In simplest terms, banks make money by investing, or lending out, the deposits of it$$s customers

at a larger interest rate than it pays those customers to keep the money in an account at the bank.

To make this arrangement work well, the bank has to have some assurance that the customers will

keep a portion of their money in their account for a given length of time. One way to do this is to

attract customers larger deposits. Typically, such customers will maintain a higher level of funds

in their accounts for longer, since there is generally limits on rate at which people spend money.

For example, just because you have $10$ times the average account balance doesn$$t imply that your

monthly expenses are $10$ times the average.

Suppose you are working for a bank as the staff modeler for investments and risk. Your boss was

recently pitched a new idea someone from the marketing department designed to attract customers

with higher levels of deposits. Basically, the marketing department wants to offer annual interest

rates on savings accounts that vary with the principle amount. You boss wants you to model this

idea to see if there are any risks associated with offering these rates to customers.

To start this modeling, we need an interest rate that varies with the principle. Consider the

following APR model:

r$($P$)=\backslash$alpha P $(1)$

a$.$ What are the units of $\backslash$alpha $?$

b$.$Use the interest rate model above in the difference equation we discussed in

class for compound interest to derive a differential equation assuming continuous

compounding.

c$.$ Solve the differential equation and comment on any potential risk this model

shows.