@Science

A response I gave to a friend who felt the QuantitativeΒ Easing used during the COVID epidemic would ultimately hurt everyone, and that an absolutely free market is the only market. While I do agree with free market sentiments the truth is QE does work when used to avoid massive market crashes and as long as it is reversed at a later point once markets recover then it does far more good than bad. Here I lay out all the math to show why.

In an ideal world I do agree that there should never be market interference. But that only works when you truly live up to that at every level, something that you can't truly have and still have a government.

Consider, the only reason we needed to handle a depression at all is because the government first interfered with the market by using law to force shutdown of businesses or their ability to accept customers inside. If the free market were not tampered with in that regard there would have been no significant depression to deal with in the first place. So we are doing cleanup at this point, so the point is somewhat moot.

That said the assessment that interfering now with QE results in more problems later isnt entirely true, only somewhat. You are correct of course that there will be a downward pressure on the market into the future for sometime, which is what you're referring to. But I think its important to understand the exponential nature of money and while earlier short term gains can far outweigh later long term loss.

Basically the dynamic plays out like this when we see QE going down in times of depression.. 1) ROI on market investments soar many times higher than what would normally be possible in the short term. 2) There is more money in the system so far more people are able to leverage that ROI than they otherwise would in the short term. These two principles combined means people see huge financial gains during a time when we would otherwise have huge losses. This is followed by a longer period of time, on the order of 4x to 6x the length of time where we see a boost where the market is now going to underperform, but at a much lower hit than the gain we saw during QE.

So lets use real world numbers to demonstrate the idea with some solid math. I will pull the numbers from the real world numbers and spitball some figures minimally and try to underestimate so as no to bias it in my favor for numbers we don't actually know but that follows the proposed premise of short term gains followed by long term downward pressure from QE that is higher but distributed over a longer time period... Here are the numbers im going with pulled from real world numbers:

Dow Jones industrial average baseline annual return when QE is not evoked, taken from the 20 year average: 7.03%

First lets look at how things would play out if you invested at the start of QE, which was after the COVID lockdowns and the initial covid drawdown (optimal time to invest):

Short term annualized pressure from QE: 93.79% (1)

Short term annualized P&L based on DJ movement (baseline + QE): 100.72%

Long term loss pressure from QE over following 20 years: -74.19% (2)

Long term annualized QE loss pressure: -6.54% (3)

Long term annualized P&L (expected DJ movement, baseline +/- QE loss): 0.49%

Portfolio P&L on 100K invested at beginning of QE after 2 years: +$290,032.80(4)

Portfolio value on 100K invested at beginning of QE after 2 years: $390,032.80 (4)

Portfolio value after 22 years when starting at 100K with QE: Β $430,088.72 (4)

Portfolio P&L over 22 years on a 100K investment with QE: +$330,088.72 (4)

Total 22 year percentage P&L with QE: +330.08% (4)

Annualized percentage P&L over 22 years with QE: +6.85% (4)

Time to DJI recovery with QE (actual observed): 275 days (0.75 years)

As you can see the gains are still quite nice even with QE slowing the market down to a crawl for 20 years. Just for comparison lets run the same numbers without QE doing our best to use the real world figures again.

Observed drawdown loss due to covid with QE: -36.77% (5)

Projected drawdown loss due to covid without QE: -85.58% (6)

Historic annualized recovery rate with QE (2009 - 2013): 18.82% (7)

Historic post-QE annualized P&L (2013 - 2018): +12.63% (7)

As you can see, even post QE for the almost 5 years that followed we saw an annualized **increase** in ROI over the baseline of 7.02% and not the decrease you propose or expected.

Time for covid recovery without QE: 10,418 days or 28.5 years (8)

So we can see the time to recover your pre-covid investments without QE is only 275 days compared to without QE we are talking 10,418, or over 28.5 years. Thats huge. So more importantly what would a pre-covid portfolio look like in 22 years with and without QE. Remember the above portfolio numbers were if you invest at the beginning of QE, an optimal point, not pre-covid drawdown, which is the least favorable point.

Loss on pre-drawdown 100K at minima with QE: -$36,770 (9)

Portfolio value on pre-drawdown 100K at minima with QE: $63,230 (10)

Portfolio value on pre-drawdown 100K after 22 years with QE: $271,939.58 (11)

Gain on 100K due to covid drawdown with QE: +$171,939.58 (12)

Now for the numbers without QE

Loss on 100K due to drawdown without QE: -$85,580 (13)

Portfolio value on pre-drawdown 100K at drawdown without QE: $14,420 (14)

Portfolio value on pre-drawdown 100K after 22 years without QE: $62,594.80 (15)

Portfolio P&L on pre-drawdown 100K after 22 years without QE: -$37,405.20 (15)

Portfolio percentage on pre-drawdown 100K after 22 years without QE: 62.59% (16)

Portfolio P&L percentage on pre-drawdown 100K after 22 years without QE: -37.4% (16)

Portfolio P&L on pre-drawdown 100K annualized from 22 years without QE: -2.1% (17)

(see notes at bottom to see how these were calculated from the real world numbers)

As you can see under a QE scheme we recover in a fraction of the time to baseline, only takes us three quarter of a year before we are back to where you started and after 22 years you've still managed to turn your original investment into 4.3x its value in 22 years, and gaining just a little below base rate per year on average at 6.85% gain annualized. By comparison without QE your original investment would actually have lost value over a 22 year period being reduced by almost a half with portfolio investment dropping by -37.4% of its starting value and with an annualized average return over that time of about half at -2.1% loss per year. So clearly QE applied early on before a recession has a chance to fully realize, even with a theoretical economic hit that applies in the 20 years that follows (though we have not observed this historically), is more profitable for everyone than it is to just let the market naturally recover.

This is, in fact, as shown, demonstrated by historic values which agree with currently observed and calculated values here. In fact I was ultra conservative and the real numbers are likely far better. QE tends to cause recoveries proportionalΒ to the rate at which the QE occurs followed by an increase in future gains rather than a decrease for at least half a decade out. In fact even when we reversed the QE from 2018 to 2020 there was no noticeableΒ hit to the economy which was still growing in part due to the fact that it was thriving from the effects of earlier QE in past recessions.

This doesn't mean QE works as a general principle outside of extreme circumstances such as a significant recession. It will effectively weaken the dollar when it comes to foreign trade as a direct result, which is reversibleΒ through reverse QE. We can see that with the current QE, the effect is relatively quick.

Pre-covid/lockdown the dollar had a value of 0.93 relative to the euro, during the economic crash of covid, but before QE was initiated the dollar crashed from 0.93 down to 0.88 relative to the euro at the point QE was initiated. QE had the initial effect of raising the dollar in the short term from 0.88 to ~0.935 but as WE progressed the dollar ultimately crashed back down from QE to 0.845. As such it went from 0.88 to 0.845 due to QE with only a temporary value boost of a few months. Essentially it reduced the value of the dollar by 4%. This value, as with past QE is realized very shortly after QE ends and stabilizes flat without any further reduction in the dollar due to QE after it ends. So while it does negatively impact the dollar, the impact is not huge. Moreover since it significantly bolsters local economic growth both long term and short term as a result ultimately the QE will/can be reversed at a later date without economic impact and the value of the dollar will gain back that 4% plus whatever value it gained as the result of the boosted economy.

So while its effect on the dollar would cause us to discourage its use in non-recession periods, and it is a tactic that has diminishing returns if used in that way, so not effective. Overall when used during recessions it is an effective tactic that overall reduces the fallout from a recession significantly and boosts economic recovery overall, rather than harming it.

==== Notes if you want the math ====

(1) to calculate the annualized gain from QE is simple. There was a 60.6% rise in the DJ from march 23rd until today, that's 248 days. The following equation will turn that into this year's annualized DJ gain: daily return = 1.606^(1/248), yearly return = (1.606^(1/248))^365. So we have a daily return of 0.19% daily or 100.82% yearly ROI. Now we already know that the baseline ROI is only 7.03%, so we can therefore calculate that at the current rate of annualized gain is 100.82-7.03 = 93.79% **over baseline**

(2) For this we don't know the actual number but we can estimate based on first principles. Let's assume that the negative effect of QE is twice that of the annual positive effect, but spread over 20 years rather than condensed into a short period like positive effects are. In terms of percentage that would be: 1 - 1/(1.937*2) = -0.7419, so basically there is an upward pressure in year one of ~93%, representing a near doubling (100% or 2x), and then a downward pressure of -74.19% the following 20 years which represents a quartering (1/4th or 0.25x) over the 20 years to follow.

(3) to go from the 20 year figure to the annualized figure we have the following equation: 1 - (1/(1.937*2))^(1/20) giving us a annualized loss pressure due to QE of -6.54%

(4) For this we have to determine how long the positive effects of QE last before the proposed negative pressure kicks in. It is hard to say for sure and we would have to do our best guessing based on historical examples of QE. What we know is the returns from QE follow a logistic curve (similar to a logarithm but restricted such that it will have a maximum attainable maximum value before leveling off). As such the initial gains you see will be sharp and appear linear at first and at about half way through you will start to see the gains slowly taper until finally leveling off. Right now we can see from the current DJI market movement we are still in the early part of seeing the GAINS from QE and must not even be half way through. So assuming the gains will last at least 2 or 3 more months before the tapering is noticeable, indicating we are about half way through the gains then a reasonable estimate is we should see these gains for at least a total of 2 years since the start of QE before the gains dissolve and a downward pressure begins to be applied instead.

Therefore assuming we see QE gains at observed rates for only 2 years before transitioning to the negative effects lets calculate what that is. We know annualized gains due to QE during the positive phase is 93.79% above baseline, taking that to a 2 year figure is:

(1.9379)^2 -1 = 2.755 or +275.5%

Add in the baseline rate to arrive at the 2 year gains during the positive period of QE:

2.755 + ((1.0702)^2 - 1) = 2.90032804 or +290.03%

Now to calculate our portfolio value n 100K after 2 years:

(100000 * 3.90032804) = $390,032.80

Subtract 100K to see profit value:

(100000 * 3.90032804) - 100000 = +$290,032.80

Now lets calculate the portfolio value after 20 more years of QE negative pressure, this would result in previously stated annualized P&L for the next 20 years of Β 0.49%.

390,032.80 * (1.0049^20) = $430,088.72

and subtract initial investment to see what that is in P&L:

390,032.80 * (1.0049^20) - 100000 = $330,088.722

The P&L as a percentage is simple now:

(430088.72/100000) - 1 = 3.3008872 or +330.08%

Lets get that to its annualized value:

4.3008872^(1/22) - 1 = 0.06855798324 or 6.85% annualized return

(5) This is the actual observed drawdown due to COVID before QE was employed, after which the stock market started to rise. Keep in mind QE was applied before covid has completely run its course so in reality we would have likely seen this trend continue throughout the lockdown period at a minimum and probably longer. The loss during max drawdown observed due to covid was from ~february 20th to march 23rd, only a 31 day period!

(6) To project the overall loss we first observe the initial downward spike was linear at a constant rate more or less. We can project that out to what it would look like if allowed to continue into the end of lockdown without intervention to artificially reverse it. Covid restrictions began easing across most states by late may and lockdown was over by late june and into july, though restrictions remain in effect for non-essential employees going back to work still by then. Being conservative so the numbers don't work in favor of my argument let's just assume the projected covid loss would have gone to the end of lockdown then the loss would have abruptly ended with a reversal and recovery. So from February 20th to July 1st which is 131 days would have been the natural drawdown period.

first with take the -36.77% loss over the observed 31 day period and turn that into our daily rate:

1 - (1-0.3677)^(1/31) = 0.01467 or -1.467% daily loss

Then we take our daily loss and apply it to a 131 day period:

1 - (((1-0.3677)^(1/31))^131) = 0.8558748085 or -85.58% total drawdown

(7) Next let's take a look at an example of QE in recent history, we used QE extensively from 2009 to 2013 for our economic recovery from the 2008 recession. This is also the most comparable drawdown to the COVID drawdown we have. The recession began 1st of october 2007 and reached its bottom point on the 2nd of february 2009 with a total drawdown of 49.43% over 490 days. The DJI recovered to the value it was at on oct 1, 2007 for the first time since then on January 15, 2013, or 1443 days.

QE began to be employed only one month prior to bottoming out from 2009 recession and was the reason we began to see recovery at this point. However in this case the QE applied was spread out over a 4 year period very slowly, as such the recovery took much longer and the damage done was much greater. In fact the total quantity of QE due to COVID is almost exactly the same as it was when applied to the 2009 recession except for two important details 1) the QE was applied over a 4 year period instead of a few months 2) Trump spent 2 years prior to 2020 from 2018 to the end of 1029 **reducing** the QE from the Obama era by 22% before employing QE again. This means that overall the total QE is in fact less than the QE used to recover from the 2009 recession but applied over a much shorter period. This is absolutely key to the success, if you are going to take a hit from QE you want the gains to be concentrated as much up front as possible to have the best long term gains. Spreading out the QE ultimately caused a lot of harm.

Still we can ue 2009 QE as a historic example of how QE can be successful, even long term, and we can also use it a bit later as a comparison against the current QE to get a sense of just how much better performance we get out of QE concentrated up front as opposed to being spread out. So lets take a minute to look at what sort of numbers we see from this.

Considering the projected drawdown due to covid is significantly higher than the earlier recession, which is understandable considering it literally halted all business everywhere, we would expect a similarly long recovery, in fact even more so since covid didn't just go away on that date, economies around the world are still crashing. What you notice if you look at the recovery post 2009 is that the market movement went higher than baseline expectations of growth at the usual 8 some percent when not in a recession though after the recovery it went back to about baseline. In fact we never really noticed much of an economic hit long term from the QE at all as you might expect as it was overshadowed by the gains of capital injection. Now covid as I stated would have likely had far more significant long term effects and and the increased rate of recover over baseline from 2009 is from the QE, a natural recession will recover at below baseline for some time before eventually recovering to a baseline recovery rate. But lets see what the historic QE based recovery rate is based on 2009's QE.

First lets calculate the percentage needed for the DJI to recover to baseline once the recession has reaching its minima:

1/(1-0.4943) - 1 = 0.97745699031 or 97.74% needed to recover from -49.43% loss

this was over 1,443 days (3.95 years) so the daily percent return seen during the recovery was:

(1.9774)^(1/1443) - 1 = 0.00047258762 or 0.0472% daily

To annualize that:

((1.0004725)^365) - 1 = 0.18817886404 or 18.82% annualized gains

That is, indeed, quite a bit above the usual 7% - 8% we calculated outside of QE. But was there a long term hit? Well lets look at the growth rate once QE ended which was beginning of april 2013 until 2018, which is when Trump began reversing the QE.

From april 1 2013 to Jan 1 2018 the DJI increased a total of 76.21% over 1,736 days or 4.76 years. We can annualize that figure and get:

(1.7621^(1/4.76)) - 1 = 0.12638560229 or +12.63%

(8) Now let's calculate the projected recovery time from COVID given a natural baseline recovery rate without QE, which we stated earlier was 7.02%. Again remember this is extremely conservative and real numbers are likely to be below baseline during recovery until we recover and only then would baseline growth be realized again. But I want conservative results that are not favorable to my assertion to prove my point.

First let's calculate the total percentage needed to recover from the covid drawdown of -85.58%:

1/(1-0.8558) - 1 = 5.93481276006 or 593.48% (yes it is that huge!)

Now assuming post-covid lockdown we have a baseline recovery rate with an annualized P&L of 7.02% how long would that take in order to compound to the point we would have the needed total percentage of gain to equal the above figure?

First we calculate the daily percentage gain off the 7.02% baseline annualized figure.

1.0702^(1/365) = 1.00018589549 or +0.01858% daily

Now lets take that and use it to figure the number of days needed to recover to pre-covid DJI:

(1.00018589549)^days - 1 = 5.9348(1.00018589549)^days = 6.9348log(6.9348) / log(1.00018589549) = 10,418.39 days or a whopping 28.5 years

(9) This is easy as we know the covid draw down of -36.77%

100000 * -0.3677 = -36,770

(10) to get total portfolio value just sum our losses and our starting value

100000 - 36770 = $63,230

(11) We know from earlier that if you had invested at the bottom of the drawdown with QE applied as it is and held for 22 years we would have the earlier stated gain of +330.08%. Therefore now that we know the amount we would have held at that dip if we invested pre-covid we simply apply that to the value to see our portfolio value after 22 years:

63230 * 4.3008 = $271,939.58

(12) for this we just subtract our initial 100K investment:

63230 * 4.3008 - 100000 = +$171,939.58

(13) We already know we have a -85.58% projected drawdown without QE so we get

100000 * -0.8558 = -$85,580

(14) subtract our loss from the last step from our initial investment to see the portfolio value at the bottom of drawdown:

100000 - 85580 = $14,420

(15) Since we know the baseline annualized growth under free market conditions is 7.02% on average, and we know the amount of money we would have on Β 100K investment at the minima of the drawdown it is trivial to calculate the amount we would have after 22 years.

Since we would hit the minima after 132 days (the length of time of the drawdown) then we subtract this from the number of days in 22 years (8030) and that is the amount of time we spend in the recovery phase.

8030 - 132 = 7898 days or 21.64 years

An annualized return of 7.02% reduced to a daily return is:

1.0702^(1/365) - 1 = 0.00018589549 or +0.0185% daily profit

Apply this across 7898 days:

1.00018589549^7898 - 1 = 3.34083241673 or +334.08%

Now apply this percentage to $14,420 and it will give us our total portfolio value after 22 years

14420 * (1.00018589549^7898) = $62,594.80

The P&L figure is just this value minus initial investment

62594.80 - 100000 = -$37,405.20 (a loss even after 22 years)

(16) As a percentage our 22 year portfolio value is:

62594.80 / 100000 = 0.6259 or 62.59%

Minus 100% to see that figure as a P&L figure: -37.4%

(17) Finally lets turn that into an annualized P&L as a percentage

0.6259^(1/22) - 1 = -0.02107Β or -2.1% annually

@General #Science #Economics #UsPol #Math #Maths #Mathematics #QOTOJournal #QE #QuantitativeEasing #Trump #COVID #COVID19