Return and Risk- Lecture

7/31/2019 Return and Risk- Lecture

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RETURN AND RISK:

PORTFOLIO THEORYAND

CAPITAL ASSETPRICING MODEL (CAPM)


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1. Individual Securities

Expected Return

Return that an individual expects a stock to earnover the next period

Actual return may be either higher or lower

Formula:

Expected return = Prob x Return

= P x Ri


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Variance () and Standard Deviation ()

Measures the variability of an individual securitys

return (squared deviations of a securitys returnfrom its expected return)

To assess the volatility or variability of anindividual securitys return on the expected return

Formula:

= P (Ri expected returni)Thus,

= P (Ri expected returni)


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Covariance andCorrelation

To measure the r/shipbetween returns onindividual securities

Covariance can bestated in terms of thecorrelation between twosecurities.

+ve cov

+ve r/ship between the 2

returns

Both returns areabove/below theiraverage returns

- ve cov

-ve r/ship between the 2

returns One return is above the

average and the other isbelow the average return

Zero cov No r/ship i.e. the 2

returns are unrelated


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Covariance is difficult to be interpretedbecause it is in squared deviation units

To solve the problem, use the correlation

Correlation is between +1 and -1 Correlation can be +ve, -ve and zero and its

interpretation is similar to covariance


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Formula:

Covariance (AB)

Prob(RAexp RA)(RBexp RB)

Correlation (AB)

AB

A X B

Therefore,

AB = AB XA XB


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Example: Individual Securities

Suppose you have invested only in two stocks,A and B. The returns on the two stocks dependon the following three states of the economywhich are:

State of economy Probability Return onStock

A B

Recession 0.25 - 2.00 5.00Normal 0.60 9.20 6.20

Boom 0.15 15.40 7.40


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a) Calculate the expected return on eachstock.

b) Calculate the standard deviation of returnson each stock.

c) Calculate the covariance and correlationbetween the returns on the two stocks.


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2. Return and Risk for Portfolios

Expected return on a portfolio (Rp)= weighted average of the expected

returns on the individual securities

= XA RA + XB RB= Rp

Where:

XA = percentage of the portfolio in security AXB = percentage of the portfolio in security B

RA and RB =expected return on security A and B respectively


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Standard Deviation (SD) of a portfolio

(p)= NOT the weighted average of the

standard deviations individual securities= XAA + 2XAXBA,B + XBB

Where:

XA = percentage of investment in securities A

XB = percentage of investment in securities B

A = SD of security AB = SD of security B

AB = Covariance of security A and B


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Covariance (AB) and Correlation (AB)

Correlation (AB) = AB

A X B

Therefore,

AB = AB X A X B


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+ve r/ship between the 2 securities increasesthe variance of the entire portfolio. Therefore,

the risk of the entire portfolio will be higher

-ve r/ship between the 2 securities decreasesthe variances (risk) of the entire portfolio.

SD of a portfolio < Weightedaverage of SD

of individualsecurities

(Due to diversification effect)


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However, it does not mean we should not invest insecurities which are positively correlated, as long asthe correlation () is less than perfect positive

correlation(


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Example: For Portfolio

Suppose you have invested only in two stocks,A and B. The returns on the two stocks dependon the following three states of the economywhich are:

State of economy Probability Return onStock

A B

Recession 0.25 - 2.00 5.00Normal 0.60 9.20 6.20

Boom 0.15 15.40 7.40


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Calculate:

expected return of a portfolio

standard deviation of a portfolio

if 40% and 60% of your funds invested in A and

B respectively.


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3. Efficient Set: Two Assets

Example:

Stock Expected return SD

A 17.5% 25.86%B 5.5% 11.50%

Portfolio of

60% in A and40% in B 12.7% 15.44%


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Graph

MV

1

1 (10% in A, 90% in B)

2

3

A (100% in A)

B (100% in B)

Expected

Return (%)

Standard deviation (%)


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1. Correlation (+1 to 1)

Diversification effect (DE) occurs when thecorrelation ()between the two securities is


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2. Minimum Variance Portfolio

At MV

The lowest possible variance or SD

3. Opportunity Set or Feasible Set

Represented by the curved line throughpoint B, MV and A

An investor can achieve any point on thecurve by selecting the appropriate mix

between the two securities Depends on the stomach of the investor

(risk averse/risk taker)


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4. Curved line between point MV and B

A portion of feasible set

SD decreases as one increases expected return

5. Point < Minimum Variance portfolio (MVP)

No investor want to hold a portfolio with expected

return below that of the MVP i.e. portfolio 1

Less expected return but higher SD than the MVPhas.

Thus, the investor only consider the curve from MVto A as the efficient set of efficient frontier.


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Question?????

How can an increase in the proportion of the risky

security lead to a reduction in the risk of theportfolio???

Answer

Diversification Effect

Correlation between the securities isve ( < 1)which represented by the backward bending curve

Thus, an addition of a small amount of risky securityacts as a hedge to a portfolio composed only onesecurity


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Two-Security Portfolios with Various Correlations

100%

bonds

return

100%

stocks

= 0.2 = 1.0

= -1.0

Relationship depends on correlation coefficient-1.0 < < +1.0

If = +1.0, no risk reduction is possible

If =1.0, complete risk reduction is possible


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4. The Efficient Set for Many Securities

Consider a world with many risky assets; we can stillidentify the opportunity setof risk-return combinationsof various portfolios.

return

P

Individual Assets

MV

X


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The shaded area represents the feasible setwhen many securities are considered.

The shaded area represents all the possiblecombinations of expected return and

standard deviation of a portfolio

No combination of securities can fall outside

the shaded area

An investor will want to be somewhere on theupper edge between MVP and X


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Given the opportunity setwe can identify theminimum variance portfolio.

return

P

minimum

varianceportfolio

Individual Assets


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The section of the opportunity set above the minimumvariance portfolio is the efficient frontier.

return

P

minimum

varianceportfolio

Individual Assets


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Relationship between the Variance of a Portfolios

Return and the Number of Securities in the Portfolio

We can never eliminate risk no matter how manysecurities we have in our portfolio.

The variance (risk) of portfolio drops, but can only

reach a floor of covariance. i.e. the lowest risk can drop is equal the covariance

Total risk of = Portfolio risk + Unsystematic

risk individual sec(diversifiable risk)


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Portfolio Risk as a Function of the Number of Stocksin the Portfolio

Nondiversifiable risk;

Systematic Risk;Market Risk

Diversifiable Risk;

Nonsystematic Risk;

Firm Specific Risk;

Unique Risk

n

In a large portfolio the variance terms are effectivelydiversified away, but the covariance terms are not.

Thus diversification can eliminate some,

but not all of the risk of individual securities.

Portfolio riskcov


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5. Optimal Risky Portfolio with a Risk-Free Asset

In addition to stocks and bonds, consider a worldthat also has risk-free securities like T-bills(Government bond)

100%

bonds

100%

stocks

rf

return


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6. Riskless Borrowing and Lending

An investor could combine a risky investmentwith an investment in a riskless or risk-freesecurity, such as an investment in Treasurybills or government bond.

SD of the risk-free security = 0


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Optimal Portfolio

Is the portfolio that will give an investor the highestreturn.

Problems associated with identifying optimalportfolio: It is unrealistic to assume that the investors can borrow at

the risk free rate.

It requires knowledge of the risk and return of all riskyinvestments.

It is expensive to construct optimal portfolio smallinvestors

Market portfolio changes over time due to changes in rfrate of return, feasible sets and efficient frontier.


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Now investors can allocate their money across theT-bills and a balanced mutual fund

100%

bonds

100%

stocks

rf

return

Balanced

fund


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With a risk-free asset available and the efficient frontieridentified, we choose the capital allocation line with thesteepest slope

return

P

rf


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Market Equilibrium

Possible when all investors are assumed to havehomogeneous expectations for expected returns,variances and covariances.

Thus, all investors have the same optimal portfolio. Homogeneous expectations:

Investors have the same expected returns, variances andcovariance.

Hence all investors will have the same efficient set. Thus, all investors have the same optimal portfolio of risky

assets and also a diversified portfolio.


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7. Market Equilibrium

With the capital allocation line identified, all investors choose a pointalong the linesome combination of the risk-free asset and themarket portfolio M. In a world with homogeneous expectations, Mis

the same for all investors.

return

P

rf

M


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Market Equilibrium

Just where the investor chooses along the Capital AssetLine depends on his risk tolerance. The big pointthough is that all investors have the same CML.

100%

bonds

100%

stocks

rf

return

Balanced

fund


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All investors have the same CML because they all havethe same optimal risky portfolio given the risk-free rate.

100%

bonds

100%

stocks

rf

return

Optimal

RiskyPortfolio

Market Equilibrium


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8. Optimal Risky Portfolio with a Risk-Free Asset

By the way, the optimal risky portfolio depends onthe risk-free rate as well as the risky assets.

100%

bonds

100%

stocksreturn

FirstOptimal

Risky

Portfolio

Second Optimal

Risky Portfolio

0

fr

1fr


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9. Definition of Risk When InvestorsHold the Market Portfolio

Researchers have shown that the bestmeasure of the risk of a security in a largeportfolio is the beta(b)of the security.

Beta measures the sensitivity orresponsiveness of a change in the return ofan individual stock to the change in return of

market portfolio. Market portfolio is proxied by broad-based

index


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Beta measures the responsiveness of a security tomovements in the market portfolio.

Concept of beta: Market beta = 0 0.5 for every 1% movement in the market, the

stock is expected to move 0.5% in the same direction(less volatile)

2.5 for every 1% movement in the market, the

stock is expected to move 2.5% in the same direction.(highly volatile) -0.5the stock is expected to move 0.5% for every

1% movement in market in different direction.

)()(

2

,

M

Mi

iR

RRCov

b =


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Beta is the best measure of risk of thesecurity when hold a large and diversified

portfolio.

Because, in a large and diversified portfolio,

the only risk left is the systematic risk andunsystematic risk is no more relevant to theportfolio.


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The Formula for Beta

)(

)(

2

,

M

Mi

i R

RRCov

b =

Clearly, your estimate of beta will depend upon yourchoice of a proxy for the market portfolio.


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It means thatve beta stock is expected to do well

when the market does poorly, vice versa.

As a result, adding ave beta security to a large ,diversified portfolio can reduce the risk of the

portfolio.

CAPM is one of the model that can be used toexplain the relationship between the risk and

required rate of return on assets when investorshold a well diversified portfolio.


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10. Relationship between Riskand Expected Return (CAPM)

Expected Return on the Market:

Expected return on an individual security:

PremiumRiskMarket= FM RR

)(F

MiF

i RRRR =

Market Risk Premium

This applies to individual securities held within well-diversified portfolios.


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Expected Return on an Individual Security

This formula is called the Capital Asset PricingModel (CAPM)

)( FMiFi RRRR =

Assume bi = 0, then the expected return isRF.

Assume bi = 1, then Mi RR =

Expected

return on

a security

=Risk-

free rate+

Beta of the

security

Market risk

premium


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Relationship Between Risk & Expected Return

Expecte

dreturn

b

)( FMiFi RRRR =

FR

1.0

MR


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Relationship Between Risk & Expected Return

Expecte

d

return

b%3=FR

%3

1.5

%5.13

5.1 =i %10=MR

%5.13%)3%10(5.1%3 ==iR


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For example Let's say that the current risk free-rate is 5%, and the

S&P 500 is expected to return to 12% next year. You areinterested in determining the return that Joe's Oyster Bar Inc (JOB)

will have next year. You have determined that its beta value is 1.9.The overall stock market has a beta of 1.0, so JOB's beta of 1.9tells us that it carries more risk than the overall market; this extrarisk means that we should expect a higher potential return than the12% of the S&P 500. We can calculate this as the following:

Required (or expected) Return = 5% + 1.9(12% - 5%)= 18.3%

What CAPM tells us is that Joe's Oyster Bar has a required rate ofreturn of 18.3%. So, if you invest in JOB, you should be getting atleast 18.3% return on your investment. If you don't think that JOBwill produce those kinds of returns for you, then you shouldconsider investing in a different company.

)( FMiFi RRRR =
http://www.investopedia.com/terms/s/sp500.asphttp://www.investopedia.com/terms/b/beta.asphttp://www.investopedia.com/terms/b/beta.asphttp://www.investopedia.com/terms/s/sp500.asp


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11. Summary and Conclusions

This chapter sets forth the principles of modernportfolio theory.

The expected return and variance on a portfolioof two securities A and Bare given by (weightedaverage)

ABAABB2

BB2

AA2P

))(x2(x)(x)(x =

)()( BBAAP RxRxR =


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By varying xA, one can trace out the efficientset of portfolios. We graphed the efficient set

for the two-asset case as a curve, pointingout that the degreeof curvature reflects thediversification effect:

the lower the correlation between the twosecurities, the greater the diversification.

The same general shape holds in a world ofmany assets.


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The efficient set of risky assetscan be combined withriskless borrowing and lending. In this case, a rational

investor will always choose to hold the portfolio of riskysecurities represented by the market portfolio.

return

P

rf

M

Then, withborrowing or

lending, the investor

selects a point along

the CML.


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The contributionof a security to the riskof a well-

diversified portfolio is proportionalto the covarianceof the

security's return with the markets return. This contributionis called the beta ().

The CAPM states that the expected return on a security ispositivelyrelated to the securitys beta:

)(

)(

2

,

M

Mi

i

R

RRCov

b =

)( FMiFi RRRR =

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Return and Risk- Lecture