## Sunday, February 17, 2013

### Deriving the 4% safe withdrawal rate for retirement planning

In a recent article I linked here, the infamous 4% withdrawal rate set out in a 1994 study by Bengen was mentioned.  In thinking about a way to derive more fundamentally where that number comes from I came up with an interesting way to look at it.  I have not verified that this is an original contribution, but here it is nonetheless:

When coming up with a safe withdrawal rate, the most conservative goal is to maintain the real value of the portfolio while spending X% of it to support retirement.

To derive what X should be, consider an all-stock portfolio, where:

• Y is the total earnings yield of the portfolio (specifically, Y = earnings / market_cap of the portfolio).
• D is the fraction of the portfolio that are distributed instead of re-invested (D usually comes from dividends but also things like stock buybacks).  Basically any part of the earnings that aren't added to book value through retention count toward D.
• Let the expected inflation be I.
• Let E be the "return on equity" (ROE) of the portfolio (ROE is earnings divided by book value).  One assumption that I am making is that the ROE is fairly constant - Warren Buffett wrote an article in 1977 showing that US stocks tended to have an ROE of 12% (link), and interestingly the S&P 500 recently has been in the ~14% range.  So let's assume E=12%.
• Let B be the price-to-book value.

For this portfolio, to meet the definition of safe withdrawal rate above, the portfolio's growth in earnings must match inflation (I).  Assuming no withdrawals, the growth in earnings is equal to:

1) Portfolio earnings growth = (D*(E/B))+((Y-D)*E) / Y

If spending (X) is less than D, the growth is:

2) Growth if X<D = ((D-X)*(E/B))+((Y-D)*E) / Y

If spending (X) is bigger than D, the growth is:

3) Growth if X>D = ((Y-X)*E) / Y

In either of #2 or #3 above, to make X safe you need to have it match inflation, or I (thereby making sure that the growth in earning power of the portfolio matches the growth in inflation).

For #2:

4) X = D - DB + YB - ((IBY)/E)

For #3, it's a bit simpler:

5) X = (Y * (E - I)) / E

Plugging in some sample values, Y = 6%, I estimate D to be ~4% even though only half of that comes from dividend yield (the rest I believe is from stock buyback although I can't prove it perfectly yet), I = 3%, E = 12% as I said above, and B = 2.2 right now.

For these values, #5 gives X = 4.5%... Since X > D I don't need to compute the more complicated #4, but for smaller spend it would matter (since companies can only re-invest at book value the money they actually retain, not dividends or repurchased stock).

4.5% is the typical safe withdrawal rate!  But now by plugging in different values for the key assumptions, you can work out what other safe rates could be.

#### 2 comments:

1. For future reference, I realized that this can be simplified since E/BY = 1 and E/Y = B. So the final formulas become:

X = E - I - (D)(E-1) {for X < D}
X = Y - I/B {for X > D}

2. Clarifying some of the units in the earnings growth formula:

Earnings growth rate (in %) = (Payout %)(Earnings yield %) + (Retain %)(ROE %)