People are worried about the VIX

"Today the VIX traded below 10 briefly intraday. A pretty rare occurrence. Since 1993, there have been only 18 days where it traded below 10 intraday and only 9 days where it closed below 10." (source: some random dude on my linkedin feed)

... indeed 18 observations is a long.... long... way from anything close to a statistically significant sample size. (my response to random dude)

You can't move on the internet these days for scare stories about the incredibly low level of the VIX, a measure of US implied stock market volatility. Notably the VIX closed below 10 on a couple of days last week, although it has since slightly ticked up. Levels of the VIX this low are very rare - they've only happened on 11 days since 1990 (as of the date I'm writing this).

The VIX in all it's glory

The message is that we should be very worried about this. The logic is simple - "Calm before a storm". Low levels of the VIX seem to presage scary stuff happening in the near future. Really low levels, then, must mean a very bad storm indeed.

Consider for example the VIX in early 2007:

Pootling around at 10 in late 2006, early 2007, the VIX responded to the failure of two Bear Stearns hedge funds which (as we know now) marked the beginning of the credit crunch. 18 months later there was a full blown panic happening.

This happened then, therefore it will happen again.

It struck me that this story is an example of what behavioural finance type people call narrative bias; the tendency of human beings to extrapolate single events into a pattern. But we need to use some actual statistics to see if we can really extend this anecdotal evidence into a full blown forecasting rule.

There has been some sensible attempt to properly quantify how worried we should be, most notably here on the FT alphaville site, but I thought it worth doing my own little analysis on the subject. Spoiler alert for the terminally lazy: there is probably nothing to be worried about. If you're going to read the rest of the post then along the way you'll also learn a little about judging uncertainty when forecasting, the effect of current vol on future price movements, and predicting volatility generally.

(Note: Explanations for the low level of the VIX abound, and self appointed finance "experts" can be found pontificating on this subject. It's also puzzling how the VIX is so low, when apparently serious sized traders are buying options on it in bucket load sized units (this guy thinks he knows why). I won't be dealing with this conundrum here. I'm only concerned about making money. To make money we just need to judge if the level of the VIX really has any predictive power. We probably don't need to know why the VIX is low.)

Does the level of VIX predict stock prices?

If this was an educational piece I'd work up to this conclusion gradually, but as it's clickbait I'll deal with the question everyone wants to know first (fully aware that most people will then stop reading).

This graph shows the distribution of rolling 20 business day (about one month) US stock returns since 1997:

(To be precise it's the return of the S&P 500 futures contract since I happened to have that lying around; strictly speaking you'd add LIBOR to these. The S&P data goes back to 1997. I've also done this analysis with actual US stock monthly returns going back to 1990. The results are the same - I'm only using the futures here as I have daily returns which makes for nicer, more granular, plots.) 

Important point here: this is an unconditional plot. It tells us how (un)predictable one month stock returns are in the absence of any conditioning information. Now let's add some conditioning information - the level of spot VIX:

I've split history in half - times when VIX was low (below 19.44%) shown in red, and when it was high (above 19.44%), which are in blue (overlaps are in purple). Things I notice about this plot are:

• The average return doesn't seem to be any different between the two periods of history
• The blue distribution is wider than the red one. In other words if spot VIX is high, then returns are likely to be more volatile. Really this is just telling us that implied vol (what the VIX is measuring) is a pretty good predictor of realised vol (what actually happens). I'll talk more about predicting vol, rather than the direction of returns, later in the post.
• Digging in a bit more it looks like there are more bad returns in the blue period (negative skew to use the jargon)

The upshot of the first bullet point is that spot VIX doesn't predict future equity returns very well. In fact the average monthly return is 0.22% when vol is low, and 0.38% when vol is high; a difference of 0.16% a month. That doesn't seem like a big difference - and it's hard to see from the plot - but can we test that properly?

Yes we can. This plot shows the distribution of the differences in averages:

This was produced by monte carlo: repeatedly comparing the difference between random independent draws from the two distributions. This is better than using something like a 't-test' which assumes a certain distribution.

A negative number here means that high VIX gives a higher return than low VIX. We already know this is true, but the distribution plot shows us that this difference is actually reasonably significant. In fact 94.4% of the differences above are below zero. That isn't quite at the 95% level that many statisticians use for significance testing, but it's close.

To put it another way we can be 94.4% confident that the expected return for a low VIX (below 20%) environment will be lower than that for days when VIX is high (above 20%).

A moments thought shows it would be surprising if we got a different result. In finance we expect that with a higher return you will get higher risk. We know that when VIX is high that returns will have a higher volatility. So it's not shocking that they also have higher risk.

So a better way of testing this is to use risk adjusted returns. This isn't the place to debate the best way of risk adjusting returns, I'm going to use the Sharpe Ratio and that is that. Here I define the Sharpe as the 20 business day return divided by the volatility of that return, and then annualised.

(You can see now why using the futures contract is better, because to calculate Sharpe Ratios I don't need to deduct the risk free rate)

Now we've adjusted for risk there is little to choose between the high VIX and low VIX environments. In fact things have reversed, with low VIX having a higher Sharpe Ratio than high VIX. But the difference in Sharpes is just 0.04, which isn't very much.

We can only be 63% confident that low VIX is better than high VIX. This is little better than chance, which would be 50% confidence.

An important point: notice that although the difference in Sharpes isn't significant, we do know it with reasonably high confidence, as each bucket of observations (high or low VIX) is quite large. We can be almost 100% confident that the difference was somewhere between -0.04 and +0.04.

"Hang on a minute!", I hear you cry. The point now is that vol is really really low now. The analysis above is for VIX above and below 20%. You want to know what happens to stock returns when VIX is incredibly low - below 10%.

The conditional Sharpe Ratio for VIX below 10 is actually negative (-0.14) versus the positive Sharpe we get the rest of the time (0.14). Do we have a newspaper story here?

Here is the plot of Sharpe Ratios for very low VIX below 10% (red), and the rest of the time (blue):

But hang on, where are the red bars in the plot? Well remember there are only a tiny number of observations where we see vol below 10. You can just about make them out at the bottom of the plot. In statistics when we have a small number of observations we can also be much less certain about any inference we can draw from them.

Here for example is the plot of the difference between the Sharpe Ratio of returns for very low VIX and 'normal' VIX.

Notice that the amount of uncertainty about the size of the difference is substantial. Earlier it was between -0.04 and 0.04, now it's between -1 and 0.5; a much larger range. To reiterate this is because one of the samples we're using to calculate the expected difference in Sharpe Ratios is very small indeed. It does look however as if there is a reasonable chance that returns are lower when VIX is low; we can be 86% confident that this is the case.

Perhaps we should do a "proper" quant investingation, and take the top and bottom 10% of VIX observations, plus the middle, and compare and contrast.That way we can get some more data. After all although statistics can allow us to make inferences from tiny sample sizes (like the 11 days the VIX closed below 10), it doesn't mean we should.

The big blue area is obviously the middle of the VIX distribution; whilst the purple (actually red on blue) is relatively low VIX, and the green is relatively high VIX.

It's not obvious from the plot but there is actually a nice pattern here. When the VIX is very low the average SR is 0.071; when it's in the middle the SR is 0.139, and when it's really high the SR is 0.20. 

Comparing these numbers the differences are actually highly significant (99.3% chance mid VIX is better than low VIX, 98.4% chance high VIX is better than mid VIX, and 99.999% chance high VIX is better than low VIX).

So it looks like there might be something here - an inverse relationship between VIX and future equity returns. However to be clear you should still expect to make money owning S&P 500 when the VIX is relatively low - just a little bit less money than normal. Buying equities when the VIX is above 30 also looks like a good strategy. It will be interesting to see if market talking heads start pontificating on that idea when, at some point, the VIX gets back to that level.

"Hang on another minute!!", I hear you unoriginally cry, again. The original story I told at the top of this post was about VIX spiking in February 2007, and the stock market reacting about 18 months later. Perhaps 20 business days is just too short a period to pick up the effect we're expecting. Let's use a year instead.

The results here are more interesting. The best time to invest is when VIX is very high (average SR in the subsequent year, 1.94). So the 'buy when everyone else is terrified' mantra is true. But the second best time to invest is when VIX is relatively low! (average SR 1.14). These are both higher Sharpes than what you get when the VIX is just middling (around 0.94). Again these are also statistically significant differences (low VIX versus average VIX is 97% confidence, the other pairs of tests are >99%).

I could play with permutations of these figures all day, and I'd be rightly accused of data mining. So let me summarise. Buying when the VIX is really high (say above 30) will probably result in you doing well, but you'll need nerves of steel to do it. Buying when the VIX is really low (say less than 15) might give you results that are a little worse than usual, or they might not.

However there is nothing special about the VIX being below 10. We just can't extrapolate from the tiny number of times it has happened and say anything concrete.

Does the level of VIX predict vol?

Whilst the VIX isn't that great for predicting the direction of equity markets, I noted in passing above that it looks like it's pretty good at predicting their future volatility. 

We're still conditioning on low, middling, and high VIX here but the response variable is the annualised level of volatility over the subsequent 20 days. You can see that most of the red (turning purple) low VIX observations are on the left hand side of the plot - low VIX means vol will continue to be low. The green (high VIX) observations are spread out over a wider area, but they extend over to the far right.

Summarising:

Low VIX (below 12.5): Average subsequent vol 8%

Medium VIX: Average subsequent vol 12.3%

High VIX: Average subsequent vol 21.9%

These numbers are massively statistically significant from each other (above 99.99%). I get similar numbers for trying to predict one year volatility. 

So it looks like the current low level of VIX means that prices probably won't move very much. 

Does the level of vol predict vol?

The VIX is a forward looking measure of future volatility, and it turns out a pretty good one. However there is an even simpler predictor of future vol, and that is recent vol. The level of the VIX, and the level of recent volatility, are very similar - their correlation is around 0.77.

Skipping to the figures, how well does recent vol (over the last 20 days) predict subsequent vol (over the next 20 days)?

Recent Vol less than 6.7%: Average subsequent vol 7.9%

Recent Vol between 6.7% and 21.7%: Average subsequent vol 13.2%

Recent Vol over 21.7%: Average subsequent vol 23.4%

These are also hugely significant differences (>99.99% probability). 

The best way of predicting volatility

Interestingly if you use the VIX to try and predict what the VIX will be in one months time you find it is also very good. Basically both recent vol and implied vol (as measured by the VIX) cluster - high values tend to follow high values, and vice versa. Over the longer run vol tends not to stay high, but will mean revert to more average levels - and this applies to both implied vol (so the VIX) and realised vol.

So a complete model for forecasting future volatility should include the following:

1. recent vol (+ve effect)
2. current implied vol (the VIX) (+ve)
3. recent vol relative to long run average (-ve)
4. recent level of spot VIX relative to long run average
5. (You can chuck in intraday returns and option smile if you have time on your hands)

However there is decreasing benefit from including each of these things. Recent vol does a great job of telling you what vol is probably going to be in the near future. Including the current level of the VIX improves your predictive power, but not very much.

Summary

The importance of the VIX to future equity returns is somewhat overblown. It's just plain silly to say we can forecast anything from something that's only happened on a handful of occasions in the past (granted that the handful in question belongs to someone with 11 fingers). Low VIX might be a signal that returns will be a little lower than average in the short term, but by no means is inevitable impending doom fast approaching. 

If there is a consistent lesson here it's that very high levels of VIX are a great buy signal. 

The VIX is also helpful for predicting future volatility - but if you have room in your life for just one forecasting rule using recent realised vol is better.

Can you eat geometric returns?

This post is about a slightly obscure, but very important, issue. Should we use geometric or arithmetic means of returns to evaluate investments?

This might seem boring, but answering this will help us with some other serious problems: Does diversification increase the expected value of your portfolio or just reduce the volatility? If so can we then afford to pay extra costs to get diversification? Does adding a small amount of bonds to an all equities portfolio increase your likely returns?

It turns out that the answer to this boils down to one of the most fundamental questions in financial economics: How should we evaluate the expected value of possible outcomes? 

A brief introduction to geometric returns

When thinking about past and future returns I'm going to be using geometric means rather than the more common arithmetic means. Geometric means reflect what you will actually earn over time.
To understand this better let's look at an example. Consider an investment in which you invest $100 and earn  30%, 30% and -30% over the next three years of returns. The arithmetic mean of returns is the sum of annual returns, 30% + 30% - 30% = 30%, divided by the number of years (3), which equals 10%. You might expect to have an extra $30 after three years: probably more with the magic of compound interest.

End of year 1: $100 + 30% * $100 = $130
End of year 2: $130 + 30% * $130 = $169
End of year 3: $169 - 30% * $169 = $118.30

Whoops. Compound interest is a wonderful thing but it magnifies losses as well as gains. Now suppose you'd made a pathetic 5.76% a year but consistently:

End of year 1: $100    + 5.76% * $100      = $105.76
End of year 2: $105.76 + 5.76% * $105.76   = $111.85
End of year 3: $111.85 + 5.76% * $111.85   = $118.30

Notice that the annual return here is much lower, just 5.76% a year, but it's consistent. The final account value after three years is exactly the same as in table 1: $118.30.  The geometric mean of a series of returns is the consistent return that gives the correct final account value. So the geometric mean of 30%, 30% and -30% is 5.76% a year.

The bit of the post where I put an obligatory equation or two

Mathematically the geometric mean is  [n√(1+ r1)(1+ r2)....(1+ rT)]-1 where rt are each of T returns. Alternatively it's exp[(1 ÷ N) Σln(1+rt))] – 1 where ln is the natural log, and exp is the exponent function.

Notice that the geometric mean is a concave function of the final value of the portfolio (1+ r1)(1+ r2)....(1+ rT). This is an important point which I'll return to later.

Some irony

I am unlikely cheerleader for geometric returns... until a couple of years ago I'd never actually used them! That's because in the hedge fund world where we rebalance to target expected risk on constant capital it's better to use non compounded curves (See this post for more). 

It's not necessary to use geometric returns since there no compounding and the volatility of different options is identical (expected risk on target), rather you can use arithmetic returns and make your life easier (you can also focus entirely on Sharpe Ratios, since you effectively have as much leverage as you need to maximise returns for a given level of risk target).

Some interesting properties of geometric returns

Geometric returns give a more realistic picture than mean returns. To take an extreme example consider the following series of returns: 100%, 100%, -100%. The arithmetic return is 100%: What a fantastic investment! But the geometric mean is easy to calculate: 0%. You will have nothing left after three years have passed.

Geometric means are always lower than the arithmetic mean, unless all annual returns are identical. The difference between the two measures is larger for more volatile assets.  In fact we can see this easily with the following which is a nice approximation for geometric means:

μg = μa – 0.5 σ_2 

...where μg is the geometric mean, μa is the arithmetic mean and σ_2 is the variance of returns. In other words the geometric mean is the arithmetic mean, less a correction for risk.

(This can be proven using Jensens Inequality which I'll return to later)

It's worth emphasising this: the benefits of diversification are greater when average returns are measured with geometric means. 

The consequences of using geometric returns

1) Diversification improves returns: so we can afford to pay for it

Geometric returns get higher as risk falls, something that never happens with arithmetic returns.

Take a group of similar assets, like equities in the same country and sector. It's not unreasonable to assume they have equal arithmetic returns, equal standard deviations (and thus equal Sharpe Ratios - and equal geometric means) and identical correlations. The optimal portfolio here has equal weights and as many assets as possible. But adding these assets doesn't affect the arithmetic mean, which is unchanged. It reduces Sharpe Ratio, rapidly. But is also improves the geometric mean, a little more gradually.

For example, assuming correlation of 0.85:

1 asset: arithmetic mean 5%, geometric mean 1.3%
5 assets: arithmetic mean 5%, geometric mean 1.8% 

We can also use geometric returns to "pay" for higher diversification costs. If you can get an extra 0.5% in geometric returns then you can pay 0.4% more in costs and still be ahead.

1 asset: arithmetic mean 5%, geometric mean 1.3%
5 assets: arithmetic mean 5% - 0.4% = 4.6%,  geometric mean 1.4% 

This explains the weird title of the post: we can "eat" higher geometric returns, or use them to pay higher costs.

2) 100% equity portfolios are bad even if you don't maximise Sharpe Ratio

Also using geometric returns is part of the way past the classic portfolio optimisation quandry: should we opt for a portfolio with higher return (more equities), or lower risk (more bonds)? The maximum Sharpe Ratio portfolio is just one possible compromise between these two options. But for those with a higher tolerance for risk it is inferior to options with more return.

The maximum geometric mean portfolio is interesting. It's the portfolio for which there is no point increasing risk further, even if doing so gives you a higher arithmetic return.

So an interesting implication is that adding a small amount of bonds to an all equities portfolio will increase geometric return: or to put it another way all investors should own some bonds.

For example given the following properties:

Bonds: arith. mean 1.6%, standard deviation 8.3%,  geo. mean 1.3% Geometric Sharpe Ratio 0.15
Equities: arith. mean 5%, standard deviation 19.8%,  geo. mean 3% Geometric Sharpe Ratio 0.15 

(these are derived from 100+ years of US real returns, adjusted to reflect more realistic forward expectations and to equalise geometric Sharpe Ratio)

A portfolio with 20% in bonds will have the following properties:

80:20 portfolio: standard deviation 16%,  geo. mean 3% Geometric Sharpe Ratio 0.188

Adding a few bonds to all equities portfolio has left geometric return the same. In fact the maximum geometric mean occurs at roughly 10% of the portfolio. Only once we add more than 20% of bonds to the portfolio does the geometric mean fall below the all equity portfolio.

Geometric mean (y axis) as bonds added to an all equity portfolio (x axis: 0= no bonds, 1.0 = 100% bonds)

The bear case for using geometric returns

The above findings are, relatively startling. So it's important that geometric return is in fact "real". But there is some debate about this. I was prompted to write this post after being shown this paper:

http://www.bfjlaward.com/pdf/25968/65-76_Chambers_JPM_0719.pdf

...h/t to Daal on elitetrader.com.

It's a very involved paper which also covers "rebalancing return" but here are the key points in relation to diversification:

• A key misconception concerning the expected geometric mean return is that it provides an accurate indication of long-term expected future wealth.
• Another potential misconception regarding geometric mean returns is that maximization of a portfolio’s expected geometric mean return is an optimal portfolio strategy
• An asset’s expected geometric mean return (i.e., the expected compounded rate of return) is the probability-weighted average of all of the potential realized geometric mean returns. 
•  ...volatility does not diminish expected value.

Fighting talk! Let's get to the core of the matter. The problem is thus:

• geometric mean scales (concavely) with final portfolio value.
• BUT expected geometric mean does not scale with expected portfolio value
• Therefore maximising expected geometric mean might not maximise expected portfolio value
• Instead maximising expected arithmetic mean will maximise portfolio value

If you're scratching your head right now, I don't blame you. To illustrate more clearly what is going on I produced some random data. The data is multiple random Gaussian returns of two sets of 10 years of daily returns, with the following properties:

High AM: Arith. mean 5%,     standard deviation 15%
Low AM:  Arith. mean 4.375%, standard deviation 10%

These apparently arbitrary values have been chosen so that both assets have the same geometric mean. 

Now I'm going to plot the distribution of statistical estimates from this little Monte Carlo exercise (with 500,000 runs; to ensure reasonably smooth results)

First the distribution of arithmetic means:

(The distribution of expected arithmetic means is Gaussian with lower standard deviation for more volatile assets)

The mean of the distribution (i.e. the expected arithmetic mean returns) are: 4.99% (High AM) and 4.38% (Low AM). This verifies that the Monte Carlo hasn't done something weird.

Now geometric means

(Again the distribution of expected geometric means is Gaussian with lower standard deviation for more volatile assets)

You can see that both assets have the same expected geometric mean, though there is more uncertainty about the estimate for the higher volatility "High AM" asset.

Finally let's have a look at the distribution of final portfolio values:

(A final value of 1.0 indicates the portfolio hasn't grown, 2 means 100% growth over 10 years and so on)

Now this distribution is more interesting. Even if you squint really hard it isn't Gaussian - it's a skewed lognormal. The bunching of values on the left hand side is happening when a lot of losses occur in a row. Because we're not using leverage the portfolio value can't go below zero; hence we get bunching.

The means of the distribution are: 1.65 (High AM) and 1.55 (Low AM). These are the key numbers. Although both assets have the same geometric mean the final portfolio value is larger for the asset with a higher arithmetic mean.

Here is clear evidence that the highest expected final value comes when the arithmetic mean is higher even with higher volatility. From the paper:

• Another potential misconception regarding geometric mean returns is that maximization of a portfolio’s expected geometric mean return is an optimal portfolio strategy

It certainly looks like the geometric mean is in trouble.

The case for the defence - a question of measuring expectations

I'm going to home in on one particular implication of the paper cited above:

• The expected final value of the portfolio  is the probability-weighted average of all of the possible portfolio final values. 

Expectation:  A word we use a lot in economics and finance without a pause. What does it mean? And also, and very importantly, which average?

When I was at school we learned about three: the mean, the median, and the mode (which I won't be using here). Remember from the figure above that the distribution of realised final values is right skewed. Hence the mean will be greater than the median. So the choice of average matters a lot.

The paper assumes that the probability weighted average of all potential portfolio values is the mean of the distribution of possible portfolio values.

To be clear then:

• The expected final value of the portfolio  is the probability-weighted average of all of the potential realized portfolio final values. In the paper this is the mean of the distribution of possible portfolio values.

The case against the geometric mean relies heavily on using the mean, not the median, to summarise the distribution of final portfolio values. All this matters because we're dealing with a distribution where the mean and the median are significantly different.

The case for using the median not the mean

Is the mean really appropriate? Personally I would say no, for two reasons:

1. Risk neutral behaviour doesn't really exist
2. The median is closer to how humans form expectations

Does risk neutral behaviour exist?

Using the mean makes sense for risk neutral investors. Let's take a simple and rather extreme example. Suppose your entire wealth is £100,000. I offer you the chance to buy a lottery ticket for £100,000, which will pay you £100 million, at odds of 999 to one. The expected value of the ticket using the weighted average mean of the outcomes is £100,100. For an economist this bet is worth taking!

The two options are:

• Don't buy the ticket. Mean of future wealth: £100,000. Median of future wealth: £100,000
• Buy the ticket. Mean of future wealth: £100,100. Median of future wealth: £0

Taking the arithmetic mean of the distribution leads us to prefer an outcome that is completely and utterly insane.  No human being would ever gamble everything they have for such a tiny expected average increase in wealth. People only take on those kinds of gambles for relatively small fractions of their wealth (so yes they do buy lottery tickets, but not for £100,000).

"Real" people require paying to take on risk: as economists like to think of it they will require the probability weighted average less a correction for risk. Classical economics tells us that people exist on a continuum:

• Risk averse who require paying more than the risk neutral mean of outcomes to take on more risk (they would want to buy the lottery ticket for less than £100,000)
• Risk neutral investors who use the weighted average mean to evaluate options
• Risk lovers who are happy to pay to take on more risk (they would happily pay more than £100,000 for the ticket)

There are definitely risk averse people: I am one myself. So if there are risk lovers then there must also be risk neutral investors; it makes no sense to have a continuum with a break in the middle.

Looking around it does look like some people seem to love risk to the point they'll happily pay for it: eg gambling in casinos when the odds are against them (which they nearly always are). If these weirdos exist it does seem more plausible that risk neutral investors also exist. However I would argue that true risk loving behaviour doesn't exist.

Instead this behaviour is a result of people misjudging probabilities due to cognitive biases in the way we think about risk. The cognitive science hadn't been incorporated into financial economics when the idea of the continuum above was proposed.

We know that people overestimate the likelihood of events with very small probabilities (which is one reason why people do buy lottery tickets which cost only a fraction of their expected value even though they always have a negative expected value, and buy insurance against terrorist attacks).

If you ask a desperate gambler who is about to put the last of their money into the slot machine if they expect to win on this spin their answer will be "of course"; probably because they suffer from gamblers fallacy and believe they are "due" a win.

Similarly the most aggressive investors invest in highly speculative portfolios of rubbish companies with almost no diversification; which on the face of it would only make sense if they were risk loving.

But I would argue - again - that is a failure of probability assessment. Yes the investors say we know that diversification is better, but we have skill and can pick the best stocks. They overestimate their probability of beating the market - all of them are above average - the Lake Wobegon effect.

Cognitive failure leading to probability mis-assessment is mistaken for risk loving behaviour by economists. This is wrong.

With no risk loving investors I also believe risk neutral behaviour is also a myth. In reality everyone requires some compensation for risk.

The median makes more sense to humans than the mean

As humans when we think about expectations it is the median that we are thinking about. If the weather forecast tomorrow is for a 10% chance of rain, and I ask someone what they expect the weather to be, they will say they expect it to be dry (the median outcome). They won't say they expect it to be a little bit wet (the mean outcome).

The £100K lottery may be an extreme example but as we've already seen future wealth is always fairly heavily right skewed; enough so that the difference between mean and median is pretty significant.

Cross sectional distribution of wealth and income in the real world is also famously right skewed. Would you want to live in a country where there is a tiny chance of being very wealthy, but you're most likely to be dirt poor? (Hint: net migration from the very equal Nordic states to much less equal America is almost zero).

Would you want to do a job where you have a miniscule chance of earning millions, but will probably barely earn a living wage? (Again a lot of kids - or their parents - want to be professional footballers, but this is a judgement error that comes from overestimating the probability that they personally will make it to the top leagues).

I think the correct way to evaluate future wealth outcomes is by using the median. To most people the idea of "I expect what will happen is what is likely to happen half the time" is a more natural concept of expectations than "probability weighted average mean".

In an ideal world you'd show people distributions and explain the uncertainty involved and then get an idea of their risk / reward payoff function. But short of that I'd say that even the least risk averse humans on the planet should use the median outcome when evaluating future investments. 

Implications for using the median rather than the mean

Returning to the plots above what figures do we get if we summarise the for the median rather than the mean? Remember that High AM has a higher arithmetic mean than Low AM, but both have the same geometric mean.

Arithmetic mean:

High AM: Mean 4.99 Median 5.0

Low AM: Mean 4.38   Median 4.38

Geometric mean:

High AM: Mean 4.06  Median 3.95

Low AM: Mean  3.95  Median 3.96

Future wealth:

High AM: Mean  1.647  Median 1.473

Low AM: Mean   1.549  Median 1.474

The important figures are those in bold. Ignoring slight differences in decimal points (because ultimately this is random data) it's clear that we can draw the following conclusion:

The expected value (using the median) of future wealth is identical when geometric returns are identical, even if the arithmetic mean is lower.

So maximising geometric mean will also maximise final portfolio value. In other words the implications of using geometric mean that I outlined above still hold:

• We can use diversification to pay for higher costs
• 100% equity portfolios are not as good as portfolios with some bonds

Summary

I think declaring the death of geometric returns is somewhat premature. It's true that using the classical economists view of expectation - the mean of the distribution of portfolio values - implies that final value isn't lowered by volatility. But this vanishes when you use the median of the distribution as your basis for expectation.

I feel personally that using the median, rather than the mean, is the correct approach. However this is an ideological debate - there is no right answer. Ultimately an economic model is a simplification of the vastly complicated reality of human behaviour.

Obligatory Brexit ex-post (Survival, Skew, Politics & Data, )

This is a two part post which blends my favourite subjects: finance, economics, and politics. It's about the stunning victory of the "leave" campaign that has left the UK facing a period of serious uncertainty.

Part one is an ex-post update on whether my portfolio was as robust to Brexit as I hoped it would be in my last post. This is based on market moves on Friday; of course things will continue to move as the full ramifications are digested.

Part two explains how Brexit was a classic negative skew event in more ways than one, how behavioural finance can help explain that, how it was mispriced by the markets, and how I used that information to make a little bit of money.

Say hello to our next Prime Minister. Source: The Sun

Did I survive Brexit?

Yes.

Let me expand.

Trading in GBPUSD

I made no money at all trading cable, the asset most exposed to Brexit. In fact I lost some. When I wrote my last post I had no position in GBPUSD IMM futures. However the swings in sterling sentiment were reflected in first a sell:

11242 GBP 201609 2016-06-15 02:38:16 -1 1.412600

And then I closed that as GBP went stronger in the post Jo-Cox* bounce in remains polling figures.

* It feels wrong to be talking about her tragic death in a grubby post which is mainly, at least at the moment, about money. However it seems likely that what happened to her was clearly linked to the referendum campaign. I had the privilege of briefly seeing her in person. She was on the select committee which I appeared before last year (16:20:00 on the timestamp of the video feed onwards), and I was very impressed with her performance. I'll be doing the politics of this later.

11296 GBP 201609 2016-06-17 18:54:08 1 1.434800

As with a typical trend following system in a range bound market I sold low and bought high. And a further opening buy almost exactly 24 hours before the result would become clear, as the world decided remain was a shoe in (more about this below).


11461 GBP 201609 2016-06-23 02:51:41 1 1.483000

Then, of course it all went wrong. The pound got smashed. With a reversing price and volatility going through the roof, the system closed out it's position.


11509 GBP 201609 2016-06-24 03:54:26 -1 1.366400

Although the price continued to collapse it to around 1.33 then recovered, and closed at pretty much the same level. As I'm writing this the pound is drifting lower, but I have no position to take advantage of this.

I lost the equivalent of nearly 2% of my portfolio on that one lousy GBP contract, about 1.25% of which was lost on the day.

The rest of my futures portfolio

Fortunately, as I said in my last post:

"It doesn't even look like a strong "risk on" or "risk off" theme, though I look to be a little bit more "risk off"."

Yeah, I got that right. I made it all back on the rest of my portfolio. In order of importantce: US bonds and Eurodollar, German bonds (makes sense), Korean bonds (who'd have though it - Korea a safe haven from the UK...), French bonds (the irony) and JPYUSD (though I gave that all back on AUDUSD).

I had no position in Gold, but you can't have everything.

I lost a bit on short positions in VIX and V2X.

Net-net on the main futures portfolio I made a little under 2.5%. Not as much as Crispin Odey is supposed to have made, but then I would imagine he would have hurt a lot more than me if the result had been as expected.

The point here isn't so much "ooh look I made money isn't my system clever" - it is pure luck how much anyone makes on a single day, but that my risk was well managed, such that a 6 sigma event in the markets translated into roughly a 2 sigma portfolio move based on my expected risk going into the vote.

Equities, currency and hedging

Again, from the post:

As I've said before my futures trading account is funded by a long stock position, which is in a mixture of UK stocks and a European ETF IDVY; hedged by a Eurostoxx futures short. This is a more interesting little portfolio; ignoring the effects on stocks if the pound gets hammered then in GBP terms this portfolio will go up in value - there will be a gain in the value of the ETF but the futures hedge won't move since it's unaffected by currency movements.

I was close here. The Eurostoxx did get hammered, more than the FTSE so I got that wrong. However it did indeed lose less in GBP terms because of the devaluation.  Also the UK shares I hold in this account didn't do half badly. Net-net the long stocks plus hedge added another 2% or so.

I also have a long only, buy and hold, portfolio of UK stocks and ETF's covering bonds and equities across the globe.

Here are some more quotes from my Brexit post:

The ETF's are not currency hedged, so a GBP devaluation following brexit would benefit them

This is exactly what happened. The only ETF in my main portfolio that fell in £ terms was a high yield Eurostoxx beast, which suffered from the huge fall in banking shares which wiped out the EURGBP revaluation benefit. All my other ETF's were up in £ terms.


The effect on stocks is a little tricker to predict, but I would imagine my UK stocks would hurt - perhaps 5% at worse? A huge one day drop which will probably be partly reversed, but not the end of the world.

The FTSE100 was down 3.2% at the close, the FTSE 250 7.2%. So on average 5% was a good guess. Some of my UK stocks did much better (Glaxo was up), others much worse (Intermediate Capital Group was done over 14%). 

Overall in my long only account I was flat.

This isn't genius trading, but at least if the book sales fall short then I could probably get a job as a risk analyst.

All of this is on a T+1 analysis. We don't really know how things will pan out over the next N years whilst this little mess gets sorted out.

Brexit as negative skew

Cameron, David

I tweeted a lot on Friday. Here is one of my first efforts:

https://twitter.com/investingidiocy/status/746240000073728000

"Cameron took on the classic short vol trade - high probability of success, small upside, huge downside. And lost. #brexit"

That is pretty self explanatory. Cameron acted like a muppet running a hedge fund.

"Muppet" is a term of abuse in the south east of England meaning "Idiot". I don't mean an actual Muppet. I have a lot of respect for Kermit. I think he'd make an excellent hedge fund manager. Source: https://autonomousmind.files.wordpress.com/

Your typical muppet HF manager will keep selling OTM options. They kept expiring without value, leaving him with the premium. He begins to confuse luck with skill; he thinks he was invincible. This is the classic behavioural finance effect of overconfidence. Then one day, he blows up.

Why was the referendum like an OTM option?

Firstly, it looked like a forgone conclusion. Promising a referendum was a cheap way to beat down the nascent UKIP vote in last years election (the vote still hit historic highs, but thanks to our bonkers electoral system they were unable to achieve concentrated support to win more than one seat in parliament). It would be so easy, just a half baked (and suspiciously quick) negotiation with the EU to get superficial concessions, hold the referendum, bish-bash-bosh with the handy side effect of destroying his main political rival (By which I mean Boris Johnson. Not the labour party).

Yes there were some tense moments, but he never looked ruffled, or looked like he ever thought he would lose. This looked like a high probability event, but the downside, which doesn't seem to have been properly considered, was enormous - both politically and personally.

Secondly, it had a small payoff if it succeeded. Without the referendum I think it is incredibly unlikely that Boris would be a serious contender for PM.

Thirdly it had a large cost if it went wrong. Both personally, politically and economically.

Overconfidence often seems to affect people from privileged backgrounds.

The whole country is paying for the fact that Cameron has had a smooth path through life with nothing going really wrong. There are many articles in the press pointing out how he's been lucky with several political gambles over the years. Even having to go into coalition with the Lib Dems was a blessing, giving him an excuse to keep the right wing of the party at bay whilst holding Nick Clegg in front of himself to take all the flak for the politics of austerity.

All this made him think he was invincible. Similar (dare I say?) to the careers of many financial traders and asset managers, many of whom come from privileged backgrounds and who sailed through life without failing at anything*, who confused luck with skill, became infected with the fatal disease of hubris; before the inevitable blow up.

* Full disclosure: The author of this blog was privately educated and was admitted to the worlds best university (the first generation in his family to go on to higher education, make of that what you will), but soon found he wasn't as clever as he thought he was, and left after one year. A series of random unconnected jobs followed. On first coming into contact with the financial sector with a job as a trader he hated it. So yes, I've been lucky, to have screwed up my life royally many many times.

Finally both hedge fund managers and David Cameron have the advantage of a fairly soft landing if things go wrong. Collecting 2 and 20 for a few years on a few hundred million or more can set you up for life. Cameron has more than enough personal wealth, his wife still works, and even with his ignominious exit could easily rack up a few million quid a year "advising" banks and private equity firms (a case of "don't do as I did, do as I say"?).

A fair payoff for negative skew

Now, taking negative skew bets isn't automatically a bad thing, if they're fairly priced. The minimum one should accept is something that gives you zero expectation. Ideally you'd get more than this, a lot more, since most reasonably people have a utility function that hates negative skew.

So if the probability of an event is P, and the payoff is a, with a loss of b if the event doesn't happen:

b>a  (definition of negative skew)

aP >  (1-P)b

NOT a black swan. A white one with one grey feather

Before I continue, I'll just point out that Brexit was NOT a black swan - an inconceivable event with a tiny probability of happening. Even at the widest odds the implied probability of the event was more than 10%.

The City, The Bookmakers and The Polls

It wasn't just Cameron who got the probability of this event wrong, it was everyone.

On the day before the referendum implied currency volatility was pricing in something like a 20% chance of Brexit happening:

"@investingidiocy on BBG, prob derived from FX options for leave is 19% - but guessing the spread is huge to do the arb trade!!!"

twitter  with thanks to Saeed Amen

Similarly on the betting markets the decimal odds on betfair were also implying roughly a 20% chance, although this was moving around (see image - not up to date, but gives you an idea of the volatility)

It was clear to me that 20% was far too low. See these opinion polls.

Source: Financial Times

Notice that:

• The vote was on a knife edge; with neither side getting a clear advantage (say 60:40) in any poll
• There is considerable variation across time
• There is considerable variation across polls
• The undecided figure, although narrowing over time, was still averaging about 6% at the end - easily enough to swing it one way or another.
• Polls have margins of error (MOE), which are shamefully absent from public discourse. Most polls of the size done in the referendum campaign would have MOE of 3-5%. Polls of polls have a smaller MOE; maybe a couple of percent.

Personally I thought on Thursday that given the final poll of polls figure of 48:46, with undecided at 6%, and an MOE of a couple of percent.... well this wasn't an event with just a 20% chance of happening. It was pretty close to a coin flip; but let's call the odds of remain P=60%. But the ratio of the two payoffs in the betting market was more than 5.0 (decimal odds of 1.2 and 6.0 respectively). So the expectation of the two options is:


0.6 x 1.2 = 0.72 
0.4 x 6.0 = 2.4

The second option (betting on leave) is clearly more attractive than the second.

Why this apparent mispricing? Well the markets (betting and FX) move depending on weight of money. Mostly wealthy well educated people of a higher social class play in both markets. Exactly the sort of people who voted remain. They ignored the polls and focused on their own prejudices. Again this is behavioural finance at work. Could be one of several effects, but lets go with narrative bias. The "story" that Brexit was terrible and nobody would vote for it was more attractive than the statistical evidence in the polls.

That is the specific effect. But generally people are really bad at understanding data when measured with uncertainty; and this is particularly dangerous when the outcome is binary - people will naturally try and assign probabilities of 100% and 0% to binary options.

My little flutter

So how best to exploit this? A straightforward way would be to sell GBP eithier in USD or EUR. Or do the same thing via options, although note that buying OTM options is an expensive business due to the spread. You could also bet on the outcome.

However although all of these bets have positive expectation, it will also on average lose money (assuming there was say a 60% chance of remain). Since we can only bet on brexit once this is somewhat annoying.

I opted for a combination trade (and here); a bet on Betfair on Brexit, and a purchase of GBP against EUR (so a bet on remain). I sized the latter trade so that my expected payout in the event of eithier remain or leave would be the same.

Actually I probably got the size wrong. I doubt that GBPEUR would have rallied enough with a remain vote to make up for my loss on Betfair. The resulting depreciation of GBPEUR was also relatively small; as it turned out Brexit is bad for Europe as well and all the capital fleeing to safe havens went to America - it was GBPUSD that really got crucified.

As a result I ended up making much more than I expected (also here) on a leave vote. However this was a fun trade done for relatively small amounts of money. This kind of global macro discretionary trading isn't my forte. It seems like a lot of work.

Summary

Generally betting on a coin toss isn't a good idea, unless the market is mispricing the chances of heads as just 20%.

However these opportunities are few and far between, and require a lot of work and analysis. Systematic trading is far easier. A properly structured (diversified, risk managed) investment portfolio or trading system will not be vulnerable to shocks like Brexit. 

So I made some money. I take absolutely no pleasure in this because I genuinely feel it was the wrong decision which will have, and already has had, terrible consequences (both economically, and in a sharp rise in racism). There is already discussion that Brexit may never actually happen; I hope this is the case.

Politics and taxes - why language matters

I thought I would post another ill conceived political blog post, in response to what is shaping up to be an election full of poor ideas.

(Those who tune into this blog for tips on systematic trading will want to look away now. Don't worry, they'll be back soon when I get this off my chest)

Over the last few days there has been a proposal to reduce the maximum payable tuition fee (for undergraduate education) from £9,000 to £6,000. This will be paid for by removing higher rate tax relief from personal pensions. I think the second idea is stupid because it doesn't go far enough, and the first idea is just plain stupid. What's more they are stupid ideas about two things that have stupid names that twist their actual meaning and make it hard to have a reasonable debate.

I have no particular axe to grind with the Labor party, and I'm equally frustrated by some of the one sided criticism of the proposal from the right wing press. In the interests of balance I am not impressed by the hysterical reaction to a series of potentially good ideas from Ed Milliband and other left leaning parties: mansion taxes (a wealth tax on houses worth more than say £2 million), other wealth taxes and a citizens income.

 The tax and benefit system in this country is complicated. Most well intentioned attempts to do good will end up making it more complicated, when the better solution would be a complete reform. Most importantly language, and specifically, names matter a lot. But we shouldn't let language cloud the issue. That is what politicians want.

(That's why there is a big fuss over a 'bedroom tax' which isn't of course a tax. But that is another story.)

The badly named graduate tax

Why do names matter? If the proposal was rewritten as "Labor plans to reduce graduate tax on high earners" it would have been much less popular with the parties supporters. But economically that is exactly what is happening. Tuition fees aren't paid when you go to university, but are part of a student loan. This loan isn't repaid until you're earning £21,000 or more. The more you earn, the more you pay back. Because the interest rate is below commercial levels the faster you earn more, the higher the economic value of what you're paying

So this is effectively a tax, and like all good taxes its progressive; the rich pay more. Weirdly though it's a tax that's only levied on graduates. Arguably that is because they have received a service and they should pay for it. The argument goes that the plumber shouldn't have to subsidise the children of the middle classes. Although this seems a rather old fashioned argument when 45% of 18 year olds are going on to higher education. The plumber will probably be subsidising their own children as well.

But we don't take that attitude with other forms of education. Nobody I know seems to have a problem with single people paying tax so that the plumbers children can go to primary and secondary school. In economics terms education generates an externality; the overall good for society is far greater than the cost.

What would I do? I'd probably abolish tuition fees, or as they should be accurately named, the graduate tax. I personally think its right that society as a whole should fund higher education. Although to be precise it won't be society as a whole; just the better off paying for it - 10% of taxpayers, or about 5% of the population, pay 55% of income tax. A plumber on £60K will contribute as much as a second year investment banker on the same salary; and why not?

Disclosure: I paid £3,000 up front (no student loan) a year to fund my own education. I have several children who could benefit from this proposal around a decade from now.

Tax relief on pensions

The pension system in the UK is one of the most complicated things I have ever had the misfortune to come across, although it is fortunately getting a little simpler this year.

The basic idea is that you put pre-tax income into your pension, so effectively paying zero tax on your contributions. The money is then taxed on the way back out of the tax shelter. Another stupid name then; there is no tax relief, just tax deferral.

Those who will benefit most are those who pay higher rates of tax when earning, and lower rates when retired. So those who are on well above average incomes benefit the most.

(This doesn't include the mega rich; with a 10 million pound pension paying out about 300K a year you'll be paying the highest rate of tax at both ends. This is one reason why the very rich don't put much money into pensions; they like their legal tax avoidance to be a bit more exotic.)

Because this doesn't seem fair, a great deal of complexity has been layered on to the product. For starters there is a limit on how much you can contribute each year. There's also a limit on the total size of the pension; the so called 'lifetime allowance'.

Part of the reason for the mess is that politicians, and the civil servants who serve them, aren't exactly familiar with personal pensions. Personal pensions are defined contribution (DC). You put in some money, your employer puts in a bit, hopefully it earns returns (though you won't be this lucky), and what you get back is down to the whims of the market.

Politicians like most public sector workers get defined benefit  (DB) pensions. They contribute a bit. Their employers contribute a bit, although often a lot more. For example in the teachers pension scheme starting teachers pay 7% into their pensions; and their employer puts in 14%. This compares to a minimum private sector employer contribution of 3% in the workplace pension scheme (after 2018. Before then it's 1 or 2%).

Their benefits are then based on a proportion of their final salaries, depending on how long they've been in service. Usually if you do a full working life in these schemes you get half your final salary.

Because of low inflation the market value of these schemes is very high. Most people with these schemes don't realise how valuable they are. If a just retired 60 year old averagely paid headteacher reads about a banker with a million pound defined contribution pension fund they'd probably scoff about how disgraceful it is. This is a bit hypocritical because their own pension of £50K a year inflation protected has a market value of about £1.2 million. The dark truth of DB pensions is they transfer wealth from younger and poorer workers, to older and wealthier ones. DC pensions do the same, but the old rich person is just an older version of the young poor one, so its fairer.

I'm not sure that Labour politicians understood this, at least initially. When it was pointed out that middle managers in the public sector - their core constituency -  would be affected the by scrapping of tax relief they said there would be 'exemptions' (although the grounds for those exemptions wasn't clear). A great example of making things more complicated through good intentions.

What would I do? In an ideal world I'd completely scrap pensions. We already have a very simple and effective saving vehicle known as an ISA, which has an annual allowance, but doesn't get tax relief at the point of funding. Most of the rules which separate pensions from ISA's, which are mainly there for paternalistic reasons to stop you blowing your retirement in one go, are going anyway. I'd increase the annual ISA allowance to say £50K a year, soaking up the pension contribution limit. These larger ISA's would still benefit the rich more than the poor, but the benefit would be considerably less than that of higher rate tax "relief" (I can show you the maths if you want to know why).

Contribution limits make more sense than lifetime allowances, which penalise those who manage to get their investments to grow more. A tax on people who are clever (or to be honest just plain lucky) doesn't seem fair.

However this doesn't solve the conundrum of defined benefit schemes. I don't have a problem with the existence of such schemes; though it would be nice if their value was properly recognised since they help close the gap between well paid private sector workers and their downtrodden fellow citizens in the private sector. In 2013, average hourly earnings in the public sector were £16.28 an hour, compared to the average £14.16 among private employees, so yes I'm being sarcastic. 

(The perception of the pay gap comes from comparing the top earners in each profession. That of course is that the media focuses our attention on. Nobody in the public sector earns more than a million quid a year, whereas about 10,000 people in the private sector do. In fact public sector workers earn more than private sector right up to the top 10% of the income distribution. Only in the top 10% is there a benefit in being in the private sector - £57K a year rather than £49K. If we add on the differential in employer pension contributions then even this gap almost vanishes. Its probably only in the top 5% that you earn more in the private sector.)
 
So what to do?

- I'd reduce public sector DB employer contributions down to average private sector levels of 3%
- increase employee contributions to get back to the same level of overall funding
- scrap all tax relief at source on these pension, to make them like ISA's
- and increase gross salaries to compensate.

Economically this amounts to the same thing, but seeing 40% of their salary leaving their payslip at source should have a sobering effect on public sector workers and help them understand where their pensions come from; as well as making it clear that public sector workers are not underpaid on the whole; at least not in middle management.

Rather than final salary, I'd make the schemes pay a set amount depending on the accumulated size of the fund at retirement; the 'fund size' rolling up and the scheme payout set at levels that would be slowly readjusted as acturial circumstances changed (the changes wouldn't affect existing employees). This would give employees the certainty of a DB scheme, without the unfairness where schemes reward the most highly paid and longest serving employees the most.

It would also make the annual limit on contribution a meaningful brake on the size of the fund, removing the need for a lifetime allowance.


Disclosure: I have contributed to a DC pension in the past, and gained higher rate 'tax relief'. I don't currently contribute, or plan to. The current lifetime allowance doesn't affect me, and probably won't unless my investments grow much faster than I'm hoping for.

Citizens income, income taxes and benefits

The complexity of pensions is only a small part of the overall mess of the tax and benefit system. Nearly all of this is as a result of successive governments trying to target specific groups of swing voters or those felt to be good political targets. To name just a few parts that are particularly stupid:

- the very high effective marginal tax paid by people on unemployment benefit who then start working
- the withdrawal of the tax allowance when you earn more than £100K (why not just have another marginal tax band?)
- not paying child benefits if one person in a household is a higher rate tax payer (even if their total income is less than another household where both earn just less than the threshold)
- lower rates of tax on dividends and capital gains income
- higher tax allowances for retired people. Why is this fair? It isn't, but the grey vote is too powerful. Language again - a proposal to withdraw this was described as a tax on the elderly; when in fact while it exists its actually an "anti-tax" or benefit if you like.
- the winter fuel allowance for pensioners. Why not just increase the total state pension by the same amount and pay it all year?

A really big mess is the false dichotomy created by the Victorian era language we use around large parts of the tax system. National insurance was called that for a reason. It is supposed to make people who didn't want to receive 'handouts', feel better. But we've moved way beyond that. Most people have no compunction about receiving benefits, whatever they are called.

National insurance is just a way of making the tax system more complicated. Unlike other insurance companies there is no big fund - its a so called pay as you go system. Typical gripe by old person "Oh but we've paid in, so we should get back what we are entitled to.". Nonsense. The amount of state pension you get is much larger than you would have got in the private sector for an equivalent NI payment. Even the weak historic link between what you pay in and what benefits you receive has almost been completely broken.

Working 'tax credits' are a more modern abuse of language. They are a benefit plain and simple, they don't reduce the amount of tax you pay, but for some reasons we don't want to call it a benefit.

Tinkering around the edges doesn't work and will probably make things worse. One radical simplifying proposal I really like is from the Green party, although its not original to them, for a citizens income. Everyone in the country gets an income. There might need to be three rates, for children, adults and those with a disability (with perhaps graduated levels for more serious conditions). This would need to be set at a level so that nobody loses out in the move from the current system. By the way this requirement makes it hard to make these simplifying proposals revenue neutral. So you need to fund them in other ways (see inheritance and wealth tax below).

You then pay tax on any income earned, regardless of the type of income. No lower rates on capital gains or dividends that mainly benefit owners of private equity businesses; and make filling in tax returns for ordinary hard working private investors a nightmare. If you are consistently earning enough you could opt to waive your citizens income in return for an equivalent tax allowance (this wouldn't be of any financial benefit, but saves you and the government some paperwork and two money transfers a month).

There would be no tax allowances or any other benefits. No state pension, job seekers allowance, "tax credits" of any sort. No means testing. It would sweep away all of the messes above, and many more, and release tens of thousands of civil servants who are needed to run the system to do more interesting and socially productive jobs, as well as saving a whole rainforest full of paper.

Notice the language again. "Citizens income" recognises that in a modern fair society everyone is entitled to a minimum standard of living, and that is something the state should pay for. It achieves the same end economically as the complicated morass of things that are supposed to be insurance but aren't really, are supposed to be funded pensions but certainly aren't, and targeted 'benefits'. Unlike them this proposal is much simpler and doesn't obfuscate the language 

A proposal I am less keen on is the 'flat tax' (which used to be UKIP policy, but isn't now). I'd keep progressive taxes (higher rates on higher incomes). Firstly because they are fairer, and secondly for pragmatic reasons the flat tax would need to be relatively high (about 35%) to make the proposal revenue neutral, as most tax is paid by the richest tenth and you'd need to hit the middle classes by much more to compensate.

Disclosure: The citizens income wouldn't affect me but higher taxes on capital gains and dividends would be bad for me; whilst freeing up some of my time currently wasted doing my tax return.

Wealth and mansion taxes

As you've probably realised I can't easily be classified as a communist or a raving capitalist lunatic. What may swing your opinion once and for all is that I am a fan of wealth taxes. Not just a specific mansion tax, but a Thomas Piketty style tax on all forms of wealth.

A level of two million pounds seem about right (and should catch enough of the current UK cabinet to be deeply satisfying). At this level you have enough that with a reasonably diversified and quite safe investment portfolio would give you an income of two to three times the UK household average. Bluntly, you have more than enough money and it seems reasonable that you contribute more to society. A rate of 3% annually on all wealth above 2 million quid doesn't seem exactly onerous, and would help pay the cost of the simplifications I've outlined above.

I will now answer all the usual criticisms. Won't this stifle entrepreneurial innovation? I doubt it. I think there are two kinds of people. Lazy people like me who would stop working if they won the lottery or earned enough money to retire early. And others who continue working hard even once they are multi millionaires, not because they need yet more money but because they are driven by other motivations. Paying a little bit more tax isn't going to stop them.

Also the tweak I would suggest is that if you already paid more than 3% tax on income derived from your wealth (dividends, capital gains) wouldn't have to pay any more. The tax is designed to hit rentiers who deliberately minimise the income from their wealth, or who own large unproductive houses in a country that is desperately short of housing, not those who invest in risky assets like start up firms.

What about all the people in London? The tax is on net wealth, so unlike a mansion tax someone with a massive but heavily mortgaged house and no other wealth won't pay anything at all - they aren't rich, just stupid. Note this is why the tax has to be on all forms of wealth; or you could just avoid a net equity housing tax by keeping your mortgage nice and high.

What about elderly people living in London with low incomes but who have mult-million pound houses owned outright 'through no fault of their own' as many people like to say.

(I love this phrase - yes I am very rich, but its through no fault of my own, so why should I be taxed? Brilliant. Yes only those who have worked hard for their money or created businesses and given jobs to people should pay tax on it.)

Solution: Anyone who owned an asset that couldn't be easily moved from the UK would be allowed to borrow money from the government secured on the asset to defer the tax until death, with an interest rate of say inflation + 1% charged on the roll up. Note that this loan including interest will count against your wealth, so you'd never be in a situation where there was no net wealth left on your death.

Isn't it easy for people to avoid it? It's true that no wealth tax will ever work properly unless its adopted globally at the same rate. Fortunately moving money offshore is getting harder. Even if you're not worried about someone leaking your Swiss bank account details to HMRC, you'll be hit by the negative interest rates. Even if the globally mobile super rich pay 3% only on their UK assets above two million quid, that is still a tidy sum (say £1.3 million a year on a modest london home).

Disclosure: I wouldn't pay the wealth tax under as it is described above.

Inheritance tax

It makes sense to finish with the final tax we will all pay. The hysteria about inheritance tax is so great that I believe no party is proposing increasing it, even though it should be a fairly soft target. This is daft. You're dead, why do you need the money? Let your kids survive on their own. Give it all to charity if you don't want the government to get it.

But older people don't seem to like this tax, and as they vote in large numbers changing it is politically toxic, hence the media's favorite name "the death tax". If you like your kids so much give them the money now. Live for seven years and its tax free. Plus you don't have to worry about your offspring putting arsenic in your tea to speed up the process.

Disclosure: Under my current will my children get the lot, and will have to pay inheritance tax. But we're planning to spend it first. Sorry kids.