CASE II: SOLUTIONS

General Situation:

Underlying Position (UP): Short JPY 200 M (T=November 17, around 132 days.)

Transaction Exposure (TE): JPY 200 M x .01247 USD/JPY = USD 2.494 M

Hedging Position (HP): Long Futures/Long JPY Calls.

Note: The optimal hedge ratio (futures position) should be (using the average interest rate –mid-quotes):h = (1+.01546425*132/360)/(1+.00706815*132/360) = 1.003071 (Very close to 1. Assuming h=1 is not a bad assumption in this example).

1) You should have found much more volume and open interest in the futures market than in the option markets. Thus, it is easier to close a position in the futures markets.

2)We generate 4-month arithmetic Returns (Excel output)

Note: The data is in JPY/USD, but the question is in terms of USD/JPY.

4-mo changes / 1-mo changes
JPY/USD / USD/JPY / JPY/USD / USD/JPY
Mean / -0.025861 / 0.02974306 / -0.00681 / 0.007716
Standard Error / 0.0070077 / 0.00755146 / 0.003822 / 0.003795
Median / -0.017104 / 0.01740315 / -0.00758 / 0.007637
Mode / #N/A / #N/A / #N/A / #N/A
Standard Deviation / 0.0542813 / 0.05849339 / 0.029602 / 0.029395
Sample Variance / 0.0029465 / 0.00342148 / 0.000876 / 0.000864
Kurtosis / 0.0412707 / 0.41337854 / 1.766956 / 1.254508
Skewness / -0.188091 / 0.52321335 / 0.878899 / -0.63346
Range / 0.2524736 / 0.27785897 / 0.16036 / 0.15887
Minimum / -0.164689 / -0.08070069 / -0.07231 / -0.08093
Maximum / 0.087785 / 0.19715828 / 0.088055 / 0.077941
Sum / -1.551668 / 1.78458335 / -0.40854 / 0.462968
Count / 60 / 60 / 60 / 60

4-month mean =0.029743 (2.97%)

4-month SD = 0.05849 (5.85%)

95% C.I. for st: [0.029743± 1.96 x 0.05849] = [-0.0849, 0.1444]

VAR(97.5%)= JPY 200 M x .01247USD/JPY x (1+0.1444) = USD 2.854134M

Note 1: You could have used USD/JPY monthly changes and approximate the 4-mo mean and 4-mo standard deviation. In this case:

4-month mean = 4*0.007716= 0.030864(3.09%)

4-month SD = sqrt(4)*0.029395= 0.05879(5.88%)

Note 2: You could have used JPY/USD monthly changes and approximate the USD/JPY changes by changing the sign.

(These are “arithmetic approximations.” Sometimes they work very well.)

3)4-Monthly Arithmetic Returns (Excel output)

a) Worst case scenario (largest 4-mo appreciation of JPY against USD): 0.1972(19.72% appreciation of JPY against USD)

Worst case = JPY 200 M x .0098290 USD/JPY x (1+0.1972) = USD 2.985817M

b) Best case scenario (largest 4-mo depreciation of JPY against USD): -0.0807 (8.07% depreciation of JPY against USD)

Best case = JPY 200 M x .008502 USD/JPY x (1-0.09042) = USD 2.292734M

4) Data

Ft=June 6,T=Dec =.01257USD/JPY

Ft=Nov. 6,T=Dec =.013062USD/JPY

U.S. short interest rates for two months or less = .2909-0.3165.

There are 41 days to maturity (3rd Wednesday of December)

Assume h=1 (HP size is JPY 200M)

Value of forward contract =FNov 6,Dec- FJune 5,,Dec=(.013062- .01257)*200M= USD 0.0983645M

[1 + iUSD x (T/360)][1 + .003165x(41/360)]

Total payment in November 6:

JPY 200 M x .01304 USD/JPY - USD 0.0983645 M =USD 2.509636 M

(If you assumed h=1.003 -i.e. HP=JPY 200.6 M-, you would have an extra 0.3%.)

5)Check the Lecture Notes, Example VIII.8.

Amount in JPY to be deposited = JPY 200 M /(1 + .0070158 x 180/360) = JPY 199.300872 M

Convert to USD = JPY 199.300872 M x .012470 USD/JPY = USD 2.4852819 M (<= amount to borrow from US bank)

Loan repayment after 180 days = USD 2.4852819 M x (1 + .0156205x 180/360) = USD 2.5046926M

The cash flows will occur in Dec 5. To compare with 4), we need to discount the cash flows back to Nov 6.

Discounted Cash flows = USD 2.5046926M/(1+0.003165x29/360) = USD 2.504054 M

6)See discussion in Chapter VI, Section 2.C.1 (Delta-hedging).