I think you are specifically interested in USD but the appreciation of a currency against let's say the basket of currencies (be stronger) is positively correlated with its index increasing. When you say strong, you should specify strong against which currency. But for USD, things have been different especially recently. USD is bought against other currencies when fear is up in markets. So it's not really clear. But generally when you want to buy the stocks at a specific country, you have to buy its currency too, which appreciates its currency. For exporting countries, the same thing applies. Good export can mean good index growth.

 
Best Response

USD represents safety for the most part, so when equities are falling, the dollar is often rallying. This doesn't always happen but in general this is what's been going on recently.

Another reason the dollar is relatively negatively correlated with equity markets is, as someone mentioned, interest rate differential. Let's use AUD as an example, since their rates are pretty high and US rates are essentially zero, a typical trade strategy is called "the carry trade" which essentially involves borrowing in low interest rate areas and lending in high interest rate areas (aka Long AUD/USD). The risk comes from significant deterioration of the AUD/USD rate. So when vol comes back into the market, you often see carry pairs like AUD/USD fall because of the unwinding of the carry trade due to increased exchange rate risk. So in general, currencies like AUD/USD are correlated strongly negative with vol and recently since a drop in the markets has caused vol to increase, AUD/USD has been generally negatively correlated with equity markets.

Hope this helps.

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