The cost of a 70 years annuity versus an eternal annuity

  

A while back, I read a sensational paper who was trying to scare us 🙀(classic, I know). He wrote that he was amazed of the cost of paying a 70 years annuity to an exec who left early with reprimand! Wow, given the cost of an annuity, how titanic would a 70 years annuity be worth? Well the journalist did not explore the exact cost 🙉, but continued to ramble on the excessiveness of this golden parachute. For an actuary, I think that the journalist left a golden nugget there. Sadly, a 70 years annuity is well not that of a high price tag versus a classic 30 years annuity.

One reason to look at this subject is that one of the fear 🙀of the FIRE, Financially Independant Retire Early, community is the fear of longevity. How much more a 30 years annuity is compared to that of an 70 years annuity... 

First, why would you want a 70 years annuity if you are not an executive? That would be like having an annuity for yourself and your children. The game “The Talos Principle” illuminated my though on the subject. The little robot🤖you play is the millionth generation of a serie of “IA”. Each previous generation died trying to escape various trap and puzzle. Each generation slowly learning to get better. At the game end, you prove that you are finally a robot🤖worth waking up. I reflected that we are also a generational species. We build on the shoulder of giants. From a financial perspective, both my grand father and father were not rich. Quite middle income. Although that is true, it is also true that if the earlier would have put 10$ in the S&P500 in 1900 (roughly 250$ in current day), I would be a millionaire 💰. (source: OfficialData.org). Why didn’t my father given me 10,000$ when I was born? I would have been already a millionaire 💰. 

Why would we want the next generation to start from nothing? The trill of the hardship? I never lived near the rich. Never seen the silver spoon effect. However, I have seen my fair share of adult with no aspiration. None of them were rich or started rich. Why really wonder how many wealthy do nothing of their lives vs poor one. It is great to have dreams and grits but I fail to see how it is related to the lack of money. Having infinite money means that you can procure what ever you want. Having a dream to do something do not require any money. Having money do not preclude someone to dream. Between having debt up to my eye ball or having a 6 figure net worth, I would select to be rich💸. Thus it make sense to try to attained a generational annuity.

From my actuarial background, I have learn the formula for calculating the worth an annuity. At 5% interest rate, a 50k yearly payment for 5 years is worth about 215k. It make sense. If you kept 250k aside, it would provide for exactly 5 year worth of 50k. At 5% return, you get a small discount of 35k. Interestingly, if you wanted a 20 year 50k fixed annuity at 5%, it would not cost 4x. It cost 620k. The trick here is that 5% interest on 1 million dollar is 50k. 1 million💰offer an eternal yearly payment of 50k. Compared to a 30 years annuity (like would you probably elect at 65 to provide a lifetime ending at age 95) cost 770k so for 30% more, instead of having a finite pension, you get an infinite one.

IQPF, which relies on historical data, expert opinion, RPC and RRQ, provide that roughly the expected rate of return in the future should be about 7% if you invest 100% in equity. Thus 1 million💰would provide either 70k annuity per year or 50k + 2% increase per year (to offset inflation).

For me, that 30% increase in cost is well worth to provide an enduring legacy for eternity. And finally a 70 years annuity cost roughly 3% less than an infinite. So it appears that the executive should have offered to pay for that extra 3% to ensure that all of his legacy will never be missing money ever again. I am not sure why the journalist or anyone would find the cost of a 70 years annuity so much greater than a 30 years one. They are in the same bulk part cost. Finally, the FIRE crowd might not be completely be appeased, but I hope this will help them have a fresh look at the real extra cost of having a longer time frame than the Retire late crowd 😆.  

Better metrics - Moving Average & Smoothed Average

Today’s main metric to inform us on how are funds are doing is the Market Value. The Market Value(MV) is highly volatile from day to day. Relying on this metric increase the likelihood of bad investment behavior like selling low, buying high. Also it magnifies, the risk of ruin of the safe withdrawal rule. Most financial planners agrees that one of the key advantage they bring to the table is to make investor stick to the plan. Instead of showing the highly volatile current value of the fund upfront, I suggest to show and use the Moving Average(MA) or the Smoothed Average(SA) as the primary metric.

At a high level, Moving Average and the Smoothed Average are the average of the last years market value adjusted for the time lag. I will explain later in details both metrics exact calculation.

These metrics, MA & SA, have 1 key advantage over the market value, they have lower volatility. Without taking into consideration cash flow both MA&SA produce near identical returns. Therefore, we can safely look at the Moving Average only for clarity at first. Here is a table summarizing the expected 1 year rate of return over the various strategy without cash flows.

Based on historical return (1918-2015) of the S&P 500 and the Moody Seasoned AAA Corporate Bond Yield

We quickly see that the volatility is substantially less in the moving average metric than the market value. The worst years would only affect the Moving average by -7% vs more than triple in Market Value change, -25%, in a 100% equity. That worst variation in Moving Average, -7%, 100% equity is less than the impact on the market value, -11%, at 50% equity/50% bonds portfolio.

If the purpose of buying bonds is to reduce the pain of losing money, why not also take a metric that automatically protect yourself from exuberance of the market.

Here’s a graphic to compare the three metrics through time. The graphics shows the Market Value in blue, Moving Average value in red and Smoothing Average in green (real$) using 100% equity allocation. In order to remove the impact of the ever increasing S&P, I divided each metrics by the average geometric return (7.1%) from 1922. In other word, 1 at 2015 means 631x a fund invested at 1922 (real$). A value of 2 means that the current value is 2x(1+7.1%)^(Yr-1922). At 1930, we can see the MV and the MA are equal at around 1.8. This means that both metric are the same value.

Overall, we can see that the Moving Average return is roughly the same as the Smoothed Average. Secondly, we see that these are reasonable metrics that are about average return like if we were doing the average of the future and the past value. Finally, we do not see the peak that is so characteristic of Market Value Highs and Lows. Instead we notice a smoother ride all along the way.

If we look at the exuberance of the market in 1928, the Market Value was 2.36x above the average return. The Moving Average was 1.64x in 1928. Thus if we were going to show the Moving Average, the investor would have a feeling that he is less rich than he truly is because the market was overwhelmingly hot. Thus, it would have led to a better predictor. Indeed, the irrational investor would have been happier to see a lower decline during the 1930s years than the sharp decline that the Market Value produce. By comparison instead of a drop of 30% over the following years in Market Value, the Moving Average would have a drop of 13%.

Obviously, the Moving Average do not protect against a drop in the market. It only spread its effect. So in the 1970s we see that the market would have drop and drop some more. The MA just trailed behind in the high end during these year, finally to settle to nearly the same value.

These lower drops using Moving Average will help the investor rationalize gain and loss over times. If the purpose of buying bonds is to reduce the pain of losing money, why not also take a metric that automatically protect yourself from exuberance of the market.

The second advantage is that it actually reduce somewhat the risk of ruin using a 4% rule of thumb. For example think of an extraordinary year, using Market value, the 4% rule of thumb would start you withdrawing from the highest peak and thus increasing risk of ruin. Instead using moving average will take a conservative view of the situation and automatically suggest to wait before pulling the trigger. It gives the investor a sense of perspective relative to time as to how well or how poor his strategy returned. However, it is not a silver bullet as much as being flexible in your spending after retirement.

The main disadvantage most widely known against Moving Average is that it hides the current value. Therefore right before the 2008 crash, the moving average would be lower than the market value and right after the 2008 crash, the moving average would be higher than the market value. I would argue that this is exactly the sort of attitude you need to take in order to refrain from doing irrational behavior.

Secondly, you cannot go sale at the Moving Average price. Ex: Market Value = 100$ and the Moving Average = 90$. You will not be able buy at 90$ market share. If you sale at 90$, than you would lose 10$ vs the market value. My counter argument to this disadvantage is that you cannot sale at the displayed market value on your statement that was produce 2 weeks ago at a price tag 1 week earlier. Market is very volatile day to day! Unless you are doing the transaction yourself (go DIY!), you will always suffer a lag. Also, if you are not going to sale the whole funds tomorrow, say 0.33% per months for foreseeable future aka 4% per year, than the price today will only impact a small portion of your wealth. It is not indicative what will be 4% next year or even 0.33% next month.

Currently at January 2020, we are experiencing a somewhat over valued market by perhaps 10%. Using this metric, I would suggest the person waiting to pull the plug to consider some level of conservatism in its estimated net worth and consider it perhaps 10% less than what it is currently. Obviously you would need to redo the math over your own fund to have a more inform consideration.

I hope that I was able to convince you of the usefulness of the metrics. In any case, here is the intricacy of the methodology. 

The Maths

Let’s start by the Moving Average. The Moving Average is a simple technique that simply takes the average of the fund value. In its more common application over a few days or month, we usually do not adjust for expected interest. Our purpose should be to have years to years less volatile portfolio funds value. This is why I would suggest to use the 5 years average. Over such a long period, I would suggest to multiply the result by twice the conservative rate of return. See below example. Using these Market Values, the 5 years average is 263,757$ without adjustment. Then assuming a 6.8% return, we get a Moving Average of 300,848$.

To see the advantage of going with a more complex metric as Smoothed Average, we need to enter some cash flow. After adding some random deposits throughout the years, here’s what the graphic looks like:

Now we can see the advantage of the Smoothed Average (in green). The Moving Average (in red) often drags its feet after we entered cash flow. Conversely, the opposite would occurs if we entered withdrawal instead.

The Smoothed Average is more complex and fits the need where there are any cash flow. Secondly it insert a bias by giving more credibility to the older balance. It will put a credibility of 80% (4/5) over the return 4 years ago, 3/5 over the return 3 years ago, 2/5 over the return 2 years ago and finally 1/5 over last year return. The Smooted Average formula is complex:

SA = MV2017 +4/5*(MV2016+CF2016/2)*(ExpectedRoR-RoR2016) +3/5*(MV2015+CF2015/2)*(ExpectedRoR-RoR2015) +2/5*(MV2014+CF2014/2)*(ExpectedRoR-RoR2014) +1/5*(MV2013+CF2013/2)*(ExpectedRoR-RoR2013).

Here’s a data example with above average return. The Smoothed average would be 298,384$ = 284,067 + 4/5*235,207*(6.8%-21%) + 3/5*264,722*(6.8%-(-11%)) + 2/5*276,271*(6.8%-(-4%)) + 1/5*258,520*(6.8%-7%)

Common adjustment include putting a corridor on these metrics. The corridor will limit the SA & MA between 120% or 80% of the MV. Ideally this corridor is unbiased and reflects the current fund. Other even more complex SA method exist which leads to slightly different average, but in essence this simple SA captures most of what we want.

On the Pros;

• It fluctuates a lot less than the fund value
• It delays the immediate impact of a sustain change which leads to lower rash decision
• It remove unsubstantial change which leads to lower folly decision
• It reduces the fund in period of high value and increase it in period of low value.

On the Cons:

• It doesn’t reflect the value of the fund directly. If you would sale the fund, no one is going to give you the Smoothed Value.
• It lags behind the real curve, if the interest rate is below real average rate of return or is in front of it if using too optimistic rate.
• It delays sustained change.
• Huge stress would be recognize over a long period. Thus the impact of the 2008 is felt even in when market were up a few years later.

Ode to Risk

Ode to Risk

Why should we invest in the company vs lending the company? In other word why should we invest in equity vs bonds?

First let’s concede that not everyone is built to withstand high level of volatility. It is far better to stay invested in a low return fund than to change strategy when things goes bad and reinvest after the market recovery. You must not be half convince of the below, but totally convince and go back to this article or to your saved thinking on the matter at the next crash to prevent you doing a yoyo.

Personally, I am convinced after research that the best way is to be 100% in equity. Here are my arguments.  

High level argument

Companies are concerned to maintain good credit level in order to have lower interest rate to pay for the future loan in order to be able to take on more profitable project. In case of bankruptcy, bonds gets repaid first and equity holders are often left with nothing. Thus bonds are safer than equity. Also, bonds have known interest rate when issued thus repayment fluctuates less than the value of the company.

However, no company would start borrowing at a rate of return higher than the project rate of return. If they did, they would get bankrupt. Thus rate of return of equity must be greater or equal over the long term than the borrowing (bonds rate). As an owner we can thus expect to receive more than the lender.

By spreading the risk throughout the entire market, you are no longer concern of any one company but to the overall risk. In a diverse index funds, bonds & Equity are protected against a catastrophic event. At this level though, there are still lot of fluctuation for equity.

In-detail argument (no withdrawal)

Why should you accept the risks that equity offers vs bonds? I have compare the return over various period (data : S&P500 and Moody AAA corporate bonds over 1968 to 2015) over various allocation.

Since the market is indifferent of investing a thousand dollar or a million, I simplified the return as a multiplier of a 1 unit of fund. The Tab 1, looks at the return (constant $) without withdrawal. The more the cell is blackened, the more the results are likely to occur. Thus we can see at time 0, all funds are at 1x (ie 1,000). After 30 years, Tab 1.a reports 2x-10.5x (ie 2,000-10,500) vs Tab 1.c which reports 3.5x-20x (ie 3,500-20,000)

First, let’s concede that we do see some drop below 0.5x in the 0 to 5 years range in the 100/0 allocation that we do not see in the 50/50 allocation. If we continue our analysis, we get the summary table

Let’s first compare what happens in the 50/50 allocation vs the 100/0 allocation:

• Let’s admit that the worst after 10 years is substantially rosier in the 50/50 than the 100/0 (0.91x vs 0.67x). However, note that this reflect a single series of event in the past 100 years. Also note that the difference is 73%.
• The worst 25% percentile of the 50/50 allocation are always similar or worse than the worst 25% percentile of a 100/0 allocation (After 1 year, 0.99x vs 0.96x). This is in-line with the fact that only about ¼ of the years returned negative return over the last 100 years.
• This means that you can expect only better result or about in 75% of the case with a 100/0 allocation than a 50/50 allocation.
• The worst 50/50 is the same as the worst 100/0 after 23 years of investing at 1.5x. This is highly interesting for the most conservative folks out there. After 20+ years, you are just losing money by having a lower allocation.
• In other words, you are certain (based on historical results) to have a lower results with a 50/50 allocation after 20 years. Thus you are trading an uncertain gain for a certain loss going for a 50/50 allocation.
• At 30 years, the average 50/50 is worse than the worst 25% of 100/0 (5.47x vs 5.78x). If you fear the worst 25%, than be happy to know that you would beat on average of 100/0 beat it after 30 years.
• I have an easy time to concede that we want to minimize our lost. However, it makes no sense to aim for it.
• At 30 years, we see that the best 75% of 50/50 is a little underneath the worst 25% of 100/0 (6.58x vs 5.78x). We could be tempted to think that the upside (75%) of a 50/50 is enough rosy. However, it is almost the same as the worst (25%) of a 100/0.
• In other word, if I’m unlucky, I will do more than a lucky conservative (50/50) would make.

Perhaps a in between method would produce better results, however they are similar

• It’s easy to see that the worst of 100/0 is lower most of the time than the 75/25. However, please note the size. Having 74% vs 64% is a difference of 10%. In other words, 100,000$ over a million. If the size difference was bigger I might concede this, but at this level, it is not enough significant. Can’t you take a 10% cut in your spending for a year?
• At the worst 25%, they are always similar or worse in the 75/25 than the 100/0 scenario.
• At 22 years, the worst 75/25 match the worst 100/0 at 1.5x.
• At 30 years, the best 75% of 75/25 is the same as average of 100/0.

In-detail argument (withdrawal)

You may raise objection that sequence of return risk or withdrawal will more than offset any gains without withdrawal. Let’s look at what happens if we adjust for a simple withdrawal strategy. Let’s withdraw 4% of the current fund and guarantee at least 75% of all previous withdrawal. This would be similar/or better than a SWR of 4% with a 3% floor. See the Tab 2 for the results.

First, let’s concede that we do see some drop below 0.5x in the 0-15 years range in the 100% Equity allocation that we do not see in the 50%/50% allocation. However, if we continue our analysis of it we get the summary table

When we compare these, we quickly see that results are well below the no withdrawal strategy. Indeed, we see the ugly head of sequence of risk. The 10-20 years worse years are more disappointing than before or after.

• We can see the difference of the worse between 50/50 to 100/0 is only 15%. However, after 10 years, the difference of the average is 22%.
• Thus you would lose more on average for a lower pain in the short term.
• At 7 years, all allocations are having similar worst 25% (0.9x). This means that on most case (75%) if would be advantageous to be in 100/0 after 7 years.
• At 22 years, all allocation are having similar worst (0.63x). It would be thus just advantageous to be in 100/0 than the other allocation.
• At 30 years, you are sacrificing more than 1x (ie your entire initial withdrawal) with going with a 50/50 rather than a 100/0.

Finally, we can go back at the question: if there’s a drop, how long does it take for us to earn back our money? The answer can be found in an expansion of Table 1. At the 14 years mark the worse real $ 50/50 would finally be over the 1x mark. The inflation is a major culprit for this. However, let’s remind us that 50/50 means that we are 50% in equity. Thus a 50% drop would still means a 25% in a 50/50 scenario. For a 75/25 allocation we need to wait 17 years so that worse hit the 1x. However, the 100/0 allocation hit the mark after 18 years, 4 years after the 50/50.

If you withdraw with the simple strategy of 4% of current fund (guaranteed 75%), the stats are worse. Only the 100/0 crosses the in the money point for the worst case scenario at the 30 years mark. The others do not have the luxury of getting even.

As my ode to risk, I hope I was able to convey the reason I think that a heavy equity allocation is superior to a 50/50 or even a 75/25 equity allocation. In the end, I would summarize my analysis as,

• Low derisking gain going at a 50/50 for a major loss often.
• Gain of a low equity is only felt in the worst scenario. Even then as time passed, the worst of 100/0 is better than all the other allocation.
• The worst 25%  results are better in a 100/0 almost all the time. All better odd returns better results in a 100/0. 
• After 30 years you would always lose out using a lower allocation.

Statistic of Ivy League millionnaire

Statistic can be deceiving. It’s even easier to deceive or genuinely fail to understand their implication. I’ve just heard such of leap of reason.

Chris Hogan have just release his new book, Everyday Millionaires(2019). I did not read it yet. I’ve heard a startling statistic that 10% of the millionaire come up from Ivy League. 90% did not gone to Ivy League in order to get millionaire.

It seems on the surface that it tells us that you don’t have to go to Ivy League to be millionaire. This is true. The reverse is not so evident. According to a simple google search, we see that the % of American who have gone to Ivy League is roughly 0.2% (Quora). The statistic is probably off. The percentage of millionaire in America is roughly about 11% (DQYDJ).

If we merge the three statistics, than we can tell that going to Ivy League has a huge correlation to becoming millionaire. They are massively over-represented! The argument goes along the line:

• If I go to any university : I got 11% chance to become millionaire.
• % of American who have gone to Ivy league and are millionaire (10% * 11% = 1.1%).
• If I go to an ivy league university : I got more than 500% chance to become millionaire (0.2% * 500% < 1.1%).

Probably some of these statistics are incorrect. 500% chance of becoming millionnaire is illogical. At least they are incoherent. Regardless, it gives a rough idea that most Ivy league graduates are going to be millionaire. They are mostly all going to be. 

Saving Rates - Including Interest

There are many ways to calculate saving rate. Saving Rate = $SAVED/$IN. Various debates exist around each terms. Items up for decision:

• Should we use Pre-tax or post-tax as the $IN?
• Should we look at the family as a whole or per person?
• Should we use House payment/debt (the portion other than the interest on loan) as $SAVED?
• Should we use Interest received as $SAVED & $IN?
• Should we recognize DB accrued as $SAVED & $IN?
• Should we include Social Security (CPP/QPP contribution)?
• Should we set a Saving Rate or set a retirement goal.

Let’s go deeper in the rabbit hole of Interest on Saving. Including Interest will have the immediate effect of increasing your saving rate.  At the basic level, including/excluding anything do not change your retirement date or how much actual earning you save. However, if by including or excluding some items some features may be highlighted or new strategy can be discovered.

Interest is define for this article as the total return of your retirement fund regardless of its origin (growth of stock, dividends or coupon). Compounding interest informs us that interest will exponentially grow as time passed. Near retirement, interest will largely outgrow our saving. Why save at all, if you save 2,000$ while the fund grows 200,000$ due to interest?

After including the interest both in the $SAVED & $IN, I’ve created ways to reduce the amount saved from earnings while the Interest grew. Given a fixed saving rate, this has the effect of delaying retirement by a few years. Thus the below strategy are more in-line for those who would like to be Financially Independant(FI), but not Retire Early. Get a higher raise than your boss wants to!

"Get a higher raise than your boss wants to!"

Tweaks & issues

The overall objective of the methods proposed below are to decrease saved earnings while reaching retirement. Thus this has the positive impact of giving you a bigger raise than your employer gives you. 

This strategy makes more sense if you increase your your replacement ratio. A 60%-70%, or (1-Saving Rate)% would not work as you expect to spend your entire earnings at retirement. Thus to keep your standard of living you would need a 100% replacement ratio.

Lowering your saved earnings would delay retirement all thing equal. As often the case things do not need to be equal. You could modify your Saving Rate after applying the below strategy to give more or less the same initial saved earnings or retirement date. In the simple strategy below, I did not adjust and thus get a higher retirement date. In the second strategy below, I did adjust. 

From an administrative issue/market volatility issue, you may want instead to use your expected withdrawal rate (3%, 4% or 5%) instead of experienced interest. This way, a bad return year or a great year will not mess up your saving strategy. Personally, I like to go with 5% withdrawal as it provide no risk of ruin and decent inflation increase using the Accumulation-Dynamic Decumulation method.

Simplest way

The first formula that come to mind is to remove the interest from what you need to save as earning.  If you have a lower than 50% saving rate, this strategy fails as you would not need any saving quickly. At 15% saving rate, this force you to stop saving at half your career.

Instead, a small tweak to this strategy is to recognize a percentage of the interest. 
Ex: if you made 20,000$ of interest this year, only recognize 10,000$ of it. The weight to use is hard to determine. At 15% Saving Rate, using a 30% weight of Interest seems appropriate as it delays retirement by 4 years and small save earnings near retirement. At 30%, you get a 4 years delays at 50% weight of interest. At 50%, you get a 4 years delays at 90% weight of interest.

Below is a table summary  of the extra available spending with/without the strategy¹ at 25% the way to retirement, 50% and 75%. It use the 4 years delays with vs without the strategy and weight discuss above and using a family salary of 100,000$.

Next, here a graphic showing how the saved earnings compared in the 15% Saving Rate simple strategy for a family with 100,000$ salary :

See google drive file, sheet Inc. Int with Weight.

More complex way

The first strategy have issues. Most of the decline in saved earnings is near the end.  It leave a difficult parameter to set up (weight of interest). Also it still have relatively large saving near retirement or no saving for a number of years prior retirement depending on the weight. To fix this, let’s look at the following formula:

Think of it this way, in the scenario where Saving Rate = 50% = (Save Earnings + Interest)/ (Earnings + Interest). When you start saving, you would need to have Save Earnings = 50% of Earnings. When you are at retirement, than you would need Interest = 50% of Earnings. Using this nice equilibrium, we can adjust any other saving rate by applying a similar accrual pattern.

This strategy ends up at retirement with almost no contribution and enjoy Spending increase 5% more than your boss give you (which decrease as you age to a spending increase of +1%).

To get similar retirement age however, we need to be more aggressive in saving in our early years than the previous strategy. Below is a summary result of comparable saving rate incl./excl. Interest and the Extra Spending along the way where Retirement age is the same for both Scenarios. 

See google drive file, sheet Inc. Int Max SR inc.

Conclusion:

Where ever you are in this journey you could think of implementing the above strategy or part of it. Otherwise, you could also adjust the strategy to say that you are going to have 10 years of higher saving and then drop back to follow these strategies. Many different path exist for the DIY.

While there is a great focus in increasing saving, we need to be conscious of why we are saving. Saving for saving sake makes no sense. Dying with millions in our bank with no plan is not a wise approach to life.

If we were immortal, than this sort of approach makes even more sense. It would be foolish to have to save for the rest of eternity for saving sake.

Note:

¹: Both scenario using a goal of 100% replacement ratio and the same Saving Rate. If you have a lower goal, you could adjust the formula. If you wanted the same retirement date, increase your saving rate after applying the strategy to get it. Based on Fund = 0$ starting saving at age 20. For example say retirement age =65, 25% of the way is (65-20)*25%=11 years after saving start (age 31). 

Retirement risk

There are two main ways to handle decumulation: annuities, self-managed(DIY) decumulation. The main advantage of annuities products is the guarantee by the sponsor. DIY decumulation have no guarantee and thus is more risky. On the up side, DIY are very flexible.

The Single life Annuity promise a stream of payment which stops at death. By design, it leave nothing to the heirs. Obviously, you could combine it with a life insurance, or guarantee in order to leave money to heirs, spouse or dependant.

By having tens of thousands of member, the sponsor will face on average the expected mortality. Member who dies later will be offset by the one who dies young. Sponsors are still at risk expected mortality to rise in the future. Future increase can be mitigated by having higher mortality based on the latest trend.

The DIY decumulation does not benefit of such risk reduction. The expected mortality is roughly age 85. However, the oldest man on earth is age 115. If you retire at age 55, that's a 20-60 years life spends. About 50% of the population will die however between -5/+5 years of the life expectancy. An 80 years old life expectancy is 86 for a Canadian female. If you survive up to 85, it grows to 89. We cannot be sure in advance if we will die early or not. In addition with you have a spouse, the chances that one of you goes on to live to 90 are great. To counter this, we need to protect for an extended life spends.

However, mortality is not the only risk. The biggest risk of DIY and annuity product is investment returns. Assuming constant rate of return is overly optimistic and naïve. The standard deviation over equity is about 18% and 7% for bonds. The classic strategy to deal with this risk is to have a conservative asset allocation with lots of bonds which have lower volatility. Going one step further, you can immunize against the bonds interest volatility using duration matching and cash flow matching. These options however, all suggest dropping the rate of return from what the equity offer to what short term bonds offers. This can means to reduce return by 2x and more over time.

Next, there is the sequence of return risk. This is the risk that a negative sequence of return prior to receiving a positive sequence of return. The annuity products are less affected by this risk as they have annuity starting each years. The law of average applies. Half will face negative sequence and half will face positive sequence. For DIY, the problem can be quite significant. The classic way to mitigate is to have a higher bonds allocation. That will reduce the volatility that you will face, but it will reduce your rate of return. Some have now recommended to have higher equity allocation after a few years though.

Finally there is the inflation risk. This is a lesser risk than the other. However, if inflation start to pick up, this may become a serious issue. Both DIY and annuity products face this risk. The solution in the annuity products is to get some bonds with inflation protection.

The shocking math of accumulation

The shocking math as to why vampires must be rich? 

There's a two main mathematical formula that determines on average the expected gains of any vampires. The composed interest accumulation formula focus on the accumulation of a Lump Sum amount for a number of years. The other is the future value of a serie of contributions.

A vampire have thus a lot of possibility to becoming rich. Either he can invest a lump sum of money or a smaller contribution over a period of years. A third possibility is anything in between. Let's explore the vampire timeframe first of a 100 years of investing.

Knowing that the past last century rate of return for a all equity was in excess of 10%, a 73$ investment would be enough to be a millionnaire. Else, the vampire with less than 10$ to spare each year would be a millionnaire also. All in all, it's super easy for a vampire to become rich.

From the above table, we can compare the lump sum strategy to a contribution strategy. The Lump Sum requires 11%/15%/21% less contribution at 10%/7%/5% interest rate. However, the lump sum is 10x/15x/21x more than the yearly contribution strategy. If you are able, it makes sense to fund early and have lower contribution in the future. I haven't read anyone doing it except when in retirement. For a person to be able to act on having lower contribution in the future, we need to have a well though out plan, to be able to live frugally at the beginning and be able to live expensively at the end. It is a rare combination. 

Over 100 years, it is staggering the impact of interest. It requires 50x more contribution to fund a million at a conservative asset allocation than an aggresive one. If we look at Lump sum, it takes 100x more lump sum to get there.

Now let's see over a human life of 35 years investment:

In order to retire a millionaire, we would require about 36k/94k/181k to compound to 1 million. The impact of interest rate is still very much felt.

The way I think about it, if you were able to split at your death 1,2 million and that you would have 3 childs and 9 grand-childs, then each grand-childs would be able to become millionnaire at age 65 easily.