Forward Valuation:

¥ 105,000,000 = $ 105,000,000 / 104.25 = $ 1,000,194.

DM 1,450,000 = $ 1,450,000 / 1.4539 = $ 997,318.

Which is correct?

How do I use the FX forward market to guarantee that I can make the DM payment?

4.4. Value At Risk (VAR)

Value At Risk (VAR) is an accepted method to measure the risk of future losses.

For large banks: VAR is mandated by bank supervisors. (Basle Commission, Federal Reserve, Comptroller of the Currency, FDIC)

Web sites:

The Basle Commission on Banking Supervision:



Especially publications no. 4, 12a, 12b, 13, 18, 21, 22, 23, 24, 25.

Comptroller of the Currency:

See the bulletin at:

Board of Governors, Federal Reserve System:

See the SR Letters, especially:





For large firms: VAR is approved by the SEC for reporting derivative exposure.

Securities & Exchange Commission's Web site is at

See document in:

VAR: What is the maximum loss, over the next n-days, that my current position would sustain with an probability of p?

[For banks, n=10, p=1%.]

The answer depends on a statistical model of the prices that affects that position.

VAR Calculations for Example 4.1

Suppose I am paying DM 1,450,000 in 6 months.

What is the risk (i.e. maximum cost) of this position?

Let St be the $ price of 1 DM at date t. Today, t=0. In 6 month, t=0.5.

Let x = ln[S0.5/S0] be the continuously compounded rate of change between today and 6 months from today.

The $ cost in 6 months

= 1,450,000 S0.5

= 1,450,000 S0 ex.

The distribution of 1,450,000 S0 ex depends on the distribution of x.

VAR Method 1: Using a Normality Model

Assume that x is iid N(μ,σ).

Estimate μ and σ from observed data. [ μ=1.38%, σ=8.99%. ]

Then z = (x-μ)/σ is iid N(0,1).

We obtain the following probabilities:

Prob z x Maximum $ Cost in 6 months

1% 2.326 0.2229 $ 1,230,977

5% 1.645 0.1617 $ 1,157,894

10% 1.282 0.1290 $ 1,120,728

25% 0.675 0.0745 $ 1,061,225

50% 0 0.0138 $ 998,759

VAR Method 2: Using The Historical Distribution

Take the distribution of x from historical data.

Prob x Maximum $ Cost in 6 months

1% 0.2056 $ 1,209,914

5% 0.1504 $ 1,145,018

10% 0.1238 $ 1,114,899

25% 0.0794 $ 1,066,414

50% 0.0165 $ 1,001,444

VAR Method 3: Simulating from A Statistical Model

We can built a statistical model on x.

The statistical model implies a distribution for x, and hence a distribution for $1,450,000 S0 ex.

For an analysis of the statistical models on exchange rates, see

David Hsieh, Journal of Business, 1989.

A popular model is the ARCH/GARCH volatility forecasting model. See:

Robert Engle, Econometrica, 1982.

Tim Bollerslev, Journal of Econometrics, 1986.

ARCH/GARCH models are covered by Steve Gray in the Advanced Futures & Option Course.

For an application of volatility forecasting models to currency futures, see:

David Hsieh, Journal of Financial and Quantitative Analysis, 1993.

Which Method to Use?

It is clear that there is no "right" or "wrong" answer to VAR.

A lot depends on the assumptions you make about the world.

There is no single statistical model that can capture all aspects of reality.

Changing the assumptions can change the results.

But VAR forces you to think in a more systematic way about potential losses than just mere "educated", or even "uneducated" guesses.

In a multi-variate setting, VAR also forces you to think about the relationships (e.g. correlations) between different variables that affect your position.

VAR References For Advanced/Highly Quantitative Readers:

D. Hsieh, Journal of Quantitative and Financial Analysis, 1993.

P. Jorion, Value at Risk.

Value At Risk Calculation for Example 4.2

Suppose I am receiving ¥ 105,000,000 and paying DM 1,450,000 in 6 months. How do I use VAR to analyze the risk of this exposure?

For historical FX data going back to the 1970s, see the Federal Reserve's Web Site:



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