Personal Finance Problem

Funding your retirement John Wick is $50$ years old and has saved nothing for re$-$

tirement. However, he has just inherited $150,000$ from his great grandfather and

wants to invest it in a retirement plan. He plans to put the money in an investment

account earning a $10\%$ return. He will let the money accumulate for $15$ years, when

he will retire at the age of $65.$ He would like to deposit enough money so that he can

begin making withdrawals of $40,000$ per year starting at age $66$ and continuing for

$24$ additional years, when he turns $90.$ He wants the balance in his account after the

last withdrawal to be $0.$ After investing in his retirement plan, John wants to spend

the rest of his money on a world tour.

a$.$ How much money must John invest to achieve his goal? Construct a timeline to

help visualize the details of this investment plan.

b$.$ John realizes that once he retires he will want to invest only in less risky assets,

like government securities that will earn a slightly lower rate of return$-5\%.$ So

he will earn $10\%$ until age $65,$ and then $5\%$ per annum from age $65$ to $90.$ How

much money does he need to set aside now to achieve his retirement goal?

c$.$ Suppose John has changed his mind and invests the entire inheritance into the

account earning $10\%.$ After making his last withdrawal at age $90,$ he wants to

leave the remaining money to his niece. How much would his niece receive when

John withdraws the last annual installment of $40,000?$