a Magic Sum

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, ... iN and similarly for the Annual Returns and Factors etc.
  • Gn is the cumulative Gain Factor over n successive years, so Gn is the product of a bunch of annual Gain Factors, like:
    Gn = g1g2g3...gn

Now, we're ready for our Magic Sum:

the N-year gMS = I1/G1 + I2/G2 + I3/G3 + ... + IN/GN

>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 = ... = iN = 0, then I1 = I2 = ... = IN = 1 and our gMS is 1/G1 + 1/G2 + ... + 1/GN

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

This Harmonic Mean will pop up ... later.

Dollar Cost Averaging

Suppose that we buy a stock whose price, initially, is P0.
Then, after 1 year, it's price is P0G1 (having grown by the Gain Factor for that first year).
Then we buy it again (at the price P0G1). After another year the price is now P0G2 (meaning the price has now grown by the 2-year Gain Factor G2) ... and we buy it again, at this price.

>You say the price has "grown". My investments always go the other way. I mean ...
Pay attention. It just means that the Gain Factor is less than "1". If G3 = 0.95 it means that, after 3 years, the price is 95% of the original value. Anyway, we're buying every year or, if you like, every month ... in which case the Gs would be the monthly Gain Factors.

>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 Pn we'd be buying A/Pn shares. Initially, we'd buy A/P0 shares but since the price, after n months, is P0Gn,
the number of units we're buying (after n months) is {A/P0} / Gn
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/GN } = 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 + ... + IN/GN }

We'll give it a place of some importance:

Number DCA units accumulated after N years is A/P0 (1 + gMS)

starting when the stock price is P0
and investing $A annually (or monthly, in which case we'd use the monthly gMS)

Sensible Withdrawal Strategies

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 + ... + IN/GN }

In order to last for N years, the annual portfolio withdrawal should be less than 1/gMS
expressed as a fraction of the initial portfolio

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

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

Immediate Annuities

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/GN-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.

If a non-indexed Life Annuity pays at the rate r, then you should spend a fraction of your portfolio on the Annuity if

    r > 1 / {1+1/G1+ 1/G2+ ... +1/GN-1}

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.