## Absolute or relative momentum?

The motivation for this comes from a table in my second book, which exposes an interesting problem. Here is the table (actually a slightly modified version of it, so you won't recognise the precise numbers), and I'll explain what it means and what the problem is:

Arithmetic mean Geometric mean Std. Deviation Sharpe Ratio

Fixed weight 8.37% 8.04% 8.14% 1.03

Relative momentum 9.26% 8.89% 8.62% 1.07

Absolute momentum 8.93% 8.61% 7.96% 1.12

**Fixed weight:**This is a portfolio with 75:25 risk weightings in US equities and US bonds (using the last 12 months of monthly returns to calculate the appropriate volatility for risk weighting; this works out to roughly 60:40 cash weightings on average).

**Relative momentum:**This portfolio tactically rebalances the strategic fixed weights using the

*relative*12 month total risk adjusted return of equities and bonds. The rebalancing is a 'tilt' to account for forecasting uncertainty; the maximum tilt is to 148% of the original portfolio weight, and the minimum is 60% of the original. The relative momentum portfolio is always fully invested.

**Absolute momentum:**This portfolio tactically rebalances the strategic fixed weights according to the

*absolute*12 month total risk adjusted return of equities and bonds, again using a 'tilt'. The absolute momentum portfolio may not be fully invested if momentum is relatively weak in one or both assets. The minimum investment is 60% (which is not unusual), and the average is 93%.

*All portfolios are rebalanced monthly, using data from January 1954 to March 2016 (I could update this, but I wanted to use the same data as in the book, and it wouldn't affect the results much). Returns shown are excess returns, net of the risk free rate.*

Let's do some basic analysis of these results. Using absolute momentum results in a slightly higher risk than for fixed weights, because equities have spent more time going up in a risk adjusted sense. I call this the

**historical volatility boost**. That more than compensates for the fact we aren't always fully invested, which drags down risk. The average cash weight to equities is 67% versus the 61% under fixed weights. But the extra risk is well rewarded with a higher arithmetic and geometric return, and a higher Sharpe Ratio.

Absolute momentum is a super popular asset allocation methodology, because people like the 'downside protection' of being partly in cash when markets are selling off.

Relative momentum has even higher risk; again it has a systematic bias towards equities and a historical volatility boost, but because we are always fully invested that all hits the 'bottom line' in the form of higher risk. The average cash weight to equities is 70%. The extra risk is rewarded with a higher arithmetic and geometric mean return, but the Sharpe Ratio is actually lower than for absolute momentum (though still better than fixed weights).

Relative momentum is less popular amongst the general public, as it seems hard to justify a big allocation to bonds just because they aren't falling quite as fast as equities. It also has a worse Sharpe Ratio, so 'theoretically' it's inferior (if you're an investor who can use leverage).

In my book I rather blandly concluded that relative momentum was better due to the higher geometric mean.

However, we're not comparing like with like. Strictly speaking we should probably compare relative momentum with an absolute version that has a higher strategic allocation to equities, so that their risk levels are comparable. To put it another way:

**is it better* to use relative momentum, or to use absolute momentum and crank up your strategic risk target to compensate for the reduction in risk?**

Already we can see that this is a variation of the classic dilemma that investors without access to leverage and high risk tolerance have: should I opt for the highest Sharpe Ratio, or for something with higher risk (and return) but a lower Sharpe Ratio? However this story is more complicated, because we have two moving parts: the original risk weights, and the choice of rebalancing strategy (fixed weights, absolute, or relative). The interaction of these will produce portfolios with different return and risk profiles.

** The dilemma would be the same** for any type of forecast, but momentum is a popular and well understood rule to establish conditional returns.*

*** Strictly speaking the idea of an 'absolute' forecast requires some kind of equilibrium value at which we have a zero position. So dividend yield as a forecast wouldn't be helpful for absolute weighting, but something like (divided yield - interest rates)*** would make sense.*

*****

*the 'Fed model'*

## The experiment

The general question we want to answer is:

**For a given risk tolerance, what is the best choice of strategic risk weights and rebalancing strategy?**

My criteria will be to judge a particular outcome by looking at the

**geometric mean**(my reasons for choosing that are documented here), and the

**standard deviation**of returns.

The range of strategic risk weights I will consider are from 10% equities 90% bonds, up to 90% equities 10% bonds. All strategic risk weight portfolios will be fully invested. Note that people with really low risk appetites will be best served by the maximum Sharpe Ratio portfolio plus a cash allocation; however I won't consider that option here. After all the problem we are exploring is most acute for investors with higher risk appetites.

To make the results starker, I'm going to allow the two tactical portfolios to 'tilt' all the way from 10% to 200% of the original strategic weight. Obviously this won't affect the fixed weights. For less aggressive tilts the relative results will be the same, but the numbers will be closer together.

First let's look at the Sharpe Ratios:

The black line is what you'd expect; the maximum SR portfolio is roughly 50:50 in equities and bonds. Absolute momentum is mostly inferior to the other options except for relatively high allocations to equities. Relative momentum shows declining performance as we increase the risk weight.

However these differences in SR might not be significant (I'll discuss this later in the post), but more importantly 'we can't eat Sharpe Ratios' if we're not leveraged investors, so let's instead focus on the geometric means and standard deviations.

Each line shows a classic 'efficient frontier', with one line for fixed weights, one for relative weights, and one for absolute weights. Each cross is a different strategic allocation, in 10% steps. So the first black cross on the bottom end of the fixed weights line is 10% risk weight in equities, the next cross is 20% in equities, and so on up to 90% on the top right end of the line.

We can safely ignore all the portfolios with lower risk than 30% equities; for these we'd be better off mixing the maximum Sharpe Ratio portfolio with cash.

It's clear from this graph that the out performance of relative momentum is pretty consistent.

**For a given risk target relative momentum is better than fixed weights or absolute momentum.**It also looks like there is no benefit from using a risk target of greater than 80% in equities.

## Which strategic portfolio weights should we use?

- a fixed risk weight of ~75% to equities
- tactical absolute weighting with a strategic risk weight of ~68% to equities
- tactical relative weighting with a strategic risk weight of ~40% to equities

That is some substantial difference!

## How robust are these results?

First let's consider the differences in geometric means. I'm extremely confident that 12 month momentum is a robust effect that has existed in the past, though we can argue about whether it will continue in the future. So I'd expect both types of momentum to beat fixed weights.

What about the out performance of relative momentum? Cross sectional momentum across asset classes is a less popular idea (though super popular within asset classes e.g. across stocks), but it would be surprising if there was a substantial difference between the two types of forecast.

However in a long only portfolio absolute momentum is operating with one hand tied behind it's back, as it cannot go short. This might explain the relatively poor performance of absolute momentum. Even when it has a slightly higher Sharpe Ratio (for relatively high equity weightings), the reduction in volatility means that absolute momentum can't compete on a geometric mean basis.

What about the differences in standard deviations? By construction the standard deviation for absolute momentum will

*always*be lower than that for relative momentum.

The reasons for the increase in standard deviation when using relative momentum is less robust (risk also rises for absolute momentum, except for very low or very high equity allocations). In theory, if both equities and bonds had the same average forecast going forward, then the standard deviation would be the same for relative momentum as it is for fixed weights.

Radically reducing your strategic weight to equities to compensate for the expected

**'volatility boost'**from your tactical overlay might not be wise. The existence of a risk boost is probably the least robust finding here - I wouldn't be 100% sure it will exist in the future.

## Conclusion

I'm reasonably happy that my superficial analysis in "Smart Portfolios" was correct when put through a more thorough test:

**relative momentum gives a higher geometric mean than absolute momentum**, except for investors with low tolerance to risk. Therefore for most investors it's preferable.

In terms of more specific advice, the graphs above suggest the optimal portfolios are:

**If you can use leverag**e, the highest Sharpe Ratio comes from using relative momentum tactical weighting with a risk weight to equities of somewhere between 30% (15:85 equity/bonds in cash weights based on current vols) and 50% (30:70 equity/bonds in cash weights). Within that range I'd err towards a higher weight in equities in case the 'risk boosting' that occurred in the past is absent.**Recommend: Strategic risk weights 50% equity 50% bond, cash weights 30% equity 70% bonds, relative momentum tactical weighting.**

**If you can't use leverage and have a high risk tolerance**, the highest geometric mean comes from using relative momentum tactical weighting with a risk weight to equities of somewhere between 60% (40:60 in cash weights based on current vols) and 90% (80:20 in cash weights). Within that range I'd err towards a higher weight in equities in case the 'risk boosting' that occurred in the past is absent.**Recommend: Strategic risk weights 90% equity 10% bond, cash weights 80% equity 20% bonds, relative momentum tactical weighting.**

**If you can't use leverage and have a modest risk tolerance (but higher than 8% standard deviation a year):**I'd use relative momentum but with a lower risk weight. If you don't buy the 'risk boosting' story then you will need between 60% and 90% risk weighting in equities; if you do buy the story and believe history will repeat itself, between 30% and 60% risk in equities.**Recommend: Strategic risk weights 60% equity 40% bond, cash weights 40% equity 60% bonds, relative momentum tactical weighting.**

**If you can't use leverage and have a low risk tolerance (lower than 8% standard deviation a year)**: I'd invest in the maximum Sharpe Ratio portfolio (see above), and blend it with cash.