optimal risky portfolio

8/11/2019 Optimal Risky Portfolio

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P . V. V I S WA N AT H

F O R A F I R S T C O U R S E I N I N V E S T M E N T S


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How does diversification help in constructing optimalrisky portfolios?How do we construct the opportunity set when there aretwo risky assets available?How do we compute the minimum variance portfolio andthe optimal portfolio when there are only two risky assetsand no other assets?How do we compute the optimal portfolio when there aretwo risky assets and a risk-free asset?The Efficient Portfolio of Risky AssetsThe Separation PropertyThe importance of covariance in diversification


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Of course, in practice, assets are not correlated in thissimplistic way. Let us look at how portfolio risk isaffected when we put two arbitrarily correlated assets ina portfolio. Let us call the two assets, a bond, D, and astock (equity), E.Then, we can write out the following relationship:

Portfolio Return

Bond Weight Bond Return

Equity Weight

Equity Return

p D E D E P

D

D

E

E

r

r

wr

w

r

w wr r

( ) ( ) ( ) p D D E E E r w E r w E r


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The expectedreturn on aportfolio consisting

of several assets issimply a weightedaverage of theexpected returnson the assetscomprising theportfolio.


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If we denote variance by s 2, then we have therelationship:

E D E D E E D D r r Covwwww ,222222 p s s s

where Cov(r D, r E) represents the covariance between thereturns on assets D and E.If we use DE to represent the correlation coefficient

between the returns on the two assets, thenCov(r D,r E) = DEs Ds EThe formula for portfolio variance can be writteneither with covariance or with correlation.


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The correlation coefficient can take values between+1 and -1.If DE = +1, there is no diversification and theportfolio standard deviation equals w Ds D + w Es E, i.e.

a linear combination of the standard deviations ofthe two assets.If DE= -1, the portfolio variance equals (w Ds D w Es E)2. In this case, we can construct a risk-free

combination of D and E.Setting this equal to zero and solving for w D and w E, we find

D

E D

D E ww 1

s s

s


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For intermediate valuesof r, the portfoliostandard deviations fallin the middle, as shown

on the graph to the right.

In this example, the stockasset has a standarddeviation of returns of20% and the bond asset,

of 12%.


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This graph shows the portfolioopportunity set for different values of .That is, the combination of portfolioE(r) and s than can be obtained bycombining the two asset.

In our example, the equity asset has anexpected return of 13%, while the bondasset has an expected return of 8%.The curved line joining the two assets Dand E is, in effect, part of theopportunity set of (E(R), s )combinations available to the investor.To get the entire opportunity set, wesimply extend this curve both beyond Eand beyond D, as is clear from the nextfigure.


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We can solve the optimization problem to computethe following useful formulas:The minimum variance portfolio of risky assets D, Eis given by the following formula:

The optimal portfolio for an investor with a risk

aversion parameter, A, is given by this formula:

= + 0.01 [ 2 , ]

0.01 [ 2 + 2 ,


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We now introduce a riskfree asset.

The expected return on a portfolioconsisting of a riskfree asset and arisky portfolio is, of course, a weighted average of the expected

returns on the component assets.But the standard deviation of theportfolio is also linear in thestandard deviation of the risky asset.Hence the CAL if there is one riskfreeasset and a risky portfolio is simply astraight line passing through the twoassets, as shown in the figure on theright.


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The slope of each of the CALsdrawn in the previous figure is areward-to-volatility (Sharpe)ratio. Since we want this ratio to

be maximized, the single CAL forthe set of risky and riskfree assetsis the CAL with the steepest slope,i.e. the highest Sharpe ratio.


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If we now superimpose theindifference curve map on theCAL, we can compute thecomplete optimal portfolio.


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The formula for the tangency portfolio (shown asportfolio C on the picture in the previous slide) is:

Note that the investor risk aversion coefficient doesnot show up in this formula.Once the tangency portfolio is available, all investors

choose a combination of this portfolio (denoted p inthe formula below) and the risk-free asset. Theformula for this, which we know already, is:

= 2


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Year

Capital

Value Fund

Green Century

Balanced Year

Capital Value

Fund

Green Century

Balanced1999 21.32% -10.12% 1996 21.48% 18.26%1998 21.44% 18.91% 1995 0.91% -4.28%1997 9.86% 24.91% 1994 10.79% -0.47%average 14.30% 7.87%stdev 8.52% 14.57%

You have available to you, two mutual funds, whose returns have a correlation of0.23. Both funds belong to the fund category Balanced Domestic. Here issome information on the fund returns for the last six years (obtained fromhttp://www.financialweb.com/funds/):In addition, you can also invest in a riskfree 1-year T-bill yielding 6.286%. Theexpected return on the market portfolio is 20%.

a.If you have a risk aversion coefficient of 4, and you have a total of $20,000 toinvest, how much should you invest in each of the three investment vehicles? (10points)b.What is the standard deviation of your optimal portfolio? (10 points)


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a. Using the formula, we can find the portfolio weights for thetangent portfolio of risky assets as follows:

which works out to 1641.13/1529.31 = 1.073. Hence w GCB

= 1-(1.073) = -0.073.In order to find the optimal combination of the tangent portfolioand the riskfree asset for our investor, we need to compute theexpected return on the tangent portfolio and the variance of

portfolio returns.E(R tgtport ) = 1.073(14.3) + (-0.073)(7.87) = 14.77% Var(R tgtport ) = (1.073) 2(8.52) 2 + (-0.073) 2(14.57) 2 +2(-0.073)(1.073)(8.52)(14.57)(0.23) = 82.47. Hence, s tgtport =9.08%


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Using the formula y* = [E(R port ) R f ]/0.01AVar(R tgtport ), we get y* = = 2.57; hence the proportion in the riskfree asset is -1.57. Inother words, the investor borrows to invest in the tangentportfolio.If the investors total outlay is $20,000, the amount borrowedequals (20000)(1.57) = $31,400. This provides a total of $51,400for investment in the tangent portfolio. However, the tangentportfolio itself consists of shortselling Green Century Balanced tothe extent of (0.073)(51,400) = 3752.20, providing a total of51,400 + 3752.2 - $55,152.20 for investment in Capital ValueFund. b. The standard deviation of the optimal portfolio is 2.57(9.08) =23.34%. The expected return on the optimal portfolio is2.57(14.77) + (-1.57)(6.286) = 28.09%


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Until now, we have dealt with the case of two risky assets. Wenow increase the number of risky assets to more than two.In this case, graphically, the situation remains the same, as we will see, except that the opportunity set instead of being asimple parabolic curve becomes an area, bounded by a

parabolic curve.However, since all investors are interested in higher expectedreturn and lower variance of returns, only the northwesternfrontier of this set is relevant, and so the graphic illustrationremains comparable.

Mathematically, the computation of the tangency portfolio is a bit more complicated, and will require the solution of a systemof n equations. We will not go further into it, here. We now look at the graphical illustration of the problem


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The first step is todetermine the risk-returnopportunitiesavailable.

All portfolios thatlie on theminimum-variancefrontier from theglobal minimum-

variance portfolioand upwardprovide the bestrisk-returncombinations


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We now searchfor the CAL withthe highest

reward-to- variability ratio


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The separation property tells us that the portfoliochoice problem may be separated into twoindependent tasks

Determination of the optimal risky portfolio is purely

technical. Allocation of the complete portfolio to T-bills versusthe risky portfolio depends on personal preference.

Thus, everyone invests in P, regardless of their degree

of risk aversion.More risk averse investors put more in the risk-freeasset. Less risk averse investors put more in P.


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25 We have seen that

If we have three assets, portfolio variance is given by:

If we generalize it to n assets, we can write the formula as:

Defining the average variance and the average covariance, we then get

That is, the portfolio variance is a weighted average of the average varianceand the average covariance.However, as the number of assets increases, the relative weight on the variancegoes to zero, while that on the covariance goes to 1.Hence we see that it is the covariance between the returns on the componentassets that is important for the determination of the portfolio variance.

E D E D E E D D r r Covwwww ,222222 p s s s

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optimal risky portfolio

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