Real Exchange Rate

Nominal Exchange Rate (GBP/CAD) = Real Exchange Rate (GBP/CAD) * (P_GBP/P_CAD)
or
Real Exchange Rate (GBP/CAD) = Nominal Exchange Rate (GBP/CAD) * (P_CAD/P_GBP)

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(e.g.)

• Nominal Exchange Rate (GBP/CAD) = 0.40 (GBP/CAD)
• The ratio of the price of the U.K. consumption basket to the Canadian consumption basket = (P_GBP/P_CAD)_t=0 = 0.3

∴Real Exchange Rate (GBP/CAD) = 0.40 * (1/0.3) = 1.33

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(e.g.)

• Real exchange rate remained constant.
• (GBP/CAD)_{Real, t=1yr} = (GBP/CAD)_{Real, t=0}
• (GBP/CAD)_{Real, t=0} = (GBP/CAD)_{Nominal, t=0} *  (P_CAD/P_GBP)_t=0
• (GBP/CAD)_{Real, t=1yr} = (GBP/CAD)_{Nominal, t=1yr} * (P_CAD/P_GBP)_t=1 = (GBP/CAD)_{Nominal, t=1yr} * (P_CAD/P_GBP)_t=0 * (1+i_CAD)/(1+i_GBP)
• ∴(GBP/CAD)_{Nominal, t=1yr}  = (GBP/CAD)_{Nominal, t=0} * (1+i_GBP)/(1+i_CAD) (exact version)
• ∴(GBP/CAD)_{Nominal, t=1yr} = (GBP/CAD)_{Nominal, t=0} * (1+i_GBP - i_CAD) (approximate version)
• i_GBP = 4%
• i_CAD = 7%

∴GBP return on the Canadian stock
=CAD Total return - (i_CAD - i_GBP) =CAD Total return + (i_GBP - i_CAD)
= 22% - (7% - 4%) = = 22% + (4% - 7%) = 19%

i_CAD - i_GBP = 7% - 4% = 3%
CAD depreciates by 3%. GBP appreciates by 3%.

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(e.g.)

• Real exchange rate changed.
• (GBP/CAD)_{Real, t=0} = 1.33
• (GBP/CAD)_{Real, t=1yr} = 1.41
• A Canadian stock's 1 year total return = 22%.
• i_GBP = 4%
• i_CAD = 7%

(GBP/CAD)_{Nominal, t=1yr}
= (GBP/CAD)_{Real, t=1yr} * (P_GBP/P_CAD)_t=1yr
= (GBP/CAD)_{Real, t=1yr} * (P_GBP/P_CAD)_t=0 * (1+i_GBP)/(1+i_CAD)
= 1.41 * 0.30 * (1+4%)/(1+7%)
≈ 0.41114

GBP appreciated while CAD depreciated.

∴GBP total return
= CAD total return + CAD appreciation
= 22% + (0.41114-0.40)/0.40
= 22% + 2.785% = 24.785%