Covered Interest Parity and Uncovered Interest Parity

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Covered Interest Parity and Uncovered Interest
Parity

• International Finance
• Dick Sweeney

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1. Covered Interest Parity

• Covered Interest Parity Has to hold in the
  absence of covered interest arbitrage profits.
  CIP is a current-time no-arbitrage condition. No
  looking to the future, only the present.
  Everything is at time t
• F1t / St (1 iusd) / (1 ieur)
• see "Covered Interest Arbitrage The Basics
  (Raul's Problem)."
• Data Current spot rate St USD 1 / EUR (iusd
  and ieur in these formulas are in decimal form)
• Example 1 1-year forward rate One year deposit
  rates iusd ? 5/year, ieur ? 4/year
• (1 iusd) / (1 ieur) (1.05) / (1.04) (1
  5/100) / (1 4/100) 1.0096154 (? 1.01)
• F1t / St (1 iusd) / (1 ieur) ?
• F1t (1 iusd) / (1 ieur) x St 1.0096154
  x USD 1 / EUR USD 1.0096154 / EUR
• ? 1-year forward rate (you lock in a higher price
  in USD/EUR one year out than the spot price)
  

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1. CIP (cont.)
Example 2 1-month forward rates One month
deposit rates iusd ? 5/year, ieur ? 4/year 1
(1 iusd) / (1 ieur) (1.0041667) /
(1.0033333) (1 5/1200) / (1 4/1200)
1.0008306 (? 1.001) F1t (1 iusd) / (1
ieur) x St 1.0008306 x USD 1 / EUR USD
1.0008306 / EUR ? 1-month forward rate (you
lock in a higher price in USD/EUR one month out
than the spot price) Approximation F1t / St
(1 iusd) / (1 ieur) ? F1t / St - 1 (1
iusd) / (1 ieur) - 1, (F1t - St) / St
(1 iusd) - (1 ieur) / (1 ieur) (iusd
- ieur) / (1 ieur) ? iusd - ieur. 1 The one
year deposit rates in USD and EUR are 5/year and
4/year. This does not imply that the one month
deposit rates in USD and EUR are 5/year and
4/year. This relationship holds only if the
yield curves are flat. If the yield curves are
normal (that is, are upwards sloping), then one
year deposit rates in USD and EUR of 5/year and
4/year imply that the one month deposit rates in
USD and EUR are less than 5/year and 4/year
respectively.
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2. Uncovered Interest Parity
Uncovered Interest Parity UIP is a forecasting
tool, or prediction. Time t versus some future
time. UIP says ESt1 / St (1 iusd) / (1
ieur). Then, ESt1 F1t from (i) CIP and
(ii) ESt1 / St (1 iusd) / (1 ieur) F1t
/ St. That is, ESt1 F1t if both UIP and CIP
hold. Approximation (in decimal
form) (ESt1 / St) - 1 (1 iusd) / (1
ieur) - 1 (ESt1 - St) / St (1 iusd) - (1
ieur) / (1 ieur) (iusd - ieur) / (1
ieur) ? iusd - ieur. Put another way, the
approximation is (ESt1 - St) / St (ieur -
iusd) (ESt1 - St) / St ieur - iusd
0.
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2. UIP (cont.)
Example 1 One year deposit rates iusd 0.05 ?
5/year, ieur 0.04 ? 4/year (ESt1 - St) / St
(1 iusd) - (1 ieur) / (1 ieur) (1
5/100) - (1 4/100) / (1 4/100) (0.05 -
0.04) / 1.04 0.01 / 1.04 0.0096154 ?
0.96154/year ? 1/year (iusd - ieur) x 100
If S goes up by 1, this is increase in the
number of USD it costs to buy a Euro (also, an
increase in the number of USD you receive for a
Euro), or (St1 - St) / St ?St1 / St is Euro
appreciation, and approximately USD depreciation.
Example 2 One month deposit rates iusd ?
5/year, ieur ? 4/year, (ESt1 - St) / St (1
iusd) - (1 ieur) / (1 ieur) (1
5/1200) - (1 4/1200) / (1 4/1200)
(5/1200 - 4/1200) / (1 4/1200)
(0.0041667 - 0.0033333) / 1.0033333
0.0008334 / 1.0033333
0.0008306 ? 0.08306/mth (? 0.1/mth)
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3. CIP, UIP Continuous Compounding
CIP, UIP and Continuous Compounding UIP is
ESt1 / St (1 iusd) / (1 ieur), where
iusd and ieur are simple rates of interests. UIP
gives (ESt1 / St) - 1 (1 iusd) / (1
ieur) - 1, (ESt1 - St) / St (1 iusd) - (1
ieur) / (1 ieur) (iusd - ieur) / (1
ieur) ? iusd - ieur where (ESt1 - St) / St is
simple expected appreciation rate. The
continuously compounded expected appreciation
rate is ln(ESt1 / St) ln ESt1 - lnSt. But
from UIP, ESt1 / St (1 iusd) / (1 ieur),
it follows ln(ESt1 /
St) ln ESt1 - lnSt ln(1 iusd) /
(1 ieur) ln(1 iusd) - ln(1 ieur), where
ln(1 iusd) is the continuously compounded rate
associated with the simple rate iusd, and ln(1
ieur) is the continuously compounded rate
associated with the ieur. Thus, in continuously
compounded terms the UIP approximation holds
exactly,
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3. Continuous Compounding (cont.)
(ESt1 - St) / St iusd - ieur. CIP is F1t
/ St (1 iusd) / (1 ieur), where iusd, ieur
are in simple terms. CIP in continuously
compounded terms is ln(F1t / St) ln(1
iusd) / (1 ieur) ln F1t - ln St ln(1
iusd) - ln(1 ieur), where ln(1 iusd) is the
continuously compounded rate associated with the
simple rate iusd, and ln(1 ieur) is the
continuously compounded rate associated with the
ieur. From UIP and CIP together, ESt1
F1t and thus ln ESt1 ln F1t, ln ESt1 -
ln St ln F1t - ln St ln(1 iusd) - ln(1
ieur), (ln ESt1 - ln St) - ln(1 iusd) - ln(1
ieur) (ln F1t - ln St) - ln(1
iusd) - ln(1 ieur) 0 (ln ESt1 - ln St) -
ln(1 iusd) - ln(1 ieur) (ln ESt1 - ln
St) - (ln F1t - ln St) 0.
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3. Continuous Compounding (cont.)
Example One-year deposits iusd ? 5, ieur ? 4,
St 1.00. Simple compounding CIP ? F1t / St
(1 iusd) / (1 ieur) 1.05 / 1.04
1.00961539 ? 1.01 FP F1t / St - 1 (F1t -
St) / St (1 iusd) / (1 ieur) - 1 (1.05 /
1.04) - 1 1.00961539 - 1 0.00961539 ? 0.01 ?
1.00 UIP ? ESt1 / St (1 iusd) / (1 ieur)
1.05 / 1.04 1.00961539 ? 1.01
ESt1 / St - 1 (ESt1 - St) / St (1 iusd)
/ (1 ieur) - 1 (1.05 / 1.04) - 1
1.00961539 - 1 0.00961539 ? 0.01 ?
1.00 Continuous compounding CIP ? F1t / St
(1 iusd) / (1 ieur) 1.05 / 1.04, ln(F1t /
St) ln(1 iusd) / (1 ieur) ln(1.05 /
1.04) lnF1t - lnSt ln(1 iusd) - ln(1
ieur) ln(1.05) - ln(1.04) 0.0487901 -
0.0392207 0.00956945 ? 0.01 ? 1.00 UIP ?
ESt1 / St (1 iusd) / (1 ieur) 1.05 /
1.04 ln(ESt1 / St) ln(1 iusd) / (1
ieur) ln(1.05 / 1.04) lnESt1 - lnSt ln(1
iusd) - ln(1 ieur) ln(1.05) - ln(1.04)
0.0487901 - 0.0392207 0.00956945 ? 0.01 ? 1.00
lnF1t ln(1 iusd) - ln(1 ieur) lnSt
ln(1.05) - ln(1.04) lnSt 0.0487901 -
0.0392207 0.00 0.00956945 ? F1t
1.00961539 ESt1 ? 1.01
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4. Suppose UIP Does Not Hold
Suppose UIP Does Not Hold If UIP holds, then
(ESt1 - St) / St (ieur - iusd) 0 in
continuously compounded terms (or as an
approximation in simple compounding terms). If
UIP does not hold, then (ESt1 - St) / St
(ieur - iusd) ? 0 or (ESt1 - St) / St
(ieur - iusd) X gt/lt 0. But recall that from
CIP the forward premium equals the interest-rate
differential in continuously compounded terms,
ln F1t - ln St - (ieur - iusd) 0 or as an
approximation in simple terms, (F1t - St)
/ St (ieur - iusd) 0. Thus, (ESt1 -
St) / St (ieur - iusd) (ESt1 - St) /
St - (F1t - St) / St X gt/lt 0.

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4. UIP Does Not Hold (cont.)
Suppose the Euro market is in equilibrium or
EReur RReur in continuous-compounding
terms. Then, in a CAPM world, E(?St1 /
St) ieur iusd ?eur,M (ERM - iusd),
E(?St1 / St) (ieur - iusd) ?eur,M (ERM -
iusd) ? 0 if ?eur,M ? 0 (assuming all
risk-factor premia are non-zero). In a more
complicated APM, E(?St1 / St) ieur iusd
?Kj1 ?eur,j RPj, E(?St1 / St) (ieur -
iusd) ?Kj1 ?eur,j RPj ? 0 if some ?eur,j ? 0
and we do not have ?eur,i RPi - ?eur,j RPj ?
0.