經濟,財務,統計學,數理科學與政治評論

寫在前面:我們會將Econometrics與Finance中常強調的幾個重要parities一起在這個部份討論!!

Hello there:

  這裡要談傳統上面所重視的投資組合建構與其相關注意的事項!!

 I. Mean-Variance Portfolio construction

 (*) Markowitz生在1952年所建構的模型,她根源於利用個別資產的預期報酬率,報酬率變動的標準差,以及資產兩兩報酬率的相關性,來計算最適的投資組合,而最適的投資組合滿足下列幾個條件:

  a.Porfolio A is mean-variane dominates Portfolio B if they have the same mean return and A has less variance than B.

  b.Portfolio A is mean-variance dominates Portfolio B if A and B have same variances but the mean return of A is larger than B.

 MVP 還強調選擇資產數量夠多時,其投資組合與Mean covariace是相近的,原因是當選擇投資組合中資產個數為n時,我們知道

  Variance(Portfolio) = [1/n] * mean Variance of individual asset + [(n-1)/n] * mean Covariance of assets

  as n gets larger, we have the variance of portfolio is nearly the same as the man Covariance of assets!!!

  Fisher與Lorie (1070)發現從1962到11965大約需要43支股票就能進行NYSE-traded stocks的風險分散與投資組合建構,但到Campbell, Lettau, Malkiel與Xu(2001)發現雖然總體市場波動度相對於過往並沒有顯著增加,但影響個別股票與特定產業的因素數目卻是增加的,隨著股市多多元性增加許多,因此在1986-1997的實證中發現大約需要50支的股票才能進行風險分散與投資組合建構!!

  但是MVP的不可行(non-feasible)來自於the instability of efficient portfolio frontier,主因在於不穩定的投資標的與市場投資組合的相關程度會於不同時點的投資是有關的,以及投資人受限於自身資金與風險屬性的不同,很難有一致性的MVP!!

   下列一些名詞得有賴您自行閱讀:CAL(heterogeneous expectation with the idea of same market price of risk),CML(Homegeneous expectation with the same market price of risk as the equilibrium condition)...

 (**) CAPM的起源 -- the market model:

   R(i) = Alpha(i) + Beta(i) * R(M) + Error(i)

   with E(Error(i)) = 0, Corr(Error(i),Error(j)) = 0 for different i and j

  有一些重要計算結果:(Unadjusted Beta)

   E(R(i)) = Alpha(i) + Beta(i) * E(R(M))

  Var(R(i)) = [Beta(i)] * [Beta(i)] * Var(R(M)) + Var(Error(i))

   Cov(R(i),R(j)) = Beta(i) * Beta(j) * Var(R(M))

   Corr(R(i),R(j)) = Cov(R(i),R(j))/[SD(R(i)) * SD(R(j))]

   因此我們才知道原來這裡面的Beta是不穩定的,主因在於Beta有mean-reverting的效果,而這也是導致MVP理論不穩定的原因,因此我們必須有adjusted beta

   adjusted Beta = A0 + A1 * unadjsted Beta

   其中 A0 + A1 = 1,一般A0 = 1/3, A1 = 2/3.

  從這裡出發開始談CAPM........

 II.CAPM(The Capital ASset Pricing Model) and APT(Arbitrage Portfolio Theory)

 (1) 在前面我們有提到MVP,而MVP是從投資組合來看,因此我們相對上重視的是CML(Capital Market Line),考量到整體的風險;而CAPM的概念主要與SML有關,指考量到與系統風險有關的部分,再下面我們會提兩個均衡式子來看這兩件事情....

  a. CML with the euilibrium market price of risk:

     [E(R(i)) - RFR]/SD(R(i)) = constant for all portfolios = Sharp Ratio 

  where

        RFR: the risk-free rate of return

        SD(.): the standard deviation of the specified portfolio

  b. SML with the equilibrium idea associated with the beta

     [E(R(i)) - RFR]/Cov(R(i),R(M)) = [E(R(M)) - RFR]/Var(R(M))

     [E(R(i)) - RFR] = Beta(i) * [E(R(M)) - RFR]

     where

         Beta(i) = Cov(R(i),R(M))/Var(R(M))

         Cov(.,.): the covariace of both....

         Var(.): the cariance of ....

         E(R(M)) - RFR : the market risk premium

  值得一提的是CAPM討論的是單期單市場風險因子的均衡條件,需要homogeneous expectation of all investor與unlimited borrowing-lending condition,背後透露出我們能夠透過投資組合的分散只需承擔系統風險而不必承擔非系統風險!!

  因此選擇新投資標的時必須滿足:

  [E(R(new)) - RFR]/SD(R(new)) > {[E(R(old Portfolio) - RFR]/SD(R(old Portfolio))} * Corr(R(new), R(old Portfolio)) 

 (2) APT(Arbitrage Pricing Theory)

  APT主要談的是non-arbitrage opportunity所產生的均衡.....

  E(R(portiolio)) = RFR + Sum of (Factor loading or sensitivity * the risk premium of portfolio to specified factor)

   通常 RFR(the risk free rate)與the risk premium 是需要計算的!!

   自行參閱:CAPM與APT的異同,與其相關解釋說明!!