11. Answer the following questions:

a)Suppose there are three assets, and the first asset has volatility 16%, the second asset has volatility 26%, and the third asset has volatility 17%. Suppose also that the first two assets' returns are correlated with each other with correlation coefficient 0.8, but the third asset is not correlated with the first two assets.

Now consider the minimum variance portfolio of these three assets. What is the weight of the first asset? (To the nearest 0.01. Express weights as a decimal, e.g. a 50% weight would be 0.50).

b) Suppose there are three assets, and the first asset has volatility 18%, the second asset has volatility 26%, and the third asset has volatility 15%. Suppose also that the first two assets' returns are correlated with each other with correlation coefficient 0.2, but the third asset is not correlated with the first two assets.

Suppose the risk free rate is 1%, and the expected returns of the three assets are 9%, 3% and 3% respectively.

Now consider the efficient portfolio of risky assets, i.e. the "one fund" F. What is the weight of the first asset? (To the nearest 0.01. Express weights as a decimal, e.g. a 50% weight would be 0.50).

c) Suppose we have a factor model with two factors f₁ and f₂, where the volatility of the factor 1 return is 12%, the volatility of the factor 2 return is 20% and the correlation between the factor returns is 0.7. Suppose also that the volatility of the error terms e₁ and e₂ are both 4%. Given two asset returns

r₁ = 1 + 0.1f₁ + 0.1f₂ + e₁

_{r2 = 2 + 0.1f1 + 0.4f2 + e2}

what is the correlation coefficient between the asset returns r₁ and r₂ (to the nearest 0.01)?

d) Suppose under arbitrage pricing theory (APT), the mean asset returns are described as

E{r} = ₀+b₁₁+b₂₂

where b₁ and b₂ are factor loadings for two factors f₁ and f₂, and ₀=3% is the risk free rate. Given two asset returns

r₁ = 0% + 0.8f₁ + 0.3f₂ + e₁

_{r2 = 2% + 0.1f1 + 0.6f2 + e2}

and mean factor returns E{f₁} = 7% and E{f₂} = 5%, what ₁ is (to the nearest 0.001)?

e) Suppose you have two strategies. Strategy 1 gains 8% with probability p, and loses 5% with probability 1-p, where p = 0.53. Strategy 2 gains 8% with probability q, and loses 5% with probability 1-q, where q = 0.51. The outcomes of the two strategies are independent. You assign weights (which sum to one) for the two strategies. What will the optimal weight be for Strategy 1 (to nearest 0.001), if you goal is to maximize your expected growth rate (i.e. maximize log utility, i.e. the Kelly criterion)?

The answer to these problems are a) 0.71 b) 0.75 c) 0.53 d) 0.037 e) 0.813.

I want the equations to get these answers.

Thank you!