Basics of Portfolio Selection Theory  Exercise 1 University of Hohenheim Chair of Banking and Financial Services Portfolio Management Summer Term 2011 Exercise 1: Basics of Portfolio Selection Theory Prof. Dr. Hans-Peter Burghof / Katharina Nau Slides: c/o Marion Schulz/ Robert Härtl

Question 1 An investor is supposed to set up a portfolio including share 1 and 2. It is E(r 1 ) =  1 = 0,2 the expected return of share 1 and E(r 2 ) =  2 = 0,3 the expected return of share 2. Moreover, it is var(r 1 ) =  1 2 = 0,04, var(r 2 ) =  2 2 = 0,08 and cov(r 1,r 2 ) =  12 = 0,02. a)Calculate the minimal variance portfolio for a given expected portfolio return of. What is the variance and the expected value of this portfolio? a)Determine the equation of the efficient frontier that can be calculated as the combination of both shares. b)Which efficient portfolio should an utility-maximizing investor with a preference function of realize? Basics of Portfolio Selection Theory: Exercise 1 1

Solution Question 1 Part a) Expected portfolio value: Calculation of the portfolio weights: 2 Basics of Portfolio Selection Theory: Exercise 1

Calculation of the portfolio variance: Standard deviation: Solution Question 1 Part b) 3 Basics of Portfolio Selection Theory: Exercise 1

What is the expected value depending on the given variance? Calculation of x 1 : c 1 ) Solution Question 1 Part c) 4 Basics of Portfolio Selection Theory: Exercise 1

Solution Question 1 Part c) Thus, on the efficient frontier we receive: This means a reduction of equation c 1 ) to: Accordingly, the equation of the efficient frontier is: 5 Basics of Portfolio Selection Theory: Exercise 1

Utility function: Maximization: Solution Question 1 Part d) 6 Basics of Portfolio Selection Theory: Exercise 1

Utility maximizing portfolio: Solution Question 1 Part d) 7 Basics of Portfolio Selection Theory: Exercise 1

Solution Question 1 Graphical solution for question 1 μPμP σPσP 8 Basics of Portfolio Selection Theory: Exercise 1

Continuation of Question 1 Stock’s portfolio risks: Firstly, the cov(r i, r p ) must be calculated: In the numerical example of part a) 9 Basics of Portfolio Selection Theory: Exercise 1

Stock’s portfolio risks: Continuation of Question 1 10 Basics of Portfolio Selection Theory: Exercise 1

Question 2 In addition to stock 1 and 2 with E(r 1 )=  1 =0,2, E(r 2 )=  2 =0,3, var(r 1 )=  1 2 =0,04, var(r 2 )=  2 2 =0,08 and cov(r 1,r 2 )=  12 =0,02, now there is a capital market providing the opportunity to invest and raise unlimited capital at a risk-free interest rate of r f = 0,1. a)Calculate the minimal variance portfolio for an expected value of the portfolio return of. What is the variance of this portfolio? b)Calculate the variance and expected value of the tangential portfolio. c)Find out the equation for the efficient frontier, which can be calculated by combining both stocks and the risk-free investment. d)How high are the portfolio-risks of stock 1 and 2 in the portfolio selected in a)? How does they correspond to each other? e)Which of the efficient portfolios should a utility-maximizing investor with a preference function of realize? 11 Basics of Portfolio Selection Theory: Exercise 1

Solution Question 2 Part a) 12 Basics of Portfolio Selection Theory: Exercise 1

Solution Question 2 Part a) 13 Basics of Portfolio Selection Theory: Exercise 1

Tangential Portfolio From Example 1c) Efficient frontier: Slope of the efficient frontier in T: Solution Question 2 Part a) 14 Basics of Portfolio Selection Theory: Exercise 1

Slope of the capital-market-line: Solution Question 2 Part b) 15 Basics of Portfolio Selection Theory: Exercise 1

Solution Question 2 Part b) 16 Basics of Portfolio Selection Theory: Exercise 1

Solution Question 2 Part b) 17 Basics of Portfolio Selection Theory: Exercise 1

2. Approach Structure of the tangential portfolio: whereas the tangential portfolio only includes stock 1 and stock 2 and there is no risk-free investment or borrowing: Solution Question 2 Part b) 18 Basics of Portfolio Selection Theory: Exercise 1

Efficient frontier: Solution Question 2 Part c) 19 Basics of Portfolio Selection Theory: Exercise 1

Comparison with the results of part 2a) Solution Question 2 Part c) 20 Basics of Portfolio Selection Theory: Exercise 1

Portfolio risks: From Exercise 1: Solution Question 2 Part d) 21 Basics of Portfolio Selection Theory: Exercise 1

Maximization of Efficient frontier: Solution Question 2 Part e) 22 Basics of Portfolio Selection Theory: Exercise 1

Solution Question 2 Part e) 23 Basics of Portfolio Selection Theory: Exercise 1

Solution Question 2 Part e) 24 Basics of Portfolio Selection Theory: Exercise 1

Graphical solution for question 2 Solution Question 2 σPσP μPμP 25 Basics of Portfolio Selection Theory: Exercise 1

Question 3 The expected return and the standard deviation of stock 1 and stock 2 are E(r 1 )=  1 =0,25,  1 =30% and E(r 2 )=  2 =0,15,  2 =10% respectively. The correlation is a)Which weights should an investor assign to stock 1 and stock 2 to set up the minimum- variance portfolio? Also compute the expected return and the variance of the portfolio. b)Assume that in addition to the above information a risk free investment with a yield of 10% exists on the capital market. Show that the investor can now realize the same expected return at a lower level of risk. For this purpose, calculate the risk of the efficient portfolio based on the expected return calculated in part a) and compare it to the minimum-variance portfolio of part a). 26 Basics of Portfolio Selection Theory: Exercise 1

Solution Question 3 Part a) 27 Basics of Portfolio Selection Theory: Exercise 1

Solution Question 3 Part b) 28 Basics of Portfolio Selection Theory: Exercise 1