Basics of Portfolio Selection Theory  Exercise 1

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

Question 1 • Question 1 • An investor is supposed to set up a portfolio including share 1 and 2. It is E(r1) = 1 = 0,2 the expected return of share 1 and E(r2) = 2= 0,3 the expected return of share 2. • Moreover, it is var(r1) = 12 = 0,04, var(r2) = 22 = 0,08 and cov(r1,r2) = 12 = 0,02. • Calculate the minimal variance portfolio for a given expected portfolio return • of . What is the variance and the expected value of this portfolio? • Determine the equation of the efficient frontier that can be calculated as the combination of both shares. • Which efficient portfolio should an utility-maximizing investor with a preference function of realize? Basics of Portfolio Selection Theory: Exercise 1

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

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

Solution Question 1 Part c) • What is the expected value depending on the given variance? • Calculation of x1: • c1) Basics of Portfolio Selection Theory: Exercise 1

Solution Question 1Part c) Thus, on the efficient frontier we receive: This means a reduction of equation c1) to: Accordingly, the equation of the efficient frontier is: Basics of Portfolio Selection Theory: Exercise 1

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

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

Solution Question 1 • Graphical solution for question 1 μP σP Basics of Portfolio Selection Theory: Exercise 1

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

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

Question 2 • Question 2 • In addition to stock 1 and 2 with E(r1)=1=0,2, E(r2)= 2=0,3, var(r1)= 12=0,04, • var(r2)= 22 =0,08 and cov(r1,r2)=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 rf = 0,1. • Calculate the minimal variance portfolio for an expected value of the portfolio return of . What is the variance of this portfolio? • Calculate the variance and expected value of the tangential portfolio. • Find out the equation for the efficient frontier, which can be calculated by combining both stocks and the risk-free investment. • How high are the portfolio-risks of stock 1 and 2 in the portfolio selected in a)? How does they correspond to each other? • Which of the efficient portfolios should a utility-maximizing investor with a preference function of realize? Basics of Portfolio Selection Theory: Exercise 1

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

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

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

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

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

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

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

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

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

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

Solution Question 2 • Graphical solution for question 2 μP σP Basics of Portfolio Selection Theory: Exercise 1

Question 3 • Question 3 • The expected return and the standard deviation of stock 1 and stock 2 are E(r1)=1=0,25, 1=30% andE(r2)= 2=0,15, 2 =10% respectively. The correlation is -0.2. • 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. • 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). Basics of Portfolio Selection Theory: Exercise 1

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

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

Basics of Portfolio Selection Theory  Exercise 1