Use the foreign exchange market to manage your fortune

devise

At the early age of my study abroad, I was trained to be an expert on economics and finance. I believed that I would be a skillful trader or quant, earning millions of money once I graduated. Nevertheless, later, I met a couple of wonderful statisticians, who have dramatically changed my life course: my interests drifted to statistics and machine learning. Now, I have graduated and am following the path I believe meaningful and searching for my ways to contribute to the world. This is a path of thorns with many difficulties.

For example, one such difficulty is to manage my fortune. Contrary to popular belief, the fortune management is not the privilege of the rich. If I want to lead a sustainable career, I have to take care of it from now on, I told myself. Moreover, it is a good opportunity for me to examine what I have learned almost a decade ago. One cannot tell its sharpness until the sword is drawn out of the sheath, as an old Chinese proverb says. That is why you are reading this post right now, where I am going to document my solution as well as my thought process.

After contemplation, I established the following two requirements:

  1. I deem myself as a scientific researcher, not an investor nor a trader. My goal of fortune management is thus to outrun the inflation, not to become a millionaire.
  2. My business requires me to handle three types of currencies: EUR, USD, and CNY. I need to be able to pay with whichever currency among these three. Naturally, I am exposed to the foreign exchange risk, which I should cover at my best.

Now let us start with the first requirement. To maximize my chance against the inflation, I need to choose a single currency as the means of my fortune storage. There must be one and only one desired currency because the underlying problem is linear programming, where the optimum lies at one of the vertices of the feasible domain.

So which one is the correct answer? One may naturally look at the interest rate and choose the currency with the highest interest rate. The problem is that the interest rate we can find is the nominal one: the one with the highest nominal interest rate may be at the same time the one entailing the highest inflation rate. Even though we can deduce the inflation rate from the interest rate with the help of the CPI, the resulted real interest rate still does not guarantee the best solution. Indeed, a higher interest rate may result in an unfavorable exchange rate, which nullifies its gain.

This problem troubled me for a couple of hours until I recalled the forward exchange rate, which allows me to walk out of the maze of the nominal-real duality. Supposing that there is currently one single currency $C$ in the portfolio, I have the following two choices:

  1. save the currency $C$ until Instant $T$, or
  2. exchange it for the currency $X$, save it until Instant $T$, and then exchange it back for the currency $C$.

If $C$ is the optimal currency, the first choice must be the better one, for any currency $X \neq C$. This question formulation is perfect, except for the exchange rate in the future, which is unknown. To my rescue, the forward exchange rate appears to be the last missing piece of the puzzle: we can replace the exchange rate in the future with the forward exchange rate. By doing so, we not only make an unknown variable known but also secure the end value of the portfolio avoiding any expose to the foreign exchange risk!

This method is amazing, and the solution shows that … each currency is equally good!

In fact, this is reasonable; otherwise, there would be an opportunity of arbitrage.

Alas, we went through all this trouble and finally found … a useless conclusion?

Just about the moment where I was going to give up and called it a day, I noticed a previously neglected fact: the interest rates I used in the formulation were the best rates in the market, to which I do not necessarily have access. If I use only the part of rates that I can lay my hand on, the two choices will make a difference. To ease the comparison, we can calculate the ratio of the exchange rates between the forward one and the current one. The comparison between this ratio and the ratio between the interest rates (you have access to) of the two currencies in question, we can tell which currency between the two is more suitable to you.

>>> # period = np.array([0, 3, 6, 12])
>>> usd_to_cny = np.array([709.02035, 710.22225, 711.05525, 712.23375])
>>> eur_to_cny = np.array([788.45120, 794.70475, 799.95015, 809.87000])

>>> usd_relative_rate = usd / usd[0]
>>> eur_relative_rate = eur / eur[0]

>>> rmb_base_rate = np.array([1, 1.0135, 1.0169, 1.0195])

>>> eur_desired_rate = rmb_base_rate / eur_relative_rate
>>> print(eur_desired_rate * 100 - 100)
[ 0.          0.55247451  0.22824863 -0.74629281]

>>> usd_desired_rate = rmb_base_rate / usd_relative_rate
>>> print(usd_desired_rate * 100 - 100)
[0.         1.1784867  1.39898326 1.49003004]

If you are still following me, the optimal currency is obvious from the calculation above.

The remaining thing is the purchase of the corresponding forward. Unfortunately, this kind of deal is available only to professionals with high reputation, not to individuals. I guess I will have to live with this exchange rate risk.

So far, I have discussed the problem in an ideal world. For instance, the exchange rates used were the middle prices. In the real world, there is the bid-ask spread and may be further constraints. Let us suppose that the currency of choice is CNY and that each time you convert CNY to other currencies, either EUR or USD, you need to pay a 2% fee. This fee is incredibly high, which pushes us to choose a second currency as the secondary means of fortune storage.

One might be tempted to use the same previous method to choose this secondary optimal currency. However, this approach is suboptimal here because of the bid-ask spread. Each time you convert the money, you pay the bid-ask spread. Naturally, you would like to limit the number of conversions. For instance, converting one-half CNY to USD and the other half to EUR is better than converting the entire to either.

Here, again, an interesting thing happened when I was investigating the bid-ask spreads of various pairs of currencies. I discovered that the bid-ask spreads between USD and other currencies are only one half of the ones between other currencies. This phenomenon suggests that the banks use USD as an intermediate between other currencies. For example, if you wish to convert CNY to EUR, the banks will first convert CNY to USD and then USD to EUR. Therefore, it is unwise to use any currency other than USD as the secondary currency.

This concludes my analysis. I protected myself against the inflation and the fee.

Written on October 26, 2019