TOPIC 2: Covered/uncovered interest rate parity and FX premium risk

Question 1

Covered interest rate parity ASSUMPTIONS

Answer 1

Default-free deposits
No counter-party default risk
no transactions costs

Question 2

CIP formula for domestic ‘d’ and any foreign currency ‘f’

Answer 2

1+i ^f (t) = [S(t, d/f) x (1 + i ^d (t))] / F(t, d/f)

Intuition: the explicit and implicit (‘f’) interest rates are equalized after removing (covering) FX risk

Question 3

IF CIP holds, we can derive:

Answer 3

F(t, d/f)−S(t, d/f) / S(t, d/f) = i^d(t)−𝐢^𝐟(t) / (1 +𝐢^𝐟(t))

Interpretation: ““F(t, d/f) < S(t, d/f)” implies “i^d(t) < i^f(t) ”

(AKA: If CIP holds today, higher interest rate currency (‘f’) should have the forward discount at time t)

Question 4

Covered Interest rate arbitrage (CIP without TC)

Answer 4

Case 1: 1+i^£ < [S($/£) x (1+i^$)] / F($/£)
‘Borrow £ and invest into $’

Case 2: 1+i^£ > [S($/£) x (1+i^$)] / F($/£)
‘Borrow $ and invest into £’

Question 5

Covered Interest rate arbitrage (CIP with TC)

Answer 5

Case 1: 1+ i£ask < [S($/£, Bid) × (1 +i$Bid)] / F($/£, Ask)
‘borrow £ and invest into $’

Case 2: 1+ i$ask < [F($/£, Bid) × (1 +i£Bid)] / S($/£, Ask)
‘borrow $ and invest into £’

Question 6

Uncovered interest rate arbitrage (UIP) - Speculating in the FX market ($ carry trade)

Answer 6

given $ interest rate < £ interest rate, “borrow $1 and convert/invest in £ deposit for 1-year and converting £ amount back to $ at t+1 at the unknown future exchange rate (i.e., without hedging FX risk)

Question 7

The excess return from $ carry trade

Answer 7

exr($, t+1)

Question 8

The expected excess return from $ carry trade

Answer 8

Et[exr($, t+1)]

Question 9

Mr. Buy’s $ profit/loss at t+1 per £1

Answer 9

£1 × [S(t+1, $/£) – F(t, $/£)]

Question 10

Buy’s $ expected profit/loss at t+1 per £1

Answer 10

£1 × {Et[S(t+1, $/£)] – F(t, $/£)}

Question 11

Forward Market Return

Answer 11

fmr($, t+1)

Question 12

Uncovered interest rate parity ASSUMPTIONS

Answer 12

Default-free deposits
no transaction costs
Risk-neutral investors

Question 13

UIP formula

Answer 13

1 + i^f(t) = [S(t, d/f)×(1 +i^d(t)] / Et[S(t+1, d/f)]

Intuitively, the explicit and implicit expected (‘f’) interest rates are equalized without removing (covering) FX risk

Question 14

Note: If UIP holds, we can derive

Answer 14

Et[S(t+1, d/f)]−S(t, d/f) / S(t, d/f) = i^d(t)−i^f(t) / (1 +i^f(t))

Interpretation: “Et[S(t+1, d/f)] < S(t, d/f)” implies “id(t) < if(t) ”

If UIP holds today, higher interest rate currency (‘f’) should be expected at time t to depreciate from t to t+1, relative to the low interest rate currency (‘d’)

Question 15

Unbiased Hypothesis (UH)

Answer 15

when the CIP and UIP hold together, the UH holds Et[S(t+1, d/f)] = F(t, d/f)

when you predict the future spot rate S(t+1, d/f) with the forward rate F(t, d/f) available at time t, the realized future spot rate will be equal to the forward rate ON AVERAGE.

Question 16

If UIP holds

Answer 16

Et[exr($, t+1) = 0

Question 17

If UH holds

Answer 17

Mr Bs $ profit/loss at t+1 per £1:

£1 x [EtS(t+1, $/£) - F(t, $/£) = 0

Question 18

$ Carry trades seem profitable in real practice, implying

Answer 18

Et[S(t+1, $/£)]– F(t, $/£) > 0

One sufficient condition:
Et[S(t+1, $/£)]– S(t, $/£) / S(t, $/£) > 0

If the high interest rate currency (£) is expected to appreciate on average, the ($) carry trade will be profitable

Question 19

$ carry trade with borrowing/lending

Answer 19

Given $ interest rate < £ interest rate, “Borrow $1 at i$ and convert/invest in £ deposit at i£ for 1-year and converting £ amount back to $ at t+1 at the unknown future exchange rate S(t+1, $/£) (i.e., without hedging FX risk)”

Question 20

CAPM

Answer 20

Et[exr($, t+1)] =
β$carry [Et(rm,t+1) – rf,t ] → β$carry >0

β$carry >0 : £ tends to depreciate relative to $ (i.e., negative exr($, t+1)) during stock market downturn (i.e., negative rm,t+1).

However, in real practice β$carry ≈ 0 (CAPM is not a good model!!)

Question 21

$ carry trade with forward contracts

Answer 21

Given $ interest rate < £ interest rate, “Buy £1 forward contract for F(t, $/£) at t and Sell £1 at t+1 using the unknown
future exchange rate S(t+1, $/£)”

Mr. Buy’s expected $ profit/loss at t+1 per buying £1 forward = (£1) × Et[S(t+1, $/£) – F(t, $/£)