Category: Risk & Style Premia Page 13 of 16

Thinking in Long/Short Portfolios

By Corey Hoffstein

On March 5, 2018

In Risk & Style Premia, Risk Management, Weekly Commentary

This post is available as a PDF download here.

Summary

Few investors hold explicit shorts in their portfolio, but all active investors hold them
We (re-)introduce the simple framework of thinking about an active portfolio as a combination of a passive benchmark plus a long/short portfolio.
This decomposition provides greater clarity into the often confusing role of terms like active bets, active share, and active risk.
We see that while active share defines the quantity of our active exposure, the active bets themselves define the quality.

Ask the average investor if they employ shorting in their portfolios and “no” is likely the answer.

Examine the average portfolio, however, and shorts abound. Perhaps not explicitly, but certainly implicitly. But what in the world is an implicit short?

As investors, if we held no particular views about the market, our default position would be a market-capitalization weighted portfolio. Any deviation from market-capitalization weighted, then, expresses some sort of view (intentional or not).

For example, if we hold a portfolio of 40 blue-chip stocks instead of a total equity market index, we have expressed a view. That view is in part determined by what we hold, but equally important is what we do not.

In fact, we can capture this view – our active bets – by looking at the difference between what we hold in our portfolio and the market-capitalization weighted index. And we quite literally mean the difference. If we take the weights of our portfolio and subtract the weights of the index, we will be left with a dollar-neutral long/short portfolio. The long side will express those positions that we are overweight relative to the index, and the short side will express those positions we are underweight.

Below is a simple example of this idea.

	Portfolio	Benchmark	Implied Long/Short
Stock A	25%	50%	-25%
Stock B	75%	50%	25%

“Dollar-neutral” simply means that the long and short legs will be of notional equal size (e.g. in the above example they are both 25%).

While our portfolio may appear to be long only, in reality it expresses a view that is captured by a long/short portfolio. As it turns out, our portfolio has an implicit short.

This framework is important because it allows us to go beyond evaluating what we hold and instead evaluate both the bets we are taking and the scale of those bets. Generically speaking, we can say:

Portfolio = Benchmark + b x Long/Short

Here, the legs of the Long/Short portfolio are assumed to have 100% notional exposure. Using the example above, this would mean that the long/short is 100% long Stock B, 100% short Stock A, and b is equal to 25%.

This step is important because it allows us to disentangle quantity from quality. A portfolio that is very overweight AAPL and a portfolio that is slightly overweight AAPL are expressing the same bet: it is simply the magnitude of that bet that is different.

So while the Long/Short portfolio captures our active bets, b measures our active share. In the context of this framework, it is easy to see that all active share determines is how exposed our portfolio is to our active bets.

We often hear a good deal of confusion about active share. Is more active share a good thing? A bad thing? Should we pay up for active share? Is active share correlated with alpha? This framework helps illuminate the answers.

Let’s slightly re-write our equation to more explicitly highlight the difference between our portfolio and the benchmark.

Portfolio – Benchmark = b x Long/Short

This means that the difference in returns between the portfolio and the benchmark will be entirely due to the return generated by the Long/Short portfolio of our active bets and how exposed we are to the active bets.

R_Portfolio – R_Benchmark = b x R_Long/Short

Our expected excess return is then quite easy to think about: it is quite simply the expected return of our active bets (the Long/Short portfolio) scaled by how exposed we are to them (i.e. our active share):

E[R_Portfolio – R_Benchmark] = b x E[R_Long/Short]

Active risk (also known as “tracking error”) also becomes quite easy to conceptualize. Active risk is simply the standard deviation of differences in returns between our Portfolio and the Benchmark. Or, as our framework shows us, it is just the volatility of our active bets scaled by how exposed we are to them.

s[R_Portfolio – R_Benchmark] = b x s[R_Long/Short]

We can see that in all of these cases, both our active bets as well as our active share play a critical role. A higher active share means that the fee we are paying provides us more access to the active bets. It does not mean, however, that those active bets are necessarily any good. More is not always better.

Active share simply defines the quantity. The active bets, expressed in the long/short portfolio, will determine the quality. That quality is often captured by the Information Ratio, which is the expected excess return of our portfolio versus the benchmark divided by how much tracking error we have to take to generate that return.

IR = E[R_Portfolio – R_Benchmark] / s[R_Portfolio – R_Benchmark]

Re-writing these terms, we have:

IR = E[R_Long/Short] / s[R_Long/Short]

Note that the active share component cancels out. The information ratio provides us a pure measure of the quality of our active bets and ignores how much exposure our portfolio actually has to those bets.

Both quantity and quality are ultimately important in determining whether the portfolio will be able to overcome the hurdle rate set by the portfolio’s fee.

b x E[R_Long/Short] > Fee_Portfolio – Fee_Benchmark

The lower our active share, the higher our expectation for our active bets needs to be to overcome the fee spread. For example, if the spread in fee between our portfolio and the benchmark is 1% and our active share is just 25%, then we have to believe that our active bets can generate a return in excess of 4% to justify paying the fee spread. If, however, our active share is 75%, then the return needed falls to 1.33%.

Through this equation we can also understand the implications of fee pressure. If the cost of the active portfolio and the cost of the benchmark are equivalent, there is zero hurdle rate to overcome. We would choose active so long as we expect a positive return from our active bets.[1]

However, through its organizational structure and growth, Vanguard has been able to continually lower the fee of the passive benchmark over the last several decades. All else held equal, this means that the hurdle rate for active managers goes up.

Thus as the cost of passive goes down, active managers must lower their fee in a commensurate manner or boost the quality of their active bets.

Conclusion

For long-only “smart beta” and factor portfolios, we often see a focus on what the portfolio holds. While this is important, it is only a piece of the overall picture. Just as important in determining performance relative to a benchmark is what the portfolio does not hold.

In this piece, we explicitly calculate active bets as the difference between the active portfolio and its benchmark. This framework helps illuminate that our active return will be a function both of the quality of our active bets as well as the quantity of our exposure to them.

Finally, we can see that if our aim is to outperform the benchmark, we must first overcome the fee we are paying. The ability to overcome that fee will be a function of both quality and quantity. By scaling the fee by the portfolio’s active share, we can identify the hurdle rate that our active bets must overcome.

[1] More technically, theory tells us we would need a positive marginal expected utility from the investment in the context of our overall portfolio.

Timing Bonds with Value, Momentum, and Carry

By Corey Hoffstein

On January 29, 2018

In Carry, Momentum, Popular, Risk & Style Premia, Value, Weekly Commentary

This post is available as a PDF download here.

Summary

Bond timing has been difficult for the past 35 years as interest rates have declined, especially since bonds started the period with high coupons.
With low current rates and higher durations, the stage may be set for systematic, factor-based bond investing.
Strategies such as value, momentum, and carry have done well historically, especially on a risk-adjusted basis.
Diversifying across these three strategies and employing prudent leverage takes advantage of differences in the processes and the information contained in their joint decisions.

This commentary is a slight re-visit and update to a commentary we wrote last summer, Duration Timing with Style Premia[1]. The models we use here are similar in nature, but have been updated with further details and discussion, warranting a new piece.

Historically Speaking, This is a Bad Idea

Let’s just get this out of the way up front: the results of this study are probably not going to look great.

Since interest rates peaked in September 1981, the excess return of a constant maturity 10-year U.S. Treasury bond index has been 3.6% annualized with only 7.3% volatility and a maximum drawdown of 16.4%. In other words, about as close to a straight line up and to the right as you can get.

Source: Federal Reserve of St. Louis. Calculations by Newfound Research.

With the benefit of hindsight, this makes sense. As we demonstrated in Did Declining Rates Actually Matter?[2], the vast majority of bond index returns over the last 30+ years have been a result of the high average coupon rate. High average coupons kept duration suppressed, meaning that changes in rates produced less volatile movements in bond prices.

Source: Federal Reserve of St. Louis. Calculations by Newfound Research.

Ultimately, we estimate that roll return and benefits from downward shifts in the yield curve only accounted for approximately 30% of the annualized return.

Put another way, whenever you got “out” of bonds over this period, there was a very significant opportunity cost you were experiencing in terms of foregone interest payments, which accounted for 70% of the total return.

If we use this excess return as our benchmark, we’ve made the task nearly impossible for ourselves. Consider that if we are making “in or out” tactical decisions (i.e. no leverage or shorting) and our benchmark is fully invested at all times, we can only outperform due to our “out” calls. Relative to the long-only benchmark, we get no credit for correct “in” calls since correct “in” calls mean we are simply keeping up with the benchmark. (Note: Broadly speaking, this highlights the problems with applying traditional benchmarks to tactical strategies.) In a period of consistently positive returns, our “out” calls must be very accurate, in fact probably unrealistically accurate, to be able to outperform.

When you put this all together, we’re basically asking, “Can you create a tactical strategy that can only outperform based upon its calls to get out of the market over a period of time when there was never a good time to sell?”

The answer, barring some serious data mining, is probably, “No.”

**This Might Now be a Good Idea**

Yet this idea might have legs.

Since the 10-year rate peaked in 1981, the duration of a constant maturity 10-year U.S. bond index has climbed from 4.8 to 8.7. In other words, bonds are now 1.8x more sensitive to changes in interest rates than they were 35 years ago.

If we decompose bond returns in the post-crisis era, we can see that shifts in the yield curve have played a large role in year-to-year performance. The simple intuition is that as coupons get smaller, they are less effective as cushions against rate volatility.

Higher durations and lower coupons are a potential double whammy when it comes to fixed income volatility.

Source: Federal Reserve of St. Louis. Calculations by Newfound Research.

With rates low and durations high, strategies like value, momentum, and carry may afford us more risk-managed access to fixed income.

Timing Bonds with Value

Following the standard approach taken in most literature, we will use real yields as our measure of value. Specifically, we will estimate real yield by taking the current 10-year U.S. Treasury rate minus the 10-year forecasted inflation rate from Philadelphia Federal Reserve’s Survey of Professional Forecasters.[3]

To come up with our value timing signal, we will compare real yield to a 3-year exponentially weighted average of real yield.

Here we need to be a bit careful. With a secular decline in real yields over the last 30 years, comparing current real yield against a trailing average of real yield will almost surely lead to an overvalued conclusion, as the trailing average will likely be higher.

Thus, we need to de-trend twice. We first subtract real yield from the trailing average, and then subtract this difference from a trailing average of differences. Note that if there is no secular change in real yields over time, this second step should have zero impact. What this is measuring is the deviation of real yields relative to any linear trend.

After both of these steps, we are left with an estimate of how far our real rates are away from fair value, where fair value is defined by our particular methodology rather than any type of economic analysis. When real rates are below our fair value estimate, we believe they are overvalued and thus expect rates to go up. Similarly, when rates are above our fair value estimate, we believe they are undervalued and thus expect them to go down.

Source: Federal Reserve of St. Louis. Philadelphia Federal Reserve. Calculations by Newfound Research.

Before we can use this valuation measure as our signal, we need to take one more step. In the graph above, we see that the deviation from fair value in September 1993 was approximately the same as it was in June 2003: -130bps (implying that rates were 130bps below fair value and therefore bonds were overvalued). However, in 1993, rates were at about 5.3% while in 2003 rates were closer to 3.3%. Furthermore, duration was about 0.5 higher in 2003 than it was 1993.

In other words, a -130bps deviation from fair value does not mean the same thing in all environments.

One way of dealing with this is by forecasting the actual bond return over the next 12 months, including any coupons earned, by assuming real rates revert to fair value (and taking into account any roll benefits due to yield curve steepness). This transformation leaves us with an actual forecast of expected return.

We need to be careful, however, as our question of whether to invest or not is not simply based upon whether the bond index has a positive expected return. Rather, it is whether it has a positive expected return in excess of our alternative investment. In this case, that is “cash.” Here, we will proxy cash with a constant maturity 1-year U.S. Treasury index.

Thus, we need to net out the expected return from the 1-year position, which is just its yield. [4]

Source: Federal Reserve of St. Louis. Philadelphia Federal Reserve. Calculations by Newfound Research.

While the differences here are subtle, had our alternative position been something like a 5-year U.S. Treasury Index, we may see much larger swings as the impact of re-valuation and roll can be much larger.

Using this total expected return, we can create a simple timing model that goes long the 10-year index and short cash when expected excess return is positive and short the 10-year index and long cash when expected excess return is negative. As we are forecasting our returns over a 1-year period, we will employ a 1-year hold with 52 overlapping portfolios to mitigate the impact of timing luck.

We plot the results of the strategy below.

Source: Federal Reserve of St. Louis. Philadelphia Federal Reserve. Calculations by Newfound Research. Results are hypothetical and backtested. Past performance is not a guarantee of future results. Returns are gross of all fees (including management fees, transaction costs, and taxes). Returns assume the reinvestment of all income and distributions.

The value strategy return matches the 10-year index excess return nearly exactly (2.1% vs 2.0%) with just 70% of the volatility (5.0% vs 7.3%) and 55% of the max drawdown (19.8% versus 36.2%).

Timing Bonds with Momentum

For all the hoops we had to jump through with value, the momentum strategy will be fairly straightforward.

We will simply look at the trailing 12-1 month total return of the index versus the alternative (e.g. the 10-year index vs. the 1-year index) and invest in the security that has outperformed and short the other. For example, if the 12-1 month total return for the 10-year index exceeds that of the 1-year index, we will go long the 10-year and short the 1-year, and vice versa.

Since momentum tends to decay quickly, we will use a 1-month holding period, implemented with four overlapping portfolios.

(Note that this backtest starts earlier than the value backtest because it only requires 12 months of returns to create a trading signal versus 6 years of data – 3 for the value anchor and 3 to de-trend the data – for the value score.)

Compared to the buy-and-hold approach, the momentum strategy increases annualized return by 0.5% (1.7% versus 1.2%) while closely matching volatility (6.7% versus 6.9%) and having less than half the drawdown (20.9% versus 45.7%).

Of course, it cannot be ignored that the momentum strategy has largely gone sideways since the early 1990s. In contrast to how we created our bottom-up value return expectation, this momentum approach is a very blunt instrument. In fact, using momentum this way means that returns due to differences in yield, roll yield, and re-valuation are all captured simultaneously. We can really think of decomposing our momentum signal as:

10-Year Return – 1-Year Return = (10-Year Yield – 1-Year Yield) + (10-Year Roll – 1-Year Roll) + (10-Year Shift – 1-Year Shift)

Our momentum score is indiscriminately assuming momentum in all the components. Yet when we actually go to put on our trade, we do not need to assume momentum will persist in the yield and roll differences: we have enough data to measure them explicitly.

With this framework, we can isolate momentum in the shift component by removing yield and roll return expectations from total returns.

Source: Federal Reserve of St. Louis. Calculations by Newfound Research.

Ultimately, the difference in signals is minor for our use of 10-year versus 1-year, though it may be far less so in cases like trading the 10-year versus the 5-year. The actual difference in resulting performance, however, is more pronounced.

Ironically, by doing worse mid-period, the adjusted momentum long/short strategy appears to be more consistent in its return from the early 1990s through present. We’re certain this is more noise than signal, however.

Timing Bonds with Carry

Carry is the return we earn by simply holding the investment, assuming everything else stays constant. For a bond, this would be the yield-to-maturity. For a constant maturity bond index, this would be the coupon yield (assuming we purchase our bonds at par) plus any roll yield we capture.

Our carry signal, then, will simply be the difference in yields between the 10-year and 1-year rates plus the difference in expected roll return.

For simplicity, we will assume roll over a 1-year period, which makes the expected roll of the 1-year bond zero. Thus, this really becomes, more or less, a signal to be long the 10-year when the yield curve is positively sloped, and long the 1-year when it is negatively sloped.

As we are forecasting returns over the next 12-month period, we will use a 12-month holding period and implement with 52 overlapping portfolios.

Again, were we comparing the 10-year versus the 5-year instead of the 10-year versus the 1-year, the roll can have a large impact. If the curve is fairly flat between the 5- and 10-year rates, but gets steep between the 5- and the 1-year rates, then the roll expectation from the 5-year can actually overcome the yield difference between the 5- and the 10-year rates.

Building a Portfolio of Strategies

With three separate methods to timing bonds, we can likely benefit from process diversification by constructing a portfolio of the approaches. The simplest method to do so is to simply give each strategy an equal share. Below we plot the results.

Indeed, by looking at per-strategy performance, we can see a dramatic jump in Information Ratio and an exceptional reduction in maximum drawdown. In fact, the maximum drawdown of the equal weight approach is below that of any of the individual strategies, highlighting the potential benefit of diversifying away conflicting investment signals.

Strategy	Annualized Return	Annualized Volatility	Information Ratio	Max Drawdown
10-Year Index Excess Return	2.0%	7.3%	0.27	36.2%
Value L/S	2.0%	5.0%	0.41	19.8%
Momentum L/S	1.9%	6.9%	0.27	20.9%
Carry L/S	2.5%	6.6%	0.38	20.1%
Equal Weight	2.3%	4.0%	0.57	10.2%

One potential way to improve upon the portfolio construction is by taking into account the actual covariance structure among the strategies (correlations shown in the table below). We can see that, historically, momentum and carry have been fairly positively correlated while value has been independent, if not slightly negatively correlated. Therefore, an equal-weight approach may not be taking full advantage of the diversification opportunities presented.

	Value L/S	Momentum L/S	Carry L/S
Value L/S	1.0	-0.2	-0.1
Momentum L/S	-0.2	1.0	0.6
Carry L/S	-0.1	0.6	1.0

To avoid making any assumptions about the expected returns of the strategies, we will construct a portfolio where each strategy contributes equally to the overall risk profile (“ERC”). So as to avoid look-ahead bias, we will use an expanding window to compute our covariance matrix (seeding with at least 5 years of data). While the weights vary slightly over time, the result is a portfolio where the average weights are 43% value, 27% momentum, and 30% carry.

The ERC approach matches the equal-weight approach in annualized return, but reduces annualized volatility from 4.2% to 3.8%, thereby increasing the information ratio from 0.59 to 0.64. The maximum drawdown also falls from 10.2% to 8.7%.

A second step we can take is to try to use the “collective intelligence” of the strategies to set our risk budget. For example, we can have our portfolio target the long-term volatility of the 10-year Index Excess Return, but scale this target between 0-2x depending on how invested we are.

For example, if the strategies are, in aggregate, only 20% invested, then our target volatility would be 0.4x that of the long-term volatility. If they are 100% invested, though, then we would target 2x the long-term volatility. When the strategies are providing mixed signals, we will simply target the long-term volatility level.

Unfortunately, such an approach requires going beyond 100% notional exposure, often requiring 2x – if not 3x – leverage when current volatility is low. That makes this system less useful in the context of “bond timing” since we are now placing a bet on current volatility remaining constant and saying that our long-term volatility is an appropriate target.

One way to limit the leverage is to increase how much we are willing to scale our risk target, but truncate our notional exposure at 100% per leg. For example, we can scale our risk target between 0-4x. This may seem very risky (indeed, an asymmetric bet), but since we are clamping our notional exposure to 100% per leg, we should recognize that we will only hit that risk level if current volatility is greater than 4x that of the long-term average and all the strategies recommend full investment.

With a little mental arithmetic, the approach it is equivalent to saying: “multiply the weights by 4x and then scale based on current volatility relative to historical volatility.” By clamping weights between -100% and +100%, the volatility targeting really does not come into play until current volatility is 4x that of long-term volatility. In effect, we leg into our trades more quickly, but de-risk when volatility spikes to abnormally high levels.

Compared to the buy-and-hold model, the variable risk ERC model increases annualized returns by 90bps (2.4% to 3.3%), reduces volatility by 260bps (7.6% to 5.0%), doubles the information ratio (0.31 to 0.66) and halves the maximum drawdown (30% to 15%).

There is no magic to the choice of “4” above: it is just an example. In general, we can say that as the number goes higher, the strategy will approach a binary in-or-out system and the volatility scaling will have less and less impact.

Conclusion

Bond timing has been hard for the past 35 years as interest rates have declined. Small current coupons do not provide nearly the cushion against rate volatility that investors have been used to, and these lower rates mean that bonds are also exposed to higher duration.

These two factors are a potential double whammy when it comes to fixed income volatility.

This can open the door for systematic, factor-based bond investing.

Value, momentum, and carry strategies have all historically outperformed a buy-and-hold bond strategy on a risk adjusted basis despite the bond bull market. Diversifying across these three strategies and employing prudent leverage takes advantage of differences in the processes and the information contained in their joint decisions.

We should point out that in the application of this approach, there were multiple periods of time in the backtest where the strategy went years without being substantially invested. A smooth, nearly 40-year equity curve tells us very little about what it is actually like to sit on the sidelines during these periods and we should not underestimate the emotional burden of using such a timing strategy.

Even with low rates and high rate movement sensitivity, bonds can still play a key role within a portfolio. Going forward, however, it may be prudent for investors to consider complementary risk-management techniques within their bond sleeve.

[1] https://blog.thinknewfound.com/2017/06/duration-timing-style-premia/

[2] https://blog.thinknewfound.com/2017/04/declining-rates-actually-matter/

[3] Prior to the availability of the 10-year inflation estimate, the 1-year estimate is utilized; prior to the 1-year inflation estimate availability, the 1-year GDP price index estimate is utilized.

[4] This is not strictly true, as it largely depends on how the constant maturity indices are constructed. For example, if they are rebalanced on a monthly basis, we would expect that re-valuation and roll would have impact on the 1-year index return. We would also have to alter the horizon we are forecasting over as we are assuming we are rolling into new bonds (with different yields) more frequently.

Factor Investing & The Bets You Didn’t Mean to Make

By Corey Hoffstein

On January 16, 2018

In Craftsmanship, Portfolio Construction, Risk & Style Premia, Risk Management, Weekly Commentary

This post is available as a PDF download here.

Summary

Factor investing seeks to balance specificity with generality: specific enough to have meaning, but general enough to be applied broadly.
Diversification is a key tool to managing risk in factor portfolios. Imprecision in the factor definitions means that unintended bets are necessarily introduced.
This is especially true as we apply factors across securities that share fewer and fewer common characteristics. Left unmonitored, these unintended bets have the potential to entirely swamp the factor itself.
By way of example, we explore a simple value-based country model.
While somewhat counter-intuitive, constraints have the potential to lead to more efficient factor exposures.

In quantitative investing, we seek a balance between generality and specificity. When a model is too specific – designed to have meaning on too few securities or in too few scenarios – we lose our ability to diversify. When a model is too generic, it loses meaning and forecasting power.

The big quant factors – value, momentum, defensive, carry, and trend – all appear to find this balance: generic enough to be applied broadly, but specific enough to maintain a meaningful signal.

As we argued in our past commentary A Case Against Overweighting International Equity, the imprecision of the factors is a feature, not a bug. A characteristic like price-to-earnings may never fully capture the specific nuances of each firm, but it can provide a directionally accurate roadmap to relative firm valuations. We can then leverage diversification to average out the noise.

Without diversification, we are highly subject to the imperfections of the model. This is why, in the same piece, we argued that making a large regional tilt – e.g. away from U.S. towards foreign developed – may not be prudent: it is a single bet that can take decades to resolve. If we are to sacrifice diversification in our portfolio, we’ll require a much more accurate model to justify the decision.

Diversification, however, is not just measured by the quantity of bets we take. If diversification is too naively interpreted, the same imprecision that allows factors to be broadly applied can leave our portfolios subject to the returns of unintended bets.

Value Investing with Countries

If taking a single, large regional tilt is not prudent, perhaps value investing at a country level may better diversify our risks.

One popular way of measuring value is with the Shiller CAPE: a cyclically-smoothed price-to-earnings measure. In the table below, we list the current CAPE and historical average CAPE for major developed countries.

	CAPE	Mean CAPE	Effective Weight
Australia	18.5	17.2	2.42%
Belgium	25.0	15.4	0.85%
Canada	22.0	21.4	3.76%
Denmark	36.5	24.5	0.73%
France	20.9	21.9	4.85%
Germany	20.6	20.6	4.36%
Hong Kong	18.2	18.3	5.21%
Italy	16.8	22.1	1.33%
Japan	28.9	43.2	11.15%
Netherlands	23.5	14.8	1.45%
Singapore	13.9	22.1	1.09%
Spain	13.4	18.3	1.58%
Sweden	21.5	23.0	1.21%
Switzerland	25.9	21.9	3.15%
United Kingdom	16.5	15.3	6.55%
United States	30.5	20.3	50.30%

Source: StarCapital.de. Effective weight is market-capitalization weight of each country, normalized to sum to 100%. Mean CAPE figures use data post-1979 to leverage a common dataset.

While evidence[1] suggests that valuation levels themselves are enough to determine relative valuation among countries, we will first normalize the CAPE ratio by its long-term average to try to account for structural differences in CAPE ratios (e.g. a high growth country may have a higher P/E, a high-risk country may have a lower P/E, et cetera). Specifically, we will look at the log-difference between the mean CAPE and the current CAPE scores.

Note that we recognize there is plenty to criticize and improve upon here. Using a normalized valuation metric will mean a country like Japan, which experienced a significant asset bubble, will necessarily look under-valued. Please do not interpret our use of this model as our advocacy for it: we’re simply using it as an example.

Using this value score, we can compare how over and undervalued each country is relative to each other. This allows us to focus on the relative cheapness of each investment. We can then use these relative scores to tilt our market capitalization weights to arrive at a final portfolio.

	Value Score	Relative Z-Score	Scaled Z-Score	Scaled Weights
Australia	-0.07	-0.13	0.88	2.31%
Belgium	-0.48	-1.50	0.40	0.37%
Canada	-0.03	0.02	1.02	4.15%
Denmark	-0.40	-1.22	0.45	0.36%
France	0.05	0.27	1.27	6.65%
Germany	0.00	0.11	1.11	5.24%
Hong Kong	0.01	0.13	1.13	6.37%
Italy	0.27	1.02	2.02	2.92%
Japan	0.40	1.45	2.45	29.59%
Netherlands	-0.46	-1.43	0.41	0.65%
Singapore	0.46	1.65	2.65	3.14%
Spain	0.31	1.15	2.15	3.68%
Sweden	0.07	0.33	1.33	1.75%
Switzerland	-0.17	-0.45	0.69	2.36%
United Kingdom	-0.08	-0.14	0.88	6.22%
United States	-0.41	-1.25	0.45	24.26%

Source: StarCapital.de. Calculations by Newfound Research. “Value Score” is the log-difference between the country’s Mean CAPE and its Current CAPE. Relative Z-Score is the normalized value score of each country relative to peers. Scaled Z-Score applies the following function to the Relative Z-Score: (1+x) if x > 0 and 1 / (1+x) if x < 0. Scaled weights multiply the Scaled Z-Score against the Effective Weights of each country and normalize such that the total weights sum to 100%.

While the Scaled Weights represent a long-only portfolio, what they really capture is the Market Portfolio plus a dollar-neutral long/short factor tilt.

	Market Weight	+ Long / Short	= Scaled Weights
Australia	2.42%	-0.11%	2.31%
Belgium	0.85%	-0.48%	0.37%
Canada	3.76%	0.39%	4.15%
Denmark	0.73%	-0.37%	0.36%
France	4.85%	1.80%	6.65%
Germany	4.36%	0.88%	5.24%
Hong Kong	5.21%	1.16%	6.37%
Italy	1.33%	1.59%	2.92%
Japan	11.15%	18.44%	29.59%
Netherlands	1.45%	-0.80%	0.65%
Singapore	1.09%	2.05%	3.14%
Spain	1.58%	2.10%	3.68%
Sweden	1.21%	0.54%	1.75%
Switzerland	3.15%	-0.79%	2.36%
United Kingdom	6.55%	-0.33%	6.22%
United States	50.30%	-26.04%	24.26%

To understand the characteristics of the tilt we are taking – i.e. the differences we have created from the market portfolio – we need only look at the long/short portfolio.

Unfortunately, this is where our model loses a bit of interpretability. Since each country is being compared against its own long-term average, looking at the increase or decrease to the aggregate CAPE score is meaningless. Indeed, it is possible to imagine a scenario whereby this process actually increases the top-level CAPE score of the portfolio, despite taking value tilts (if value, for example, is found in countries that have higher structural CAPE values). We can, on the other hand, look at the weighted average change to value score: but knowing that we increased our value score by 0.21 has little interpretation.

One way of looking at this data, however, is by trying to translate value scores into return expectations. For example, Research Affiliates expects CAPE levels to mean-revert to the average level over a 20-year period.[2] We can use this model to translate our value scores into an annualized return term due to revaluation. For example, with a current CAPE of 30.5 and a long-term average of 20.3, we would expect a -2.01% annualized drag from revaluation.

By multiplying these return expectations against our long/short portfolio weights, we find that our long/short tilt is expected to create an annualized revaluation premium of +1.05%.

The Unintended Bet

Unfortunately, re-valuation is not the only bet the long/short portfolio is taking. The CAPE re-valuation is, after all, in local currency terms. If we look at our long/short portfolio, we can see a very large weight towards Japan. Not only will we be subject to the local currency returns of Japanese equities, but we will also be subject to fluctuations in the Yen / US Dollar exchange rate.

Therefore, to achieve the re-valuation premium of our long/short portfolio, we will either need to bear the currency risk or hedge it away.

In either case, we can use uncovered interest rate parity to develop an expected return for currency. The notion behind uncovered interest rate parity is that investors should be indifferent to sovereign interest rates. In theory, for example, we should expect the same return from investing in a 1-year U.S. Treasury bond that we expect from converting $1 to 1 euro, investing in the 1-year German Bund, and converting back after a year’s time.

Under uncovered interest rate parity, our expectation is that currency change should offset the differential in interest rates. If a foreign country has a higher interest rate, we should expect that the U.S. dollar should appreciate against the foreign currency.

As a side note, please be aware that this is a highly, highly simplistic model for currency returns. The historical efficacy of the carry trade clearly demonstrates the weakness of this model. More complex models will take into account other factors such as relative purchasing power reversion and productivity differentials.

Using this simple model, we can forecast currency returns for each country we are investing in.

	FX Rate	1-Year Rate	Expected FX Rate	Currency Return
Australia	1.2269	-0.47%	1.2546	-2.21%
Belgium	1.2269	-0.47%	1.2546	-2.21%
Canada	0.8056	1.17%	0.8105	-0.60%
Denmark	0.1647	-0.55%	0.1685	-2.29%
France	1.2269	-0.47%	1.2546	-2.21%
Germany	1.2269	-0.47%	1.2546	-2.21%
Hong Kong	0.1278	1.02%	0.1288	-0.75%
Italy	1.2269	-0.47%	1.2546	-2.21%
Japan	0.0090	-0.13%	0.0092	-1.88%
Netherlands	1.2269	-0.47%	1.2546	-2.21%
Singapore	0.7565	1.35%	0.7597	-0.42%
Spain	1.2269	-0.47%	1.2546	-2.21%
Sweden	0.1241	0.96%	0.1251	-0.81%
Switzerland	1.0338	-0.72%	1.0598	-2.46%
United Kingdom	1.3795	0.43%	1.3981	-1.33%
United States	1.0000	1.78%	1.0000	0.00%

Source: Investing.com, XE.com. Euro area yield curve employed for Eurozone countries on the Euro.

Multiplying our long/short weights against the expected currency returns, we find that we have created an expected annualized currency return of -0.45%.

In other words, we should expect that almost 50% of the value premium we intended to generate will be eroded by a currency bet we never intended to make.

One way of dealing with this problem is through portfolio optimization. Instead of blindly value tilting, we could seek to maximize our value characteristics subject to currency exposure constraints. With such constraints, what we would likely find is that more tilts would be made within the Eurozone since they share a currency. Increasing weight to one Eurozone country while simultaneously reducing weight to another can capture their relative value spread while remaining currency neutral.

Of course, currency is not the only unintended bet we might be making. Blindly tilting with value can lead to time varying betas, sector bets, growth bets, yield bets, and a variety of other factor exposures that we may not actually intend. The assumption we make by looking at value alone is that these other factors will be independent from value, and that by diversifying both across assets and over time, we can average out their impact.

Left entirely unchecked, however, these unintended bets can lead to unexpected portfolio volatility, and perhaps even ruin.

Conclusion

In past commentaries, we’ve argued that investors should focus on achieving capital efficiency by employing active managers that provide more pure exposure to active views. It would seem constraints, as we discussed at the end of the last section, might contradict this notion.

Why not simply blend a completely unconstrained, deep value manager with market beta exposure such that the overall deviations are constrained by position limits?

One answer why this might be less efficient is that not all bets are necessarily compensated. Active risk for the sake of active risk is not the goal: we want to maximize compensated active risk. As we showed above, a completely unconstrained value manager may introduce a significant amount of unintended tracking error. While we are forced to bear this risk, we do not expect the manager’s process to actually create benefit from it.

Thus, a more constrained approach may actually provide more efficient exposure.

That is all not to say that unconstrained approaches do not have efficacy: there is plenty of evidence that the blind application of value at the country index level has historically worked. Rather, the application of value at a global scale might be further enhanced with the management of unintended bets.

[1] For example, Predicting Stock Market Returns Using the Shiller CAPE (StarCapital Research, January 2016) and Value and Momentum Everywhere (Asness, Moskowitz, and Pedersen, June 2013)

[2] See Research Affiliate’s Equity Methodology for their Asset Allocation tool.

Are Market Implied Probabilities Useful?

By Nathan Faber

On November 27, 2017

In Risk & Style Premia, Weekly Commentary

This post is available as a PDF download here.

Summary

Using historical data from the options market along with realized subsequent returns, we can translate risk-neutral probabilities into real-world probabilities.
Market implied probabilities are risk-neutral probabilities derived from the derivatives market. They incorporate both the probability of an event happening and the equilibrium cost associated with it.
Since investors have the flexibility of designing their own portfolios, real-world probabilities of event occurrences are more relevant to individuals than are risk-neutral probabilities.
With a better handle on the real-world probabilities, investors can make portfolio decisions that are in line with their own goals and risk tolerances, leaving the aggregate decision making to the policy makers.

Market-implied probabilities are just as the name sounds: weights that the market is assigning an event based upon current prices of financial instruments. By deriving these probabilities, we can gain an understanding of the market’s aggregate forecast for certain events. Fortunately, the Federal Reserve Bank of Minneapolis provides a very nice tool for visualizing market-implied probabilities without us having to derive them.[1]

For example, say that I am concerned about inflation over the next 5 years. I can see how the probability of a large increase has been falling over time and how the probability of a large decrease has fallen recently, with both currently hovering around 15%.

Historical Market Implied Probabilities of Large Moves in Inflation

Source: Minneapolis Federal Reserve

I can also look at the underlying probability distributions for these predictions, which are derived from the derivatives market, and compare the changes over time.

Market Implied Probability Distributions for Moves in Inflation

Source: Minneapolis Federal Reserve

From this example, we can judge that not only has the market’s implied inflation forecast increased, but the precision has also increased (i.e. lower standard deviation) and the probabilities have been skewed to the left with fatter tails (i.e. higher kurtosis).

Inflation is only one of many variables analyzed.

Also available is implied probability data for the S&P 500, short and long-term interest rates, major currencies versus the U.S. dollar, commodities (energy, metal, and agricultural), and a selection of the largest U.S. banks.

With all the recent talk about low volatility, the data for the S&P 500 over the next 12 months is likely to be particularly intriguing to investors and advisors alike.

Historical Market Implied Probabilities of Large Moves in the S&P 500

Source: Minneapolis Federal Reserve

The current market implied probabilities for both large increases and decreases (i.e. greater than a 20% move) are the lowest they have been since 2007.

Interpreting Market Implied Probabilities

A qualitative assessment of probability is generally difficult unless the difference is large. We can ask ourselves, for example, how we would react if the probability of a large loss jumped from 10% to 15%. We know that the latter case is riskier, but how does that translate into action?

The first step is understanding what the probability actually means.

Probability forecasts in weather are a good example of this necessity. Precipitation forecasts are a combination of forecaster certainty and coverage of the likely precipitation.[2] For example, if there is a 40% chance of rain, it could mean that the forecaster is 100% certain that it will rain in 40% of the area. Or it could mean that they are 40% certain that it will rain in 100% of the area. Or it could mean that they are 80% certain that it will rain in 50% of the area.

Once you know what the probability even represents, you can have a better grasp on whether you should carry an umbrella.

In the case of market implied probabilities, what we have is the risk-neutral probability. These are the probabilities of an event given that investors are risk neutral; these probabilities factor in the both the likelihood of an event and the cost in the given state of the world. These are not the real-world probabilities of the market moving by a given amount. In fact, they can change over time even if the real-world probabilities do not.

To illustrate these differences between a risk-neutral probability and a real-world probability, consider a simple coin flip game. The coin is flipped one time. If it lands on heads, you make $1, and if it lands on tails, you lose $1.

The coin is fair, so the probability for the coin flip is 50%. How much would you pay to play this game?

If you answer is nothing, then you are risk neutral, and the risk neutral probability is also 50%.

However, risk averse players would say, “you have to pay me to play that game.” In this case, the risk neutral probability of a tails is greater than 50% because of the downside that players ascribe to that event.

Now consider a scenario where a tails still loses $1, but a heads pays out $100. Chances are that even very risk-averse players would pay close to $1 to play this game.

In this case, the risk neutral probability of a heads would be much greater than 50%.

But in all cases, the actual likelihoods of heads and tails never changed; they still had a 50% real-world probability of occurring.

As with the game, investors who operate in the real world are generally risk averse. We pay premiums for insurance-like investments to protect in the states of the world we dread the most. As such, we would expect the risk-neutral probability of a “bad” event (e.g. the market down more than 20%) to be higher than the real-world probability.

Likewise, we would expect the risk-neutral probability of a “good” event (e.g. the market up more than 20%) to be lower than the real-world probability.

How Market Implied Probabilities Are Calculated

Note (or Warning): This section contains some calculus. If that is not of interest, feel free to skip to the next section; you won’t miss anything. For those interested, we will briefly cover how these probabilities are calculated to see what (or who), exactly, in the market implies them.

The options market contains call and put options over a wide array of strike prices and maturities. If we assume that the market is free from arbitrage, we can transform the price of put options into call options through put-call parity.[3]

In theory, if we knew the price of a call option for every strike price, we could calculate the risk-neutral probability distribution, f^RN, as the second derivative with respect to the strike price.

where r is the risk-free rate, C is the price of a call option, K is the strike price and T-t is the time to option maturity.

Since options do not exist at every strike price, a curve is fit to the data to make it a continuous function that can be differentiated to yield the probability distribution.

Immediately, we see that the probabilities are set by the options market.

Are Market Implied Probabilities Useful?

Feldman et. al (2015), from the Minneapolis Fed, assert that market-based probabilities are a useful tool for policy makers.[4] Their argument centers around that fact that risk-neutral probabilities encapsulate both the probability of an event occurring – the real-world probability – and the cost/benefit of the event.

Assuming broad access to the options market, households or those acting on behalf of households can express their views on the chances of the event happening and the willingness to exchange cash flows in different states of the world by trading the appropriate options.

In the paper, the authors admit two main pitfalls:

Participation – An issue can arise here since the people whose welfare the policy-makers are trying to consider may not be participating. Others outside the U.S. may also be influencing the probabilities.
Illiquidity – Options do not always trade frequently enough in the fringes of the distribution where investors are usually most concerned. Because of this, any extrapolation must be robust.

However, they also refute many common arguments against using risk-neutral probabilities.

These are not “true” probabilities – The fact that these market implied probabilities are model-independent and derived from household preferences rather than from a statistician’s model, with its own biased assumptions, is beneficial, especially since these market probabilities account for resource availability.
No Household is “Typical” – In equilibrium, all households should be willing to rearrange their cash flows in different states of the world as long as the market is complete. Therefore, a policy-maker aligns their beliefs with those of the households in aggregate by using the market-based probabilities.

We have covered how policymakers often do not forecast very well themselves[5], which Ellison and Sargent argue may be intentional, stating that the FOMC may purposefully forecast incorrectly in order to form policy that is robust to model misspecification.[6]

Where a problem could arise is when an individual investor (i.e. a household) makes a decision for their own portfolio based on these risk-neutral probabilities.

We agree that having a financial market makes a “typical” investor more relevant than the “average fighter pilot” example in our previous commentary.[7] But what a central governing body uses to make decisions is different from what may be relevant to an individual.

The ability to be flexible is key. In this case, an investor can construct their own portfolio. It would be like a pilot constructing their own plane.

Getting to Real World Probabilities

Using the method outlined in Vincent-Humphreys and Noss (2012), we can transform risk-neutral probabilities into real-world probabilities, assuming that investor risk preferences are stable over time.[8]

Without getting too deep into the mathematical framework, the basic premise is that if we have a probability density function (PDF) for the risk-neutral probability, f^RN, with a cumulative density function (CDF), F^RN, we can multiply it by a calibration function, C, to obtain the real-world probability density function, f^RW.

The beta distribution is a suitable choice for the calibration function.[9] Using a beta distribution balances being parsimonious – it only has two parameters – with flexibility, since it allows for preserving the risk-neutral probability distribution by simply shifting the mean and adjusting the variance, skew, and kurtosis.

The beta distribution parameters are calculated using the realized value that the market implied probability represents (e.g. change in the S&P 500, interest rates commodity prices, etc.) over the subsequent time period.

Deriving the Real-World Probability for a Large Move in the S&P 500

We have now covered what market-implied probabilities are and how they are calculated and discussed their usefulness for policy makers.

But individual investors price risk differently based on their own situations and preferences. Because of this, it is helpful to strip off the market-implied costs that are baked into the risk-neutral probabilities. The real-world probabilities could then be used to weigh stress testing scenarios or evaluate the cost of other risk management techniques that align more with investor goals than option strategies.

Using the framework outlined above, we can go through an example of transforming the market implied probabilities of large moves in the S&P 500 into their real-world probabilities.

Statistical aside: The options data starts in 2007, and with 10 years of data, we only have 10 non-overlapping data points, which reduces the power of the maximum likelihood estimate used to fit the beta distribution. However, with options expiring twice a month, we have 24 separate data sets to use for calculating standard errors. Since we are concerned more about the potential differences between the risk-neutral and real-world distributions, we could use the rolling 12-month periods and still see the same trends. As with any analysis with overlapping periods, there can be significant autocorrelation to deal with. By using the 6-month distribution data from the Minneapolis Fed, we could double the number of observations.

Since the Minneapolis Fed calculates the market implied (risk neutral) probability distribution and the summary statistics (numerically), we must first translate it into a functional form to extend the analysis. Based on the data and the summary statistics, the distribution is neither normal nor log-normal. It is left-skewed and has fat tails most of the time.

Market Implied Probability Distributions for Moves in the S&P 500

Source: Minneapolis Federal Reserve

We will assume that the distribution can be parameterized using a skewed generalized t-distribution, which allows for these properties and also encompasses a variety of other distributions including the normal and t-distributions.[10] It has 5 parameters, which we will fit by matching the moments (mean, variance, skewness and kurtosis) of the distribution along with the 90^th percentile value, since that tail of the distribution is generally the smaller of the two.[11]

We can check the fits using the reported median and the 10^th percentile values to see how well they match.

Fit Percentile Values vs. Reported Values

Source: Minneapolis Fed. Calculations by Newfound Research.

There are instances where the reported distribution is bi-modal and would not be as accurately represented by the generalized skewed t distribution, but, as the above graph shows, the quantiles where our interest is focused line up decently well.

Now that we have our parameterized risk-neutral distribution for all time periods, the next step is to input the subsequent 12-month S&P 500 return into the CDF calculated at each point in time. While we don’t expect this risk-neutral distribution to necessarily produce a good forecast of the market return, this step produces the data needed to calibrate the beta function.

The graph below shows this CDF result over the rolling periods.

Cumulative Probabilities of Realized 12-month S&P 500 Returns using the Risk-Neutral Distribution from the Beginning of Each Period

Source: Minneapolis Fed and CSI. Calculations by Newfound Research.

The persistence of high and low values is evidence of the autocorrelation issue we discussed previously since the periods are overlapping.

The beta distribution function used to transition from the risk-neutral distribution to the real-world distribution has parameters j = 1.64 and k = 1.00 with standard errors of 0.09 and 0.05, respectively.

We can see how this function changes at the endpoints of the 95% confidence intervals for each parameter as a way to assess the uncertainty in the calibration.

Estimated Calibration Functions for 12-month S&P 500 Returns

Source: Minneapolis Fed and CSI. Calculations by Newfound Research. Data from Jan 2007 to Nov 2017.

When we transform the risk-neutral distribution into the real-world distribution, the calibration function values that are less than 1 in the left tail reduce the probabilities of large market losses.

In the right tail, the calibration estimates show that real-world probabilities could be higher or lower than the risk-neutral probabilities depending on the second parameter’s value in the beta distribution (this corresponds to k being either greater than or less than 1).

With the risk-neutral distribution and the calibrated beta distribution, we now have all the pieces to calculate the real-world distribution at any point in the options data set.

The graph below shows how these functions affect the risk-neutral probability density using the most recent option data. As expected, much more of the density is centered around the mode, and the distribution is skewed to the right, even using the bounds of the confidence intervals (CI) for the beta distribution parameters.

Risk Neutral and Real-World Probability Densities

Source: Minneapolis Fed and CSI. Calculations by Newfound Research. Data as of 11/15/17. Past performance is no guarantee of future results.

	Risk Neutral	Real-World	Real World (Range Based on Calibration)
Mean Return	0.1%	6.4%	4.2% – 8.3%
Probability of a large loss (>20%)	10.6%	2.6%	1.5% – 4.3%
Probability of a large gain (>20%)	1.7%	2.8%	1.7% – 4.4%

Source: Minneapolis Fed and CSI. Calculations by Newfound Research. Data as of 11/15/17. Past performance is no guarantee of future results.

Based on this analysis, we see some interesting things occurring.

The mean return is considerably higher than the values suggested by many large firms, such as JP Morgan, BlackRock, and Research Affiliates.[12] Those estimates are generally for 7-10 years, so this doesn’t rule out having a good 2018, which is what the options market is showing.
The real-world probability of a large loss is considerably lower than the 10.6% risk-neutral probability. When firms like GMO are indicating that 10% levels are already unreasonably low, their assessment of complacency in the market would only get stronger.[13]
The lower end of the range for the real-world probability of a large gain is in line with the risk-neutral probability, suggesting that investors are seeking out risks (lower risk aversion) in a market with depressed yields on fixed income and low current volatility.

This also shows how looking at market implied probabilities can paint a skewed picture of the chances of an event occurring.

However, we must keep in mind that these real-world probabilities are still derived from the market-implied probabilities. In an efficient market world, all risks would correctly be priced into the market. But we know from the experience during the Financial Crisis that that is not always the case.

Our recommendation is to take all market probabilities with a grain of salt. Just because having a coin land on heads five times in a row has a probability of less than 4% doesn’t mean we should be surprised if it happens once. And coin flipping is something that we know the probability for.

Whether the market probabilities we use are risk-neutral or real-world, there are a lot of assumptions that go into calculating them, and the consequences of being wrong can have a large impact on portfolios. Risk management is important if the event occurs regardless of how likely it is to occur.

As with the weather, a 10% chance of a large loss versus a 4% chance is not a big difference in absolute terms, but a large portfolio loss is likely more devastating than getting rained on a bit should you decide not to bring an umbrella.

Conclusion

Market implied probabilities are risk-neutral probabilities derived from the derivatives market. If we assume that the market is efficient and that there is sufficient investor participation in these markets, then these probabilities can serve as a tool for governing organizations to adjust policy going forward.

However, these probabilities factor in both the actual probability of an event and the perceived cost to investors. Individual investors will attribute their own costs to such events (e.g. a retiree could be much more concerned about a 20% market drop than someone in the beginning of their career).

If individuals want to assess the probability of the event actually happening in order to make portfolio decisions, then they have to focus on the real-world probabilities. Ultimately, an investor’s cost function associated with market events depends more on life circumstances. While a bad state of the world for an investor can coincide with a bad state of the world for the market (e.g. losing a job when the market tanks), risk in an individual’s portfolio should be managed for the individual, not the “typical household”.

While the real-world probability of an event is typically dependent on an economic or statistical model, we have presented a way to translate the market implied probabilities into real-world probabilities.

With a better handle on the real-world probabilities, investors can make portfolio decisions that are in line with their own goals and risk tolerances.

[1] https://www.minneapolisfed.org/banking/mpd

[2] https://www.weather.gov/ffc/pop

[3] https://en.wikipedia.org/wiki/Put%E2%80%93call_parity

[4] https://www.minneapolisfed.org/~/media/files/banking/mpd/optimal_outlooks_dec22.pdf

[5] https://blog.thinknewfound.com/2015/03/weekly-commentary-folly-forecasting/

[6] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2160157

[7] https://blog.thinknewfound.com/2017/09/the-lie-of-averages/

[8] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2093397

[9] The beta distribution takes arguments between 0 and 1, inclusive, and has a non-decreasing CDF. It was also used in Fackler and King (1990) – https://www.jstor.org/stable/1243146.

[10] https://cran.r-project.org/web/packages/sgt/vignettes/sgt.pdf

[11] Since we have 5 unknown parameters, we have to add in this fifth constraint. We could also have used the 10^th percentile value or the median. Whichever, we use, we can see how well the other two align with the reported values.

[12] https://interactive.researchaffiliates.com/asset-allocation/

[13] https://www.gmo.com/docs/default-source/research-and-commentary/strategies/asset-allocation/the-s-p-500-just-say-no.pdf

The Frustrating Law of Active Management

By Corey Hoffstein

On October 2, 2017

In Risk & Style Premia, Weekly Commentary

A PDF version of this post is available for download here.

Summary

In an ideal world, all investors would outperform their benchmarks. In reality, outperformance is a zero-sum game: for one investor to outperform, another must underperform.
If achieving outperformance with a certain strategy is perceived as being “easy,” enough investors will pursue that strategy such that its edge is driven towards zero.
Rather, for a strategy to outperform in the long run, it has to be hard enough to stick with in the short run that it causes investors to “fold,” passing the alpha to those with the fortitude to “hold.”
In other words, for a strategy to outperform in the long run, it must underperform in the short run. We call this The Frustrating Law of Active Management.

A few weeks ago, AQR published a piece titled Craftsmanship Alpha: An Application to Style Investing[1], to which Cliff Asness wrote a further perspective piece titled Little Things Mean a Lot[2].

We’ll admit that we are partial to the title “craftsmanship alpha” because portfolio craftsmanship is a concept we spend a lot of time thinking about. In fact, we have a whole section titled Portfolio Craftsmanship on the Investment Philosophy section of our main website.[3] We further agree with Cliff: little things do mean a lot. We even wrote a commentary about it in May titled Big Little Details[4].

But there was one quote from Cliff, in particular, that inspires this week’s commentary:

Let’s just make up an example. Imagine there are ten independent (uncorrelated) sources of “craftsmanship alpha” and that each adds 2 basis points of expected return at the cost of 20 basis points of tracking error from each (against some idea of a super simple “non-crafted” alternative.) Each is thus a 0.10 Sharpe ratio viewed alone. Together they are expected to add 20 basis points to the overall factor implementation inducing 63 basis points of tracking error (20 basis points times the square-root of ten). That’s a Sharpe ratio of 0.32 from the collective craftsmanship (in addition to the basic factor returns).
[…]
But, as many have noted in other contexts, a Sharpe ratio like 0.32 can be hard to live with. Its chance of subtracting from your performance in a given year is about 37%. Its chance of subtracting over five years is about 24%. And, wait for it… over twenty years the chance it subtracts is still about 8%. That’s right. There’s a non-trivial chance your craftsmanship is every bit as good as you think, and it subtracts over two full decades, perhaps the lion’s share of your career. Such is the unforgiving, uncaring math.

Whether it is structural alpha, style premia, or craftsmanship alpha: we believe that the very uncertainty and risk that manifests as (expected) tracking error is a necessary component for the alpha to exist in the first place.

The “unforgiving, uncaring math” that is a result – the fact that you can do everything right and still get everything wrong – is a concept that in the past we have titled The Frustrating Law[5] of Active Management.

Defining The Frustrating Law of Active Management

We define The Frustrating Law of Active Management as:

For any disciplined[6] investment approach to outperform over the long run, it must experience periods of underperformance in the short run.

As if that were not frustrating enough a concept – that even if we do everything right, we still have to underperform from time-to-time – we add this corollary:

For any disciplined investment approach to underperform over the long run, it must experience periods of outperformance in the short run.

In other words, even if a competing manager does everything wrong, they should still be rewarded with outperformance at some point. Talk about adding insult to injury.

For the sake of brevity, we will only explore the first half of the law in this commentary. Note, however, that the second law is simply the inverse case of the first. After all, if we found an investment strategy that consistently underperformed, we could merely inverse the signals and have a strategy that consistently outperforms. If the latter is impossible, so must be the former.

For it to work, it has to be hard

Let’s say we approach you with a new investment strategy. We’ve discovered the holy grail: a strategy that always outperforms. It returns an extra 2% over the market, consistently, every year, after fees.

Ignoring reasonable skepticism for a moment, would you invest? Of course you would. This is free money we’re talking about here!

In fact, everyone we pitch to would invest. Who wouldn’t want to be invested in such a strategy? And here, we hit a roadblock.

Everyone can’t invest. Relative performance is, after all, zero sum: for some to outperform, others must underperform. Our extra return has to come from somewhere.

If we do continue to accept money into our strategy, we will begin to approach and eventually exceed capacity. As we put money to work, we will create impact and inform the market, driving prices away from us. As we try to buy, prices will be driven up and as we try to sell, prices will be driven down. By chasing price, our outperformance will deteriorate.

And it needn’t even be us trading the strategy. Once people learn about what we are doing – and how easy it is to make money – others will begin to employ the same approach. Increasing capital flow will continue to erode the efficacy of the edge as more and more money chases the same, limited opportunities. The growth is likely to be exponential, quickly grinding our money machine quickly to a halt.

So, the only hope of keeping a consistent edge is in a mixture of: (1) keeping the methodology secret, (2) keeping our deployed capital well below capacity, and (3) having a structural moat (e.g. first-mover advantage, relationship-driven flow, regulatory edge, non-profit-seeking counter-party, etc).

While we believe that all asset managers have the duty to ensure #2 remains true (we highly recommend reading Alpha or Assets by Patrick O’Shaughnessy[7]), #1 pretty much precludes any manager actually trying to raise assets (with, perhaps, a few limited exceptions in the hedge fund world that can raise assets on brand alone).

The takeaway here is that if an edge is perceived as being easy to implement (i.e. not case #3 above) and easy to achieve, enough people will do it to the point that the edge is driven to zero.

Therefore, if an edge is known by many (e.g. most style premia like value, momentum, carry, defensive, trend, etc), then for it to persist over the long run, the outperformance must be difficult to capture. Remember: for outperformance to exist, weak hands must at some point “fold” (be it for behavioral or risk-based reasons), passing the alpha to strong hands that can “hold.”

This is not just a case of perception, either. Financial theory tells us that a strategy cannot always outperform its benchmark with certainty. After all, if it did, we would have an arbitrage: we could go long the strategy, short the benchmark, and lock in certain profit. As markets loathe (or, perhaps, love) arbitrage, such an opportunity should be rapidly chased away. Thus, for a disciplined strategy to generate alpha over the long run, it must go through periods of underperformance in the short-run.

Can We Diversify Away Difficulty?

Math tells us that we should be able to stack the benefits of multiple, independent alpha sources on top of each other and simultaneously benefit from potentially reduced tracking error due to diversification.

Indeed, mathematically, this is true. It is why diversification is known as the only free lunch in finance.

This certainly holds for beta, which derives its value from economic activity. In theory, everyone can hold the Sharpe ratio optimal portfolio and introduce cash or leverage to hit their appropriate risk target.

Alpha, on the other hand, is explicitly captured from the hands of other investors. Contrary to the Sharpe optimal portfolio, everyone cannot hold the Information ratio optimal portfolio at the same time[8]. Someone needs to be on the other side of the trade.

Consider three strategies that all outperform over the long run: strategy A, strategy B, and strategy C. Does our logic change if we learn that strategy C is simply 50% strategy A plus 50% strategy B? Of course not! For C to continue to outperform over the long run, it must remain sufficiently difficult to stick with in the short-run that it causes weak hands to fold.

Conclusion

For a strategy to outperform in the long run, it has to be perceived as hard: hard to implement or hard to hold. For public, liquid investment styles that most investors have access to, it is usually a case of the latter.

This law is underpinned by two facts. First, relative performance is zero-sum, requiring some investors to underperform for others to outperform. Second, consistent outperformance violates basic arbitrage theories.

While coined somewhat tongue-in-cheek, we think this law provides an important reminder to investors about reasonable expectations. As it turns out, the proof is not always in the eating of the pudding. In fact, track records can be entirely misleading as validators of an investment process. As Cliff pointed out, even if our alpha source has a Sharpe ratio of 0.32, there is an 8% chance that it subtracts from performance over the next 20-years.

Conversely, even negative alpha sources can show beneficial performance by chance. An alpha source with a Sharpe ratio of -0.32 has an 8% chance that it adds to performance over the next 20-years.

And that’s why we call it The Frustrating Law of Active Management. For investors and asset managers alike, there is little more frustrating than knowing that to continue working over the long run, good strategies have to do poorly, and poor strategies have to do well over shorter timeframes.

[1] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3034472

[2] https://www.aqr.com/cliffs-perspective/little-things-mean-a-lot

[3] https://www.thinknewfound.com/investment-philosophy

[4] https://blog.thinknewfound.com/2017/05/big-little-details/

[5] To be clear that we don’t mean a “law” in the sense of an inviolable, self-evident axiom. In truth, our “law” is much closer to a “theory.”

[6] The disciplined component here is very important. By this, we mean a strategy that applies a consistent set of rules. We do not mean, here, a bifurcation of systematic versus discretionary. Over the years, we’ve met a large number of discretionary managers who apply a highly disciplined approach. Rather, we mean those aspects of an investment strategy that can be codified and turned into a set of systematically applied rules.

Thus, even a discretionary manager can be thought of as a systematic manager plus a number of idiosyncratic deviations from those rules. The deviations must be idiosyncratic, by nature. If there was a consistent reason for making the deviations, after all, the reason could be codified itself. Thus, true discretion only applies to unique, special, and non-repeatable situations.

Note that the discipline does not preclude randomness. You could, for example, flip a coin and use the result to make an investment decision every month. So long as the same set of rules is consistently applied, we believe The Frustrating Law of Active Management applies.

[7] http://investorfieldguide.com/alpha-or-assets/

[8] Well, technically they can if everyone is a passive investor. In this case, however, the information ratio would be undefined, with zero excess expected return and zero tracking error.

Category: Risk & Style Premia Page 13 of 16

Thinking in Long/Short Portfolios

Summary­

Conclusion

Timing Bonds with Value, Momentum, and Carry

Summary­­

Historically Speaking, This is a Bad Idea

This Might Now be a Good Idea

Timing Bonds with Value

Timing Bonds with Momentum

Timing Bonds with Carry

Building a Portfolio of Strategies

Conclusion

Factor Investing & The Bets You Didn’t Mean to Make

Summary­­

Value Investing with Countries

The Unintended Bet

Conclusion

Are Market Implied Probabilities Useful?

Summary­­

Interpreting Market Implied Probabilities

How Market Implied Probabilities Are Calculated

Are Market Implied Probabilities Useful?

Getting to Real World Probabilities

Deriving the Real-World Probability for a Large Move in the S&P 500

Conclusion

The Frustrating Law of Active Management

Summary­­

Defining The Frustrating Law of Active Management

For it to work, it has to be hard

Can We Diversify Away Difficulty?

Conclusion

Summary

Summary

**This Might Now be a Good Idea**

Summary

Summary

Summary