Discount & Interest rates

They are related but they are not same.

An interest rate is broad term for what you pay for the use of assets as a percentage of the value of the asset or principal. See for example definition here.

A discount rate can be the interest rate that Fed charges at its discount window, or it is often in theory used more broadly as the rate at which people discount future cash flows. The latter is also in a sense an interest rate but not interest rate charged by any institution but the one that person would expect to receive to be compensated for lending/investing their resources and postponed consumption.

A discount rate is a form of a rate of interest. The lender lends against a debt instrument at a reduced price relative to par value - a discount.

For example, a firm may have an accounts receivable from a customer that implies a need to pay $100 in three months. A bank may buy the receivable at $98 from the firm, which generates $2 of “interest” on the outlay of $98 (price paid for the receivable). One can then consult money market instrument manuals to determine a quoted rate of interest associated with this transaction. (Different conventions give slightly different results.)

However, it can refer to a particular posted interest rate. The most well-known discount rate is the policy discount rate that is set by the U.S. Federal Reserve. (Other central banks may have discount rates as well.) Banks that are short liquidity can go to the “discount window,” and borrow against their assets (discounting them) at the discount rate. Note that this is normally discouraged, since it implies that the bank was not able to raise funding in the private markets. However, the Federal Reserve encouraged discount window borrowing during the Financial Crisis of 2008 to help break funding problems in the money markets.

Overview: interest rate is a broad term, discount rate is a specific type of interest

An interest rate is a very broad term that applies to a host of assets. Interest rates are paid on mortgages, bonds, CDs, savings account, etc.

The discount rate is a specific type of interest rate that applies to the rate of interest charged to banks borrowing from a central bank like the Federal Reserve. The discount rate is also the term used for the interest rate in discounted cash flow analysis which is a way of valuing companies.

In economics, as opposed to finance, people often use discount factor and interest rate sometimes interchangeably. This is specially when defining inter-temporal preferences. The idea stems from the fact that discount rate and interest rate measures a similar kind of trade-off: current utlity/wealth vs future utility/wealth.

Interest rate, simply put, is the rate at which you borrow against your future wealth. The discount factor determines the rate at which you 'borrow' against your future utility. Hence the comparison.