Probability and the timing of hedging transactions
A successful foreign exchange hedging policy must rely on an understanding of the probabilities of currency fluctuations over periods of time, writes Walter Ochynski.
A successful foreign exchange hedging policy must rely on an understanding of the probabilities of currency fluctuations over periods of time, writes Walter Ochynski.
Hedging decisions can be very complex. A multinational company can either hedge everything, do totally without hedging, or apply a selective hedging. Any hedging strategy must be derived from the corporate strategy and support it. I do not want to engage in a discussion about hedging in general. This article is just about analyzing the timing issue of hedging transactions. Should hedging transaction be implemented for the maturity of the exposure or for shorter periods and then swapped forward if still appropriate?
In January 1976 the US dollar (USD) was against the Deutsche Mark (DM) at the same level as it is now against the euro (EUR). Expressed as the EUR/USD rate, in 1976 it was 1.14 and in August 2018 is 1.14 as well. In between, the EUR/USD rate experienced huge swings, trends which lasted between a few quarters and many years.
No finance manager can base a hedging decision on the expectation that someday in the future the USD will be exactly at the same level as it was 10 or 20 years ago
A hedging transaction that looked very promising at the beginning of a budget year could become a total flop at the end of the same year. It could become a very profitable transaction again, if maintained, but hedging without underlining transaction is a speculation and does not belong in the toolbox of most finance managers.
No finance manager can base a hedging decision on the expectation that someday in the future the USD will be exactly at the same level as it was 10 or 20 years ago.
How should we implement hedging transactions under the concept of the probability of appreciation at different time scales? Nassim Taleb in his excellent book Fooled by Randomness uses a similar example explaining investment decisions under different time scales.
Let’s use the same concept for hedging decisions. Assume that the EUR/USD spot rate is 1. For the year from now the expected increase is 15% and the expected standard deviation is 10%. I use at first the same numbers as Taleb in his example; they have only a slightly different meaning here since we do not think about investment but about hedging of the exposure – we see it from a hedging point of view.
When the expected average change in the forex rate is 15% with a standard deviation of 10% we would like to know what the probability of an unfavorable exchange movement is. In statistical terms we look for the zscore. A zscore always reflects the number of standard deviations above or below the mean of a particular distribution. For instance, if the expected increase is 0.15 and the standard deviation is 0.10 we look for the zscore for 0 – representing no change at all. Applying the formula:
zscore of 0 is (0.15)/.10=1.5 standard deviation below the expected average change (mean) and now with the help of the standard normal distribution curve we can answer the question of how probable it is that the EUR develops favorably (from a US exporter point of view) or unfavorably (from a US importer point of view)
Statisticians have determined that 6.7% of the scores in a normal distribution are lower than a score one and half standard deviations below the mean. Therefore, the proportion of the values above 0 (zscore equals 1.5) is 93.3%.
With the probability of 93.3% the EUR/USD rate will be higher in one year than it is today. It is a very high probability for the appreciation of euro against USD. Using the same starting expectation of an average increase by 15% and standard deviation of 10% on the annual basis, we transform now mean and standard deviation into different time scales, especially half year, quarter, month etc. To convert annual mean we simply divide annual value by time involved – half year by 2, month by 12, etc. (To obtain the same results as Taleb, I assume a year has 265 working days and a day has eight working hours).
To convert annual standard deviation we divide annual deviation by a square root of the period involved. For example to calculate standard deviation per hour, we divide annual standard deviation by square root of 265*8. The results can be seen in Table 1.
Table 1: The probability of EUR/USD appreciation at different scales  
Scale  Zvalue for x bigger than zero  mean  deviation  Probability of zValue*  
1 year  1.500  15.0000  10.000  93.32%  
1 half  1.061  7.5000  7.071  85.56%  
1 quarter  0.750  3.7500  5.000  77.34%  
1 month  0.433  1.2500  2.887  66.75%  
1 day  0.092  0.0566  0.614  53.67%  
1 hour  0.033  0.0071  0.217  51.30%  
1 minute  0.004  0.0001  0.028  50.17%  
1 second  0.001  0.0000  0.004  50.02% 
* =1NORM.S.DIST(Zvalue, TRUE) – Excel formula
We see from Table 1 that a 15% annual expected appreciation of EUR/USD rate with a 10% volatility per annum has a 93% probability in the year but on the shorter time scale this transforms into a substantially lower probability, for example of 50.02% per second.
Every foreign exchange dealer very often assumes a position for a very short period of time, often just a few seconds, where he is only slightly more than 50:50 convinced that the position will be profitable.
However, overall foreign exchange dealers are very successful without having very strong expectations. Can this also apply to a finance manager who implements hedging decisions? I think yes.
In an Excel spreadsheet we can work with expectations with arbitrary precision but in real life our expectations are not so exact. We might expect euro to deteriorate but to be sure with 93% or 59% probability is almost impossible. We rather see it as less or more likely that a currency will deteriorate.
In an Excel spreadsheet we can work with expectations with arbitrary precision but in real life our expectations are not so exact
In Table 1 we see that this kind of expectation is more compatible with shorter periods. According to Table 1 we just require a probability of 66.75% to be successful on a monthly basis – somewhat more than 50%.
Most hedging decisions occur in a current budget with a time scale of up to one year. Since we never know whether our expectations are accurate and correct, and probability requirements lessen with shorter time scales, we will be more successful if we implement shortterm hedging decisions.
Let’s say we hedge only for one month, although we have for example a ninemonth exposure, and before the maturity of our hedging transaction we either swap the hedging transaction forward for another month or we terminate the hedging transaction if our expectations have changed. With this strategy a company will never have a huge amount of hedging transactions underwater, which can not only be very detrimental for the company (for example press reports about huge foreign exchange losses) but for the treasurer as well.
Taleb showed with the probability of success analysis at different scales, that it is more prudent to analyze an investment portfolio on an annual basis, because with the expected return of 15% and variability of 10%, you would be 93% certain that investment portfolio is profitable at the end of each year, and over the next 20 years you could expect to experience 19 profitable years and one unpleasant one.
However, looking at the same portfolio daily or monthly you would have much more negative unpleasant surprises.
Applying the same concept to hedging, we came to the opposite conclusion. As finance managers we do not have the leisure of waiting for better results in the future years. Every year is a new budget year and we have to protect the current budget with our hedging measures.
But due to the variability of foreign exchange rates and the possible negative impact of underwater hedging transactions, the assumption that all hedging transactions will be successful is very risky. Therefore monthly hedging with the possibility of undoing hedging (just letting the transaction expire) or swapping forward when expectations are still valid is much more appropriate.
Over the 42 years between 1976 and 2018, the value of the USD against the EUR (or DM as substitute for EUR before 1999) almost did not change.
However, we have experienced huge swings – in some years around 30%. There was no year in which the EUR/USD rate remained unchanged. Therefore the assumption that EUR/USD changed on average 15% might be exaggerated. 10% or 7% might be more appropriate.
As far as volatility is concerned, the assumption of 10% supports the data quite well. The current implied volatility as shown by option pricing amounts to around 8%.
Therefore in Table 2 I show the probability of EUR/USD appreciation at different scales using different expected appreciation rates. The annual 10% error rate (volatility) remains the same. We can now see that the weaker the trend, the expected mean appreciation goes down, here from 15% to 3%. The probability requirements become less stringent and somebody with expectation slightly over 50% can execute successful hedging decisions when time scale is reduced to one month.
More frequent decisions about hedging transactions can be profitably executed when we have a chance to try them more times during the year and the time scale analysis supports this procedure. Nobody has a crystal ball that can predict the future, but with help of statistics we can substantially improve our chances of providing hedging when necessary, and avoid huge losses resulting from underwater hedging transactions.
Table 2: The probability of EUR/USD appreciation at different scales with an average appreciation from 3% to 15% 

Scale  Expected Change 15% annual Volatility 10% annual 
Expected Change 10% annual Volatility 10% annual 
Expected Change 7% annual Volatility 10% annual 
Expected Change 3% annual Volatility 10% annual 

1 year  93.32%  84.13%  75.80%  61.79%  
1 half  85.56%  76.02%  68.97%  58.40%  
1 quarter  77.34%  69.15%  63.68%  55.96%  
1 month  66.75%  61.36%  58.01%  53.45%  
1 day  53.67%  52.45%  51.71%  50.74%  
1 hour  51.30%  50.87%  50.61%  50.26%  
1 minute  50.17%  50.11%  50.08%  50.03%  
1 second  50.02%  50.01%  50.01%  50.00% 
The analysis of probability of foreign exchange change at different scales also showed that hedging for shorter periods require less precise expectations, which better represents the reality.
About the author