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Reading from this and other sources, it appears to me that there are three different ratios between any two currencies, namely

  1. the bid rate
  2. the ask rate
  3. the real rate

My understaning is that the two primal ratios are 1 and 2, while 3 is just a derived quantity, specifically the mid-point between the former two. In other word, I understand that 1 and 2 are the two real, market-driven numbers, and that 3 is just a convenience to know where 1 and 2 roam in that specific historical moment.

Is my understanding correct?

In the following I just report my line of reasoning.

For instance, if I have an amount $A_1$ in one currency and I want to exchange it with another currency, I would receive the amount $A_2$ in this other currency, defined as

$ A_2 = E_{1 \rightarrow 2} A_1$

where $E_{1 \rightarrow 2}$ is the exchange rate.

If I change my mind and decide to convert $A_2$ back to the previous currency, then I will be given the amount $\overline{A_1}$,

$ \overline{A_1} = E_{2 \rightarrow 1} A_2 = E_{2 \rightarrow 1} E_{1 \rightarrow 2} A_1$

I've understood that, if we leave the time elapsed between the two operations aside (or if we do the two exchanges at exactly the same moment), we are sure that $\overline{A_1} < A_1$, because we are sure that in any given moment in time the following holds:

$E_{2 \rightarrow 1} < \displaystyle\frac{1}{E_{1 \rightarrow 2}} \tag{1}$

Is this true?

To me it seems obvious, since if $(1)$ had an equal sign, that would mean that I can freely exchange between currencies at zero cost (hence I could lose or earn money only as a consequence of the change of that unique rate over time due to other factors).

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  • $\begingroup$ I'm not sure what you mean by "real". Let's say I have some USD and my friend has some EUR. We decide to exchange some USD and EUR with each other. We use the mid-market rate, i.e. the average of the buy and sell rates that we see on some website. Is this real enough for you? $\endgroup$ – Kenny LJ Jul 14 '19 at 8:14
  • $\begingroup$ Well, in that case you're both agreeing none of you two wants to earn at the expense of the other one. By real I mean determined directly by the market (if this makes any sense; I'm not an economist). For instance the exchange rate you and your friend agreen on is not real, in that you and your friend have seen some rate on the web and decided you want to use it (so it's indirectly determined by the market). $\endgroup$ – Enrico Maria De Angelis Jul 14 '19 at 8:21
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    $\begingroup$ none of you two wants to earn at the expense of the other one -- this is another statement that has no meaning in economics. When two parties enter into a voluntary trade, both parties gain (or "earn" as you put it), without either party losing. You seem to be creating some artificial distinction between me trading with my friend vs me trading with say a bank. $\endgroup$ – Kenny LJ Jul 14 '19 at 8:24
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    $\begingroup$ we are sure that in any given moment in time the following holds -- No this is false, because at any moment in time, I (or anyone else in the world) can offer exchange rates such that E2→1 and 1/E1→2 are equal. There is no law (legal or economic) forbidding me or anyone else in the world from doing so. $\endgroup$ – Kenny LJ Jul 14 '19 at 8:41
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    $\begingroup$ I think the key here is to understand that there do not exist some "real" or "true" prices. Prices are formed by individuals and any individual is always free to set any price she wants. It makes no sense to think of some prices as being less "real" or "true" than others. $\endgroup$ – Kenny LJ Jul 14 '19 at 8:44
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Indeed the bid and ask spread is very real. The liquidity taker cannot buy and sell at the same price. The market maker extracts a "liquidity premium" for providing liquidity.

When you buy with a market order, you pay the ask price. When you sell with a market order, you get the offer price.

(OTOH, what you mean by "real rate" is not at all clear.)

So, in your hypothetical example, assuming all else constant---yes, after making the round trip trade, you will end up with less than you started. This already happens at, for example, the currency exchange booths at an airport. The booth is acting as the market maker.

Now you can also buy with a limit order. That is, post a offer price and wait. If your order gets hit by someone's sell market order, then you get to buy cheap. Downside is execution risk---you may have to wait a while, possibly a very long while.

Some further comments:

  1. The mid-quote between the best bid and ask could be a vacuous quantity when the spread is equal to the minimal price increment.

  2. The spread is the difference between the best bid and best ask. If an market order size is larger than number of limit orders at those best prices, it will deplete those orders at the best prices and hit orders at the second best price and so on. So if an order size is large or market is thin at the best prices, the spread is a conservative indication of transaction cost. Professional investors split their large orders into smaller ones in order to minimize price impact.

  3. How much the spread matters depends on the type of trader. For a retail trader who occasionally transacts small orders, the cost is probably negligible. On the other hand, slippage is an important issue for institutional investors. Also, on inter-dealer platforms where orders are placed in units of million, spread is closely watched.

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Yes, bid/offer prices are what are directly observed. But that does not mean too much.

In a liquid market, market participants have a target mid rate, and bid/offer prices are as spreads off mid. The bid/offer spread is normally set by convention, and is normally low during stable market conditions.

This means that if you do a sequence of transactions, you normally model it as being at mid rates, and then you have expected slippage from that due to transaction costs: the bid/offer spread on each transaction.

And it is only on an electronic trading platform that “bid/offer” are real. Many markets trade over the phone, and bid/offer are indicative pricing only: the actual price paid is negotiated. In fact, entry into derivatives positions can often be negotiated at mid: it is the unwind where costs show up.

In illiquid markets, bid/offer spreads can be very wide. (For example, some Canadian small caps I used to look at in my personal portfolio would have bid/offers like of \$5-\$6 back in the pre-high frequency trading world.) In this case, the mid is somewhat meaningless. However, they are also an indication of a lack of a market; nobody is trading, so how real are those prices? The only sensible way to trade is to throw in a new order near mid, and see what happens.

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