Wednesday, February 7, 2018

Stock Gyrations

Is this 1929, the beginning of the end? Or 2007? Is it 1974, annus horribilis in which the stock market drifted down 40% having something to do with stagflation, and did not recover until the 1980s? Is it 1987, a quick dip followed by recovery in a year? Or just an extended version of the flash crash, when the market went down 10% in a few hours, and bounced back by the end of the day? Are we in a "bubble" that's about to burst? How much does this have to do with the Fed? Of course I don't know the answer, but we can think through the logical possibilities.

(Note: This post has equations, graphs and quotes that tend to get mangled when it gets picked up. If it's mangled, come back to the original here.)

Why do prices fall?

Stock prices fall when there is bad news about future profits, or when the discount rate rises.

The discount rate is the rate investors require, looking forward, to get them to buy stocks. If people require a better rate of return, with no change in their expected cashflows, prices drop.

Stop and think about that a second, as it's counterintuitive. Yes, the only way to get a better return out of the same profits or dividends is for today's price to drop.

Another way to think about it: Suppose all of a sudden there are good profitable opportunities for your money -- bond interest rates rise, or it's a good time to take money out of markets and invest in your company. Well, people will try to move money to those alternatives. But the stock market is a hot potato; someone has to hold the stocks. So stock prices must decline until the rate of return going forward matches the other attractions on a risk-adjusted basis. Good news about returns going forward is bad news about a downward jump in stock prices.

Bad news about cashflows is, well, bad news. The dashed line shifts down. Your stocks are not going to pay off as well as before. Higher required returns are neutral, really, for long-term investors. The price drops today, but the higher returns mean the price will slowly recover, just as long-term government bonds do.

So is this a moment of bad cashflow news or higher discount rate?  Most commentary suggests it's not bad news about cashflows. The economy seems finally to be growing, and there isn't anything like a brewing subprime or other problem, as there was in 2007. Maybe we don't know about it, but one certainly doesn't read about it.

So let's think about discount rates. Why might investors require a higher return on stocks? Is it interest rates, a risk premium, and is the Fed behind it all?

Where are we? 

Are we  in a "bubble'' that is about to pop? Let's start by reviewing some facts. Here is the cumulative return on the NYSE since 1990. (This is the CRSP NYSE. Sadly the data stops 12/31/2017 so I don't show the recent drop. The larger index including NASDAQ shows a larger rise and fall in the tech boom and bust, but is otherwise about the same.)
Cumulative return NYSE since 1990. Source CRSP
This graph does not show anything terribly unusual about the recent period. Stocks drift up during expansions, and take a beating in recessions. There are also little blips like the Krugman election panic of November 2016. (Sorry, I couldn't resist.)  Why have stocks gone up so much? Well, mostly because the expansion has gone on so long. The recent period is also notable in that the little wiggles are much smaller -- less volatility. That ended last week too.

Update: Thanks to Torsten Slok at DB the last year follows. His point, it's sharp but not all that big.

Next, look at the price-dividend ratio. (For a variety of reasons this is a better valuation measure than the commonly used price earnings ratio. This (CRSP) measure of dividends includes all cash payment to shareholders. No, repurchases don't cause a problem.)

Price / dividend ratio, NYSE. Source CRSP
You can see prices were high relative to dividends in the booming 1960s; they really rose in the late 1990s before the big 2000 bust. Then you see the 2008 crisis and recovery, and more recent wiggles. You can see prices fall in recessions, even relative to dividends which also fall in recessions.

Where is the booming stock market? Stock prices relative to dividends have not grown at all since the end of the recession.  Well, evidently, dividends have been rising just as fast as prices in the current expansion -- which again weren't rising all that fast anyway. So the main reason stock prices are high is that dividends are high, and people expect that slow growth to continue.

So here we were before the recent drop. Are prices too high? Well, not as much as in 1999 for sure! But P/D is a lot higher than historical norms.  Is this the beginning of a drop back to historic levels like 30, or even 20? Or is this a new normal? There is way too much commentary lately that whatever we remember from 20 years ago was "normal" and that things have to go back to that. Not without a reason.

Interest rates and stock prices 

To think about this question we need some basics of what determines price-dividend ratios. Over long time periods, the return you get on stocks is the dividend yield -- how many dividends they pay per dollar invested plus the growth in dividends. Over short time periods you also get price appreciation per dividend, but over long time periods, the ratio of price to dividends comes back and price growth is the same as dividend growth.  In sum,

return = dividend yield + growth rate
\[ r = \frac{D}{P} + g \]This is also where price comes from. The price you're willing to pay depends on the expected return going forward, and expected  dividend growth (Prices are high relative to current dividends if people expect a lot more dividends in the future.)\[  \frac{D}{P}  = r - g = r^l + r^e - g \]\[  \frac{P}{D}  =\frac{1}{r - g} =\frac{1}{ r^l + r^e - g}  \] Here I broke apart the expected return into components. First, the expected return on stocks is equal to the long-term real risk-free rate \(r^l\) plus the risk premium \(r^e\). This is just a definition -- the risk premium is \(r^e=r-r^l\).

So, looked at either as D/P or P/D, we now have the tools to think about what pushes stock prices around.

(There is nothing inherently ``rational'' or ``efficient markets'' about this. Behavioral finance just says the expectations are wrong, for example that people think \(g\) is big when in fact \(r\) is small.)

Stock prices are very sensitive to real interest rates, risk premiums, and growth expectations. At our current P/D of 40, for example, this means \(r-g=1/40=0.025\) or 2.5%. Just half a percent change  in expected return or growth rate, \(r-g=0.02\) would mean \(P/D=1/0.02=50\), a 25% rise in stock prices. Conversely, a half percent rise in real interest rates would mean \(r-g=0.03\), a decline to \(P/D=33\) a 16.7% fall.  No wonder stocks are (usually) volatile!

Now, to what's going on? If we take growth rate expectations off the table, then stock prices are moving because of changes in interest rates. And small interest rate changes do indeed imply big stock valuation changes -- though, again, take heart because it means the rate of return is higher, as in the first picture.

Does this relationship hold historically? Here is the D/P ratio (P/D upside down) and a measure of long-term real interest rates.
NYSE D/P, Cleveland Fed 10 Year real rate, and 10 year TIPS

(The problem with 10 year real rates is knowing what 10 year expected inflation is, given that we did not have TIPS. There are lots of other problems too, such as unwinding the liquidity premium in government bonds. Here I used the Cleveland Fed's real rate model, which is in part based on survey expectations. I added the 10 year TIPS yield where we have it to confirm the general pattern of the Cleveland Fed's calculation.)

This is a remarkable graph: The entire rise in valuations from 1980 to 2008 corresponds exactly to the decline in real interest rates.

By this measure, the decline in real rates was huge, from 7% to essentially 0%. Plug that in to \(P/D=1/(r-g)\) and we're done. Stock prices are exactly where they should be.

In fact, by this measure, stock prices are too low. In 2008, real rates kept right on trundling down another two percentage points, but the dividend yield stabilized.

Well, I was careful to say "corresponds to" not "caused by" for a reason. The risk premium and growth expectations changed as well. Arguably the move to a low-growth economy starting in 2000, cutting one to two percentage points off \(g\), offset the decline in real rate. Or perhaps the risk premium isn't as low as we think it is. This isn't just waffling -- the relationship is basically an identity. One of those options must be true. If the dividend yield is 2.5%, and the real interest rate is 0%, then \(r^e-g\) is 2.5%, and has grown since 2000. Either the risk premium has grown 2.5% -- so much for the ``low risk premium'' -- or growth expectations have fallen 2.5%. Or the long-term real rate is profoundly mismeasured here.

More on all this in a minute. But the graph reminds us 1) Real rates have come down a lot, and 2) persistent changes in real rates really are an important part of stock market valuations. Oh, and they have nothing to do with ``risk appetite'' and all that other blather. Stocks are valued like bonds plus risk. We are noticing here that the bond-like component got much more valuable. That alone, not the risk component or the growth component, accounts for two decades of huge price rises.

This actually updates significantly some of my own work, and the asset pricing consensus. The great question why do price-dividend ratios vary so much occupied us a lot in the late 1980s and early 1990s, including myself, John Campbell, Bob Shiller, Gene Fama and Ken French. The rough answer we came to -- pretty much all variation in D/P or P/D comes down to variation in risk premiums, the \(r^e=r-r^l\) term. The underlying fact is that times of high P/D are not reliably followed by higher dividend growth (Shiller), and they are reliably followed by low  excess returns (Fama and French). If you add it up, the risk premium effect neatly accounts for all the variation in P/D (Campbell and Shiller, me).

Well, in the data up to 1990, we didn't see much persistent variation in real rates of interest, and what we did see was not correlated well with stock prices. Well, that was 1990, and now is now. This graph suggests that in fact a lot of the recent variation in P/D corresponds to lower real interest rates.  Also, it's the low frequency, decade to decade movement in P/D that is not well accounted for by any models. An academic version of this observation needs to be written.

Practical bottom line: The stories that the recent stock price decline comes from rising long-term real interest rates make sense. They might be wrong, but they make sense. That's saying a lot more than most of the other stories being bandied around right now.

Interest rates, growth,  stock prices, and the Fed. 

The story is not that easy however. We have to think about real interest rates \(r\) and growth \(g\) together. And there is this puzzle to answer -- how can it be that good news about the economy sends the market down? If \(P/D=1/(r-g)\) more \(g\) should raise \(P\), no? It should shift up the dashed line in my first graph?

No. We have to think about where real interest rates come from. One of the most basic relationships in economics is that higher growth means higher real interest rates.  If everyone will be richer in the future -- growth -- they need an incentive to save and not blow it all today. And growth means a higher marginal product of capital, and hence higher interest rates. As a simple equation, \[ \text{real rate} = \gamma g \]where \(\gamma\) is a parameter, usually between about 1/2 and 2, and get ready for a bar fight at the AEA convention over just what value to use. 1% higher growth means about a half percent to two percent higher real interest rate.

(There is a second term too, important in understanding things like the financial crisis. More uncertainty means lower interest rates. Not today.)

If \(\gamma=1\), if one percent growth means one percent higher real interest rates, then higher growth has no effect at all on stock prices or price dividend ratios. (\(D/P = r^l + r^e - g.\) Raise \(r^l\) and \(g\) by the same amount.) If, as I think is more likely right in this case, \(\gamma>1\), then higher growth lowers stock prices. Yes. Higher growth means a higher discount rate as well as more dividends. The discount rate effect can overwhelm the cashflow effect.

This has nothing to do with the Fed. There is a natural human tendency to look for Agency, for some man or woman behind the curtain pulling all the strings, and these days that means the Fed. For example, the WSJ Editorial on stocks:
"The paradox of the equity-market correction is that it’s taking place even as the real economy looks stronger than it’s been since at least 2005 and maybe 1999. "
"So why are stocks falling amid all the good news? The best answer we’ve heard is that stocks are reflecting a return to volatility and risk after years of the Fed’s financial repression. With its quantitative easing bond purchases, the Fed has for a decade suppressed market price signals in bonds."
"Investors may finally be figuring out that the global quantitative-easing monetary party is ending."
Look back at my graph. Real interest rates have been on a slow downward trend since 1980. That trend is unbroken since 2008. There is not a whiff that QE or anything else has budged that trend. (Lots of good graphs on this point in 8 heresies of monetary policy here. ) If the Fed has anything to do with it, it is the slow victory over inflation expectations, not QE and a lot of talk.

Yes, the papers like to say that higher growth will induce the Fed to raise rates. The Fed can put a finger in this dike for a bit if they want to, but even the Fed cannot long fight the positive or negative relationship between real growth and real interest rates.

So it makes perfect sense, at least as a logical possibility, that more growth lowers stock prices! Again, this is like my lower line in the first picture -- and actually a bit better because we also raise the terminal point. If this is what happened, well, regret that you didn't see it happening and stay out during the dip, but be reassured the market will make it back.

Risk premiums 

What about the ``unusually low risk premium''? Aren't the Fed's ``massive QE and abnormally low interest rates distorting risk premiums and causing asset price bubbles?'' (The best definition of ``bubble'' I can muster is a risk premium that is too low, distorted somehow.)

Here is the contrary view. We are at the late summer of the business cycle. The economy is relatively healthy, at least if you're a stock market investor. (Many of these own companies.) Economic volatility is still at an all time low. Bonds are still giving pretty atrocious real returns. Yeah, stocks look pretty healthily priced -- as you contemplate your \(P/D = 1/(r^l + r^e - g)\) it looks like the extra return from stocks \(r^e\) is pretty low. But what else are you going to do with the money? You can afford a little risk. Contrariwise, the same investors in the bottom of the great recession, with very low \( P/D\) signaling a high risk premium \(r^e\), said to themselves or their brokers, yes, this is a buying opportunity, stocks will likely bounce back. But my business is in danger of closing, my house might get foreclosed, I just can't take any risk right now.

In short, it is perfectly rational for investors to be more risk-averse, and demand a higher risk premium \(r^e\) in the bottom of recessions, and to hold stocks despite a low risk premium in quiet good times like right now. And this has nothing to do with the Fed, QE, or anything else.

John Campbell and I wrote a simple model of this phenomenon a long time ago, and I've reviewed it several times since, most recently here. Sorry for flogging the same ideas, but this possibility still hasn't made it to, say, the Fed-obsessed WSJ editorial pages, to say nothing of the Trump-obsessed pages at other outlets.

John and I tied risk aversion to consumption trends. If consumption is high relative to the recent past, in good times, you more willing to hold risk. If consumption is declining relative to the recent past, you get more scared. Lots of other mechanisms, including debt, work much the same way. If you don't like the precise model, consumption relative to recent past is a good general business cycle indicator.

Let's look historically. Here is consumption less a moving average (I used \(x_t = \sum_{j=0}^\infty 0.9^j c_{t-j})\), plotted with the log of the price/dividend ratio. The two series have different scales. The point is to see the correlation.

Consumption minus a moving average, and log P/D on NYSE. 

The pattern is longstanding. In good times, when consumption rises relative to recent past, stock valuations go up. In bad times, such as the great recession, consumption falls and so do stock valuations. People are scared. The same pattern happens regularly in the past.

The two lines drift apart, but as we saw above real interest rates account for that. Then the business-cycle related risk premium here accounts for the rises and dips.

And, if I may belabor the point, there was no QE, zero interest rates, and so forth going on in all these past instances when we see exactly the same pattern. Higher real interest rates are a regular, simple, utterly normal part of expansions, and lower risk premiums are a regular, simple, utterly normal part of expansions. 

I was interested to read Tyler Cowen at Bloomberg back in to this view, based entirely on intuition:
In a volatile and uncertain time politically, we have observed sky-high prices for blue-chip U.S. equities. Other asset prices also seem to be remarkably high: home values and rentals in many of the world’s top-tier cities, negative real rates and sometimes negative nominal rates on the safest government securities, and the formerly skyrocketing and still quite high price of Bitcoin and other crypto-assets.
Might all of those somewhat unusual asset prices be part of a common pattern? Consider that over the past few decades there has been a remarkable increase of wealth in the world, most of all in the emerging economies. Say you hold enough wealth to invest: What are your options?
In relative terms, the high-quality, highly liquid blue-chip assets will become expensive. So we end up with especially high price-to-earnings ratios and consistently negative real yields on safe government securities. Those price patterns don’t have to be bubbles. If this state of affairs persists, with a shortage of safe investment opportunities, those prices can stay high for a long time. They may go up further yet.
These high asset prices do reflect a reality of wealth creation. They are broadly bullish at the global scale, but they don’t have to demonstrate much if any good news about those assets per se. Rather there is an imbalance between world wealth and safe ways of transferring that wealth into the future
To sum this all up in a single nerdy finance sentence, in a world where wealth creation has outraced the evolution of good institutions, the risk premium may be more important than you think.
Except for this business about "shortage of safe assets," that's pretty much the intuition. (Tyler: all assets are in fixed supply in the short run. Prices adjust. This isn't really a ``shortage.'') The point that high valuations extend to homes, bonds, bitcoins, and global stocks is a good indicator that the phenomenon is generalized risk aversion rather than something specific to one market or economy. 

This view should not necessarily make you sleep at night however.  It means that a downturn will be accompanied by higher risk aversion again, and not only will dividends fall, prices will fall further. Moreover, historically, asset price falls have been preceded by periods of higher volatility. Alas, many periods of higher volatility have just faded away, so it's a warning sign not a signal.  Sure, this mechanism means they will bounce back, but if you are clairvoyant enough to see it coming it will be better to avoid the fall! If not, well, be read to buy when everyone else is scared -- if you are one of the lucky few who can afford not to be scared.

The VIX, volatility, technical factors 

There is another kind of ``discount factor variation,'' including 1987 and the flash crash. Sometimes the machinery of markets gets in the way, and prices fall more than they should. They quickly bounce back. If you can buy at the bottom you can make a fortune, but the prices fell precisely because it's hard to buy.

There were scattered report on Monday of hours long delays for retail customers to trade. (Can't find link.) But I do not get a sense this was a big clog in the markets. I would be curious to hear from people closer to markets.

The bigger news is the return of volatility -- big daily changes. To put this in historical perspective, here are two plots

The surprise, really, is just how low low volatility had become. Historically the stock market index has had a volatility around 15-20% per year -- a typical year saw a 15-20% change, and a typical day saw a \(15-20 / \sqrt{250} \approx \) one percent change. But, as you see in the top graph, volatility also declines in the late summer of the business cycle. Volatility has many occasional little eruptions, typically around price drops, and then washes away. Except when volatility rises in advance of the next recession and market decline. Which is this? I wish I knew.

Volatility is not about "fear" nor is it about "uncertainty." Volatility occurs when options change quickly. Constant bad news or good news just leads to constantly low or high prices. This is a sign of a time when either a lot of real information is hitting the market, or a lot of people are trying to process what's going on ahead of everyone else.

The "VIX bust" is hot in the news. A lot of people bet that the graph you saw above would not rise. To be ``short volatility'' means basically that you write insurance to people who worry about markets going down, (volatility is a big part of the value of put options) and you write insurance to people who are worried about events like right now in which markets start to move a lot. Hello, when you write insurance, occasionally you have to pay up.

As the graphs make clear, writing volatility insurance, or betting that volatility will continue to go down,  is like writing earthquake insurance. Not much happens for many years in a row, and you can post nice profits. Then it jumps and you lose big time. Anyone who did this based just on historical returns is now crying the tears of the greedy neophyte. But they have lots of company. Back in the 1990s, Long Term Capital Management went under, basically for betting that similar looking graphs would continue to go down.

Well, if after all these years people are at it, P.T. Barnum had a good word for them. But did this have something to do with the stock market crash? How Bets Against Volatility Fed the Stock Market Rout in WSJ is an example of this train of thought.

On first glance, sure, a lot of people lost a ton of money, and then sold out other risky positions. But Econ 101: for every buyer there is a seller.  Derivatives contracts are pure cases of this fact -- the net supply really is zero, for everybody who lost a dollar shorting VIX somebody else made a dollar buying it.

To get a story like this to go you need all sorts of market discombobulations. Somehow the people who lost money must be more important to markets than the people who made money. This can happen -- if a bunch of traders in a complex obscure security all lose money, and all try to sell, there is nobody to buy. But I don't really see that case here, and stocks are not a complex obscure security.

A trader friend also tells me that he has seen lots of people stop hedging -- so sure low volatility would continue that they don't cover the downside. He said many have lost a ton, and now are frantically selling to cover their positions. Such price pressure can have short run impacts, but does not last long.

Inflation and real interest rates 

So we're back at hints of higher long-term real interest rates as the main likely culprit behind this week's decline and gyrations.

Here too most of the stories don't make much sense. Inflation per se should not make much difference. If expected inflation rises, interest rates rise, but real interest rates are unaffected. Inflation may make the Fed act more quickly, but there is not much correlation between what the Fed does with short term rates and the behavior of 10 year or more rates that matter to the stock market -- or to corporate investment.

Yes, there is some correlation -- especially at the end of expansions, short and long rates rise together. But the correlation is a whole lot less than the usual Wizard of Oz behind the curtain stories. And even the Fed cannot move real rates for very long. There is a good chicken-and-egg question whether the Fed can hold short rates down for long when long rates want to rise. The Fed pretty much has to jump in front of the parade and pretend to lead it.

Inflation does seem finally to be rising. The fact that higher rates are associated with the dollar falling suggests that a lot of the higher rates are due to inflation, and TIPs have not moved (top graph.)

So, the question before us is, are long-term real rates finally rising -- back to something like the historical norm that held for centuries, and if so why?

The good story is that we are entering a period of higher growth. Depending on your partisan tastes, point to tax cuts and deregulation, or state that Obama medicine is finally kicking in. This would raise real growth, with \(\gamma>1\) lead to a small stock price decline, but higher stock returns and bond returns going forward.

There is a bad story too. Having passed a tax cut that left untouched will lead to trillion-dollar deficits, Congressional leaders just agreed to $300 billion more spending. The Ryan plan that tax cuts would be followed by entitlement reform may be evaporating. Publicly held debt is $20 trillion. At some point bond markets say no, and real rates go up because the risk premium goes up. The US is in danger that higher interest rates mean higher interest costs on that debt, which means higher deficits, which means higher interest rates. $20 trillion times 5% interest = $1 trillion in interest costs.

The former leads to some inflation if you believe in the Phillips curve. The latter leads to stagflation in a tight fiscal moment.

Which is it? I don't know, I'm an academic not a trader.

One consolation of the stock market decline: I hope we don't have to hear how all the corporate tax cut did was to boost the stock market!

Well, two days ago this was going to be a short post responding to the WSJ's view that the Fed is behind it all, and Tyler's nice intuition. It got a bit out of hand, but I hope it's still interesting.


Data Update (Geeks only).

P/D isn't really "better" than P/E or other measures. A measure is what it is, you have to specify a question before there is an answer. Ideally, we want a measure that isolates expected returns, and tells us if prices are higher or lower given the level of expected dividends.  So ideally, we would account for expected future dividends and the result would be a pure measure of expected returns (rational or not). P/D works pretty well that way because dividends are not very forecastable. Price divided by this year's dividends turns out to be a decent approximation to price divided by anyone's forecast of future dividends. But not perfect. P/E is less good that way because earnings bat about a bit more than dividends. For individual companies you can't use P/D, because so many of them do not pay dividends. Following Fama and French, the ratio of market value to book value is better there, because book value is usually positive, or not so frequently zero.

I use the CRSP definitions. I start from the CRSP return with and without dividends and infer the dividend yield. ''Dividends" here includes not only cash dividends but all cash payments to shareholders. So, if your small company gets bought by Google, and the shareholders get cash, that is a "dividend" payout. I suspect this accounts for the difference noted by WC Varones below. As others point out, earnings has all sorts of measurement issues, and also does not control for leverage.

Dividends are very seasonal, so you can't divide price by this month's dividends or you get a lot of noise. I use the last year's worth of dividends, brought forward by reinvesting them. This introduces some "return" into the dividend series. If you just sum dividends, though, identities like \(R_{t+1} = (P_{t+1}+D_{t+1})/P_t \) no longer hold in your annual data.

x = load('crsp_nyse_new_2018.txt');

caldt = x(:,1);
totval = x(:,2);
usdval  = x(:,3);
sprtrn = x(:,4);
spindx = x(:,5);
vwretd = x(:,6);
vwretx = x(:,7);

[yr,mo,day,crsp_date_number] = decode_date(caldt);

T = size(vwretd,1);
vwretda = (1+vwretd(1:T-11)).*(1+vwretd(2:T-10)).*(1+vwretd(3:T-9)).*...

vwretxa = (1+vwretx(1:T-11)).*(1+vwretx(2:T-10)).*(1+vwretx(3:T-9)).*...
vwdp = vwretda./vwretxa-1; %D_t+1/P_t+1 = [(P_t+1+D_t+1)/P_t] / [P_t+1/P_t] -1;
vwdda = vwdp(13:end)./vwdp(1:end-12).*vwretxa(13:end);   % D_t+1/D_t = D_t+1/P_t+1 / D_t/P_t * P_t+1/P_t
vwdda = [ones(23,1)*NaN; vwdda];
cumval = cumprod(1+vwretd);
vwdp = [ ones(11,1)*NaN; vwdp]; % keep length of series the same

I get stock data from CRSP via WRDS. This is the NYSE only. I can't post the full data, as it belongs to CRSP. Here is an excerpt that will let you calculate the last year, and check that things are right if you download the whole thing.

%   caldt            totval                  usdval                 sprtrn         spindx        vwretd        vwretx
19260130       27624240.80       27412916.20      0.022472       12.74      0.000561     -0.001395
19260227       26752064.10       27600952.10     -0.043956       12.18     -0.033046     -0.036587
19260331       25083173.40       26683758.10     -0.059113       11.46     -0.064002     -0.070021
19260430       25886743.80       24899755.60      0.022688       11.72      0.037019      0.034031
20160129    17059005700.00    17976992500.00     -0.050735     1940.24     -0.050111     -0.051700
20160229    16986848800.00    17001893900.00     -0.004128     1932.23      0.005104      0.002251
20160331    18122913200.00    16951468600.00      0.065991     2059.74      0.072190      0.069562
20160429    18503144900.00    18082712100.00      0.002699     2065.30      0.023324      0.021716
20160531    18479138100.00    18410229900.00      0.015329     2096.96      0.006124      0.003392
20160630    18613173100.00    18422135300.00      0.000906     2098.86      0.011175      0.008957
20160729    19054705700.00    18557630600.00      0.035610     2173.60      0.028433      0.026872
20160831    18993464300.00    19049575300.00     -0.001219     2170.95      0.000196     -0.002532
20160930    18829544800.00    18880924600.00     -0.001234     2168.27     -0.003876     -0.005878
20161031    18404742600.00    18802632900.00     -0.019426     2126.15     -0.021331     -0.022944
20161130    19220882900.00    18383296300.00      0.034174     2198.81      0.048548      0.045701
20161230    19568491300.00    19178151000.00      0.018201     2238.83      0.021566      0.019486
20170131    19824534000.00    19526674900.00      0.017884     2278.87      0.014007      0.012623
20170228    20355248600.00    19781803200.00      0.037198     2363.64      0.031422      0.028905
20170331    20237616500.00    20334429600.00     -0.000389     2362.72     -0.003961     -0.006103
20170428    20286715000.00    20194157100.00      0.009091     2384.20      0.003950      0.002468
20170531    20299003900.00    20276905500.00      0.011576     2411.80      0.002199     -0.000507
20170630    20602218600.00    20256933000.00      0.004814     2423.41      0.018204      0.016235
20170731    20747539100.00    20488018000.00      0.019349     2470.30      0.015290      0.013394
20170831    20593088100.00    20742392900.00      0.000546     2471.65     -0.005133     -0.007874
20170929    21147810200.00    20381001300.00      0.019303     2519.36      0.030435      0.028662
20171031    21343546700.00    21130998500.00      0.022188     2575.26      0.011831      0.010360
20171130    21904734200.00    21302790800.00      0.028083     2647.58      0.030537      0.027670
20171229    22016063100.00    21683038400.00      0.009832     2673.61      0.015914      0.014117

If I screwed up, let me know and I'll fix it!


  1. "Next, look at the price-dividend ratio. (For a variety of reasons this is a better valuation measure than the commonly used price earnings ratio. This (CRSP) measure of dividends includes all cash payment to shareholders. No, repurchases don't cause a problem.)"

    Hi John, excellent blog as usual. Given the pervasive use of the P/E ratio in the industry, an elaboration of this point would be extremely valuable and informative. Thanks very much.

    1. Earnings are GAAP accounting measures which are massaged unitl they are soft pizza dough. Dividends and stock repurchases are money out the door.

  2. I just have to say that this just may be the best article from you and one of the best economic articles I've ever read.

    1. It is an interesting article. But really, the cause of Volatility was Goldma Sachs. Politics can destroy that which seems foolproof. They set out to bludgeon the New Normal when the Goldman Men came on board. That was their plan. Will it work without creating systemic risk and economic disaster? If not they will all be hung, or wish they had been hung.

    2. Professor Cochrane: This is a great article. Thanks for posting. You make financial economics look easy. I wish!

  3. Consumption minus a moving average, and log P/D on NYSE.

    I get this graph, when consumers queue up at my store, then my rate of surplus growth increases. Now, when this relation applies to an aggregate of firms, the the firms are acting like one firm, more likely they are acting like one value added chain. What is the third common variable? Basically distributing the amount of transaction space left over after the government books are balanced. What does the consumer have left after federal taxes. The public corporation participates in keeping government balanced and government returns the favor with zombie insurance.

  4. How are you getting a P/D of 40 (div yield of 2.5%)?

    Div yield on S&P 500 is 1.8%, and it's even lower for smaller stocks.

    1. Thanks.

      Initial reaction is that it doesn't make sense to include cash acquisitions as dividends, because they are one-time, and a return of principal / zeroing out of equity, while the whole point of a DDM is about recurring and growing dividends.

      Does it make theoretical sense to you?

    2. No. This is the return on the market portfolio. So, you pay a price, and you receive cashflows. We want to get all the cashflows that you receive. If a company in the portfolio gets bought and investors get cash, this is exactly like a dividend. Delisting returns are also important. CRSP data is really good because it researches these issues. Lots of strategies look good until you realize they leave out these effects.

    3. Thanks.

      So then you have to adjust "g" by the expected average amount of acquisition/delisting.

      Example: Qualcomm makes up a large portion of my dividend portfolio. It's likely to be acquired for mostly cash, which will cause a huge "dividend" this year but a large negative "g" going forward.

      Of course investors can price this in implicitly in your model, but it's important to note that this "g" has a big deviation from any growth rate derived from ROE or GDP growth.

    4. Maybe it can be explained, at least in part, by the choice of the NYSE as representative for the stock market.

      A rough calculation on the S&P 500 stocks shows that two thirds (by market cap) are listed on the NYSE and the (weighted) average yield is 2.2% while one third is listed on the NASDAQ Stock Market with an average yield of just 1.1%.

  5. Market cap has nothing to do with asset value since the 70s or so.

    There was no bombing or earthquake that destroyed two trillion in factories or housing or retail assets since Feb 1.

    Instead a few thousand people decided to not buy shares, reducing buying by maybe a billion, but maybe as little as a hundred million. That forced speculators to cover shorts of maybe a billion. The marginal price reductions for a small number of shares does not change the share value.

    Eg, stores running sales on toilet paper don't result in a loss of billions by households on the TP they have on hand.

  6. How is consumption measured? Is it "Personal Consumption Expenditures" more than it is GDP?

    1. It is personal consumption expenditures, per capita.

  7. Great post! I am imagining a semester long undergraduate course on asset pricing built around this essay.

    1. Phillip, I purchased Cochrane's text, years ago. An excellent monograph on asset pricing. It's dense with math, but well worth the journey. I refer to it often, even when I managed a hedge fund!

  8. Awesome write up!

  9. Asset prices are expected cash flows discounted at the risk free rate, plus a risk premium that depends on the covariance of a payoff with the discount factor. If covariance is negative, assets will have lower prices and higher returns. P(t) =[E(X(t+1)/R(f)] + COV [(M(t+1),X(t+1)]. the part after the plus sign is the risk premium. Implied in this equation, the volatility of an assets cash-flow is irrelevant. Asset returns covary negatively with the discount rate. This reasoning suggests this market selloff resulted inversely to an increase in... (anticipated inflation) rates. We know bonds adjust inversely and instantaneously to changes in interest. January S&P 500 return was 2.5%, 30% annualized. Huge risk premium suggested buying puts with VIX at 10.

  10. Up top you "take growth rate expectations off the table", but rising rates not only increase the discount rate, they also threaten growth by constraining companies' ability to finance their operations. So rising rates have a double whammy impact to both r and g.

  11. So if it's interest rates, how do you explain how the 5-year TIPs yield was either down or flat on the recent days when the market has fallen more than -3%.

    1. Good point. I looked this morning and TIPS seemed to be up about half as much as 10 year treasuries. But still, my story was about real rates rising. If it's just inflation, even if the story is inflation makes the Fed tighten, then we should see long-term real rates rise. Maybe it is the beginning of the end after all, and markets know something about a looming economic problem that we don't know. Will be fun to watch.

    2. Is this an arbitrage opportunity? We'll see if bond traders are all over the spread.

    3. How do you explain the Nikkei falling so much?

  12. "So ideally, we would account for expected future dividends and the result would be a pure measure of expected returns (rational or not). "

    Why don't you use dividend futures?

    1. The price of a dividend future is not its expected value. There is a risk premium everywhere.


    2. John, the model you are using is the "Gordon Growth Model" formulation for the price of equities. The model is named for Myron J. Gordon (U. of Toronto) who, along with E. Shapiro, presented it in a 1956 publication. The variable D in the equation is next year's dividend (i.e., in period k=1, if the current period is k=0). The variable g is the expected compound annual growth rate in the dividends, assumed constant for all k=1,...,infinity. The variable r is the discount rate, in nominal terms, because the dividend D is in nominal dollars. The discount rate, r, is assumed constant for k=1,...,infinity. These assumptions limit the application of the model. A further restriction applies: i.e., r > g. If r and g are determined by independent processes, then it is not certain that r- g >0 for all time t. One can fudge it, by stating that r is the sum of two processes, i.e., the real return and a ‘risk premia’ (without specifying either one, or both, noting that the 'risk premia' may include more that one factor, i.e., it may include the expected inflation rate).

      In your construction of the chart of P/D vs time, P=P(t), where t equals the current date. But D is a time-lagged average, exponentially-smoothed, of past dividend payments, i.e., D= D(t)= a*DD(t)+ (1-a)*D(t-1), where DD(t) is the current period dividend payment, not the future dividend payment. The Gordon Growth Model requires that D= DD(t+1), i.e., a future value. Because DD(t+1) is given as a single value, it is the expected value of DD(t+1) which is a stochastic variable, i.e., uncertain. Consequently, the inference that P/D from your charts is equal to 1/(r- g) does not follow. At best it is a loose approximation suitable for academic class room discussion, but it is not appropriate for valuation purposes. Which is to say, one cannot prove that the market index value P is determined by the quotient of D and (r- g).

      If r and g vary with time t, then the simple Gordon Growth Model no longer applies, i.e., P≠ D/(r-g). In that case, one must move to another model, e.g., CAPM, or any one of a number of asset pricing models for which the economic theory is on a sounder footing than the Gordon Growth Model.

  13. Great article. I have been commenting in wsj articles "the stock market is the discounted present value of future earnings at the discount rate" (learned at UCLA's no nonsense Finance/Economics department a while back) for the last two years because most of the wsj authors don't seem to know that. The wsj guys/gals say "there is QE (and I don't think there has been QE in the U.S. for the last few years) therefore stock prices are high". So my bunny has a good nose.

  14. I think the average investor is not as forward-looking as you are so they think higher inflation causing higher nominal bond yields is enough of an excuse to sell. 10 year TIPS implied inflation went up the past few days.

  15. Lots of great blogging.

    So the Fed is selling trillions of bonds while the lobby-controlled DC is selling trillions of bonds. What could go wrong?

    I sense the income-tax collection system is broken, while spending is runaway. If DC cannot balance the budget after a 10-year economic expansion, when will they?

    The only hope is the Fed (central bank) conducts QE and that it is not harmful. That seems to be the case in Japan.

    Otherwise, learn the speak Greek.

    1. The real question, Benjamin, is whether the Fed will be serious about trying to protect the New Normal. Higher rates help some things. But they don't help collateral bonds which are in widespread use in clearing houses near you. Or near the UK. :)

  16. "On first glance, sure, a lot of people lost a ton of money, and then sold out other risky positions. But Econ 101: for every buyer there is a seller. Derivatives contracts are pure cases of this fact -- the net supply really is zero, for everybody who lost a dollar shorting VIX somebody else made a dollar buying it."

    I don't think this is quite right. These markets often contain market makers who once they buy/sell immediately hedge their risk in other products. So say person A trades with person B, a market maker.

    A more concrete example would be S&P futures - whilst it is true that in the futures themselves the next supply is zero, typically one side is hedged in the cash. (The same is being done with the VIX, but it's trickier to fully understand the hedging vehicles).

    I appreciate your response will be 'well the people who've sold/bought the hedge to the market makers will be make the money which person A has lost', but it's easy to see this might not be the case. If A goes long on the S&P future, B goes short the future, long cash. C has sold some of his stocks.

    After the sell-off:
    - A has lost money
    - B has neither made, nor lost money
    - C has a paper gain on the trade, but realistically he wont have been net short to really profit.

    Derivative contracts are closely tied to 'real' products with non-zero net supply. If it's possible for everyone to lose money on a 'real' product, then it's equally possible for everyone to lose money on a derivative.

  17. Correction to my post. I stated, S&P returns for January 2018 were 2.5%, 30% annualized. Those were my hedged returns. The S&P gained 4.7% for January, 56.4% annualized. Greater risk premium. VIX at 10, buy puts.

  18. Here's another data point. A risky bond is also a bond plus a risk premium. The chart below shows the rate on a 10-year Treasury vs the BofA Merrill Lynch BBB effective yield. If anything, the 10 year is rising faster.

    1. I should have used this link:

  19. This is my first "Grumpy Economist" article - how did I miss it all these years? Only one quibble, you note:

    "There is another kind of discount factor variation,'' including 1987 and the flash crash. Sometimes the machinery of markets gets in the way, and prices fall more than they should. They quickly bounce back. If you can buy at the bottom you can make a fortune, but the prices fell precisely because it's hard to buy."

    I think it would be more fair to say prices fell because it was hard sell - that is there are no willing buyers, so one has to continually drop prices to entice buyers. I recall the pits at the CBOT and big down days with all the traders palms out indicating the desire to sell and no palms in, no buyers.

    I look forward to more articles!!

  20. great post. two comments:

    While the Fed may not be able to sustainably raise real rates, that can damage 'g' while trying to, yes?

    Also, re: options hedging - while its true that for every derivatives winner these is an equal and opposite loser, they tend to have different risk tolerances and relevant horizons. If the buyer of a put is hedging a portfolio of equities, he is likely happy and pretty much non-responsive in terms of short-term trading. The dealer who sold him the put, however, is "dynamically hedging" by selling equities into the hole in an attempt to keep up with his expanding (and increasingly off-sides) risk stemming from the short put position. This can continue until either the put is fully hedge (i.e. "100 delta") or until the put owner decides he wants to re-establish some outright long risk and starts buying stocks (or liquidates the put).

  21. Question regarding relation of real interest rates to economic growth. My understanding is that real interest rates in pre-industrial world of essentially zero growth (say antiquity & medieval Europe) were higher - all loans including to sovereign states were riskier, etc., and that they've fallen over the last 2.5 centuries. So, with safer assets, more reliable institutions such as U.S. treasury, and greater worldwide wealth we expect a long-term trend of falling real rates, everything else being equal. On the other hand, is the above positive association between real rates and growth. These two forces move in opposite directions, no? Any way to disentangle?

    1. P.S. You get at this, I think, when you express concern at the debt - & whether this will push rates higher on Treasuries due to a perception of higher risk - a lower trust in the institution. In which case, interest rates could rise due to a worsened perception of the political/economic system as well as from economic growth - correct? Is there any way to get at analyzing these opposed stories when observing interest rate changes?

  22. People look at the numerator (earnings/dividends) and growth thereof, but neglect the possibility of the cost of capital changing. This is why when you get higher growth expectations, real rates rise, it leads to a rise in the cost of capital, and stock prices often fall.

    Stocks are not GDP futures. Stocks often do best slightly after unemployment peaks, because the cost of capital is typically dropping in a weak economic environment.

    That said, the model for the stock market over at my blog fits the market better, and yes, the market is still overvalued.

    One last note: people were playing weird hedging games between short volatility ETFs and stocks. It is not that the trading produces losses, aside from commissions, it is that the expected cash flows from the derivative contracts collapsed as the market fell, the VIX rose, and then rose more as some hustled to sell before the rest realized that the intrinsic value of the short vol ETFs had fallen for more than the market price had. Then, after the close, the carnage was revealed, with one fund closing, and another w/the NAV down more than 80%. I warned people away from the short vol trade, but no one listens in a bull market.

  23. So John, why, in your opinion, did "real" interest rates go so low, compared to historical trends? Was it a global savings glut (Summers)? Was it all the central banks buying bonds with money that didn't previously exist? Does the Fed control the 30 year Treasury rate?

  24. Just to doublecheck, your dividend metric D includes all stock buybacks? (This is implied by your definition but not expressly stated).

    Thanks for the clarification.

    1. No. A stock buyback is not the same as a dividend. Conceptually, D includes all the things it should so that price = present value of dividends continues to hold. For that, if the company is bought by Google, you need that cash payment. If the company buys back some of its own shares, you don't have to tender, so you can ignore it. The CRSP documentation online says just what they do and don't include, especially how they treat stocks that get delisted. That's important. If you just drop the stock, then you assume people sell on the last day, which requires clairvoyance. Delisting returns make many fun strategies fall apart.

  25. Thank you for an excellent post. It's rare to see such complex issues explained so succinctly. My one problem with your analysis is that it does not explain the speed in which these price changes are happening. If this is about discount rates adjusting you would expect the changes to be continuous and slow, rather then erratic and fast. While this could be a shock to liquidity premium the latest TED spread report from the SL Fed was still lower then one year ago.

  26. I think you are missing the point that the long trend of disinflation in the US has allowed the Fed to act as a shock absorber, reducing economic volatility and therefore decreasing the equity risk premium. In a world where inflation is above the fed's target, they could no longer perform this function and we should see lower equity valuations as a result of higher risk premia. Also I think its important to note that the g in your equation is the growth rate of dividends, not economic growth. While the two are related, the long uptrend in corporate profits as a % of GDP has likely meant g>rl which has led to higher valuations.

  27. "The economy seems finally to be growing,... "

    I would love an explanation of this. Didn't the economy grow for the last 6 years of the Obama Presidency?

    -- Jonathan Goodman

    1. That seems to be an excellent question. If we are going to have a fair minded discussion about macro policy surely we should recognize that there was recovery. We should probably also recognize that the sequester, now recently abandoned, may have constrained the recovery.

  28. I think you are saying the real rate = gamma times growth and gamma > 1 but is gamma being > 1 an empirical observation or is there a rational explanation or both?

    As for the "good" reason for a higher real interest rate: Most of the value of a company is assumed to depend upon cash flows that occur beyond the term of the current administration that gave us some deregulation and tax cuts. It seems likely that was already priced in given that there was the expectation that in the next 100 years sometimes there would be administrations of that bent.

    As for the "bad" reason for a higher real interest rate: The tax cut passed in December so it should have been priced in. I don't know about the $300B you referred to but people expected big spending news (infrastructure) eventually so I wonder if there really was any late-January or early-February budget/debt news that could be considered to be "new news" caable of changing expectations?

    The news I think mattered: the fact that equities went up a lot in January and that inflation expectations went up also in the context of a weakening immigration outlook.

  29. > I start from the CRSP return with and without dividends and infer the dividend yield. ''Dividends" here includes not only cash dividends but all cash payments to shareholders. So, if your small company gets bought by Google, and the shareholders get cash, that is a "dividend" payout.

    Is it? I'm not a CRSP user but looking at the definitions of "Returns" and "Returns Without Dividends" available at I understand that "dividends" doesn't even include extra-ordinary dividends.

    If the stock price is $100, a $1 dividend is distributed and then the company is acquired by Google at $200 per share I would expect the "returns" to be 101% and the "returns without dividends" to be 100%.


  30. Prof. Cochrane,
    I have the following questions:
    Question 1
    How are the ideas presented in this blog post different from what we already know through a basic application of the “Fed Mode” that compares the stock market's earnings yield (E/P) to the yield on long-term government bonds?
    Question 2
    Is the Dividend Discount Model a suitable tool to value the aggregate equity market? Would a simple and common measure like P/E be more suitable since it has tended to mean revert in the long term?
    To be clear, DDM is certainly suitable for single names. Also DDM can never be disproved since Projected Cash Flows and the discount rate are both unobservable at the time of estimation and price is the only truth.
    Why applying DDM in the current context feels like trying to fit a square peg in a round hole.
    DDM relies on the following for valuation: Projected Cash Flows, Risk Free Rate, Equity Risk Premium and Growth.
    3 out of the above 4 factors are assumptions and the 4th one has been controlled through central bank intervention recently. The control exercised by central banks is evident in the historic magnitude of central bank balance sheet growth and in the outsized growth in total USD denominated financial assets relative to GDP(discussed later). In my view, the price control exercised by the major central banks has effectively changed the numeraire for financial asset. From the perspective of financial assets, the effect of central bank actions is akin to a National Bank declaring a “currency split”, i.e. declaring that all holder of the currency will receive a 2 units (or a multiple) of the currency for every unit of the same currency. If consider such as scenario, in a closed system, all assets, prices and wages would theoretically double and the announcement would have no economic impact what so ever.
    The difference is that the dollar system is not a closed system and the change of numeraire has had an uneven effect on wages, prices and financial asset, favoring financial assets disproportionately. This may be attributable to many factors including:
    - The reserve currency status of the dollar that allows US importers to pay their bills to foreign suppliers in the USD.
    - The inability of the labor markets in US and Europe to negotiate a proportional wage.
    - The reluctance and inability of the major exporters to the US such as China to transmit the benefits to their factors of production enabled through structural and political factors (such as oligopolies of SOEs).
    In the above outlined context one could argue that deriving a risk premium out of prevailing prices which are observable (given a certain assumption for growth and cash flows) is equivalent to deriving prices from assuming a risk premium that is unobservable.

    A two factor model independent of discount rate assumptions
    In the current context where the price of money, a key macro variable that is an input to DDM and the only one that is observable, is controlled, the equity valuation model can be reduced to 2 factors: total financial assets and the allocation preference (bonds versus equity). This ratio has remarkably tended to mean revert to an even greater extent than P/E ratios. The following table is illustrative:

    Financial Assets Corporate Equities Ratio (CE/FA) GDP Adjusted GDP
    2017 220345 38588 17.51% 19000 17000
    2006 132001 20909 15.84% 14000 14000
    Growth 166.93% 184.55% 135.71% 121.43%

  31. continued...

    When equity valuation is viewed through the lens of these 2 macro factors, one can quickly make the following useful observations:

    1. The Ratio of Market Value of Corporate Equities to Total USD denominated Financial Assets has not moved much between now and the last equity market high before the 2008 crash.

    2. The total USD denominated Financial Assets have grown substantially faster than GDP between 2006 and 2017.

    3. Purely from the demand / supply perspective, the total supply of outstanding listed shares on NYSE has remained relatively flat during this period. Even if one were to assume a constant allocation between equity and debt, the math would dictate a proportionally higher share price. The market value of total debt has, of course, increased due to new issuance as well as capital appreciation due to much lower interest rates despite the (minimal) write-offs.

    Further, the following anomalies stand out:

    A large part of the GDP increase may be described as financially engineered, which is especially important if it turns out that the economy cannot face higher interest rates due to a high debt base and hence debt servicing costs. To be precise, the disproportionate increase in rental income, resulting from lower mortgage servicing costs is an extremely illustrative example:

    Based on other factors such as these, I have approximated an Adjusted GDP figure to be $2T lower. This takes out the fed induced low interest rate effects on GDP.

    Assuming that one of the 2 factors namely the allocation preference (bonds versus equity) is mean reverting and varies within a narrow band, the task to ascertain the aggregate value of US equity market distills down to the other remaining factor: total value of USD denominated financial assets. The following important question arises:

    Is the current growth in USD financial assets sustainable?
    If US cannot run a budget surplus after approx. 9 years into an economic expansion, does it mean that US government debt is on a perpetual growth path? Historically, the total credit in the US economy (i.e. Total Financial Assets) has never shrunk without severe economic downturns. Within the classic Keynesian model, the US government has used its credit at times when the creditworthiness of the private borrowers has been impaired. Two scenarios may be envisioned:
    Scenario 1: The credit worthy borrowers (government and private) of the reserve currency (USD) have infinite capacity to borrow at low interest rates (independent of activity i.e. GDP)
    The equity valuation model can then certainly be reduced to 2 factors: total financial assets and the allocation preference (bonds versus equity). The ratio of Market Value of Corporate Equities to Total USD denominated Financial Assets has mean reverted remarkably well in the medium term well across regimes.

    Scenario 2: US treasury funding rates rise, i.e. Fed loses control of long term nominal interest rates
    The following trifecta of factors that can cause this:
    - Treasury’s continued need to fund record deficits
    - Shrinking US trade deficit, driven by export oriented policies, i.e. less money in the hands of foreign central banks the traditional buyers of US Treasuries other than the Fed.
    - Fed has flipped to the supply side of govt. bonds
    In such a scenario, while a DDM approach will have fairly limited predictive power due to the difficulty in forecasting risk premiums. The aggregate MV of Equities will ultimately be determined by the amount of write-off in the bond markets which is where, arguably, the real bubble resides.

  32. continued...

    Following almost a decade of unconventional monetary policies by the major central banks, we may be entering a regime where macro risk are paramount in the determination of aggregate equity market valuation in major international equity markets. I have made a preliminary case that in such a regime, the equity valuation model can be reduced to 2 factors: total financial assets and the allocation preference (bonds versus equity). This ratio has remarkably tended to mean revert to an even greater extent than P/E ratios.

  33. Please find a link to a graph that I could not includ ein the post:

    A large part of the GDP increase may be described as financially engineered, which is especially important if it turns out that the economy cannot face higher interest rates due to a high debt base and hence debt servicing costs. To be precise, the disproportionate increase in rental income, resulting from lower mortgage servicing costs is an extremely illustrative example:


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