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:

1. Starting salary, at age 25, is \$30,000.
2. Salary increases each year (until retirement) of 4%.
3. Each year (until retirement), 10% of the salary is invested. Maybe 5% from your salary, matched by employer
4. Return on investments is 8% per year. Too optimistic? Mebbe.
5. 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?
6. 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 seventy-eight. 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?

1. A is the initial salary
2. f is the fraction which is invested (To invest 9% of your salary, put f = 0.09)
3. x is the salary increase factor (For a 4% annual increase, x = 1.04)
4. y is the investment increase factor (For an 8% return on investments, y = 1.08)
5. F is the starting, after-retirement, salary fraction (If you want 47% of your final working salary, put F = 0.47)
6. i is the inflation factor ... and after-retirement 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 Ax2 (fAy+fAx)y+fAx2 = fA(x2+xy+y2) = fA(y3-x3)/(y-x)
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 x2+xy+y2.

Okay, let's continue:
 Year Current Salary Investment Portfolio 4 Ax3 fA(x2+xy+y2)y + fAx3 = fA(y4 - x4)/(y-x) ... ... ... n Axn-1 fA(yn - xn)/(y-x)
After n years, our investment portfolio is now worth:

 fA(yn - xn)/(y-x)

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)(1.0830 - 1.0430)/(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 Axn-1.
Our portfolio decreases by this amount, namely by FAxn-1, leaving fA(yn - xn)/(y-x) - FAxn-1.

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 = FAxn-1 and P = fA(yn - xn)/(y-x)
and we can generate a table, again, like so:
 Year Current Income Investment Portfolio 0 S P 1 Si Py-Si 2 Si2 (Py-Si)y - Si2 = Py2-Si(i+y) 3 Si3 {Py2-Si(i+y)}y - Si3 = Py3-Si(i2+iy+y2) 4 Si4 {Py3-Si(i2+iy+y2)}y - Si4 = Py4-Si(i3+i2y+iy2+y3)
It's time to use our magic formula (for things like i2+iy+y2 and i3+i2y+iy2+y3)
 Year Current Income Investment Portfolio m Sim Pym-Si(ym-im)/(y-i)

>zzzZZZ
Pay attention. Now we determine how long before our portfolio runs dry.
That is, for what value of m does Pym - Si(ym-im)/(y-i) = 0?

Substituting P = fA(yn - xn)/(y-x) and S = FAxn-1 we get:

fA(yn - xn)ym/(y-x) = FAxn-1i(ym-im)/(y-i)

We can rearrange this somewhat and get:

(i/y)m = 1 - (f/iF){(y/x)n-1}(y-i)/{(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, after-retirement, salary fraction example: 0.6 (meaning you can live on 60% of your working salary)
i is the inflation factor ... and after-retirement 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):

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 before-retirement 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 life-style. 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?

 fA(yn - xn)/(y-x)
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:
 Years of Investing = (until retirement) Current Salary = \$ Portfolio Required = \$ (at retirement) Salary Increases = % Return on Investments =% Percentage of Salary that you need to invest = %

Of course, y'all may just be interested in How Big Must My Portfolio Be?
Then try out this calculator, where you ask yourself:

1. "How much would I need today? meaning, in today's dollars, realizing that it'd be more in umpteen years
2. "How long until I retire?
3. "What annual inflation should I expect? over the next umpteen years, until retirement
4. "What Withdrawal Rate will I use? 4% or 5% or x% of your starting portfolio, at retirement, increasing with inflation
 Desired Salary = \$ (at retirement, in today's dollars) Years of Investing = (until retirement) Annual Inflation = % Withdrawal Rate =% Portfolio Required = \$ (at retirement)
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 ...?

... and/or you can play with a Monte Carlo spreadsheet described here.

Or, we can continue in Part II. Are you game?

>zzzZZZ