For most people, employer pensions, government pensions and the tax situation in their country are important factors, typically taken account

For most people, employer pensions, government pensions and the tax situation in their country are important factors, typically taken account of in calculations by actuaries. Ignoring those significant nation-specific factors but not necessarily assuming zero real interest rates, a 'not to be relied upon' calculation of required personal savings rate zprop can be made using a little mathematics. It helps to have a dimly-remembered acquaintance with geometric series, maybe in the form

1+r+r2+ r3 + + r n-1 = (1-rn)/(1-r)

You work for w years, saving a proportion zprop of pay at the end of each year. So the after-savings purchasing power is (1-zprop) of pay while you are working. You need a pension for p years. Let’s say that at retirement you are earning S per year and require to replace a ratio Rrepl of your pre-retirement living standard. So you need a pension of (1 – zprop ) Rrepl S, indexed to price inflation.

Let’s assume that the investments, after price inflation fprice, earn a real rate ireal in real terms where


(1+ ireal ) = ((1+inominal))/((1+fprice ) ) (Ret-01)


Let’s assume that the investments, after wage inflation fpay, earn a real rate i rel to pay where


(1+ i rel to pay ) = ((1+inominal))/((1+fpay ) ) (Ret-02)


[edit] Size of lump sum you need: Is a million enough?
To pay for your pension, assumed for simplicity to be received at the end of each year, and taking discounted values in the manner of a net present value calculation, you need a lump sum available at retirement of:




(1 – zprop ) R repl S {(1+ ireal ) -1+(1+ ireal ) -2 +… ….+ (1+ ireal ) -p}




= (1-zprop ) R repl S {(1 – (1+ireal)-p )/ireal}


Above we have used the standard mathematical formula for the sum of a geometric series. (Or if ireal =0 then the series in curly parentheses sums to p since it then has p equal terms). As an example, assume that S=60,000 per year and that it is desired to replace Rrepl=0.80, or 80%, of pre-retirement living standard for p=30 years. Assume for current purposes that a proportion z prop=0.25 (25%) of pay was being saved. Using ireal=0.02, or 2% per year real return on investments, the necessary lump sum is given by the formula as (1-0.25)*0.80*60,000*annuity-series-sum(30)=36,000*22.396=806,272 in the nation's currency in 2008-2010 terms. To allow for inflation in a straighforward way, it is best to talk of the 806,272 as being '13.43 years of retirement age salary'. It may be appropriate to regard this as being the necessary lump sum to fund 36,000 of annual supplements to any employer or government pensions that are available. It is common to not include any house value in the calculation of this necessary lump sum, so for a homeowner the lump sum pays primarily for non-housing living costs.

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