Lecture 5

Optimal portfolios

Learning outcomes

By the end of this lecture you should: Be familiar with the separation theorem Know why this implies that every investors optimal risky portfolio is the market portfolio

Be able to solve the portfolio problem in the presence of a risk free asset and the market portfolio

Understand how this implies the CAPM

The optimal portfolio

Last time we found the optimal portfolio by picking the portfolio on the efficient frontier that touched our highest indifference curve (gave the highest utility). E(r)

Introduce a risk free asset

Suppose that in addition to the risky assets that we talked about last lecture, we could also invest in some risk free asset

Well call the return of this asset the risk free return, rf

By definition of risk free, we have During our first three lecture we were only concerned with these kind of assets

For simplicity we will assume that we have a flat and constant term structure of interest rates

ff rrE 0 frVar

Combining the risk free asset with a risky portfolio into a complete portfolio

Suppose we want to combine some risky portfolio, P, with the risk free asset

Lets denote the fraction invested in the risky asset y and the fraction invested in the risk free asset (1-y)

The return of our complete portfolio, C, is defined as before:

PfC yrryr 1

The expected return of a complete portfolio

As usual, we are interested in the risk and expected return of this portfolio. Lets start with the expected return:

Like before the portfolio expected return is a weighted average of the asset expected returns

It will be convenient to express this equation as:

By varying y, we can choose our expected return

> @ PfC

PfC

ryEryrEyrryErE

1

1

> @fPfC rrEyrrE

The risk of a complete portfolio

We figured out in the last lecture how to calculate the variance of a two asset portfolio:

The neat thing here is that rf is a constant, so Var(rf) = 0 and Cov(rf, rP) = 0:

We find that C is linear in P

> @ PfPfC

PfC

PfC

rryCovyrVaryrVaryrVar

yrryVarrVaryrryr

,121

1

1

22

PPC

PPC

yy

yrVaryrVar

VVVV

22

222

These are linear combinations

We have found that:

So y determines what fraction of the distance between the risk free asset and P is covered in both dimensions

When y = 0 we are in the risk free asset, e.g. C = 0 and E(RC) = rf

When y = 0.5 we are halfway between the risk free asset and P, e.g. C = 0.5P and E(RC) = rf + 0.5[E(RC) - rf]

When y = 1 we are in portfolio P, e.g. C = P and E(RC) = E(RP)

> @PC

fPfC

yrrEyrrE

VV

Lets plot it

Graphically, this means that all our complete portfolios plot on a straight line between the risk free asset and P

Lets call the line associated with the risky portfolio P for CALP (for reasons that will become clear later)

E(r)

rf

P CALP

Lets plot it

Last lecture, we learned only to consider risky portfolios on the efficient frontier, so lets chose P from that set

E(r)

rf

P CALP

Apply the mean variance criterion

We see that some of our complete portfolios dominate some portfolios on the efficient frontier

Which ones are dominated depends on the risky portfolio we choose

E(r)

rf P1

P2

CALP1

CALP2

Apply the mean variance criterion

Were interested in choosing the P that dominates the most portfolios This turns out to be the portfolio where the line of complete portfolios is

tangent to the efficient frontier

We call this P the optimal risky portfolio, P*

The associated line CALP* is often simply denoted CAL

E(r)

rf

P*

CALP* = CAL

Apply the mean variance criterion

A different way to phrase this is to note that we only consider risky portfolios on the efficient frontier

We can then forget about the efficient frontier and only compare CALs We note that for any portfolio on CAL1 there is a dominating portfolio just above it on CAL2 Portfolios on CALs with higher slopes will always dominate portfolios on CALs with lower

slopes

The optimal risky portfolio is the portfolio associated with the CAL that has the highest slope

E(r)

rf

CAL1

CAL2

The optimal risky portfolio

All portfolios other than P* on the efficient frontier are dominated by some combination of the optimal risky portfolio, P*, and the risk free asset

This means that all efficient portfolios consist of some such combination

The reason we call the corresponding line CAL is that all capital will be allocated along it

CAL is an acronym for the Capital Allocation Line

The separation theorem

All efficient portfolios (in the presence of a risk free asset) are on the CAL

The CAL is determined by the optimal risky portfolio We pick the portfolio on the CAL that offers the

amount of risk we want to take. This is expressed in our choice of y.

Thus, our entire portfolio choice problem can be separated into two parts: Find the optimal risky portfolio, P* Choose how much risk we want by choosing the fraction of

our wealth that we invest in that portfolio, y

Step 1: Choosing P* (and implicitly the CAL)

The CAL is a line in risk-return space Its slope, S, determines how much reward in terms of

E(r) we get for taking on one more unit of risk We can easily calculate this slope from our two known

points on the line:

We usually refer to the slope of a CAL associated with a given portfolio as the Sharpe ratio of that portfolio

*

*

*

*

0 PfP

P

fP rrErErExyS VV

'

'

Step 1: Choosing P*

Recall from the construction of our utility function that we only care about E(r) and

This means we always prefer a higher Sharpe ratio, S We find the optimal portfolio, P*, by choosing its

portfolio weights, wP, so as to maximizes SP:

This gets messy (at least when we can choose between many assets)

You can do this using the Excel solver or some other suitable computer program

P

fPPw

rrES

P V max

Step 2: Choosing the risky share, y

We choose y according to our risk preference, which is modeled in our utility function by A

We may again illustrate this choice using indifference curves:

E(r)

rf

P*

C

Step 2: Choosing the risky share, y

Once we have determined P*, we know from before that our risk and return will be:

We also know that our utility will depend on these quantities in the following manner:

Lets combine these equations: > @ > @ 2*221*2*21* PfPfPfPf AyrrEyryArrEyrU VV

221 VArEU

> @PC

fPfC

yrrEyrrE

VV

Step 2: Choosing the risky share, y

By choosing y, we choose where to end up on the CAL Lets choose y so as to maximize our utility: We set the first derivative equal to zero and solve for y:

is known as the reward-to-risk ratio (and has an interpretation that is very similar to the Sharpe ratio)

y* is increasing in this ratio, meaning that the more rewards in terms of E(r) we get for taking on extra risk, the more we invest in the risky portfolio

y* is decreasing in A, meaning that the more risk-averse we are, the less we invest in the risky portfolio

> @ 2*221*max PfPfy AyrrEyrU V

> @

2*

*2*

**

2**

1

0

P

fP

P

fP

PfP

rrEAA

rrEy

AyrrEyU

VV

V

ww

2*

*

P

fP rrEV

Leveraged positions

There is nothing in principle that prevents us from choosing y > 1

This means that well take a short position in the risk-free asset, i.e. (1 - y) < 0

The interpretation of this is that we borrow money

We say that we take a leveraged position in P*

Borrowing constraints

In practice, we must borrow at a higher rate than we can invest at

This is because lending money to us is not really risk free

Graphically we get a kink in the CAL when y = 1

Since wed have higher default risks for more leveraged positions, the CAL may also be concave when y > 1

Implications of the separation theorem

The rational way to increase risk taking is to increase leverage (not to buy more tech stocks)

All investors will end up holding the same risky portfolio

Since prices adjust to set the supply of stocks equal to the demand for stocks, the portfolio demanded must be the portfolio supplied

P* is the market portfolio, M

Implications of the separation theorem

The attractiveness of a stock is determined by its risk and return effects on this portfolio

We saw last lecture that the expected return effect of a stock on a portfolio is linear and that the risk effect depends crucially on its covariance with the other stocks in the portfolio

A note on expected returns A company that issues a stock is basically selling a claim to its future

profits These profits are determined by the operations of the company Since the profits are risky, the company has to sell the claims at a price

that is lower than their expected value If the price is lower, the expected return of the investors is higher Since these high expected returns are used to induce investors to hold

unattractive risky stocks, high expected returns signify unattractive stocks, which may seem counterintuitive

Of course, high expected returns are not themselves unattractive In equilibrium, expected returns are set so as to make all stocks equally

attractive Some times we emphasize this by referring to a stocks expected return as

its required return (to make it as attractive as all other stocks)

The market portfolio

Recall that the market portfolio is the optimal risky portfolio for all investors

Each investor buys a small fraction of the portfolio

The entire market portfolio simply consists of all assets

Just like in other portfolios, the weight of each asset in the market portfolio is the assets total market value divided by the total value of the portfolio:

j

j

ii V

Vw

The expected return of the market portfolio

The return of the market portfolio, rM, is

We calculate expectations just like with any other portfolio

It is clear that the contribution of asset i to the expected return of the

market portfolio is

It will be useful to express this as the contribution of asset i to the market portfolios excess return, i.e. its return over and above the risk-free rate

The contribution of asset i to the market excess return is

N

iiiM rwr

1

> @

N

ifiifM rrEwrrE

1

ii rEw

N

iiiM rEwrE

1

> @fii rrEw

The variance of the market portfolio

We calculate the variance just like any other portfolio variance, i.e. by setting up the covariance matrix and summing the elements:

Note that every asset corresponds to one row in the matrix

w1r1 w2r2 wNrN

w1r1 Cov(w1r1,w1r1) Cov(w1r1,w2r2) Cov(w1r1,wNrN)

w2r2 Cov(w2r2,w1r1) Cov(w2r2,w2r2) Cov(w2r2,wNrN)

wNrN Cov(wNrN,w1r1) Cov(wNrN,w2r2) Cov(wNrN,wNrN)

The variance of the market portfolio

The contribution of each asset to the variance of the market portfolio is captured by the sum of the elements in its row

Lets view the row for asset i in isolation:

This matrix corresponds to the matrix we would set up to calculate

w1r1 w2r2 wNrN

wiri Cov(w1r1,w1r1) Cov(w1r1,w2r2) Cov(w1r1,wNrN)

N

jjjii rwrwCov

1,

The risk-return ratio of the market portfolio

We see that the contribution of asset i to the variance of the market portfolio is

Note that the reward-to-risk ratio of the market portfolio is:

The contribution of asset i to this ratio is:

MiiMiiN

jjjii rrCovwrrwCovrwrwCov ,,,

1

2M

fM rrEV

> @

Mi

fi

Mii

fii

rrCovrrE

rrCovwrrEw

,,

The risk-return ratio of the market portfolio

Since the market portfolio is the portfolio with the best risk-return ratio, it cannot be improved by changing the portfolio weights

This means that no isolated investment can make a larger contribution to the risk-return ratio than any other investment:

This is also true for the market portfolio itself:

Mj

fj

Mi

fi

rrCovrrE

rrCovrrE

,,

2,, M

fM

MM

fM

Mi

fi rrErrCovrrE

rrCovrrE

V

The CAPM

We can rewrite this equation as

This equation expresses a relation that must hold between an assets expected return and its covariance with the market

We call this model the Capital Asset Pricing Model or the CAPM

It will be the focus of our coming lectures

2

2

,

,

i f M f

i M M

i Mi f M f

M

E r r E r rCov r r

Cov r rE r r E r r

V

V

Lecture 5Learning outcomesThe optimal portfolioIntroduce a risk free assetCombining the risk free asset with a risky portfolio into a complete portfolioThe expected return of a complete portfolioThe risk of a complete portfolioThese are linear combinationsLets plot itLets plot itApply the mean variance criterionApply the mean variance criterionApply the mean variance criterionThe optimal risky portfolioThe separation theoremStep 1: Choosing P* (and implicitly the CAL)Step 1: Choosing P*Step 2: Choosing the risky share, yStep 2: Choosing the risky share, yStep 2: Choosing the risky share, yLeveraged positionsBorrowing constraintsImplications of the separation theoremImplications of the separation theoremA note on expected returnsThe market portfolioThe expected return of the market portfolioThe variance of the market portfolioThe variance of the market portfolioThe risk-return ratio of the market portfolioThe risk-return ratio of the market portfolioThe CAPM