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# Advanced Investment Theory 28E00300

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Was the course worth attending? Were the lectures useful? What questions did the exam have? Give us your honest opinion on the course!

Exam 23.5.2018

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Question 1 (30 points)

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PART A:

1) Suppose that S follows the process dS = μSdt + σSdz. What process does G follow if G is a function of S and t so that G = Se^(r(T-t)) ?

2) Consider a variable S that follows the process dS = μdt + σdz where μ = 3 and σ = 4. If S = 1 when t = 0, what is the probability distribution of the value of the variable at time t = 2?

3) A stock has the price $50 today, and has a variance of 0.3 per annum and expected yearly return μ = 0.16. What is the 90% confidence interval for the stock price tomorrow (z = -1.645 for 5%). Assume there are 250 trading days in the year.

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Question 2 (40 points)

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PART A:

1) Derive the risk-return trade-off equation in CAPM. Start by constructing a portfolio where proportion a is invested in a risky asset and (1-a) in the market portfolio.

2) Derive the CAPM model from the risk-return trade-off equation. Explain the assumptions in the derivation of the equation and of the model.

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PART B:

1) Derive the following Black-Scholes differential equation (∂f/∂t) + rS(∂f/∂S) + (1/2)*(σ^2)*(S^2)*(∂2f/∂S2) = rf

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PART C:

1) Construct a risk-less portfolio using two assets f and g that follow the processes:

df = μ(f)*fdt + σ(f)*fdz

dg = μ(g)*gdt + σ(g)*gdz

[μ(f), σ(f), μ(g), σ(g) are constants. I just had to mark them like this because you can’t put subscripts into the shadow study guide, i.e. μ subscript f]

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Question 3 (30 points)

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PART A:

1) Explain Kyle’s paper (Continuous Auctions and Insider Trading, 1985). How does this compare to Grossman and Stiglitz (On the Impossibility of Informationally Efficient Markets, 1980)?

2) What is the momentum anomaly and what are its causes?

3) Explain what a currency carry trade is. What may cause crashes to the returns of this trade?