204 Mathematics for Finance
pounds):
sales 5 , 000 , 000
cost of sales − 3 , 000 , 000
earnings before tax 2 , 000 , 000
tax − 400 , 000
earnings after tax 1 , 600 , 000
dividend − 1 , 250 , 000
result 350 , 000
The surplus income will be 0.35 million pounds.
However, if the exchange ratedbecomes 2 dollars to a pound, the com-
pany will end up with a loss of 0.45 million pounds (and the dividend will
in fact have to be reduced):sales 4 , 000 , 000
cost of sales − 3 , 000 , 000
earnings before tax 1 , 000 , 000
tax − 200 , 000
earnings after tax 800 , 000
dividend − 1 , 250 , 000
result − 450 , 000Let us compute VaR. We assume that the rate of exchange has log-
normal distribution with mean return equal to the difference between the
interest rates, 8%−11% =−3%.^1 With the volatility of the return on
the exchange rate at 15%, the return on the investment will exceed−3% +
1. 65 ×15% = 21.75% with probability 95%. This corresponds to an exchange
rated=1. 6 ×e^21 .75%∼= 1 .9887 dollars to a pound, for which the income
statement will be as follows (all amounts rounded to the nearest pound):sales 4 , 022 , 728
cost of sales − 3 , 000 , 000
earnings before tax 1 , 022 , 728
tax − 204 , 546
earnings after tax 818 , 182
dividend − 1 , 250 , 000
result − 431 , 818(^1) This assumption can be justified as follows: If a pound is invested without risk
for one year and then converted to dollars at a ratedknown in advance, to avoid
arbitrage we should haved×e11%=1. 6 ×e8%,sod=1. 6 ×e−3%.Thisgives−3%
logarithmic return on the exchange rate. For a random exchange rate it is therefore
natural to assume the mean logarithmic return to be−3%.