The other day, while browsing my tutorials ... and correcting typos
I kept running across a certain sum.

It showed up in so many different scenarios I thought I should summarize ... but first, some terminology:

• We'll use i to represent an Inflation Rate. (Example: if inflation is 3%, then i = 0.03)
• We'll call I = 1 + i an Inflation Factor. (Example: 1.03)
• We'll call r an Annual Return on some investment. (Example: Annual Return is 8.9% so r = 0.089)
• Then g = 1 + r is the corresponding Gain Factor. (Example: g=1.089 means a $1.00 investment grows to $1.089 over the year)
• For years 1, 2, 3, ... up to year N, we'll call the inflation rates i₁, i₂, ... i_N and similarly for the Annual Returns and Factors etc.
• G_n is the cumulative Gain Factor over n successive years, so G_n is the product of a bunch of annual Gain Factors, like:
  G_n = g₁g₂g₃...g_n

Now, we're ready for our Magic Sum:

>gMS? I assume that MS stands for Magic Sum, eh?
Yes, and g stands for grand or maybe good or maybe great or ...

>Or maybe gummy. So why is it magic?
Suppose we ignore inflation, putting i₁ = i₂ = ... = i_N = 0, then I₁ = I₂ = ... = I_N = 1 and our gMS is 1/G₁ + 1/G₂ + ... + 1/G_N

      gMS = 1/{the Harmonic Mean of the Gs}

This Harmonic Mean will pop up ... later.

>We're buying every month? That's DCA, eh?
Dollar Cost Averaging. Yes, and we'd buy, each month, a fixed dollar amount, say $A worth of stock or mutual fund.
Then at a price P_n we'd be buying A/P_n shares. Initially, we'd buy A/P₀ shares but since the price, after n months, is P₀G_n,
the number of units we're buying (after n months) is {A/P₀} / G_n
and the total number of shares we'd buy after N years is ...

>Don't tell me! It's the gMS!
Pay attention! It's A/P₀{ 1 + 1/G₁ + 1/G₂ + 1/G₃ + ... + 1/G_N } = A/P₀ (1 + gMS)

>Well, I was close. Besides, you ignored inflation and ...
Suppose we invest $A initially, but increase that amount with inflation. After all, we can afford to buy more since our salary is going up with inflation and ...

>Salary going up? You're kidding, right?
... in which case the amounts invested go up, like $A, $A I₁, $A I₂ etc., and the number of units we'd have after N years is
A/P₀{ 1 + I₁/G₁ + I₂/G₂ + I₃/G₃ + ... + I_N/G_N }

We'll give it a place of some importance:

In a tutorial on Safe and Sensible Withdrawals we got a related Magic Formula:

We're retired and start to withdraw a fraction f of our portfolio, this amount increasing with inflation. Will our portfolio last for N years?
Answer? YES, provided f isn't too large.
(f is a fraction: if we withdraw 3.5% each year, increasing with inflation, then f = 0.035.)

In fact f should be no greater than 1/{ I₁/G₁ + I₂/G₂ + I₃/G₃ + ... + I_N/G_N }

Note:
The above magic formula assumes your first withdrawal is at the end of the first year and your N^th withdrawal reduces your portfolio to $0.
Note that your N^th withdrawal is then at the end of the N^th year.

If your first withdrawal is at the start of the first year (and your N^th withdrawal reduces your portfolio to $0), the magic formula is:
1/{1 + I₁/G₁ + I₂/G₂ + I₃/G₃ + ... + I_N-1/G_N-1 }
Note that your N^th withdrawal is then at the start of the N^th year ... or the end of the (N-1)^st year.

In a tutorial on Annuities, we considered the following:

We're retired and we want to decide whether we should devote a fraction of our portfolio to buying a Life Annuity which pays a fixed amount each year, the amount being a certain percentage of the cost of the annuity. For example, if you buy a Life Annuity for $100,000 and it pays you $5,000 every year until you drop dead, then we'll say it's a 5% annuity.

>A 5% annuity because it pays $5K ... which is 5% of $100K?
Exactly. So we'll look at ...

>Wait! Won't that 5% get bigger as you get older?
It'll depend upon long term interest rates and your age. Anyway, you spend a fraction of your portfolio on an annuity and invest the balance (which is now a "reduced" portfolio) and you ask: "Is my "reduced" portfolio likely to last longer, having bought an annuity?"

>Of course it won't last as long! It's smaller!
But we're withdrawing less each month, because we now have an annuity. Who knows? Maybe ...

>But you said it was a fixed amount! What about inflation? It'll kill you, right?
Interesting question. That's what that tutorial was all about. Anyway, the result of the analysis was to conclude that, if the Annuity Rate was large enough, then you're better off spending a portion of your portfolio on a Life Annuity. In fact, the Annuity Rate should satisfy:
    Annuity Rate > 1 / {1+1/G₁+ 1/G₂+ ... +1/G_N-1}

>What's N?
You drop dead after N years.

>Where's I₁ and I₂ ... the inflation guys?
That's why it's magic. Inflation doesn't enter the picture.

See? The Harmonic Mean pops up.

>But those Gs are future Gain Factors. How would I know ...?
You can borrow this  

>Very funny, but you don't have a single picture. A picture is worth a thousand ...
Here's your picture:

>That's funny. For DCA I'd want a small gMS. That means small inflation and big portfolio gains.
Yes. That's also true for the Maximum Withdrawal Rate. In order to increase our withdrawal rate, we'd want big gains. But ...

>But for the Annuity Rate, we'd want a big   gMS. Isn't that strange? It just means that we'd be better off buying an annuity with a fraction of our portfolio only if our portfolio gains are small ... and that means a big gMS. >But the Annuity Rate normally increases as you get older, right? Yes, so you should keep your eye on these rates, as you get older. When they get large anough, then ... >Then buy an annuity! Why not? Check out Annuity Estimates. In the meantime, here's a picture of several possible gMS evolutions, picking random portfolio gains and inflation rates

>Can I play?
Sure. Here's a picture of a spreadsheet where you can generate random Returns and Inflation (with prescribed Means and Standard Deviations) and see what happens to the resultant gMS:

Just RIGHT-click on the picture and Save Target to download a .ZIPd version of the spreadsheet.

For a more on the Magic Sum, see this.