Why are average interest rates often weighed by loan (and deposit) sizes?
Since the size and interest rate of a loan are function of each other, I expect the resulting statistic to be hard to interpret.
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When a bank lends money at different rates for different purposes the weighted average interest rate measures how well the 'average dollar' is being utilized by the bank. (A very small amount lent at a very high rate for example does not impact overall bank profitability very much). The average is a kind of 'overview' statistic for the bank as a whole, but I agree with you that if you want to understand WHY Bank A has a higher average than Bank B you would have to look at more detail on the loans themselves, such as their size, how long they have been on the books (years ago interest rates were higher) etc. etc.