2024-05-27-Portfolio Theory

Types of risks

• Specific risks: associated with a specific asset
• Systematic risks: common to all securities

Diversification and Portofolio theory

1. Argument:
   The theroy assumes that the risk-return profile of a portfoliocan be optimized, where an optimal portfolio displays the lowest possible level of risk for its level of return.
   • Variance calculation:
     $${\sigma }^2=(\sum _{i=1}^n{w}_i{\sigma }_i{)}^2$$
     $$\sum _{i=1}^n{w}_i=1$$
2. Strategy:
   • Uncorrelated returns:
     Select securities whose performance is uncorrelated or negatively correlated
   • Portfolio with short sales:
     Allow short sales: sell short sells of the lower return asset and use the proceeds to nuy the higher return one(Applying leverage)
   • The Portfolio Frontier(visualization):
     Represents whether the current portfolio can be optimized to reach the frontier
   • Competitive Analysis:
     1. Motivation:
        • typical statistical method: only access to all the relevant information
        • online problemL continually produces new input and requires answers in response
     2. Effectiveness analysis:
        Comparison object: offers a worst measure of the quality of the behavior of an algorithm which predicts the fufure
     3. Competitive Ratio: Determine the linear coefficient
        ALF(I) $\le$ c$\ast$OPT (I) + $\alpha$
     4. Examples:
        • Price searching:
          Sell when the price reach the case either the it goes to max or min are OK
          RPP(reservation price policy) :
          global fluctuation ratio: $\varphi =\frac{M}{m}$
          competitive ratio: $c=\frac{\sqrt{Mm}}{m}=\sqrt{\varphi }$
        • Randomized Algorithms:
          Make it difficult for an adversary to design a future that is bad for you
        • One-way trading
          Liquidate my position in a stock -> more flexibility, less market impact, less risks
        • Threat-based strategy:
          The threat-based strategy sells only when the price hits a new maximum. It sells just enough to ensure that we achieve a competitive ratio of c if the price drops to m for the rest of the game
        • Randomized Strategies:
        • Rebalancing Algorithms(adjust the weights of different stocks):
        • Two-way trading:
          • Two-way trading is a special case of online portfolio selection where you have only cash and one other security you can hold
          • put all your assets into the security on any day it offers positive returns. Otherwise, put everything into cash.
        • The (n,$\varphi$) Adversary: