With discount mortgages, lenders offer a discount off their standard variable rate for an initial period, which then reverts to the standard variable rate when the set period ends. This has the effect of making mortgages appear more affordable when you've just bought the property.
Taking an example of a lender which has a standard variable rate of 6.5%, a 2 year discount of 2% would mean you would pay 4.5% for the first 2 years. If interest rates went up, the lender's standard variable rate would probably also go up, and with it your discounted rate. The same would happen if interest rates fell.
As with fixed and capped rates, the discount period can last from 6 months to 5 years to the whole length of the mortgage. Some lenders will offer a stepped discount, which means that rather than the payments jumping back up when the discount period ends, the size of the discount decreases over a set period. For example, the discount could be 2% in the first year, dropping to 1% in the second, and then 0.5% in the third. This gives you time to make changes to the way you budget your outgoings.
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