Overview

• 4 back tests with positive alpha in all of them on the testing is goal
• \(\color{#D7A84B}\tau = [t_i,t_f]\) is the time period
• \(\color{#D7A84B}u\) is the set of assets
• Series for an asset \(\color{#D7A84B}i\in u\) over \(\color{#D7A84B}\tau\) has value series, return series, rolling mean, rolling std, rolling z-score, cross sectional mean, cross sectional z-score, and other statistical quantities defined for \(\color{#D7A84B}t\in \tau\)
  • \(\color{#D7A84B}v_t:[t_i,t_f]\rightarrow [0,\infty )\) is the value/price of an asset
  • \(\color{#D7A84B}r_t:[t_i+1,t_f]\rightarrow [0,\infty )\) is the return series of the asset
  • \(\color{#D7A84B}q_t:[t_i,t_f]\rightarrow \mathbb Z[0]\) is the amount of held quantities of the asset
    • Notation indicates \(\color{#D7A84B}\mathbb Z[a,b]\) is the set of integers from \(\color{#D7A84B}a\) to \(\color{#D7A84B}b\) inclusive, and \(\color{#D7A84B}\mathbb Z[0]\) is the set of positive integers,
  • \(\color{#D7A84B}w_t:[t_i,t_f]\rightarrow [-1,1]\) is the weight of the asset
• Portfolios are defined by a vector of assets with weights \(\color{#D7A84B}w=(w_1,...,w_{N}){}\) for \(\color{#D7A84B}N\) assets.
• Number of assets held \(\color{#D7A84B} q=(q_1,...,q_{ N })\) also describe it
• For \(\color{#D7A84B}U\) assets, \(\color{#D7A84B}w\) represents the vector of all the related weights \(\color{#D7A84B}w\in \color{#D7A84B}\mathcal{W}= \bigoplus_{i\in u}[-1,1]\) , the set of all possible weights.
  • \(\color{#D7A84B}\mathcal W \subset \mathbb R ^U\)
  • Alternatively, it can be viewed as will be defined the decision series itself, \(\color{#D7A84B}O_t\) .
• For a fully invested portfolio, \(\color{#D7A84B}\color{#D7A84B}\sum _{i\in \mathcal u } |w_i| = 1\) . Negative weights represent borrowed assets, meaning shorts, while positive weights represent bought assets, meaning longs.
  • Note this is the \(\color{#D7A84B}L^1\) -norm defined on \(\color{#D7A84B}\mathbb R^U\) .

Portfolio Quantities

• \(\color{#D7A84B}p\subset u\) is a portfolio
• \(\color{#D7A84B}v_p\) is the value of the portfolio.
• \(\color{#D7A84B}v_{p,t} =\sum_{i=1}^ U q_{i,t } v_{i,t}\)
• \(\color{#D7A84B}r_{p,t} =\sum_{i=1}^ U r_{i,t}\) is the portfolio return
• Then, definitions
  • \(\color{#D7A84B}r_{p,t}\) is the portfolio returns defined \(\color{#D7A84B}r_{p,t } =\frac{ v_{ p,t } - v_ {p,t-1 } }{v _ {p, t-1 } }{}\)
  • \(\color{#D7A84B}r_{i,t}\) is the returns of asset \(\color{#D7A84B} i\)
    • \(\color{#D7A84B}r_{i,t} \rightarrow P_ {i,t}\) is the probability distribution of the asset return at time \(\color{#D7A84B}t\)
  • \(\color{#D7A84B}P_i\) is the probability that the asset will have the return value \(\color{#D7A84B}r_i\) .
    • If \(\color{#D7A84B}r_i\in \mathbb R\) , \(\color{#D7A84B}P(r_i ) dr_i\) is the probability of asset having the value in the neighborhood of the value \(\color{#D7A84B}r_i\) with radius \(\color{#D7A84B}dr_i\) .
    • Then, note \(\color{#D7A84B}P(r_i) = \mathcal N(0,\sigma)\) , a normal distribution by assumption. (Most situations)
• Actions are done on a series and can be classified as filters, strategies, or market actions, meaning a stochastic evolution of a Markov process
  • Actions are mappings from time to decisions, which are represented as operators.
  • Actions are done by users or by the market.
  • An user can do actions \(\color{#D7A84B}o\in \mathcal O\) defined as follows
    • \(\color{#D7A84B}q_{i,t+1} = q_{i,t} +k=l_k(q_{i,t})\) is a long of size \(\color{#D7A84B}k\) of asset \(\color{#D7A84B}i.\)
      • \(\color{#D7A84B}{l_k}_{k\in \mathbb Z[0 )]} =\mathcal L\) is the set of all possible long operators
    • \(\color{#D7A84B}q_{i,t+1} = q_{i,t} -k=p_k(q_{i, t} )\) is a short of size \(\color{#D7A84B}k\) of asset \(\color{#D7A84B}i.\)
      • \(\color{#D7A84B}{k _k}_{k\in \mathbb Z[-\infty , 0 ]} = \mathcal S\) is the set of all possible short operators
    • If \(\color{#D7A84B}k=0\) , \(\color{#D7A84B}q_{i,t+1}=q_{i,t}=I(q_{i, t})\) is identity operator
  • \(\color{#D7A84B}\mathcal O_i= \mathcal L\cup \mathcal S\cup I\) is the set of all possible operators which can act on an asset.
  • \(\color{#D7A84B}O=(o_1,...,o_U )\) is the vector of operators which acts on the set of assets at a given time, and \(\color{#D7A84B}O \in \bigotimes_ {i=1 } ^ U \mathcal O\)
  • \(\color{#D7A84B}O\) \(\color{#D7A84B}:\mathbb R^U\rightarrow \mathbb R ^U\) for all \(\color{#D7A84B}O\)
  • A sequence \(\color{#D7A84B}O_t\) of such operators is a decision series for the portfolio denoted \(\color{#D7A84B}d_p\) while the market can act on the portfolio through stochastic evolution defined by operators \(\color{#D7A84B}M_t\) .
  • Then, \(\color{#D7A84B}d = \{O_t\}_ { t\in \tau}\) is a decision series with \(\color{#D7A84B}O_t(w_t )=w_{t+1}\)
    • Note \(\color{#D7A84B}d = O_{t_f}...O _ {t_i}\) means \(\color{#D7A84B}d(w_{t_i }) =w_{t_f}\) . This is net change of a portfolio over a time period
• To optimize is to choose \(\color{#D7A84B}d\) such that \(\color{#D7A84B}g(w,\lambda)\) is optimized, where \(\color{#D7A84B}g\) represents a desired goal
  • Markowitz optimization is \(\color{#D7A84B}g_t(w,\lambda) =r_p^Tw -\lambda w^T\Psi_ p w\) , \(\color{#D7A84B}\Psi =\text{Var}(r_p)\) is the covariance matrix of the assets, which is the net variance of the portfolio return.
  • \(\color{#D7A84B}O_t\) can be chosen such that \(\color{#D7A84B}g_t\) is maximized at every \(\color{#D7A84B}t\) , but the goal overall is to maximize \(\color{#D7A84B}r_\tau\) the total return over a time period \(\color{#D7A84B}\tau\)
• \(\color{#D7A84B}\{M_t\}_{t=t_i}^ {t_f}\) is the set of market operators which represent the action of the market on a weight vector.
• All possible changes to \(\color{#D7A84B}w\) arise from a combination of decision operators and market.
  • Evolution of a system \(\color{#D7A84B}w_{t+1}=M_t O_ t w_ t,\forall \,t\in \tau\) .
• Time filtration in terms of acting on different periods is done through indicator functions over the set of possible times \(\color{#D7A84B}1_{[t_i,t_f]\in \tau}{}\)
• \(\color{#D7A84B}w_{t_i}=0\) , where \(\color{#D7A84B}0\) is the \(\color{#D7A84B}U-\) dimensional zero vector, usually.
  • The weight of asset \(\color{#D7A84B}i\) is \(\color{#D7A84B}w_{i,t} = \frac{q_{i,t}v_{i,t}}{v_{p,t }}\) also.
  • Note \(\color{#D7A84B}o\in \mathcal O\) , \(\color{#D7A84B}o(w _{i,t{ }}) =\frac{o(q_{i,t })v_{i,t}}{v_{p,t }}\) changes only the quantities of the asset while \(\color{#D7A84B}w_{i,t+ 1} = M(w_{i ,t })=q_{i,t}M(\frac{v_{i,t}}{v_{p,t }})\) changes only prices

Filters/Strategies

• \(\color{#D7A84B}\eta ={\eta _i}_{i\in \mathbb Z[1,C ]}\) is the set of filter classes if there are \(\color{#D7A84B}C\) types of filters available.
• \(\color{#D7A84B}\gamma ={\gamma _i}_{i \in \mathbb Z[1, X ]}\) is the set of strategy classes if there are \(\color{#D7A84B}X\) types of strategies can implement
  • Strategies act on a single weight vector while filters act on the power set of weights.
  • The net filter is process \(\color{#D7A84B}w_{t_i}\rightarrow \mathcal F (w_{t_i})\) where \(\color{#D7A84B}\mathcal F=\prod_{t\in \tau}{\eta _t }\) with \(\color{#D7A84B}\eta _t\) being the filter at time \(\color{#D7A84B}t\) .
    • \(\color{#D7A84B}\eta _t \in \eta,\forall \,t\) where a filter from the given set is chosen at each step.
    • \(\color{#D7A84B}\eta _t:\mathcal{W}\rightarrow \mathcal{O}\) where \(\color{#D7A84B}\mathcal{W}\) is the space of weight vectors and \(\color{#D7A84B}\mathcal{O}\) is the space of operators
    • \(\color{#D7A84B}2^\mathcal{W }\subset2^{\mathbb { R}^U}\) is the powerset of this space, meaning all possible subsets of possible weights
• Assets have weights \(\color{#D7A84B}w_i\) , but strategies also have weights \(\color{#D7A84B}\alpha _j\) distributed over the strategies themselves
  • The weight vector for strategies \(\color{#D7A84B}\alpha_i\) is the decided trust in those strategies \(\color{#D7A84B}\gamma _i\)

• The net strategy acting on the portfolio is \(\color{#D7A84B}\gamma _p= \sum_{i=1} ^X \alpha _ i \gamma _i\)

• A signal time series \(\color{#D7A84B}z_t\) is any series which is derived from \(\color{#D7A84B}r_t\) for an asset.

• Each \(\color{#D7A84B}z_t\) can be associated to a strategy which belongs in the class \(\color{#D7A84B}\gamma _i\)
• A strategy is a mapping from a signal time series to a decision time series
• If \(\color{#D7A84B}o_{i,t},\,\forall \,i\in u\) is the decision series for an asset,
  • A strategy for a portfolio \(\color{#D7A84B}F = \oplus _{i=1}^ U o _{i ,t }\) is the direct sum of disjoint decision time series.
  • In general, \(\color{#D7A84B}o _{i,t} =f({ \eta_t}_{t=t_i} ^ {t } )\) meaning a combination of different strategies, which are executed in terms of their respective signals
• \(\color{#D7A84B}w\) is restricted sometimes to reduce the overall space of possible weights
• This is done by a projection operator \(\color{#D7A84B}\mathcal P_{ N }\in \mathbb R(N \times U)\) with the matrix elements \(\color{#D7A84B}\mathcal P_{ij}=\delta _{ij}\mathcal 1_{i \in \mathbb Z[0,N ]}\) , where \(\color{#D7A84B}n\subset u\) .
• Assets can have qualities which can be used to rank them or cluster them
• Asset \(\color{#D7A84B}i\) has quality \(\color{#D7A84B}\mathcal J\) such that \(\color{#D7A84B} i <j\) means \(\color{#D7A84B}J_i < J_j\) , where \(\color{#D7A84B}<\) is the ordering of the real numbers. \(\color{#D7A84B}u_J={\min_J u,...,\max_Ju}={u_{iJ}}_{i\in u}\) is the ordered set, where \(\color{#D7A84B}<\) induces an ordering on \(\color{#D7A84B}u\) . This is defined as the set \(\color{#D7A84B}u\) ordered by quality \(\color{#D7A84B}\mathcal J\)

Rank Operators

• For \(\color{#D7A84B}K < U\) , a rank operator can be defined which returns the assets that are within rank \(\color{#D7A84B}K\) as defined by the ordering implied by quality \(\color{#D7A84B}\mathcal J\)
  • A rank operator \(\color{#D7A84B}R_{ J,k} =\mathcal P_{{j:j<k}}{}\) is the selection of assets \(\color{#D7A84B}k\) with respect to quality \(\color{#D7A84B}\mathcal J\)
  • The ordering is from least to greatest, meaning the bottom \(\color{#D7A84B}k\) are returned by the above operator.
  • The top \(\color{#D7A84B}k\) to be returned \(\color{#D7A84B} \color{#D7A84B}R_{J,n-k} =\mathcal P_{{j: j \geq n-k}}{}\) are defined by reversing the ordering defined by \(\color{#D7A84B}\mathcal J\)
• Similarly, portfolios themselves can be acted on through orderings induced by a metric \(\color{#D7A84B}m\)
• For a set of \(\color{#D7A84B}k'\) portfolios, \(\color{#D7A84B}R_{m,k}\) returns the \(\color{#D7A84B}k \subset k '\) portfolios as ranked by the metric \(\color{#D7A84B}m\)

• Note threshold filters are just indicator functions over the set of assets/portfolios \(\color{#D7A84B}\mathcal 1_{i < J_0}\) for a given value of the aspect/metric.

  • Notice ranks and thresholds are equivalent to each
  • \(\color{#D7A84B}I_j = x, x\in \mathbb Z[0,u-1{]}\) is the indexing function of the ordered set and \(\color{#D7A84B}\mathcal 1_{I_j<J_0} \equiv\mathcal P_{i<j'}\) where \(\color{#D7A84B}j',J_{I_j'} < J_0 <J_{I_{j' }+1 }\) by definition and is unique.
    • Both are isomorphic to each other, since, \(\color{#D7A84B} i\Leftrightarrow J_i\) .

• A strategy is tested on many portfolios through a direct sum \(\color{#D7A84B} \eta ( \omega) \equiv \oplus_{i=1}^n\eta\) for \(\color{#D7A84B}n\) portfolios.

  • Assets have aspects while portfolios have metrics which induce an ordering respectively
  • The whole operation is a projection onto the space \(\color{#D7A84B}2^\tau \times 2^{\mathcal{R ^u}}\)

Strategy Classes

• The following strategies can be defined to act on the time period.

  • \(\color{#D7A84B}f_{mt ,R } (z_0) (k_{i,t})=1_{z_{i,R,t} > z_0}l_k+\color{#D7A84B}1_{z_{i,R,t} < -z_0} s _k+ I(1_{t<t_i + R})\) is momentum trending
  • \(\color{#D7A84B}{f_{mt , R }(z_0 )}_{R\in \mathbb Z[0 ],z_0 \in \mathbb R }\) is the set of all possible momentum trending strategies.
    • Note this is a decision time series \(\color{#D7A84B}o _{i,t}=f_{mt ,z_0,R } (k_{i,t})\) for asset \(\color{#D7A84B}i\) at time \(\color{#D7A84B} t\) .
  • \(\color{#D7A84B}\color{#D7A84B}f_{mv, R }(z_0 )=1_{z_{i,R,t} < -z_0}l_k+\color{#D7A84B}1_{z_{i,R,t} > z_0} s _k+ I(1_{t<t_i + R})\) is mean reversion
    • \(\color{#D7A84B}{f_{mv ,R }(z_0 )}_{R\in \mathbb Z[0],z_0 \in \mathbb R }\) is the set of all possible mean reversion strategies.
  • \(\color{#D7A84B}z_0\) is the threshold z-score, but it can be defined as 4 different quantities if the long/short entry/exit thresholds are required to be different.
    • \(\color{#D7A84B}z_0 \in \mathbb R^4\) .
  • Cross sectional quantities are defined for assets with time series \(\color{#D7A84B}k_t:[t_i,t_f]\rightarrow \mathbb R\) , such that \(\color{#D7A84B}C_t=f(\cup _{i\in \mathbb Z[U]}k_{i,t })\)
  • The cross sectional mean is \(\color{#D7A84B} \color{#D7A84B} \mu _{ t }= \sum_{i=1}^{U} \frac{ k _ { i,t}}{U}\) , the cross sectional standard deviation is \(\color{#D7A84B}\sigma _ { t } =\sqrt{\sum_{i=1} ^ {U} (k _ {i, t} - \mu _ {i, t} )^2}\) , and the cross sectional z score of asset \(\color{#D7A84B}i\) is \(\color{#D7A84B}z_{ i, t } =\frac{( k _ {i, t } - \mu _ { t } )}{\sigma _{ t }}\) . Note there is only one such quantity for a given \(\color{#D7A84B}t\) .
  • \(\color{#D7A84B}f_{c mv }(z_0)=1_{z_{i,R,t} < -z_0}l_k+\color{#D7A84B}1_{z_{i,R,t} > z_0} s _k\) is the cross mean reversion
    • \(\color{#D7A84B}{f_{cm v }(z_0 )}_{z_0 \in \mathbb R }\) is the set of all possible cross-sec mean reversion strategies.
  • Related rolling quantities are \(\color{#D7A84B}\color{#D7A84B}\text{Cov}(r _{i,R ,t}) =\sqrt{\sum_{n = t - {R }}^ t(k _ {i, n} - \mu _ {i, n , R} )(m _ {i, n} - \mu _ {m, n , R} )}\) and \(\color{#D7A84B}\beta _ {i,R ,t}=\frac{Cov_{i,R,t}}{\sigma _{ {i, R, t } }}\) .
  • There is also pairwise beta of assets with others. The beta of asset \(\color{#D7A84B}i\) with \(\color{#D7A84B}j\) is \(\color{#D7A84B}\beta _{ij}\) .

  • Taking as a rolling quantity, the residual of \(\color{#D7A84B}i\) with \(\color{#D7A84B}j,\epsilon _{ij}\) is defined \(\color{#D7A84B}v_{i,t}= \beta _{ij,R,t}v _ {j ,t} + \epsilon_{i,j ,R, t}\)

    • Cointegration is defined \(\color{#D7A84B}f_{co, R',R }(z_0,r_{i,t},r_{j,t} )\color{#D7A84B}\color{#D7A84B}=1_{z_{\epsilon_{ij} ,R,t} < - z_0}l_k+\color{#D7A84B}1_{z_{\epsilon _{ij },R,t} > z_0} s _k+ I(1_{t<t_i + R+R'})\)

  • The market portfolio implements strategy \(\color{#D7A84B}f_{I,t} = l_ k(1 _{t= t_i }) + I(1 _{t > t _ i })\) .

    • It acts on \(\color{#D7A84B}q_i =q_m\) with \(\color{#D7A84B}w_i=w_m=1\) . Note \(\color{#D7A84B}q = (q_m), w =(1)\) .

  Neutrality Conditions

  • In general, projections \(\color{#D7A84B}\mathcal P\) can be used to define neutrality conditions.
  • If \(\color{#D7A84B}A\in \mathbb R^{N\times { U } }\) where \(\color{#D7A84B}n \subset u\) , then \(\color{#D7A84B}A \lambda = \lambda a\) has solutions \(\color{#D7A84B}a _j\) such that \(\color{#D7A84B}\color{#D7A84B} A =\sum_i \lambda _ i a{a^T}\)
  • If \(\color{#D7A84B}a=0,P_ {{0}} =\sum_{i }^{|u |- | n }a_ia_i^T\) is the projection operator which describes the projection of the vector space on its null space.
  • \(\color{#D7A84B}\ker(A ) \neq 0\) is the condition for a non-trivial solution.
    • \(\color{#D7A84B} \ker (A) \,\cup \, n = u\)
• The interpretation is \(\color{#D7A84B}u\) assets have \(\color{#D7A84B}K=\dim\ker A\) const equations
• If \(\color{#D7A84B}w\) is the vector which is returned from the strategies acting on the market, it is constrained with a projection onto \(\color{#D7A84B} w^T a_i =\color{#D7A84B}w _p\) , gives the coefficients of \(\color{#D7A84B}w\) on that space
  • It is the best approximation of the raw strategies weights subject the constraints defined upon the portfolio

• Strategies give direction, not the weights.

• Any linearized operator \(\color{ }\color{#D7A84B}L(\sum _{i\in { }\mathbb Z[U]}c_i w_ i)= \sum _{i\in { }\mathbb Z[U]} c_ i L ( w_i )\) can be used to define a condition.

  • \(\color{#D7A84B} L( w _ P) = L(\sum _{i\in { }\mathbb Z[U]}c_i w_ i)=0\) if this condition is required. This is a demand on the portfolio.
    • If \(\color{#D7A84B}L = \begin{bmatrix}l_1 \ ...\ l_n\end{bmatrix}\) are the horizontal vectors of the matrix representation of this operator, \(\color{#D7A84B}l_i^T w=0\) is the requirement, written in terms of the inner product on this space.
  • \(\color{#D7A84B}r_P = r^T w\) is the return of the portfolio and \(\color{#D7A84B}r_M\) is the return of the market
  • \(\color{#D7A84B}\beta _P = \text{Cov}(r_P,r_M)=\text {Cov}(\sum _{i\in { }\mathbb Z[U]}r_i,r_M)=\sum _{i\in { }\mathbb Z[U]}w_i\text{Cov}(r_i, r_M) =\sum _{i\in { }\mathbb Z[U]}w_i\beta _i = \beta ^T w\)
  • To make the portfolio \(\color{#D7A84B}\beta -\) neutral means to make its correlation with respect to the market \(\color{#D7A84B}0\) , meaning \(\color{#D7A84B}\beta ^T w = 0\) .
  • Another is PC neutrality \(\color{#D7A84B}\Psi \lambda = \psi \lambda\) , with the covariance matrix, where \(\color{#D7A84B}\psi_1^Tw=0\) is the condition, \(\color{#D7A84B}\psi_1\) is the first element in the reverse ordered set of \(\color{#D7A84B}\psi_i\) , meaning the largest element.
  • Another is dollar neutrality \(\color{#D7A84B}1^Tw=0\) , where \(\color{#D7A84B}1 = (1,...,1)\) .

Metrics and Statistical Indicators

• All metrics are defined in terms of time periods.
  • The time period \(\color{#D7A84B}\tau=[t_i,t_f]\) is an element of the set of all possible closed intervals \(\color{#D7A84B}\tau_\infty\) defined on \(\color{#D7A84B}\mathbb {R}\) .

• Then, \(\color{#D7A84B}r_\tau:\tau_\infty \rightarrow \mathbb R\) , where \(\color{#D7A84B}r_{\tau}=\prod_{t=1}^T(1+r_{p,t} )\) where \(\color{#D7A84B}T=t_f - t_i\) , is the return over the time period.

  • \(\color{#D7A84B}\color{#D7A84B}\alpha _{\tau}=\prod_{t=1}^T (1+r_{p,t} )-1\) is the alpha.
  • \(\color{#D7A84B}\mu _{r,\tau}=(1+\alpha _\tau)^{\frac{1}{T}}-1\) is the average increase per time step
  • \(\color{#D7A84B}SR_{\tau } =\frac{\mu _\tau}{\sigma _\tau }\) with \(\color{#D7A84B}\mu _\tau = \sum_{t=t_i+ { }1}^{t_f}\frac{r_t}{ t_f-t_i}\) and \(\color{#D7A84B}\color{#D7A84B}\sigma _ { t } =\sqrt{ \sum_{t=t_i+ { }1}^{t_f} (r _ { t} - \mu _ { t} )^2}\) is the Sharpe

• Rolling quantities, \(\color{#D7A84B}\forall \,R\in \mathbb Z[ T]\) , are defined for a time series \(\color{#D7A84B}k_t:[t_i, t_f]\rightarrow \mathbb {R}\) for \(\color{#D7A84B}h_ t:[t_i+R,t_f]\) . In general, \(\color{#D7A84B}h_t = f(\cup _{t-R}^tk_t )\) , meaning it depends on the \(\color{#D7A84B}R\) quantities of the time series before \(\color{#D7A84B}t,\,\forall \,t\) .

  • The rolling mean for quantity \(\color{#D7A84B}i\) is a member of \(\color{#D7A84B}{\mu _{i, t , R}} _R\) with \(\color{#D7A84B} \color{#D7A84B} \mu _{i, t , R}= \sum_{n=t-R}^t\frac{ k _ {n, i}}{R}\)
  • There is a similarly defined set of standard deviation series \(\color{#D7A84B}\sigma _ {i, R, t } =\sqrt{\sum_{n = t - {R }}^ t(k _ {i, n} - \mu _ {i, n , R} )^2}\) and a z-score series \(\color{#D7A84B}z_{ i, t } =\frac{( k _ {i, t } - \mu _ {i, t , R } )}{\sigma _{{i, t , R }}}\)

• There is also exp weighted averages which are based on all time steps before \(\color{#D7A84B}t,\,\forall \,t\)

  • \(\color{#D7A84B}\text{ewm}( x_t ) =(1-\gamma ) x_ t + \gamma\ \text{ewm}(x _{t- 1})\) where \(\color{#D7A84B}\gamma \in [0,1]\) is the exponential weighted moving average.
    • \(\color{#D7A84B}x _ 0\) is given, \(\color{#D7A84B}\text{ewm}(x_0 ) = x_0\) .
    • This means \(\color{#D7A84B}\text{ewm}(x _1 ) = (1-\gamma )x_1 + \gamma x_0\Rightarrow\) \(\color{#D7A84B} \color{#D7A84B}\text{ewm}(x _2 ) = (1-\gamma )x_2 + \gamma x_1 = \color{#D7A84B} (1-\gamma )x_ 2 + \gamma ( (1-\gamma )x_1 + \gamma x_ 0) = (1-\gamma )x_ 2 + \gamma ( x_1-\gamma x_1 + \gamma x_ 0) =(1-\gamma )x_ 2 + (\gamma -\gamma^2) x_1 + \gamma^2 x_ 0)\) means \(\color{#D7A84B}\sum_{t= 1}^i w_t= \gamma ^i + (1-\gamma ) + (1-\gamma )\gamma + ... =\sum_{t=1}^i{}{}\gamma ^ {i-t}{{}}(1-\gamma )=\sum_{t=1}^i{}{}-\gamma ^ {i+1-t}{{}} + \sum_{t=1}^i{}{}\gamma ^ {i-t}{{}}= -\gamma ^ {i }{{}} +\gamma ^0 = 1-\gamma ^i\)
    • As \(\color{#D7A84B}i\rightarrow \infty ,\gamma ^i\rightarrow 0,\sum_{t } w_ t = 1\) .

• The market is also represented as different states and exhibits significantly different behavior dependent on the state.

• Along with the return series, the probability distribution at a given time can be defined as \(\color{#D7A84B}P(r_t|r_{t-1} \,\cap \, S_t )\) where \(\color{#D7A84B}S_t\) is the state of the market. This represents the return as a conditional distribution
• \(\color{#D7A84B}IC_{t+1} = \text {Cov}{}(z_t,r_{t+1 } )\) is the information coefficient of the signal \(\color{#D7A84B}z_t\) , meaning its predictive power.