Question

Pat is the type to plan everything in advance, and she is already starting to plan for retirement. She currently has $0 in her retirement account. She plans to work for 40 years, and then retire for the following 30 years. She expects to spend $120,000 in your first year of retirement, with a 4% growth rate. She will increase her working-years savings rate by 7.5% per year. While working, she expects her account will have a 6.75% return, and during retirement it will have a 4.75% return. After the 30th year of retirement, she wants to have $600,000 in her account for safety net and bequest reasons (It is fine if the amount is not exactly $600,000 due to rounding error). To solve the problem, determine how much Pat needs to save in year 1 to accomplish her retirement goals (use Solver or Goal Seek).

Answer 1

Using PV of growing annuity formula, we calculate the amount required after 40 years to fund retirement expense

PV = P / (r - g) x [1 - ((1 + g) / (1 + r))^n]

= 120,000 / (4.75% - 4%) x [1 - (1.04/1.0475)^30]

= $3,102,688

Now, add the present value of $600,000 at that time, PV = FV / (1 + r)^n = 600,000 / (1 + 4.75%)^30 = $149,118

Total amount needed at retirement = 3,102,688 + 149,118 = $3,251,806

Now, using payment for growing annuity formula

P = FV x (r - g) / [(1 + r)^n - (1 + g)^n]

= 3,251,806 x (6.75% - 7.5%) / (1.0675^40 - 1.075^40)

= $5,533.61 is the amount Pat needs to save in year 1