(1+ i rel to pay )+ 1 } = zprop S ((1+i rel to pay)w- 1)/i rel to pay Equate and derive necessary saving proportion To make the accumulation match with the lump sum needed to pay pension: zprop S (((1+i rel to pay )) w – 1)/i rel to pay = (1-zprop ) R repl S (1 – ((1+i real)) −p )/i real Bring zprop to the left hand side to give the answer, under this rough and unguaranteed method, for the proportion of pay that should be saved: zprop = R repl (1 – ((1+i real )) −p )/i real / [(((1+i rel to pay )) w – 1)/i rel to pay + R repl (1 – ((1+i real )) −p )/i real ] (Ret-03) Note that the special case i rel to pay =0 = i real means that the geometric series should be summed by noting that there are p or w identical terms and hence z prop = p/(w+p). This corresponds to the graph above with the straight line real-terms accumulation. Sample results The result for the necessary zprop given by (Ret-03) depends critically on the assumptions made. As an example, one might assume that price inflation will be 3.5% per year forever and that one's pay will increase only at that same rate of 3.5%. If a 4.5% per year nominal rate of interest is assumed, then (using 1.045/1.035 in real terms) pre-retirement and post-retirement net interest rates will remain the same, irel to pay = 0.966 percent per year and ireal = 0.966 percent per year. These assumptions may be reasonable in view of the market returns available on inflation-indexed bonds, after expenses and any tax. Equation (Ret-03) is readily coded in Excel and with these assumptions gives the required savings rates in the accompanying picture. Monte Carlo: Better allowance for randomness Finally, a newer method for determining the adequacy of a retirement plan is Monte Carlo simulation. This method has been gaining popularity and is now employed by many financial planners. Monte Carlo retirement calculators allow users to enter savings, income and expense information and run simulations of retirement scenarios. The simulation results show the probability that the retirement plan will be successful. Early retirement Retirement is generally considered to be "early" if it occurs before the age (or tenure) needed for eligibility for support and funds from government or employer-provided sources.