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					International Finance                       Dr. Angela Ng
FINA 342                                          HKUST



                        Class Notes 6

 INTERNATIONAL PARITY CONDITION II:
  COVERED AND UNCOVERED INTEREST
           RATE PARITY

I. COVERED INTEREST RATE PARITY

EXAMPLE

 Mr. David Sylvian, an arbitrageur for the Hong Kong &
  Shanghai Banking Corporation in Hong Kong, arrives at
  work on Tuesday morning to be faced with the following
  quotations on his Reuters’ screen:
  Spot rate: ¥125.00/$
  One year forward rate: ¥130.00/$
  Euroyen one year interest rate: 10%
  Eurodollar one year interest rate: 5%

   What should Mr. Sylvian do?

 Solution:




                              1
DERIVATION OF A GENERAL EXPRESSION FOR
COVERED INTEREST RATE PARITY

 Notations:

  it = U.S. interest rate

  it* = “comparable” foreign interest rate

  St = spot exchange rate ($/FC)

  Ft+1 = forward exchange rate ($/FC)

 An investor has two options:

  1. Invest in U.S.  $ terminal wealth = 1+ it

  2. Invest abroad  $ terminal wealth = (1/St)(1+it*)Ft+1

  To avoid arbitrage opportunities, we need equality of dollar
  returns so that one statement of interest rate parity, with
  exchanges rates expressed in “American terms”, is
                                     Ft 1 (1  i* )
                        (1  i t )              t
                                            St

  Dividing both sides of the above expression by (1+it*) gives
                               Ft 1 1  i t
                                    
                               St     1  i*
                                           t



  Now subtracting 1 from both sides gives
                            Ft 1  St i t  i*
                                              t
                                St      1  it*




                                     2
 Alternatively, the covered interest rate parity can be written
  as:
                         Ft  1  i 
                                         t

                                   
                         S0  1  i* 
  where i and i* are the geometric mean US and foreign
  interest rates over the t periods.

 Forward premiums and discounts are entirely determined by
  interest rate differentials.

 Suppose StFC/$ is the spot rate in European terms and Ft+1FC/$
  is the forward rate in European terms. How would the interest
  rate parity formula look like?




 Exercise: Spot (FF/$) = 7.4825
            90-day Forward = 7.5650
            90-day Eurodollar interest rate = 8.2% p.a.
            90-day Eurofranc interest rate = 12.7% p.a.
  Does the covered interest rate parity hold?




                               3
 If Ftd/f/S0d/f > [(1+id)/(1+if)]t, then Ftd/f must fall, S0d/f must rise,
  id must rise, and/or if must fall.
   Sell FC at Ftd/f. Buy FC at S0d/f. Borrow at id. Lend at if.

   If Ftd/f/S0d/f < [(1+id)/(1+if)]t, then Ftd/f must rise, S0d/f must fall,
   id must fall, and/or if must rise.
    Buy FC at Ftd/f. Sell FC at S0d/f. Lend at id. Borrow at if.

   Example: i$ = 7%             i£ = 3%
            S0$/£ = $1.20/£     Ft$/£ = $1.25/£
   How could one make a riskless arbitrage profit?




 Covered interest arbitrage is the profit-seeking activity that
  forces interest rate parity to hold.

 Interest rate parity line (see Exhibits 6.1 & 6.2)




                                     4
RELAXING THE BASIC ASSUMPTIONS

 Transaction costs

  If you are holding a dollar asset, foreign investment requires
  four transaction costs:
  1. Brokerage fees on sale of dollar security  t
  2. Buy spot exchange (1/2 bid-ask spread)  tS
  3. Buy foreign security – brokerage fee  t*
  4. Sell forward (1/2 bid-ask spread)  tF

  How do you construct the neutral band within which covered
  interest arbitrage transactions will not occur, when
  transaction costs are present?

  Illustration: Exhibit 6.3

 Taxes

  Consider the case where income is taxed at a rate of y and
  capital gains at k. On an after-tax basis, the parity condition
  is modified as
                 Ft 1  St              it  it*
                            (1   k )           (1   y )
                     St                  1  it*
  or
                       Ft 1  St it  it* 1   y
                                 
                           St      1  it* 1   k

  Illustration: Exhibit 6.4

 Uncertainty




                                     5
IRP WITH BID-ASK SPREAD

 Notations: Spot rate ($/¥)          St b – St a
             Forward rate ($/¥)       Ft,Tb – Ft,Ta
             Interest on $            iTb – iTa
             Interest on ¥            iTb* – iTa*

 To see whether there is a riskless profit, there are two
  approaches.

  (1) Borrow $
      Borrow $ at the rate of iTa.
        Get $ 1/(1+ iTa)
      Convert this $ amount into ¥
        Have ¥ 1/[(1+ iTa)Sta]
      Invest this ¥ amount
        Obtain ¥ (1+ iTb*)/[(1+ iTa)Sta]
      Sell all of the ¥ amount in the forward market
        End up with $ [(1+ iTb*) Ft,Tb]/[(1+ iTa)Sta]

      So, if this final amount is great than 1, then there is a
      profit opportunity. To eliminate arbitrage, we require
                                 1  ia
                      F S 
                        b
                       t,T
                             a
                             t
                                      T
                                          *
                                 1  iT
                                      b




                                 6
  (2) Borrow ¥
      Borrow ¥ at the rate of iTa*.
        Get ¥ 1/(1+ iTa*)
      Convert this amount into $
        Have $ Stb/(1+ iTa*)
      Invest this $ amount in the US
        Obtain $ Stb(1+ iTb)/(1+ iTa*)
      Sell this amount of $ forward
        End up with ¥ Stb(1+ iTb)/ Ft,Ta(1+ iTa*)

      To eliminate arbitrage, we require
                                  1  iT
                                       b
                       F S 
                         a
                        t,T
                              b
                              t            *
                                  1  ia
                                       T



IMPORTANCE OF IRP

 One-way arbitrage:

  One-way arbitrage is simply picking the lowest-priced
  alternative when buying or the highest-priced alternative
  when selling.

  Example: Consider an investor who holds US$ cash and
  wants to make a DM payment in 3 months. The investor is
  given the following quotations:
  Spot $/DM: 0.6793 – 0.6803
  3-month Forward $/DM: 0.6800 – 0.6830
  3-month Euro DM: 4.00 – 4.125% p.a.
  3-month Euro $: 6.0625 – 6.1875% p.a.

  Can the investor make a one-way arbitrage profit?




                                  7
   The search for one-way arbitrage opportunities will limit
    the possibility of round-trip, covered interest arbitrage
    profits.

 Cost of hedging for firms:

  Consider a manager wishes to own the Japanese yen in 6
  months from now. The following two hedging strategies are
  possible:
  1. Bank forward contract
  2. Money market hedge

  Firms with lower credit rating and firms with lower
  borrowing capacity prefer to hedge using a bank forward.

 Country risk:

  IRP provides a market determined measure of political risk
  differences between countries. To evaluate risky cash flows
  from foreign countries, one needs a risk-adjusted discount
  rate. IRP provides a market determined measure of this
  discount rate.




                               8
II. INTERNATIONAL FISHER EFFECT

 Fisher Relation: (1  i)  (1  r)(1  p)

   Nominal returns are part real return and part inflation.

 Uncovered Interest Rate Parity or International Fisher Effect
  states that returns in different countries’ money markets
  should be equalized “in an expected sense.”

DERIVATION OF A GENERAL                             EXPRESSION           FOR
INTERNATIONAL FISHER EFFECT

 Notations:

   it = U.S. interest rate

   it* = “comparable” foreign interest rate

   St = spot exchange rate, in American terms ($/FC)

 An investor has two options:

   1. Invest in U.S.  $ terminal wealth = 1+ it

   2. Invest abroad and covert at future spot  $ terminal
      wealth = St+1(1+ it*)/St

   International Fisher Effect (in American terms) states:
                                   S                  E[St 1 ]
              1  i t  E (1  i* ) t 1   (1  i* )
                                 t                  t
                                    St                  St

               E[St 1 ] 1  i t              E[St 1 ]  St i t  i*
                                   or                             t
                 St       1  i*
                               t                   St         1  it*




                                       9
 What would be the formulation for the International Fisher
  Effect in European terms?



 Foreign investment proposed in the IFE contains exchange
  rate risk.

 When the U.S. interest rates are higher than foreign interest
  rates, the dollar is expected to depreciate, and vice versa.

 International Fisher and PPP combined: real interest rate
  parity

  Based on PPP, the expected percentage change in the
  exchange rate is
      E[St 1 ]  St E[Ptus1 ]  Ptus E[Ptf1 ]  Ptf
                        
                             us
                                             f
                                                       p us1  p ft 1
                                                          t
           St             Pt               Pt
  where pt+1us and pt+1f are the expected inflation rates.

  Combining with the International Fisher Effect gives
                             p us  p f  i  i*
  Thus, the nominal interest differential is equal to the expected
  inflation differential.

 Alternatively, if we assume that real interest rates are equal
  across countries, then the Fisher relation implies
                              1  i 1  p us
                                    
                              1  i* 1  p f
  Measured over t periods, the relation becomes
                                                   t
                          1  i  1  p 
                                  t        us
                                          
                          1  i*   1  p f 
                                             


                                      10
 The IFE does not hold at any particular point in time, but
  does hold in the long run.

THE FORWARD RATE UNBIASED CONDITION

 Recall that the International Fisher Effect is given by
                                         1  it
                          E[S t 1 ]           St
                                         1  i*
                                              t

  This is equivalent to
                              Ft  E[St ]
  Thus, the forward rate is an “unbiased predictor” of the future
  spot rate.

 If speculators think the spot rate will close above the forward
  rate, they can buy the forward contract and settle in the spot
  market at a positive expected profits. This type of speculative
  activity forces the forward price to rise and ensures that
  forward exchange rates are close to expected future spot
  rates.

 Consider a 90-day forward exchange contract to purchase the
  British pounds. Think of an investor as intentionally
  speculating, that is, as wanting to bear the risk. If you
  contract to buy pound forward at $1.50/£ and you do not hold
  any pounds, then you are taking on exchange risk. You will
  profit if the future spot exchange rate is greater than $1.50/£,
  because you could sell the pounds for more than it takes to
  buy them. Your profit at time t+90 from a contract entered at
  time t is St+90 - Ft,90. (See Exhibit 6.5)




                                    11
 If the forward rate is an unbiased predictor of the future spot
  exchange rate, the forecast error made by the forward rate has
  a mean of zero.

  This has two important practical implications:

  1. When you make a sequence of purchases in the forward
     market over a long time period, profits average out to zero.

  2. At any time t, we can describe the possible future spot
     rates in any 90 days by a probability distribution. This
     gives us the probability of profits and losses.




                               12
III. INTERNATIONAL PARITY CONDITIONS: WRAP-
     UP

 Summary

   PPP

      Absolute version: link exchange rate and price ratio

      Relative version: link inflation rates and exchange rate
        changes

   Interest Rate Parity

      Covered Interest Rate Parity: link forward premium and
        interest differential

      Uncovered Interest Rate Parity or International Fisher
        Effect: link expected exchange rate changes and interest
        rates

   What about links between interest rates and inflation?

 The PPP, CIRP and IFE relations are sometimes referred to as
  the International Parity Conditions. (See Exhibit 6.6)

  If they all hold simultaneously, they imply equal real interest
  rates across countries.




                               13
                                                       Exhibit 6.1
                                         The Interest Rate Parity Line
                                      Equilibrium and Disequilibrium Points




Note: Points A and B are on the interest rate parity line. Points off the parity line generate economic incentives for capital
to flow out of investments in one currency and into another currency. The arrows indicate the marginal impact of
arbitrage transactions on prices of F, S is, and iforeign.




                                                       Exhibit 6.2
                            Covered Interest Rate Parity in European Terms

                                                   i*  i t
                                                    t                                             45° line
                                                   1 it



                          Region of Arbitrage




                                                                                                           Ft  S t
                                                                                                              St

                                                                     Region of Arbitrage




                                                              14
                                                     Exhibit 6.3
                                         The Interest Rate Parity Line
                                     Transaction Costs and the Neutral Band




Note: Points A, A”, and B are inside the neutral band and considered equilibrium points with no arbitrage profit
possibility. For points A’, B’ and B”, the profit opportunity is greater than the cost of executing arbitrage transactions.


                                                     Exhibit 6.4
                                           Interest Rate Parity Lines
                                      Pretax (PT) and After-Tax (AT) Lines




                                                            15
                          Exhibit 6.5
Speculative Long Position of a £ Forward Contract

Profit per unit of £




 0.10

                       1.40
                                                    Future Spot
                                                    Rate ($/£)
                                   1.50   1.60

 -0.10




                              16
                              Exhibit 6.6
                     International Parity Conditions


                          Forecast Change in
                          Spot Exchange Rate



          (E)                                            (A)




Forward Premium on                                     Forecast Difference
 Foreign Currency                  (C)                  in Inflation Rates




           (D)                                            (B)



                          Difference of Nominal
                              Interest Rates




                                   17

				
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