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Finance Exercises 6 - Risk and the Cost of Capital - LBS, Solutions, Exercises of Finance

London Business School (LBS)Finance

Various solved exercises for the Finance exam on: Session 6: Risk and the Cost of Capital Read: Chapter 10: Risk and Return Chapter 11: Portfolio Choice and Diversification Risk, 2 stock portfolio, standard deviation, variance, correlation

Typology: Exercises

2010/2011

Uploaded on 09/15/2011

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Session 6: Risk and the Cost of Capital
Read: Chapter 10: Risk and Return
Chapter 11: Portfolio Choice and Diversification
1. Answer the following questions:
a. Briefly explain the difference between beta as a measure of risk and variance
as a measure of risk.
b. What is the correlation coefficient between the two stocks that gives the
maximum reduction in risk for a two-stock portfolio?
c. Historical nominal return for stock A is –8%, +10% and +22%. The nominal
return for the market portfolio is +6%, +18% and 24%. Calculate the beta for
stock A.
d. The correlation coefficient between stock B and the market portfolio is 0.8. The
standard deviation of the stock B is 35% and that of the market is 20%.
Calculate the beta of the stock.
Solution
a. Variance measures the total risk of a security and is a measure of stand-
alone risk. Total risk has both unique risk and market risk. In a well-diversified
portfolio, unique risks tend to cancel each other out and only the market risk
is remaining. Beta is a measure of market risk and is useful in the context of a
well-diversified portfolio. Beta measures the sensitivity of the security returns
to changes in market returns. Market portfolio has a beta of one and is
considered the average risk.
b. The correlation coefficient that gives the maximum reduction in risk for a two-
stock portfolio is -1.
c. Mean A = 8%, Mean M=16%, Cov(Ra, Rm)=0.0138 (138%2),
Var(Rm)=0.0084, Beta=138/84=1.643.
d. Cov(Rb,Rm)= (0.8)(20)(35) = 560, Beta = 560/400 = 1.4.
2. Answer the following questions:
a. How many variance terms and how many covariance terms do you need to
calculate the risk of a 100-share portfolio?
b. Suppose all stocks had a standard deviation of 30 percent and a correlation
with each other of .4. What is the standard deviation of the returns on a
portfolio that has equal holdings in 50 stocks?
c. What is the standard deviation of a fully diversified portfolio of such stocks?
Solution
a.
Stock
Stock
1 2 3 4 5 6 7 N
1
2
3
4
5
6
7
N
pf3

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Session 6: Risk and the Cost of Capital

Read: Chapter 10: Risk and Return

Chapter 11: Portfolio Choice and Diversification

  1. Answer the following questions: a. Briefly explain the difference between beta as a measure of risk and variance as a measure of risk. b. What is the correlation coefficient between the two stocks that gives the maximum reduction in risk for a two-stock portfolio? c. Historical nominal return for stock A is – 8%, +10% and +22%. The nominal return for the market portfolio is +6%, +18% and 24%. Calculate the beta for stock A. d. The correlation coefficient between stock B and the market portfolio is 0.8. The standard deviation of the stock B is 35% and that of the market is 20%. Calculate the beta of the stock. Solution a. Variance measures the total risk of a security and is a measure of stand- alone risk. Total risk has both unique risk and market risk. In a well-diversified portfolio, unique risks tend to cancel each other out and only the market risk is remaining. Beta is a measure of market risk and is useful in the context of a well-diversified portfolio. Beta measures the sensitivity of the security returns to changes in market returns. Market portfolio has a beta of one and is considered the average risk. b. The correlation coefficient that gives the maximum reduction in risk for a two- stock portfolio is - 1. c. Mean A = 8%, Mean M=16%, Cov(Ra, Rm)=0.0138 (138% 2 ), Var(Rm)=0.0084, Beta=138/84=1.643. d. Cov(Rb,Rm)= (0.8)(20)(35) = 560, Beta = 560/400 = 1.4.
  2. Answer the following questions: a. How many variance terms and how many covariance terms do you need to calculate the risk of a 100-share portfolio? b. Suppose all stocks had a standard deviation of 30 percent and a correlation with each other of .4. What is the standard deviation of the returns on a portfolio that has equal holdings in 50 stocks? c. What is the standard deviation of a fully diversified portfolio of such stocks? Solution a. Stock Stock 1 2 3 4 5 6 7 N 1 2 3 4 5 6 7 N

Refer to the figure above. With 100 securities, the box is 100 by 100. The variance terms are the diagonal terms, and thus there are 100 variance terms. The rest are the covariance terms. Because the box has (100 times

  1. terms altogether, the number of covariance terms is: 1002 – 100 = 9, Half of these terms (i.e., 4,950) are different. b. Once again, it is easiest to think of this in terms of figure above. With 50 stocks, all with the same standard deviation (0.30), the same weight in the portfolio (0.02), and all pairs having the same correlation coefficient (0.40), the portfolio variance is: σ^2 = 50(0.02)^2 (0.30)^2 + (50)^2 – 50^2 (0.40)(0.30)^2 =0. σ = 0.193 = 19.3% c. For a fully diversified portfolio, portfolio variance equals the average covariance: σ^2 = (0.30)(0.30)(0.40) = 0. σ = 0.190 = 19.0%
  1. Your eccentric Aunt Gerlinda has left you €50,000 in Deutsche Bank shares plus €50,000 cash. Unfortunately, her will requires that the shares not be sold for one year and the €50,000 cash must be entirely invested in one of the securities listed below. What is the safest attainable portfolio under these restrictions? Correlation Coefficients Alcan BP Deutsche Bank KLM LVMH Nestle Sony Standard Deviation (%) Alcan 1 0.39 0.55 0.54 0.61 0.26 0.36 30. BP 1 0.23 0.29 0.22 0.3 0.14 23. Deutsche Bank 1 0.36 0.48 0.16 0.39 38. KLM 1 0.49 0.32 0.19 54. LVMH 1 0.02 0.5 42. Nestle 1 0.1 15. Sony 1 4 7. Solution “Safest” means lowest risk. In a portfolio context this means lowest variance of return. Half of the portfolio is invested in DB stock, and half of the portfolio must be invested in one of the other securities listed. Thus, we calculate the portfolio variance for six different portfolios to see which is the lowest. The safest attained portfolio is comprised of DB and Nestle. Corr. Coef. with DB Std dev Variance = Alcan 0.55 0.302 0.5^2((σ(alcan))^2+(σ(db))^2+2ρ(alcan,db)σ(alcan)σ(db)) 0. BP 0.23 0.239 0.5^2((σ(bp))^2+(σ(db))^2+2ρ(bp,db)σ(bp)σ(db)) 0. KLM 0.36 0.545 0.5^2((σ(klm))^2+(σ(db))^2+2ρ(klm,db)σ(klm)σ(db)) 0. LVMH 0.48 0.42 0.5^2((σ(lvmh))^2+(σ(db))^2+2ρ(lvmh,db)σ(lvmh)σ(db)) 0. Nestle 0.16 0.155 0.5^2((σ(nestle))^2+(σ(db))^2+2ρ(nestle,db)σ(nestle)σ(db)) 0. Sony 0.39 0.475 0.5^2((σ(sony))^2+(σ(db))^2+2ρ(sony,db)σ(sony)σ(db)) 0.