RISK AND RETURN

INTRODUCTION

What is an investment?Investment is current commitment of dollars for a period of time in order to derive future payment that will compensate the investor for:

1. the time funds are committed2. the expected rate of inflation3. the uncertainty of the future payments

What are investment returns?Investment returns measure the financial results of an investment.Returns may be historical or prospective (anticipated).

Calculating the rate of return

Rate of return= Amount received−Amount investedAmount invested

¿¿

Example: (Tool Kit 2.1)Suppose you pay $500 for an investment that returns $600 in one year. What is the annual rate of return?

Amount invested $500Amount received in one year $600

Dollar return $100

Rate of return 20%

If you have a known amount of the initial market price (P0), realized or planned dividend (D1) and realized or planned selling price (P1), and a holding period of security is not exceeding one year, then the rate of return on investment in this securities can be calculated as follows:

Rate of return=D1+(P1−P0)

P1

So, this formula can be used to determine the historical rate of return for the holding period of one year (based on the historical data) as the expected rate of return for the holding period of one year (based on expected future dividends and prices).

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For holding periods longer than one year we have to calculated internal rate of return, which is consistent with the principles of time value of money.

The term in brackets is called a capital gain or loss during the period.

Examples:1. Suppose you buy 10 shares of a stock for $ 1,000. The stock pays no dividends, but at the end of 1 year, you sell the stock for $ 1,100. What is the return on your $ 1,000 investment?P0=1,000 $P1=1,100 $D1=0 $n=1

Rate of return=1,100−1,0001,000

=0.1=10 %

2. At the begin of the year, the investor has bought a share of BH Telecom on the Sarajevo Stock Exchange at a price of 25 KM. BH Telecom paid during the year a dividend in the amount of 0.90 KM. The investor sold the share at the end of the year for 39 KM. What is the rate of return on this investment?

P0=25 KMD1=0,90 KMP1=39 KMR=?

First way: since the holding period is one year we can use the above mentioned formula

R=0,9+(39−25)

25=0.596 →59.6 %

Second way: we calculate the internal rate of return or we discount all cash flows from investments

P0=D1

1+k+

P1

1+k

25= 0.91+k

+ 391+k

25=39.91+k

1+k=39.9/25

1+k=1.596

k=0.596→ 59.6%

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3. Another invester has also bought a share of BH Telecom at a price of 25 KM. He kept the share for two years and sold it after two years for 55 KM. BH Telecom paid during the first year a dividend in the amount of 0.90 KM and during the second year a dividend in the amount of 1.60 KM. What is the rate of return on this investment?

P0=25 KMP2=55 KMD1=0.90 KMD2=1.6 KMk=?

P0=D1

1+k+

D 2

(1+k)2 +P2

(1+k )2

25= 0.91+k

+ 1.6

(1+k )2+ 55

(1+k )2

25= 0.91+k

+ 56.6

(1+k )2

By introducing the following replacement (1+ k) = t we have

25=0.9t

+ 56.6

t2

After multiplying the equation with t2 we have:

25t2=0.9t + 56.6

25t2-0.9t-56.6=0

t 1=−b+√b2−4 ac

2 a

t 1=0.9+√0.81+5,660

50=0.9+75.238355

50=1.522767 ≈ 1.52

1+k=t k=1.52-1=0.52→ 52%

What is investment risk?For all securities, except for the risk free securities, the return expected may differ from the return realized. So, we defined the risk as the chance that some unfavorable event will occur. An event's probability is defined as the chance that the event will occur. If all possible events, or outcomes, are listed, and if a probability is assigned to each event, the listing is called a probability distribution. The probabilities must sum to 1,0 or 100%. For risky securities the actual rate of return can be viewed as a random variable that has a probability distribution.

The risk of an assets cash flow can be considered:a) on a stand-alone basis (each asset by itself)Stand−alone Risk

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b) in a portfolio context, where the investment is combined with other assets and its risk is reduced through diversification Risk∈a portfolio context

INVESTMENT IN ONE ASSET

Expected rate of return ( r̂ ¿

r̂i=P1 ∙ r1+P2 ∙ r2+…+Pn ∙ rn=∑i=1

n

Pi ∙r i

r̂i = the expected rate of return on the ith stockri=¿ the ith possible outcomePi=¿the probability of the ith outcomen=¿ the number of possible outcomes

The tighter the probability distribution, the more likely is that the actual outcome will be close to the expected value and consequently, the less likely it is that the actual return will end up far below the expected return. Thus, the tighter the probability distribution of expected return, the smaller the risk of a given investment.The measure of tightness of the probability distribution is the standard deviation, which we calculate as follows:

Variance (σ 2¿

σ 2=∑i=1

n

¿¿

Standard deviation (σ ¿

σ=√∑i=1

n

¿¿¿

The smaller the standard deviation, the tighter the probability distribution and accordingly, the less risky the stock. The standard deviation provides an idea how far above or below the expected value the actual value is likely to be.

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-30 -15 0 15 30 45 60

Stock A

Returns (%)

If a probability distribution is a normal distribution1, then we can apply six-sigma rule:- the actual return will be within ± 1 standard deviation of the expected return 68.26% of the time

p [ r̂−σ ≤ X ≤ r̂+σ ]≅ 68.26 %- the actual return will be within ± 2 standard deviation of the expected return 9.,46% of the time

p [ r̂−2 ∙ σ ≤ X ≤ r̂+2 ∙ σ ]≅ 95.46 %- the actual return will be within ± 3 standard deviation of the expected return 99.74% of the time

p [ r̂−3 ∙ σ ≤ X ≤ r̂+3 ∙ σ ] ≅ 99.74 %

Example: (Tool Kit 2.2)An investment has a 30% chance of producing a 25% return, a 40% chance of producing a 10% return, and a 30% chance of producing a -15% return. What is its expected return? What is its standard deviation?

Probability Return

30% 25%40% 10%30% -15%

r̂i=P1 ∙ r1+P2 ∙ r2+P3 ∙ r3= (0.3∙ 0.25 )+( 0.4 ∙0.1 )+ (0.3 ∙ (−0.15 ) )=0.07 → 7 %

σ=√(0.25−0.07)2 ∙ 0.30+(0.10−0.07)2 ∙ 0.40+(−0.15−0.07)2∙ 0.30=0.157 →15.7 %

σ 2=2.46 %

Note:

Standard deviation can be used as an absolute measure of risk meaning that if the standard deviation is larger, the uncertainty of the actual outcome is greater and consequently. So if a choice has to be made between two investments that have the same expected returns but different standard deviations, most people would choose the one with lower standard deviation and therefore the lower risk.

In order to perform comparisons of alternative investments with different expected rates of return, you cannot use the standard deviation. In that case we use another measure of risk called the coefficient of variation (CV) which shows the risk per unit of return.

Coefficient of variation (CV ¿

CV =σr̂

Example: (Tool Kit 2.2)An investment has an expected return of 15% and a standard deviation of 30%. What is its coefficient of variation?

1 Rate of return is continuous variable, we have large number of iterations and because of that we will take approximation with normal distribution.

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CV =30 %15 %

=2

Using Historical data

We described the procedure for finding mean and standard deviation when data are in the form of a known probability distribution.But in case that we have available only data on returns realized in some past period (rt ¿available we use following formula:

Annual return (r Avg ¿¿

r Avg=∑t=1

n

rt

nEstimated σ (S )

S=√∑t=1

n

(rt−r Avg)2

n−1

Example: (Tool Kit 2.2)A stock's returns for past 3 years are 10%, -15% and 35%. What is the historical average return? What is the historical sample standard deviation?

r Avg=10 %+ (−15 %)+35 %

3=10 %

S=√(10 %−10 %)2+(−15 %−10 %)2+(35 %−10 %)2

3−1=25 %

Exercise: A stock's return has the following distributions:

Demand for the Company's Products

Probability of This Demand Occurring

Rate of Return if This Demand Occurs (%)

Weak 0.1 50Below average 0.2 5

Average 0.4 16Above average 0.2 25

Strong 0.1 60 1.0

Calculate the stock's expected return, standard deviation and coefficient of variation.

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Solution:

r¿

= (0.1) (-50%) + (0.2) (-5%) + (0.4) (16%) + (0.2) (25%) + (0.1) (60%)∙ ∙ ∙ ∙ ∙ = 11.40%.

σ2 = (-50% - 11.40%)2 (0.1) + (-5% - 11.40%)∙ 2 (0.2) + (16% - 11.40%)∙ 2 (0.4)∙ + (25% - 11.40%)2 (0.2) +∙ (60% - 11.40%)2 (0.1) = 712.44; σ= 26.69%∙

CV =

26 . 69%11. 40% = 2.34

The average investor is risk averse, which means that he or she must be compensated for holding risky assets. Therefore, riskier assets have higher required returns than less risky assets.

INVESTMENT IN THE PORTFOLIO

Portfolio is a combination of two or more securities, currencies, real estate or other assets held by individuals or companies. The goal of creating a portfolio is to minimize risk through diversification.

So, portfolio of securities consists of two and / or more securities in which the investor invests money in specific ratios to reduce the risk. Diversification of risk is achieved by the successful combination of securities. The successful combination of securities is done by selecting the securities that are each weakly correlated with each, and whose yields move inversely. So we can use the benefits of diversification in terms of risk reduction until the securities are not perfectly positively correlated (ρ ≠ 1).

Expected return on a portfolio (r̂ p ¿The expected return on a portfolio is weighted mean of the expected returns of individual investments that make up the portfolio, where weights are the shares of the money invested in each security.

r̂ p=w1 ∙ r̂1+w2 ∙r̂ 2+…+wn ∙ r̂n=∑i=1

n

wi ∙r̂i

r̂ p = the expected return on a portfolior̂i=¿ the expected returns on the individual stocksw i=¿the weightsn=¿ the number of stocks in the portfolio

Standard deviation of a portfolio (σ p¿

The risk of the portfolio is as with the individual securities expressed by the variance and standard deviation of returns. However, the risk of the portfolio is not simply the weighted mean of individual securities standard deviations, because the risk of a portfolio depends not only on the riskiness of the securities that make up the portfolio, but also the relationships that exist between these securities.

σ p=√∑i=1

n

∑j=1

n

w i ∙ w j ∙COV ij

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w i=¿ share of total funds invested in security jw j=¿ share of total funds invested in security iCOV ij=¿ covariance between the yield of the i-th and j-th security

Covariance between stock i and the stock j (COV ij ¿

This is a statistical measure that indicates the degree to which two variables, in this case securities' rates of return, are moving together. Positive value means that, on average, they are moving in the same direction.

COV ij=∑t=1

m

Pi ∙ (ri ,t−r̂i ) ∙ (r j , t− r̂ j )

COV ij=∑t=1

m

(ri , t− r̂ i )∙ (r j ,t−r̂ j )

m

COV ij= ρij ∙ σ i ∙ σ j

where we work with m-period sample.

Correlation coefficient (ρij ¿

This is a standardized statistical measure of linear relationship between two variables. The correlation coefficient (ρij ¿ can range from +1,0, denoting that two variables move up and down in perfect synchronization (perfect positive correlation), to -1,0, denoting that the variables always move in exactly opposite directions (perfect negative correlation). A correlation coefficient of zero indicates that two variables are not related to each other.

ρij=COV ij

σ i ∙ σ j

Thus, the risk of the portfolio or portfolios or standard deviation of a portfolio (σ p¿ depends on:- variances of individual securities and the- covariance between different pairs of securities (which are crucial for large portfolios)

The goal of a successful diversification is the combination of securities which are each slightly with each dependent (as measured by covariance and correlations) and whose rates of returns move inversely.

The effect of diversification is present when the ρij≠1, and we have the maximum effect of diversification when ρij=-1.

TWO-STOCK PORTFOLIOS

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Expected return on a portfolio (r̂ p ¿

r̂ p=w1 ∙ r̂1+w2 ∙r̂ 2

Standard deviation of a portfolio (σ p¿

σ P=√w12 σ1

2+2 w1 (1−w1 ) Cov1,2+(1−w1)2σ2

2

Covariance between stock 1 and the stock 1 (COV 1,2¿

COV 12=ρ12 ∙ σ1∙ σ2

Exercise:Stocks A and B have the following historical returns:

Year r A (%) r B (%)2005 18.00 -14.502006 33.00 21.802007 15.00 30.502008 -0.50 -7.602009 27.00 26.30

a) Calculate the average rate of return for each stock during the 5-year periodb) Assume that someone held a portfolio consisting of 50% of Stock A and 50% of Stock B. What would have been the realized rate of return on the portfolio in each year? What would have been the average return on the portfolio during this period?c) Calculate the standard deviation of returns for each stock and for the portfoliod) Calculate the coefficient of variation for each stock and for the portfolio.e) If you are a risk-averse investor, would you prefer to hold Stock A, Stock B, or the portfolio? Why?

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The answers to a, b, c, and d are given below:

¯A ¯B Portfolio

2005 (18.00%) (14.50%) (16.25%)

2006 33.00 21.80 27.40

2007 15.00 30.50 22.75

2008 (0.50) (7.60) (4.05)

2009 27.00 26.30 26.65

Mean 11.30 11.30 11.30

Std Dev 20.79 20.78 20.13

CV 1.84 1.84 1.78

e. A risk-averse investor would choose the portfolio over either Stock A or Stock B alone, since the portfolio offers the same expected return but with less risk. This result occurs because returns on A and B are not perfectly positively correlated (ρAB = 0.88).

Exercise:The market and Stock J have the following probability distributions:

Probability rM rj

0,3 15 % 20%0,4 9% 5%0,3 18% 12%

a) Calculate the expected rates of return for the market and Stock Jb) Calculate the standard deviations for the market and Stock Jc) Calculate the coefficients of variation for the market and Stock J

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a.

M= (0.3) (15%) + (0.4) (9%) + (0.3) (18%) = 13.5%∙ ∙ ∙

J= (0.3) (20%) + (0.4) (5%) + (0.3) (12%) = 11.6%∙ ∙ ∙

b. σM = [(0.3) (15% - 13.5%)∙ 2 + (0.4) (9% - 13.5%)∙ 2 + (0.3) (18% -13.5%)∙ 2]1/2

= √14 . 85% = 3.85%

σJ = [(0.3) (20% - 11.6%)∙ 2 + (0.4) (5% - 11.6%)∙ 2 + (0.3) (12% - 11.6%)∙ 2]1/2

= √38 .64% = 6.22%

c. CVM = 3. 85%13 .5% = 0.29

CVJ =

6 .22%11. 6% = 0.54

REQUIRED RATE OF RETURN

The average investor is risk averse, which means that he or she must be compensated for holding risky assets, in a form of a minimum rate of return which is called the required rate of return. Therefore, riskier assets have higher required returns than less risky assets.

An asset's risk consists of diversifiable risk (unsystematic risk or company-specific risk), which can be eliminated by diversification, plus market risk (systematic risk), which cannot be eliminated by diversification.

Diversifiable risk is caused by such random events as lawsuits, strikes, winning or losing a major contract, successful and unsuccessful marketing programs and other events that are unique to a particular firm. Because the events are random, their effects on a portfolio can be eliminated by diversification-bad events in one firm will be offset by good events in another.

Market risk, on the other hand, stems from factors that systematically affect most firms: war, inflation, recession, and high interest rates. Since most stocks are negatively affected by these factors, market risk cannot be eliminated by diversification.

A stock's beta coefficient, b, is a measure of its market risk. Beta measures the extent to which the stock's returns move relative to the market. Since a stock's beta coefficient determines how stock affects the risk of a diversified portfolio, beta is the most relevant measure of any stock's risk. If b equal 1.0 then the stock is about as risky as the market, if held in a diversified portfolio. If b is less than 1.0, the stock is less risky than the market. If beta is greater than 1.0, the stock is more risky than a market. The beta of a portfolio is a weighted average of the betas of the individual securities in the portfolio.

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Since market risk cannot be eliminated by diversification, investors must be compensated for bearing it. So, the relevant risk of an individual asset is its contribution to the risk of a well diversified portfolio, which is the asset's market risk. That logic is explained by the Capital Asset Pricing Model (CAPM).

CAPM gives us the answer to the question of the size of the required rate of return on risky asset. On the other hand, if we have assessed the expected rate of return, then comparing the expected and the required rate of return implied by CAPM we can determine whether a property is undervalued, overvalued or fair valued.

Assuming that the unsystematic risk is completely remove by diversification, the required rate of return on a stock i is equal to risk-free rate plus the stock's beta times the market risk premium:

ri=rRF+bi ∙ (RPm)

where RPm=r M−rRF

b i=( σ i

σ M) ∙ ρℑ=

COV ℑ

σ2M

= slope of regression (SML)

COV ℑ=ρℑ ∙σ i ∙ σ M

ri=¿the required return on a stock ir M=therequired return on stock market (or on a portfolio consisting of all stocks, which is called the market portfolio)b i = the beta of stock ir RF= the risk-free rateRPm= the market risk premiumCOV ℑ=thecovariance between stock i and the market

Exercise:Assume that the risk-free rate is 6% and the expected return on the market is 13%. What is the required rate of return on a stock that has a beta of 0.7?

rRF = 6%;

rM = 13%;

b = 0.7;

rs = ?

ri=rRF+bi ∙¿

Exercise:Assume that the risk-free rate is 6% and the market risk premium is 6%. a) What is the required rate of return for the overall stock market?

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b) What is the required rate of return on a stock that has a beta of 1.2?

rRF = 5%;

RPM = 6%;

rM = ?ž

a) rM = 5% + (6%)∙1 = 11%.

b) rs = 5% + 6%(1.2) = 12.2%.

Exercise:Suppose rRF=5%, rM=10% and rA=12%a) Calculate Stock A's betab) If Stock A's beta were 2.0, what would be A's new required rate of return?

a) rA = rRF + (rM - rRF) b∙ A

12% = 5% + (10% - 5%)∙bA

12% = 5% + 5%(bA)

7% = 5%(bA)

1.4 = bA

b) rA = 5% + 5%∙bA

rA = 5% + 5% (2)∙

rA = 15%

Exercise:Suppose rRF=9%, rM=14% and bi=1.3a) What is ri, the required rate of return on Stock i?b) Now suppose b.1. rRF increases to 10%b.2. rRF decreases to 8%The slope of the SML remains constant.How would this affect rM and ri?c) Now assume rRF remains 9% butc.1. rM increases to 16% orc.2. rM falls to 13%The slope of the SML doe not remains constant.How would these changes affect ri?

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a.ri = rRF + (rM - rRF)∙bi = 9% + (14% - 9%)∙1.3 = 15.5%.

b.1.rRF increases to 10%:

rM increases by 1 percentage point, from 14% to 15%.

ri = rRF + (rM - rRF)∙bi = 10% + (15% - 10%)∙1.3 = 16.5%

b.2.rRF decreases to 8%:

rM decreases by 1%, from 14% to 13%.

ri = rRF + (rM - rRF)∙bi = 8% + (13% - 8%)∙1.3 = 14.5%

c.1.rM increases to 16%:

ri = rRF + (rM - rRF)∙bi = 9% + (16% - 9%)∙1.3 = 18.1%

c.2.rM decreases to 13%:

ri = rRF + (rM - rRF)∙bi = 9% + (13% - 9%)∙1.3 = 14.2%

Exercise:Stock R has a beta of 1.5, Stock S has a beta of 0.75, the expected rate of return on an average stock is 1.3% and the risk-free rate of return is 7%. By how much does the required return on the riskier stock exceeds the required return on the less risky stock?

We know that bR = 1.50, bS = 0.75, rM = 13%, rRF = 7%

ri = rRF + (rM - rRF)∙bi = 7% + (13% - 7%)∙bi

rR = 7% + 6%(1.50) = 16.0%rS = 7% + 6%(0.75) = 11.5 4.5%

The required rate of return on a portfolio due to the CAPM would be:

r p=rRF+bp ∙(RPm)

Where

b p=∑i=1

n

w i ∙ bi

Exercise:An individual has $ 35,000 invested in a stock which has a beta of 0.8 and $ 40,000 invested in a stock with beta of 1.4. If these are the only two investments in her portfolio, what is her portfolio's beta?

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Investment Beta

$35,000 0.8

40,000 1.4

Total $75,000

b p=w1 ∙ b1+w2 ∙ b2=¿ ($35,000/$75,000) (0.8) + ($40,000/$75,000) (1.4) = 1.12∙ ∙

Exercise: (Tool Kit 2.3) An investor has a 3-stock portfolio with $25,000 invested in Dell, $50,000 invested in Ford, and $25,000 invested in Wal-Mart. Dell’s beta is estimated to be 1.20, Ford’s beta is estimated to be 0.80, and Wal-Marts beta is estimated to be 1.0. What is the estimated beta of the investor’s portfolio?

b p=w1 ∙ b1+w2 ∙ b2+w3 ∙ b3=¿=($25,000/$100,000) (1.2)+($50,000/$100,000) (0.8)+($25,000/$100,000) (1.0) = 0.95∙ ∙ ∙

Exercise:Suppose you hold a diversified portfolio consisting of a $7,500 investment in each of 20 different common stocks. The portfolio beta is equal to 1.12. Now, suppose you have decided to sell one of the stock in your portfolio with a beta equal to 1.0 for $7,500 and to use these proceeds to buy another stock for your portfolio. Assume the new stock's beta is equal 1.75. Calculate your portfolio's new beta.

Old portfolio beta =

$142,500$150,000 (b) +

$7,500$150,000 (1.00)

1.12 = 0.95∙b + 0.05 1.07 = 0.95∙b 1.13 = b

New portfolio beta = 0.95 (1.13) + 0.05 (1.75) = 1.16∙ ∙

Alternative Solutions:

1.Old portfolio beta = 1.12 = (0.05)∙b1 + (0.05)∙b2 +...+ (0.05)∙b20

1.12 = (Sbi)(0.05)Sbi = 1.12/0.05 = 22.4

New portfolio beta = (22.4 - 1.0 + 1.75) (0.05) = 1.1575 = 1.16∙

2.Sbi excluding the stock with the beta equal to 1.0 is (22.4 - 1.0) = 21.4, so the beta of the portfolio excluding this stock is b = 21.4/19 = 1.1263. The beta of the new portfolio is:

1.1263 (0.95) + 1.75 (0.05) = 1.1575 = 1.16∙ ∙

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Example:Suppose you are the money manager of a $4 million investment fund. The fund consists of four stocks with the following investments and betas:

Stock Investment ($) BetaA 400,000 1.50B 600,000 -0.50C 1,000,000 1.25D 2,000,000 0.75

If the marker required rate of return is 14% and the risk-free rate is 6%, what is the fund's required rate of return?

Portfolio beta = (1.50) + (-0.50) + (1.25) + (0.75)= = (0.1) (1.5) + (0.15) (-0.50) + (0.25) (1.25) + (0.5) (0.75)∙ ∙ ∙ ∙ = 0.15 - 0.075 + 0.3125 + 0.375 = 0.7625

rp= rRF + (rM - rRF) (∙ bp) = 6% + (14% - 6%) (0.7625) = 12.1%.∙

Alternative solution: First compute the return for each stock using the CAPM equation [rRF + (rM - rRF) (∙ bp)], and then compute the weighted average of these returns.

rRF = 6% and (rM - rRF )= 8%.

Stock Investment Beta r = rRF + (rM - rRF)∙b Weight A $ 400,000 1.50 18% 0.10 B 600,000 (0.50) 2 0.15 C 1,000,000 1.25 16 0.25 D 2,000,000 0.75 12 0.50Total $4,000,000 1.00

rp = 18% (0.10) + 2% (0.15) + 16% (0.25) + 12% (0.50) = 12.1%∙ ∙ ∙ ∙

Exercise:You have a $2 million portfolio consisting of a $100,000 investment in each of 20 different stocks. The portfolio has a beta equal to 1.1. You are considering selling $ 100,000 worth of one stock that has a beta equal to 0.9 and using the proceeds to purchase another stock that has a beta equal to 1.4. What will be the new beta of your portfolio following this transaction?

First, calculate the beta of what remains after selling the stock:

bp = 1.1 = ($100,000/$2,000,000) 0.9 + ($1,900,000/$2,000,000)∙ ∙bR

1.1 = 0.045 + (0.95)∙bR

bR = 1.1105

bN = (0.95) 1.1105 + (0.05) 1.4 = 1.125∙ ∙

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Example:You have observed the following returns over time:

Year Stock X (%) Stock Y (%) Market (%)2005 14 13 122006 19 7 102007 -16 -5 -122008 3 1 12009 20 11 15

Assume that the risk-free rate is 6% and the market risk-premium is 5%a) What are the betas of Stocks X and Y?b) What are the required rates of return for Stocks X and Y?c) What is the required rate of return for a portfolio consisting of 80% of Stock X and 20% of Stock Y?d) If Stock X's expected return is 22%, is Stock X under-or overvalued?

a. bX = 1.3471; bY = 0.6508. These can be calculated with a spreadsheet.

b. rX = 6% + (5%) 1.3471 = 12.7355%∙rY = 6% + (5%) 0.6508 = 9.2540%∙

c. bp = 0.8 (1.3471) + 0.2 (0.6508) = 1.2078∙ ∙rp = 6% + (5%) 1.2078 = 12.04%∙Alternatively,rp = 0.8 (12.7355%) + 0.2 (9.254%) = 12.04%∙ ∙

d. Stock X is undervalued, because its expected return exceeds its required rate of return.

Security Market Line (SML)

Security Market Line is a line that in a state of market equilibrium shows the relationship between the required rate of return on individual securities and systematic risk expressed by beta.

The line that reflects the combination of risk and return availabe on alternative investments is referred as the security market line (SML). The SML refelcts the risk-return combinations available for all risky assest in the capital market at a given time. Investors would select investmnet that are consistent with their risk prefercences: some would consider only low-risk investments, whereas other welcome high-risk inevstments.

The SML intersects the ordinate, which shows the expected rate of return, at the level of risk-free rate of return ¿¿=NRFR), because the beta of risk-free securities is zero.

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SML line and its changes

Beginning with an initial SML, three changes can occur

1. Movement along the SML

A movement along the SML demonstrate a change in the risk characteristics of a specific investment, such as a change in its business risk, its financial risk, or its systematic risk (its beta).

This change affects only the individual investment.

2. Changes in the Slope of the SML

A change on the slope of the SML occurs in response to a change in the attitudes of inevstors toward risk. Such a change demonstrates that investors want either higher or lower rate of return for the same risk. This is also described as a change in the market risk premium(RPm ¿ .

A changes in the market risk premim will affecst all risky investments.

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RiskNRFR

Original SML

New SML

R m

R m’

Expected Return

ExpectedReturn

Risk(business risk, etc., or systematic risk-beta)

NRFR

SecurityMarket LineLow

RiskAverageRisk

HighRisk

The slope indicates therequired return per unit of risk

3. Changes in Capital Market Conditions or Expected Inflation

A shift in the SML reflects a change in expeced real growth, a change in market conditins (such as ease or tughtness of money), or a change in the expected rate of inflation.

Such a change will affect all investments.

Example: Tool Kit (Chapter)We want to predict stock i's required return using SML if we know:

rRF 6% << Varies over time, but is constant for all companies at any given time.

rM 11% << Varies over time, but is constant for all companies at any given time.

bi 0.5 << Varies over time, and varies by company.

rp = rRF + bp∙ (RPM)rp= rRF + (rM - rRF) (∙ bp) rp= 6% + 0.5 (11% - 6%)=8.5∙The SML shows the relationship between the stock's beta and its required return, as predicted by the CAPM.With the data about different levels for beta and appropriate levels of required return (according to previous model), we can generate a Security Market Line that will be flexible enough to allow for changes in:

BetaRequired ReturnrRF + (rM - rRF) (∙ bp)

0.00 6.0%0.50 8.5%1.00 11.0%1.50 13.5%2.00 16.0%

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Risk

NRFR

Original SML

New SML

Expected Return

NRFR'

The Security Market Line shows the projected changes in expected return, due to changes in the beta coefficient. We will use graph to present this relationship:

0.00 0.50 1.00 1.50 2.00 2.50

0%

6%

12%

18%

Security Market Line

Beta

Re

qu

ire

d R

etu

rn

However, we can also look at the potential changes in the required return due to variation of other factors, namely the market return and risk-free rate. In other words, we can see how required returns can be influenced by changing inflation and risk aversion. The level of investor risk aversion is measured by the market risk premium (rM-rRF), which is also the slope of the SML. Hence, an increase in the market return results in an increase in the maturity risk premium, other things held constant.We will look at two potential conditions as shown in the following columns:

rp= rRF + (rM - rRF) (∙ bp)

Scenario 1. Inflation Increases:Scenario 2. Investors become more risk averse:

Risk-free Rate 6% Risk-free Rate 6%Change in inflation 2% Old Market Return 11%Old Market Return 11% Increase in MRP 2,5%New Market Return 13%

New Market Return 13,5%

Beta 0,50 Beta 0,50

Required Return 10,5% Required Return 9,75%

Now, we can see how these two factors can affect a Security Market Line, by creating a data table for the required return with different beta coefficients.

Beta Original Situation Inflation increasesRisk aversion

increases0,00 6,00% 8,00% 6,00%0,50 8,50% 10,50% 9,75%1,00 11,00% 13,00% 13,50%1,50 13,50% 15,50% 17,25%2,00 16,00% 18,00% 21,00%

Or graphically:

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0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.000%

5%

10%

15%

20%

25%

The SML Under Different Conditions

Original

Inflation up

Risk aversion up

Beta

Req

uire

d R

etur

ns

The graph shows that as risk as measured by beta increases, so does the required rate of return on securities. However, the required return for any given beta varies depending on the position and slope of the SML.

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CALCULATING BETA COEFFICIENT

Homework 1:Calculate beta for an actual company.Apply all three methods presented in the example below:1. I way – Excel function2. II way – Regression3. III way – Scatter plot

Example: Tool Kit (Chapter)

Acquire data

We downloaded stock prices and dividends from http://finance.yahoo.com for General Electric, using its ticker symbol GE. We also downloaded data for the S&P 500 Index (^SPX), which contains 500 actively traded large stocks. For example, to download the GE data, enter its ticker symbol and click Go. Then select Historical Prices from the left side of the page. After the daily prices come up, get monthly prices by entering the start date and the end date. Yahoo provides monthly prices as of the first trading day of the month. Note that these prices are "adjusted" to reflect any dividends or stock splits.

DateMarket Level

(S&P 500 Index) Market Return (rM)GE Adjusted Stock Price GE Return (rp)

February 2008 1330,63 -3,5% 33,14 -5,4%January 2008 1378,55 -6,1% 35,04 -4,6%

December 2007 1468,36 -0,9% 36,74 -2,3%November 2007 1481,14 -4,4% 37,62 -7,0%

October 2007 1549,38 1,5% 40,44 -0,6%September 2007 1526,75 3,6% 40,68 7,2%

August 2007 1473,99 1,3% 37,94 0,3%July 2007 1455,27 -3,2% 37,83 1,3%June 2007 1503,35 -1,8% 37,36 2,6%May 2007 1530,62 3,3% 36,41 1,9%

April 2007 1482,37 4,3% 35,72 4,3%March 2007 1420,86 1,0% 34,26 1,3%

February 2007 1406,82 -2,2% 33,83 -2,4%January 2007 1438,24 1,4% 34,66 -3,1%

December 2006 1418,3 1,3% 35,78 6,3%November 2006 1400,63 1,6% 33,67 0,5%

October 2006 1377,94 3,2% 33,51 -0,5%September 2006 1335,85 2,5% 33,69 4,4%

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August 2006 1303,82 2,1% 32,27 4,2%July 2006 1276,66 0,5% 30,98 -0,8%June 2006 1270,2 0,0% 31,23 -3,1%May 2006 1270,09 -3,1% 32,22 -1,0%

April 2006 1310,61 1,2% 32,53 -0,6%March 2006 1294,87 1,1% 32,71 5,8%

February 2006 1280,66 0,0% 30,92 1,1%January 2006 1280,08 2,5% 30,57 -6,6%

December 2005 1248,29 -0,1% 32,72 -1,2%November 2005 1249,48 3,5% 33,11 5,3%

October 2005 1207,01 -1,8% 31,43 0,7%September 2005 1228,81 0,7% 31,21 0,8%

August 2005 1220,33 -1,1% 30,95 -2,6%July 2005 1234,18 3,6% 31,77 -0,4%June 2005 1191,33 0,0% 31,91 -4,4%May 2005 1191,5 3,0% 33,39 0,8%

April 2005 1156,85 -2,0% 33,13 0,4%March 2005 1180,59 -1,9% 33 2,5%

February 2005 1203,6 1,9% 32,21 -2,0%January 2005 1181,27 -2,5% 32,86 -1,0%

December 2004 1211,92 3,2% 33,2 3,8%November 2004 1173,82 3,9% 31,97 3,6%

October 2004 1130,2 1,4% 30,85 1,6%September 2004 1114,58 0,9% 30,36 3,0%

August 2004 1104,24 0,2% 29,47 -1,4%July 2004 1101,72 -3,4% 29,88 2,6%June 2004 1140,84 1,8% 29,12 4,7%May 2004 1120,68 1,2% 27,8 3,9%

April 2004 1107,3 -1,7% 26,76 -1,9%March 2004 1126,21 -1,6% 27,27 -6,1%

February 2004 1144,94 NA 29,05 NA

Beta is slope for regression line where Market Return is independent variable and GE Return is dependent variable.

a) I way – Excel function

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24

Beta for this case is 0.7.

b) II way – Regression

Excel – Data – Data Analysis

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SUMMARY OUTPUT

Regression Statistics

Multiple R0,5030009

9R Square 0,25301Adjusted R Square

0,23677109

Standard Error0,0302065

1Observations 48

ANOVA

  df SS MS FSignificance

F

Regression 10,01421

6 0,01421614415,5804

8 0,000268704

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Residual 460,04197

2 0,000912433

Total 470,05618

8      

 Coefficient

sStandard Error t Stat P-value Lower 95%

Upper 95%

Intercept0,0008980

90,00440

3 0,2039534890,83928

9 -0,0079654940,00976

2

X Variable 10,7099301

70,17985

6 3,9472112090,00026

9 0,3478985421,07196

2

Well, beta for this case is 0.7.

c) III way – Scaterrplot (tool kits)

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0 2 4 6 8 10 120

2

4

6

8

10

12

#REF!

#REF!

With right mouse we will click on points on graph and:

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29

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

f(x) = 0.709930165352151 x + 0.00089808867172776R² = 0.253010000198748

rp

rpLinear (rp)

Therefore, beta for this case is 0.7.

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