ELEMENTARY PORTFOLIO MATHEMATICS

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1 QRMC06 9/7/0 4:44 PM Page 03 CHAPTER SIX ELEMENTARY PORTFOLIO MATHEMATICS 6. AN INTRODUCTION TO PORTFOLIO ANALYSIS (Backgroud readig: sectios 5. ad 5.5) A ivestor s portfolio is the set of all her ivestets. The ivestor s wealth ad ability to sped is a fuctio of her etire portfolio of ivestets. Thus, it is reasoable for a ivestor to be cocered with the perforace of idividual securities oly to the extet that their perforace affects overall portfolio perforace. Sice the returs of idividual ivestets ted to be related to oe aother, the copositio of the ivestor s portfolio is of priary iportace. We ca deostrate that ivestors ca ore effectively cotrol their ivestet risk by selectig appropriate securities i appropriate cobiatios. The retur o a ivestor s portfolio is a siple weighted average of the copoet idividual security returs. Oe calculates the expected retur of a portfolio based o either a fuctio of potetial portfolio returs ad their associated probabilities (as coputed i chapter 5) or fidig the weighted average of expected returs o idividual securities. I the vast ajority of cases, portfolio returs variace or stadard deviatio will be less tha the weighted average of the idividual security retur variaces or stadard deviatios. The extet to which overall portfolio risk is less tha the weighted average of copoet asset risk levels will deped o the ature of cooveet betwee the assets. 6. PORTFOLIO RETURN (Backgroud readig: sectios.8, 5.5, ad 6.) I sectio 5.5, we coputed the expected retur of a security as a fuctio of potetial retur outcoes ad associated probabilities. The expected retur of a portfolio is calculated siilarly, usig equatio (6.), where the subscript p desigates the portfolio ad the subscript j is a couter desigatig a particular outcoe out of potetial outcoes:

2 QRMC06 9/7/0 4:44 PM Page Eleetary portfolio atheatics E p = p, j j [ R ] R P. (6.) For ay portfolio aageet applicatios, it is useful to express portfolio retur as a fuctio of the returs of the idividual securities that coprise the portfolio: E[ R ] = w E[ R ]. p i= i i (6.) The subscript i desigates a particular security out of i the portfolio. The weights w i are the portfolio proportios. Thus, a security weight w i specifies how uch oey is ivested i security i relative to the total aout ivested i the etire portfolio: w i = $ ivested i security i. Total $ ivested i portfolio p Thus, portfolio retur is siply a weighted average of idividual security returs. Cosider a portfolio ade up of two securities, oe ad two. The expected retur of security oe is 0% ad the expected retur of security two is 0%. If 40% of the dollar value of the portfolio is ivested i security oe (that is, [w ] = 0.40), ad the reaider is ivested i security two ([w ] = 0.60), the expected retur of the portfolio ay be deteried as follows by equatio (6.): E[R p ] = (w E [R ]) + (w E [R ]), E[R p ] = ( ) + ( ) = PORTFOLIO VARIANCE (Backgroud readig: sectios.9, , ad 6.) Portfolio retur variace ay also be defied as a fuctio of potetial portfolio returs ad associated probabilities: σ p = p j E p ( R [ R ]) Pj. (6.3) Agai, it is usually helpful to express portfolio variace as a fuctio of idividual security characteristics. However, we stress that the variace of portfolio returs is ot siply a weighted average of idividual security retur variaces. We will actually be able to deostrate that we ca cobie a set of high-risk assets ito a low-risk portfolio. This is due to the iportat relatioship betwee portfolio risk ad retur covariaces betwee pairs of securities. The portfolio returs variace ca be coputed by solvig the followig fuctio:

3 QRMC06 9/7/0 4:44 PM Page 05 Portfolio variace 05 σ = wwσσ ρ = wwσ. p i j i j i, j i j i, j i= i= (6.4) Sectio.9 discussed the geeral process for workig with double suatios. We would solve equatio (6.4) to copute the variace of a two-security portfolio as follows: σ p = w w σ σ ρ, + w w σ σ ρ, + w w σ σ ρ, + w w σ σ ρ,. (6.5) Notice that we start the suatio operatios at i = ad j =. This eas that i the first stage of operatio, both i ad j refer to security. I the secod stage, i still refers to security but j refers to security. Sice, the uber of securities, equals, the third stage starts j over agai at ad i ow refers to security. I the fourth stage, both i ad j refer to security. We siplify the result to coplete the coputatioal process. Equatio (6.5) ca oly be used for a two-security portfolio. Equatio (6.4) ad variatios of it to be discussed later ay be used to copute portfolio variaces whe exceeds (see equatio (6.9)). Hece, equatio (6.5) is oly a special case of equatio (6.4). Cosider the portfolio costructed i sectio 6.. The weights associated with securities ad are 0.4 ad 0.6, respectively. Assue that the stadard deviatios of returs for securities oe ad two are 0.0 ad 0.30, respectively, ad that the correlatio coefficiet ρ ij betwee returs o the two securities is 0.5. We should also ote that ρ, = ρ, ad that ρ, = ρ, = because the correlatio coefficiet betwee aythig ad itself equals. Followig equatios (6.4) ad (6.5), we copute the variace ad stadard deviatio of portfolio returs as follows: σ p = = (6.6) Thus, the stadard deviatio of portfolio returs would be 0.3, the square root of its variace level. By siplifyig the expressios i the first ad fourth sets of paretheses, cobiig the ters i the secod ad third sets, ad rearragig, oe ca siplify equatios (6.5) ad (6.6) for two-security portfolios: σ p = w σ + w σ + (w w σ σ ρ, ), σ p = ( ) = A careful exaiatio of these expressios will reveal that equatio (6.4) ca also be rewritte as follows: σ = w σ + w wσ. p i i i= i= i j i j i, j (6.7)

4 QRMC06 9/7/0 4:44 PM Page Eleetary portfolio atheatics Whe a portfolio cosists of oly two securities, its variace ca be deteried by equatio (6.8): σ p = w σ + w σ + w w σ,. (6.8) Larger portfolios require the use of equatios (6.4) or (6.7), accoutig for all products of security weights ad stadard deviatios squared ad all possible cobiatios of pair-wise security covariaces ad weight products. For exaple, equatio (6.8) ca be rewritte for a three-security portfolio as σ p = w σ + w σ + w 3σ 3 + w w σ, + w w 3 σ,3 + w w 3 σ,3. (6.9) Notice that equatio (6.9) ivolves workig with three idividual security variaces (oe for each security) ad three sets of covariaces (betwee ad, ad 3, ad ad 3). Also otice the siilarities betwee equatio (6.8) for a two-security portfolio ad security 9 for a three-security portfolio. As the uber of securities i the portfolio icreases, the aout of coputatioal effort icreases disproportioately. The uber of sets of covariaces betwee oidetical pairs of securities equals ( )/. If 50 securities were to be icluded i the ivestor s portfolio,,5 sets of covariaces would be required to cobie with 50 variace ters i order to solve equatio (6.7). Obviously, as the uber of securities i the portfolio becoes large, coputers ad coputer spreadsheets becoe quite useful i workig through the repetitive calculatios. The equatios are ot difficult to solve; they are erely repetitive ad tie-cosuig. 6.4 DIVERSIFICATION AND EFFICIENCY The iportat cotributio of the covariace ters i equatios (6.4) (6.9) is that portfolio risk is a fuctio of the extet to which security returs are related to oe aother. Security risk ca be diversified away by costructig portfolios of urelated assets. The stateet Do t put all your eggs i oe basket has a strog basis i reality. Ivestet i a variety of differet securities with differet retur structures really does result i portfolio risk reductios. Oe should expect that portfolio risk levels will be lower tha the weighted average security risk levels. Diversificatio is ost effective whe the returs of the idividual securities are iversely correlated. Lower covariaces σ i,j result i lower portfolio risk. Portfolio risk is quite depedet o the correlatio coefficiet of returs ρ ij betwee securities icluded i the portfolio. Lower correlatio levels iply lower risk levels. Because the covariace betwee security returs σ i,j equals the product σ i σ j ρ i,j, reduced covariaces iply reduced correlatio coefficiets. Thus, lower correlatio coefficiets betwee securities iply lower portfolio risk. Portfolio risk should be expected to declie wheever ρ ij is less tha oe ad diversificatio icreases. We will orally expect ay radoly selected pair of securities to have a correlatio coefficiet less tha oe. Hece, we should orally expect that addig securities to a radoly selected portfolio will ted to reduce portfolio risk.

5 QRMC06 9/7/0 4:44 PM Page 07 Diversificatio ad efficiecy 07 I our first exaple with securities ad, the weighted average of the stadard deviatio of returs of the two securities is 6%: Weighted Average σ i = = 0.6. However, recall that the stadard deviatio of returs of the portfolio that they cobie to ake is oly 3%: σ p = ( ) = 03.. Clearly, soe risk has bee diversified away by cobiig the two securities ito the portfolio. I fact, the risk of a portfolio will alost always be lower tha the weighted average of the stadard deviatios of the securities that coprise that portfolio. The oly two exceptios occur whe: The returs correlatio coefficiet betwee all pairs of securities equals. Oe of two assets i a two-asset portfolio has a zero stadard deviatio of returs. For a ore extree exaple of the beefits of diversificatio, cosider two securities, 3 ad 4, whose potetial retur outcoes are perfectly iversely related. Data relevat to these securities is listed i table 6.. If outcoe oe occurs, security three will realize a retur of 30%, ad security four will realize a 0% retur level. If outcoe two is realized, both securities will attai returs of 0%. If outcoe three is realized, securities three ad four will attai retur levels of 0% ad 30%, respectively. If each Table 6. A portfolio retur with perfectly iversely correlated securities: w 3 = w 4 = 0.5 i R 3i R 4i R pi P i Give: The: B 3 = 0.0, B 4 = 0.0, σ 3 = , σ 4 = , w 3 = 0.50, w 4 = 0.50, ρ 3,4 =. B p = w 3 B 3 + w 4 B 4 = ( ) + ( ) = 0.0, σ = w σ + w σ + w w σ σ ρ p ,, σ p = ( ) = 0.

6 QRMC06 9/7/0 4:44 PM Page Eleetary portfolio atheatics outcoe is equally likely to occur (P i is for all outcoes), the expected retur level of each security is 0%; the stadard deviatio of returs for each security is The expected retur of a portfolio cobiig the two securities is 0% if each security has equal portfolio weight (w 3 = w 4 = 0.5), yet the stadard deviatio of portfolio returs is zero. Thus, two relatively risky securities have bee cobied ito a portfolio that is virtually risk-free. Applicatio 3.5 i chapter 3 provides a geeral forat for costructig a riskless portfolio i the presece of two perfectly iversely correlated assets. Notice i the previous paragraph that we first cobied securities 3 ad 4 ito a portfolio ad the foud that portfolio s retur give each outcoe. The portfolio s retur is 0% regardless of the outcoe; thus, it is risk-free. The sae result ca be obtaied with the variaces of securities three ad four, the correlatio coefficiet betwee their returs, ad solvig for portfolio variace with equatio (6.8) as i table 6.. The iplicatio of the two exaples provided i this chapter is that security risk ca be diversified away by cobiig the idividual securities ito portfolios. Spreadig ivestets across a variety of securities does result i portfolio risk that is lower tha the weighted average risks of the idividual securities. This diversificatio is ost effective whe the returs of the idividual securities are at least soewhat urelated; or, better still, iversely related, as were securities three ad four i the previous exaple. For exaple, returs o a retail food copay stock ad o a furiture copay stock are ot likely to be perfectly positively correlated; therefore, icludig both of the i a portfolio ay result i a reductio of portfolio risk. Fro a atheatical perspective, the reductio of portfolio risk is depedet o the correlatio coefficiet of returs ρ ij betwee securities icluded i the portfolio. Thus, the lower the correlatio coefficiets betwee these securities, the lower will be the resultat portfolio risk. I fact, as log as ρ ij is less tha oe, which realistically is always the case, soe reductio i risk ca be realized fro diversificatio. Cosider the Portfolio Possibilities Frotier displayed i figure 6.. This frotier aps out portfolio retur ad stadard deviatio cobiatios as security weights vary. The correlatio coefficiet betwee returs of securities C ad D is oe. Reeber that portfolio retur is always a weighted average of idividual security returs. The stadard deviatio of returs of ay portfolio cobiig these two securities is a weighted average of the returs of the two securities stadard deviatios, but oly because the correlatio coefficiet betwee returs o these securities equals. Thus, both portfolio returs ad portfolio stadard deviatio are liearly related to the proportios ivested i each of the two securities. Diversificatio here yields o beefits. I figure 6., the correlatio coefficiet betwee returs o securities A ad B is 0.5. Portfolios cobiig these two securities will have stadard deviatios less tha the weighted average of the stadard deviatios of the two securities. Hece, the portfolio possibilities frotier for these two securities arches toward the vertical axis. Give this lower correlatio coefficiet, which is ore represetative of real-world correlatios, there are clear beefits to diversificatio. I fact, we ca see i figures 6.3 ad 6.4 that decreases i correlatio coefficiets result i icreased diversificatio beefits. Lower correlatio coefficiets result i lower risk levels at all levels of expected retur. The portfolio possibilities frotier will exhibit a ore sigificat arch toward the vertical axis as the correlatio coefficiet betwee security returs decreases. Thus, a ivestor will beefit by costructig his portfolio of securities with low correlatio coefficiets.

7 QRMC06 9/7/0 4:44 PM Page 09 Diversificatio ad efficiecy 09 R p 0.0 W D = W C = W D = 0 W C = W D = 0.5 W C = 0.5 R C = 0.0 R D = 0.0 σ C = 0.0 σ D = 0.30 ρcd = σ p Figure 6. The relatioship betwee portfolio retur ad risk whe ρ CD =. R p 0.0 W B = W A = W B = 0.5 W A = 0.5 W B = 0 W A = W B = 0.6 W A = 0.4 R A = 0.0 R B = 0.0 σ A = 0.0 σ B = 0.30 ρab = σ p Figure 6. The relatioship betwee portfolio retur ad risk whe ρ AB = 0.5. To this poit, we have focused o usig correlatio coefficiets to aage the diversificatio of a portfolio. However, we will also cosider a secod powerful diversificatio tool. The risk of a portfolio will ted to declie as, the uber of securities i the portfolio, icreases. This result will hold as log as the securities are ot perfectly correlated. It is perfectly reasoable to expect that securities will ot be perfectly correlated. Thus, two factors gover the level of diversificatio i a portfolio: This result also requires that as securities are added to the portfolio, their idividual variaces are ot icreasig eough to offset the diversificatio beefits that they provide. If variaces aog all securities are approxiately equal or are radoly distributed with a costat ea, this result will hold. This result will be deostrated atheatically i applicatio 8.5.

8 QRMC06 9/7/0 4:44 PM Page 0 0 Eleetary portfolio atheatics R p W F = 0.5 W E = 0.5 W F = 0 W E = W F = W E = 0 R E = 0.0 R F = 0.0 σ E = 0.0 σ F = 0.30 ρef = σ p Figure 6.3 The relatioship betwee portfolio retur ad risk whe ρ EF = 0. R p 0.0 W H = W G = W H = 0 W G = R G = 0.0 R H = 0.0 σ = 0.0 σ G H = 0.30 ρgh = σ p Figure 6.4 The relatioship betwee portfolio retur ad risk whe ρ GH =. The covariaces betwee pairs of securities i the portfolio. Saller retur covariaces of icluded securities lead to reduced portfolio risk. The uber of assets icluded i the portfolio. Larger ubers of icluded securities lead to decreased portfolio risk. 6.5 THE MARKET PORTFOLIO AND BETA The arket portfolio is the collective set of all ivestets that are available to ivestors. That is, the arket portfolio represets the cobiatio or aggregatio of all securities (or other assets) that are available for purchase. Ivestors ay wish to cosider the perforace of this arket portfolio to deterie the perforace of securities i geeral. Sice portfolio retur is a weighted average of idividual security returs,

9 QRMC06 9/7/0 4:44 PM Page Derivig the portfolio variace expressio the retur o the arket portfolio is the average of returs o securities. Thus, the retur o the arket portfolio is represetative of the retur o the typical asset. A ivestor ay wish to kow the arket portfolio retur to gauge perforace of a particular security or ivestet portfolio relative to the perforace of the arket or a typical security. The arket retur is also very useful for costructig additioal risk easures such as a security or ivestet portfolio beta: σi σσ i ρi, COV(, i) βi = ρi, = =. σ σ σ (6.0) Cosider a stock A whose stadard deviatio of returs is 0.4 ad assue that the arket portfolio stadard deviatio equals 0.. Further assue that the correlatio coefficiet betwee returs o security A ad the arket equals The the beta (β A ) of security A would be.5: σi 04. βi = ρi, = 075. = 5.. σ 0. The beta of a stock easures the risk of a stock relative to the risk of the arket portfolio. Part of a stock s risk derives fro diversifiable sources (fir-specific) ad other risks are udiversifiable (arket-related). The lower the correlatio of a stock with the arket, the greater is the risk that ca be diversified away. Thus, lower-risk stocks ad portfolios will have lower betas. Because beta oly accouts for udiversified risk, the beta of a portfolio equals the weighted average beta of its copoet securities. Deteriatio of the retur o the arket portfolio requires the calculatio of returs o all of the assets available to ivestors. Because there are hudreds of thousads of assets available to ivestors (icludig stocks, bods, optios, bak accouts, real estate, ad so o), deteriig the exact retur of the arket portfolio ay be ipossible. Thus, ivestors geerally ake use of idices such as the Dow Joes Idustrial Average or the Stadard ad Poor s 500 to gauge the perforace of the arket portfolio. These idices erely act as surrogates for the arket portfolio; we assue that if the idices are icreasig, the the arket portfolio is perforig well. For exaple, perforace of the Dow Joes Idustrials Average depeds o the perforace of the 30 stocks that coprise this idex. Thus, if the Dow Joes arket idex is perforig well, the 30 securities, o average, are probably perforig well. This strog perforace ay iply that the arket portfolio is perforig well. I ay case, it is easier to easure the perforace of 30 or 500 stocks (for the Stadard ad Poor s 500) tha it is to easure the perforace of all of the securities that coprise the arket portfolio. 6.6 DERIVING THE PORTFOLIO VARIANCE EXPRESSION (Backgroud readig: sectio 6.3) We first discussed variace as a fuctio of potetial squared deviatios fro expected retur outcoes. This is cosistet with defiitios of variace i ost other applicatios.

10 QRMC06 9/7/0 4:44 PM Page Eleetary portfolio atheatics For practical purposes, it is orally ore useful to defie portfolio variace i ters of idividual security variaces ad covariaces betwee pairs of securities. This eables us to characterize risk as a fuctio of portfolio weights. Kowig appropriate portfolio weights eables us to deterie appropriate aouts to ivest i each security. A uderstadig of the relatioship betwee the two portfolio risk easures will help us uderstad ore coplex cocepts cocerig portfolio risk. We will derive the variace of give portfolio p as a fuctio of security variaces, covariaces, ad weights as i equatio (6.4). First, we start with the variace expressed as a fuctio of potetial portfolio retur outcoes j ad associated probabilities: σ p = p j E p ( R [ R ]) Pj. (6.3) For the sake of siplicity, we will work with = securities i our portfolio. It will be easy to geeralize this procedure fro to securities. Portfolio variace ay be rewritte fro equatio (6.3) as follows: σ p = ( wr j + wrj we[ R] we[ R]) Pj. (A) The first two ters iside the paretheses i cobiatio refer to returs for securities ad give outcoe i. The last two ters i cobiatio refer to expected returs for securities ad. Next, we coplete the square for equatio (A) ad cobie ters ultiplied by w ad w : σ p + w (R j E[R ]) = [ w ( Rj E[ R]) Pj P j + w w (R j E[R ])(R j E[R ])P j ]. (B) We ext ove the suatio operatio iside the brackets: σ p = w ( R E[ R ]) P + w ( R E[ R ]) P j j j j + ww ( R j E[ R ])( R j E[ R ]) Pj. (C) The derivatio is copleted by substitutig i equatio (C) variaces ad covariaces as defied i chapter 5: σ p = w σ + w σ + w w σ,, (6.8) which is the two-security equivalet for equatio (6.4).

11 QRMC06 9/7/0 4:44 PM Page 3 Exercises 3 EXERCISES 6.. Sevety-five percet of a portfolio is ivested i Hoeybell stock ad the reaiig 5% is ivested i MBIB stock. Hoeybell stock has a expected retur of 6% ad a expected stadard deviatio of returs of 9%. MBIB stock has a expected retur of 0% ad a expected stadard deviatio of 30%. The coefficiet of correlatio betwee returs of the two securities is expected to be 0.4. Deterie the followig: (a) (b) (c) the expected retur of the portfolio; the expected variace of the portfolio; the expected stadard deviatio for the portfolio. 6.. What is the stadard deviatio of returs for a equally weighted portfolio coprisig two idepedet securities with retur variaces equal to 0.09? 6.3. Each of the pairs of stock listed below will be cobied ito two-security portfolios. I each case, the first stock will coprise 60% of the portfolio ad the secod stock will coprise the reaiig 40%. Copute the stadard deviatio of returs for each portfolio. (a) σ = 0.60, σ = 0.60, σ, = 0.36; (b) σ = 0.60, σ = 0.60, σ, = 0.8; (c) σ = 0.60, σ = 0.60, σ, = 0; (d) σ = 0.60, σ = 0.60, σ, = 0.8; (e) σ = 0.60, σ = 0.60, σ, = A equally weighted portfolio will cosist of shares fro AAB Copay stock ad ZZY Copay stock. The expected retur ad stadard deviatio levels associated with the AAB Copay stock are 5% ad %, respectively. The expected retur ad stadard deviatio levels for ZZY Copay stock are 0% ad 0%. Fid the expected retur ad stadard deviatio levels of this portfolio if returs o the two stocks are: (a) (b) (c) perfectly correlated; idepedet; perfectly iversely correlated How do the coefficiet of correlatio betwee returs of securities i a portfolio affect the expected retur ad risk levels of that portfolio? 6.6. A ivestor will place oe-third of his oey ito security, oe-sixth ito security, ad the reaider (oe-half ) ito security 3. Security data is give i the table below:

12 QRMC06 9/7/0 4:44 PM Page 4 4 Eleetary portfolio atheatics Security, i E[R] σ (i) COV(i,) COV(i,) COV(i,3) Fid the expected retur ad variace of this portfolio The expected variace of returs o y two-security portfolio is The variace of y oly risky security is 0.0; y other security is riskless ad has a expected retur of 0.0. The expected retur of the risky security is 0.5. What is the expected retur of y portfolio? 6.8. There exists a arket where all securities have a retur stadard deviatio equal to 0.8. All securities are perceived to have idepedet retur outcoes; that is, returs betwee pairs of securities are ucorrelated. (a) (b) (c) (d) (e) What would be the retur stadard deviatio of a two-security portfolio i this arket? What would be the retur stadard deviatio of a four-security portfolio i this arket? What would be the retur stadard deviatio of a eight-security portfolio i this arket? What would be the retur stadard deviatio of a 6-security portfolio i this arket? Suppose that all securities i this arket have a expected retur equal to 0.0. How do the expected returs of the portfolios i parts (a) through (d) differ?

Models of Asset Pricing

Models of Asset Pricing APPENDIX 1 TO CHAPTER 4 Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see