Location Arbitrage

1
Location Arbitrage

• arbitrage in the trading of two currencies
• example Canadian dollar quote (U.S./Cdn)
• Bank A Bank B
• Bid Ask Bid Ask
  
• 0.6875 0.6879 0.6871 0.6874
• buy Canadian dollars from Bank B and sell to Bank
  A

2
Location Arbitrage

• with 1m U.S. your profit is
• demand for Canadian dollars from Bank B will
  pressure the bid-ask up
• supply of Canadian dollars from Bank A will
  pressure the bid-ask down
• differences may still persist but generally
  competition and arbitrage keep rates in line

3
Location Arbitrage

• arbitrage exists when bid-ask bands do not
  intersect
• arbitrage works to re-align quotes

0.6879
0.6875
0.6874
0.6871
4
Location Arbitrage

• the following is not an arbitrage opportunity
• competition (through supply and demand) works to
  re-align quotes

0.6879
0.6876
0.6875
0.6872
5
Triangular Arbitrage

• arbitrage in the trading of three currencies
• cross-rate arbitrage
• obtain implied cross-rate bid-ask
• compare to quoted cross-rate bid-ask
• if bands do not intersect there is an arbitrage
  opportunity
• arbitrage works to re-align quotes
• if bands do not align competition works to
  re-align quotes

6
Triangular Arbitrage

• Bank A Canadian dollar quote
  (U.S./Cdn)
• Bid Ask
• 0.6875 0.6879
• Bank A British pound quote (U.S./pound)
• Bid Ask
• 1.5375 1.5385

7
Triangular Arbitrage

• determining implied cross-rate bid-ask
• we want Cdn/pound
• bid
• sell Cdn and buy pounds
• ask
• buy Cdn and sell pounds

8
Triangular Arbitrage

• given the implied cross-rate bid-ask we can
  compare it to the quoted bid-ask
• if quote is 2.2320-2.2348 an arbitrage
  opportunity exists
• arbitrage re-aligns the quotes
• if quote is 2.2340-2.2368
• competition re-aligns the quotes

9
Covered Interest Arbitrage

• arbitrage in the following three markets
• spot and forward FX market
• domestic money and bond market
• foreign money and bond market
• leads to interest rate parity (IRP)
• indifferent to investing in domestic or foreign
  market
• covered currency adjusted returns are identical

10
Covered Interest Arbitrage

• definitions
• S spot units of domestic currency per unit of
  foreign currency
• F forward units of domestic currency per unit
  of foreign currency
• iH domestic interest rate commensurate with
  forward
• iF foreign interest rate commensurate with
  forward

11
Covered Interest Arbitrage

• we have derived the following
• note if quote is U.S./Cdn home is U.S.
• arbitrage ensures the three markets are aligned
• with bid-ask spreads we end up with a band
• arbitrage is precluded if quotes intersect with
  the band
• neutral band quotes do not produce arbitrage

12
Interest Rate Parity

• we have derived the following
• divide both sides by S
• subtract 1.0 from both sides

13
Interest Rate Parity

• forward premium p
• if home interest rate is greater than foreign
  interest rate the premium is positive
• F gt S
• if home interest rate is less than foreign
  interest rate the premium is negative
• F lt S

14
Interest Rate Parity

• example 6-month forward
• S 0.6875 U.S./Cdn
• iH 0.04 (8.16 annual) U.S. interest rate
• iF 0.05 (10.25 annual) Canadian interest rate
• forward price

15
Interest Rate Parity

• forward premium
• this is the premium required to keep investments
  in the two countries consistent
• plot interest rate differential against the
  forward premium
• intercept should be zero
• slope should be one

16
Interest Rate Parity
Int. differential
arb
arb
premium
arb
arb
17
Empirical Results

• testing interest rate parity requires a test of
  whether the condition holds
• we know there are bid-ask spreads
• this results in a band of acceptable quotes
• if the premium falls within the band interest
  rate parity holds
• check for deviations outside band
• it should hold if markets are aligned

18
Empirical Results
Int. differential
arb
arb
premium
arb
arb
19
Empirical Results

• early tests
• used T-bill data of two countries
• found many deviations
• perhaps band was too tight
• did not take into consideration all costs
• more likely risks were different
• country risk (expropriation)
• liquidity risk
• credit risk

20
Empirical Results

• modern tests
• used Eurocurrency deposits
• country risk?
• liquidity risk?
• credit risk?
• found most results fell within neutral band
• much better adherence to theory than T-bills

21
Interest Rate Parity

• interest rate parity should always hold
• if countries have open financial mobility
• country risk is minimized
• if countries have similar credit risk
• if financial markets are liquid
• otherwise differences in parity relation is due
  to risk
• Eurocurrency markets help to equalize risk
• do they eliminate the risk?

22
Purchasing Power Parity

• in one year we have
• price in Canada is PAF1 105
• price today in U.S. is PAH1 70.72
• for there to be one price the new exchange rate
  is
• S1 0.6735 U.S./Cdn

23
Purchasing Power Parity

• percentage change in exchange rate

24
Purchasing Power Parity

• if IH gt IF then e gt 0 and foreign currency
  appreciates
• if IH lt IF then e lt 0 and foreign currency
  depreciates
• from above example
• IH lt IF 0.04 lt 0.05
• e -0.0096
• foreign goods became more expensive
• exchange rate falls to keep prices equal

25
Purchasing Power Parity

• tests of purchasing power parity usually fail
• problem 1 relies on tests of CPI
• if consumption baskets were the same in two
  countries this makes sense
• but there are significant differences
• comparing two indices will not yield results
  wanted
• in the long-run baskets become more homogeneous

26
Purchasing Power Parity

• problem 2 trade barriers
• price may not equalize if there are trade
  barriers
• quotas and subsidies cause frictions as well
• transportation costs also keep prices from
  equalizing
• in the long-run PPP should overcome these
  frictions

27
Purchasing Power Parity

• problem 3 wealth and interest rates
• capital flows can also influence exchange rates
• inflation may push in one direction but capital
  flow could offset the pressure
• in long-run again we expect some move to
  equalization
• PPP is more a long-run phenomena

28
International Fisher Relation

• attempts to correct for problem 3
• this is an equilibrium condition
• not a forward rate
• tells what we should expect to happen to the
  exchange rate not what rate we can lock in
• based on the premise that real rates of return
  are equal across countries

29
International Fisher Relation

• suppose there is only type of investment risk
• current exchange rate is S0 0.68 U.S./Cdn
• suppose Cdn inflation over the year is IF 0.05
• suppose US inflation over the year is IH 0.04
• these are expected inflation figures
• let r equal the real rate of return for this
  investment risk

30
International Fisher Relation

• the interest rate in the U.S. is 7.5
• the interest rate in Canada is 9.1
• consider a U.S. investor
• if they invest in the U.S. the real return is

31
International Fisher Relation

• if they invest in Canada the real return is
• of course real returns must equalize

32
International Fisher Relation

• rearranging we get
• in our example we have e -0.01467
• this is different than the PPP calculation
• here capital flows drive equilibrium
• in PPP goods prices drive equilibrium
• which would prevail?

33
International Fisher Relation

• empirical tests are not conclusive wrt IFR
• are the two securities consistent
• are the risks consistent
• both PPP and IFR are equilibrium arguments
• long-run scenarios are more consistent

34
LIBOR

• London Interbank Offered Rate
• interest offered by banks to other banks on
  deposits
• Eurocurrency market
• rate reflects average riskiness of banks
• different rate for each maturity
• one-month LIBOR
• three-month LIBOR
• six-month LIBOR

35
Floating-Rate Notes

• consider a two year floating rate note issued at
  par
• coupons are semi-annual and based on LIBOR
• principal value is 100
• we need to understand the cash flows

36
Floating-Rate Notes

• suppose the following rates materialize
• 6-month 6-month cash
• date LIBOR flow
• 0.0 0.0506
• 0.5 0.0517 2.530
• 1.0 0.0537 2.585
• 1.5 0.0558 2.685
• 2.0 0.0527 102.790

37
Floating-Rate Notes

• how do we value floating rate notes
• look at problem recursively
• first consider value at reset dates
• start with last reset date (t1.5)
• note spot and LIBOR are as observed at time 1.5

38
Floating-Rate Notes

• at reset date 1.5 the note is worth 100
• now consider second last reset date (t1.0)
• we get the general result that on each reset date
  the floating rate note is worth par

39
Floating-Rate Notes

• how do we value floating rate notes between
  resets
• with above note we know that at time 0.5
• note is worth 100 at reset
• note pays coupon 2.530
• suppose we are at time 0.15
• we observe

40
Floating-Rate Notes

• the value of the floating-rate note is
• we always use the following facts between reset
  dates
• value of note at next reset par
• we know the exact cash flow for next coupon
• valuation is straightforward
• think of note as zero-coupon instrument

41
Summary

• completed chapter 7
• good questions 2,5,8,14,16,17
• completed ch. 8
• good questions 9, 23, 24, 25
• next lecture
• futures Ch. 5
• swaps