Question

The U.S. dollar is trading at $1.2565/£ and exhibiting volatility of 13%. The one-month continuously compounded spot rates in $ and £ are 0.72% and 0.25% p.a. A U.K. firm is considering whether to hedge its USD receivables:

a. Plot the probability distribution of the spot exchange rate one month hence.

b. What’s the probability the USD will weaken by 5% or more in the month?

c. What are the 95% confidence bounds on the exchange rate?

d. What’s the probability of earning a return of less than -20% p.a. over the next month?

e. What are the 99% confidence bounds on the return earned over the next month?

Answer 1

The exchange rate in time t, will be determined as follows,

Where r = 0.72% pa

rf = 0.25% pa

t: time in years

t = 1month = 1/12 years

s: standard deviation = 13% pa

z: standard normal variable

This means that the monthly change in exchange rate is dependent on a normal distribution,

mean = (0.72%-0.25%)/12 = 0.0392% per month

standard deviation = 13%*(1/12)^0.5 = 3.753% per month

(a) Probability distribution of the spot exchange rate 1 month hence

b) Weaken by 5%

x = ln1.05 = 4.879%

Z score = (4.879%-0.0392%)/3.753% = 1.289

normsdist(1.289) = 90.14%

Probability of 5% or more weakening = 1-90.14% = 9.86%

c) 95% bounds

z lower value = 0.0392%-1.96*3.753% = -7.317%

z upper value = 0.0392%+1.96*3.753% = 7.395%

Confidence interval = [1.2526*e^(-7.317%),1.2526*e^(7.395%)] = [1.1642,1.3487]

d) return = -20%

x = ln(1-20%) = -22.314%

z value = (-22.314%-0.0392%)/3.753% = -5.956

Probability = normsdist(-5.956) =

0.00000013%

e. 99% confidence interval

z lower value = 0.0392%-2.232*3.753% = -8.337%

z upper value = 0.0392%+2.232*3.753% = 8.416%

Confidence interval = [1.2526*e^(-8.337%),1.2526*e^(8.416%)] = [1.1524,1.3626]