markoqitz optimization

Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion

Lecture 4: Optimal Risky PortfoliosSAPM [Econ F412/FIN F313 ]

Ramana Sonti

BITS Pilani, Hyderabad Campus

Term II, 2014-15

1/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti


Agenda

1 PreliminariesIntroduction and review

2 Analysis with no risk-free assetTwo risky assetsThree (and more) risky assetsMore on efficient frontiers

3 Analysis with risk-free assetTwo risky assets and a risk-free assetThree (and more) risky assets and a risk-free asset

4 ConclusionOptimal complete portfoliosFinal thoughts



Introduction and review

The basic problem

Investor has available investment alternatives n risky assets: random returns 1 risk-free asset: constant return

Question: How should the investor allocate money amongst theavailable choices?

Basic assumptions: Investors care about

Average return: Expected return, E(r) Risk: Variance of return, 2(r), or Standard deviation (r)

Investors are greedy: like more return with less risk Investors are risk averse: as the risk level increases, the extra return

needed to compensate them for taking an unit of extra risk increases



Introduction and review

Portfolio math review

For a portfolio of two assets (with weights w1 and w2), Portfolio expected return: E(rp) = w1E(r1) + w2E(r2) Portfolio variance: 2p = w2121 + w2222 + 2w1 w21,2

In general, for a portfolio of n assets, Portfolio expected return: E(rp) =

ni=1

wiE(ri)

Portfolio variance: 2p =ni=1

w2i 2i +

ni=1

nj=1i,j

wiwji,j

In more elegant matrix notation, Portfolio expected return: E(rp) = w Portfolio variance: 2p = ww Note: is the vector of expected returns, and is the

variance-covariance matrix of asset returns of the set of assets heldaccording to weights in w



Two risky assets

Two risky assets: Example

Two assets X and Y with expected returns, variances, andcovariance (assumed correlation is ) as follows:

=[0.100.20

] =

[0.0049 (0.07)(0.10)

(0.07)(0.10) 0.01

] What is the expected return and standard deviation of a w : 1 w

portfolio of X and Y? Portfolio expected return: E(rp) = w(0.10) + (1 w)(0.20) Portfolio variance:2p = w

2(0.0049) + (1 w)2(0.01) + 2w(1 w)(0.07)(0.10) For a given value of , we can trace out the expected return and

standard deviation of different portfolio combinations of X and Y,e.g., 100:0, 60:40,



Two risky assets

Diversification in action: Two risky assets

0 0.02 0.04 0.06 0.08 0.1 0.12

0.1

0.12

0.14

0.16

0.18

0.2

Portfolio standard deviation

Port

folio

expe

cted

retu

rn

X

Y

=1.0=0.5=0=-0.5=-1.0



Two risky assets

Mean-Standard deviation diagram

The mean-standard deviation diagram shows the risk-returncharacteristics of all portfolios that can be formed using the twoassets. In other words, it traces out the investment opportunity set

When correlation is +1.0 or -1.0, the mean-standard deviationdiagram is linear

When correlation is between the two extremes, the mean-standarddeviation diagram is a hyperbola

As correlation decreases, the diagram curves in to the left, i.e., wehave portfolios which have lesser risk for the same expected return

As we change the correlation, only portfolio variance (standarddeviation) changes, not the expected return

Note that we cannot choose the correlation between any pair ofrisky assets, it is what it is



Three (and more) risky assets

Three risky assets

Lets fix the correlation between X and Y at = 0.10, and add a thirdrisky asset, Z, with the following properties

= 0.100.200.15

=

0.0049 0.0007 00.0007 0.01 0.01080 0.0108 0.0144

What is the investment opportunity set now? With two risky assets, the opportunity set was a curve With three risky assets, the opportunity set is a whole area

Question: Is there an optimal combination of the three assets? Yes, the mean variance frontier plots the optimal combinations of the

three assets



Three (and more) risky assets

Mean variance frontier: Three risky assets

0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.130

0.05

0.1

0.15

0.2

0.25


Port

folio

expe

cted

retu

rn

X

Y

Z

Efficient Frontier

MVP



More on efficient frontiers

Efficient frontier: Analytical solution Question: Of the infinite number of portfolios that can be formed ofX, Y and Z, which ones are optimal? For a given expected return m, what combination of the assets results

in the lowest variance (standard deviation)?

minw 2p = w

w

subject to E(rp) = w = m andni=1

wi = 1

Alternatively, for a given level of variance s, what combination of theassets results in the highest expected return?

maxw E(rp) = w

subject to 2p = ww = s and

ni=1

wi = 1

The bad news: General analytical solution is quite complicated The good news: we can use Excels Solver quite effectively to

numerically generate the efficient frontier




Two fund separation

Two fund separation : Any efficient portfolio can be obtained as thecombination of two efficient portfolios

The efficient portfolio with E(rp) = 16% consists of wX = 0.54,wY = 0.74, wZ = 0.28, and that with E(rp) = 22% consists ofwX = 0.2625, wY = 1.6625, wZ = 0.925. Using this information, howdo we derive the efficient portfolio with E(rp) = 20%? Step 1: Given that 20 = 13 (16) + 23 (22), we deduce weights as 13 and 23 Step 2: Now we get the weights of the 20% expected return efficient

portfolio as wX = 13 (0.54) + 23 (0.2625) = 0.355 wY = 13 (0.74) + 23 (1.6625) = 1.355 wZ = 13 (0.28) + 23 (0.925) = 0.71

Step 3: The mean and std. devn. of this portfolio are 20% and 0.0783try this calculation!), which puts it right on the efficient frontier, asadvertised!




Two fund separation: Example

0.06 0.07 0.08 0.09 0.1

0.14

0.16

0.18

0.2

0.22

0.24


Port

folio

expe

cted

retu

rn

EP 1: 0.54,0.74,-0.28

EP 2: 0.2625,1.6625,-0.925

EP 3: 0.355,1.355,-0.71

Efficient Frontier

MVP

This suggests the following recipe: Step 1: Use Solver to solve for any two efficient portfolios Step 2: Form combinations of the two efficient portfolios to trace the

entire efficient frontier




Constraints on investment Say we want to solve a constrained version of the problem, e.g., no

short sales allowed for mutual funds The constrained efficient frontier lies completely within the

unconstrained efficient frontier (why?)

0.06 0.07 0.08 0.09 0.1

0.1

0.12

0.14

0.16

0.18

0.2


Port

folio

expe

cted

retu

rn

Unconstrained frontier

Constrained frontier




Adding a fourth...and a fifth...asset

0.04 0.06 0.08 0.1 0.120

0.05

0.1

0.15

0.2

0.25


Port

folio

expe

cted

retu

rn

More assets




A second look at diversification In general, as we keep adding assets to a portfolio, the efficient

frontier moves to the left Question: Will the efficient frontier ever touch the vertical axis, i.e., is

there a combination of the risky assets that is completely risk-free? Answer: Generally, no

Start with portfolio variance:2p =ni=1

w2i 2i +

ni=1

nj=1i,j

wiwji,j

Assume all assets have a variance v, and each pair of assets acovariance c

This means 2p = n 1n2 v + n(n 1) 1n2 c = 1n v + n1n c As n , 1n 0, and n1n 1, which implies 2p c

Conclusion 1: For a well-diversified portfolio, covariances matter, notso much the variances

Conclusion 2: If c is not zero, we cannot diversify away all risk: leadsto the idea of diversifiable (idiosyncratic) vs. undiversifiable(systematic) risk




Diversification in well-diversified portfolios

0 20 40 60 80 100No. of stocks

Port

folio

var

ianc

e

Diversifiable risk

Systematic risk



Two risky assets and a risk-free asset


Let us add a risk-free asset to our two assets X and Y. Further,assume rf = 0.05. What happens? The opportunity set now expands dramatically The investor can now combine the risk-free asset and X, and choose

any portfolio along the line CALX The investor can also combine the risk-free asset and Y, and choose

any portfolio along the line CALY Better yet, the investor can combine the risk-free asset and any

portfolio of X and Y pivoting about the risk-free asset until the CALwhich passes through the risk-free asset and the portfolio labelledMVE

The portfolio of X and Y at the point of tangency is called the MeanVariance Efficient (MVE) portfolio. It provides the maximumreward-to-risk ratio, a.k.a. Sharpe ratio




Two risky assets: Tangency and MVE

0 0.02 0.04 0.06 0.08 0.1 0.120

0.05

0.1

0.15

0.2

0.25


Port

folio

expe

cted

retu

rn

X

Y

CAL X

CAL Y

MVE

CAL




Analytics of the MVE (2 asset case) The problem to be solved is

maxw

E(rp) rfp

,

where E(rp) = wE(rX) + (1 w)E(rY )and p =

[w22X + (1 w)22Y + 2w(1 w)X,Y

] 12

The solution to this problem is

wX =

[E(rX) rf

]2Y

[E(rY ) rf

]X,Y[

E(rX) rf]2Y +

[E(rY ) rf

]2X

[E(rX) rf + E(rY ) rf

]X,Y

wY = 1 wX

Try using this formula in our example to get wX = 0.3607 andwY = 0.6393



Three (and more) risky assets and a risk-free asset

Three risky assets and a risk-free asset Let us add the risk-free asset with rf = 0.05 to our three risky assetsX, Y and Z

Now, Sharpe ratio is maximized at the point where the CAL istangent to the efficient frontier

The MVE, or tangency portfolio iswX = 0.2274,wY = 1.7793,wZ = 1.0067

0 0.02 0.04 0.06 0.08 0.1 0.120

0.05

0.1

0.15

0.2

0.25


Port

folio

expe

cted

retu

rn

MVE

CAL

H0.2274, 1.7793,-1.0067L




Analytical solution (3 asset case) Set up the following equation system

0.0049x + 0.0007y + 0.0000z = 0.10 0.05 = E(rX) rf0.0007x + 0.0100y + 0.0108z = 0.20 0.05 = E(rY ) rf0.0000x + 0.0108y + 0.0144z = 0.15 0.05 = E(rZ) rf

Solve: x = 4.8186, y = 37.6984, z = 21.3294 Rescale weights to add up to 1, i.e., divide by the sum of the three

numbers, 21.1876

wX =4.818621.1876

= 0.2274,

wY =37.698421.1876

= 1.7793,

wZ =21.329421.1876

= 1.0067,

which are the same values as before!21/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti



The MVE portfolio

It turns out that the MVE portfolio has an interesting propertyE(rX) rf

Cov(rX, rMVE)=

E(rY ) rfCov(rY , rMVE)

=E(rZ) rf

Cov(rZ , rMVE)

Intuition: In the MVE portfolio, the marginal reward-to-risk ratio isequalized across all risky assets, and there is nothing to be gainedby changing the portfolio composition further

It is this important property underlies our earlier analytical method




The MVE portfolio...Contd. Sketch of proof:

Say we start with the MVE portfolio T of X, Y, and Z (expected returnof T and variance of 2T )

Suppose we try to improve upon the Sharpe ratio of this portfolio asfollows: For every $1 invested in T, buy $X amount of X (financed byborrowing at rf ), and, sell $Y amount of Y from this portfolio (proceedsinvested at rf )

New (dollar) expected return N = T + X(X rf ) Y (Y rf ) New (dollar) variance 2N 2T + 2XCov(rX , rT ) 2YCov(rY , rT ) Set 2N 2T = 0, by choosing X = Y Cov(rY ,rT )Cov(rX ,rT ) , i.e., we have rigged this

to make sure the portfolio variance does not change due to themodifications we made with X and Y

But since we started with the MVE portfolio, we must haveX(X rf ) Y (Y rf ) = 0, otherwise Portfolio T must not have beenMVE to begin with

Combining the last two equations, we get the desired propertyE(rX )rfCov(rX ,rT )

=E(rY )rfCov(rY ,rT )



Optimal complete portfolios


Suppose we have two investors: one with a coefficient of riskaversion A = 15, and another with A = 40. How will their money beallocated among the risk-free asset and the three risky assets?

Key insight 1: Risk aversion does not play any role in determiningthe optimal risky portfolio, a.k.a. the MVE portfolio.

Key insight 2: All investors, irrespective of risk aversion, choose tobe on the CAL with the highest slope. In other words, all investorshold portfolios of the risk-free asset and the MVE portfolio. This istwo-fund separation in the presence of a risk-free asset

In summary, all investors follow a two step process Step 1: Come up with the MVE portfolio (common to all) the

investment decision Step 2: Decide on the split of their money between T-bills and the MVE

portfoliothe financing decision




Investor with A = 15

The MVE portfolio consists ofwX = 0.2274,wY = 1.7793,wZ = 1.0067. The expected return andstandard deviation of this portfolio are 0.2276 and 0.0916respectively

Recall that the relative split between the risk-free asset and the MVEportfolio is given by our old formulaw = E(rMVE)rfA2MVE

= 0.22760.0515(0.0916)2 = 141.25%

If the investor had a capital of $1 M to start with, she would borrow$412,500 (at the risk-free rate), and invest the total, i.e. $1.4125 M inthe MVE portfolio, i.e. split this amount among X, Y, and Z accordingto the proportions wX = 0.2274,wY = 1.7793,wZ = 1.0067

Exercise: Work out the allocation for an investor with A = 40, andcompare it to this allocation. It is useful to understand thedifferences between the two allocations




The big picture

0 0.04 0.08 0.12 0.16 0.2 0.24 0.280

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45


Port

folio

expe

cted

retu

rn

MVE

Risky asset efficient frontier

CAL

A=40

A=15

T-bills

Both investors choose portfolios of the same two assets: T-bills and the MVE Locate on the CAL according to their risk aversion



Final thoughts

Final thoughts Markowitz optimization is elegant and easy to implement if we are

willing to assume that all investors care about are the mean andvariance of risky assets This is strictly true if (a) all investors really have quadratic utility

functions, or if (b) asset returns are always normally distributed Both assumptions are a bit problematic. Nevertheless, as a first

approximation, they will do

This procedure suggests that an investment advisor shouldrecommend the same optimal risky portfolio to different investors,irrespective of their risk aversion Ignores real-world issues such as taxation This is a one-period problem,and not a multi-period formulation. Here,

the long run is essentially a series of short runs

We have still not talked about the inputs for this optimization, i.e.,expected returns and variances/covariances This turns out to be the most problematic thing to do, and is the stuff of

asset pricing, which we shall study next


PreliminariesIntroduction and review

Analysis with no risk-free assetTwo risky assetsThree (and more) risky assetsMore on efficient frontiers

Analysis with risk-free assetTwo risky assets and a risk-free assetThree (and more) risky assets and a risk-free asset

ConclusionOptimal complete portfoliosFinal thoughts

markoqitz optimization

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