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xccfApr 2 20152

Using 0-interest special offers from credit card companies is one crazy idea: https://news.ycombinator.com/item?id=3026230 You might do it just once, carefully building up your credit over years and then totally destroying it in the course of 24 hours. I'm not sure if this idea still works or if it could be made to scale.

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Best way to invest with leverage?

by Brian_Tomasik

Apr 2 20151 min read 51

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Investing

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Altruists who don't care too much about risk (and young people in general) should plausibly use leveraged investing. What's the best way to get leverage?

Margin borrowing seems like the default solution. I might try it if there's nothing better.
Theoretically options could be used, but I'm unsure whether they work in practice.
Supposedly futures offer massive leverage, but I haven't explored the details, and they seem hard to trade yourself. I'd like something I can just buy and hold for a long time.
Something else?

Ideally, there should be a fund that you just buy into to get leverage, with someone else handling the details. But leveraged ETFs don't work because they're optimized for day trading and as a result lose money for buy-and-hold investors.

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Tom_AshApr 8 20155

It's worth mentioning that there's a mailing list where EAs can ask for and give investment advice, which has some savvy investors on it: https://groups.google.com/forum/#!forum/ea-d-investment

BenGilbertApr 4 20154

It depends what you are investing in but, if it's equities and especially equity indices, one way that I know of (for those who have access) is spread betting. In the UK, you can effectively buy and sell futures with low spreads, and no taxes or commissions.

Today, for example, I can sell or buy the Dow at a price of 17320/17344 to settle in January 2016. On the site I use, I believe I need to have £50 of margin for each point I buy, so I can effectively buy a position of £173,000 by just having £500 deposited (though a very small movement against me will lead to having to put in more money or having my position closed).

If you can put in £1,750, you can have, say, a position of £173,000 which will stay open unless the index goes more than 1% below where it is now, or £86,500 open until it goes more than 2% below where it is now, etc.

The idea of buy-and-hold doesn't work well with the idea of leverage, since when you've got leverage you will always have the risk of being closed out, but you can increase the risk of your position by, say, 10 times and only be closed out with a 10% move against you.

I'm not a financial advisor and nothing I write should be considered advice of any sort.

Brian_TomasikApr 4 2015-3

Thanks! That's helpful.

Paul_ChristianoApr 4 20152

If you are e.g. maximizing log wealth and prices are a gaussian brownian motion, then leveraged ETFs and leveraged positions are equivalent for small amounts of leverage, and the ETF is better for large amounts of leverage.

ETFs take their losses in small doses, a day at a time, while a leveraged position takes its losses with the probability of wipeout. If you scale down the leverage position when things get bad (or get margin-called), then you make the leveraged position closer to the leveraged ETF. Both losses are quadratic in expected volatility (for small leverage), and expected volatility is linear in the number of days for a GBM.

In this setting, the leveraged position is a bet on the anti-correlations of day-to-day movements, i.e. that volatility scales sublinearly with time. I think empirically this hasn't been true recently on the scale of months, but has been true on the scale of years---that said, there are more direct ways to bet on this question if you feel like betting.

Presumably your issue is not day trading vs. buy-and-hold, but that you aren't risk-averse (or at least you don't care so much about a wipe-out). If philanthropy as a whole is risk averse, then the same analysis applies for philanthrpists in aggregate, and this is coming down to your personal efforts to drag philanthropy towards an optimal aggregate allocation.

Brian_TomasikApr 4 20151

Both losses are quadratic in expected volatility (for small leverage), and expected volatility is linear in the number of days for a GBM.

Let me know if you have a reference to understand this better. :)

It doesn't seem intuitive to me because with a leveraged ETF, you lose from volatility alone. In contrast, if you bought stocks on margin and those stocks went up (with whatever degree of volatility you want so long as it didn't trigger a margin call), then you wouldn't lose from volatility because you're not buying and selling. Maybe you're suggesting that the expected losses if you do have to make a margin call are big enough to make margin buying comparable to leveraged ETFs?

Another reason I'm confused is that almost everyone advises against long-term investing in leveraged ETFs, while it's common to encourage long-term margin investing. So there must be a difference between them.

Paul_ChristianoApr 4 20151

It doesn't seem intuitive to me because with a leveraged ETF, you lose from volatility alone. In contrast, if you bought stocks on margin and those stocks went up (with whatever degree of volatility you want so long as it didn't trigger a margin call)

You can exactly simulate a leveraged position using a leveraged ETF by increasing your position whenever the market goes down and decreasing it whenever the market goes up. If you think that price changes between periods are a martingale, then it's pretty obvious that changing your investment ilike this won't increase your net profits.

This strategy is only a good idea if the market is more likely to go up after it goes down. In practice, this has been true on some time scales, while the reverse is true on others. In theory, this shouldn't predictably happen, because a savvy investor can make money. But whether or not you think this is true, there are more direct ways to earn money if you think that you can predict which way the market will move.

Note that one reason that this strategy looks good on paper is that it's a thinly-veiled version of the martingale, taking place on a log scale. If you keep doubling down every time you lose, then you will either win a bit or you will lose everything. I.e., you can keep borrowing more to avoid a margin call, but if you do that it just makes an eventual margin call worse.

It's pretty easy to do the math for a GBM and log utility. I don't know a reference. An easy way to think about it is to consider a leveraged ETF that rebalances every day vs. every other day. Both have losses that depend on volatility. Which one experiences more volatility on average? The answer is that they are basically the same, because the volatility over two days is double the volatility over one day: half of the time the moves are in opposite directions and the volatility is 0, and half of the time they are in the same direction and the reallized volatility is 4x. Similarly if we go up to every 4 days, or every 8 days. Your leveraged positioan is just rebalancing every few years.

it's common to encourage long-term margin investing

I haven't encountered much of this (at least not at 2x leverage), most people caution against the risk of a margin call during a downturn.

Brian_TomasikApr 4 20150

It's going to take me a while to grok all of your arguments. :)

You can exactly simulate a leveraged position using a leveraged ETF by increasing your position whenever the market goes down and decreasing it whenever the market goes up.

This is a key point that I don't understand. It seems like the two are different, as the following example shows.

Leveraged ETF: Buy a 2:1 leveraged ETF for $50. They borrow $50. They buy $100 in stocks. During the day, stocks increase 10%, so that the value is $110. To restore a 2:1 leverage ratio, the fund borrows another $10 and uses it to buy stocks, resulting in $120 of stocks and $60 of debt.

Regular margin investing: Starting with $50, borrow another $50 to buy $100 in stocks. Stocks increase 10% to $110. The ratio of assets to debt is now 2.2:1.

If you had sold off some of the ETF to get assets down to $110, the leverage ratio would have remained 2, not 2.2. So how can buying/selling the ETF replicate regular leverage?

Paul_ChristianoApr 5 20150

You care about your leverage debt ratio, not the leverage : debt ratio of a fund in which you have invested some of your money.

If you sell $5 of the leveraged ETF in your scenario, then you have exactly the same position as the margin investor, just it’s $55 in a 2x leveraged ETF instead of $110 of the equity. You have $5 in your pocket and $55 of debt from the ETF, rather than $0 in your pocket and $50 of debt. Of course, if you liquidated the whole position, you would just end up with $60 either way. If there is any difference, I don’t see it.

Brian_TomasikApr 5 20150

Thanks for elaborating. :)

You have $5 in your pocket and $55 of debt from the ETF,

If you sell $5 of the ETF, it seems like the sold $5 would get rid of half equity and half debt, leaving $5 in your pocket and $115 in the ETF, of which $57.5 is debt.

Various articles seem to suggest a difference in performance characteristics between daily rebalancing vs. buy-and-hold. For example, Figure 3 on p. 10 of this paper.

AGB 🔸Apr 5 20151

After that day, the ETF is worth $60. If you sell $5 of it, you hold $55 of 2x exposure and $5 of cash, which is equivalent to what you would have if you had done no trading; on net you are long $110 of stocks and short $50 cash.

I think the mistake in your reasoning is saying that you 'get rid of half equity and half debt'. You actually reduce your equity by 2x, your debt by 1x, and then receive 1x cash also.

Note that the paper you link to says several things that basically echo Paul:

"The above suggests that a leveraged ETF with a positive expected raw return but negative expected growth, increasingly resembles a lottery ticket over time. As time passes, the chances of the lottery player ending up with zero approach certainty, but the payoff if he wins continues to increase to ensure that the lottery itself has favorable odds."

"We find that the expected raw return of the levered ETF is the highest simply because the investor borrows at 2% to invest at 7.5%; however, the distribution of return outcomes is arguably unattractive. The investor in a levered ETF achieves negative expected growth and also has the lowest median portfolio return."

As Paul pointed out, this is close to the opposite of the martingale strategy; in the martingale strategy you eliminate your losses with near-certainty (where how near depends on how much you can afford to keep doubling up) but your losses should you incur them get larger and larger, here you eliminate your gains with near-certainty but the gains get larger and larger.

Something that is explicitly ignored in that paper is the presence of fees and transaction costs. Daily leveraged ETFs do more trading than, say, one that rebalanced monthly would, because standard deviations scale with the square root of time. In the non-theoretical world, there are costs to this which should be considered, and they get bigger as the ETF(s) in question get bigger.

Finally, I would note that there is a surprising amount of nonsense written about ETFs online. I could point to much clearer-cut examples of incorrect or highly misleading statements.

Brian_TomasikApr 5 20150

After that day, the ETF is worth $60. If you sell $5 of it, you hold $55 of 2x exposure and $5 of cash, which is equivalent to what you would have if you had done no trading; on net you are long $110 of stocks and short $50 cash.

Thanks for this! Now I understand.

BTW, if you have time:

What's your opinion on non-rebalancing leverage done manually?
What if you pay for the leverage interest using new income in a similar way as people pay off home loans? (This doc encourages such an approach.)

Paul_ChristianoApr 4 20152

I don't think your treatment of risk for altruists is right. What matters is the correlation of your investment with available philanthropic capital: it's not necessarily a big deal if you lose 100% of your investment, but it is a big deal if your portfolio underperforms when the market underperforms, because all of the donors are losing money at the same time.

So you are basically making a judgment call about whether other donors to your cause are over- or under-exposed to the market. It's a fine call to try to make, but you should be aware that that's what it is.

In the world of rational donors, big charities, and CAPM, the result is that the risk-adjusted returns to the risky asset and the risk-free asset become equal for altruists, regardless of their portfolio composition. Then other idiosyncratic considerations kick in, and you end up with widely varying individual allocations. But the overall allocation is just the same as if there were one giant philanthropist.

Brian_TomasikApr 4 20150

Thanks, Paul!

but it is a big deal if your portfolio underperforms when the market underperforms, because all of the donors are losing money at the same time.

Doesn't it depend how much the charity is risk-averse? If the charity's value as a function of wealth were completely linear (which isn't true in practice), then these correlations wouldn't matter, only expected income.

I don't understand your last paragraph. Feel free to clarify if it's worth the trouble. :)

Paul_ChristianoApr 4 20151

Yes, it depends on whether there are diminishing marginal returns to charity overall. But you have made arguments based on the small size of individual donors relative to the large size of the charities they support, and those don't settle the issue.

xccfApr 2 20152

Benjamin_ToddApr 8 20151

Note that with using leverage, there's always margin calls, so calculating expected return becomes more complicated.

e.g. if you're at 5x leverage, a 20% loss in the market will wipe you out, and 20% losses happen pretty frequently in stocks.

I'm not sure what's the best quick and dirty way to adjust your expected return estimates to take account of this.

Maybe you could take your regular estimate (i.e. asset expected return * leverage ratio), then multiply it by the probability of wipe out in the relevant time period?

e.g. If you think the S&P has a 10% expected return, and you invest at 5x leverage for a year, the overall expected return is very roughly:

(1 - p ) 50% + p 0%

Where p is the probability of wipeout occurring in the period. In this case, p is the probability of a 20% loss occurring at some point in the next year.

In this case, it's only worth investing with the leverage if p < 0.8, and that's if you're fully risk neutral. I'm not sure what p is, but it could easily be ~80%, which would mean using 5x leverage has lower expected returns than unleveraged investing. Which could help to explain why so few people use that much leverage.

Paul might have a better way of thinking about this!

Brian_TomasikApr 11 20151

If you buy index ETFs instead of stocks, the chance of a 20% loss is much smaller. (There have been few 20%-per-year losses in stock-market history.)

I wrote a simulation of 2X margin investing. I'll be writing up a description of it more formally once it's debugged and tweaked, but here are the current results for the present value of accumulated savings in US$ assuming an investor contributes $30K/year to buy an S&P 500 ETF at 2X margin over 30 years:

Mean regular = 937563

Mean margin = 1060336

0th percentile regular = 194161

0th percentile margin = 185719

10th percentile regular = 524319

10th percentile margin = 563046

25th percentile regular = 527721

25th percentile margin = 566873

50th percentile regular = 876835

50th percentile margin = 965868

75th percentile regular = 1212209

75th percentile margin = 1366938

90th percentile regular = 2042918

90th percentile margin = 2504675

100th percentile regular = 2042918

100th percentile margin = 2504675

The expected returns from leverage are much less than double ordinary returns but are still nontrivial.

Note that since I haven't fully tested the program, these results may not be correct. :)

EDIT: A draft of the write-up about these resutls now here.

Benjamin_ToddMay 1 20151

Very interesting, thank you.

Impressive you only end up with 10% more over 30 years! Not much gain for far more risk. Explains why so few people invest with that much leverage.

Just eyeballing the data, -20% annual returns seem to occur about once every 20 years.

What's the frequency of peak-trough losses of -20% though? That's what actually matters for getting wiped out.

(And does your analysis take account of those? You'd need to be using daily data rather than annual to pick them up).

It also seems like ex ante returns should be lower than historical returns, because the last 50yr or so has in the US been unusually good for equities, and there are various reliable indicators that predict lower returns (e.g. Shiller PE).

Brian_TomasikMay 20 20150

Thanks for those caveats. :)

The full write-up is now available here. Comments are welcome. The numbers have changed somewhat since before, and I trust them more now because they generally agree with theory.

I used 5.6% as the ex ante annual expected return and included black swans in my simulation. The simulation uses daily returns.

LarksMay 2 20150

I'm not sure what p is

Over what period are you measuring these drawdowns? I can look it up for you.

Benjamin_ToddApr 8 20151

Rather than just buying out of the money call options, you could make a synthetic long position. http://www.theoptionsguide.com/synthetic-long-stock.aspx

With the margin requirements for a typical broker like Interactive Brokers, you can achieve about 10x leverage like that.

With margin borrowing, you'll only be able to get up to about 1.5x. Moreover, since you're not buying OTM calls (you're buying them at the money), you avoid the problem that OTM options seem to have low expected returns.

Futures are similar - you can get about 10x leverage. On a liquid index I'd expect you'd be able to buy one year futures, so you'd only need to trade your position once every 6-12 months.

Brian_TomasikApr 10 20151

Thanks!

With margin borrowing, you'll only be able to get up to about 1.5x.

Why not 2X? Ordinary margin accounts have an initial margin requirement of only 50%.

I'm wary of options because

the OTM pricing problems make me at least nervous that the same could be true to a lesser degree for ATM options
options need to be sold or exercised, whereas stocks bought on margin could, if prices don't drop substantially, be held for many years until you donate them, so that you never pay capital-gains tax (and can possibly deduct margin interest).

Futures are similar - you can get about 10x leverage. On a liquid index I'd expect you'd be able to buy one year futures, so you'd only need to trade your position once every 6-12 months.

But you'd have to watch them closely, because if the index declined 10%, you'd have to sell your whole account.

Do you have thoughts on more than 2:1 leverage? Does the implied interest rate change with higher leverage ratios?

Benjamin_ToddMay 1 20150

Sorry, my mistake, I mean about 2x.

My understanding is that the same isn't true of OTM options. Also, you know your costs up-front as Paul explains below. Another benefit is that spreads are much smaller.

On more than 2:1 leverage unfortunately I haven't done the analysis. I'd start by looking at 2yr option prices on interactive brokers and working out the implied costs to construct a synthetic long.

Peter McIntyreApr 2 20151

If I remember correctly, CEA et al. decided against pursuing this strategy due to risk adversity. Due to the large downsides which may be unique to EA, it's not clear - to me at least - that our personal strategy should differ from this. I'd be interested in seeing some more thoughts on this.

Brian_TomasikApr 2 20152

I agree the situation would be different for a single small organization or if the charity you're donating to depends sensitively on your donations.

But if you're just an individual earning to give to relatively big charities (e.g., MIRI, which has a budget >$1m/year), then if you lose, say, ~$20K due to leverage, you can just make it up again with another ~2-3 months of work, and no major harm is done.

Paul_ChristianoApr 4 20150

Could you just use normal calls instead? Calls on the S&P500 a year out, at the current price, are on the order of $100, which seems like enough to get plenty leverage for most of us. If you think that the market is too afraid of the downside, then you can sell the corresponding put.

Benjamin_ToddApr 8 20152

I thought there was some evidence that out of the money call options tend to underperform on average. It's covered in "expected returns" by Ilmanen. He thinks it's due to lottery-ticket biases, causing OTM calls to be systematically overvalued.

LarksApr 9 20151

That would be suprising as many income strategies (invest in high dividend yield stocks) run covered call strategies (effectively selling out-of-the-money calls) because their investors prefer downside protection. These strategies do this in a largely price-inelastic manner so I would expect their to be positive expected value in buying these options.

I'll re-read Ilmanen on the subject when I get a chance.

Benjamin_ToddMay 1 20150

Selling calls benefits from OTM calls being overvalued. I'm talking about going long OTM calls. Am I missing something?

LarksMay 2 20150

Right, I'm saying there is a conflict between these two facts that I don't know how to resolve.

Benjamin_ToddMay 2 20150

Ah, you're saying that because a lot of funds sell OTM calls no matter the price, you'd expect the returns to be positive.

I think the explanation is just that there's an even larger group of people who buy OTM calls due to lottery-ticket biases, and this effect wins.

LarksMay 4 20150

Hmm, maybe. Do you know who does this? Is this retail investors?

Brian_TomasikApr 10 20150

Brian_TomasikApr 4 20150

Buying calls is what this book advises, but I'm very skeptical because of all the studies suggesting that call-option returns deviate heavily from theoretical predictions. Maybe it's different for one-year calls than shorter-term calls, but I'm still nervous. Glad to hear your thoughts.

Paul_ChristianoApr 4 20152

They deviate heavily in the thinly-traded, far-out-of-the-money regime. I don't see such evidence in the normal regime. I don't expect big divergences because they would lead to arbitrage opportunities. And what's more, if there is such a disparity, then you can personally make a lot of money from it.

(They are worse 1 year out than over shorter time periods.)

The easy way to verify this is just buying a call at $X and selling a put at $X. You will lose 1-2% of the value of the underlying asset (if you go a year out on the S&P500; the loss is mostly in the dividends you won't receive), and then receive a payoff equal to the change in that asset's price. So you are basically borrowing at 1-2% interest. If you are more careful about the analysis and the procedure, you can get quite close to market interest rates.

This loss is the sticker price, it doesn't depend on any not-immediately-verifiable claims about option pricing.

(I don't recommend investing in the S&P 500).

Brian_TomasikApr 4 20150

They deviate heavily in the thinly-traded, far-out-of-the-money regime. I don't see such evidence in the normal regime.

Do you mean in my "Do Call Options Have High Expected Returns?" piece or elsewhere?

(They are worse 1 year out than over shorter time periods.)

What are worse? Theoretical predictions?

I don't recommend investing in the S&P 500

Why not? Shouldn't all capital markets have about equal expected returns? Or do you mean that some markets have higher returns due to higher systemic risk?

Paul_ChristianoApr 5 20151

Do you mean in my "Do Call Options Have High Expected Returns?" piece or elsewhere?

I haven't seen such evidence anywhere.

Shouldn't all capital markets have about equal expected returns? Or do you mean that some markets have higher returns due to higher systemic risk?

On the efficient market story it doesn't matter for returns whether you invest in the US or abroad. But (1) the non-capital investments of philanthropists with your values are particularly tied up in the US, and so if it make no difference it seems bad to take on the extra correlation, (2) prices really do look high right now in the US, and there are plausible stories about how that could happen in the existing financial system, so even if you only assign those stories a modest probability, it seems worth moving in that direction.

LarksApr 9 20151

prices really do look high right now in the US, and there are plausible stories about how that could happen in the existing financial system

Do you have a short summary? I could probably fill in the details from even a one line description. I know of several such stories but would be interested in hearing which you find plausible. Certainly there are also several popular but false ones (e.g. Schiller PE)

Paul_ChristianoApr 20 20151

My main reason for pessimism is the comparison between US equities and international equities; I guess that forward P/E's are higher in the US than elsewhere. This is largely based on the high Schiller PE / current profit margins in the US though (along with comparable P/E and high P/B ratios), so it would be good to know if you think this is a bad basis for extrapolation.

The "plausible stories" I was referring to were about how mispricings could persist. My understanding is that many investors' allocations between US and international equities isn't very flexible. Such investors could make up a large enough majority and short-selling could be unattractive enough that a moderate mispricing could persist. There are also principal-agent problems related to benchmarking, and well-documented market optimism about continuing growth vs. regression to the mean, that seem to point in the same direction.

But I haven't thought about this angle very much either, so it would be good to know if you think these mispricings would get fixed.

Brian_TomasikMay 22 20150

My understanding is that many investors' allocations between US and international equities isn't very flexible

Do you know why not? I moved my 401k to all international equities, and I assume many retirement investors have this option.

That said, it does seem that US investors don't invest enough internationally: 27% of US mutual-fund equity is international, while international equity accounts for 51% of the total market.

LarksApr 25 20150

The US market is slightly more expensive on a forward PE basis. However, Schiller PE is nonsense. For example, buybacks have become much more commonplace since the early 1980s, which increase the secular EPS growth rate (by reducing the dividend yield). The schiller PE does not adjust for this however; it assumes EPS will revert to their previous level, rather than profits reverting to their previous level. It also ignores the effects of dilution. Many companies issued a lot of equity during the crisis (especially banks). These companies now have much lower EPS as a result, even if profits returned, yet schiller PE implicitly assumes that their EPS will magically revert to their previous level. Hopefully this is clear; if not I can explain when we skype.

(There are several other problems with schiller PE).

Yes many investors can't re-allocate between markets but there are some whose entire job is this. I'm not sure about the end result of this.

I agree with your other points.

Benjamin_ToddMay 1 20150

Larks, could you explain a bit more?

1) If the EPS growth rate is higher but dividend yield has been lowered a corresponding amount, then aren't expected returns unchanged?

2) If you make the adjustment you propose, is that enough to show normal valuations? My understanding is that according to Shiller PE valuations are about 2x historical norms.

3) Also, there's many other alternative valuation models that currently give similar results to Shiller PE e.g. P/R, Tobins Q ratio, P/E with normalized profit margins.

4) These models are all correlated ~0.8 with 10yr returns, so you'd need to think something pretty substantial had changed for them to break down. [1yr forward P/E, by contrast, has much less correlation with long-run returns]

LarksMay 2 20151

1) Yes but the Shiller PE will be higher, thereby (incorrectly) reducing its estimate of forward returns.

2) No, I agree the market is at above average valuation, just less extremely so than shiller PE would suggest. Though it looks cheap on Equity Risk Premium measures.

3) Yeah my objection is to the methodology not the conclusion. However I think many of those other methodologies are silly as well; for example, P/R does not make sense from an accounting standpoint (it should be EV/R). Changes in the structure of the economy have made book value metrics less relevant than historically, but I agree they are somewhat concerning.

4) Looking at historical correlations with returns introduces lookahead bias. If the market valuation doubled from here, and then remained flat for 100 years while earnings grew at their historic rate, schiller PE would remain correlated with returns, except it would retrospectively advise us to buy here.

Also, I'm having trouble replicating your numbers. Using the schiller data, I get correlations of -0.54 for CAPE vs 10yr return (real or nominal), and -0.49, -0.47 for PE vs 10yr return (real and nominal respectively). This is a small enough difference that we should prefer to use the more theoretically justified measure.

Also forward earnings estimates are not available that far back so I am sceptical of any research into their ability to forecast long-run returns! For 1-year holding periods they do about as well as trailing earnings though.

However, if we use a lookahead bias free measure, and instead of using the absolute level of PE / CAPE, we instead use the percentile of that metric, relative to its own history, the results basically reverse. Using this better measure I get -0.54 and -0.51 for PE vs -0.48 and -0.43 for CAPE. (in both cases I started the correlation in 1900 so we had a few decades of data for the percentiles to stabilize in.)

I will send you the excel spreadsheet.

Benjamin_ToddMay 2 20150

Also see this for a summary of the 'risk-aversion' measure:

http://www.hussmanfunds.com/wmc/wmc150413.htm

LarksMay 7 20150

Interesting. My worry with credit-spread metrics is no-one cared about them pre-2008, so their performance is all in-sample basically. People always add new explanatory variables that explain the last crisis. However I am not an expert on this.

Benjamin_ToddMay 2 20150

Very interesting. For the correlations I was just going with Hussman's analysis. I know it's simple, but it's the best I'm aware of.

This post has a summary of a couple of methods which he says have 84% correlation with long-run returns, and currently predict returns of just a couple of percent for the next decade.

http://www.hussmanfunds.com/wmc/wmc130318.htm

Why do these disagree with your figures?

Paul_ChristianoMay 2 20151

Philosophical economics has a some critical discussion of this and similar graphs, e.g. here and more directly here. There is also a lot of discussion of current elevation of profit margins and CAPE, which I found useful.

Benjamin_ToddMay 3 20150

Thanks, that looks like a great blog.

LarksMay 7 20150

Why do these disagree with your figures?

Not sure. Did you get the excel sheet I sent?

Benjamin_ToddMay 1 20150

Larks, what do you think of Hussman's analysis? He combines valuation with a model of risk aversion, based on credit spreads, market breadth, and bull-bear ratio.

Who do you think publishes the best analysis of expected stock returns?

LarksMay 2 20150

I don't know of anyone who does it well. I intend to work on it myself once I'm done with CFA.