Optimal Risky Portfolios

Transcription

1 Otimal Risky Portolio Otimal Risky Portolios When choosing the otimal allocation between a risk-ree asset and a risky ortolio, we have assumed that we have already selected the otimal risky ortolio In this section, we learn how to determine the otimal risky ortolio We start rom two risky assets Most o the intuition carries to the case o more than two risky assets Portolio o Two Risky ssets Suose you hold a roortion w in asset and (1- in asset The ortolio exected return and risk is given by ( R ) = w( R ) + (1 ( R ) P = w +(1 + w(1 ρ Return and Risk o Portolio Given the exected returns o and, variances o and, and the covariance (correlation) between and, we comute the exected return and variance o the ortolio or a series o ortolio weights Then we lot the exected returns against variances n xamle n xamle Suose you hold two assets in your ortolio, G and IM Let the ortolio weight o G be w, and then the ortolio weight o IM be (1- I w = 1, you hold only G, I w = 0, you hold only IM, I w = 05, you have an equally weighted or naively diversiied ortolio ased on data or , we ind that The average monthly return is 168% or G, and 1% or IM The standard deviation is 649% or G and 810% or IM The correlation between G and IM is

2 qually weighted ortolio The exected return o the equally weighted ortolio is: 05*168%+05*1%=145% The standard deviation o this ortolio is more comlicated qually weighted ortolio = w = % % % 810% 0377 = = 607% + (1 + w(1 ρ This ortolio is less risky than either o G and IM!!! nother Portolio Calculate the exected return and standard deviation o the ortolio consisting o 80% G and 0% IM Diversiication eneit The ortolio risk is lower than either o the individual stocks This is called the beneit o diversiication I reeat the above stes or other ortolio weights, w=0, 01, 0,, 09, 10 I lot the exected return against standard deviation Portolios o G and IM Diversiication and Risk 0018 Variance 0017 $1,000 G, $0 IM Standard Deviation $800 G, $00 IM $600 G, $400 IM $400 G, $600 IM $00 G, $800 IM $0 G, $1000 IM Investment Oortunity Set Non- Systematic Systematic xected Return Number o ssets

3 Feasible Portolios With N Risky ssets xected Return (R) Investment Oortunity Set or Feasible Set C Std dev icient rontier icient Frontier You can construct the eicient rontier using one o two equivalent aroaches For given exected return, ind the minimum variance For given variance, ind the maximum exected return You can do this using the in xcel Utility Maximization What i a risk-ree asset is available? xected Return (R) Higher Utility C Indierence Curves is the Utility maximizing risky-asset ortolio We have covered the caital allocation roblem between a risk-ree asset and a risky asset Recall that the caital allocation line is the straight line through the risk-ree asset and the risky asset Std dev The Otimal Risky Portolio has the Highest Share Ratio xected Return (R) Riskless sset Otimal Risky Portolio C Caital Market Line D CL Std dev CL What i a risk-ree asset is available? The easible set o ortolios becomes more attractive We can identiy an otimal risky ortolio which dominates all other risky ortolios (irresective o risk reerences) The otimal (tangency) ortolio has the highest Share ratio among all easible ortolios 3

4 Utility Maximization with a Risk-ree sset Otimal Risky Portolio is the Market Portolio xected Return (R i ) Riskless sset Otimal Risky Portolio D Caital Market Line icient rontier verybody holds a combination o risk-ree asset and the otimal risky ortolio This otimal risky ortolio is the same or everybody regardless o how risk averse you are It must be the market ortolio I not, then there must be some assets that no one holds, which cannot be true Std dev i Passive Strategy is icient The otimal risky ortolio is the same or every investor, and is the market ortolio No need or stock selection Investors need only to adjust the mix o riskree asset and the market ortolio based on risk aversion The Otimal Risky Portolio Key Question: How do we ind the otimal risky ortolio? y choosing asset weights w i that maximize the Share Ratio: Max wi S = ( R ) R The Otimal Risky Portolio (with risky assets) For two risky assets, we know that the ortolio return and standard deviation are given by ( R ) = w( R ) + (1 ( R ) P = w + w(1 +(1 The Otimal Risky Portolio (with risky assets) Thereore, we need to maximize the ratio S = w w ( R ) + (1 ( R ) + w(1 by choosing w aroriately This can be done using in Microsot xcel R + (1 4

5 W The Otimal Risky Portolio (with risky assets) Or, by the ollowing ormula which gives the weights or the otimal ortolio comrised o only two assets: = ( ( R ) ( ( R ) [ ( R ] + ( R R R [ ] [ ( ) + ( R ] The Otimal Risky Portolio (with N risky assets) Stes to solve or the otimal risky ortolio weights using in Microsot xcel: Identiy all o the risky assets to be included in the investment universe Comute return series (rom rices) or each risky asset and the risk-ree asset and determine the average return or each W = ( 1 W ) The Otimal Risky Portolio (with N risky assets) Comute the covariance matrix o the risky assets nter the ormula or the Share ratio into a cell Set u a column o cells or the ortolio weights Use to maximize the Share ratio by changing the weights, subject to the constraint that the weights sum to one 5

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