FINS2624Portfolio Management

Lecture 5: Optimal Portfolios

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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)

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𝑼 = 𝑬 𝒓 − 𝟏 𝟐𝑨𝝈𝟐 as general representation

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YPOSSIBLE OUTCOMES

Returns Under Uncertainty

Conditions of UncertaintyMeasuring Return and Risk

Expected return ⇒ 𝐸 𝑟𝑖 = 𝑠 𝑝𝑠𝑟𝑖𝑠

as a measure of central tendency

Easily transferred to portfolios

𝐄 𝒓𝒑 =

𝒊=𝟏

𝑵

𝒘𝒊𝑬 𝒓𝒊

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YPOSSIBLE OUTCOMES

Returns Under Uncertainty

Variance ⇒ 𝜎𝑖2 = 𝐸 𝑟𝑖 − 𝐸 𝑟𝑖

2

as a measure of dispersion

Not so easily transferred to portfolios

𝝈𝑷𝟐 =

𝒊=𝟏

𝑵

𝒘𝒊𝟐𝝈𝒊𝟐 +

𝒊=𝟏

𝑵

𝒋=𝟏

𝑵

𝒘𝒊𝒘𝒋𝑪𝒐𝒗 𝒓𝒊, 𝒓𝒋

i≠j

⨯

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

𝑤𝐴𝑟𝐴 − 𝐸 𝑤𝐴𝑟𝐴 𝑤𝐵𝑟𝐵 − 𝐸 𝑤𝐵𝑟𝐵 … 𝑤𝑁𝑟𝑁 − 𝐸 𝑤𝑁𝑟𝑁

𝑤𝐴𝑟𝐴 − 𝐸 𝑤𝐴𝑟𝐴𝑤𝐵𝑟𝐵 − 𝐸 𝑤𝐵𝑟𝐵

⋮𝑤𝑁𝑟𝑁 − 𝐸 𝑤𝑁𝑟𝑁

𝑤𝐴2𝑉𝑎𝑟 𝑟𝐴 𝑤𝐴𝑤𝐵𝐶𝑜𝑣 𝑟𝐴, 𝑟𝐵 … 𝑤𝐴𝑤𝑁𝐶𝑜𝑣 𝑟𝐴, 𝑟𝑁

𝑤𝐵𝑤𝐴𝐶𝑜𝑣 𝑟𝐵 , 𝑟𝐴 𝑤𝐵2𝑉𝑎𝑟 𝑟𝐵 … 𝑤𝐵𝑤𝑁𝐶𝑜𝑣 𝑟𝐵, 𝑟𝑁

⋮𝑤𝑁𝑤𝐴𝐶𝑜𝑣 𝑟𝑁 , 𝑟𝐴

⋮𝑤𝑁𝑤𝐵𝐶𝑜𝑣 𝑟𝑁, 𝑟𝐵

⋮𝑤𝑁2𝑉𝑎𝑟 𝑟𝑁

𝑤𝐴𝑟𝐴𝑤𝐵𝑟𝐵⋮𝑤𝑁𝑟𝑁

𝑤𝐴𝑟𝐴 𝑤𝐵𝑟𝐵 … 𝑤𝑁𝑟𝑁

𝑤𝐴2𝑉𝑎𝑟 𝑟𝐴 𝑤𝐴𝑤𝐵𝐶𝑜𝑣 𝑟𝐴, 𝑟𝐵 … 𝑤𝐴𝑤𝑁𝐶𝑜𝑣 𝑟𝐴, 𝑟𝑁

𝑤𝐵𝑤𝐴𝐶𝑜𝑣 𝑟𝐵 , 𝑟𝐴 𝑤𝐵2𝑉𝑎𝑟 𝑟𝐵 … 𝑤𝐵𝑤𝑁𝐶𝑜𝑣 𝑟𝐵, 𝑟𝑁

⋮𝑤𝑁𝑤𝐴𝐶𝑜𝑣 𝑟𝑁 , 𝑟𝐴

⋮𝑤𝑁𝑤𝐵𝐶𝑜𝑣 𝑟𝑁, 𝑟𝐵

⋮𝑤𝑁2𝑉𝑎𝑟 𝑟𝑁

OR

Investment Decision & Way Forward

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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:

𝐸 𝑟𝑓 = 𝑟𝑓 and 𝑉𝑎𝑟 𝑟𝑓 = 0

This could be thought of as a short term US Government obligation

(Treasury Bill) or the like – 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:

𝑟𝑐 = 1 − 𝑦 𝑟𝑓 + 𝑦𝑟𝑝. From that we can deduce:

𝐸 𝑟𝐶 = 1 − 𝑦 𝑟𝑓 + 𝑦𝐸(𝑟𝑃) or:

𝐸 𝑟𝐶 = 𝑟𝑓 + 𝑦[𝐸 𝑟𝑃 − 𝑟𝑓]

𝑉𝑎𝑟(𝑟𝐶) = 1 − 𝑦2𝑉𝑎𝑟 𝑟𝑓 + 𝑦

2𝑉𝑎𝑟 𝑟𝑃+2 1 − 𝑦 𝑦𝐶𝑜𝑣(𝑟𝑓, 𝑟𝑃 )

= 𝑦2𝑉𝑎𝑟 , 𝑟𝑃 = 𝑦2𝜎𝑃2 (since 𝑟𝑓 is fixed)

Hence 𝜎𝐶 = 𝑦𝜎𝑃

Both the expected return and the standard deviation are linear functions increasing

in y with ∆𝐸(𝑟𝐶)

∆𝑦= [𝐸 𝑟𝑃 − 𝑟𝑓] and

∆𝜎𝑐

∆𝑦= 𝜎𝑃

The Risk Free Asset & the Capital Allocation Line

σ

𝐸 𝑟𝐶 = 𝑟𝑓, 𝜎𝐶 = 0

When 100% invested in 𝑟𝑓

σP

𝐸 𝑟𝐶 = 𝐸(𝑟𝑃)𝜎𝐶 = 𝜎𝑃

When 100% invested in risky PP

Points on line represent

allocations between 𝑟𝑓 & 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)

σP

[𝐸 𝑟𝑃 − 𝑟𝑓]

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

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rf

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P1

P2

The slope of each line, 𝑬 𝒓𝑷 −𝒓𝒇

𝝈𝑷,

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

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rf

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PN

P1

P2

Combinations beyond PP

are associated with y > 1

This implies more the 100% of

allocation is going into risky

portfolio P ⇒ 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 𝑬 𝒓𝑷 −𝒓𝒇

𝝈𝑷

& slope of CAL kinks lower

Portfolio CALs and Investment ChoiceInterplay with the Efficient Frontier

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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 – it is possible to

achieve a zero risk with a still positive return

CALP

The Optimal Risky PortfolioConceptualised

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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 – 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

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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

σ

rf

E(r)

P*

P1

Recall that:

𝑆𝑙𝑜𝑝𝑒 𝐶𝐴𝐿𝑃 =𝑬 𝒓𝑷 − 𝒓𝒇

𝝈𝑷

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

𝒊=𝟏𝑵 𝒘𝒊 𝒓𝒊 amongst universe of

assets according to

max 𝑆𝑝 𝑤𝑃 =𝑬 𝒓𝑷 − 𝒓𝒇

𝝈𝑷via a suitable computer program

P2

Efficient

Frontier

The Optimal Risky PortfolioInterpretation

σ

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 “CAL”

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 – eg. S&P 500

Efficient

Frontier

Alternatives Narrowed Down but More Work to Do

σ

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

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Bringing Back PreferencesThe Separation Theorem

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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

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U

Different Preferences – Different Outcomes

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Recall that each indifferent curve represents

combinations of risk and return that give the

same level of satisfaction/utility

And the generalised utility function:

𝑼 = 𝑬 𝒓 − 𝟏 𝟐𝑨𝝈𝟐, where A is a risk

aversion parameter than will differ by investor

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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: 𝐸 𝑟 = 𝑟𝑓 + 𝑦[𝐸 𝑟𝑃 − 𝑟𝑓] and σ2 = 𝑦2𝜎𝑃2 into our utility

function to give 𝑈 = 𝑟𝑓 + 𝑦 𝐸 𝑟𝑃∗ − 𝑟𝑓 − 12𝐴𝑦2𝜎𝑃∗2 as our representation

of utility for any point on the CAL

Our solution then becomes:

𝑚𝑎𝑥𝑈 𝑦 = 𝑟𝑓 + 𝑦 𝐸 𝑟𝑃∗ − 𝑟𝑓 − 12𝐴𝑦

2𝜎𝑃∗2

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 𝜕𝑈

𝜕𝑦= 𝐸 𝑟𝑃∗ − 𝑟𝑓 − 𝐴𝑦

∗𝜎𝑃∗2 = 0

⇒ 𝑦∗ =𝐸 𝑟𝑃∗ −𝑟𝑓

𝐴𝜎𝑃∗2 =

1

𝐴×𝐸 𝑟𝑃∗ −𝑟𝑓

𝜎𝑃∗2

𝑬 𝒓𝑷∗ −𝒓𝒇

𝝈𝑷∗𝟐 is known as the reward to risk ratio and is closely related to

the Sharpe ratio, 𝐸 𝑟𝑃∗ −𝑟𝑓

𝜎𝑃∗2

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 𝑬 𝒓𝑴 = 𝒊=𝟏𝑵 𝒘𝒊𝑬 𝒓𝒊 and in particular the contribution,

𝒘𝒊𝑬 𝒓𝒊 , coming from each asset, i

If we rearrange in terms of portfolio excess return (over rf) we get:

𝑬 𝒓𝑴 − 𝒓𝒇 =

𝒊=𝟏

𝑵

𝒘𝒊[𝑬 𝒓𝒊 − 𝒓𝒇]

with the contribution of each asset being 𝒘𝒊[𝑬 𝒓𝒊 − 𝒓𝒇]

Breaking Down Market PortfolioVariance

We calculate the variance of the market portfolio in the usual way

ie. we set up the covariance matrix:

𝑤𝐴𝑟𝐴𝑤𝐵𝑟𝐵⋮𝑤𝑁𝑟𝑁

𝑤𝐴𝑟𝐴 𝑤𝐵𝑟𝐵 … 𝑤𝑁𝑟𝑁

𝑤𝐴2𝑉𝑎𝑟 𝑟𝐴 𝑤𝐴𝑤𝐵𝐶𝑜𝑣 𝑟𝐴, 𝑟𝐵 … 𝑤𝐴𝑤𝑁𝐶𝑜𝑣 𝑟𝐴, 𝑟𝑁

𝑤𝐵𝑤𝐴𝐶𝑜𝑣 𝑟𝐵 , 𝑟𝐴 𝑤𝐵2𝑉𝑎𝑟 𝑟𝐵 … 𝑤𝐵𝑤𝑁𝐶𝑜𝑣 𝑟𝐵, 𝑟𝑁

⋮𝑤𝑁𝑤𝐴𝐶𝑜𝑣 𝑟𝑁 , 𝑟𝐴

⋮𝑤𝑁𝑤𝐵𝐶𝑜𝑣 𝑟𝑁, 𝑟𝐵

⋮𝑤𝑁2𝑉𝑎𝑟 𝑟𝑁

And then sum the components

[𝑤𝑖𝑟1]

𝑤𝐴𝑟𝐴 𝑤𝐵𝑟𝐵 … 𝑤𝑁𝑟𝑁

𝐶𝑜𝑣(𝑤𝑖𝑟𝑖 , 𝑤𝐴𝑟𝐴) 𝐶𝑜𝑣(𝑤𝑖𝑟𝑖 , 𝑤𝐵𝑟𝐵)… 𝐶𝑜𝑣(𝑤𝑖𝑟𝑖 , 𝑤𝑁𝑟𝑁)

The contribution of each asset can be seen as the sum of the elements in its row

Therefore contribution asset i = 𝐶𝑜𝑣 𝑤𝑖𝑟𝑖 , 𝑗=1𝑁 𝑤𝑗𝑟𝑗 = 𝐶𝑜𝑣 𝑤𝑖𝑟𝑖 , 𝑟𝑀 or 𝑤𝑖𝐶𝑜𝑣 𝑟𝑖 , 𝑟𝑀

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, 𝑬 𝒓𝑴 −𝒓𝒇

𝝈𝑴𝟐 is given by

𝒘𝒊[𝑬 𝒓𝒊 −𝒓𝒇]

𝑤𝑖𝐶𝑜𝑣 𝑟𝑖,𝑟𝑀=[𝑬 𝒓𝒊 −𝒓𝒇]

𝐶𝑜𝑣 𝑟𝑖,𝑟𝑀

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 [𝑬 𝒓𝒊 −𝒓𝒇]

𝑪𝒐𝒗 𝒓𝒊,𝒓𝑴=[𝑬 𝒓𝒋 −𝒓𝒇]

𝑪𝒐𝒗 𝒓𝒋𝒓𝑴=𝑬 𝒓𝑴 −𝒓𝒇

𝝈𝑴𝟐

Importance of Contributions Result: The Capital Asset Pricing Model

Taking our [𝑬 𝒓𝒊 −𝒓𝒇]

𝑪𝒐𝒗 𝒓𝒊,𝒓𝑴=𝑬 𝒓𝑴 −𝒓𝒇

𝝈𝑴𝟐 result and expressing in terms of E(𝑟𝑖)

we get 𝑬 𝒓𝒊 = 𝒓𝒇 +𝑪𝒐𝒗 𝒓𝒊,𝒓𝑴

𝝈𝑴𝟐 [𝑬 𝒓𝑴 − 𝒓𝒇]

This gives the required

(expected) return for a given

asset when these optimal

conditions hold

It suggests this goes up with the

asset’s 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