risk and rates of return updated

8/9/2019 Risk and Rates of Return Updated

1/23

Risk and Return


2/23

Overview

1. Portfolio Returns and Portfolio Risk

Calculate the expected rate of return and volatility fora portfolio of investments and describe how

diversication aects the returns to a portfolio ofinvestments.

. !ystematic Risk and the "arket Portfolio

#nderstand the concept of systematic risk for an

individual investment and calculate portfoliosystematic risk $beta%.

FIN3000, Liuren Wu2


3/23

Portfolio Returns and PortfolioRisk

&y investin' in many dierent stocks to form aportfolio( we can lower the risk without lowerin'the expected return.

)he eect of lowerin' risk via appropriate portfolioformulation is called diversifcation.

&y learnin' how to compute the expected returnand risk on a portfolio( we illustrate the eect of

diversication.

FIN3000, Liuren Wu3


4/23

)he *xpected Return of aPortfolio

)o calculate a portfolio+s expected rate of return( weweighteach individual investment+s expected rate ofreturn usin' the fraction of money invested in each

investment. *xample ,.1 - f you invest /0of your money in the

stock of Citi bank $C% with an expected rate of returnof 20 and 3/0 of your money in the stock of 4pple$44P5% with an expected rate of return of 160( what

will be the expected rate of return on this portfolio7 *xpected rate of return 8 ./$20% 9 .3/ $160% 8

82%

FIN3000, Liuren Wu4


5/23

Calculating a Portfolios Execte! "ate of "eturn

Penny !impson has her rst fulltime :ob and is considerin' how to

invest her savin's. ;er dad su''ested she invest no more than /0 of

her savin's in the stock of her employer( *merson *lectric $*"R%( so she

is considerin' investin' the remainin' 3/0 in a combination of a risk

free investment in #.!. )reasury bills( currently payin'


6/23

FIN3000, Liuren Wu$


7/23

*valuate the expected return for Penny+sportfolio where she places 1@0.

FIN3000, Liuren Wu


8/23

*valuatin' Portfolio Risk

#nlike expected return( standard deviation is not'enerally eAual to the a wei'hted avera'e of thestandard deviations of the returns of investments

held in the portfolio. )his is because ofdiversication eects.

)he diversication 'ains achieved by addin' moreinvestments will depend on the de'ree of

correlation amon' the investments.

)he de'ree of correlation is measured by usin'the correlation coe&cient ' (.

FIN3000, Liuren Wu8


9/23

Correlation and diversication

)he correlation coeBcient can ran'e from 1.6 $perfectne'ative correlation%( meanin' two variables move inperfectly opposite directions to 91.6 $perfect positivecorrelation%( which means the two assets move exactlyto'ether.

4 correlation coeBcient of 6 means that there is norelationship between the returns earned by the two assets.

4s lon' as the investment returns are not perfectlypositively correlated( there will be diversication benets.

;owever( the diversication benets will be 'reater whenthe correlations are low or ne'ative.

)he returns on most stocks tend to be positively correlated.

FIN3000, Liuren Wu)


10/23

!tandard eviation of aPortfolio

Dor simplicity( let+s focus on a portfolio of stocks-

FIN3000, Liuren Wu*0


11/23

iversication eect

nvesti'ate the eAuation-

Ehen the correlation coeBcient 81( the portfoliostandard deviation becomes a simple wei'hted avera'e-

f the stocks are perfectly movin' to'ether( they areessentially the same stock. )here is no diversication.

Dor most two dierentstocks( correlation is less thanperfect $F1%. ;ence( the portfolio standard deviation isless than the wei'hted avera'e. G )his is the eect ofdiversication.

FIN3000, Liuren Wu**

portfolio =|W11 +W22 |, when =1


12/23

*xample

etermine the expected return and standarddeviation of the followin' portfolio consistin'of two stocks that have a correlation

coeBcient of .3/.Portfolio Weig+t Execte!

"eturntan!ar!-e.iation

/le #0 *4 20

Coca1Cola #0 *4 20



13/23

4nswer

*xpected Return 8 ./ $.1


14/23

E.aluating a Portfolios "is an! "eturn

!arah plans to invest half of her


15/23

Verify the answer: 13%, 23.5%

*valuate the expected return and standard

deviation of the portfolio( if the correlation is .6 instead of 6.3/.

FIN3000, Liuren Wu*#


16/23

4nswer

)he expected return remains the same at 120.

)he standard deviation declines from 2./0 to1>.L0 as the correlations declines from 6.3/ to

6.6.

)he wei'ht avera'e of the standard deviation of thetwo funds is /0( which would be the standarddeviation of the portfolio if the two funds areperfectly correlated.

?iven less than perfect correlation( investin' in thetwo funds leads to a reduction in standard deviation(as a result of diversication.

FIN3000, Liuren Wu*$


17/23

,. !ystematic Risk and "arketPortfolio

t would be an onerous task to calculate the correlationswhen we have thousands of possible investments.

Capital 4sset Pricin' "odel or the C4P" provides a

relatively simple measure of risk. C4P" assumes that investors choose to hold the

optimally diversied portfolio that includes all riskyinvestments. )his optimally diversied portfolio thatincludes all of the economy+s assets is referred to as the

aret ortfolio 4ccordin' to the C4P"( the relevant risk of an investment

relates to how the investment contributes to the risk ofthis market portfolio.

FIN3000, Liuren Wu*


18/23

Risk classication

)o understand how an investment contributes to the risk ofthe portfolio( we cate'oriMe the risks of the individualinvestments into two cate'ories-

N !ystematic risk( and

#nsystematic risk( or idiosyncratic risk

)he ssteatic ris component measures thecontribution of the investment to the risk of the market.Dor example- Ear( hike in corporate tax rate.

)he unssteatic ris is the element of risk that doesnot contribute to the risk of the market. )his component isdiversied away when the investment is combined withother investments. Dor example- Product recall( laborstrike( chan'e of mana'ement.



19/23

!ystematic versusdiosyncratic Risk

4n investment+s systematic risk is far moreimportant than its unsystematic risk.

f the risk of an investment comes mainlyfrom unsystematic risk( the investment willtend to have a low correlation with thereturns of most of the other stocks in theportfolio( and will make a minor contributionto the portfolio+s overall risk.

FIN3000, Liuren Wu*)


20/23

iversication and !ystematicRisk

Di'ure , illustrates that as the number of securities in aportfolio increases( the contribution of the unsystematic ordiversiable risk to the standard deviation of the portfoliodeclines.

!ystematic or nondiversiable risk is not reduced even aswe increase the number of stocks in the portfolio.

!ystematic sources of risk $such as ination( war( interestrates% are common to most investments resultin' in a

perfect positive correlation and no diversication benet.

5ar'e portfolios will not be aected by unsystematic riskbut will be inuenced by systematic risk factors.

FIN3000, Liuren Wu20


21/23

!ystematic Risk and &eta

!ystematic risk is measured by 5eta coe&cient, whichestimates the extent to which a particular investment+sreturns vary with the returns on the market portfolio.

n practice( it is estimated as the slope of a strai'ht line$see 'ure ,2%-

&eta could be estimated usin' excel or nancial

calculator( or readily obtained from various sources onthe internet $such as Qahoo Dinance and "oneyCentral.com%

FIN3000, Liuren Wu2*

Ri = a+ R

m+ e


22/23

Portfolio &eta

)he beta of a portfolio measures the systematicrisk of the portfolio and is calculated by takin' asimple wei'hted avera'e of the betas for the

individual investments contained in the portfolio. *xample ,. Consider a portfolio that is comprised

of four investments with betas eAual to 1./( .3/(1., and .L6. f you invest eAual amount in each

investment( what will be the beta for the portfolio7 Portfolio beta8

1./$1@


23/23

!olution

&eta $4*P% 8

risk and rates of return updated

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