Artem Tsvetkov & Lammertjan Dam FTM Week 1: Luenberger, Chapters 1 and 6 Luenberger 6.3 (Two correlated assets) The correlation ρ be...

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Week 1: Luenberger, Chapters 1 and 6 Luenberger 6.3 (Two correlated assets) The correlation ρ between assets A and B is 0.1, and other data are given in Table. [Note: ρ = σAB / (σA σB )]. Two Correlated Cases Asset r σ A 10.0% 15% B 18.0% 30% a. Find the proportions α of A and (1 − α ) of B that deﬁne a portfolio of A and B having minimum standard deviation. b. What is the value of this minimum standard deviation? c. What is the expected return of this portfolio? Using the analytical solution from exercise 6.4: a. α =

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b. ¯σ ≈ 13.92% c. ¯r ≈ 11.39%

Luenberger 6.4 (Two stocks) Two stocks are available. The corresponding expected rates of return are ¯r1 and ¯r2 ; the corresponding variances and covariances are σ21 , σ 22 , and σ12. What percentages of total investment should be invested in each of the two stocks to minimize the total variance of the rate of return of the resulting portfolio? What is the mean rate of return of this portfolio? For two stocks, the Markowitz problem is 1 2 2 2 2 min w1 σ1 + 2 w1 w2 σ12 + w 2 σ2 w1,2 2 s.t. w1 + w2 = 1.

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Using Lagrange multipliers, the problem is reduced to a system of three linear equations for w1 , w2 , and µ ∂L = w1 σ 21 + w 2 σ12 − µ = 0 ∂w1 ∂L = w1 σ 12 + w 2 σ22 − µ = 0 ∂w2 w 1 + w2 = 1 The solution for w1 is

σ22 − σ12 . w1 = 2 σ1 − 2 σ12 + σ22

The mean rate of return of the portfolio is simply r¯ = w 1 ¯r1 + w 2 ¯r2 , and minimal variance is

σ21 σ22 − σ212 . min σ = 2 σ1 − 2σ12 + σ 22 2

Luenberger 6.5 (Rain insurance) Gavin Jones’s friend is planning to invest $1 million in a rock concert to be held 1 year from now. The friend ﬁgures that he will obtain $3 million revenue from his $1 million investment – unless, my goodness, it rains. If it rains, he will lose his entire investment. There is a 50% chance that it will rain the day of the concert. Gavin suggests that he buy rain insurance. He can buy one unit of insurance for $0.50, and this unit pays $1 if it rains and nothing if it does not. He may purchase as many units as he wishes, up to $3 million. a. What is the expected rate of return on his investment it he buys u units of insurance? (The cost of insurance is in addition to his $1 million investment.) b. What number of units will minimize the variance of his return? What is this minimum value? And what is the corresponding expected rate of return? [Hint: Before calculating a general expression for variance, think about a simple answer.] (a) The invested amount is 106 + 0.5 u million dollars. The expected return is R=

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· 3 · 106 + 12 u 106 + 0.5u

(b) It is easy to see that buying 3 million units of insurance removes any uncertainty in 2