Saving for Retirement: how much of our salary should we save?

Once upon a time I was talking to my daughter about saving for retirement:
"Put away 5% of your salary," I said, "and invest it, and you'll be in good shape
when y'all retire in, say, thirty years."
After some cogitation, I wasn't sure if she would be in good shape. It depends upon the return she'd get on her investments
and her salary increases and what annual salary she'd need after retirement and how that
would have to increase to keep up with inflation and ... and ...
Anyway, to make a long story short, I decided to analyze this scenario, assuming:
 Starting salary, at age 25, is $30,000.
 Salary increases each year (until retirement) of 4%.
 Each year (until retirement), 10% of the salary is invested. Maybe 5% from your salary, matched by employer
 Return on investments is 8% per year. Too optimistic? Mebbe.
 After thirty years you retire, with an annual salary just 40% or 50% or maybe 60%
of the last year's working salary.
One can live on less, right?
 This retirement salary is taken from the investment portfolio and
increases with inflation, at 3% per year.
Okay, so this is what I got:
Remember the parameters:
Starting Salary: $30,000
Investments: 10% of salary.
Salary growth: 4% per year.
Return on Investments: 8%
Retire afer 30 years  at age 55.
Retirement Salary reduced to 40%, 50% or 60% .
Retirement Salary grows: 3% per year.
If you're satisfied with just 40% of your final working salary, be sure to drop dead at
age seventyeight. If you want 60% of your final working salary
(increasing 3% per year, to be sure),
then be sure to drop dead at ...
Well, you get the idea.
>I'm surprised, especially at the shape of the curve
after retirement; it's concave DOWN ... dropping like a rock!
Me too.


There are so many parameters ... so let's analyze with a wee bit o' math:
>Can I skip this?
Sure. Just click here.
 A is the initial salary
 f is the fraction which is invested
(To invest 9% of your salary, put f = 0.09)
 x is the salary increase factor
(For a 4% annual increase, x = 1.04)
 y is the investment increase factor
(For an 8% return on investments, y = 1.08)
 F is the starting, afterretirement, salary fraction
(If you want 47% of your final working salary, put F = 0.47)
 i is the inflation factor ... and afterretirement income increases.
(For 3% inflation, put I = 1.03)
Okay, we consider the annual salaries, year after year:
Year  Current Salary  Investment Portfolio 
1  A  fA 
2  Ax  fAy+fAx 
3  Ax^{2}  (fAy+fAx)y+fAx^{2} = fA(x^{2}+xy+y^{2}) = fA(y^{3}x^{3})/(yx) 
Note: The investment portfolio is obtained by multiplying last years's portfolio by y
(the investment increase factor) and adding another contribution, namely a fraction f of
the current salary. Further, we've used a magic formula for things like x^{2}+xy+y^{2}.
Okay, let's continue:
Year  Current Salary  Investment Portfolio 
4  Ax^{3}  fA(x^{2}+xy+y^{2})y + fAx^{3} = fA(y^{4}  x^{4})/(yx) 
...  ...  ... 
n  Ax^{n1}  fA(y^{n}  x^{n})/(yx) 
After n years, our investment portfolio is now worth:
For the example considered earlier, A = $30,000, f = 0.1, x = 1.04,
y = 1.08 so we'd get a final portfolio of:
(.1)(30000)(1.08^{30}  1.04^{30})/(1.08  1.04) = $511,444
so we've managed to put away $511,444 from which we can withdraw, during retirement.
Now comes the withdrawal phase:
We begin by withdrawing a fraction F of the final salary of Ax^{n1}.
Our portfolio decreases by this amount, namely by FAx^{n1}, leaving
fA(y^{n}  x^{n})/(yx)  FAx^{n1}.
After a year, this investment portolio has increased by a factor y after which we withdraw
another income which, in the meantime, has increased by the factor i and ...
Hmmm ... for sanitary reasons, maybe we should call the first retirement income S and the
initial retirement portfolio P, so:
S = FAx^{n1} and
P = fA(y^{n}  x^{n})/(yx)
and we can generate a table, again, like so:
Year  Current Income  Investment Portfolio 
0  S  P 
1  Si  PySi 
2  Si^{2}  (PySi)y  Si^{2} = Py^{2}Si(i+y) 
3  Si^{3}  {Py^{2}Si(i+y)}y  Si^{3} = Py^{3}Si(i^{2}+iy+y^{2}) 
4  Si^{4}  {Py^{3}Si(i^{2}+iy+y^{2})}y  Si^{4} = Py^{4}Si(i^{3}+i^{2}y+iy^{2}+y^{3}) 
It's time to use our magic formula (for things like i^{2}+iy+y^{2} and i^{3}+i^{2}y+iy^{2}+y^{3})
Year  Current Income  Investment Portfolio 
m  Si^{m}  Py^{m}Si(y^{m}i^{m})/(yi) 
>zzzZZZ
Pay attention. Now we determine how long before our portfolio runs dry.
That is, for what value of m does
Py^{m}  Si(y^{m}i^{m})/(yi) = 0?
Substituting P = fA(y^{n}  x^{n})/(yx)
and S = FAx^{n1}
we get:
fA(y^{n}  x^{n})y^{m}/(yx) = FAx^{n1}i(y^{m}i^{m})/(yi)
We can rearrange this somewhat and get:
(i/y)^{m} = 1  (f/iF){(y/x)^{n}1}(yi)/{(y/x)1}
or (are you ready for this?):
Notes:
 The parameters are:
A is the initial salary example: 30,000 (meaning you start at $30K per year)
n is the number of years of investing example: 30
f is the fraction which is invested example: 0.1 (meaning 10% of your salary is invested)
x is the salary increase factor example: 1.05 (meaning salary increases of 5% per year)
y is the investment increase factor example: 1.08 (meaning 8% per year return on investments)
F is the starting, afterretirement, salary fraction example: 0.6 (meaning you can live on 60% of your working salary)
i is the inflation factor ... and afterretirement income increases example: 1.03 (meaning 3% annual increase)
 The initial, starting salary ($A, originally assumed to be $30,000) is irrelevant (and doesn't affect the years until "portfolio = zero")
 Any logarithm can be used ... to base "10" or base "2" or base "e" or whatever.
 For our earlier example, with x = 1.04, y = 1.08, f = 0.1, F = 0.4, i = 1.03 and n = 30, we get m = 23
years.
That means, if we started at age 25 and retired at age 55,
we'd run out of money 23 years later, at age 78 ... as indicated in the chart, above.
 The formula for m has, in the numerator, log[1  something]
and that something is (normally) positive but less than 1.
As it gets closer to 1, the value of m becomes infinite (and our portfolio will last
forever)! We'll return to this later.
You can play with the terrible formula (above) with this calculator (below):
If you get a result NaN, it means your portfolio will last forever :^)
Okay, we're now in a position to plot some graphs ...





The most interesting chart (to me, at least)
is Fig. 3 where the number of years (before your portfolio drops to zero)
decreases if you get big beforeretirement salary increases!
The value of x, which measures your salary increases, occurs only in the ratio
y/x which decreases as x increases, making that magic
something, noted above, smaller.
It means, of course, that our retirement income (as a fraction of our final working salary)
will be larger ... and our portfolio will suffer!
I guess the moral is:
Get big salary increases while yer workin'.
Expect a smaller fraction of that big salary
after retirement!

One last thing:
We talked about that magic something which, if it got to be
as large as 1, would guarantee that your retirement portfolio would last forever.
If we set something = 1 we get this formula:
It tells us how much to expect, upon retiring; F = the fraction of our final
working salary. To make F big (so we can live high off the hog, after retirement)
we want f and y/x big and i small and ... well, a picture is worth a thousand bucks:
For example: if we can live with just 35% of our final, working salary (after 35 years of working & investing)
... then our portfolio would last forever!
"Forever" means "until we drop dead"; that's the red point
on the chart
If'n y'all want more than 35%, count on some government pension (?) or old age security (?)
... or a rich uncle drops dead
Above, I said one last thing? I lied. Here's another last thing:
Suppose we need, say, a $1,000,000 portfolio today, to live according to some
lifestyle. If we invested 10% of our annual salary each year at 8%, for thirty years, and
that salary increased by, say, 4% each year, what must our initial salary have been, thirty
years ago?
Remember the Magic Formula above?
That's our final $1,000,000 portfolio and we put f = 0.1, y = 1.08 and
x = 1.04 and n = 30 and we get:
17.048 A = 1,000,000
so our original salary (thirty years ago!) must have been:
A = 1,000,000/17.048 = $58,657
Thirty years ago? Fat chance!
In fact, if we assume that notorious "Safe Withdrawal Rate" of 4%, after retirement, then
we must expect to withdraw 4% x $1,000,000 = $40,000 per year. That's
presumably what we need to live comfortably. Suppose we can live on half of our final
working salary.
Then our final working salary must be 2 x $40K = $80K. If this final salary grew at 4% per year,
then it must have been $80,000/(1.04)^{30} =
$24,665, thirty years ago.
Our initial salary must have been $58,657 ?
Our initial salary must have been $24,665 ?
What's going on here?
With an unrealistic starting salary of $58,657 (30 years ago) we'd have made our $1M portfolio.
With the more realistic starting salary of $24,665 (30 years ago) we would not make the $1M portfolio.
So here's the Big question:
If we need umpteen dollars in our portfolio, today, then what percentage of our salary
must we have invested thirty (or forty years) ago?

I think you'll find the answer scary.
>I can hardly wait.
For example:
 If we are currently making $50K and our salary increased
at the rate of 3% per year for the past 30 years, then (30 years ago) we were making $20,599.
 If we wanted a portfolio of $1,000,000 today, then we would have had to invest
over 25% of our salary at 9%
... for the past 30 years
Of course, if your current salary is larger you needn't have invested quite such a large percentage
of your salary.


If you're satisfied with using the Magic Formula,
you can play with this calculator to estimate investment requirements in order to achieve
some (future) portfolio ... remembering that $1,000,000 thirty years from now is worth MUCH less
than it would be today, due to inflation. So, if you'd like the equivalent of today's
$1M (thirty years from now at 3% inflation) ask for $2.4M:
Of course, y'all may just be interested in How Big Must My Portfolio Be?
Then try out this calculator, where you ask yourself:
 "How much would I need today? meaning, in today's dollars, realizing that it'd be more in umpteen years
 "How long until I retire?
 "What annual inflation should I expect? over the next umpteen years, until retirement
 "What Withdrawal Rate will I use? 4% or 5% or x% of your starting portfolio, at retirement, increasing with inflation
Note: If you pick a BIG withdrawal rate you may run out of money before you drop dead!
>So, how can I save that much from my salary in order to ...?
All of the above assumes that everything stays constant ... for years and years.
There's no change in inflation or return on investment or salary increases or etc. etc. To see a more
realistic scenario, see Investing
... and/or you can play with a Monte Carlo spreadsheet described
here.
Or, we can continue in Part II. Are you game?
>zzzZZZ
Click for Part II
>But what if I start with an existing portfolio, say $B?
Good question! When the withdrawal phase begins, your initial portfolio (which we've called
P, above) will be increased by n years of gains (for your initial $B) and using the annual
gain factor of y, we get:
P = fA(y^{n}  x^{n})/(yx) + B y^{n}
where we've added the final value of the initial portfolio,
namely B y^{n}
We can still use the above terrible formula by changing the value of f, like so:
P = fA(y^{n}  x^{n})/(yx) + B y^{n}
= f^{*} A(y^{n}  x^{n})/(yx)
where
f^{*} =
f + (B/A)(yx)/(1(x/y)^{n})
In other words, the effect of starting with a portfolio which is, say,
K times your
initial salary (so B/A = K) is to increase the effective
saving rate from f to
f^{*} = f + K(yx)/(1(x/y)^{n})
and that can be quite significant, for example:


