I am given the following problem from a tutorial in my course:
(Portfolio Optimization with Recourse). You have £10,000 to invest (without short selling) in a portfolio composed of eight leading stock market indices (think of investing in a market index as investing equally in all securities that form the index, e.g., if your decision is to invest 8% in a index formed by 10 securities, then each security is getting 0.8% of your total investment) in the US stock market: S&P100, S&P500, S&P SmallCap 600, Dow Jones, NASDAQ Composite, Russell 2000, Barron’s 400, Wilshire 5000.
You have to invest while considering the performance of these indices over four quarters of a calendar year,
Q1 : 1 Jan – 31 Mar, Q2 : 1 Apr – 30 Jun, Q3 : 1 Jul – 30 Sep, Q4 : 1 Oct – 31 Dec.
We are allowed recourse actions on our investment via having shortfall on target return rates. Let $\Omega$ denote the scenario set with probabilities $p(\omega)$ for $\omega \in \Omega$. Let $r_{i,t}(\omega)$ denote the random return rate of index $i$, for $i = 1, . . . , n$ where $n = 8$, in time period t, for quarters $t = 1, . . . , T$ where $T = 4$. Let $R_t$ denote the target return rate in time period t, for $t = 1, . . . , T$. You incur investment cost of $c_i$ for each index $i$, for $i = 1, . . . , n$, and also a penalty cost of $b_t$ for unit shortfall in time $t$, for $t = 1, . . . , T$.
The two-stage stochastic program that minimises the total cost is formulated as follows, where, $\omega_{·t}$ is the vector of scenarios $(\omega_{it})i=1,...,n$ formed by scenarios for returns of asset $i$ in time $t$,
$$\min_x c^Tx + \mathbb{E} \left[\sum_{t=1}^T b_ty_t(\omega_{.t})\right]$$
$$\text{s.t } \sum_{i=1}^n r_{i,t}(\omega)x_i + y_t(\omega_{.t}) \geq R_t, \;\;\; t = 1, ..., T$$
$$\sum_{i=1}^n x_i = 1, \;\;\; x \geq 0, y \geq 0$$
Then the problem is to solve the stochastic program where there are 5 scenarios given, that we assume apply to all assets for all quarters. They are $\Omega = \{ \omega_1, ..., \omega_5 \}$ where the probabilities of the scenarios are given by the vector $p = [0.12,0.22,0.33,0.18,0.15]$ and the returns are $r = [\mu_{i,t}-1.5\sigma_{i,t},\mu_{i,t}-\sigma_{i,t},\mu_{i,t},\mu_{i,t}+\sigma_{i,t},\mu_{i,t}+\sigma_{i,t}]$ and the means and standard deviations are calculated by sampling from historical data.
The investment costs for the indices respectively are given as $c = [0.45,1.15,0.65,0.8,1.25,1.1,0.9,0.7]$ and the penalty costs for the quarters are $3,4.2,2.75,5$ respectively.
Now, to solve this problem, I must construct the deterministic equivalent of the LP to be able to solve. The deterministic equivalent program (from my course) is given as,
$$\min_{x,y} c^T x + \sum_s \pi_s q_s^T y_s$$
$$\text{s.t } Ax = b$$
$$T_sx + W_s y_s \quad \quad \forall s$$
$$x \geq 0, y_s \geq 0$$
My questions here are:
- I know that the recourse decision here is a form of rebalancing the portfolio, does that mean the vector $y_s$ is a vector of length $8$ with components corresponding to a buy/sell decision on each of the indices? In this case why do we restrict $y_s \geq 0$, how do we represent a sell decision in an asset that we already hold in $x$?
For example, if the optimal portfolio composition is $x^* = [1,0,0,0,0,0,0,0]$ and after witnessing $\omega_1$, it is optimal to transfer all holdings to asset 4, would the recourse decision not be $y=[-1,0,0,1,0,0,0,0]$ corresponding to selling all of asset 1 and allocating all funds to asset 4?
- If $y_s$ is indeed a vector of length 8, then how do we construct the objective function of the Stochastic program, since it will not be a numerical value, but rather a vector?
- When does the recourse rebalancing take place in relation to the quarters? It is my understanding that the model above is a two-stage stochastic program. Then, is it the case that we make our initial portfolio selection $x$ and the outcome is revealed immediately after the decision, allowing us to rebalance before the end of Q1?
- Once we have the return rates and standard deviations sampled from historical data, is the problem not completely deterministic and the optimal composition will be of the asset with the highest overall return? So that $x^*$ will have $1$ in one component and $0$ in all others?
- In the deterministic equivalent's objective function, how do we construct the vector $q_s$? Since it should correspond to the expected future cost in that scenario, that is the sum of the portfolio's performance over all quarters related to the holdings.
- Should we have a total of $4 \times 5 + 3 = 23$ constraints? For each quarter we have $5$ scenarios where the target return must be met, and the other three constraints ($\sum x = 1, x \geq 0, y \geq 0$).
