Question

How do you know if a currency is overvalued or undervalued

I have 2 questions with answers that don't seem to be matching:

If real exchange rate qUS/qforeign is more than 1, then US currency is said to be undervalued correct?

Question 1: You are given the following information. The current dollar-pound exchange rate is $2 per pound. A U.S basket that costs $100 would cost $120 in the United Kingdom. For the next year, the Fed is predicted to keep U.S. inflation at 2% and the Bank of England is predicted to keep U.K inflation at 3%. The speed of convergence to absolute PPP is 15% per year.

Answer: The dollar is overvalued because real exchange rate is 1.29 (=$120/$100). shouldn't it be undervalued since it is more than 1?

Question 2:

Suppose a new car costs 210,000 Mexican pesos in Mexico, while the same car costs $19,500 in the United States. The nominal exchange rat is currently at E$/Peso = $.10/Peso. If we assume PPP to hold, is the dollar under or overvalued? If so, by how much?

qUS/qPESO= (210,000)*(.1)/(19,500)=1.0769. The answer is that the US is overvalued as well.

Why is it that qUS/qForeign are both more than one but US currency is overvalued in one of them but undervalued by another. Is the answer wrong or am i missing something?

Answer 1

The way you have determined that US currency is overvalued than the Mexican Peso in Question 2 is correct (based on the nominal sport rate).

In question 1 however, the current nominal spot exchange rate is $2 per Pound while the real exchange rate is certainly less than that as that. Here's why

Per spot rate: qUS/qGBP = 2

Per real rate: qUS/qGBP = $120/$100 = 1.2

(Theoretically, should ideally cost GBP 50 or $100 but costs $120, so the US dollar is commanding a premium)

So right now, the spot or the nominal rate for for US/GBP is higher than the real rate and hence USD is overvalued.

Alternatively, like you calculated in question 2, applying the exact same logic in question 1, the same $100 (in the US) item costs $120 in the UK, so qUS/qGBP = $120/$100 which gives a value greater than 1 resulting in the USD being overvalued.

Where you are getting confused in Question 1, is that you are assuming that qUS is in the denominator but it is not. IT is qUS/qGBP which is greater than 1 so US currency is overvalued.