To manage this risk, what we want to do is make a calculated guess to estimate the potential loss involved.
So for example, suppose a trader is long 1000 EUR/USD and they have a stop loss at 1000 points. They could assume that as being a risk of USD 10, because that’s the theoretical maximum they can lose.
While this gives an absolute dollar value, it doesn’t answer certain questions that are necessary for proper risk management. It doesn’t say under what conditions that loss could occur. Depending on the market it could be a very low risk or a very high risk. It could happen with high probability after 5 minutes or perhaps after three months. We don’t really have any idea.
In other words, knowing the absolute loss from stop levels alone is of little use. This is because a trader can’t frame that against any time period or understand the level to which other positions might be affected.
Value at Risk
Value at Risk or VAR as it’s known for short is a calculation that helps you to judge exposure to market risk. It’s helpful because it can answer questions like this:
- If I hold positions A, B and C, what is the likelihood that I’ll lose X dollars within the next 7 days?
- What is the chance of my account losing X percent of its value in the next 3 days?
- If A falls in value, what will happen to B and C?
- If I close position C, how will that change my overall risk?
- What position can I add to my basket to lower my overall risk?
- Where should I set a stop loss to have a probability Y% of it being hit?
The answers to the above will give far more insight than simply saying that a stop loss is at 1000 points, so that must be the total risk.
This is why VAR is an important tool in risk management. The rest of this article describes using simple VAR for risk analysis. For more details on stop loss settings see here.
Simple VAR for One Currency Pair
A basic VAR estimate is done as follows.
Let’s say I’ve done a spot trade in 1000 EUR/USD and the price is 1.10. I calculate the 1-day volatility of EUR/USD to be 0.5%. On average then I would expect the position’s value to change by 1.2 x 1000 x 0.5% – that is ±6 USD over any given day. Over 7 days, the average position change would be: would be: ±√7 × 6 USD.
If the daily changes are normally distributed, then a 99% confidence interval lies at 2.33 standard deviations. So using this gives:
1 day VAR = 6 x 2.33 = 13.96 USD
7 day VAR = √7 × 6 × 2.33 = 36.93 USD
This tells me that I can be 99% confident that I won’t lose more than 13.96 USD in 1 day. And I can be 99% confident that I won’t lose more than 36.93 USD in 7 days.
VAR for Multiple Currency Pairs
The above is for a simple case where a trader has one EUR/USD spot position. What about a more realistic scenario when multiple positions are held – for example long 1000 EUR/USD and long 2000 GBP/USD?
We can’t just add the VARs together for the simple reason that most currency pairs are correlated to one another. When EUR/USD goes down, this might be because the US dollar has strengthened.
So, if I hold long 1000 EUR/USD and long 2000 GBP/USD there’s a high chance that both of these positions would fall in value if the dollar were to rise.
Here is the basic VAR again, now for a basket of two currencies:
The second position in the account is long 2000 GBP/USD and the spot price is 1.30. The 1-day volatility of GBP/USD is 0.6%. On average then we would expect the position value to vary by 1.3 x 2000 x 0.6% or ±15.60 USD each day. And over 7 days, the average position change would be: ±√7 ×15.60 USD. This means
1 day VAR = 15.60 x 2.33 = 36.29 USD
7 day VAR = √7 × 15.60 × 2.33 = 96.02 USD
This now gives the VAR separately for the two positions. But this isn’t very helpful because as said above EUR/USD and GBP/USD are not independent of one another because of their exposure to US dollar among other things.
At the present time, the average correlation between EUR/USD and GBP/USD over 1 week is +70%. For that reason we need to take the VAR of the two positions under the assumption that these positions are not independent of one another:
This now gives the covariance VAR:
1 day VAR is √(36.29^2+13.96^2+2×36.29×13.96×0.7)=47.12 USD
7 day VAR is √(96.02^2+36.93^2+2×96.02×36.93×0.7)=124.69 USD
This tells me that if I hold both EUR/USD and GBP/USD, there’s a 99% chance that my total loss in 1 day will not be bigger than 47.12 USD. And there’s a 99% chance that my total loss in 7 days will not be above 124.69 USD.
Notice that GBP/USD provided a small amount of risk reduction but hardly very much. If the two positions were perfectly correlated with r=1, the VARs would simply have been additive. That means the 7 day value at risk would have been 132.95 (from 96.02+36.93) and not 124.69. The 1 day VAR would be 50.25 and not 47.12.
That means as a diversification the second position only reduced the relative risk by about 6%.
Using VAR to Lower Risk
Adding a long 2000 GBP/USD, didn’t provide very much in the way of risk reduction because the two currency pairs are strongly dependent on one another. What is the outcome if we use another currency pair which has a lower correlation, such as AUD/CAD?
The correlation between AUD/CAD and EUR/USD is 0.5. The spot price for AUD/CAD is 1.01 and the 1 day volatility is 0.6%. That means by adding long 2000 AUD/CAD, the combined VAR of the account would now be:
1 day VAR = 37.19 USD
7 day VAR = 98.40 USD
Even though the AUD/CAD position does have a lower dollar value than GBP/USD it does provide more diversification.
By using the VAR method it’s easy to do “what-if” experiments by adding or removing positions to see what the account risk will be in actual dollar terms. This VAR calculator does all of the calculations, and will greatly speed up this task.
How to Estimate Account Loss
The other question VAR can answer is: what is the chance of my account losing X percent over a certain time period?
Let’s say a trader’s account has an equity value of $500 and they want to know the chance of losing 10% of value in the next 7 days. The two positions held are long 1000 EUR/USD and long 2000 GBP/USD.
This means we need to know what is the probability of this account losing 10% x 500 or 50 USD in the next 7 days?
The 7 day VAR is already known from the above and that’s 124.69 USD.
This tells us there is a 1% probability of losing this amount. We then work backwards and find that the 82% VAR over 7 days is 50 dollars. This means there’s an 18% chance of this account losing 50 USD (10% of its value) or more over the next 7 days.
Using VAR in Practice
While simple VAR isn’t perfect by any means it is a highly useful technique because it’s easy to calculate and lets us understand risk in practical cash terms.
Limitations of VAR
When using VAR it’s always worth keeping in mind the limitations as well. One of the main criticisms of VAR is that it doesn’t weigh up abnormal market risk. By abnormal market risk this means:
- Rare disasters such as black swan events
- Counterparty risk
- Liquidity risks
The other limitation is that simple VAR assumes that averages such as volatility, correlation and so on are fixed over the time period analysed. While in real market situations these variables are changing all of the time and have more complex outcomes than simple normal curves. Of course there are plenty of extensions to VAR that work around these problems.
Weekends and Holidays
It’s often assumed that when the market is closed there’s zero volatility. But this isn’t quite true.
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Information arriving over the weekend does influence markets even though the price doesn’t react while markets are closed. That means volatility is happening over the weekend even though we can’t see it.
It’s important to factor this in when using volatility estimates in VAR calculations.
For example, if I use 1-day volatility, there’ll be a difference between whether that averages across the entire month including weekends and holidays or if it just covers periods when the market is open.
VAR is a simple, yet powerful tool in risk management and provided the assumptions are understood it can be used to good effect. VAR’s biggest advantage is that it gives a single cash figure that tells you how much you could lose in the worst case.
Here is the VAR calculator: