CHAPTER 5 Risk and Rates of Return

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CHAPTER 5Risk and Rates of Return

Stand-alone risk Portfolio risk Risk & return: CAPM / SML

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Return & Risk Return is the annual income received

plus any change in market price of an asset or investment.

Risk is the variability of actual return from the expected return associated with a given asset.

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Rate of Return The rate of return on an investment for

a period (which is usually a period of one year) is defined as follows,

Annual Income + (Ending price – Beginning price)

Beginning price

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Rate of ReturnPrice at the beginning of the year = Tk.

60.00Dividend paid toward the end to the year

= Tk. 2.40Price at the end of the year = Tk. 66.00

2.40 + (66.00 – 60.00)60.00

= 0.14 or 14%.

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Rate of Return Current Yield & Capital gains/Losses

Current Yield Capital gains/Losses Yield

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Current Yield & Capital gains/LossesAnnual Income Ending – Beginning

Price+

Beginning Price Beginning Price

(Current Yield) (Capital Gain/Losses)

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Current Yield & Capital gains/Losses

2.40(66.00 – 60.00)+60.00 60.00

= 4% + 10%

Current Yield Capital gains/Losses

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Current Yield & Capital gains/LossesIf the price of a share on April 1 is TK. 25,

the annual dividend received at the end of the year is TK. 1 and the year end price on March 31 is TK 30.

Find the Rate of ReturnFind the Current YieldFind the Capital gains/Losses Yield.

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1 + (30.00 – 25.00)Rate of return = 25.00

= 0.24 or 24%.Current & Capital gain/Losses

1 (30.00 – 25.00)+

25 25.00 = 4% + 20%

Current Yield Capital gains/Losses

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What is investment risk? Two types of investment risk

Stand-alone risk Portfolio risk

Investment risk is related to the probability of earning a low or negative actual return.

The greater the chance of lower than expected or negative returns, the riskier the investment.

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Measurement of Risk: Single Asset

The risk associated with single asset is assessed from both,

Behavioral point of view Sensitivity Analysis Probability Distribution

Statistical point of view Standard Deviation Coefficient of Variation

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Behavioral Point of View This approach is to estimate the worst

(pessimistic), the expected (most likely) and the best (optimistic) return associated with the asset.

The level of outcome may be related to the economic conditions namely, recession, growth and Boom.

The difference between pessimistic and optimistic outcome is the RANGE which is the measurement of RISK.

The greater the RANGE, the more RISKY the Asset.

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Sensitivity Analysis Particular Asset X Asset Y

Initial Outlay 50 50Annual Return (%) Pessimistic 14 8 Most Likely 16 16 Optimistic 18 24

RANGE = 4 16(optimistic – Pessimistic)

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Measurement of Risk: Single AssetBehavioral Point of View Probability Distribution

The probability of an event represent the % chance of its occurrence.

Probability Distribution is model that relates probabilities to the associated outcome.

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Probability Distribution : Asset X

Possible Outcome

(1)

Probability(2)

Returns(3)

Expected Returns

(2)X(3)=4

PessimisticMost LikelyOptimistic

0.200.600.201.00

141618

2.89.63.6

16.00

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Probability Distribution : Asset Y

Possible Outcome

(1)

Probability(2)

Returns(3)

Expected Returns

(2)X(3)=4

PessimisticMost LikelyOptimistic

0.200.600.201.00

81624

1.69.64.8

16.00

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Statistical Point of ViewStandard Deviation

Risk refers to the dispersion of returns around an expected value.

The most common statistical measure of risk of an asset is the standard deviation from the mean/expected value of return.

= (R-R)2 X pr

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Standard DeviationAsset X

i R R R - R (R – R)2 Pr (R – R)2x Pr

1 14%

16% -2% 4% .20 .80

2 16%

16% 0% 0% .60 0

3 18%

16% 2% 4% .20 .801.6

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Standard Deviation

= (R-R)2 X pr= 1.6= 1.26%

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Standard DeviationAsset Y

i R R R - R (R – R)2 Pr (R – R)2x Pr

1 8% 16% -8% 64% .20 12.80

2 16%

16% 0% 0% .60 0

3 24%

16% 8% 64% .20 12.8025.6

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Standard Deviation

= (R-R)2 X pr= 25.6= 5.06%

The greater the Standard Deviation of Returns, the greater the risk.

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Measurement of Risk: Single AssetStatistical Point of ViewCoefficient of Variation

it is the measure of relative dispersion used in comparing the risk of assets with differing expected returns.

CV =

R

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Measurement of Risk: Single AssetStatistical Point of ViewCoefficient of Variation

The coefficient of variation of assets X & Y are respectively, Asset X = ( 1.26% / 16%) = 0.079Asset Y = (5.06 / 16% ) = 0.316

The larger the CV, the larger the relative risk of the asset.

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Risk & Return of PORTFOLIOPortfolio means a combination of two or more Assets. Each portfolio has risk return characteristics of its own.

Portfolio theory developed by Harry Markowitz, shows that portfolio risk, unlike portfolio return, is more than simple aggregation of the risks of individual assets. This depends on the interplay between the returns on assets comprising the portfolio.

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Portfolio Expected return

E (rp) = wi E(ri)

E (rp) = Expected return from portfolio

Wi = Proportion invested in asset i

E(ri) = Expected return for asset i

n = number of assets in portfolio

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Portfolio Expected returnThe expected return on two assets L and H are 12% & 16% respectively. If the corresponding weights are 0.65 & 0.35. Calculate Portfolio Expected return

E (rp) = wi E(ri)

= [0.65 x 0.12 + 0.35 x 0.16]= 0.134= 13.4%.

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Portfolio Risk:Two Asset portfolio2p = w2

121 + w2222 + 2 w1 w2 (12)

Alternatively, 2p = (w11)2 + (w22 )2 + 2 w1 w2 (P 12 1 2)

W1 = fraction of total portfolio invested in Asset 1W2 = fraction of total portfolio invested in Asset 221 = Variance of asset 11 = Standard deviation of Asset 122 = Variance of asset 22 = Standard deviation of Asset 212 = Covariance between returns of two assets (P 12 1 2)P 12 = Coefficient of correlation between the returns of two asset.

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Portfolio Risk:Two Asset portfolio

The expected return on two assets L and H are 12% & 16% respectively. The standard deviations of assets L & H are 16% and 20% respectively. If the coefficient of correlation between their returns is 0.6 and the two assets are combined in the ratio of 3:1.(1) Calculate expected rate of return(2) variance of Portfolio(3) Standard Deviation

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Portfolio Expected return

E (rp) = wL E(rL) + wH E(rH)

= (0.75 x 0.12) + (0.25 x 0.16)= 9%+4%= 13%.

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The Variance of the Portfolio[2] 2p = (w11)2 + (w22 )2 + 2 w1 w2 (P 12 1 2)

= (0.75 x 16)2 + (0.25 x 20) 2 + 2 (0.75) (0.25) [(0.06) (16 x 20) = 144 + 25+ (0.375)(192)= 144 + 25+72= 241

[3] p = 241

= 15.52

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Portfolio RiskThe above discussion shows that the portfolio risk depends on 3 factors[1] Variance or Standard deviation of each asset in portfolio.[2] Relative importance or weight of each asset in the portfolio[3] Interplay between returns on two assets

Among these only weights can be controlled by the portfolio managers. Therefore his/her primary task is to decide the proportion of each security in the portfolio.

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Investor attitude towards risk Risk aversion – assumes investors

dislike risk and require higher rates of return to encourage them to hold riskier securities.

Risk premium – the difference between the return on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities.

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Breaking down sources of risk

Stand-alone risk = Market risk + Firm-specific risk

Market risk – portion of a security’s stand-alone risk that cannot be eliminated through diversification. Measured by beta.

Firm-specific risk – portion of a security’s stand-alone risk that can be eliminated through proper diversification.

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Capital Asset Pricing Model (CAPM)

Model based upon concept that a stock’s required rate of return is equal to the risk-free rate of return plus a risk premium that reflects the riskiness of the stock after diversification.

It is the logical & major extension of the portfolio theory of Markowitz by william Sharpe (1964), John Linter ( 1965) & Jan Mossin (1967).

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CAPM is a theory that explains how asset prices are formed in the market place. CAPM provides the framework for determining the equilibrium expected return for risky return. It uses the results of capital market theory to derive the relationship between expected return and systematic risk of individual assets/securities and portfolio.

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The CAPM has implication for Risk-Return relationship for an efficient

Portfolio Risk-Return relationship for an individual

asset Identification of over valued or under valued

assets traded in in the market Pricing of assets not yet traded in the market Effect of leverage on cost of equity.

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Capital budgeting decision & cost of capital

Risk of the firm through diversification of the project portfolio.

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Capital Asset Pricing Model (CAPM) : Assumption

All investors are price takers. There number is so large that no single investor can affect prices

Assets/securities are perfectly divisible All investors plan for one identical

holding period Investors can lend or borrow at an

identical risk-free rate. There is no transaction costs & income

Tax

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The elements of the model:K = K RF + (KM - K RF) β

Where, K RF = Risk Free Return KM = required rate of return of marketβ = Beta (systematic risk of the asset)

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BetaIt measure the risk of an individual asset relative to the market portfolio. Beta shows how the price of securities responds to market force. In practice, the more responsive the price of security is to changes in the market, the higher will be its beta. The beta for the overall market is equal to 1.00 Beta can be positive or negative. Investors will find beta helpful in assessing systematic risk and understanding the impact the market movement can have on the return expected from a share or stocks.

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Calculating betasThe ABC Company is considering a new capital investment proposal. The project’s risk structure is very similar to that of the company’s existing business. Return for this company’s stocks for the past ten years are given in the following table together with returns for a country’s stock market index. The Govt. Treasury Bill return (Risk Free Return) was around 5.6% per annum.

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Calculating betasYear Company’s Stock

Return Stock Market Index

Return 1992 0.09 0.07 1993 0.10 0.09 1994 0.10 0.10 1995 0.11 0.12 1996 0.10 0.11 1997 0.11 0.10 1998 0.11 0.10 1999 0.10 0.09 2000 0.09 0.08 2001 0.07 0.07

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Calculating betasRequired Calculate the [1] Beta[2] Required Return according to

CAPM model

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Comments on beta If beta = 1.0, the security is just as risky

as the average stock. If beta > 1.0, the security is riskier than

average. If beta < 1.0, the security is less risky

than average. Most stocks have betas in the range of

0.5 to 1.5.

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Problem: Assume a security with beta of 1.2

being considered at a time when the risk free rate is 4% and the market return is expected to be 12%. Substitute those data by using CAPM equation.Calculate Expected Return according to CAPM model

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Problem: There are three assets- X, Y & Z

with beta value of 0.5, 1.0 & 1.5 respectively. The risk free rate is assumed to be 5% and the market return is expected to be 15%. calculate the expected return

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Illustrating the calculation of beta

.

.

.ki

_

kM

_-5 0 5 10 15 20

20

15

10

5

-5

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Regression line:ki = -2.59 + 1.44 kM^ ^

Year kM ki 1 15% 18% 2 -5 -10 3 12 16

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The Security Market Line (SML):Calculating required rates of return

SML: ki = kRF + (kM – kRF) βi

Assume kRF = 8% and kM = 15%. The market (or equity) risk

premium is RPM = kM – kRF = 15% – 8% = 7%.

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Illustrating the Security Market Line

..Coll.

.HT

T-bills

.USR

SML

kM = 15

kRF = 8

-1 0 1 2

.

SML: ki = 8% + (15% – 8%) βi

ki (%)

Risk, βi

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Factors that change the SML What if investors raise inflation expectations

by 3%, what would happen to the SML?

SML1

ki (%)SML2

0 0.5 1.0 1.5

181511 8

I = 3%

Risk, βi

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Factors that change the SML What if investors’ risk aversion increased,

causing the market risk premium to increase by 3%, what would happen to the SML?

SML1

ki (%) SML2

0 0.5 1.0 1.5

181511 8

RPM = 3%

Risk, βi

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Verifying the CAPM empirically The CAPM has not been verified

completely. Statistical tests have problems that

make verification almost impossible. Some argue that there are

additional risk factors, other than the market risk premium, that must be considered.

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More thoughts on the CAPM Investors seem to be concerned with both

market risk and total risk. Therefore, the SML may not produce a correct estimate of ki. ki = kRF + (kM – kRF) βi + ???

CAPM/SML concepts are based upon expectations, but betas are calculated using historical data. A company’s historical data may not reflect investors’ expectations about future riskiness.

CHAPTER 5 Risk and Rates of Return

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