Question

A recent college graduate has taken a new job at Work LLC, and since the company does not offer a traditional pension plan, she plans to take advantage of a tax-free investment account backed by a reputable financial institution that offers a guaranteed 8% annual return for as long as she lives. The graduate plans on working for 45 years before retiring and will save a fixed amount each year until retirement, starting at the end of this year and continuing for all 45 years of work. Once she is retired, she expects to be able to live on the equivalent of $30,000/year in today’s terms in addition to expected social security payments. She expects annual inflation to be 4% per year over her life. She doesn’t know how long she will live, but knows that with medical advancements, it could be for a very long time. Since one of her great fears is that she will outlive her savings, she plans to arrange retirement funding that will be in place if she were to live “forever” with the understanding that her heirs will inherit the remainder when she dies. If she wants to save a fixed amount each year, starting with one year from now until her 45th work anniversary, how much does she need to save each year? Identify the problem – All information needed to identify the problem is included in the case, and the student must select the needed information and represent it in a simplified form, which could be a combination of words and graphical representations. Explain the strategy for solving the problem – Although there may be more than one way to solve a problem, the student must choose a method and explain how it will be applied. Application of solution tools – The student must apply mathematical and logical tools to solve the problem. These may include formulae, logical deductions, and other appropriate devices. Describe the solution – The student must provide a solution to the problem. The quality of the solution will be evaluated.

- I'm specifically curious about the bolded questions, not the final answer. Thank you.

Answer 1

Soln : Here in this case, 1st of all we need to understand , how much yearly amount is required in terms of today's $30000/year, inflation rate = 4% per annum

Value of 30000 after 45 years = 30000*(1.04)^45 = 175235 .27

So, this amount is to be paid to her every year. Now, this amount is to be paid forever as per the given condition.

let's calculate the value of all the payments, each year, done after 45 years of working, at t = 45 years working

PV = Amount/rate of interest (infinite annuity formula )

PV = 175235.27/0.08 = 2190441

Amount required to be saved after 45 years of working = 2190441

Let X be the amount to be saved each year for 45 years of working.

Now, the summation of this amount , FV of this yearly amount to be summed up to the value calculated above

X*(F/A,8%,45) = 2190441

X*((1.08)⁴⁵-1)/0.08 = 2190441

On solving we get X = 5667

Hence the amount to be saved each year = $5667