What Exchange Rate?

I recently got the following question from a reader:

I have a question. I am a little confused when I read about the "strong Turkish Lira". In 2003, the dollar was about 1.45 TL and in 2008, right before the crash, it was about 1.20 TL. Now it is around 1.58. What am I missing??

The question is: Which exchange rate? As a an investor, the reader is interested in the nominal exchange rate (and the return on asset she has invested), which you can see below:

But in terms of the country's competitiveness, the relative rates of inflation matter as well. That's the concept of the real exchange rate, which adjusts for the inflation differentials between countries. As you can see, it then matters which countries you are comparing Turkey to, and the Central Bank computes RER indices comparing the country to the whole world, developed countries, or developing countries:

Note that purchasing power parity and The Economist's Big Mac index are variations of this concept.  Note also that until of late, Turkish inflation was well above many of its peers, and it is still above the developed world. And most of the RER appreciation, as a result, was coming from the inflation differential rather than the nominal exchange rate. That's why many economists, including yours truly, have argued that the best way of increasing Turkey's competitiveness sort of structural reforms is to keep inflation under tab.

But the big question of the nominal and the real exchange rates is the following: What is the equilibrium value of the exchange rates? In other words, what is the fair value of the lira? There are many different methodologies, and I regularly use (although never at the same time) never less than half a dozen. 

So the reader is after all right on questioning the "strong lira". But that was before the recent bout of weakness. Although I have not done my calculations for a while (lack of incentives, when no one pays me to do it), the analyst reports I have been reading, as well as my own intuition, tell me that the lira is not "that overvalued" anymore...