How do different FIs offer different interest rates for deposits?

I understand that they’re all tied to the repo rate/base rate set by the RBI - what are the other factors that influence the FD interest rate finally offered to customers?

Broadly, all banks and some Non-Bank Financial Companies (NBFCs) can take deposits. For both banks and NBFCs, deposits are a key input (raw material) for giving out loans (finished goods). Like any product pricing decision, FD interest rate is based on a combination of short term demand/supply as well as longer term strategy. In general, banks that have ample current account/saving account balances which are not increasing their loan book in a big way offer lower FD rates.

The interest rate offered for deposits are typically decided by the Asset Liability Management (ALM) Committee, that meets regularly.

Some of the inputs that go into what interest rate to offer depends on
a. What is the current RBI repo rate? How much can the bank borrow from RBI? (Not applicable to NBFCs)
b. What are the other sources of funding available and what are their costs ? (Commercial paper, bonds)
c. How is the demand for loans? What can the bank earn by lending out money? As a result, what is the margin available?
d. What are the recent trends in the customer base? Are there strategic reasons that need to be considered from a customer perspective (customer acquisition/retention etc)
e. What are the broader strategic considerations (diversification of sources of funding, ability to cross-sell other products to new deposit customers)
f. What is the maturity profile of existing loans and deposits? Are there any gaps that need to be addressed? (Eg. Between 9-12 m from now, we expect a need for funds because of a bond maturing)

Throwing some data into the mix for points (a) and (b) . This is from a recent Credit Suisse report on how different banks have been passing through cuts in RBI borrowing rates.