asset & the Capital Allocation Line
ฯ
๐ธ ๐๐ถ = ๐๐, ๐๐ถ = 0
When 100% invested in ๐๐
ฯP
๐ธ ๐๐ถ = ๐ธ(๐๐)๐๐ถ = ๐๐
When 100% invested in risky PP
Points on line represent
allocations between ๐๐ & risky P
rf
E(r)
E(rP)
The linear property makes it easy to map the different possible allocations
between the risk free and the risky portfolio out
We call the line formed by
joining the combinations of
return and risk from allocations
between the risk free asset
and some risky portfolio a
Capital Allocation Line (CALP)
ฯP
[๐ธ ๐๐ โ ๐๐]
Capital Allocation Lines for All Risky Portfolios
Moreover we need not stop with just the one risky portfolio. We can map
out CALs for all possibilities
ฯ
rf
E(r)
PN
P1
P2
The slope of each line, ๐ฌ ๐๐ท โ๐๐
๐๐ท,
gives reflects the implied trade off
between risk and return
Hence the steeper the CAL, the
better the additional return
expected from adding more risk
and the more attractive the portfolio
Flexibility & Portfolio CALs: Adding Some Scope for Borrowing
ฯ
rf
E(r)
PN
P1
P2
Combinations beyond PP
are associated with y > 1
This implies more the 100% of
allocation is going into risky
portfolio P โ the risk free
weight is negative, and we are
borrowing to invest
In practice though borrowing
rates (for investors) > lending
(deposit) rates
rf beyond PP such that ๐ฌ ๐๐ท โ๐๐
๐๐ท
& slope of CAL kinks lower
Portfolio CALs and Investment ChoiceInterplay with the Efficient Frontier
ฯ
rf
E(r)
P1
Efficient Frontier
Recall the Efficient Frontier as the
list of superior risky portfolio
combinations from which we will
select. It is therefore only these
we are concerned with
Including a risk free asset effectively excludes everything
to the left/below where the CAL intersects. This is due
the risk/return trade off being better โ it is possible to
achieve a zero risk with a still positive return
CALP
The Optimal Risky PortfolioConceptualised
ฯ
rf
E(r)
P*
P1
P2
The portfolio whose CAL
intersects Efficient Frontier
at the highest point allows
for the exclusion the most
amount of portfolios
Note that in this case that
even Efficient Frontier
portfolios to right are
dominated as well โ the
capacity to borrow at the risk
free rate allows improvement
on expected return outcomes
by taking on less additional
risk than we would be
possible otherwise
Best set of combinations
that can be attained
The Optimal Risky PortfolioThe Mechanics
There are 2 deductions that can help us here:
Deduction 1: That the best achievable risk
& returns combinations are captured by the
efficient frontier. Hence it is not possible to
go further to the left and higher at the same
time in risk return space
Deduction2: That introducing a risk free
asset and associated risky portfolio CALs
allows us to choose portfolio P* (at highest
possible intersection point with the Efficient
Frontier) which dominates all others
E(r)
ฯ
Implication: Can identify optimal portfolio
by simply considering the universe of
CALs. The one with the highest slope
will by definition run tangential to the
Efficient Frontier and so be associated
with the optimal risky portfolio
The Optimal Risky PortfolioThe Mechanics
ฯ
rf
E(r)
P*
P1
Recall that:
๐๐๐๐๐ ๐ถ๐ด๐ฟ๐ =๐ฌ ๐๐ท โ ๐๐
๐๐ท
This represents the risk reward
trade-off on the portfolio
involved and is otherwise
known at the Shape Ratio (SP)
Hence our problem amounts to
solving for portfolio weights in
๐=๐๐ต ๐๐ ๐๐ amongst universe of
assets according to
max ๐๐ ๐ค๐ =๐ฌ ๐๐ท โ ๐๐
๐๐ทvia a suitable computer program
P2
Efficient
Frontier
The Optimal Risky PortfolioInterpretation
ฯ
rf
E(r)
P*
P* is the portfolio that gives the
best risk return trade-off
amongst all alternatives. CAL* is
called The Capital Allocation
Line and shortened to just โCALโ
It is derived on an objective basis
on the assumption that all asset
variances and returns are known.
Therefore it is the optimal choice
for all investors and since all will
hold it the weights in the aggregate
will be the same too. That is why it
is also termed the Market Portfolio.
Approximations in reality would be
the market indices โ eg. S&P 500
Efficient
Frontier
Alternatives Narrowed Down but More Work to Do
ฯ
rf
E(r)
P*
The analytical framework so
far has delivered us with an
optimal risky portfolio, P*, and
the associated CAL - which
represents its different
possible combinations with
the risk free asset
Hence it has narrowed down
our list of choices to just rf
and P*. However, we are yet
to consider how best to
allocate between them
U
Bringing Back PreferencesThe Separation Theorem
ฯ
rf
E(r)
Having recognised our optimal Complete Portfolio lies on the CAL belonging to P*,
the objective criteria for narrowing down our choices has gone as far as it can go
To determine the end allocation
we need to incorporate
preferences for risk and return.
This is the Separation Theorem
The idea is to select the
implied combination of risk and
return which maximises
investor utility
This is represented by the point
at which the CAL intersects with
the highest possible
indifference curve
PC
U
U
Different Preferences โ Different Outcomes
E(r)
ฯ
Recall that each indifferent curve represents
combinations of risk and return that give the
same level of satisfaction/utility
And the generalised utility function:
๐ผ = ๐ฌ ๐ โ ๐ ๐๐จ๐๐, where A is a risk
aversion parameter than will differ by investor
E(r)
ฯ
Less Risk Averse Investor
More
Risk
Averse
Investor
Hence utility curves can have
different contours leading
different investors to vary in
their complete portfolio choice
Completing the Portfolio
In mathematical terms we are choosing the risk allocation (y) on the
CAL which maximises utility
Hence we substitute: ๐ธ ๐ = ๐๐ + ๐ฆ[๐ธ ๐๐ โ ๐๐] and ฯ2 = ๐ฆ2๐๐2 into our utility
function to give ๐ = ๐๐ + ๐ฆ ๐ธ ๐๐โ โ ๐๐ โ 12๐ด๐ฆ2๐๐โ2 as our representation
of utility for any point on the CAL
Our solution then becomes:
๐๐๐ฅ๐ ๐ฆ = ๐๐ + ๐ฆ ๐ธ ๐๐โ โ ๐๐ โ 12๐ด๐ฆ
2๐๐โ2
y*
U
y
It can help to picture this in y, U space
Max U with respect to y
Completing the Portfolio
y*
U
y
To determine the turning point we find the
derivative, set to zero and solve for y
So ๐๐
๐๐ฆ= ๐ธ ๐๐โ โ ๐๐ โ ๐ด๐ฆ
โ๐๐โ2 = 0
โ ๐ฆโ =๐ธ ๐๐โ โ๐๐
๐ด๐๐โ2 =
1
๐ดร๐ธ ๐๐โ โ๐๐
๐๐โ2
๐ฌ ๐๐ทโ โ๐๐
๐๐ทโ๐ is known as the reward to risk ratio and is closely related to
the Sharpe ratio, ๐ธ ๐๐โ โ๐๐
๐๐โ2
More gets allocated to the risky portfolio at higher reward to risk ratios while the
allocation is lower with the degree of risk aversion (A)
Breaking Down Market Portfolio Expected Value
Given its prominence the properties of the market portfolio are worth
considering in more detail, Some useful applications come from it as well
Consider ๐ฌ ๐๐ด = ๐=๐๐ต ๐๐๐ฌ ๐๐ and in particular the contribution,
๐๐๐ฌ ๐๐ , coming from each asset, i
If we rearrange in terms of portfolio excess return (over rf) we get:
๐ฌ ๐๐ด โ ๐๐ =
๐=๐
๐ต
๐๐[๐ฌ ๐๐ โ ๐๐]
with the contribution of each asset being ๐๐[๐ฌ ๐๐ โ ๐๐]
Breaking Down Market PortfolioVariance
We calculate the variance of the market portfolio in the usual way
ie. we set up the covariance matrix:
๐ค๐ด๐๐ด๐ค๐ต๐๐ตโฎ๐ค๐๐๐
๐ค๐ด๐๐ด ๐ค๐ต๐๐ต โฆ ๐ค๐๐๐
๐ค๐ด2๐๐๐ ๐๐ด ๐ค๐ด๐ค๐ต๐ถ๐๐ฃ ๐๐ด, ๐๐ต โฆ ๐ค๐ด๐ค๐๐ถ๐๐ฃ ๐๐ด, ๐๐
๐ค๐ต๐ค๐ด๐ถ๐๐ฃ ๐๐ต , ๐๐ด ๐ค๐ต2๐๐๐ ๐๐ต โฆ ๐ค๐ต๐ค๐๐ถ๐๐ฃ ๐๐ต, ๐๐
โฎ๐ค๐๐ค๐ด๐ถ๐๐ฃ ๐๐ , ๐๐ด
โฎ๐ค๐๐ค๐ต๐ถ๐๐ฃ ๐๐, ๐๐ต
โฎ๐ค๐2๐๐๐ ๐๐
And then sum the components
[๐ค๐๐1]
๐ค๐ด๐๐ด ๐ค๐ต๐๐ต โฆ ๐ค๐๐๐
๐ถ๐๐ฃ(๐ค๐๐๐ , ๐ค๐ด๐๐ด) ๐ถ๐๐ฃ(๐ค๐๐๐ , ๐ค๐ต๐๐ต)โฆ ๐ถ๐๐ฃ(๐ค๐๐๐ , ๐ค๐๐๐)
The contribution of each asset can be seen as the sum of the elements in its row
Therefore contribution asset i = ๐ถ๐๐ฃ ๐ค๐๐๐ , ๐=1๐ ๐ค๐๐๐ = ๐ถ๐๐ฃ ๐ค๐๐๐ , ๐๐ or ๐ค๐๐ถ๐๐ฃ ๐๐ , ๐๐
Breaking Down the Reward to Risk Ratio
With the asset contributions to market portfolio expected return and
variance worked out it follows that the contribution of asset i to the reward
to risk ratio, ๐ฌ ๐๐ด โ๐๐
๐๐ด๐ is given by
๐๐[๐ฌ ๐๐ โ๐๐]
๐ค๐๐ถ๐๐ฃ ๐๐,๐๐=[๐ฌ ๐๐ โ๐๐]
๐ถ๐๐ฃ ๐๐,๐๐
But the market portfolio is the portfolio with the best return to risk ratio so
it is not possible to improve on this by varying asset weights
The implication is that the asset reward to risk ratios must match that of
the market portfolio as well as each other
Therefore [๐ฌ ๐๐ โ๐๐]
๐ช๐๐ ๐๐,๐๐ด=[๐ฌ ๐๐ โ๐๐]
๐ช๐๐ ๐๐๐๐ด=๐ฌ ๐๐ด โ๐๐
๐๐ด๐
Importance of Contributions Result: The Capital Asset Pricing Model
Taking our [๐ฌ ๐๐ โ๐๐]
๐ช๐๐ ๐๐,๐๐ด=๐ฌ ๐๐ด โ๐๐
๐๐ด๐ result and expressing in terms of E(๐๐)
we get ๐ฌ ๐๐ = ๐๐ +๐ช๐๐ ๐๐,๐๐ด
๐๐ด๐ [๐ฌ ๐๐ด โ ๐๐]
This gives the required
(expected) return for a given
asset when these optimal
conditions hold
It suggests this goes up with the
assetโs market covariance risk and
the excess return on the market
portfolio
Hence in this framework it is only systematic risk that matters. The
expression is known as the Capital Asset Pricing Model