The basic equation for calculating net present value $($NPV$)$ is:

t: time of the cash flow

N: total customer lifetime r: discount rate

Ct: net cash flow $($the profit$)$ at time t $($The initial cost of acquiring a customer would be a negative net cash flow at time $0.)$

$(1)$ Assume that a customer shops at a local grocery store, spending an average of $$200$ a week and resulting in a retailer profit of $$10$ each week from this customer. Assuming the shopper visits the store all $52$ weeks of the year, calculate the customer lifetime value if this shopper remains loyal over a $10-$year life span. Also, assume a $5$ percent annual interest rate and no initial cost to acquire the customer.