# Correctly Accounting for Inflation?

For the purpose of 401k retirement planning, suppose:

1. An individual has 20 working years left.
2. Upon retiring, the individual wants to withdraw \$60k per year (in today's dollars) from a nest egg for 30 years.

Assuming the 3% rule, a \$2 million nest egg would last for 30 years? But that's \$2 million in today's dollars. How do I estimate what the monthly contribution should be to build this nest egg over 20 years?

Do you reduce the predicted annual rate of return by inflation or do you increase the minimum size of the nest egg?

I've seen multiple articles and blogs that say the average return of the stock market is 10% over 30 years (citation needed). I assume this doesn't include inflation?

• Have you tried plugging the numbers into a spreadsheet and just playing with the starting number? This is a fairly straightforward way to get your answer as you already have all the variables worked out. – Lawrence Oct 23 '18 at 5:39
• Did none of the articles and/or blogs you read show math for the extremely easily researched 30 year stock marker average return? What do you mean includes inflation? 3% inflation means \$1 today is worth \$0.97 in a year. IF the stock market raises 10% that means your \$1 became \$1.10, but the effect of inflation means your \$1.10 is actually worth \$1.067 of last year's stuff. If you ask me, and I've written answers to this effect, inflation is the boogyman, it happens and is felt broadly over long periods of time and it's not really worth the mental bandwidth to prepare for to this extent. – quid Oct 26 '18 at 7:41

You can use formulas presented here: https://money.stackexchange.com/a/94307/11768

First calculating the size of the pension fund required.

``````p = ((1 + i)^o (1 + m)^-n ((1 + i)^n - (1 + m)^n) w)/(i - m)  (formula 2)

where

n is the number of payments to be received
o is the number of the period at the end of which the first payment is received
w is the payment amount
m is the pension fund's periodic rate of return
i is the periodic inflation rate
``````

With the following criteria

• time now is start of year 1
• final payment into pension pot at the end of year 20
• first retirement payment received at end of year 21
• final retirement payment received at end of year 50, totaling 30 payments

For example, with fund interest at 3% pa and inflation at 2% pa.

``````n = 30
o = 21
w = 60000
m = 0.03
i = 0.02

p = ((1 + i)^o (1 + m)^-n ((1 + i)^n - (1 + m)^n) w)/(i - m) = 2307538
``````

Upon final payment into the fund, at the end of year 20, the fund should total \$2307538 to support 30 subsequent payments of \$60000 at today's value.

The first payment received, at the end of year 21, will be \$90940, and the final payment, at the end of year 50, will be \$161495.

``````w (1 + i)^21 =  90940
w (1 + i)^50 = 161495
``````

Calculating monthly payments into the fund

Formula 4 is designed to expect an immediate first payment, i.e. at the start of period 1, and thereafter for `q` further periods, with the payments increasing to offset inflation.

``````d = ((i - m) p)/((1 + i)^(1 + q) - (1 + m)^(1 + q))           (formula 4)

where

d is the initial payment amount
m is the pension fund's periodic rate of return
i is the periodic inflation rate
q is the number of payment periods
``````

The period is now monthly so converting interest and inflation.

``````m = (1 + 0.03)^(1/12) - 1 = 0.00246627
i = (1 + 0.02)^(1/12) - 1 = 0.00165158
q = 20*12 = 240

d = ((i - m) p)/((1 + i)^(1 + q) - (1 + m)^(1 + q)) = 5835.30
``````

The first payment at the start of month 1 should be \$5835.30.

There are 241 payments in this scheme.

The payments increase like so

``````end of month 1:     d (1 + i)     = 5844.94
end of month 2:     d (1 + i)^2   = 5854.59
end of month 3:     d (1 + i)^3   = 5864.26
. . .
end of month 240:   d (1 + i)^240 = 8670.95
``````

Alternatively, for 241 equal payments (also starting immediately) a simpler formula can be used

``````g = (m p)/((1 + m)^(1 + q) - 1) = 7021.04
``````
• For '2% pa', what is pa? – Seth Oct 23 '18 at 16:11
• 'pa' is per annum. 2% per annum (per year) is equivalent to 2% effective annual interest, or 2% nominal interest compounded annually. ref. link – Chris Degnen Oct 23 '18 at 17:53
• How does formula 4 change if the starting value of the nest egg isn't zero? – Seth Oct 24 '18 at 19:47
• It changes to `d = ((i - m) (p - a (1 + m)^q))/((1 + i)^(1 + q) - (1 + m)^(1 + q))` where `a` is the starting value. There is still a payment at the start of the first period, so the fund starts with `a + d`. As before the final payment into the fund is `d (1 + i)^q`, bringing the fund value to `p` at the end of period `q`. – Chris Degnen Oct 25 '18 at 11:49
• It is similar to the Excel PMT function but includes the inflation term to increase the payments. It is a solution for `d` of the closed-form equation of `nest egg = sum of future values of payments` i.e. `p = (d + a) (1 + m)^q + Σ d (1 + i)^(q - k) (1 + m)^k` for `k = 0 to k = q - 1`. The Excel PMT function does the same kind of thing but is simpler, keeping the payment size the same. PMT would find a solution for `d` from `p = Σ d (1 + m)^k` for `k = 0 to k = q - 1` (for payments at the end of each period). – Chris Degnen Oct 25 '18 at 22:34

It’s easiest just to ignore inflation and do all of your planning in 2018 dollars, looking at real returns rather than nominal returns on your investments. But don’t forget that you then have to increase your monthly contributions each year by the rate of inflation — they can’t be a fixed nominal amount. So your \$2 million in twenty years will probably be closer to \$3 million in 2038 dollars when you come to retire.

• But suppose I want a fixed contribution that I don't have to increase each year? – Seth Oct 22 '18 at 19:40
• @Seth Why not just make it a fixed percentage of your salary? Then it increases automatically as your income increases (hopefully at least on pace with inflation) – D Stanley Oct 22 '18 at 21:49
• @Seth If you really want a fixed contribution, you can use nominal investment returns and quote your required nest egg in 2038 dollars. – nanoman Oct 23 '18 at 2:41
• @Seth Why do you want a contribution that’s fixed in nominal terms rather than in real terms? It will represent an ever-declining proportion of your income, which can generally be assumed to be going up at least in line with inflation. – Mike Scott Oct 23 '18 at 6:28

I use a spreadsheet. The basics are simple. Each row is a year.

• Use today's current income and a reasonable guess for inflation to model how your income will change each year. Assume you slightly outpace inflation.

• For the 401-K/IRA set a dollar figure or percentage for each year. Keep in mind there are annual limits for these funds.

• Estimate the rate of growth. 7% to 10% seems to be the range. I suggest using the lower number.

At the moment of retirement:

• Set a percentage of your final years income that you need to replace.

• Get an estimate from the Social security administration for an estimate of your monthly benefit at different ages of retirement. They update their numbers each year.

• The rest is from retirement funds.

For the retirement years:

• Don't forget inflation.

• your investments might be more conservative so a lower estimated growth may be in order.

Then play with the numbers. See what make sense and is achievable.