Optimal Risky Portfolios: Efficient Diversification

Transcription

1 Prof. Alex Shapiro Lecture Notes 7 Optimal Risky Portfolios: Efficiet Diversificatio I. Readigs ad Suggested Practice Problems II. Correlatio Revisited: A Fe Graphical Examples III. Stadard Deviatio of Portfolio Retur: To Risky Assets IV. Graphical Depictio: To Risky Assets V. Impact of Correlatio: To Risky Assets VI. Portfolio Choice: To Risky Assets VII. Portfolio Choice: Combiig the To Risky Asset Portfolio ith the Riskless Asset VIII. Applicatios IX. Stadard Deviatio of Portfolio Retur: Risky Assets X. Effect of Diversificatio ith Risky Assets XI. Opportuity Set: Risky Assets XII. Portfolio Choice: Risky Assets ad a Riskless Asset XIII. Additioal Readigs Buzz Words: Miimum Variace Portfolio, Mea Variace Efficiet Frotier, Diversifiable (Nosystematic) Risk, Nodiversifiable (Systematic) Risk, Mutual Fuds.

2 I. Readigs ad Suggested Practice Problems BKM, Chapter Suggested Problems, Chapter 8: Ope the Portfolio Optimizer Programs ( ad 5 risky assets) ad experimet ith those. II. Correlatio Revisited: A Fe Graphical Examples A. Remider: Do t get cofused by differet otatio used for the same quatity: Notatio for Covariace: Cov[r,r ] or [r,r ] or or, Notatio for Correlatio: Corr[r,r ] or ρ[r,r ] or ρ or ρ, B. Recall that covariace ad correlatio betee the radom retur o asset ad radom retur o asset measure ho the to radom returs behave together. C. Examples I the folloig 5 figures, e Cosider 5 differet data samples for to stocks: - For each sample, e plot the realized retur o stock agaist the realized retur o stock. - We treat each realizatio as equally likely, ad calculate the correlatio, ρ, betee the returs o stock ad stock, as ell as the regressio of the retur o stock (deoted y) o the retur o stock (x). [Note: the regressio R equals ρ ]

3 . A sample of data ith ρ 0.60: y 0.948x R % 0% 5% Retur o Stock 0% 5% 0% 5% 0% -5% -0% -5% 0% -5% 5% 0% 5% 0% 5% -0% Retur o Stock. A sample of data ith ρ -0.74: 0% 5% 0% Retur o Stock 5% 0% -5% 0% 5% 0% 5% 0% 5% -5% -0% -5% y -0.86x R 0.5-0% Retur o Stock

4 . Sample ith ρ +: Retur o Stock 6% 6% 5% 5% 5% 5% 5% 5% y 0.0x R 5% -0% -5% 0% 5% 0% 5% 0% 5% 0% Retur o Stock 4. Sample ith ρ -: 5% 0% Retur o Stock 5% 0% -0% 0% 0% 0% 0% 40% -5% -0% -5% -0% -5% y -0.8x R Retur o Stock 5. Sample ith ρ 0: Retur o Stock 5% 0% y 0.009x R % 0% -5% 0% 5% 0% 5% 0% 5% 0% -5% -0% Retur o Stock 4

5 D. Real-Data Example Us Stocks vs. Bods , A sample of data ith ρ 0.8: STB Stocks ad Bods (Aual returs o S&P 500 ad log term US govt bods.) Ra Data Excess over T-bill S&P500 LT Gov t T-bill Iflatio S&P500 LT Gov t % -0.0% 0.5% 8.6% -8.4% -0.45% % -.6% 0.50% 9.0% 5.% -.% %.40% 0.8%.7% 4.69%.59% % 6.45%.0% -.80% 7.69% 5.5% 950.7% 0.06%.0% 5.79% 0.5% -.4% % -.9%.49% 5.87%.5% -5.4% %.6%.66% 0.88% 6.7% -0.50% %.64%.8% 0.6% -.8%.8% % 7.9% 0.86% -0.50% 5.76% 6.% % -.9%.57% 0.7% 9.99% -.86% % -5.59%.46%.86% 4.0% -8.05% % 7.46%.4%.0% -.9% 4.% % -6.09%.54%.76% 4.8% -7.6% % -.6%.95%.50% 9.0% -5.% %.78%.66%.48% -.9%.% % 0.97%.% 0.67% 4.76% -.6% % 6.89%.7%.% -.46% 4.6% 96.80%.%.%.65% 9.68% -.9% %.5%.54%.9%.94% -0.0% % 0.7%.9%.9% 8.5% -.% %.65% 4.76%.5% -4.8% -.% % -9.8% 4.%.04% 9.77% -.9% % -0.6% 5.% 4.7% 5.85% -5.47% % -5.07% 6.58% 6.% -5.08% -.65% %.% 6.5% 5.49% -.5% 5.59% 97 4.%.% 4.9%.6% 9.9% 8.84% % 5.69%.84%.4% 5.4%.85% % -.% 6.9% 8.80% -.59% -8.04% % 4.5% 8.00%.0% -4.47% -.65% % 9.0% 5.80% 7.0%.40%.40% % 6.75% 5.08% 4.8% 8.76%.67% % -0.69% 5.% 6.77% -.0% -5.8% % -.8% 7.8% 9.0% -0.6% -8.6% % -.% 0.8%.% 8.06% -.6% 980.4% -.95%.4%.40%.8% -5.9% %.86% 4.7% 8.94% -9.6% -.85% 98.4% 40.6% 0.54%.87% 0.87% 9.8% 98.5% 0.65% 8.80%.80%.7% -8.5% % 5.48% 9.85%.95% -.58% 5.6% 985.6% 0.97% 7.7%.77% 4.44%.5% % 4.5% 6.6%.%.% 8.7% % -.7% 5.47% 4.4% -0.4% -8.8% % 9.67% 6.5% 4.4% 0.46%.% % 8.% 8.7% 4.65%.% 9.74% % 6.8% 7.8% 6.% -0.98% -.6% % 9.0% 5.60%.06% 4.95%.70% % 8.05%.5%.90% 4.6% 4.54% % 8.4%.90%.75% 7.09% 5.4% 994.% -7.77%.90%.67% -.59% -.67% %.67% 5.60%.74%.8% 6.07% Retur o S&P % 50% 40% 0% 0% 0% -0% -0% -0% y 0.59x R % -0% -5% 0% 5% 0% 5% 0% 5% 0% 5% 40% 45% Retur o US Gov t Bods N Mea.6% 5.8% 4.84% 4.4% 8.% 0.99% Std.Dev. 6.57% 0.54%.8%.8% 7.0% 0.% Std.r.Mea.4%.49% 0.45% 0.54%.4%.4% Corr(Stocks, Bods)

6 III. Stadard Deviatio of Portfolio Retur: To Risky Assets A. Formula [ r p ( t)],p [ r( t)] +,p [ r ( t)] +,p, p [ r( t),r ( t)] [ r p ( t)] [ r p ( t)] here [r (t), r (t)] is the covariace of asset s retur ad asset s retur i period t, i,p is the eight of asset i i the portfolio p, [r p (t)] is the variace of retur o portfolio p i period t. B. Example Cosider to risky assets. The first oe is the stock of Microsoft. The secod oe itself is a portfolio of Small Firms. The folloig momets characterize the joit retur distributio of these to assets. E[r Small ].9, E[r Msft ].6, [r Small ].7, [r Msft ] 8.0, r Msft, r Small ].00 A portfolio formed ith 60% ivested i the small firm asset ad 40% i Microsoft has stadard deviatio ad expected retur give by: [r p ] Small,p [r Small ] + Msft,p [r Msft ] + Small,p Msft,p [r Small,r Msft ] [r p ] [ ] r p E[r p ] Small,p E[r Small ] + Msft,p E[r Msft ]

7 IV. Graphical Depictio: To Risky Assets A. Represetatio i the Mea-Variace Space The stadard deviatio, p, of a retur o a portfolio cosistig of asset ad asset, ad the portfolio s expected retur, E p, ca be expressed i terms of, the eight of asset. Whe plottig i the Mea-Variace plae p ad E p for all possible values of, e get a curve. The curve is ko as the portfolio possibility curve -, or as the portfolio frotier -, or as the set of feasible portfolios-, or as the opportuity set - ith to risky assets. A Algorithm to Plot the Portfolio Frotier:. Pick a value for (ad the - ). Compute expected retur ad stadard deviatio: E[ r p ] E[ r ] + E[ r ] E[ r ] + ( )E[ r ] p + +, + ( ) + ( ),. Plot a sigle poit { p, E[r p ]} 4. Repeat - for various values of 7

8 B. Example (cot.) To get the portfolio possibility curve usig the small-firm portfolio ad Microsoft equity (i.e., to get all possible p s ), the stadard deviatio of retur o a portfolio cosistig of the small firm portfolio (asset ) ad Microsoft equity (asset ) ad its expected retur ca be idexed by the eight of the small firm portfolio ithi portfolio p: Small,p. Small,p Msft,p [r p (t)] E[r p (t)] %.69% %.6% %.88% %.64% %.98% %.55% %.9% %.670% Figure here 8

9 Note: he oe asset is risk-free the set of feasible portfolios is described by the CAL discussed i Lecture Notes 6 (i other ords, the CAL together ith its mirror image obtaied he shortig the risky asset is the portfolio frotier of oe risky asset ad oe riskless asset). V. Impact of Correlatio: To Risky Asset Case A. Stadard Deviatio Formula Revisited The stadard deviatio formula ca be reritte i terms of correlatio rather tha covariace (usig the defiitio of correlatio): [ r p ( t)],p [ r( t) ] +,p [ r ( t)] +,p, p ρ[ r( t),r ( t)] [ r( t)] [ r ( t)] here [r (t), r (t)] is the correlatio of asset s retur ad asset s retur i period t. For a give portfolio ith,p >0,,p >0, ad [r (t)] ad [r (t)] fixed, [r p (t)] decreases as [r (t), r (t)] decreases. B. Example (cot.) Suppose the E[r] [r] for the small firm asset ad for Microsoft remai the same but the correlatio betee the to assets is alloed to vary: 9

10 Figure here VI. Portfolio Choice: To Risky Assets A. A risk averse ivestor is ot goig to hold ay combiatio of the to risky assets o the egative sloped portio of the portfolio frotier.. So the egative-sloped portio is ko as the iefficiet regio of the curve.. Ad the positive-sloped portio is ko as the efficiet regio of the curve, or as the efficiet frotier, or as the miimumvariace frotier. A portfolio is efficiet if it is o the efficiet frotier (i.e., achieves the maximum expected retur for a give level of stadard deviatio). 0

11 B. The exact positio o the efficiet frotier that a idividual holds depeds o her tastes ad prefereces. C. Example (cot.) The portfolio possibility curve for the small firm portfolio ad Microsoft ca be divided ito its efficiet ad iefficiet regios. Ay risk averse idividual combiig the small firm portfolio ith Microsoft ats to lie i the efficiet regio: so ats to ivest a positive fractio of her portfolio i Microsoft. Figure here

12 VII. Portfolio Choice: Combiig the To Risky Asset Portfolio ith the Riskless Asset To-stage Decisio Process. Stage I: Asset Selectio Stage II: Asset Allocatio Stage I: Asset Selectio What are the preferred eights of the to risky assets i the risky portfolio? a. all risk averse idividuals at access to the CAL ith the largest slope; this ivolves combiig the riskless asset ith the same risky portfolio ( i the figure belo). b. this same risky portfolio is the oe hose CAL is taget to the efficiet frotier; this is hy is ko as the tagecy portfolio, deoted T. We o ko ho to select the optimal portfolio of risky assets for asset allocatio betee risky ad riskless assets: The portfolio, deoted P i the previous lecture, should be chose as simply the portfolio T o the efficiet frotier (like the oe labeled by i the figure belo), ith a CAL taget to the frotier. Note: The optimal determiatio of P ad that of the associated CAL is doe simultaeously. The best P is the tagecy portfolio T.

13 c. Ca calculate the eight of risky asset i the tagecy portfolio T usig the folloig formula: [ r [ r ] E[ R] - [ r,r] E[ R] ] E[ R] - [ r,r] E[ R] + [ r] E[ R] - [ r,r,t ] E[ R] here R i r i - r f is the excess retur o asset i (i excess of the riskless rate). Stage II: Asset Allocatio What are the preferred eights of the risky portfolio T ad the riskless asset i the idividual s portfolio? As e discussed i the previous lecture, the eight of T ( ) i a idividual s portfolio T,p depeds o the idividual s tastes ad prefereces. Figure here

14 VIII. Applicatios A. Asset Allocatio betee To Broad Classes of Assets The to-risky-asset formulas ca be used to determie ho much to ivest i each of to broad asset classes. Example: The Wall Street Joural articles at the ed of the previous Lecture Notes sho recommedatios for a composite portfolio C. The risky portfolio ithi C, ca be thought of as the oe hich each strategist believes to be the taget portfolio T. The eights ithi T of the to broad asset classes Stocks ad Bods ca be determied as above. (The eights of Stocks relative to Bods differ across strategists possibly because each oe of them sees a differet efficiet frotier, ad hece recommeds to its cliets a differet T ). B. Iteratioal Diversificatio The to-risky-asset formulas ca also be used he decidig ho much to ivest i a iteratioal equity fud ad ho much i a U.S. based fud. 4

15 IX. Stadard Deviatio of Portfolio Retur: Risky Assets A. Portfolios of may assets There are risky assets, i,,, Basic data ( + ( ) / iputs ): Expected returs : Stadard deviatios : ( ) coef. of corr. : P roblems:. Give p defied by ho to compute E[ r. Ho do e form efficiet portfolios (those hich miimize p E[ r ], E[ r,, ], but hat [ r p, ρ,,,, ρ, ρ, e ko ] give ], E[ r,, about? E[ r p p ] ])? B. Formula p ( t)] i,p j,p [ ri ( t),r j ( t)] i,p j,p [ ri ( t)] [ r j ( t)] ρ i j i j [ r [ ri ( t),r here [r i (t)] is the stadard deviatio of asset i s retur i period t, [r i (t), r j (t)] is the covariace of asset i s retur ad asset j s retur i period t, ρ[r i (t), r j (t)] is the correlatio of asset i s retur ad asset j s retur i period t; i,p is the eight of asset i i the portfolio p; [r p (t)] is the variace of retur o portfolio p i period t. j ( t)] 5

16 6 C. Example: A -stock portfolio X. Effect of Diversificatio ith Risky Assets To uderstad ho to form efficiet portfolios, e eed to uderstad first the effect of diversificatio. A. The Case of Ucorrelated Risky Assets Suppose all assets have the same expected retur ad same stadard deviatio [r] ad have returs hich are ucorrelated: Sice stocks are idetical, ca a portfolio be better tha each stock??? Sice stocks are idetical, there is othig to be lost by puttig a equal eight o each stock; so e cosider a equally eighted portfolio, here i,p / for all i. Example: he, a equally eighted portfolio has 50% i each asset.,,, p p ρ ρ ρ ,,, ρ ρ ρ

17 The: With stocks (): E[r p (t)] E[r (t)] + E[r (t)] [r p (t)] ( ) [r (t)] + ( ) [r (t)] With stocks (): E[r p (t)] E[r (t)] + E[r (t)] + E[r (t)] [r p (t)] ( ) [r (t)] + ( ) [r (t)] + ( ) [r (t)] Arbitrary : E[r p (t)] [r p (t)] / P As icreases:. the variace of the portfolio declies to zero. (all the risk is diversifiable!). the portfolio s expected retur is uaffected. This is ko as the effect of diversificatio (ca thik of it as risk reductio, or as the isurace priciple). 0 7

18 B. The case of idetical positively correlated assets. ρ, ρ, ρ, ρ > 0 P ρ I this case the equally E[ r P P ] + ( ( ) ρ ρ ) + ρ eighted p has ρ 0 (-ρ)/ is the uique / ideosycratic / firm specific / diversifiable / osystematic risk. It ca be reduced by combiig securities ito portfolios. As e diversify ito more assets, the risk reductio orks for the specific-risk compoet. ρ is the market / odiversifiable / systematic risk. This portio of risk e caot diversify aay. The loer is the correlatio betee assets, the loer is the odiversifiable compoet. 8

19 XI. Opportuity Set: Risky Assets A. Set of Possible Portfolios Because, i geeral, there is a limit to diversificatio, it follos that ith assets, although e have a ifiite set of curves (each as i the to asset case), these are combied ito the folloig geeral shape: Efficiet set of risky assets B. Miimum Variace (Stadard Deviatio) Frotier Sice idividuals are assumed to have Mea-Variace (MV) prefereces, ca restrict attetio to the set of portfolios ith the loest variace for a give expected retur (as e did ith assets). This set is a curve, ad it is the miimum variace frotier (MVF) for the risky assets. Every other possible portfolio is domiated by a portfolio o the MVF (loer variace of retur for the same expected retur). 9

20 C. Addig risky assets. Addig risky assets to the opportuity set alays causes the miimum variace frotier to shift to the left i { [r],e[r]} space. Why? -- For ay give E[r], the portfolio o the MVF for the subset of risky assets is still feasible usig the larger set of risky assets. Further, there may be aother portfolio hich ca be formed from the larger set ad hich has same E[r] but a loer [r].. Example (cot. igorig DP) a. MVF for IBM, Apple, Microsoft, Nike ad ADM is to the left of the MVF for IBM, Apple, Microsoft ad Nike excludig ADM. This happes eve though ADM has a { [r],e[r]) deoted by hich lies to the right of the MVF for the 4 stocks excludig ADM. Figure here 0

21 XII. Portfolio Choice: Risky Assets ad a Riskless Asset A. The aalysis for the to risky asset ad a riskless asset case applies here:. A Mea Variace ivestor combies the riskless asset ith the risky portfolio hose Capital Allocatio Lie has the highest slope.. That risky portfolio is o the efficiet frotier for the risky assets ad is i fact the tagecy portfolio T. Calculatig the eights of assets i the tagecy portfolio ca be performed via computer (see the Spreadsheet Model i BKM ch. 8, pp. 9-5).. Ivestors at to hold this tagecy portfolio i combiatio ith the riskless asset. The associated Capital Allocatio Lie is the efficiet frotier for the risky assets ad the riskless asset. 4. Oly the eights of the tagecy portfolio ad the riskless asset i a idividual s portfolio deped o the idividual s tastes ad prefereces.

22 XIII. Additioal Readigs The articles about Gold as a ivestmet, illustrate that eve though it may be a bad ivestmet i isolatio, ivestig i gold makes sese as a hedge, i.e., as a isurace. This meas that i some scearios, perhaps very ulikely oes (like the YK computer problem discussed i oe article), the gold fractio of the portfolio ill help to maitai favorable returs at times of recessio. Overall, addig gold improves the efficiet frotier, aalogously to ho addig ADM improved the frotier of IBM, Apple, Microsoft, ad Nike i our Example. The article about Mutual Fuds explais, i layma terms, that it is the risk reductio through diversificatio, hich is the major reaso to hold mutual fuds. Differet cliets of moey maagers may have differet costraits, requiremets, tax cosideratios, etc. Still, our class discussio suggests that a limited umber of portfolios may be sufficiet to serve may cliets. This is the theoretical basis for the mutual fud idustry. This is hy fuds ere itroduced i the first place, ad this is hy they are idely popular. There are more articles about fuds: I particular Idex Fuds (the Fast Trades... article may be of iterest to those ho at to lear more about taxissues related to mutual fuds -- although e are ot focusig o these i class); Total-Market Fuds, Bod Fuds, ad Exchage Traded Fuds (ETFs). Take a look at the article that illustrates that eve Uiversities( Emory ) make ivestmet mistakes, hich could be easily avoided give hat e leared i class! A Busiess Week article further elaborates o the Asset Selectio ad Asset Allocatio problems. Aother article illustrates that decisio makers i Washigto are payig attetio to the beefits of diversificatio, ad hece are cosiderig ivestig Social Security fuds i the market. The debate is regardig the appropriately diversified portfolio.. Ad there are OTHER iterestig articles to READ!