Wednesday, July 13, 2011

The Highest Sustainable Withdrawal Rate Comes From 100% Bonds?

I'm now in the process of reading through some of the classic studies about retirement withdrawal rates. 

One of the papers I read again tonight is Rory Terry's article from the May 2003 issue of Journal of Financial Planning, "The Relationship Between Portfolio Composition and Sustainable Withdrawal Rates."  

Recently in investigating about retirement withdrawal rates I have been growing sympathetic to the idea that retirees should not feel obliged to maintain a high stock allocation of 50-75% as seemingly suggested by the Trinity study and others.  I've been finding that the sweet spot is actually somewhere around 40%. [Update: My July 16 post updates my reviews about asset allocation]

So I am sympathetic to the message in Prof. Terry's article, and his article is cited frequently for providing a loud voice against the notion of using high stock allocations in retirement.  In fact, he concludes that retirees will be best served with 100% bonds.

However, in re-reading the article this evening, I am concerned that his main conclusions are based on a mistake.

He sets up a simple scenario.  An investor will be retired for 30 years and wants to withdraw inflation-adjusted amounts from the portfolio for this length of time.  Bonds provide a nominal return of 6% and no volatility.  In real terms, this means bonds provide a return of about 3%.  Of course investors must deal with reinvestment risk and interest rate risk with bonds, but I will give him the benefit of the doubt that retirees could set up a portfolio with zero-coupon bonds that will provide precisely the necessary spending power with no risk.  Since inflation is constant, there is no inflation risk either.  If the investor lives longer than 30 years, wealth is guaranteed to be gone.  But okay, this is not my main concern.

Next, consider stocks.  Prof. Terry assumes that stocks will have an expected nominal return of 12 percent [or about a 9% real return] and a standard deviation of 10%.  Actually, this is remarkably optimistic!  High returns and a relatively low standard deviation. 

The problem comes with the next calculation, in which Prof. Terry uses Monte Carlo simulations with a normal distribution to find that with this stock portfolio, a retiree can only use a withdrawal rate of 1.85% and still face a 10% chance for retirement failure.  Leaving a 1% chance for failure, the maximum sustainable withdrawal rate is only 0.19%.  With such low withdrawal rates, and since bonds are assumed to have no volatility, it is natural that 100% bonds provides the best outcome for retirees, a withdrawal rate of 4.9%.

The problem is... these Monte Carlo simulation results for the stock portfolio must simply just be wrong.  A standard deviation of 10% really isn't so high to get such terrible results.

When I replicate such a calculation assuming a lognormal distribution with a 9% real return and 10% standard deviation, I find that a withdrawal rate of 6.63% (not 1.85%) is successful with a 10% chance of failure.  For a 1% chance of failure, I get a withdrawal rate of 5.32% (not 0.19%).

To have a withdrawal rate of around 1.85% with a 10% chance of failure, I'm looking at something more like a real return of 2% with a standard deviation of 16%.  I'm using a table which I made a few weeks ago, and it doesn't provide me with any way to get to the 0.19% number.  The worst case scenario I had considered was a 0% real return with a 20% standard deviation and that still gives a 0.45% withdrawal rate with a 1% chance for failure.

I'm sure that I can find some others to also investigate these results and make sure that I am not the one making a mistake.  Pending any further notice, it does seem that there is a serious problem with the results of this article.  Really, the whole basis of the article is centered around that 1.85% withdrawal rate for 100% stocks and how it is lower than the 4.9% withdrawal rate for bonds.  If that 1.85% number is wrong, then it is back to the drawing board.


Update, I made this post at the Bogleheads Forum:

The issue of normal vs. lognormal distribution isn't going to matter. As Dick Purcell wrote, his results are based on the normal distribution, but at multiple points in this thread I confirmed getting very similar results as him with my simulations based on the lognormal distribution.

The two distributions are quite similar and won't make much difference when discussing the US. I used to use the normal distribution, but ran into some problems when doing simulations for more volatile emerging market countries. With a high enough standard deviation, it can become common to see returns from the normal distribution of less than -100%. I had people saving for retirement and not using any leverage who were ending up with negative wealth. But that is impossible. The worst that can happen is that your wealth is wiped out. It can't go negative. The lognormal distribution corrects for this, and I've never looked back.

The Terry (2003) article mentions the importance of using 100,000 simulations. So I just tried that. My results above were based on only 1,000 simulations.

With a 9% real return and a 10% standard deviation, for 100,000 simulations I get the following withdrawal rates:

Worst-case scenario out of 100,000 tries: 3.09%
1% failure rate: 5.16%
5% failure rate: 6.02%
10% failure rate: 6.52%

My earlier calculations were for 1,000 simulations.

Again, Prof. Terry calculated a 1.85% withdrawal rate with a 10% failure rate. This invalidates the whole premise of his article.


Thank you to Mike Piper for linking to this post from his Oblivious Investor blog.


  1. Dear Prof. Pfau

    I have been doing my own research into drawdowns using monte carlo simulations based on a normal distribution. To generate returns I have been using the following formula: =NORMINV(RAND(),$C$1,$D$1. I would like to have a look at my results when basing returns on a lognormal distribution.

    How would I go about entering this in excel as just replacing NORMINV with LOGINV yields very unrealistic results.

    Many thanks

    Jonathan Brummer

  2. Just realised the formula in my question refers to the cells. C1 and D1 have the mean and standard deviation for the distribution.



  3. Dear Jonathan,

    There are a couple more steps.

    Let me call the mean A1, and the standard deviation A2. These are what you can use for NORMINV

    For the lognormal make sure these are defined in decimal form. i.e. a 5% return with 20% standard deviation would be .05 and .2

    You need to calculate 3 more cells:


    Then you can use:


    The results shouldn't be too different, but an advantage of lognormal distributions is that they prevent you from having a return of less than -100%. This could sometimes happen with volatile returns for the normal distribution.

  4. "I've been finding that the sweet spot is actually somewhere around 40%."

    Wade, You and Jim Otar say the same thing.

    My issue is that I can't figure out how you/Jim come up with 40% when most of the studies recommend 60-80% stocks as optimal.

    Any thoughts? This is the last critical piece of my own AA.


  5. Hi,

    I have a couple of recent blog posts which I hope can help with this.

    First, see this one about the explanation for higher stocks:

    Then, this one, based on MOnte Carlo simulations, which tends to show numbers closer to 40%:

    this one also includes link to another article I wrote which explains how the historical data includes a bias against bonds

  6. Hi Wade,
    Thanks for posting your research for the benefit of all, very kind of you.
    Two points here. I was under the impression the journal is referreed and this article is nine years old. Does no one check the calculations before publication? Gives one pause for thought I think.

    Referring to your January article, Table 3, we see that for a thirty year withdrawal period and 1% failure rate, the rate is only 3.4%. Not far from professor Terry's 3.33% for cash.
    If one felt that a 5% failure rate was not bearable.... why not put all one's wealth in I Savings Bonds with a zero percent (real) yield and withdraw 3.33% annually for the next thirty years?

    Bill Thompson

    1. Thanks Bill,

      About journal refereeing, the Journal of Financial Planning is pretty good and I've gotten helpful comments from referees. But sometimes things get through the cracks, and usually referees do not try to re-create the results to see if they are right. It just seems that the completely unrealistic numbers in that article escaped close scrutiny. Fortunately, such errors don't show up too often, though they can only really be found if people go through the effort of trying to replicate the findings from existing articles.

      About your point on building a ladder of I Bonds. Yes, what you are saying is fine. My article didn't include bond ladders, just spending from volatile assets such a bond mutual fund. Do keep in mind though that with the strategy you are describing, your wealth will be gone at the end of the 30th year with 100% probability, which could be a problem if you live longer than that.

    2. A good point Wade and one I considered back in 1998. So the average rate of return would now sustain 4% for at least 30 more years.

      I would be interested to see more about professor Terry's article because it influenced my thinking in more than one way. It seems to me that if you encounter someone who isn't accustomed to working with numbers and tell them a plan has a 5% chance of failure you'll get a far different reaction than saying "There is once chance in twenty you'll run out of money!" Something for financial planners to consider I guess.

      Best wishes Bill Thompson

    3. Bill,

      If you would like a copy of the article, send me an email and I can reply with it.

      I probably won't discuss it more here. It has an error. My own take on asset allocation and safe withdrawal rates can be found here:

      I do like your one in twenty idea. That might make more sense for people.