Why borrow? Rational Expectations Redux and why nominal prices matter

Unrealized capital gains were a particularly important factor in the increase in net worth over the 2004–07 period. The share of total assets attributable to unrealized capital gains from real estate, businesses, stocks, or mutual funds rose 5.1 percentage points, to 35.8 percent in 2007. Although the level of debt owed by families rose noticeably, debt as a percentage of assets was little changed. The largest percentage change in debt was in borrowing for residential real estate other than a primary residence.

With median and mean debt advancing faster than income, payments relative to income might be expected to increase substantially. In fact, total payments relative to total income barely increased, and the median of payments relative to income rose at a slower pace than it did between 2001 and 2004. Nonetheless, the share of families with high payments relative to their incomes increased notably.

From the 2007 Survey of Consumer Finances. Emphasis added.

Yet more evidence, together with the Permanent Income Hypothesis, that people look at their earnings power over time as an asset, just as any other asset, and their willingness to incur debt is due to how they price that earning-asset (together with their financial assets). A house is a 30 year commitment -- so you need to estimate 30 years of wages, discounted by both inflation and the risk of volatility in those wages. Same thing for other durables, and this percolates into day-to-day spending as well. "Flexible wages" are deeply destabilizing, as assets with volatile earnings are priced lower than assets with reliable earnings. Flexible wages would cause a further decrease in demand in excess of the actual change in wages.

Inequality and Debt growth

The following chart, taken from census data, shows the growth of income inequality since 1967:

The chart shows the share of national income for each cohort: bottom fifth, second fifth, third fifth, fourth fifth, top fifth and top 5%. You would expect the bottom fifth to collect a smaller share of income than the top fifth -- these ratios have been normalized so that 1967 =1. I.e. We assume 1967 is a "sustainable" level of inequality and track the increase that occurred since then.

Focusing on the third fifth, or "middle America", we can see the income gap from the decline in it's share of income from 1967 to 2008:

The next chart shows the growth of household debt/GDP since 1967 (i.e. we set the 1967 level to zero, and look at the increase that has occurred since then.) At the same time, we plot the running total of the income gap for the third fifth:

The growth in debt was a re-cycling mechanism, allowing the middle class to continue consuming (and hence maintain GDP growth) all while incomes fell. In other words, wages shifted from the middle incomes to top management/asset holders, while at the same time the purchasing power of the middle incomes was maintained by debt-financing. The increase in household debt funded the increased gains for the top incomes.

The total amount spent on wages did not change very much, but the distribution changed, effectively the top 1% earned much more, the top decile net of the top 1% stayed the same, and everyone else's wages collapsed.

In such an environment, if workers are paid less, they wont be able to purchase goods, demand will decrease, and therefore prices will fall to adjust, meaning lower profits for the owners of capital, and the income balance will be restored in real terms.

Debt-financing can be used as a mechanism to prevent this adjustment. As long as the debt is rolled over -- as long as consumers are willing to take on more debt -- inequality can grow and prices/profits will not adjust to reflect the lowered wages of average employees.

In this case, the asset base will swell, interest rates will fall, the debt stock will swell, and consumption will remain constant even as wages fall. That is, until a debt crisis occurs.

A similar dynamic can be seen with the current account. The Chinese government confiscates dollars from exporters, prints up RMB and "buys" those dollars at prescribed rates. The end result is that the people who produce goods suffer inflation as the goods are loaded on ships to disappear. As a result, the real purchasing power and consumption of the Chinese workers is suppresed, while U.S. workers are put under wage discipline. There is transfer of technology which helps China, but you can have technology transfer without suppressing wages. Again, debt-financing is used to bypass the market adjustment, this time of currencies as well as real wage rates in China.

This is the key macro-economic and political problem of our era -- debt-financing is being used to suppress wages, preventing necessary price adjustments from bursting the income concentrations, and this dynamic is fundamentally de-stabilizing.

But until we understand the root of the problem, we wont be able to address it. The problem isn't "Debt" in some abstract sense, and it certainly isn't "too much borrowing" in and of itself, since the borrowing was only an attempt to keep wages at their 1967 share of GDP. In other words, that middle quintile only borrowed enough to try buy their own domestic output. The problem is not paying workers enough money to maintain demand for the goods they produce. This is why we are suffering from chronic demand shortages, asset bubbles, abnormally low interest rates, record inequality, and crushing private debt.

For every invested dollar...

... almost 80 cents goes to pay the salaries, bonuses, and other benefits of employees in the Financial Services and Insurance Industries.

"Domestic Investment" is defined as gross investment exenditures by both the government and private sectors net of depreciation. "Finance and Insurance" compensation consists of BEA FIRE compensation data net of real estate sector compensation.

Whose government is it?

You can tell the priorities of government by looking at what it is trying to optimize.

China is printing RMB like no tomorrow, so that they can sterilize dollar inflows and keep exporting. They are interested in developing domestic capital: skills, a trained workforce, technology transfers, research and development, viability of the business sphere and full employment.

Reading Keynes, the emphasis was always on maintaining full employment. Then the emphasis shifted to total output growth. And since then the emphasis has shifted to protecting bondholders and keeping inflation and interest rates low.

We are protecting and bailing out bondholders, not investing in our capital base or fighting to build the tangible things that create wealth for the future: better infrastructure, education, research, stable business profits and high wages.

Any government that is more concerned with inflation and interest rates than with employment and business development has been taken over by the illusion that financial wealth is equivalent to real wealth.

The Price/Earnings Ceiling

From the valuation approach described previously, we can look at the earnings yield of a broad index such as the SP 500 and invert the growth formula:

earnings yield = dividend yield + growth rate

to obtain:

P/E = 1/(dividend yield + growth rate)

To put an upper bound on the P/E ratio, we want a lower bound on the yield + growth rate. Generally speaking, the total returns on capital (for equities, that would be growth rate and dividends) have been the GDP growth rate over the period. That is one of Kaldor's "stylized facts", which actually says that returns to capital as a fraction of GDP are the same over time, and that therefore the growth rate will be the same. Now, that doesn't need to be in equity, it could be in other forms of capital, but over time, it's hard for the equity returns to outgrow the total returns.

If you don't like the Kaldor argument, then notice that GDP = GNI, so that if national incomes are growing at g, people will be willing to accept effective bond rates of a little below g, so that the premium for equity should be a little higher than that. So that gives g as the expected returns from capital.

So that gives a long run upper bound (actually approximate equality) for the P/E multiple for the equity market as a whole. Obviously the variance is large :) Going forward, output growth may continue to slow.

Suppose you think a GDP growth rate of only 4% (nominal) is warranted over the next 20 years. In that case, the average P/E multiple would be at most 25, but there is a good chance that the equity markets wont like such a low yield, and money will shift to other forms of capital.

This surprises people who associate "high multiples" with growth stocks, but that is only to the degree that their multiple is higher than that of the index. The economy as a whole grows with GDP, as does the return to equity capital as a whole. Some specific returns may be more highly valued, but if the entire index is more highly or lowly valued than the yield available for the economy, then you are looking at a bubble or arbitrage opportunity. Historically 10 years has been long enough for this to unwind, and 20 years certainly has.

The larger the deviation, the more difficult it is to maintain earning growth as the company grows. For the SP 500 as a whole, if it is a constant share of capital, then the multiple will increase in absolute terms as yields fall.

It is usually better to buy when earnings are collapsing, but in this case you look at "smoothed" 10 year earning multiples (i.e. the current price divided by the average real earnings over the last 10 years). This would allow you to not be fooled into thinking that a high multiple is expensive, when in reality the denominator is abnormally low. Alternately, smoothing will not let you think that anything goes -- for example the July 2009 multiple was 117 -- and the market kept climbing.

FYI, in July '09, the smoothed P/E multiple was about 18. Not cheap by any means, and certainly not sustainable, unless you are assuming something along the lines of a dividend yield of 2% and nominal earning growth rates of 3.5% going forward. In that case, it might not have been a bad idea to buy. Just be aware that a high multiple is only justified on a long term basis only if we experience a secular growth collapse that also would also result in long term bond yields falling to well below 3.5% as well. Such an environment might not fare well for equities.

Negative Re-Investment Rates

When the cost of equity capital is low, then the re-investment rate is high. This is dangerous for firms because the market is fickle, and periods of complacency quickly turn into panics, at which point the capital is re-priced, and now higher returns are demanded. Companies respond to this by trying to defend their dividends and thus end up liquidating a portion of their capital stock in the form of layoffs, shutdowns, and other asset sales.

You can see a historic negative re-investment rate in this graph, corresponding to capital liquidation for the market as a whole.

Note that this is different from lowering the re-investment rate as you would expect in a regular recession. As long as the re-investment rate is positive, then productive capacity is still being increased, but at a lower rate. I.e., the growth of productive capacity is "slowing down" as opposed to shrinking.

Layoffs are always occuring as is hiring, but when the market as a whole has a negative re-investment rate, then productive capacity as a whole is decreasing. You can think of the difference as that between laying off a few workers versus closing a factory. It is harder for employment in the latter case to recover because it requires a greater investment commitment on the part of management as they emerge from the recession. But this is exactly when management is still skittish. This is not to say that employment can't or wont recover -- but the hurdle will be higher.

Moreover, the graph suggests that the current market downturn is fundamentally different from the crashes and recessions in earlier periods. You have to go back to Great Depression, the 1921 recession, or the Long Depression (1893-1898) to find such periods of negative re-investment, and all of those periods were dwarfed by the negative re-investment that we see today. Note that all three of the historical precedents were deflationary recessions.

Valuing Equities without Discounting

In the diagram above, each time a firm generates profits (earnings), a portion are paid out as a dividend, and the remainder are re-invested, adding to the firm's capital stock. Assuming that the firm is on a stable growth trajectory, then the various multiples (price to earnings or price to dividend) should be constant over time, while the earnings, dividend, and market cap should grow with some real growth rate, g. If this were not the case, these multiples would go to zero or to infinity over time.

By "real growth rate", I mean the logarithmic derivative (i.e. (1/x)dx/dt = g). Now let r be the re-investment ratio:

r*earnings = retained earings = earnings - dividend

In particular, if r = 1, then all earnings are re-invested, and if r is negative, then the firm is selling off capital, since it cannot meet the owners' required rate of return.

Then the growth rate of capital (dC/dt) is just rE -- the amount re-invested. Therefore the "real growth rate" is rE/C, so that we have

g = rE/C = r*(E/C) = r*(earnings yield)

Which we can re-arrange to read:

dividend yield = ((1-r)/r)*g= (1/r-1)*g

earnings yield = g/r

earnings yield – dividend_yield = g

earnings yield = dividend yield + g ("dividend capitalization model")

• g = growth rate of earnings, dividends, market-cap
• r = re-investment ratio

This allows us to value equities without discounting infinite sums -- by defining steady-state ratios. The assumption is only one of stability (which the discounted sums approach also requires). In order to value a company, you need to estimate the earnings growth rate as well as the dividend yield (or the re-investment ratio). This gives the standard "dividend discount" valuation approach, although the method here is a bit more general, in that no dividends need to be paid -- full re-investment is also allowed.

Now, in the real world, the market doesn't know how much to value capital -- that is a difficult problem and so the market value oscillates around the "true" value, as investors constantly adjust their expectations based on optimism and pessimism, excessively marking down the value of capital during busts, and over-valuing capital during booms. To see these mood swings, consider the following:

This chart graphs the stable growth path, g, obtained by taking the 10 year average earnings-yield and the present dividend yield. It then graphs the 10-year trailing average of the observed earnings growth.

You can see that there are turning points, in which the actual earnings growth (blue) wildly overshoot and undershoot. These are equivalent to "earnings booms" and "earnings busts", in which the earnings grow much faster or slower than the required growth rate as determined from a 10 year backwards-looking window. These correspond to secular booms and busts in corporate profits, and are influenced not just by changes in output, but by debt-fueled demand, tax law changes, access to new markets, etc.

In general, expectations formation is backwards looking but there are real turning points -- changes in mood about the future (such as the Great Depression or the post-war boom) that a purely backward looking metric will not detect. Also note that the booms and busts are secular -- they last much longer than simple recessions, so that large multiples seem normal, until the next collapse and re-pricing, or until the next boom occurs.

Chart of the Day: Earnings Arbitrage in Financial Yields/Private Sector Yields

When financial yields are low, price to book ratios are high, so the businesses represented in the financial markets are expensive. This should encourage investment in things like proprietorships and non-corporate business. When yields are high, then price-to-book ratios are low, and the reverse is true -- why futz around with Sally's Shoe Store when Johnson and Johnson is yielding 6%?

The following chart from the NIPA table 2.1 plots personal income received from financial assets (dividend and interest payments) against personal income received from proprietorships, both farm and non-farm. The latter attempts to exclude wages and operating income and tries to account for depreciation and inventory changes -- this is "return on capital" for proprietorships, which are classified and non-corporate farm and non-farm non-corporate businesses that must file certain tax forms that I don't care about. Note that rental income received is a separate type of income and is not shown (it moves uniformly up, from left to right, but there are owner-equivalent rent issues there). Both time series are presented as ratios of GNI.

The sum of both time series is shown below:

See how these returns began to decline during the debt-deflation and climbed during the debt-inflation starting in the 1980s, but there is at least evidence for not a lot of variation for the sum of earnings for both corporate and non-corporate businesses.

Gloomy Chart of the Day

It seems that you can only pay down one sector's debt to GDP at the expense of increasing another's -- that is, except for mass defaults. This chart omits a few sectors, such as farm debt, and nonfarm non-corporate business debt, but the sum of non-financial private sector debt/GDP + the sum of government debt/GDP has always been growing, too -- ever since "Fed Independence Day", when the U.S. stopped forcing the Fed to monetize whenever treasury yields rose above a certain level.

In particular, the chart suggests that the paydown of Federal debt that occurred in the post-war period came at the expense of an increase in household debt, and to a lesser degree business debt. The current miniscule drop in household debt is coming at the expense of a significant increase in Federal debt, and to a lesser extent, (corporate) business debt. The source is The Federal Reserve's Flow of Funds, and the time period is from Q1 1952 to Q1 2009.

A Tale of Nine Cities: Rents 1930-2007

The following continues the historical exploration of relative house prices by looking at the ratio of median rents paid in nine selected California cities to the ratio of median rents throughout the state. The state ratio takes into account urban areas only (rural median rents are lower). All data comes from the census.

Rare voice of reason

Simple, honest, well-reasoned and bound-to-be-ignored post:

http://www.winterspeak.com/2009/08/looking-back-at-bailouts.html

I will paraphrase from an influential article by Clay Shirky, with my changes in italics :)

Revolutions create a curious inversion of perception. In ordinary times, people who do no more than describe the world around them are seen as pragmatists, while those who imagine fabulous alternative futures are viewed as radicals. The last couple of decades haven’t been ordinary, however. Inside the papersnation, the pragmatists were the ones simply looking out the window and noticing that the real world was increasingly resembling the unthinkable scenario. These people were treated as if they were barking mad. Meanwhile the people spinning visions of popular walled gardens and enthusiastic micropayment adoption schemes to repackage bad assets into good assets, visions unsupported by reality, were regarded not as charlatans but saviors.

Another Look at Long Run Equity Returns

Earlier I mentioned how the market cap growth of companies cannot outpace GDP growth -- except for brief periods of multiple expansion. Here is a chart from 1929 to 2008 showing the total net operating profits after taxes together with dividend payouts of U.S. Corporations divided by Gross National Income (the data comes from the BEA):

Over this time period, the ratio of profits available to corporate business was around 5% of the economy. There is what seems to be obvious mean-reversion. More interesting is the stability of the dividend payouts except for the two credit bubble periods, one deflating since 1929, and one inflating from the 1980s until just recently. Companies are loathe to increase dividends without appropriate future coverage, and so the dividend payout/GNI contains meaningful information about the estimates of management for sustainable future returns, and is much less volatile than the equity multiple. A dividend payout of 2% of GNI was remarkably stable between the two credit bubbles.

Early data about market capitalization is hard to find, but we can look at total market capitalization in the second half of the 20th century. The following chart, from the Federal Reserve Flow of Funds, shows total market capitalization of U.S. corporations divided by the Gross National Income, reported on a quarterly basis. You can also see the total dividend payouts divided by the total market cap (i.e. the dividend yield of the entire market).

By this historical metric, the present market is not cheap. It would need to fall by 50% in order to reach the post-war lows from 1974-1984 -- and that is a decade long period! We can also see the same shape that is a common refrain in most of the graphs: the 20 year run-up culminating to the twin-peak blow off of 2000 and 2007 that at least seem to suggest the beginning of a downtrend. From this long-run perspective, the period from 2003-2007 seems like an enormous bear market rally.

Don't let anyone fool you into believing that the total equity value of U.S. corporations can consistently outpace the size of the economy. As can be seen from the historical record, periods of multiple expansion for both corporate profits/GNI as well as market-cap/GNI tend to mean-revert, even though they can last long enough to convince the public that "this time it's different". Those who cut their teeth in the period from 1980-2000 did not experience a representative time in the history of U.S. equity values.

Historically, the Market Cap of U.S. Corporations averaged approximately the size of GNI (actually 0.94xGNI), although it did swing around this value. The following chart shows the actual 10 year returns of purchasing a slice of equity, together with the predictions of 2 estimates.

The 10-year forward equity returns have been the worst on record in the postwar period.

The first estimate assumes 100% reversion in a 10 year time period, with the assumption that GDP growth will be the value of the current 10 year treasury. The second line assumes 50% reversion to the 0.94xGDP value, also assuming that 10 year forward GDP growth will be equal to the 10 year treasury rate. Of course, no one can tell when or to what extent mean-reversion will occur. This type of analysis is a crude version of John Hussman's analysis, with the notable exception that I am using the 10 year treasury estimate of GDP as opposed to a constant 6% GDP growth trend, I'm excluding dividends in the total return calculation, and am selecting the parameters not by fixed P/E terminal multiples, but by proportion of trend reversion. The interested reader can make each of these adjustments themselves -- the purpose here is not to predict market values, as it is hard to out-guess the crowd, but to provide a sense of valuation for the market as a whole -- what does a dollar of equity buy today in comparison to what it bought yesterday? Moreover, if the present multiples were to mean-revert, then what would the market cap be? These types of considerations should inform those who don't always believe that we are in a new era.

Both estimates currently predict annualized returns of approximately 3% (excluding dividends), over a 10 year period. These are all nominal returns, but the Old Barkeep is not too worried about inflation over the next decade. I hope to return to this theme with some more sophisticated models.

In Search Of The California "Desirability Premium"

Housing is more expensive in more "desirable" areas -- that's a bit of a tautology, of course, since we don't have a way of measuring how desirable an area is, and having high paying jobs will certainly attract more people, in which case they will be willing to bid up the cost of housing in order to access that labor market.

But in more desirable areas, are owners willing to pay a greater proportion of their income on housing? Let's take a look:

Source: American Community Survey 2005-2007 3-Year Estimate

Our universe consists of all census "places" -- cities or townships -- in California.  The x-axis shows the per-capita income for each place, while the Y-axis measures the median percentage of income that mortgage holders are willing to pay. "Mortgage costs" here also include other ownership costs such as property taxes and utilities.

FYI, owners are willing to devote a median value of 31% of their household income for mortgage costs in San Francisco, 35% in Oakland, 34% in Los Angeles, 29% in Vacaville, 44% in Beverly Hills, and 40% in Lennox. In general, the graph suggests a slight downtrend, in that owners in wealthier areas are willing to devote a slightly smaller share of their income for mortgage costs, however the R^2 value does not give a lot of confidence.

But ownership rates vary -- in San Francisco, only about 1/3 of the housing stock is owner occupied, whereas the state average is much higher. Let's look at the relationship between ownership rates and mortgage burdens:

Source: American Community Survey 2005-2007 3 Year Estimate

Here there is some evidence that ownership burdens increase somewhat as ownership rates fall -- by 0.65% for every 10% decline in ownership rate -- but again the R^2 doesn't give confidence and the drop is slight.

Perhaps it's more a question of inequality -- in areas with high income inequality, owners might be willing to pay a premium to showcase their status:

Source: American Community Survey 2005-2007 3 Year Estimate

In this case, there is an extremely weak correlation that suggests that the greater the inequality, the less owners are willing to pay, proportionately -- but the relationship is weak.

So it turns out that the "desirability premium" is very hard to find when looking at data. Generally speaking, areas with higher household incomes command higher house prices, but not as a percentage of income.  There is evidence that as incomes increase, people are willing to spend less, proportionate to their income, on monthly mortgage costs. 

Visualizing Housing Subsidies

A Tale of Six Cities: 1930-2007

Source: Census

The chart shows the ratio of the median house price to the median annual rent paid for six california cities form 1930-2007, using census data together with the American Community Survey. Rents are calculated as median monthly contract rent x12.

You can see both the wage compression period, together with a secular period of expanding multiples (and declining yields). This is due to effects such as Proposition 13, the mortgage interest deduction, longer repayment periods and the introduction of additional subsidies. Don't forget the boom/bust pattern common to California, particularly coastal California, as well as the parabolic multiple increase starting with the 2000 census. The time series ends in 2007, but we can watch it collapse as subsequent surveys are released.

A Tale of Seven Cities: 1930-2007

The following tracks the ratio of city median rents to state median rents for seven California cities. Note that "median rent" means median contract rent paid -- not asking rent. The data is from the decadal census, together with the American Community Survey.

Source: Census

Note the rent uptick during the dot-com period followed by collapse for San Francisco and San Jose, although Oakland had a very brief mini-spike as well. Also note the effects of wage compression during the post war period, as rents crowded around 1 and then diverged. Also note the increasing volatility of these ratios, as the higher income cities tend to be more volatile after the period of wage compression.

Can you guess when rent control was instituted in some of these cities?

Long Run San Francisco House Prices: 1930-2007

This chart shows the ratio of the median price of a San Francisco owner occupied house to the median price of a California owner occupied house. Census data for selected years 1930-2007.

Source: Census

The spike peaking in 2001 represents the dot-com era -- the first hump of the income bubble. The fall from the spike is not due to San Francisco prices adjusting downward, but to California prices shooting upward. By 2006, prices in California caught up with San Francisco, but by 2007, prices in California began to fall again as the housing bubble deflated in the suburban areas of the state. Unfortunately there is a 2 year lag for releasing the American Community Survey, so we will see what happens to that spike going forward. I predict another spike, followed by a reversion to the long run trend.

The San Francisco Income Bubble

The Old Barkeep is fascinated by the insanity of San Francisco's housing market, and demographic shifts generally. Always a rich town, the city has recently experienced a big income boost -- can you recognize the pattern from 1997-2007?

Source: BEA Regional Accounts

There is a 2 year lag for the local data, but I don't think this bump is durable.  Let's see what that graph looks like once the 2009 numbers come out.

Goosing Equity Returns via Dividend Re-investment?

The Old Barkeep is not impressed with various book peddlers and optimists advertising the possibility of high "total" returns in excess of GDP growth.

Here is the algorithm:

1. Start with a notional account. E.g. $1 in the DJIA index, invested in 1900
2. Over each time period, if the index changes by X%, assume that your equity does, too.
3. Over each time period, if a dividend payout of D% is given, add back that amount to the equity value, via a "dividend re-investment model".
4. Watch your holdings increase at a compound annual growth rate in excess of the growth rate of the index.
5. Peddle books, or drive a pension fund into the ground.

Do you see the flaw?

First, some background.

The market cap of a firm reflects investors' beliefs about the present value of all future earnings of that firm. Some of those earnings are retained -- re-invested by the management --  and the remainder are classified as "surplus profits" and are distributed as dividends.

The decision to make a distribution amounts to a belief by the firm that it cannot re-invest those profits while maintaining the expected growth rate -- so it distributes those dividends to investors.

On the ex-dividend date, the equity value falls by the distribution amount -- since the "value" of the company is now less (it has one less future earning!), and on the distribution date, the owners receive a cash payment.

The total value of the owners' stake has not changed, they just have some of that value in the form of cash that they control as opposed to having that cash locked up in their equity accounts. For this reason, you do need to include cash-holdings in any calculation of total return. A total return calculation should  include all holdings.

In the real world, some of those dividends are re-invested in equities, and to the degree that they are, this drives up the equity value of target investments as investors compete to bid up the share price of the target firms. So the net effect is to lower the cost of capital for some firms while raising it for others as investors continually engage in a process of valuation.

This process supposedly results in no company being without sufficient capital to give investors the return they demand. So for the market as a whole, the dividends flowing out of the entire market are surplus profits for all the firms in the market. In other words, over time, investors believe that no company can increase its capital base while maintaining the expected yield;  otherwise they would bid up the share price and drive down yield.

Therefore when looking at a historical time series of prices, whatever equity re-investments actually occurred, they are already recorded in the time series.  It is double-counting to include the observed market cap increases of those firms that are the targets of investment, and to not count the market cap declines of firms that made distributions. Moreover, any dividends thrown out by the entire index should be assumed to be surplus capital for the index as a whole. This means, if those dividends were re-invested, they would come at the expense of a fall in yield.

Equity prices are everywhere and always subject to a process of valuation, and this means that you should not always re-invest. You cannot always throw surplus capital back into the hands of management and maintain the same rate of return. You cannot always bid the price of equity up and expect profits to rise. At some point, additional investment results in stock price bubbles followed by crashes, because the earnings growth rate is constrained by GDP growth rates, and you cannot goose GDP sustainably by bidding up the price of equity.

At some point, you are better off buying a bond or even holding the dividend in a cash-equivalent, rather than overpaying for low yield.

Now, the specific flaw in the algorithm: the mistake is in Step 3.

The book-peddler is inserting a fake transaction into the historical time series, by assuming that you will be able to increase your holdings of the company without bidding up the price. All these backwards looking "what-if" scenarios suffer from the same flaw, which is that they are minority strategies. If enough people followed these strategies, the price would shoot up, the dividend yield would collapse, and the strategy would underperform the historical time series. Any time you see someone peddling a "total return" strategy that is greater than long-run GDP growth (which was 6%, in nominal terms, over the last 100 years), then you are being sold snake oil.

How much does the fake transaction that allows for purchases of shares without bidding up the share price distort the historical time series? One way to measure this is that if the "average" investor's equity holding could grow, by any series of transactions at the rate of the dividend payout + the observed rate of equity growth, then the total equity value of the whole market would also grow faster than the observed rate (by the same amount). Moreover, the synthetic model assumes that the dividend yield on this larger equity value would be the historical yield. This means that there would be a divergence between the observed profits and the profits as predicted in the model:

Given that aggregate earnings grow at the rate of GDP growth, a strategy that claims to outperform GDP growth by 3%, will after 100 years predict

• dividend payments 19 times the observed payments
• market caps 19 times greater than the observed market caps 
• "total returns" 19 times greater than the total returns of the average investor.

Long run indeed!

But, what about our investor, who suffered an equity loss of $1 and has $1 in his cash account? Can't we add $1 to his total returns? Absolutely! If by "total returns" you mean a real-time mark-to-market of all the holdings of a particular investor, then this is valid (and necessary) to gauge performance. And you should add total dividends paid to the cash account of any historical time-series. The only flaw is when you insert fake transactions into the historical time series, by assuming re-investment without assuming that this will change the historical prices  --  i.e. assume it is possible to buy or sell shares without altering the historical prices. That is fundamentally a minority strategy, and will not succeed if engaged by the public at large, or even by a sizeable minority of the public. Certainly it will fail if large players such as pension funds use the strategy.

So, then, what are the expected "total returns" of the market?

• A first order long run estimate is the long run GDP growth estimate. This has been the observed returns over 50 year time periods, at least for equities.

• John Hussman authored an analysis that takes the "gap" between current stock values and the long run GDP growth rate, taken to be 6%, with the assumption that mean-reversion will occur in the longer run. There is evidence for this over 10 year time periods

Note that, in general, there is nothing wrong with specifying an asset allocation strategy in an attempt to outperform. Most try to outperform! Just be aware that any strategy promising returns in excess of GDP growth will be a minority strategy, that if followed by enough people, will fail to outperform.