經濟,財務,統計學,數理科學與政治評論

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  這裡要介紹常用的parities與其背後的套利機制!!一般說來當用外國貨幣換算成計價貨幣時,這種轉換被稱為直接報價(a direct quote);而當我們將計價貨幣換成外國貨幣時,這種轉換被稱為間接報價(an indirect quote)!!一般在外匯市場中因為參與者多為銀行,都有買賣價差,因此我們知道(A:B)(bid or ask)為A貨幣轉換成B貨幣的買價/賣價!!!

(1)匯率買價/賣價計算的基本元素與三角套利(triangular arbitrage)

  a.已知(A:B)(bid or ask),(B:C)(bid or ask),則(A:C)(bid) = (A:B)(bid) * (B:C)(bid)與(A:C)(ask) = (A:B)(ask) * (B:C)(ask)

  b.今天(A:B)(bid or ask),(A:C)(bid or ask)為已知,則我們知道(B:C)(bid) = (A:C)(bid)/(A:B)(ask)與(B:C)(ask) = (A:C)(ask)/(A:B)(bid)

  c.triangular arbitrage的條件為今天(A:B)(bid or ask),(A:C)(bid or ask),(B:C)(bid or ask)為已知,

    1.若(A:C)(bid)/(A:B)(ask) > (B:C)(bid)且(A:C)(ask)/(A:B)(bid) > (B:C)(ask),則B貨幣匯率被高估,可進行三角套利使其回復均衡!!

    2.若(A:C)(bid)/(A:B)(ask) < (B:C)(bid)且(A:C)(ask)/(A:B)(bid) < (B:C)(ask),則C貨幣匯率被高估,可進行三角套利使其回復均衡!!

  d.bid/ask spread = (ask - bid)/ask

  e.(A:B)(bid) = 1/(B:A)(ask), (A:B)(ask) = 1/(B:A)(bid)

(2)Forward Rate issues

  a.Forward rate premium or discount = [Forward rate of n months - spot rate] * (12/n) * 100%

  b.indirect quote of (A:B)(bid or ask) is given, the risk-free rate of A is RaL-RaU and the risk-free rate of B is RbL-RbU,

    Bid Forward exchange rate = (A:B)(bid) * [(1 + RbL)/(1 + RaU)]

    Ask Forward exchange rate = (A:B)(ask) * [(1 + RbU)/(1 + RaL)]

(3)Covered interest rate parity

  (採用Forward contract,bounded by the arbitrage)

  先用direct quote取得spot rate的資訊,然後求算forward rate..

  Forward rate

  = Spot rate*[1+r(dosmestic)*(n/360)]/[1+r(foreign)*(n/360)]

  where

    n is the expiration days for the forward contract

    r(.) is the risk-free fate of respective country

<-> [Forward rate]/[Spot rate]

  = [1+r(dosmestic)*(n/360)]/[1+r(foreign)*(n/360)]

<-> (Forward rate - Spot rate)/Spot rate

  = [r(domestic) - r(foreign)]*(n/360)/[1+r(foreign)*(n/360)]

i.e., if the equality for the spot-forward relation does not hold, then the riskless arbitrage exists!!

(4)Relative Purchasing power parity

  Absolute PPP指的是一籃子的物品在兩個不同國家或地區的價格是一致的!!(the law of one price)

  Relative PPP則指的是direct quote spot rate S,計價國家預期通膨率EI(domestic),外國預期通膨率EI(foreign),則T天後的期望匯率為

  E(S(T)) = S * [1+EI(dosmestic)*(T/365)]/[1+EI(foreign)*(T/365)] (or other equivalent expression)

(5)International Fisher Effect

   傳統的Fisher effect: (1+名目利率) = (1+實質利率) * (1+通貨膨脹率)

   因此  (1+名目利率)/(1+通貨膨脹率) = (1+實質利率)

   與 實質利率大約為(名目利率扣除通貨膨脹率)!!!

   而國際Fisher effect : 實質利率(本國) = 實質利率(外國)

   (1+名目利率(本國))/(1+通貨膨脹率(本國)) = (1+名目利率(外國))/(1+通貨膨脹率(外國))

    i.e., R(domestic) - R(foreign) = E(I(domestic)) - E(I(foreign))

   當internaitonal Fisher effect確實為真時,當一個國家的名目利率比其他國家高很多很多的時候,因為各國的實質利率是相同的,表示這個國家的通貨膨脹率非常非常高,因此我們預期它的direct-quote匯率會降低很多!!!

(6)Uncovered interest rate parity

  (採用expected spot rate in the future time, not bounded by the arbitrage)

  已知 S 為direct-quote spot rate,

       Inflation(foreign)為外國的通膨率,

       Inflation(domestic)為本國的通膨率,

       R(domestic)為本國的名目利率,

       R(foreign)為外國的名目利率,

  則T天後的期望匯率為

 E(S(T)) = S * [1+E(Inflation(domestic))]/[1+E(Inflation(foreign))]

          (by the relative PPP)

         = S * [1 + R(domestic)]/[1 + R(foreign)]

          (by the international Fisher effect)

注意: Uncovered interest rate parity

      = International Fisher effect + relative PPP

因此當外國的通膨率比本國高時,我們會預期direct-quote的匯率是下跌的,亦即一塊錢外幣能夠換到的參考貨幣是減少的!!

值得一提的是相對於Covered interest rate parity,我們知道Uncovered interest rate parity中隱含Forward rate is the unbiased estimate of future spot rate, F = E(S(T))!!!