Interest Rate Arbitrage

def. Interest Rate Arbitrage. If a foreign country has a highe risk-free rate:

1. Borrow money (USD) from domestic bank with interest $i_D$
2. Exchange to foreign currency at spot rate $S\text{USD}=1\text{EUR}$
3. Lend money (EUR) to domestic bank with interest $i_F$
4. Get money back, convert it back at future rate $F$ or $\mathbb{E}(S_{t+1})$ Total profit for investing \x$:

$$\text{Profit}=\left(\stackrel{\text{ conversion }}{\overbrace{\frac{x}{S}}}\underset{\text{ foreign lend }}{\underbrace{(1+i_F)}}\right)\underset{\text{ conversion back }}{\underbrace{F}}-\underset{\text{ domestic borrow }}{\underbrace{x(1+i_D)}}$$

You can hedge (=cover) or not hedge for future fluctuations:

• Covered: Enter a futures contract at $t_0$ for a rate $F$ at time $t+1$
• Uncovered: Just take whatever future spot price $\mathbb{E}[S_{t+1}]$ (also called currency carry trade)

No-Arbitrage §

Motivation. Since there should be no arbitrage in the economy the profit from this risk-free trate should be zero. This holds either for the covered or the uncovered. Simply from setting $\text{Proft}=0$:

$$\begin{array}{rlr}1+i_D& =\frac{F}{S}(1+i_F)& \text{Covered}\\ 1+i_D& =\frac{\mathbb{E}(S_{t+1})}{S}(1+i_F)& \text{Uncovered}\end{array}$$