## The Mathematics of Retirement Investing

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longinvest
Posts: 2891
Joined: Sat Aug 11, 2012 8:44 am

### The Mathematics of Retirement Investing

Here's an interesting series of questions and answers with a surprising conclusion at the end.

I'm sure that many would not have believed me if I simply stated the conclusion without first showing the complete calculations.

Enjoy!

Q-1: What is the inflation-indexed amount that must be invested yearly, over a period of 30 year, to accumulate an inflation-adjusted \$1,000,000 portfolio, at a nominal growth rate of 10.1% and a 3% inflation rate?

A-1: \$10,778.40

This was calculated as follows:
• Inflation-adjusted growth rate = (1 + 10.1%) / (1 + 3%) - 1 = 6.9%
• Using a financial calculator: n=30, i=6.9, PV=0, FV=1000000 => PMT = -10778.40

Q-2: What is the inflation-indexed amount that must be invested yearly, over a period of 30 year, to accumulate an inflation-adjusted \$1,000,000 portfolio, at a nominal growth rate of 8.7% and a 3% inflation rate?

A-2: \$13,805.39

This was calculated as follows:
• Inflation-adjusted growth rate = (1 + 8.7%) / (1 + 3%) - 1 = 5.5%
• Using a financial calculator: n=30, i=5.5, PV=0, FV=1000000 => PMT = -13805.39

Q-3: How much more must be invested yearly, over a period of 30 year, to accumulate an inflation-adjusted \$1,000,000 portfolio, at a growth rate of 5.5% real instead of 6.9% real?

A-3: 28%

This was calculated as follows:
• Using the answers of Q-2 and Q-1: \$13,805.39 / \$10,778.40 - 1 = 28%

Q-4: If future annual returns in the upcoming 30 years were exactly 6.9% real for a 100% stocks portfolio, and 5.5% real for a 60/40 stocks/bond portfolio, how much more should an investor save, assuming he makes an inflation-adjusted \$80,000 yearly salary and will be getting an inflation-adjusted \$25,000 yearly Social Security pension during retirement?

A-4: 21%

This was calculated as follows:
• Our objective is for work-years salary minus savings to be equal to total retirement income. We're assuming that savings are invested into a tax-deferred account, and that tax rates will be similar on equal taxable total-income during work years and retirement.
• Assuming that a portfolio at retirement supports approximately 4% in withdrawals, our objective is to split the excess in work-years salary over future Social Security income, that is (\$80,000 - \$25,000) = \$55,000, between savings and taxable income such that work-years excess taxable income is equal to retirement-years excess taxable income (on top of Social Security).
• At a growth rate of 6.9%, every \$1 of annual savings accumulates to \$92.78, 30 years later, supporting \$92.78 X 4% = \$3.71 in retirement spending.
• As a consequence, we split the excess \$55,000 according to the ratio 1:3.71 between savings and taxable spending. This results into \$11,674.49 in tax-deferred yearly savings. This leaves \$80,000 - \$11,674.49 = \$68,325.51 for paying taxes and spending.
• At a growth rate of 5.5%, every \$1 of annual savings accumulates to \$72.44, 30 years later, supporting \$72.44 X 4% = \$2.90 in retirement spending.
• As a consequence, we split the excess \$55,000 according to the ratio 1:2.90 between savings and taxable spending. This results into \$14,111.90 in tax-deferred yearly savings. This leaves \$80,000 - \$14,111.90 = \$65,888.10 for paying taxes and spending.
• The additional ratio of savings is thus: \$14,111.90 / \$11,674.49 - 1 = 21%

Q-5: What is the impact of the additional 21% in tax-deferred savings on after-tax net spending, for the example shown in Q-4?

A-5: 3.3% or \$153.17 per month

This was calculated as follows:
• We approximate after-tax income using the 2017 marginal tax rates in this Wikipedia table.
• The first \$9,325 of income attract 10% in taxes. That's \$9,325 X 10% = \$932.50.
• Additional income up to \$37,950 attracts 15% in additional taxes. That's (\$37,950 - \$9,325) X 15% = \$4,293.75.
• The 100/0 investor has a pre-tax after-savings income of \$68,325.51. It attracts (\$68,325.51 - \$37,950) X 25% = \$7,593.88 in additional taxes. He is left with: \$68,325.51 - \$932.50 - \$4,293.75 - \$7,593.88 = \$55,505.38 for spending.
• The 60/40 investor has a pre-tax after-savings income of \$65,888.10. It attracts (\$65,888.10 - \$37,950) X 25% = \$6,984.53 in additional taxes. He is left with: \$65,888.10 - \$932.50 - \$4,293.75 - \$6,984.53 = \$53,677.32 for spending.
• The impact on net spending, in terms of ratio, is thus: \$53,677.32 / \$55,505.38 - 1 = -3.3%.
• As an absolute amount, this is: (\$55,505.38 - \$53,677.32) / 12 = \$153.17 per month.

Yes, that's it. By reducing total spending by a mere 3.3%, an investor could use a less volatile 60/40 portfolio all lifelong and still retire with dignity.

Added (1): Forum member #Cruncher has developed an awesome little spreadsheet encoding the calculations of this post, along with improvements and additional flexibility. It can be found here: viewtopic.php?p=3778872#p3778872.

Added (2): Forum member #Cruncher has provided a summary of how the "cost" of the 60:40 portfolio computed in this post compares to other ways of computing its "cost": viewtopic.php?p=3494999#p3494999
Last edited by longinvest on Mon Feb 12, 2018 3:25 pm, edited 4 times in total.
Bogleheads investment philosophy | Lifelong Portfolio: 25% each of (domestic/international)stocks/(nominal/inflation-indexed)bonds | VCN/VXC/VLB/ZRR

Grt2bOutdoors
Posts: 18588
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Location: New York

### Re: The Mathematics of Retirement Investing

Outstanding! The examples should be added to wiki on Retirement Planning.
"One should invest based on their need, ability and willingness to take risk - Larry Swedroe" Asking Portfolio Questions

Iliketoridemybike
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### Re: The Mathematics of Retirement Investing

Very nice!

longinvest
Posts: 2891
Joined: Sat Aug 11, 2012 8:44 am

### Re: The Mathematics of Retirement Investing

Original poster (OP) here.

OK, far from me to suggest that the future real returns of 100/0 and 60/40 portfolios will be 6.9% and 5.5%, respectively. I was simply using the numbers (10.1% and 8.7% nominal) given in the post that preceded mine in that other thread, to which I added a 3% inflation assumption.

Actually, when sketching plans for myself, I use lower numbers. Something like 5.5% real for stocks and 1.5% real for bonds*, assuming that actual future returns will quite possibly be lower, but could also be higher. I'll take whatever markets give in the future but, for now, I need some numbers to run planning calculations.

* That would be 0.6 X 0.055 + 0.4 X 0.015 = 3.9% for a 60/40 portfolio.

The main point I was trying to make is that, even if we knew for sure that a 100/0 would beat a 60/40 portfolio, the impact on net spending all lifelong would be pretty small smaller than generally believed* if we chose the less volatile portfolio.

In the above example, we see that the pre-tax savings rate of the 100/0 investor is \$11,674.49/\$80,000 = 14.6% while that of the 60/40 investor is \$14,111.90/\$80,000 = 17.6%. As I calculated, we're talking of barely more than a \$150/month difference in after-tax spending.

I would not save as little as 15% or 18% on a \$80,000 salary; I would save more! Yet, it is important to look at the entire picture and not be mislead into taking reckless risk in our portfolios because someone somewhere shows a simplistic calculation where using a 100/0 stocks/bonds portfolio, \$10,000 left to grow for 30 years results into almost 50% more than with a 60/40 portfolio (e.g. 1.069^30 / 1.055^30 - 1 = 48.5%). This is simply not a relevant calculation.

* Modified at the suggestion of forum member #Cruncher.
Last edited by longinvest on Wed Aug 16, 2017 1:34 pm, edited 1 time in total.
Bogleheads investment philosophy | Lifelong Portfolio: 25% each of (domestic/international)stocks/(nominal/inflation-indexed)bonds | VCN/VXC/VLB/ZRR

longinvest
Posts: 2891
Joined: Sat Aug 11, 2012 8:44 am

### Re: The Mathematics of Retirement Investing

NPT wrote:
longinvest wrote:Q-5: What is the impact of the additional 21% in tax-deferred savings on after-tax net spending, for the example shown in Q-4?

(...)

Yes, that's it. By reducing total spending by a mere 3.3%, an investor could use a less volatile 60/40 portfolio all lifelong and still retire with dignity.
In Canada (where I live), we can save up to 18% of our annual salary in a tax deferred account (the equivalent of an IRA) up to a certain ceiling (currently \$26,010). Contributions to the equivalent of a 401K, as well as amounts related to defined-benefit pension plans, are subtracted from this 18%.

We can also contribute up to \$5,500 per year to a tax-free account (the equivalent of a Roth IRA).

Having all the above tax-advantaged accounts maxed out is a first world problem, as they say. I should know; I face this problem, too, because my workplace defined-benefit pension plan reduces my annual tax-deferred contributions down to next to nothing.

Even when all tax-advantaged accounts are filled, the general principles of retirement planning calculations remain the same. But, as soon as investing starts in taxable accounts, it becomes important to worry about tax-efficient investing as to minimize taxes.

Here's what I do. I know that an exact calculation involving taxable accounts would require more complexity. But calculations like those in the first post are already grossly approximate; I simply don't know the actual future real growth rate of my portfolio. So, I simply don't take into account taxes on taxable investments, in my planning calculations; I simply assume that these taxes will be paid off of net income (e.g. subtracted from spending). In other words, I worry about the savings rate, then pay taxes out of what remains. It's not perfect, but it's good enough for me.
Bogleheads investment philosophy | Lifelong Portfolio: 25% each of (domestic/international)stocks/(nominal/inflation-indexed)bonds | VCN/VXC/VLB/ZRR

North Texas Cajun
Posts: 186
Joined: Sun Jul 02, 2017 8:56 am

### Re: The Mathematics of Retirement Investing

longinvest wrote:Yet, it is important to look at the entire picture and not be mislead into taking reckless risk in our portfolios because someone somewhere shows a simplistic calculation where using a 100/0 stocks/bonds portfolio, \$10,000 left to grow for 30 years results into almost 50% more than with a 60/40 portfolio (e.g. 1.069^30 / 1.055^30 - 1 = 48.5%). This is simply not a relevant calculation.
Longinvest,

I have enjoyed reading your posts and comments, and greatly appreciate the VPW method you developed. We usually agree on issues.

Why do you believe a 100% equity portfolio to be a "reckless risk" for a long term investor?

Two of my favorite financial authors are Dr. Jeremy Siegel, from my alma mater, and Ken Fisher, founder of Fisher Investments. Both have made what I believe are convincing arguments that bonds are less risky over the short term but more risky over the long term (periods of 15 years or more). They believe, as I do, that stocks will not only outperform bonds for a 30 year period but run a much greater risk of multiple years of negative returns.

As I see it, retirement investors would be better served by a 100% or near 100% equity portfolio for at least the first two decades - and possibly the first 25 years - of their accumulation phase of investing.

Can you help me understand why you apparently disagree with Professor Siegel, Ken Fischer, and me?
Last edited by North Texas Cajun on Fri Aug 11, 2017 5:41 pm, edited 1 time in total.

NPT
Posts: 23
Joined: Sat Jul 01, 2017 12:25 pm

### Re: The Mathematics of Retirement Investing

NPT wrote:
longinvest wrote:Q-5: What is the impact of the additional 21% in tax-deferred savings on after-tax net spending, for the example shown in Q-4?

(...)

Yes, that's it. By reducing total spending by a mere 3.3%, an investor could use a less volatile 60/40 portfolio all lifelong and still retire with dignity.
In Canada (where I live), we can save up to 18% of our annual salary in a tax deferred account (the equivalent of an IRA) up to a certain ceiling (currently \$26,010). Contributions to the equivalent of a 401K, as well as amounts related to defined-benefit pension plans, are subtracted from this 18%.

We can also contribute up to \$5,500 per year to a tax-free account (the equivalent of a Roth IRA).

Having all the above tax-advantaged accounts maxed out is a first world problem, as they say. (...)
These limits are roughly similar to those in the U.S. (where I live) and they are really not very high if you live in a HCOL area and intend to stay there. I would not call that a luxury if that's where you feel at home, that's where your family and friends are etc. The notion of first world problems sounds a bit derogatory to me (no offense taken, just pointing it out ), and in any case, it doesn't help with analyzing the numbers.

The topic here seems to be that a 60% stock portfolio could possibly replace a 100% stock portfolio under certain conditions. It's an intriguing idea but I think it's important to realize that the math simply does not work for a large number of people who have no more room left in tax-advantaged accounts. No matter how cleverly you account for, or manage, taxes on taxable investments, the particular trick you introduced in the opening post may simply not be available to you.

North Texas Cajun
Posts: 186
Joined: Sun Jul 02, 2017 8:56 am

### Re: The Mathematics of Retirement Investing

Here's a link to what I believe is convincing support for the argument that bonds are risky over the long run:

https://seekingalpha.com/article/398171 ... ier-stocks

Bond returns seem to be very episodic with decade-long super-cycles

You can have many decades in a row of very strong returns, but also multi-decade phases of sideways or even strong negative real returns. There was an 80+ year time window from 1898 to 1981 when bonds had zero real returns. We consider that the mother of all risks! Bonds also had a 40Y return window with -2.08% average real return, compared to +2.62% as the worst equity 40Y return window. The Bond return upside is limited. Only under 5% real return over the best 4-Y window, compared to over 10% for equities.

longinvest
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Joined: Sat Aug 11, 2012 8:44 am

### Re: The Mathematics of Retirement Investing

North Texas Cajun,

North Texas Cajun wrote: Two of my favorite financial authors are Dr. Jeremy Siegel, from my alma mater, and Ken Fischer, founder of Fischer Investments. Both have made what I believe are convincing arguments that bonds are less risky over the short term but more risky over the long term (periods of 15 years or more). They believe, as I do, that stocks will not only outperform bonds for a 30 year period but run a much greater risk of multiple years of negative returns.
Here are fundamental facts about investment-grade stocks and bonds, true regardless of past returns or future returns:
• A common stock is certificate of ownership in a company which entitle its holder to his fair share of future dividends. It usually comes with a voting right at shareholder meetings.
• A bond is a contract to make specific payments on specific future date. It has a par value (principal), a coupon rate, and a maturity date. An inflation-indexed bond (such as a TIPS) pays its coupons and (at maturity) principal in CPI-adjusted terms.
Nowhere, on a stock certificate, do we find any promise of specific payments on specific date. A stock certificate doesn't even have a redemption value. Of course, the stock holders of a company hope that it will make profits and use them to grow its assets or distribute generous dividends. But, there is no guarantee that this will happen. Bond holders will be first to be on the list of creditors, if the company gets in trouble. Common stock holders are usually last on the list.

It is thus simply impossible, using mathematics, to make any proof that an investment-grade stock will have higher returns between now and a specific future date than an investment-grade bond. We don't even know the exact future total-return of the bond, between now and its maturity date, if this bond makes coupon payments before maturity, as we don't know the reinvestment yield.

In Stocks for the Long Run, Dr. Jeremy Siegel never compared stocks to a balanced portfolio. The widely known popular chart of Figure 1.1 (below) compares the performance of money 100% invested into stocks to money invested 100% invested into another single asset in 1800, for a little over two centuries. Worse, that comparison does not take into account periodic investments (or flexible withdrawals, during retirement).

Figure 1.1 from Dr. Jeremy Siegel's Stocks for the Long Run:

Actually, his claims about bonds, based on the chart of Figure 1.1, are generally unfair, as he couldn't include TIPS which simply did not exist. I've recently written specifically about the backtesting against underperforming of bonds during the 1940-1980 period, in the following post: viewtopic.php?f=10&t=224979&p=3480314#p3480314.

The arguments, in Stocks for the Long Run (I've read the 5th edition), are based on the observation of past returns, and making assumptions (not specifically written in black ink on stocks certificates or in bond contracts) about the future. There's no sufficient fundamental material, in this book, for making a solid mathematical proof that will guarantee that I would be best to avoid diversifying my portfolio; that for my specific lifetime, I would be best to go 100% (or why not 140%, based on the Kelly criterion) in stocks.

Let me insist on this. I have but a single life. Even if the odds were 99% in favor of stocks, over all the possible retirement years of the past and the future, I don't know whether I'll retire within one of those 1%-probability awful retirement years for stock-heavy portfolios. I don't get to live 1,000 lives, then try to maximize the number of lives with an outstanding outcome. That is not how my unique life works. I've got a single shot at this.

Why should I take the risk of concentrating my portfolio into a single asset class, when the cost of diversifying my portfolio and avoiding disastrous outcomes is low enough? I don't care about maximizing my portfolio at death; I care about accumulating enough money to retire with dignity.
North Texas Cajun wrote: As I see it, retirement investors would be better served by a 100% or near 100% equity portfolio for at least the first two decades - and possibly the first 25 years - of their accumulation phase of investing.
It's easy to dream of the perfect scenario, that is, while we're accumulating, we buy stocks on the cheap, with part of our regular salary, which then grow to a gigantic amount of money by the time we retire. The perfect dollar-cost averaging scenario. How did author William Bernstein write about it in my favorite investing book of all times, The Four Pillars of Investing? Let me go look... OK, here it is:
In The Four Pillars of Investing, William Bernstein wrote: What this says is that a young person saving for retirement should get down on his knees and pray for a market crash, so that he can purchase his nest egg at fire sale prices. For the young investor, prolonged high stock prices are manifestly a great misfortune, as he will be buying high for many years to invest for retirement.

It's easy to forget about the unlikely(?) but possible reverse scenario, one where stocks remain expensive all along during our work years, only to become cheaper on the day we retire or soon after.

There's also a huge risk that we tend to forget about; it's personal risk in our life. Life doesn't always pan out the way we planned it. In other words, plans change. Things like divorce, disease, and other milder things: getting married and wanting to use part of the portfolio as down payment. A 100% stocks portfolio is fine for somebody who will never, ever, need to withdraw from it at the wrong time.

For the rest of us, adding enough bonds to our portfolios, including inflation-indexed bonds, seems like a prudent course of action.
Last edited by longinvest on Thu Apr 12, 2018 6:55 am, edited 1 time in total.
Bogleheads investment philosophy | Lifelong Portfolio: 25% each of (domestic/international)stocks/(nominal/inflation-indexed)bonds | VCN/VXC/VLB/ZRR

longinvest
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### Re: The Mathematics of Retirement Investing

NPT wrote: The notion of first world problems sounds a bit derogatory to me (no offense taken, just pointing it out ), and in any case, it doesn't help with analyzing the numbers.
NPT, I'm sorry what I wrote offended you. It was meant as a joke, specially that I included myself in the set of people struggling with the problem of taxable investing.
Bogleheads investment philosophy | Lifelong Portfolio: 25% each of (domestic/international)stocks/(nominal/inflation-indexed)bonds | VCN/VXC/VLB/ZRR

longinvest
Posts: 2891
Joined: Sat Aug 11, 2012 8:44 am

### Re: The Mathematics of Retirement Investing

North Texas Cajun,
North Texas Cajun wrote:Here's a link to what I believe is convincing support for the argument that bonds are risky over the long run:

https://seekingalpha.com/article/398171 ... ier-stocks

Bond returns seem to be very episodic with decade-long super-cycles

You can have many decades in a row of very strong returns, but also multi-decade phases of sideways or even strong negative real returns. There was an 80+ year time window from 1898 to 1981 when bonds had zero real returns. We consider that the mother of all risks! Bonds also had a 40Y return window with -2.08% average real return, compared to +2.62% as the worst equity 40Y return window. The Bond return upside is limited. Only under 5% real return over the best 4-Y window, compared to over 10% for equities.
Have you looked at the returns of inflation-indexed bonds (TIPS), over that period? Probably not, as they did not exist.

I would not recommend projecting past returns in the future.
Bogleheads investment philosophy | Lifelong Portfolio: 25% each of (domestic/international)stocks/(nominal/inflation-indexed)bonds | VCN/VXC/VLB/ZRR

North Texas Cajun
Posts: 186
Joined: Sun Jul 02, 2017 8:56 am

### Re: The Mathematics of Retirement Investing

longinvest,

I think you are misunderstanding or misrepresenting both Dr. Siegel and me. I don't think either of us has recommended a 100% equity portfolio either right up to retirement or during retirement.

My comment was about your categorizing as reckless a 100% equity retirement portfolio during the early accumulation portion of one's life. I still believe that to be an incorrect categorization. Perhaps it would be reckless for you, butcI do not see how.

I do not see a 100% equity portfolio as undiversified. There are certainly a variety of asset classes within the broad category of equities.

I do not foresee bonds being a complement to equities any time soon. I think the negative correlation between equities and bonds which was realized at times during the past three decades had much to do with the sustained decline in interest rates.

I think we will continue to disagree on whether a 100% equity portfolio for younger savers is reckless.

forkhorn
Posts: 129
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### Re: The Mathematics of Retirement Investing

Why is 10.1% nominal and 3% inflation equal to 6.9% real? Isn't that 7.1% real? Is it more complex than subtracting inflation from nominal? Maybe I have had too much wine?

rocket354
Posts: 182
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### Re: The Mathematics of Retirement Investing

forkhorn wrote:Why is 10.1% nominal and 3% inflation equal to 6.9% real? Isn't that 7.1% real? Is it more complex than subtracting inflation from nominal? Maybe I have had too much wine?
Yes, it is more complex, although subtracting is a quick'n'dirty way of estimating it. If you gain 10.1%, you multiply by 1.101. If inflation is 3%, you divide by 1.03 to keep the value a real value*. If both happen, you multiply by (1.101/1.03) = 1.0689 => 6.89% real return.

*A little more specifically, if inflation is 3%, then \$100 today would be the equivalent of \$103 next year. 103 = 100*1.03. To convert \$103 next year to today's dollars, you then do \$103/1.03 = \$100.

#Cruncher
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Location: New York City
Contact:

### Re: The Mathematics of Retirement Investing

Nice work, longinvest!
longinvest in original post wrote:Our objective is for work-years salary minus savings to be equal to total retirement income.
It took me a while to understand just what your objective means. Normally one posits a desired retirement income and then backs into the savings needed to attain it. Instead you posit that one wants to spend the same in retirement as one does while working (net of savings). I like this fresh way of posing the problem.
longinvest in same post wrote:The 100/0 investor ... is left with: \$68,325.51 - \$932.50 - \$4,293.75 - \$7,593.88 = \$55,505.38 ...
The 60/40 investor ... is left with: \$65,888.10 - \$932.50 - \$4,293.75 - \$6,984.53 = \$53,677.32 ...
You're overstating the taxes by neglecting the deduction and exemption a US taxpayer takes in arriving at taxable income. For 2017 the standard deduction and exemption for a single taxpayer total \$10,400 (6350 + 4050). So the tax should be \$2,600 less (10400 * 25%) than what you show. But this affects the 6.9% and 5.5% cases the same so the difference you report is unaffected.

Code: Select all

``````Row         Col A                                    Col B        Col C
---  -------------------------------                ------       ------
1  Years                                              30
2  Withdrawal rate                                    4%
3  Salary                                         80,000
4  Social Security                                25,000
5  Retirement % of working spending                 100%
6  Base tax                                        5,226
7  Plus marginal tax rate of                         25%
8  On excess over                                 48,350
9  Growth rate                                      6.9%          5.5%``````

Code: Select all

`````` 10  \$1 grows to                                   92.7782       72.4355
11  Withdrawal @ 4%                                3.7111        2.8974
12  Needed savings per year                        11,674        14,112
13  Salary less savings                            68,326        65,888
14  Tax                                            10,220         9,611
15  Salary less savings less tax                   58,105        56,277  (\$152 / mo difference)

16  After 30 years savings grow to              1,083,138     1,022,202
17  Withdrawal @ 4%                                43,326        40,888
18  Plus Social Security                           68,326        65,888
19  Tax                                            10,220         9,611
20  Withdrawal plus Social Security less tax       58,105        56,277
21  Retirement percent of working                100.000%      100.000%``````
In row 5 I added the option to make the retirement spending something other than 100% of working spending. Here are the results when this is set to 80%:

Code: Select all

``````Row         Col A                                    Col B        Col C
---  -------------------------------                ------       ------
1  Years                                              30
2  Withdrawal rate                                    4%
3  Salary                                         80,000
4  Social Security                                25,000
5  Retirement % of working spending                  80%
6  Base tax                                        5,226
7  Plus marginal tax rate of                         25%
8  On excess over                                 48,350
9  Growth rate                                      6.9%          5.5%``````

Code: Select all

`````` 10  \$1 grows to                                   92.7782       72.4355
11  Withdrawal @ 4%                                3.7111        2.8974
12  Needed savings per year                         8,240        10,053
13  Salary less savings                            71,760        69,947
14  Tax                                            11,079        10,625
15  Salary less savings less tax                   60,681        59,321  (\$113 / mo difference)

16  After 30 years savings grow to                764,464       728,197
17  Withdrawal @ 4%                                30,579        29,128
18  Plus Social Security                           55,579        54,128
19  Tax                                             7,033         6,671
20  Withdrawal plus Social Security less tax       48,545        47,457
21  Retirement percent of working                 80.000%       80.000%``````
• Select All, Copy, and Paste the following at cell A1 of a blank Excel sheet:

Code: Select all

``````Years
Withdrawal rate
Salary
Social Security
Retirement % of working spending
Base tax
Plus marginal tax rate of
On excess over
Growth rate
\$1 grows to
="Withdrawal @ "&TEXT(B2,"#0%")
Needed savings per year
Salary less savings
Tax
Salary less savings less tax
="After "&B1&" years savings grow to"
="Withdrawal @ "&TEXT(B2,"#0%")
Plus Social Security
Tax
Withdrawal plus Social Security less tax
Retirement percent of working``````
• Select All, Copy, and Paste the following at cell B1:

Code: Select all

``````30
0.04
80000
25000
1
=10%*9325+15%*(37950-9325)
0.25
=37950+10400
0.069
=FV(B9,\$B1,-1,0,0)
=\$B2*B10
=(\$B5*(\$B3-\$B6-\$B7*(\$B3-\$B8))-\$B4+\$B6+\$B7*(\$B4-\$B8))/(B11-\$B7*B11+\$B5*(1-\$B7))
=\$B3-B12
=\$B6+\$B7*(B13-\$B8)
=B13-B14
=FV(B9,\$B1,-B12,0,0)
=\$B2*B16
=\$B4+B17
=\$B6+\$B7*(B18-\$B8)
=B18-B19
=B20/B15``````
• Copy cells B9:B21 right to column C.
• Format as needed and alter the assumptions as desired.

longinvest
Posts: 2891
Joined: Sat Aug 11, 2012 8:44 am

### Re: The Mathematics of Retirement Investing

North Texas Cajun,

First, on a theoretical basis, I have yet to see any mathematical proof that a stock with dividends reinvested, any stock, will deliver a higher total return than a TIPS maturing in 30 years with coupons reinvested, in the next 30 years. I have not, either, seen any mathematical proof that a total-market stock portfolio will outperform a balanced portfolio of total-market stocks and bonds over the next 30 years.

All I've seen so far, in my various readings about investing, is a lot of arguments based on statistical analysis of the past performance of (mostly) US bond and stock markets, where the survival bias in older data is often downplayed.

I've learned, in The Four Pillars of Investing, that the longer-term known history of investing (beyond the relatively short history of US market returns) is full of examples where the past was no prelude to the future. So, why should we assume, today, that the future will look anything like the past for US and International stocks and bonds, when there is no guarantee that it will be similar? Why should we project the past vulnerability of nominal bonds to inflation onto future inflation risk for TIPS? And why should we only consider single-asset portfolios?

Second, in my first-post example, I intentionally used an annual salary of \$80,000, not an annual salary of \$1,000,000. I'm not trying to address the case of ultra-rich people who have way more money than they'll ever be able to spend*. In this context, I do think that it is fair to consider a 100% stocks portfolio (without an equal size "emergency fund" in cash on the side, or some other mental accounting trick) as reckless. Life is just too unpredictable, especially when one is young, to be 100% sure that one will never need to withdraw money at the wrong time. I remember reading Prof. William Sharpe describing stock risk as doing badly in bad times.

* Yes, this is a joke. I'm not trying to insult anybody. I know that it's very easy to spend lots of money. Please give me some literary leeway to write down my ideas, even if not always perfectly politically correct!

I think that it is OK for bogleheads to steer young investors away from taking more risk than they realize with a 100% stock portfolio, when more reasonable alternatives exist to achieve one's financial goals.
Bogleheads investment philosophy | Lifelong Portfolio: 25% each of (domestic/international)stocks/(nominal/inflation-indexed)bonds | VCN/VXC/VLB/ZRR

longinvest
Posts: 2891
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### Re: The Mathematics of Retirement Investing

#Cruncher wrote:Nice work, longinvest!
Thanks, #Cruncher. Coming from you, it means a lot to me.
#Cruncher wrote:You're overstating the taxes by neglecting the deduction and exemption a US taxpayer takes in arriving at taxable income.

As a Canadian, I know very little about the US tax system. But, in all fairness, I also do a relatively simple calculation to estimate Canadian (and provincial) taxes, somewhat overstating them, even though I could easily do a more precise, but more complex calculation.

As a weak defense, my goal, here, was not tax planning. It was simply to vaguely estimate how much I should minimally save, given a known to be wrong future portfolio growth rate. In other words, even if I was to get a perfect tax calculation, my overall calculation would still result in a known to be wrong answer. There's no escaping it. So, I try keep the calculations simple, but good enough, to avoid giving the false illusion of an accurate minimal savings rate.

The resulting known to be wrong minimal savings rate is still useful, in real life. I can use it as an indicator to help me detect whether I am living below my means or whether I am getting "near my means" in my spending.

It's not like if I knew exactly what my salary will be between now and retirement, or when I'll need to spend money on unanticipated things. Life, or at least my life, is not that predictable. So, I try to come up with good enough approximate plans, and I also try to remain flexible.

I really like the option you added in your spreadsheet to make the retirement spending something other than 100% of working spending. That can be particularly useful for people paying a mortgage during working years and retiring with a paid off house.

Bogleheads investment philosophy | Lifelong Portfolio: 25% each of (domestic/international)stocks/(nominal/inflation-indexed)bonds | VCN/VXC/VLB/ZRR

gilgamesh
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### Re: The Mathematics of Retirement Investing

If only taxable account is left, as per #Cruncher data (retirement income = current income-savings), assuming 25% tax bracket from \$48,350 to \$80k, this person would need to save 4.42% more of their monthly wages, correct? For a portfolio with 5.5% real vs 6.9%.

sschullo
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### Re: The Mathematics of Retirement Investing

gilgamesh wrote:If only taxable account is left, as per #Cruncher data (retirement income = current income-savings), assuming 25% tax bracket from \$48,350 to \$80k, this person would need to save 4.42% more of their monthly wages, correct? For a portfolio with 5.5% real vs 6.9%.
For us Bogleheads, and for some of us nonmath folks (me), this is great.
However, the 99% of the population who are non Bogleheads (many friends) who unfortunately pay up to 1.0% to 1.25% AUM (not to mention the hourly fee) for a genuine fiduciary financial adviser this not complete but I realize that is not the OP's purpose anyway. However, just reducing the assuming return to take into account FA fees (and investment costs) could be easily included in these calculations.

Simply comparing the two calculations the current one and one with a 1.0% AUM cost for a FA and showing the difference to my friends would be enlightening. This could be used to isolate just how lucky we are not to need a FA and select the lowest cost investments in our 401k, 403bs or 457bs get to our goal that much faster! Good job.
Public School K-12 Educators: "Ask NOT what your annuity sales person can do for you, ask what you can do to be a Do-It-Yourselfer (DIY)."

NPT
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### Re: The Mathematics of Retirement Investing

longinvest wrote:
NPT wrote: The notion of first world problems sounds a bit derogatory to me (no offense taken, just pointing it out ), and in any case, it doesn't help with analyzing the numbers.
NPT, I'm sorry what I wrote offended you. It was meant as a joke, specially that I included myself in the set of people struggling with the problem of taxable investing.
longinvest, it's OK, really.

It's interesting to consider, as gilgamesh mentioned, how much more would those need to save who only have taxable space left.

willthrill81
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### Re: The Mathematics of Retirement Investing

longinvest wrote:
Fri Aug 11, 2017 10:55 pm
First, on a theoretical basis, I have yet to see any mathematical proof that a stock with dividends reinvested, any stock, will deliver a higher total return than a TIPS maturing in 30 years with coupons reinvested, in the next 30 years. I have not, either, seen any mathematical proof that a total-market stock portfolio will outperform a balanced portfolio of total-market stocks and bonds over the next 30 years.
Considering that there have only been two 30 year periods where bonds (long-term bonds at that, which few recommend) beat stocks, it seems that you are advocating hedging your bet against something that has historically been very unlikely.

You seem to be trying to cast doubt on historic performance, but the return numbers used in the OP are derived in no small part from historic performance.

While not a mathematical proof, there are very sound, logical arguments to be made that being an owner of thousands of value-producing entities (i.e. companies) over the long-term will result in a higher return than lending money to a government or those same companies.

Might a balanced portfolio outperform an all stock portfolio over the span of multiple decades? Certainly. Is it likely? Most experts don't think so at all, as evidenced by the fact that virtually all of the target date funds by Vanguard and everyone else are ~90% stocks for the distant future. About the only reason they aren't 100% stocks is because there would be little point in holding the target date fund instead of an all stock fund.

That being said, it is interesting to note how relatively small the savings rate difference would be, if the assumptions hold true, given different AAs.
“It's a dangerous business, Frodo, going out your door. You step onto the road, and if you don't keep your feet, there's no knowing where you might be swept off to.” J.R.R. Tolkien,The Lord of the Rings

longinvest
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### Re: The Mathematics of Retirement Investing

Willthrill81,

I think that I made myself quite clear in a previous post:
longinvest wrote:
Fri Aug 11, 2017 7:21 pm
Let me insist on this. I have but a single life. Even if the odds were 99% in favor of stocks, over all the possible retirement years of the past and the future, I don't know whether I'll retire within one of those 1%-probability awful retirement years for stock-heavy portfolios. I don't get to live 1,000 lives, then try to maximize the number of lives with an outstanding outcome. That is not how my unique life works. I've got a single shot at this.

Why should I take the risk of concentrating my portfolio into a single asset class, when the cost of diversifying my portfolio and avoiding disastrous outcomes is low enough? I don't care about maximizing my portfolio at death; I care about accumulating enough money to retire with dignity.
I also wrote about non-portfolio risks which are enough, in themselves, to justify against a portfolio 100% concentrated in stocks:
longinvest wrote:
Fri Aug 11, 2017 7:21 pm
There's also a huge risk that we tend to forget about; it's personal risk in our life. Life doesn't always pan out the way we planned it. In other words, plans change. Things like divorce, disease, and other milder things: getting married and wanting to use part of the portfolio as down payment. A 100% stocks portfolio is fine for somebody who will never, ever, need to withdraw from it at the wrong time.

For the rest of us, adding enough bonds to our portfolios, including inflation-indexed bonds, seems like a prudent course of action.
Bogleheads investment philosophy | Lifelong Portfolio: 25% each of (domestic/international)stocks/(nominal/inflation-indexed)bonds | VCN/VXC/VLB/ZRR

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### Re: The Mathematics of Retirement Investing

It does not matter if bonds (or any other component) of your nest egg is riskier.

Because the risk of a component don't necessarily add risk to your nest egg. It might subtract risk.

You have to focus on the overall risk.

gilgamesh
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### Re: The Mathematics of Retirement Investing

sschullo wrote:
Sat Aug 12, 2017 8:20 am
gilgamesh wrote:If only taxable account is left, as per #Cruncher data (retirement income = current income-savings), assuming 25% tax bracket from \$48,350 to \$80k, this person would need to save 4.42% more of their monthly wages, correct? For a portfolio with 5.5% real vs 6.9%.
For us Bogleheads, and for some of us nonmath folks (me), this is great.
However, the 99% of the population who are non Bogleheads (many friends) who unfortunately pay up to 1.0% to 1.25% AUM (not to mention the hourly fee) for a genuine fiduciary financial adviser this not complete but I realize that is not the OP's purpose anyway. However, just reducing the assuming return to take into account FA fees (and investment costs) could be easily included in these calculations.

Simply comparing the two calculations the current one and one with a 1.0% AUM cost for a FA and showing the difference to my friends would be enlightening. This could be used to isolate just how lucky we are not to need a FA and select the lowest cost investments in our 401k, 403bs or 457bs get to our goal that much faster! Good job.w
I know my number 4.42% is wrong in many ways....firstly my quick rough calculations did not keep the current after-tax and retirement after-tax constant. It did not take into account tax drag in a taxable account. Just two factors I can see. I cannot operate a spreadsheet. However, with trial and error I could come up with a better number that will keep current and retirement after-tax the same.

Hopefully OP or #Cruncher with their prowess can come with a spreadsheet accounting for all this and possible AUM fee too.
Last edited by gilgamesh on Sat Aug 12, 2017 10:43 am, edited 1 time in total.

willthrill81
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### Re: The Mathematics of Retirement Investing

longinvest wrote:
Sat Aug 12, 2017 10:20 am
I also wrote about non-portfolio risks which are enough, in themselves, to justify against a portfolio 100% concentrated in stocks:
longinvest wrote:
Fri Aug 11, 2017 7:21 pm
There's also a huge risk that we tend to forget about; it's personal risk in our life. Life doesn't always pan out the way we planned it. In other words, plans change. Things like divorce, disease, and other milder things: getting married and wanting to use part of the portfolio as down payment. A 100% stocks portfolio is fine for somebody who will never, ever, need to withdraw from it at the wrong time.

For the rest of us, adding enough bonds to our portfolios, including inflation-indexed bonds, seems like a prudent course of action.
If the fear is that you don't want to sell your stake in stocks when they are down (I can't imagine why anyone would have a problem selling stocks when they're at or near highs), then historically the odds are high that a bear market will not be underway when you need the money, unless the bear market is due to a recession that also caused your own long-term unemployment to occur. I would argue that's what EFs are for, but that's just me.

Some try to say that the decision to be or not to be 100% stocks is strictly logical, but I believe that it's more based on personal factors than anything else.
“It's a dangerous business, Frodo, going out your door. You step onto the road, and if you don't keep your feet, there's no knowing where you might be swept off to.” J.R.R. Tolkien,The Lord of the Rings

longinvest
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### Re: The Mathematics of Retirement Investing

Willthrill81,

This is one last attempt at making my general idea understood (note, in particular, the text that I've highlighted in red):
longinvest wrote:
Fri Aug 11, 2017 10:55 pm
Second, in my first-post example, I intentionally used an annual salary of \$80,000, not an annual salary of \$1,000,000. I'm not trying to address the case of ultra-rich people who have way more money than they'll ever be able to spend*. In this context, I do think that it is fair to consider a 100% stocks portfolio (without an equal size "emergency fund" in cash on the side, or some other mental accounting trick) as reckless. Life is just too unpredictable, especially when one is young, to be 100% sure that one will never need to withdraw money at the wrong time. I remember reading Prof. William Sharpe describing stock risk as doing badly in bad times.

* Yes, this is a joke. I'm not trying to insult anybody. I know that it's very easy to spend lots of money. Please give me some literary leeway to write down my ideas, even if not always perfectly politically correct!

I think that it is OK for bogleheads to steer young investors away from taking more risk than they realize with a 100% stock portfolio, when more reasonable alternatives exist to achieve one's financial goals.
Bogleheads investment philosophy | Lifelong Portfolio: 25% each of (domestic/international)stocks/(nominal/inflation-indexed)bonds | VCN/VXC/VLB/ZRR

North Texas Cajun
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### Re: The Mathematics of Retirement Investing

OK, longinvest, I can see why you believe a 100% equity portfolio would not be advisable for some long term investors. I disagree with you. I think our main disagreement is about whether long term investors should expect a retirement portfolio to also serve as an emergency fund. I would certainly advise most people to set up an emergency fund before they start saving for retirement.

We also disagree about how large an emergency fund long term investors will require. You seem to be implying that 40% of a long term investor's retirement portfolio should be available for emergency use. I would not advise long term investors to accept zero real return on 40% of their retirement portfolio in order to be certain that unknown risks could be funded.

I would suggest to younger long term investors that various forms of insurance, plus a dedicated emergency fund, should handle the unknown risks they might encounter.

Because I consider a dedicated emergency fund and various forms of insurance to be entirely separate from a retirement portfolio, I would not consider that portfolio to be recklessif it was 100% equities.

We may not be as far apart as I first thought, as my recommended total portfolio of retirement savings, emergency fund, and various forms of insurance is certainly not 100% equities.

longinvest
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### Re: The Mathematics of Retirement Investing

Artificially separating money into different accounts based on intended use is often considered a behavioral pitfall: wiki: Mental accounting. Understanding that money is fungible is very liberating and provides lots of flexibility. While I'm not recommending an allocation to cash in a portfolio, the following article illustrates how to eliminate artificial mental fences between accounts: wiki: Placing cash needs in a tax-advantaged account.

As for cash and bonds, they are distinct asset classes: wiki: Money Markets (also called cash equivalents), wiki: Bonds.
Last edited by longinvest on Sat Aug 12, 2017 4:26 pm, edited 2 times in total.
Bogleheads investment philosophy | Lifelong Portfolio: 25% each of (domestic/international)stocks/(nominal/inflation-indexed)bonds | VCN/VXC/VLB/ZRR

Random Walker
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### Re: The Mathematics of Retirement Investing

I've looked at Monte Carlo simulations for myself using numbers similar to what longinvest uses: 5.5% for equities and 1.5% for bonds. I've been impressed by the tremendous sensitivity of the results to changes in spending and the relative lack of sensitivity of results to changes in asset allocation. Although a more aggressive allocation may maximize the expected Mean terminal wealth, it may yield an overall lower likelihood of meeting one's goals than a less aggressive allocation. That's because the dispersion of possible outcomes is so much greater with more aggressive allocations.

Dave

North Texas Cajun
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### Re: The Mathematics of Retirement Investing

I'm sorry, but I think you made it on topic when you argued that a retirement portfolio should also be used as an emergency fund.

I very much disagree with what you believe to be the basic stuff - that separating retirement accounts from emergency funds and insurance is a behavior flaw.

In any case, I don't want to argue this any longer. I'm fairly certain we will change no one's mind.

longinvest
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### Re: The Mathematics of Retirement Investing

North Texas Cajun wrote:
Sat Aug 12, 2017 4:24 pm
I'm sorry, but I think you made it on topic when you argued that a retirement portfolio should also be used as an emergency fund.
You're right. I modified my post accordingly.
North Texas Cajun wrote:
Sat Aug 12, 2017 4:24 pm
I'm sorrI very much disagree with what you believe to be the basic stuff - that separating retirement accounts from emergency funds and insurance is a behavior flaw.
I wrote "often", implying "not always". Having a small stash of cash of a few thousand dollars can be very useful, especially when one is starting his financial life and has not yet mastered his cash flows. But, as soon as the "emergency fund" starts being big enough to cover many months (and sometimes a year or two) of expenses, I see no reason for this money to be kept away from a portfolio which could be used, during a period of unemployment, to draw a temporary income.

Here's how Taylor Larimore puts it:
Taylor Larimore wrote:
Fri Aug 11, 2017 7:34 pm
I do not have a separate, low-yielding, "emergency fund" that may never be used.

In the event I should need cash in hurry, I know I can get it from my bank account, credit card, portfolio, bank loan, family, etc..

It is a myth that everyone needs an "emergency fund."
"Our life is frittered away by detail. Simplify. Simplify." -- Henry David Thoreau
Last edited by longinvest on Sat Aug 12, 2017 4:42 pm, edited 1 time in total.
Bogleheads investment philosophy | Lifelong Portfolio: 25% each of (domestic/international)stocks/(nominal/inflation-indexed)bonds | VCN/VXC/VLB/ZRR

North Texas Cajun
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### Re: The Mathematics of Retirement Investing

Random Walker wrote:
Sat Aug 12, 2017 4:18 pm
I've looked at Monte Carlo simulations for myself using numbers similar to what longinvest uses: 5.5% for equities and 1.5% for bonds. I've been impressed by the tremendous sensitivity of the results to changes in spending and the relative lack of sensitivity of results to changes in asset allocation. Although a more aggressive allocation may maximize the expected Mean terminal wealth, it may yield an overall lower likelihood of meeting one's goals than a less aggressive allocation. That's because the dispersion of possible outcomes is so much greater with more aggressive allocations.

Dave
Researchers I've read - including Professors Jeremy Siegel and Michael Kitces - have pointed out that random walk assumptions in most Monte Carlo simulations are not consistent with bond and market returns. Specifically, they showed that most such simulations do not adjust for mean reversion for equity returns and mean aversion for bond returns.

Is it possible that your Monte Carlo simulations might suffer from the flaw which these two professors independently identified?

North Texas Cajun
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### Re: The Mathematics of Retirement Investing

Random Walker wrote:
Sat Aug 12, 2017 4:18 pm
I've looked at Monte Carlo simulations for myself using numbers similar to what longinvest uses: 5.5% for equities and 1.5% for bonds. I've been impressed by the tremendous sensitivity of the results to changes in spending and the relative lack of sensitivity of results to changes in asset allocation. Although a more aggressive allocation may maximize the expected Mean terminal wealth, it may yield an overall lower likelihood of meeting one's goals than a less aggressive allocation. That's because the dispersion of possible outcomes is so much greater with more aggressive allocations.

Dave
Here's the link to Michael Kitces' discussion about how Monte Carlo assumptions might overstate tail risk for equities:

https://www.kitces.com/blog/monte-carlo ... l-returns/
Michael Kitces at Nerd's Eye View blog wrote:Yet by default, Monte Carlo analysis assumes each year is entirely independent, and that the risk of a bear market decline is exactly the same from one year to the next, regardless of whether the market was up or down for the past 1, 3, or 5 years already. In other words, Monte Carlo analysis (as typically implemented in financial planning software) doesn’t recognize that bear markets are typically followed by bull markets (as stocks get cheaper and eventually rally), and this failure to account for long-term mean reversion ends out projecting the tails of long-term returns to be more volatile than they have ever actually been!

Random Walker
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### Re: The Mathematics of Retirement Investing

North Texas,
doesn't mean reversion for equities argue for the use of a realistic mean? The MCS simulations incorporate historical standard deviations into the calculations. Yes the simulations make no effort to predict sequence of returns including momentum and reversion to mean, but to me those seem impossible to incorporate. The MCS results are always a mean with associated wide dispersion surrounding the Mean.
I think a big issue is change in valuations, Bogle's speculative component. I think MCS generally assumes no change in valuations.

Dave

Random Walker
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### Re: The Mathematics of Retirement Investing

I'll peek at Kitce's article. Of course when valuations change, expected returns change as well. Over time, one can run repeat simulations over time as assumptions (including future expected returns based on current valuations) change.

Dave

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### Re: The Mathematics of Retirement Investing

North Texas Cajun wrote:
Sat Aug 12, 2017 4:39 pm

Researchers I've read - including Professors Jeremy Siegel and Michael Kitces - have pointed out that random walk assumptions in most Monte Carlo simulations are not consistent with bond and market returns. Specifically, they showed that most such simulations do not adjust for mean reversion for equity returns and mean aversion for bond returns.

Is it possible that your Monte Carlo simulations might suffer from the flaw which these two professors independently identified?
Backtests like the Trinity study capture what ever mean reversion occurred during the test period. These don't support a 100% stock allocation as superior either.

The Monte Carlo simulations at financialengines.com include mean reversion: "This mean reversion property is built into Financial Engines’ model of stock volatility for both forecasting and portfolio optimization." http://financialenginesadvisorteam.com/pdf/invmeth.pdf

The notion that all Monte Carlo simulations are random walks is out of date.

financialengines.com analysis is free to Vanguard clients, by the way.
Last edited by tadamsmar on Sat Aug 12, 2017 5:34 pm, edited 2 times in total.

willthrill81
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### Re: The Mathematics of Retirement Investing

North Texas Cajun wrote:
Sat Aug 12, 2017 3:11 pm
OK, longinvest, I can see why you believe a 100% equity portfolio would not be advisable for some long term investors. I disagree with you.
There are many out there who wholeheartedly agree that 100% equities is fine for those who (1) can handle the volatility and (2) have enough time or money on their side. Someone with a \$10M portfolio will still have \$5M after a 50% market drop, still more than adequate for a very comfortable retirement just about anywhere, while the same may not be true of the 66 year old who just saw their \$1M drop to \$500k.

I've heard many on this forum (not necessarily anyone in this thread) argue that 100% stocks is too volatile for most investors, but they have no problem with an 80/20 split (or even 90/10 as most target date funds whose date is decades out are allocated). Behaviorally, I think that a lot of people are going to react very similarly to a 40% drop as to a 50% drop. But for those investors who know this can happen and are financially and emotionally prepared for it, 100% stocks with their retirement portfolio (most Bogleheads who aren't already FI treat their EF as separate from their retirement assets) has been, historically speaking, the best allocation for maximum long-term return. Jeremy Siegel has demonstrated that this has not just been the case for U.S. investors.

Some argue that it is emotionally comforting to know that some percentage of one's retirement assets is 'safe', but when you look at the historic difference of returns, that emotional comfort has come with a big (in my estimation) price tag.

Despite the performance of stocks across time and geography and the known risks of bonds, the bottom line is that stocks' volatility and uncertainty drive many investors to emotionally fear them, even when they cognitively know that they need them. They then try to find cognitive ways to justify their emotional reaction. Note that this isn't just the case with stocks; it's a common mental tactic.
“It's a dangerous business, Frodo, going out your door. You step onto the road, and if you don't keep your feet, there's no knowing where you might be swept off to.” J.R.R. Tolkien,The Lord of the Rings

willthrill81
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### Re: The Mathematics of Retirement Investing

Sat Aug 12, 2017 5:24 pm
North Texas Cajun wrote:
Sat Aug 12, 2017 4:39 pm

Researchers I've read - including Professors Jeremy Siegel and Michael Kitces - have pointed out that random walk assumptions in most Monte Carlo simulations are not consistent with bond and market returns. Specifically, they showed that most such simulations do not adjust for mean reversion for equity returns and mean aversion for bond returns.

Is it possible that your Monte Carlo simulations might suffer from the flaw which these two professors independently identified?
Backtests like the Trinity study capture what ever mean reversion occurred during the test period. These don't support a 100% stock allocation as superior either.
They don't support a 100% allocation to stocks due to sequence of returns risk in the decumulation phase. This risk is far less consequential to an investor in the early stages of the accumulation phase (i.e. a young investor who is decades away from retirement).

Look at Vanguard's portfolio models with different AAs. The trend is undeniable: a great allocation to stocks, all the way up to 100%, has led to higher long-term returns, though admittedly this increase is not linear.
Last edited by willthrill81 on Sat Aug 12, 2017 5:38 pm, edited 1 time in total.
“It's a dangerous business, Frodo, going out your door. You step onto the road, and if you don't keep your feet, there's no knowing where you might be swept off to.” J.R.R. Tolkien,The Lord of the Rings

willthrill81
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### Re: The Mathematics of Retirement Investing

Random Walker wrote:
Sat Aug 12, 2017 5:11 pm
North Texas,
doesn't mean reversion for equities argue for the use of a realistic mean? The MCS simulations incorporate historical standard deviations into the calculations. Yes the simulations make no effort to predict sequence of returns including momentum and reversion to mean, but to me those seem impossible to incorporate. The MCS results are always a mean with associated wide dispersion surrounding the Mean.
I think a big issue is change in valuations, Bogle's speculative component. I think MCS generally assumes no change in valuations.

Dave
That is a fatal flaw to most MC simulations. Mean reversion in stocks is well documented (i.e. Siegel's work and many others). Statistically speaking, the best and worst returns predicted by MC simulations are outside the range expected given the historic data (i.e. they are more extreme at both ends than what the data says is within the margin of error).

Kitces showed that long-term returns have not been substantially impacted by valuations. He showed that for 30 year periods starting with high valuations, subsequent 30 year returns were only 1% lower than average, while the reverse was true of starting with low valuations. A 1% average difference that itself came with a fair amount of dispersion is hardly worth worrying over. Valuations are indeed helpful for determining starting withdrawal rates, but, interestingly, not for predicting long-term returns.
“It's a dangerous business, Frodo, going out your door. You step onto the road, and if you don't keep your feet, there's no knowing where you might be swept off to.” J.R.R. Tolkien,The Lord of the Rings

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### Re: The Mathematics of Retirement Investing

financialengines.com, a Monte Carlo analysis available for free to Vanguard clients, includes mean reversion.

willthrill81
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### Re: The Mathematics of Retirement Investing

Sat Aug 12, 2017 5:41 pm
financialengines.com, a Monte Carlo analysis available for free to Vanguard clients, includes mean reversion.
A few MC simulations do include mean reversion, but this is a relatively new development. Most MC simulations, such as those typically used by financial advisers, do not include this important feature.

I would be interested to see how the predictions of this new analysis compare to the historic record.
“It's a dangerous business, Frodo, going out your door. You step onto the road, and if you don't keep your feet, there's no knowing where you might be swept off to.” J.R.R. Tolkien,The Lord of the Rings

Posts: 7681
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### Re: The Mathematics of Retirement Investing

willthrill81 wrote:
Sat Aug 12, 2017 5:45 pm
Sat Aug 12, 2017 5:41 pm
financialengines.com, a Monte Carlo analysis available for free to Vanguard clients, includes mean reversion.
A few MC simulations do include mean reversion, but this is a relatively new development. Most MC simulations, such as those typically used by financial advisers, do not include this important feature.

I would be interested to see how the predictions of this new analysis compare to the historic record.
A new development? Financial Engines has been around for 21 years.

willthrill81
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### Re: The Mathematics of Retirement Investing

Sat Aug 12, 2017 6:37 pm
willthrill81 wrote:
Sat Aug 12, 2017 5:45 pm
Sat Aug 12, 2017 5:41 pm
financialengines.com, a Monte Carlo analysis available for free to Vanguard clients, includes mean reversion.
A few MC simulations do include mean reversion, but this is a relatively new development. Most MC simulations, such as those typically used by financial advisers, do not include this important feature.

I would be interested to see how the predictions of this new analysis compare to the historic record.
A new development? Financial Engines has been around for 21 years.
I wonder why, then, most MC simulations don't incorporate this feature.
“It's a dangerous business, Frodo, going out your door. You step onto the road, and if you don't keep your feet, there's no knowing where you might be swept off to.” J.R.R. Tolkien,The Lord of the Rings

North Texas Cajun
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### Re: The Mathematics of Retirement Investing

tadasmar wrote:The notion that all Monte Carlo simulations are random walks is out of date.
Did anyone assert that all Monte Carlo simulations are random walks? I didn't see that in the Michael Kitces piece and I don't think I wrote that in my comment. But I know that Michael Kitces and his team do a lot of work with current practitioners, and I suspect he is aware of Monte Carlo simulations being used which are random walks.

Random Walker
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### Re: The Mathematics of Retirement Investing

I assumed MCS was random walk. I didn't know it was possible to model anything else. My own naive intuition makes me think it would be very hard to beat the assumption of random yearly returns with a Mean, SD, and a strong appreciation of financial history and its many fat tails. I think assuming random returns from an expected mean and SD is an excellent initial approximation. Moreover, if one has a multi factor portfolio where the correlations between the various sources of returns is very low, zero, or negative, the random returns assumption I think must be very good. The lack of correlation between factors I think may well override any modeled effects of Momentum and reversion. What matters is the portfolio as a whole, not just one factor or asset class. Diversifying across sources of return perhaps smoothes out the yearly returns. I believe thats the main reason to diversify across factors: keep expected return constant and minimize SD. Perhaps a more efficient factor portfolio can be better modeled because errors with regard to any one factor (like market beta) don't have as large an effect on outcomes.

Dave

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### Re: The Mathematics of Retirement Investing

willthrill81 wrote: Despite the performance of stocks across time and geography and the known risks of bonds, the bottom line is that stocks' volatility and uncertainty drive many investors to emotionally fear them, even when they cognitively know that they need them. They then try to find cognitive ways to justify their emotional reaction. Note that this isn't just the case with stocks; it's a common mental tactic.
Well, I think many investors are influenced greatly by the mainstream media. When investors lose almost all their life savings to a Bernie Madoff or some other fraud, that's big news and everyone heard about it. When the stock market dropped 37% in 2008 that was big news and everyone heard about it. But I don't think everyman hears much about the years and years of positive returns. Or that markets recover relatively quickly.

So when a Fidelity or Schwab financial planner tells them that they need 40% bonds at age 40 or they may run out of money, they'll believe it. They remember Bernie Madoff and Enron and the 2008-2009 recession. And forget how much the typical long term equity investor has made over the past 40 years.

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### Re: The Mathematics of Retirement Investing

Random Walker wrote:Moreover, if one has a multi factor portfolio where the correlations between the various sources of returns is very low, zero, or negative, the random returns assumption I think must be very good. The lack of correlation between factors I think may well override any modeled effects of Momentum and reversion
Is there an asset which has consistently held a low or negative correlation with U.S. equities? Consider this quote I read in a PIMCO quantitative research document:

https://www.pimco.com/en-us/insights/vi ... rrelation/
PIMCO Quantitative Research wrote:The correlation between stocks and bonds is one of the most important inputs to the asset allocation decision. However, it is difficult to estimate reliably, and can change drastically with macroeconomic conditions.1 From 1927 to 2012, the correlation between the S&P 500 and long-term Treasuries – as calculated by calendar year based on monthly data – has changed sign 29 times, and has ranged from −93% to +86%.

willthrill81
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### Re: The Mathematics of Retirement Investing

Random Walker wrote:
Sat Aug 12, 2017 7:28 pm
I assumed MCS was random walk. I didn't know it was possible to model anything else. My own naive intuition makes me think it would be very hard to beat the assumption of random yearly returns with a Mean, SD, and a strong appreciation of financial history and its many fat tails.
One of the problems with the random walk is that it often assumes that there are multiple bear (or bull) markets immediately following one another. This is quite illogical and has not happened in the historic record to the extent that the random walk used in many MC simulations assumes.
The key point is that, despite the common criticism that Monte Carlo understates volatility relative to using rolling historical scenarios, the reality, at least relative to historical scenarios, is actually the opposite – the Monte Carlo projections show more long-term volatility, resulting in faster and more catastrophic failures (to the downside), and more excess wealth in good scenarios (to the upside). In fact, the 0th percentile historical scenario (worst case scenario, running out of money at the end of 30 years) was actually the 6.5th percentile in Monte Carlo, while the 50th percentile historically (finishing with \$1.1M remaining) was the 37th percentile in the Monte Carlo projection, and the 100th percentile (best case scenario) historically was “just” the 93.7th percentile result with the Monte Carlo projection.
https://www.kitces.com/blog/monte-carlo ... l-returns/
“It's a dangerous business, Frodo, going out your door. You step onto the road, and if you don't keep your feet, there's no knowing where you might be swept off to.” J.R.R. Tolkien,The Lord of the Rings

Random Walker
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### Re: The Mathematics of Retirement Investing

North Texas Cajun,
I think it's fair to say overall bonds uncorrelated to stocks. Now, the longer the duration and the lower the credit quality, the more equity like characteristics the bonds will have. I'm sure the correlation between stocks and treasuries does change over time, BUT there is a strong tendency for the correlation to turn strongly negative just when you need it to the most, in equity bears. I strongly believe in using bonds to dampen portfolio risk, so I keep quality high and duration fairly short. Also the sign of the correlation may change, but the absolute value of the short term bonds won't change all that much. SD of equities in the range of 20ish and for bonds in range of 5ish.

Dave

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### Re: The Mathematics of Retirement Investing

Random Walker wrote:
Sat Aug 12, 2017 9:19 pm
North Texas Cajun,
I think it's fair to say overall bonds uncorrelated to stocks. Now, the longer the duration and the lower the credit quality, the more equity like characteristics the bonds will have. I'm sure the correlation between stocks and treasuries does change over time, BUT there is a strong tendency for the correlation to turn strongly negative just when you need it to the most, in equity bears. I strongly believe in using bonds to dampen portfolio risk, so I keep quality high and duration fairly short. Also the sign of the correlation may change, but the absolute value of the short term bonds won't change all that much. SD of equities in the range of 20ish and for bonds in range of 5ish.

Dave
For those investing for the long term - the ones for whom I've recommended 100% equities - short term volatility should not matter. For the long term - 20 years or longer - the SD of of equities has not been nearly that high. And there is little evidence for consistent long term negative correlations between equities and bonds.

So why hold bonds in a retirement fund only needed for the long term and accept lower returns for nearly equivalent volatility? The only two explanations I've heard:

1. one might need the funds for disease or real estate down payment or loss of income or some unknown need;

2. many investors cannot control their fears and act rationally.

I've already explained why I disagree with 1.

Of course, once a retiree investor gets close to the start of his decumulation period, the short term negative correlation during bear markets becomes important. And a 100% equity retirement portfolio would be risky.