December 26, 2008 Letters to the Editor

Updated: January 5, 2008.

Less a well thought out post but more of a hypothesis...

I received this letter from Michael.

Hi John - thank you again for your recent response to my earlier letter...I believe I read somewhere on the site that you are a retired engineer, so let me speak for a second in math terms...more of a hypothesis than anything empirical yet, but it SEEMS to me that the partial derivative of the "ideal" stock allocation (let's assume for now this means the equity allocation that maximizes the SWR) with respect to changes in PE10 is less sensitive to changes in PE10 the longer your time horizon and/or the higher your target terminal balance.... perhaps a better way to say what I have been observing thru my recent analysis...i.e., if you have a 50 year horizon and a 100% (real) terminal balance, are you less sensitive to PE10 levels than someone with a shorter time frame? So, not doubting VII's superiority so even in VII STILL means you came out ahead that fixed allocation on average, does it perhaps matter less?

Curious if you can give me some sense of this sensitivity mathematically...or logically...

HERE IS MY RESPONSE

I ran a sensitivity study. At favorable valuations, you are much better off with a 100% stock allocation. At high valuations, you are better off with a lower stock allocation.

The reason is simple: the fixed income portion (2% TIPS) has a low return when compared to stocks when stocks are priced reasonably. When stock prices are sky high, the TIPS help out or do better.

A higher stock allocation does better if the final balance needs to equal the initial balance. This is because stocks eventually provide strong growth. TIPS provide a constant, low level of growth (the coupon rate).

Stock volatility reduces Safe Withdrawal Rates, which are defined by unfavorable outcomes.

The Year 15 SWR Retirement Risk Evaluator shows a very rapid reduction in the stock allocation as P/E10 increases. This is because it has no knowledge of what has happened immediately before. It “forgets” history. If you place two Year 15 results back to back, you do not know what valuation to assume at the midpoint. You know the starting valuation, but not the likely Year 15 valuation given the starting point. (That is, you do not know conditional probabilities.)

In contrast, the Year 30 SWR Retirement Risk Evaluator automatically has the correct valuations at the midpoint of a 30-year interval. It has the same issue, however, if you try to connect two 30-year segments to determine the Year 60 outcomes.

Fixed Stock Allocations and Valuations

If you look at Historical Surviving Withdrawal Rates HSWR, you will see that smaller and smaller changes in the withdrawal rate result in longer and longer survival periods. For example, the amount that it takes to go from a 30-year HSWR to a 40-year HSWR reduces the withdrawal rate by 0.2% of the original balance. To extend this another decade, the 50-year HSWR requires a further reduction of 0.1%.

The reason is that stocks grow dramatically over the long term. A busted retirement is usually caused by some unfavorable returns early in one’s retirement. Such periods tend to cluster together. Over a long period of time, a retirement portfolio faces a variety of hurdles (unfavorable stock returns) clustered together with long favorable periods in between.

Because of your interest in the mathematics, I have added this formalism.

Safe Withdrawal Rate Formalism

The fractional balance fbal(n) is the balance at Year n divided by the initial balance.

The Safe Withdrawal Rate formula is

fbal(n) = TR(n)*(1-[w/wfail(n)]) where TR(n) is the total return at Year n. It is the product of the gain multipliers through Year n.

TR(n) = G1*G2*..*Gn, where Gk=the balance at the end of Year k divided by the balance at the beginning of Year k=the balance at the end of Year k divided by the balance at the end of Year (k-1).

From Gummy’s [Professor Peter Ponzo] Safe Withdrawal Rate formula, with constant annual withdrawals of w, the Safe Withdrawal Rate at Year n is wfail(n) and

1/wfail(n) = (1/G1) + (1/[G1*G2]) +..+ (1/[G1*G2*..*Gn])

When w = wfail(n), the balance is zero at the end of Year n.

Observe that 1/wfail(n+1) = (1/wfail(n)) + (1/[TR(n)*Gn+1])

From the formula:

fbal(n) = TR(n)*(1-[w/wfail(n)])

fbal(n+1) = TR(n+1)*(1-[w/wfail(n+1)]) = (TR(n)*Gn+1)*(1-[w/wfail(n+1)])
= Gn+1*TR(n)*(1-[w/wfail(n+1)])
= Gn+1*TR(n)*(1-[w/wfail(n)]-[w/[TR(n)*Gn+1]])
= Gn+1*TR(n)*(1-[w/wfail(n)] - Gn+1*TR(n)*[w/[TR(n)*Gn+1]])
= Gn+1*fbal(n) - Gn+1*TR(n)*[w/[TR(n)*Gn+1]])
= Gn+1*fbal(n) – w

fbal(n+1)-fbal(n) = [(Gn+1)-1]*fbal(n) – w

Stated differently,
fbal(n+1) – fbal(n) = return(n+1)*fbal(n) – w
where return(n) is the fractional return for year n = -1+Gn

Here is a link to Gummy’s web site and his derivation of the Safe Withdrawal Rate formula.

Gummy’s (Peter Ponzo's) Web Site
Gummy's (Peter Ponzo's) Equation

Extending the formalism:

If the withdrawal at Year k is wk and if you allow it to vary, the formulas remain the same. Replace w/wfail with the formula:

ratio(n)= (w1/G1) + (w2/[G1*G2]) +..+ (wn/[G1*G2*..*Gn])

The revised formula becomes:

fbal(n+1)-fbal(n) = return(n+1)*fbal(n) – [w(n+1)]

Shiller, P/E10

I received this letter from Raymond.

I found your comments and link to your web site on the Bogleheads web site.

In your section "What Should You Do: Addendum?" you discuss Prof. Shiller's P/E10 information and state it is available at his web site. I went there but could not find anything specific and there is no search function to help.

Can you provide a specific link or some details on what I'm looking for at Prof. Shiller's web site, e.g., an excel data set, a research paper or some other source?

Also, your referenced section is dated December, 2005; has there been any subsequent work or publication since then, given the recent and dramatic market changes?

Many thanks

HERE IS MY RESPONSE

Thank you. Welcome aboard.

Here are a variety of links to Professor Robert Shiller’s web site. What you are looking for is found in his Online Data. He maintains an Excel file that you can download.

Professor Shiller’s Web Site
Professor Shiller's Online Data
Professor Shiller's Online Papers

I keep the P/E10 updated in my Stock Returns (see button on the left) Predictor. It shows the current relationship between P/E10 and the S&P500 index.

Unfortunately, you cannot put the latest S&P500 index value into the calculator and read today’s P/E10. You must put in various values of P/E10 and press Calculate until you see something very close to today’s value.

January levels of P/E10 have varied from 5 to 44. Before the run up in the late 1990s, the highest was 29 just before the Great Depression. It reached 24 in 1966, which was the worst stock market for retirees who depended upon selling stocks for income.

You can read my NOTES for my latest thoughts and introductions to my latest articles. Take advantage of my NOTES and NOTE INDEX.

You can practice on the Scenario Surfer. It is a great learning tool for retirees. It includes all sorts of markets. Today, you would select P/E10=14 Bear Market. You will quickly learn how to do better than mechanically defined algorithms.

For something out of the ordinary, read How about 8%? in my Notes starting from December 13, 2008. I recommend that you plan on a lower withdrawal rate until we can gain more confidence of continued success. However, it should hold up. The worst case outcome is that the withdrawal amount would drop temporarily to 6% of the original balance (plus inflation). Still, this is a continuing withdrawal rate, not one with a limited time frame.

Notes starting from December 13, 2008

Letters to the Editor in 2008

Letters to the Editor in 2007

Letters to the Editor in 2006

Letters to the Editor in 2005