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    Treasurers have the technology available to them to help quantify and model potential risks for hedging as well as for optimising payments and minimising the impact of foreign exchange risk. Utilising this technology well could help uncover surprising insights about risk management and help identify best practices for hedging. For example, a simulation exercise on a simple coin toss game below would reveal something more than meets the eye.

    The 50-50 game

    Let's play a game. You have to flip a coin 263 times. You make GBP1 if the coin returns head and lose GBP1 if it returns tail. You will have to keep track of your cumulative profit and loss (P&L) as you do so. After 263 coin flips, which of the following P&L scenarios is the most likely?

    A. You spent roughly 50 per cent of the time in the positive (including par) and roughly 50 per cent of the time in the negative.

    B. You spent roughly 70 per cent of the time in the positive (including par) and roughly 30 per cent of the time in the negative.

    C. You spent roughly 90 per cent of the time in the positive (including par) and roughly 10 per cent of the time in the negative.

    You might be surprised to learn that the correct answer is actually C and by a long mile. Probabilistically speaking, C is more than twice as likely as either A or B (to be exact, they are 2.53 times and 2.25 times, respectively). In other words, even B is 12.5 per cent more likely than A.

    Note that the arc sine law is true of any time horizon

    Intuition would have told us that a fair, random process should average out in the long run. However, statistical theory tells us the exact opposite is happening in this particular context.

    Arc sine law

    The above exercise can be used to depict an unhedged exposure over a one-year period. For this particular exposure, every business day can be analogous to a fair coin flip. It has a 50-50 chance to either go up or down in value. After a year, or 263 business days, the treasurer would most likely find that the same exposure has been in a gain (or in a loss) almost all the time throughout the year. This is mathematically guaranteed by the Lévy's arc sine law.

    Note that the arc sine law is true of any time horizon. Consider long-term exposures such as net investment exposures, which we have observed a lot of treasurers tend to regard as perpetual exposures and therefore do not normally hedge. To illustrate, let's assume the asset will be held for 100 years. The arc sine law states that the company is most likely to report 99 out of 100 years of outstanding net investment gains, or 99 out of 100 years of outstanding net investment losses, than it is to report 50 years of outstanding gains plus 50 years of outstanding losses.

    Implications for treasury management

    Arc sine law can be highly counter-intuitive to many.

    While markets may behave in a random fashion, the arc sine law tells us that random processes do not tend to even things out, unlike what our intuition would have us believe.

    Hedging cannot be overlooked on the basis of mean reversion or market randomness

    This intuition that we wrongly inferred from a random process may also be giving us a false sense of hope should there be any existing loss position: that it could one day revert back to par. But the arc sine law tells us that such loss position may be stickier than one would assume. In fact, the correct intuition should be that the loss position could become twice as large as it is likely for the loss position to disappear.

    Corporates often need to report cumulative gains or losses of specific items on their balance sheet, including their net investment gains or losses. The arc sine law reminds us that this gain or loss number will more likely be persistent than it is to change signs. This is why hedging cannot be overlooked on the basis of mean reversion or market randomness, lest we fall into the heuristics of evenness.

    Given the complexities of treasury risk management, the HSBC Thought Leadership has put together a dedicated team of experts to help corporates optimise their strategy.

    Next month's topic

    Next month, we will show how people tend to overestimate their ability and/or luck and how this hardwired optimism could affect treasury risk management.

    For more information, contact us:
    thought.leadership.global@hsbc.com

    Find out more markets insights in our New Future series.

    Two ways to identify best practices for hedging - Simplified Chinese

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