11.8 Covered interest parity: derivation and arbitrage

Two strategies turn ₹1 today into a riskless rupee amount at $T$.

• (i) Lend ₹1 in INR, receiving $1+r_dT$ at $T$.
• (ii) Convert ₹1 to $1/S$ dollars, lend at $r_f$ to get $(1+r_fT)/S$ dollars at $T$, and sell these forward today at $F$. That guarantees $F(1+r_fT)/S$ rupees at $T$.

Both are riskless and cost ₹1. If (ii) paid more, borrow rupees, do (ii) and repay (i): a riskless profit with zero investment. If (i) paid more, run the reverse. Absence of arbitrage forces equality.∎

(a) Borrowing ₹1 costs $1+r_d^{\text{bor}}T$ at $T$. Buying dollars at the ask gives $1/S^a$ USD, which grows to $(1+r_f^{\text{lend}}T)/S^a$, and selling forward at the bid yields $F^b(1+r_f^{\text{lend}}T)/S^a$ rupees. Subtracting gives $\Pi_A$. Then $\Pi_A>0 \iff F^b > S^a(1+r_d^{\text{bor}}T)/(1+r_f^{\text{lend}}T) = U$. (b) Borrow $1/S^b$ USD and sell it at the bid for ₹1. Lend at $r_d^{\text{lend}}$. At $T$, repay $(1+r_f^{\text{bor}}T)/S^b$ USD, bought forward at $F^a$. Then $\Pi_B>0\iff F^a<L$. (c) $S^b\le S^a$, $1+r_d^{\text{lend}}T\le 1+r_d^{\text{bor}}T$ and $1+r_f^{\text{bor}}T\ge 1+r_f^{\text{lend}}T$, so $L\le U$. Both profitable would need $F^b>U\ge L>F^a$, contradicting $F^b\le F^a$.∎