vendredi 31 janvier 2020

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.



vendredi 17 janvier 2020

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.



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 d...