There are several ways to figure out PPP. The Organization for Economic Cooperation and Development (OECD) does it as described above. They calculate that purchasing power parity for U.S. dollar/yen is now around 106, meaning the yen is still a bit overvalued. Normally, though it's much more overvalued.
On this methodology the yen has been overvalued by around 40 percent on average since 1985 (ranging from 1.4 percent to 113 percent on a monthly basis).There's nothing to say that couldn't turn around, though. Currencies often fluctuate between over- and under-valued according to economic fundamentals and market sentiment.
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Fundamentals push currencies towards fair value over time, but the market usually overshoots fair value, because people tend to move in herds.
In any case, the situation with the yen has changed dramatically from the days when it was overvalued. Japan used to have a trade surplus; now it has a trade deficit. The current account surplus is narrowing too.
The consensus forecast is that Japan's current account surplus will stabilize over the next several years, but I wonder. The current account surplus is the difference between a country's savings and its investment.
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The reason Japan has a current account surplus is because it's been saving more than it's been investing. But those days are coming to an end. Savings, which peaked in 1991, seem likely to decrease further as the country's population ages and the government continues to run a huge deficit.
Retired people spend their savings, while "austerity" doesn't seem to be in the Japanese government's dictionary. On the other hand, I would expect investment to pick up as the yen weakens and Japan becomes a much better place to export from. Lower savings plus higher investment would mean an even smaller current account surplus or even a deficit.
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Moreover, until recently the Japanese were selling off their foreign assets and bringing the money back home. But as they get used to the idea of the yen being weak and as inflation gains a foothold, I expect them to move more money abroad in search of higher returns. That should weaken the yen.
If the yen averaged 40 percent overvalued before, why couldn't it now be 40 percent undervalued under these conditions? That would be quite a turnaround, but nonetheless U.S. dollar/yen has sustained such a mis-evaluation before. (Euro/yen has been as much as 20 percent undervalued on this metric.) That would take U.S. dollar/yen to around 140.
Another way of calculating PPP is to take a time when trade between the two countries was largely in balance – neither side had a big deficit or surplus with the other – and assume that the prevailing rate then was the correct level to balance trade. Then you use the difference in prices in the two countries since then to calculate what the exchange rate should have been to keep the trade balanced.