Now, it's clear that when the withdrawal rate is smaller, our Portfolio lasts longer. That means that there is a greater chance of having our Portfolio last for thirty years or forty or ...

>Your point?
That taking a portion of your Portfolio to buy an annuity which pays a fixed amount every year may be something to consider.

>The annuity amount increases with inflation?
No, we consider only an annuity which pays a fixed percentage of the amount devoted to the annuity. If you buy an annuity for $100,000 and it pays 8% then you get 8% x $100K = $8K every year ... until you drop dead.

>But it isn't a good strategy, right? I mean, inflation could kill you, right?
It would appear, at first blush, that devoting a portion of your money to a fixed annuity would be a terrible thing to do ... if inflation gets out of hand. But, let's see:

• Suppose you need $A from your Portfolio, currently worth $P.
• The withdrawal rate is W = A/P which may be something like W = 0.04 (using the notorious 4% Safe Withdrawal Rate).
• You use a fraction f of your Portfolio to buy an annuity. (f = 0.25 means 25% devoted to the annuity)
• Assume the annuity pays a fixed fraction r every year. (r = 0.08 means 8%)
• You then get an annuity income worth r f P, every year (a fraction r of the amount   f P which was used to buy the annuity.)
• Instead of needing A from your Portfolio, you now only need A - r fP.
• Of course, your Portfolio is now worth only P (1 - f) ... after using a fraction f to buy the annuity.
• The withdrawal rate has now changed to {A - r fP}/ {P(1 - f)} (Withdrawal Amount)/(Portfolio Amount)
  which can be written W (1 - r f/W)/(1 - f)
• This withdrawal rate (with an annuity) is smaller than W if (1 - r f/W)/(1 - f) < 1
• This means (after some jiggling of terms): r > W

>So?
So, if we use that Safe Withdrawal Rate of 4%, then we'd get a smaller withdrawal rate if the annuity pays more than 4%.

>And will an annuity pay more than 4%?
Most likely.

>And with the annuity, our reduced Portfolio will last longer?
Perhaps.

>So we can withdraw more from the reduced Portfolio?
Maybe.

>Isn't there anything else you can say?
Uh ... we can consider the sequence of withdrawals. Remember, what we did above was consider the withdrawal rate at the beginning of our retirement. It'll be a smaller withdrawal rate if we buy an annuity ... provided r > W which is likely. However, if our TOTAL income is to grow with inflation and the Annuity is FIXED, then the amount we withdraw must grow faster than inflation. That could be a problem.

Suppose that:

• Without an annuity you need a certain fraction, W , of your initial Portfolio, P(0).
  (That means an initial income of A = WP(0).)
• If inflation is constant at i (i = 0.03 means 3% annual inflation), after N years, you'll want a total income of A(1+i)^N.
• With an annuity which pays a fixed r f P(0) each year, you need only withdraw A(1+i)^N - r f P(0) from your Portfolio.
• After using a fraction f of your Portfolio to buy the annuity, the initial Portfolio is reduced to P(0)(1-f).
• ...

>Can you skip the math?
Okay. Here's a calculator for quick-and-dirty estimates:

Initial Portfolio =$ Required Income =$   made up of portfolio withdrawals and annuity Return on Investments =%   assumed constant (!) Fraction of Portfolio used to purchase an Annuity =% Annuity Rate =%   annual income for each $100 paid for annuity Inflation Rate =% Number of Years = Final Portfolio = $   after umpteen years

>When I put 100% of my Portfolio into an Annuity, I still get a Final Portfolio. How can ...?
If your Annuity Income is greater than your Required Income, the excess is invested at the specified Return on Investments.

>Why quick-and-dirty?
Everything remains constant for umpteen years. If you use the calculator you'll find that if the Annuity Rate is greater than the Return on Investments, you'll be advised to put more into an Annuity. However, in the real world ...

>So, how about pictures, some sample portfolio evolutions ... with real market returns?
Sure. Here are some pictures.
What's left of our Portfolio (after buying a 9% Annuity) is invested in the S&P 500 (starting in 1960) and we need an income of $60K per year (increasing at 3%, with inflation):

>So buying an annuity can be good, eh?
Here's another with 4% inflation:

>Yeah, but when will buying an Annuity be a worthwhile thing to do??
When the Annuity Rate, r, is large enough. In fact, if the investment Gain Factors over 1 year, 2 years, 3 years, etc. are denoted by G(1), G(2), G(3), etc., then buying an Annuity improves your Final Portfolio, after N years, if:

>Remind me. What are these Gs?
If your Portfolio grows by a factor 3.45 after 7 years, then G(7) = 3.45 and if it grows by a factor ...

>Are you saying that inflation has nothing to do with it? That whether or not one should buy an annuity doesn't depend upon inflation? That ...?
I'm saying that it's only the returns on your investments that count, NOT inflation.

>Okay, I get it. Any idea what this sum 1/G(1) + 1/G(2) etc. looks like?
Like so, where we start in 1928 and go on and on ... for seventy years or so:

>But what about starting in 1947 or 1970, or using actual inflation instead of 3%, or ...?
Well, actually, the spreadsheet described in Part I, actually does that ... if you ask nicely
The spreadsheet has a part which looks like this.

There's also a spreadsheet to play with (comparing your portfolio WITH and WITHOUT an annuity).
Just RIGHT-click here and Save Target file.

See also this article in the Financial Planning Journal.

See also this post on Morningstar: El Toro

Check out "typical" annuitites here.

Check out Life Annuities.