(Portfolio optimization problems) We consider a single-period optimization problem involving n assets, and a decision vector x ∈ Rn which contains our position in each asset. Determine which of the following objectives or constraints can be modeled using convex optimization.
1. The level of risk (measured by portfolio variance) is equal to a given target t (the covariance matrix is assumed to be known).
2. The level of risk (measured by portfolio variance) is below a given target t.
3. The Sharpe ratio (defined as the ratio of portfolio return to portfolio standard deviation) is above a target t ≥ 0. Here both the expected return vector and the covariance matrix are assumed to be known.
4. Assuming that the return vector follows a known Gaussian distribution, ensure that the probability of the portfolio return being less than a target t is less than 3%.
5. Assume that the return vector r ∈ Rn can take three values r(i), i = 1, 2, 3. Enforce the following constraint: the smallest portfolio return under the three scenarios is above a target level t.
6. Under similar assumptions as in part 5: the average of the smallest two portfolio returns is above a target level t.
7. The transaction cost (under a linear transaction cost model, and with initial position xinit = 0) is below a certain target.
8. The number of transactions from the initial position xinit = 0 to the optimal position x is below a certain target.
9. The absolute value of the difference between the expected portfolio return and a target return t is less than a given small number Є (here, the expected return vector ˆr is assumed to be known).
10. The expected portfolio return is either above a certain value tup, or below another value t low.