Problem $3$

Your grandma just retired at the age of $70.$ She has a tax$-$free retirement annuity coming due soon. She is asked to choose between the following options. She can either receive a lump sum of $$295,000$ now or receive $$2,300$ per month for the rest of her life. She is expecting to live at least $88$ years but not more than $90$ years. She is asking you to find out which is the wiser choice between the two options, if she lives:

a$.$ for $88$ years, and

b$.$ for $90$ years.

Her assumption is that if she accepts the lump sum, she can deposit it in an account earning a market annual interest rate of $7$ percent, compounded monthly and withdraw equal monthly amounts from the account. The annual inflation rate is estimated to be $4$ percent. Using these values, show complete work to help her make the right choice.