A variable APR on a credit card serves two purposes. For the lender, the variable rate insures that the money it has lent or will lend is always being paid back at the current market interest rates plus a profit margin. For the borrower, the variable rate may allow the card to have a lower starting rate than what is available on a fixed rate card. It may also allow the rate to drop if interest rates fall.
If the borrower can get the same rate on a fixed interest rate card as on a variable rate card, then it is advantageous to choose the fixed card when interest rates are low and the variable rate card when rates are high. If the borrower can get a cheaper rate on a variable rate card, then it is usual to their advantage to go with the variable rate card.
With a variable APR rate on a credit card, if interest rates rise, the cost of the interest on the balance will rise too. This may raise the minimum payment, which can make it difficult to pay off a credit card bill each month. Falling behind on such payments will adversely impact your credit score.
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All credit cards are required to disclose the terms of the card account up front. While the front page may be full of color and catchy headlines, the legally required information will be listed on the back of the application. One of the items on that page will say either "Variable" or "Fixed" so the borrower knows up front how the APR on their card will work.
Variable APR credit cards tend to have their interest rates rise quickly when market interest rates rise, but fall much more slowly when rates fall, so it is not often in the borrower's best interest to have a variable rate card even if rates are expected to decline.
Virtually all variable APR cards have a "floor" interest rate, which is the minimum amount of interest that will be charged regardless of how low interest rates go. This "floor" can negate the entire benefit of having a variable rate card, since lower interest rates may not be able to offset higher rates.