Oct 8, 2020 CO 372: Assignment 2 DUE: Oct 22 (10 PM Wat. Time)


Assignment Guidelines: The current TA, the current course instructor, the course notes, and the
lectures, are your sources for help. Do not seek help from other written sources. This assignment
is to be completed individually

Background: In class we have discussed the 1-period investment problem. Given a set of n
risky assets and a certain wealth, the 1-period problem is to split the investment amongst the
risky assets to minimize the risk over this period while meeting a target return for the portfolio in
an expectation sense. We have allowed for shorting of assets (i.e., sell $x of an (unowned) asset
at the current price to be purchased at a future date (at the new price)). If the asset has increased
in value then your portfolio value has correspondingly decreased in value; if the shorted asset has
decreased in value then your portfolio value has a corresponding increase in value). Finally, our
model to date allows for additional linear equality constraints (e.g., sector constraints).
This problem is concerned with the multi-period problem: i.e., your portfolio is rolled over to
the next time period, and the next,… . If you have a portfolio x (a vector representing dollar
amounts in each asset) at the end of time period t, and your optimization problem says to use
portfolio x in the next period then you must rebalance x to get x, which means buying and
selling asset investments. These are transactions and typically transactions have a corresponding
costs related to the amount bought and sold (the relationship can be complicated but here we
assume the cost is just a linear tax on the amount bought and sold ., e.g.,
1
_ cos *
n
i i
i
tr t ttax x x

  where ttax is the unit transaction cost (or tax).
Your data: Your data is available in PS2. The matrix R1 is the matrix of 108 monthly historical
returns of 12 stocks. R1 is ordered such that the oldest record is in the first row and the newest
record is in the last row The matrix is 108x12 and each row corresponds to the stock returns over
that month. So for example if a dollar invested in stock j at the beginning of month i is worth
$1.02 at the end of month i then R1(i,j) = 1.02.

The matrix R2 is similar to R1 but represents 12 months of ‘future’ returns. Row 1 of R2
represents the first future month, row 2 represents the 2nd future month, etc.
wealth is the initial money to be invested.

Your problem: Track your optimal portfolio value (wealth) over the 12 future months. Track
also a uniform ‘hold’ portfolio. (i.e., invest wealth/n in each asset at the beginning of the year
and hold the investment (i.e., no rebalancing).
Oct 8, 2020 CO 372: Assignment 2 DUE: Oct 22 (10 PM Wat. Time)
a. Compute the n-by-n covariance matrix H based on R1 – the methodology was discussed
in class. You will need to compute erv, the expected return vector for the assets based on
matrix R1.. [10 pts]
b. Use a monthly portfolio target return : mtarget = 1.02. Assume the unit transaction cost is
5%, i.e., ttax = .05. For each month:
i. Determine the wealth at the beginning of this month: adjust your portfolio
$-investment x based on the returns for the just-ended month. (using the
next row of futures data in R2). [10 pts]
ii. Use one of your QPLE solvers to determine the optimal portfolio for the
next period using the just-computed wealth. (Do not use any futures data
in R2 in this step) [10 pts]
iii. Calculate and record the corresponding transaction costs. [10 pts]
iv. Update the return matrix used to compute erv and H. That is, remove the
row corresponding to the oldest data in R1(row 1). Shift R1 up by 1 row;
set the new last row of R1 to be the most recent data used in R2: R2(k,:).
Compute the new expected return vector erv and the new covariance
matrix H. Note: the dimensions of R1 remain the same. [10 pts]
v. Carry forward the ‘hold’ portfolio as well. In this case the initial
investment is evenly divided among the stocks and there is no rebalancing
(of course the monthly wealth varies as asset returns fluctuate). [10 pts]
c. Your MATLAB output should include final wealth values, final transaction costs and net
values, and a graph of the monthly wealth of the 2 portfolios (‘optimal’ and ‘hold’) over
the 12 months. [20 pts]
d. Next modify your wealth calculation at each month – subtract the transactions costs from
the previous month. Plot this transaction-cost adjusted portfolio monthly wealth with the
portfolio that does not include transaction costs (above) and the hold portfolio. Note that
the ‘hold’ portfolio incurs no transaction costs except for the initial investment. [20 pts]




________
[100 pts]


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