投资组合优化

General Idea

  1. Financial optimization:
    • Variables: Amounts invested in each asset
    • Constraints: budget, investment per set, minimum return
    • Objective: Maxize proft, Minimize risk
  2. Machine learning:
    • Variables: Model parameters
    • Constraints: Prior information, parameter limits
    • Objective: Minimize prediction error
  3. Method:
    • least squares
    • linear optimization
    • convex optimization
  4. Basic terms:
    • allocation weights:
      • 总资产:V
      • w: n维向量->记录给每个资产分配的权重
      • Property:
        $V_(w_j)$ dollar value hold in asset j
        w中分量线性和为1（齐次）
        $w_j < 0$ means short positions(borrow)
    • Return over a period:
      • Asset return: $\hat{r_t}$ the farctional return vector of each asset
      • Portfolio return: $r_t$ = $\hat{r_t^T}w$ the return for the entire portfolio over period t
      • Total porofolio value after a period
        $$ V_{t+1} = V_t + V_t\hat{r_t^T}w = V_t(1+r_t)$$

Linear optimization

  1. Average return:
    $$ avg(r) = \frac{\sum_{t=1}^T{R_t}}{T} = \mu^Tw $$
    • Average return calculate the average return value over period T;
    • Additionally, $\mu$ is n-dimensional vector, representing the average return of each asset,it can be calculated by $$\mu = \frac{R^T}{T} \text{and $R^T$ is a n*T matrix, including all assets’ return rate over the whole period} $$

  2. 1-nrom Risk Approximation(1-范数风险近似):
    Notice: It is no longer standard deviation(标准差)
    $$||r-avg(r)1||_{\frac{1}{T}}$$

  3. Risk-Return Objective:
    Objective = $\mu^Tw - \frac{\sum_{t=1}^T(R_tw-\mu^Tw)^2}{T} + \text{(trade off parameter)}$
    $\frac{\sum_{t=1}^T(R_tw-\mu^Tw)^2}{T}$ is the risk evaluation over time