Q3. Online algorithm for portfolio investment (25 points). In this problem we are going to see...
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Q3. Online algorithm for portfolio investment (25 points). In this problem we are going to see how you can use online learning to make money from stocks! To do so, we are going to look at the portfolio optimization problem. It involves choosing one out of n assets a,...,an to invest in at time t. The goal is to choose assets over time to maximize your financial returns. Each asset has a monetary return at round t (given by z; (t) for asset i) that reflects the multiplicative growth of the asset; e.g. if r(t) = 2, it means that asset i grew in value by a factor of 2; as an example, if the value of an asset was 125 at time step t - 1, then grew to 250 at time step t, then you would get 2i(t)=250/125=2. Our goal is to adaptively choose to invest in different assets over a time horizon (given by T) to maximize our overall profit. We start with an initial amount of cash Coc, and we choose to invest all our cash in one of the assets at every time step. This can and will be randomized; for example, we can invest in asset i at time t with probability p.(t) (a) (5 points) Remember that we want to maximize our total profit. Consequently, a good loss function that captures our objective turns out to look like: n L=-p;(s)logr(s). 8-0 i 1 Our goal for an online learning algorithm is to minimize the total loss LT at the end of the horizon. Use this loss function to write the portfolio optimization problem as an instance of prediction with expert advice. What are the experts and the corresponding loss functions for each expert? Q3. Online algorithm for portfolio investment (25 points). In this problem we are going to see how you can use online learning to make money from stocks! To do so, we are going to look at the portfolio optimization problem. It involves choosing one out of n assets a,...,an to invest in at time t. The goal is to choose assets over time to maximize your financial returns. Each asset has a monetary return at round t (given by z; (t) for asset i) that reflects the multiplicative growth of the asset; e.g. if r(t) = 2, it means that asset i grew in value by a factor of 2; as an example, if the value of an asset was 125 at time step t - 1, then grew to 250 at time step t, then you would get 2i(t)=250/125=2. Our goal is to adaptively choose to invest in different assets over a time horizon (given by T) to maximize our overall profit. We start with an initial amount of cash Coc, and we choose to invest all our cash in one of the assets at every time step. This can and will be randomized; for example, we can invest in asset i at time t with probability p.(t) (a) (5 points) Remember that we want to maximize our total profit. Consequently, a good loss function that captures our objective turns out to look like: n L=-p;(s)logr(s). 8-0 i 1 Our goal for an online learning algorithm is to minimize the total loss LT at the end of the horizon. Use this loss function to write the portfolio optimization problem as an instance of prediction with expert advice. What are the experts and the corresponding loss functions for each expert?
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Related Book For
Fundamentals of Financial Management
ISBN: 978-1337395250
15th edition
Authors: Eugene F. Brigham, Joel F. Houston
Posted Date:
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