Question

11. The spot exchange rate is 110.0JPN/USD and the semiannually compounded risk-free rate of USD is 5.06%. You observe a exchange forward with 9-month maturity is 112.9JPN/USD. a. What risk-free rate of JPN is implied by this forward price? b. Suppose you believe the risk-free rate of JPN over the next 9 months will be only 0.5%. What arbitrage would you undertake? c. Suppose you believe the risk-free rate of JPN will be 3 % over the next 9 months. What arbitrage would you undertake?

Answer 1

a) Risk free rate on USD for 9 months = (1+0.0506/2)*(1+0.0506/4) -1 = 0.03827

So, let the Risk free rate for 9 months in JPN be r

then by Interest rate Parity

Forward exchange rate/Spot Exchange rate  = (1+interest rate of Japan for 9 months)/(1+interest rate in USD for 9 months)

112.9/110 = (1+r)/1.03827

=>  (1+r) =1.065643

=> r =0.065643

So, Risk free rate per year x is given by

(1+x/2)*(1+x/4) = 1.065643

=> x = 0.086283 or 8.63% semiannually compounded

b) If the risk free rate is only 0.5% then $ rate in Forward is costly and hence $ must be sold in forward

Steps of Arbitrage

1. Today, Borrow 110 JPN at risk free rate for 9 months. After 9 months , the amount to be returned is JPN 110*(1+0.005/2)*(1+0.005/4) = JPN 110.4128

2. Today, convert the 110 JPN to $1 using the spot exchange rate and invest this amount in USD at 5.06%,

3. Today, Sell $1.03827 in forward after 9 months

4. After9 months, get $1.03827 from US bank, Use forward to convert it to 112.9*1.03827 = JPN117.2207 and repay the JPN loan of JPN 110.4128 . Get the remaining JPN 6.81 as arbitrage income

c) If the risk free rate is 3% then $ rate in Forward is costly and hence $ must be sold in forward

Steps of Arbitrage

1. Today, Borrow 110 JPN at risk free rate for 9 months. After 9 months , the amount to be returned is JPN 110*(1+0.03/2)*(1+0.03/4) = JPN 112.4874

2. Today, convert the 110 JPN to $1 using the spot exchange rate and invest this amount in USD at 5.06%,

3. Today, Sell $1.03827 in forward after 9 months

4. After9 months, get $1.03827 from US bank, Use forward to convert it to 112.9*1.03827 = JPN117.2207 and repay the JPN loan of JPN 112.4874 . Get the remaining JPN 4.73 as arbitrage income