Minimum Required Nest Egg Formula?

Question

As a follow-up to this question, what is the minimum size of a nest egg to support 30 years of $5,583.33 monthly distributions? I want the $5,583.33 to increase with inflation each month.

The present value of the first payment should be $5,583.33.

First withdrawal will be in 20 years: $5,583.33*(1 + 0.0033)^240 = $12,310.86

Here is what I've pieced together from this question:

Total withdrawals:          n = (30 years)(12 months) = 360 payments
Inflation per period:       i = 4.0% per year / 12 = 0.3333% per period)
Return per period:          m = 8.0% per year / 12 = 0.6666% per period)
Periods until 1st payment:  o = (20 years)(12 months) = 240 periods
First payment amount:       w = $67,000 / 12 = $5,583.33 (today's dollars)

p = ([(1 + i)^o]*[(1 + m)^-n]*((1 + i)^n - (1 + m)^n)*w)/(i - m)  
p = ([(1 + 0.0033)^240]*[(1 + 0.00667)^-360]*((1 + 0.0033)^360 - (1 + 0.00667)^360)*5583.33)/(0.00333 - 0.00667)
p = $2,594,790.06

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

Is this the right equation? From what I could find on Google, this calculation is called the present value of a growing or graduated annuity. Is this correct?

Is it correct to say that $2.5 million is the nest egg balance 20 years from now on the day the first withdrawal is made? And that $2.5 million is not in today's dollars but in equivalent dollars 20 years from now?

Answer 1

If you want the first payment amount to be $5583.33 (unadjusted for inflation), o should be set to zero because o sets the number of inflation periods prior to the first payment received, (so that adjustment can be set from within the saving period).

To illustrate with a simple example, showing 4 deposits and 3 withdrawals.

Planning to retire in 4 months and draw monthly income of $1000 for 3 months, adjusted for inflation starting from the first withdrawal. APR is 8% and inflation is 4%, both nominal rates, compounded monthly. What should the pot be?

Calculating the monthly rates.

inf = 0.04
i = inf/12 = 0.00333333

apr = 0.08
m = apr/12 = 0.00666667

There are to be 3 payments received in total, at the end of periods 4, 5 & 6. The first payment should be $1000 unadjusted for inflation. The second and third payments are to be adjusted for inflation.

Calculating the pot at the end of period 3 (using formula 2).

w = 1000
n = 3
o = 0

p = ((1 + i)^o (1 + m)^-n ((1 + i)^n - (1 + m)^n) w)/(i - m) = 2970.28

Checking the result

at the end of month 3, p = 2970.28
at the end of month 4, p = p (1 + m) - w (1 + i)^0 = 1990.59
at the end of month 5, p = p (1 + m) - w (1 + i)^1 = 1000.12
at the end of month 6, p = p (1 + m) - w (1 + i)^2 = 0

So by the end of month 6 the pot is empty.

The three payment amounts are

w (1 + i)^0 = 1000
w (1 + i)^1 = 1003.33
w (1 + i)^2 = 1006.68

Returning to your figures.

w = 5583.33
n = 30*12 = 360
o = 0

p = ((1 + i)^o (1 + m)^-n ((1 + i)^n - (1 + m)^n) w)/(i - m) = 1167478.60

The pot should be $1,167,478.60 at the start of the month prior to the first withdrawal, which will be $5583.33.

With adjustment for inflation the final payment will be $18,438.89.

w (1 + i)^(360 - 1) = 18438.89

To illustrate what type of calculation this is, let inflation be zero. Then all the payments are $5583.33 and the required pot is only $760,915.72.

i = 0

p = ((1 + i)^o (1 + m)^-n ((1 + i)^n - (1 + m)^n) w)/(i - m) = 760915.72

Demonstrating with Excel.

PV(0.08/12, 360, -5583.33, 0, 0)

$760,915.72

PMT(0.08/12, 360, 760915.72, 0, 0)

-$5,583.33

Excel correctly calculates the present value and payment amount. However, it does not have the facility to add in an inflation factor.

The Excel PMT calculation with cashflow at the end of each period uses the calculation of present value of an ordinary annuity, where the present value is p.

https://www.investopedia.com/retirement/calculating-present-and-future-value-of-annuities/

Derivations

The Excel PMT-type function can be derived from the sum of the present value of payments by induction.

∴ w = m (1 + 1/((1 + m)^n - 1)) p

E.g.

m = 0.08/12
n = 360
p = 760915.72

w = m (1 + 1/((1 + m)^n - 1)) p = 5583.33

With an inflation terms added: i and o, the sum of the present value of the payments becomes this, (formula 2).