Risk and Return

1
Risk and Return

• Learning Module

2
Expected Return

• The future is uncertain.
• Investors do not know with certainty whether the
  economy will be growing rapidly or be in
  recession.
• Investors do not know what rate of return their
  investments will yield.
• Therefore, they base their decisions on their
  expectations concerning the future.
• The expected rate of return on a stock represents
  the mean of a probability distribution of
  possible future returns on the stock.

3
Expected Return

• The table below provides a probability
  distribution for the returns on stocks A and B
• State Probability Return On Return
  On
• Stock A
  Stock B
• 1 20 5
  50
• 2 30 10
  30
• 3 30 15
  10
• 4 20 20
  -10
• The state represents the state of the economy one
  period in the future i.e. state 1 could represent
  a recession and state 2 a growth economy.
• The probability reflects how likely it is that
  the state will occur. The sum of the
  probabilities must equal 100.
• The last two columns present the returns or
  outcomes for stocks A and B that will occur in
  each of the four states.

4
Expected Return

• Given a probability distribution of returns, the
  expected return can be calculated using the
  following equation
• N
• ER S (piRi)
• i1
• Where
• ER the expected return on the stock
• N the number of states
• pi the probability of state i
• Ri the return on the stock in state i.

5
Expected Return

• In this example, the expected return for stock A
  would be calculated as follows
• ERA .2(5) .3(10) .3(15) .2(20)
  12.5
• Now you try calculating the expected return for
  stock B!

6
Expected Return

• Did you get 20? If so, you are correct.
• If not, here is how to get the correct answer
• ERB .2(50) .3(30) .3(10) .2(-10)
  20
• So we see that Stock B offers a higher expected
  return than Stock A.
• However, that is only part of the story we
  haven't considered risk.

7
Measures of Risk

• Risk reflects the chance that the actual return
  on an investment may be different than the
  expected return.
• One way to measure risk is to calculate the
  variance and standard deviation of the
  distribution of returns.
• We will once again use a probability distribution
  in our calculations.
• The distribution used earlier is provided again
  for ease of use.

8
Measures of Risk

• Probability Distribution
• State Probability Return On Return
  On
• Stock A
  Stock B
• 1 20 5
  50
• 2 30 10
  30
• 3 30 15
  10
• 4 20 20
  -10
• ERA 12.5
• ERB 20

9
Measures of Risk

• Given an asset's expected return, its variance
  can be calculated using the following equation
• N
• Var(R) s2 S pi(Ri ER)2
• i1
• Where
• N the number of states
• pi the probability of state i
• Ri the return on the stock in state i
• ER the expected return on the stock

10
Measures of Risk

• The standard deviation is calculated as the
  positive square root of the variance
• SD(R) s s2 (s2)1/2 (s2)0.5

11
Measures of Risk

• The variance and standard deviation for stock A
  is calculated as follows
• s2A .2(.05 -.125)2 .3(.1 -.125)2 .3(.15
  -.125)2 .2(.2 -.125)2 .002625
• sA (.002625)0.5 .0512 5.12
• Now you try the variance and standard deviation
  for stock B!
• If you got .042 and 20.49 you are correct.

12
Measures of Risk

• If you didnt get the correct answer, here is how
  to get it
• s2B .2(.50 -.20)2 .3(.30 -.20)2 .3(.10
  -.20)2 .2(-.10 - .20)2 .042
• sB (.042)0.5 .2049 20.49
• Although Stock B offers a higher expected return
  than Stock A, it also is riskier since its
  variance and standard deviation are greater than
  Stock A's.
• This, however, is still only part of the picture
  because most investors choose to hold securities
  as part of a diversified portfolio.

13
Portfolio Risk and Return

• Most investors do not hold stocks in isolation.
• Instead, they choose to hold a portfolio of
  several stocks.
• When this is the case, a portion of an individual
  stock's risk can be eliminated, i.e., diversified
  away.
• From our previous calculations, we know that
• the expected return on Stock A is 12.5
• the expected return on Stock B is 20
• the variance on Stock A is .00263
• the variance on Stock B is .04200
• the standard deviation on Stock A is 5.12
• the standard deviation on Stock B is 20.49

14
Portfolio Risk and Return

• The Expected Return on a Portfolio is computed as
  the weighted average of the expected returns on
  the stocks which comprise the portfolio.
• The weights reflect the proportion of the
  portfolio invested in the stocks.
• This can be expressed as follows
• N
• ERp S wiERi
• i1
• Where
• ERp the expected return on the portfolio
• N the number of stocks in the portfolio
• wi the proportion of the portfolio invested in
  stock i
• ERi the expected return on stock i

15
Portfolio Risk and Return

• For a portfolio consisting of two assets, the
  above equation can be expressed as
• ERp w1ER1 w2ER2
• If we have an equally weighted portfolio of stock
  A and stock B (50 in each stock), then the
  expected return of the portfolio is
• ERp .50(.125) .50(.20) 16.25

16
Portfolio Risk and Return

• The variance/standard deviation of a portfolio
  reflects not only the variance/standard deviation
  of the stocks that make up the portfolio but also
  how the returns on the stocks which comprise the
  portfolio vary together.
• Two measures of how the returns on a pair of
  stocks vary together are the covariance and the
  correlation coefficient.
• Covariance is a measure that combines the
  variance of a stocks returns with the tendency
  of those returns to move up or down at the same
  time other stocks move up or down.
• Since it is difficult to interpret the magnitude
  of the covariance terms, a related statistic, the
  correlation coefficient, is often used to measure
  the degree of co-movement between two variables.
  The correlation coefficient simply standardizes
  the covariance.

17
Portfolio Risk and Return

• The Covariance between the returns on two stocks
  can be calculated as follows
• 
  N
• Cov(RA,RB) sA,B S pi(RAi - ERA)(RBi -
  ERB)
• 
  i1
• Where
• sA,B the covariance between the returns on
  stocks A and B
• N the number of states
• pi the probability of state i
• RAi the return on stock A in state i
• ERA the expected return on stock A
• RBi the return on stock B in state i
• ERB the expected return on stock B

18
Portfolio Risk and Return

• The Correlation Coefficient between the returns
  on two stocks can be calculated as follows
• sA,B
  Cov(RA,RB)
• Corr(RA,RB) rA,B sAsB SD(RA)SD(RB)
• Where
• rA,Bthe correlation coefficient between the
  returns on stocks A and B
• sA,Bthe covariance between the returns on stocks
  A and B,
• sAthe standard deviation on stock A, and
• sBthe standard deviation on stock B

19
Portfolio Risk and Return

• The covariance between stock A and stock B is as
  follows
• sA,B .2(.05-.125)(.5-.2) .3(.1-.125)(.3-.2)
  
• .3(.15-.125)(.1-.2) .2(.2-.125)(-.1-.2)
  -.0105
• The correlation coefficient between stock A and
  stock B is as follows
• -.0105
• rA,B (.0512)(.2049) -1.00

20
Portfolio Risk and Return

• Using either the correlation coefficient or the
  covariance, the Variance on a Two-Asset Portfolio
  can be calculated as follows
• s2p (wA)2s2A (wB)2s2B 2wAwBrA,B sAsB
• OR
• s2p (wA)2s2A (wB)2s2B 2wAwB sA,B
• The Standard Deviation of the Portfolio equals
  the positive square root of the the variance.

21
Portfolio Risk and Return

• Lets calculate the variance and standard
  deviation of a portfolio comprised of 75 stock A
  and 25 stock B
• s2p (.75)2(.0512)2(.25)2(.2049)22(.75)(.25)(-1)
  (.0512)(.2049) .00016
• sp .00016 .0128 1.28
• Notice that the portfolio formed by investing 75
  in Stock A and 25 in Stock B has a lower
  variance and standard deviation than either
  Stocks A or B and the portfolio has a higher
  expected return than Stock A.
• This is the purpose of diversification by
  forming portfolios, some of the risk inherent in
  the individual stocks can be eliminated.

22
Capital Asset Pricing Model (CAPM)

• If investors are mainly concerned with the risk
  of their portfolio rather than the risk of the
  individual securities in the portfolio, how
  should the risk of an individual stock be
  measured?
• In important tool is the CAPM.
• CAPM concludes that the relevant risk of an
  individual stock is its contribution to the risk
  of a well-diversified portfolio.
• CAPM specifies a linear relationship between risk
  and required return.
• The equation used for CAPM is as follows
• Ki Krf bi(Km - Krf)
• Where
• Ki the required return for the individual
  security
• Krf the risk-free rate of return
• bi the beta of the individual security
• Km the expected return on the market portfolio
• (Km - Krf) is called the market risk premium
• This equation can be used to find any of the
  variables listed above, given the rest of the
  variables are known.

23
CAPM Example

• Find the required return on a stock given that
  the risk-free rate is 8, the expected return on
  the market portfolio is 12, and the beta of the
  stock is 2.
• Ki Krf bi(Km - Krf)
• Ki 8 2(12 - 8)
• Ki 16
• Note that you can then compare the required rate
  of return to the expected rate of return. You
  would only invest in stocks where the expected
  rate of return exceeded the required rate of
  return.

24
Another CAPM Example

• Find the beta on a stock given that its expected
  return is 12, the risk-free rate is 4, and the
  expected return on the market portfolio is 10.
• 12 4 bi(10 - 4)
• bi 12 - 4
• 10 - 4
• bi 1.33
• Note that beta measures the stocks volatility
  (or risk) relative to the market.