Hybrid units of wealth (and a surprising guarantee)

Suppose you have some New Zealand dollars (\mathrm{NZD}) and some other valuable, fungible, relatively divisble items called, say, \mathrm{XYZ}. If the value of \mathrm{XYZ} increases, then the value of your portfolio (measured in \mathrm{NZD}) increases.

But the value of \mathrm{XYZ} increasing is indistinguishable from the value of \mathrm{NZD} decreasing, so if you measure the value of your portfolio in \mathrm{XYZ}, then an increase in the value of \mathrm{XYZ} appears as a decrease in the value of your portfolio.

So which is correct? Has your portfolio increased in value, or decreased in value? Well, it depends on whether \mathrm{XYZ} or \mathrm{NZD} is more useful to you. But what if they’re equally useful to you? What if \mathrm{XYZ} is another currency that you use as often as \mathrm{NZD}?

Well, we could try to measure the value of the portfolio in hybrid units. Suppose

1\,\mathrm{XYZ} = x\,\mathrm{NZD}

Take the square root of both sides. I’m serious.

1\,\sqrt{\mathrm{XYZ}} = \sqrt{x}\,\sqrt{\mathrm{NZD}}

Now multiply both sides by \sqrt{\mathrm{XYZ}}:

1\,\mathrm{XYZ} = \sqrt{x}\,\sqrt{\mathrm{NZD}\,\mathrm{XYZ}}

Similarly, 1\,\mathrm{NZD} = \frac{1}{\sqrt{x}}\,\sqrt{\mathrm{NZD}\,\mathrm{XYZ}}. So if you have m\,\mathrm{NZD} and n\,\mathrm{XYZ}, then your portfolio is worth (\frac{m}{\sqrt{x}} + n\sqrt{x})\,\sqrt{\mathrm{NZD}\,\mathrm{XYZ}}.

Now here comes the surprising part.

There’s a well-known mathematical theorem called the inequality of arithmetic and geometric means, or the AM–GM inequality, to its friends. For now, we only need to consider the means of two numbers, in which case, the AM–GM inequality tells us that, given non-negative real numbers a and b, it is the case that \frac{a + b}{2} \geq \sqrt{ab}.

If a = \frac{m}{\sqrt{x}} and b = n\sqrt{x}, then the inequality tells us that (\frac{m}{\sqrt{x}} + n\sqrt{x}) \geq 2\sqrt{mn}; that is, the value of your portfolio cannot fall below 2\sqrt{mn}\,\sqrt{\mathrm{NZD}\,\mathrm{XYZ}}.

Now is probably a good time to note that the above assumes that neither the value of \mathrm{NZD} nor the value of \mathrm{XYZ} drops to 0; otherwise you end up trying to divide by 0. Also, if the value to you of \mathrm{NZD} or \mathrm{XYZ} drops to 0 (or close to it), then the value to you of \sqrt{\mathrm{NZD}\,\mathrm{XYZ}} also drops to 0 (or close to it).

The AM–GM inequality carries a coda that says that equality holds if and only if a = b. So if you ensure that \frac{m}{\sqrt{x}} = n\sqrt{x} (that is, the value of your \mathrm{NZD} holdings matches the value of your \mathrm{XYZ} holdings), then the value of your portfolio is exactly 2\sqrt{mn}\,\sqrt{\mathrm{NZD}\,\mathrm{XYZ}}; furthermore, any fluctuation in the exchange rate between \mathrm{NZD} and \mathrm{XYZ} must increase the value of your portfolio (as measured in \sqrt{\mathrm{NZD}\,\mathrm{XYZ}})!

Once this fluctuation has happened, you can trade so that you again have the same value in \mathrm{NZD} as in \mathrm{XYZ}, and any subsequent fluctuation will again increase the value of your portfolio (as measured in \sqrt{\mathrm{NZD}\,\mathrm{XYZ}}).

If someone tried to tell me this, I’d be skeptical about “guaranteed profits”, and so you should be, too. Check my reasoning carefully. And remember that it assumes that \mathrm{NZD} and \mathrm{XYZ} are equally valuable to you, and that the value of neither will ever drop to 0; if these assumptions don’t hold, then you might end up worse off by following this strategy. It’s not a get-rich-quick scheme. It’s a get-rich-quite-slowly-if-both-\mathrm{NZD}-and-\mathrm{XYZ}-are-valuable-to-you-and-the-value-of-neither-drops-to-0 scheme.

You’ll get rich faster if the exchange rate between \mathrm{NZD} and \mathrm{XYZ} is very volatile, and you’ll do so by very slightly dampening the volatility. Essentiallly, you have stockpiles of \mathrm{NZD} and \mathrm{XYZ}, and you build up your stockpiles in times of plenty, in order to supply them to people who want them in times of scarcity; you profit by providing this service.

I wouldn’t be surprised if I found I’m legally obliged to tell you that this is not investment advice. So: This isn’t investment advice. It’s theoretical musing about a possible trading strategy, and should not be construed as a recommendation to use that strategy, or to avoid using that strategy.

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