Risk and Return: The Security Markets Line

Transcription

1 FIN 614 Rsk and Return 3: Markets Professor Robert B.H. Hauswald Kogod School of Busness, AU 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 1 Rsk and Return: The Securty Markets Lne From securtes ndvdual rsk and return characterstcs to portfolos: rsk dversfcaton to securty markets: systematc/unsystematc rsk, beta Two fundamental rules prcng rsk: return compensates for rsk bearng there s not free lunch (NFL): no arbtrage Two fundamental results: portfolo dversfcaton: modern portfolo theory Captal Asset Prcng Model: market equlbrum 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 2

2 Rsk and Dversfcaton Surprses come n two flavors and turn nto rsk: systematc or market rsk: a surprse that affects a large number of assets to varyng degrees common shock unsystematc (unque) rsk: a surprse that affects at most a small number of assets dosyncratc shock Dversfcaton prncple: portfolos reduce total rsk varablty of multple assets held together (more, less, equal?) than varablty of typcal stock. dversfable, unque rsk: the porton of an asset s varablty not present n a large group of assets (portfolo) held together undversfable rsk: the level of varance that s present n collectons of assets (portfolo rks) 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 3 Portfolo Rsk as a Functon of the Number of Stocks n the Portfolo σ In a large portfolo the varance terms are effectvely dversfed away, but the covarance terms are not. Dversfable Rsk; Nonsystematc Rsk; Frm Specfc Rsk; Unque Rsk Portfolo rsk Nondversfable rsk; Systematc Rsk; Market Rsk N Thus dversfcaton can elmnate some, but not all of the rsk of ndvdual securtes. 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 4

3 Systematc and Unsystematc Return Components Total return = Expected return + Unexpected return: snce Unexpected return = (nvestor) surprses come n two flavors systematc (market, economy) returns: unsystematc, dosyncratc frm-specfc returns: In portfolos, securtes unsystematc rsks tend to cancel each other out: what s left? dversfable rsk s synonymous wth unsystematc rsk large portfolos have lttle or no unsystematc rsk 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 5 Systematc and Unsystematc Rsk σ ε We can break down the rsk, U, of holdng a stock nto two components: systematc rsk and unsystematc rsk: Total rsk; U Nonsystematc Rsk; ε Systematc Rsk; m R = R + U becomes R = R + m + ε where m s the systematc rsk ε s the unsystematc rsk 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 6 N

4 Average covarance of the securtes wth the portfolo domnates any ndvdual securty s unque rsk: squared weghts become neglgble For N large, left wth As N Gets large can be rearranged nto Cov(,p): measure of how much rsk any one securty contrbutes to portfolo covarance: statstcal measure of co-movement of returns Proporton of rsk any one asset contrbutes to overall portfolo rsk can be expressed as: 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 7 Varance of Completely Dversfed Portfolos Varance of a portfolo s composed of 2 parts: As N becomes large (const. σ) only market rsk of a securty remans and contrbutes to portfolo rsk: 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 8

5 Effcent Fronter Effcent fronter: composed of portfolos such that No other portfolo offers a hgher return for a gven level of rsk No other portfolo offers less rsk for a gven level of return Plot Standard Devaton on x-axs, and Expected Return on the y-axs The lne connectng rsk free rate and curve s called Captal Market Lne 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 9 Many Securtes return Indvdual Assets Consder a world wth many rsky assets; we can stll dentfy the opportunty set of rsk-return combnatons of varous portfolos. 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 10 σ P

6 The Effcent Set for Many Securtes return mnmum varance portfolo Indvdual Assets The secton of the opportunty set above the mnmum varance portfolo s the effcent fronter. 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 11 σ P Optmal Rsky Portfolo wth a Rsk- Free Asset return 100% stocks r f 100% bonds σ In addton to stocks and bonds, consder a world that also has rsk-free securtes lke T-blls 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 12

7 Rskless Borrowng and Lendng: The Captal Market Lne return Balanced fund 100% stocks r f 100% bonds Now nvestors can allocate ther money across the T-blls and a balanced mutual fund σ 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 13 Why Invest n Mutual Funds? 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 14

8 Market Equlbrum return M r f Wth the captal market lne dentfed, all nvestors choose a pont along the lne some combnaton of the rsk-free asset and the market portfolo M. In a world wth homogeneous expectatons, M s the same for all nvestors. 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 15 σ P Optmal Rsky Portfolo wth a Rsk- Free Asset return 100% stocks 1 r f 0 r f Frst Optmal Rsky Portfolo 100% bonds Second Optmal Rsky Portfolo The optmal rsky portfolo depends on the rsk-free rate as well as the rsky assets. 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 16 σ

9 Sharpe Rato A portfolo number used to compare the rskreturn trade-off of dfferent portfolos based on standard devaton allows comparson of undversfed portfolos [ ] E r p r f Sharpe Rato = σ p E(r p ) = expected return of portfolo r f = rsk free rate, σ p = standard dev. of portfolo 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 17 Systematc Rsk: Beta The Systematc Rsk prncple: the reward for bearng rsk depends only upon the systematc or undversfable rsk of an nvestment what about unsystematc or dversfable rsk? Measurng Systematc Rsk: Beta coeffcent, β β measures of how much systematc rsk an asset has relatve to an average rsk asset (defned how?) Cov( r,rm ) σ m β = = Var r σ ( ) 2 2 where σ m : the return varance of the market portfolo, (typcally we use S&P 500), r m : return on S&P 500 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 18 m m

10 Estmatng β wth Regresson Securty Returns Slope = β Return on market % R = α + β R m + e 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 19 β and Portfolo Varance Recall that the varance of a portfolo s composed of two parts: substtutng n from the prevous slde nto earler expresson we obtan because only market rsk of a securty remans as N becomes large: 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 20

11 Portfolo Betas Contrary to portfolo varance, portfolo betas are equal to the weghted sum of ndvdual betas Example: (1) (2) (3) (4) Amount Portfolo Beta Product Stock Invested Weght Coeffcent (3) x (4) IBM $ % General Motors $ % Dow Chemcal $ % Portfolo 100%.91 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 21 Beta and the Rsk Premum Recall: rsky nvestment return - rsk-free rate A rskless asset has a beta of 0; beta of portfolo s a weghted average of Betas of ndvdual assets. Let a portfolo be comprsed of an nvestment n Portfolo A wth a beta of 1.2 and expected return = 18%, and T-blls wth 7% return. Proporton Proporton Portfolo Portfolo nvested Invested Expected return beta n Portfolo A n Rf 0% 100% 7% 0 25% 75% 9.75%.30 50% 50% 12.50%.60 75% 25% % % 0% 18% % -25% % 1.5 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 22

12 Reward-to-Rsk Rato Reward-to-Rsk Rato The combnatons of portfolo expected return and beta n the prevous example, f plotted, le on a straght lne wth slope: Rse = E(r A ) - r f = ( ) =.092 = 9.2% Run β Α 1.2 The lne s the securty market lne ts slope s sometmes called the Reward-to-Rsk rato t s the expected return per "unt" of systematc rsk. Fundamental Result: the Reward-to-rsk rato must be the same for all assets n the market E[ ra ]- rf E[ rb ] - rf = f t were not what would happen? A B 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 23 β β The Securty Market Lne Securty market lne: the lne whch gves the expected return - systematc rsk combnatons (trade-offs) Market Portfolos: "average" systematc rsk,.e., beta of 1. snce all assets must le on the securty market lne when approprately prced, so must the market portfolo Denote the market portfolo s expected return by E(r m ) E[ ra ] - rf E[ rm ] - rf = β A 1 = slope of SML NB: ths s NOT the captal market lne, whch s what? 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 24

13 Captal Asset Prcng Model Celebrated result n fnance tyng t all together snce the expected return to any asset, E(r ), must satsfy the same reward-to-rsk rato as the market portfolo, E [ r ] - r f E [ rm ] - r f = β 1 Rearrangng terms, addng a tme subscrpt E CAPM, Captal Asset Prcng Model equaton ( ) [ r ] =r [ ] + β E r t ft mt - rft 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 25 Relatonshp between Rsk and Expected Return (CAPM) Expected Return on the Market: R M = R F + Market Rsk Premum Expected return on an ndvdual securty: R = R + β F ( R M RF ) Market Rsk Premum Ths apples to ndvdual securtes held wthn welldversfed portfolos. 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 26

14 Expected Return on an Indvdual Securty Ths formula s called the Captal Asset Prcng Model (CAPM) Expected return on a securty R = R + β F = Rskfree rate ( RM RF + Beta of the securty ) Market rsk premum Assume β = 0, then the expected return s R F. Assume β = 1, then R = R M 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 27 Relatonshp Between Rsk and Expected Return: SML Expected return R M R = R + β F ( R M RF ) R F 1.0 β R = R + β F ( RM RF 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 28 )

15 Relatonshp Between Rsk and Expected Return: SML Expected return 13.5% 3% β =1.5 R F = 3% R M =10% R 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald = 3 % (10% 3%) = 13.5% β Summary and Conclusons From securtes over rsk-return to portfolos va systematc and unsystematc rsk to SML return needs to compensate for rsk bearng portfolos dversfy away unsystematc rsk securty market lne and beta: return and systematc rsk The CAPM states that the expected return on asset depends upon: 1. the tme value of money, as measured by r f 2. the reward per unt of systematc rsk, E(r m ) - r f 3. the asset's systematc rsk as measured by Beta, β How well do securtes markets conform to theory? 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 30

16 Estmatng Beta Appendx lnear regresson estmaton Graphcal representaton of lnear regresson estmaton Betas of selected stocks examples 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 31 Estmatng Beta E(r ) = r f + β (r m - r f ) = r f + β r m - β r f = (1 - β ) r f + β r m If we assume that β s constant, then E(r ) = α + β r m For expected returns, substtute realzed returns to get a market model r = α + β r m + e where e : error n the regresson model 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 32

17 Regresson Estmate of β Securty Returns Slope = β Return on market % R = α + β R m + e 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 33 Estmates of β for Selected Stocks Stock Beta Bank of Amerca 1.55 Borland Internatonal 2.35 Travelers, Inc Du Pont 1.00 Kmberly-Clark Corp Mcrosoft 1.05 Green Mountan Power 0.55 Homestake Mnng 0.20 Oracle, Inc /25/2011 Rsk and Return: Markets Robert B.H. Hauswald 34