You have recently been hired as a consultant for a personal financial planning firm. One of your first projects is creating a retirement plan for a young couple, Henry and Samantha Carlisle. They have just celebrated their $40$th birthdays and have decided to get serious about saving for retirement.

Henry and Samantha hope to retire $25$ years from now $($on their $65$th birthdays$),$ and they expect to live until age $85.$ Their hope is to be able to withdraw $$155,000$ a year from their retirement account $$ the first withdrawal will occur on their $65$th birthdays, and the $20$th and final withdrawal will occur on their $84$th birthdays. They expect to leave their children a total inheritance of $$1,500,000$ to split amongst themselves. So$,$ on their $85$th birthdays, the account is expected to have a $$1,500,000$ value $($i$.$e$.,$ they expect to leave their children a total inheritance of $$1,500,000).$

Henry and Samantha currently have $$72,000$ saved in a retirement account, which consists of a portfolio of mutual funds that is expected to produce an annual return of $6\%.$ To accomplish their goals, they would like to deposit an equal annual amount into their account starting one year from today $($on their $41$st birthdays$)$ and continue to make those deposits through age $65.($Again$,$ the account has an expected annual return of $6\%.)$ Thus, they will make $25$ annual end$-$of$-$year deposits to this account.

a$.$ How much do Henry and Samantha need to deposit into the account at the end of each of the next $25$ years to accomplish their goals?

b$.$ Henry and Samantha recognize that the value of their $$155,000$ annual withdrawals during retirement will steadily decline because of expected inflation. Assume that they want to have the value of these withdrawals increase by $4\%$ a year for the first $9$ years after their first withdrawal and then increase the withdrawals by $5\%$ a year for the last $10$ years of retirement to account for expected inflation. In other words, they want to withdraw $$155,000$ at age $65$ and $$161,200$ at age $66($$$155,000\backslash$times $1\mathrm{.}04).$ The last withdrawal at the $4\%$ inflation rate will be $$220,613\mathrm{.}33$ at age $74.$ After then, the retirement withdrawals will grow at $5\%$ a year $--$ so the retirement withdrawal at age $75$ will be $$231,644\mathrm{.}00($$$220,613\mathrm{.}33\backslash$times $1\mathrm{.}05).$ From that point forward, the withdrawals will increase $5\%$ a year up until age $85.$

How much would they need to deposit into the account at the end of each of the next $25$ years to meet this revised goal, which protects them against rising inflation? Assume they still plan to leave their children a total inheritance of $$1,500,000.$ Set up this problem using Excel.