*This post is available as a PDF download here.*

# Summary

- In this commentary we explore the application of several quantitative signals to a broad set of fixed income exposures.
- Specifically, we explore value, momentum, carry, long-term reversals, and volatility signals.
- We find that value, 3-month momentum, carry, and 3-year reversals all create attractive quantile profiles, potentially providing clues for how investors might consider pursuing higher returns or lower risk.
- This study is by no means comprehensive and only intended to invite further research and conversation around the application of quantitative styles across fixed income exposures.

In *Navigating Municipal Bonds with Factors*, we employed momentum, value, carry, and low-volatility signals to generate a sector-based approach to navigating municipal bonds.

In this article, we will introduce an initial data dive into applying quantitative signals to a broader set of fixed income exposures. Specifically, we will incorporate 17 different fixed income sectors, spanning duration, credit, and geographic exposure.

**U.S. Treasuries**: Near (3-Month), short (1-3 Year), mid (3-5 Year) intermediate (7-10 Year), and long (20+ Year).**Investment-Grade Corporates**: Short-term, intermediate-term, and Floating Rate corporate bonds.**High Yield**: Short- and intermediate-term high yield.**International****Government Bonds**: Currency hedged and un-hedged government bonds.**Emerging Market**: Local and US dollar denominated.**TIPs**: Short- and intermediate-term TIPs.**Mortgage-Backed**: Investment grade mortgage-backed bonds.

In this study, each exposure is represented by a corresponding ETF. We extend our research prior to ETF launch by employing underlying index data the ETF seeks to track.

The quantitative styles we will explore are:

**Momentum:**Buy recent winners and sell recent losers.**Value:**Buy cheap and sell expensive.**Carry:**Buy high carry and sell low carry.**Reversal:**Buy long-term losers and sell long-term winners.**Volatility:**Buy high volatility and sell low volatility.^{1}

The details of each style are explained in greater depth in each section below.

Note that the analysis herein is by no means meant to be prescriptive in any manner, nor is it a comprehensive review. Rather, it is meant as a launching point for further commentaries we expect to write.

At the risk of spoiling the conclusion, below we plot the annualized returns and volatility profiles of dollar-neutral long-short portfolios.^{2} We can see that short-term Momentum, Value, Carry, and Volatility signals generate positive excess returns over the testing period.

Curiously, longer-term Momentum does not seem to be a profitable strategy, despite evidence of this approach being rather successful for many other asset classes.

*Source: Bloomberg; Tiingo. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all management fees, transaction fees, and taxes, but net of underlying fund fees. Total return series assumes the reinvestment of all distributions.*

However, these results are not achievable by most investors who may be constrained to a long-only implementation. Even when interpreted as over- and under-weight signals, the allocations in the underlying long/short portfolios differ so greatly from benchmark exposures, they would be nearly impossible to implement.

For a long-only investor, then, what is more relevant is how these signals forecast performance of different rank orderings of portfolios. For example, how does a portfolio of the best-ranking 3-month momentum exposures compare to a portfolio of the worst-ranking?

In the remainder of this commentary, we explore the return and risk profiles of quintile portfolios formed on each signal. To construct these portfolios, we rank order our exposures based on the given quantitative signal and equally-weight the exposures falling within each quintile.

**Momentum**

We generate momentum signals by computing 12-, 6- and 3- month prior total returns to reflect slow, intermediate, and fast momentum signals. Low-ranking exposures are those with the lowest prior total returns, while high ranking exposures have the highest total returns.

The portfolios assume a 1-month holding period for momentum signals. To avoid timing luck, four sub-indexes are used, each rebalancing on a different week of the month.

Annualized return and volatility numbers for the quintiles are plotted below.

A few interesting data-points stand out:

- For 12-month prior return, the lowest quintile actually had the highest total return.However, it has a dramatically lower Sharpe ratio than the highest quintile, which only slightly underperforms it.
- Total returns among the highest quintile increase by 150 basis points (“bps”) from 12-month to 3-month signals, and 3-month rankings create a more consistent profile of increasing total return
*and*Sharpe ratio. This may imply that short-term signals are more effective for fixed income.

*Source: Bloomberg; Tiingo. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all management fees, transaction fees, and taxes, but net of underlying fund fees. Total return series assumes the reinvestment of all distributions.*

**Carry**

Carry is the expected excess return of an asset assuming price does not change. For our fixed income universe, we proxy carry using yield-to-worst minus the risk-free rate. For non-Treasury holdings, we adjust this figure for expected defaults and recovery.

For reasonably efficient markets, we would expect higher carry to imply higher return, but not necessarily higher *risk-adjusted *returns. In other words, we earn higher carry as a reward for bearing more risk.

Therefore, we also calculate an alternate measure of carry: carry-to-risk. Carry-to-risk is calculated by taking our carry measure and dividing it by recent realized volatility levels. One way of interpreting this figure is as forecast of Sharpe ratio. Our expectation is that this signal may be able to identify periods when carry is episodically cheap or rich relative to prevailing market risk.

The portfolios assume a 12-month holding period for carry signals. To avoid timing luck, 52 sub-indexes are used, each rebalancing on a different week of the year.

We see:

- Higher carry implies a higher return as well as a higher volatility. As expected, no free lunch here.
- Carry-to-risk does not seem to provide a meaningful signal. In fact, low carry-to-risk outperforms high carry-to-risk by 100bps annualized.
- Volatility meaningfully declines for carry-to-risk quintiles, potentially indicating that this integrated carry/volatility signal is being too heavily driven by volatility.

*Source: Bloomberg; Tiingo. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all management fees, transaction fees, and taxes, but net of underlying fund fees. Total return series assumes the reinvestment of all distributions.*

**Value**

In past commentaries, we have used real yield as our value proxy in fixed income. In this commentary, we deviate from that methodology slightly and use a time-series z-score of carry as our value of measure. Historically high carry levels are considered to be cheap while historically low carry levels are considered to be expensive.

The portfolios assume a 12-month holding period for value signals. To avoid timing luck, 52 sub-indexes are used, each rebalancing on a different week of the year.

We see not only a significant increase in total return in buying cheap versus expensive holdings, but also an increase in risk-adjusted returns.

* *

**Reversal**

Reversal signals are the opposite of momentum: we expect past losers to outperform and past winners to underperform. Empirically, reversals tend to occur over very short time horizons (e.g. 1 month) and longer-term time horizons (e.g. 3- to 5-years). In many ways, long-term reversals can be thought of as a naive proxy for value, though there may be other behavioral and structural reasons for the historical efficacy of reversal signals.

We must be careful implementing reversal signals, however, as exposures in our universe have varying return dynamics (e.g. expected return and volatility levels).

To illustrate this problem, consider the simple two-asset example of equities and cash. A 3-year reversal signal would sell the asset that has had the best performance over the prior 3-years and buy the asset that has performed the worst. The problem is that we expect stocks to outperform cash due to the equity risk premium. Naively ranking on prior returns alone would have us out of equities during most bull markets.

Therefore, we must be careful in ranking assets with meaningfully different return dynamics.

(Why, then, can we do it for momentum? In a sense, momentum is explicitly trying to exploit the relative time-series properties over a short-term horizon. Furthermore, in a universe that contains low-risk, low-return assets, cross-sectional momentum can be thought of as an integrated process between time-series momentum and cross-sectional momentum, as the low-risk asset will bubble to the top when absolute returns are negative.)

To account for this, we use a time-series z-score of prior returns to create a reversal signal. For example, at each point in time we calculate the current 3-year return and z-score it against all prior rolling 3-year periods.

Note that in this construction, high z-scores will reflect higher-than-normal 3-year numbers and low z-scores will reflect lower-than-normal 3-year returns. Therefore, we negate the z-score to generate our signal such that low-ranked exposures reflect those we want to sell and high-ranked exposures reflect those we want to buy.

The portfolios assume a 12-month holding period for value signals. To avoid timing luck, 52 sub-indexes are used, each rebalancing on a different week of the year.

Plotting the results below for 1-, 3-, and 5-year reversal signals, we see that 3- and 5-year signals see a meaningful increase in both total return and Sharpe ratio between the lowest quintile.

**Volatility**

Volatility signals are trivial to generate: we simply sort assets based on prior realized volatility. Unfortunately, exploiting the low-volatility anomaly is difficult without leverage, as the empirically higher risk-adjusted return exhibited by low-volatility assets typically coincides with lower total returns.

For example, in the tests below the low quintile is mostly comprised of short-term Treasuries and floating rate corporates. The top quintile is allocated across local currency emerging market debt, long-dated Treasuries, high yield bonds, and unhedged international government bonds.

As a side note, for the same reason we z-scored reversal signals, we also hypothesized that z-scoring may work on volatility. Beyond these two sentences, the results were nothing worth writing about.

Nevertheless, we can still attempt to confirm the existence of the low-volatility anomaly in our investable universe by ranking assets on their past volatility.

The portfolios assume a 1-month holding period for momentum signals. To avoid timing luck, four sub-indexes are used, each rebalancing on a different week of the month.

Indeed, in plotting results we see that the lowest volatility quintiles have significantly higher realized Sharpe ratios.

Of the results plotted above, our eyes might be drawn to the results in the short-term volatility measure. It would appear that the top quintile has both a lower total return and much higher volatility than the 3^{rd }and 4^{th }quintiles. This might suggest that we could improve our portfolios risk-adjusted returns without sacrificing total return by avoiding those top-ranked assets.

Unfortunately, this is not so clear cut. Unlike the other signals where the portfolios had meaningful turnover, these quintiles are largely stable. This means that the results are driven more by the composition of the portfolios than the underlying signals. For example, the 3^{rd }and 4^{th }quintiles combine both Treasuries and credit exposure, which allows the portfolio to realize lower volatility due to correlation. The highest volatility quintile, on the other hand, holds both local currency emerging market debt and un-hedged international government bonds, introducing (potentially uncompensated) currency risk into the portfolio.

Thus, the takeaway may be more strategic than tactical: diversification is good and currency exposure is going to increase your volatility.

Oh – and allocating to zero-to-negatively yielding foreign bonds isn’t going to do much for your return unless currency changes bail you out.

**Conclusion**

In this study, we explored the application of value, momentum, carry, reversal, and volatility signals across fixed income exposures. We found that value, 3-month momentum, carry, and 3-year reversal signals may all provide meaningful information about forward expected returns and risk.

Our confidence in this analysis, however, is potentially crippled by several points:

- The time horizon covered is, at best, two decades, and several economic variables are constant throughout it.
- The inflation regime over the time period was largely uniform.
- A significant proportion of the period covered had near-zero short-term Treasury yields and negative yields in foreign government debt.
- Reversal signals require a significant amount of formation data. For example, the 3-year reversal signal requires 6 years (i.e. 3-years of rolling 3-year returns) of data before a signal can be generated. This represents nearly 1/3
^{rd }of the data set. - The dispersion in return dynamics (e.g. volatility and correlation) of the underlying assets can lead to the emergence of unintended artifacts in the data that may speak more to portfolio composition than the value-add from the quantitative signal.
- We did not test whether certain exposures or certain time periods had an outsized impact upon results.
- We did not thoroughly test stability regions for different signals.
- We did not test the impact of our holding period assumptions.
- Holdings within quantile portfolios were assumed to be equally weighted.

Some of these points can be addressed simply. Stability concerns, for example, can be addressed by testing the impact of varying signal parameterization.

Others are a bit trickier and require more creative thinking or more computational horsepower.

Testing for the outsized impact of a given exposure or a given time period, for example, can be done through sub-sampling and cross-validation techniques. We can think of this as the application of randomness to efficiently cover our search space.

For example, below we re-create our 3-month momentum quintiles, but do so by randomly selecting only 10 of the exposures and 75% of the return period to test. We repeat this resampling 10,000 times for each quintile and plot the distribution of annualized returns below.

Even without performing an official difference-in-means test, the separation between the low and high quintile annualized return distributions provides a clue that the performance difference between these two is more likely to be a pervasive effect rather than due to an outlier holding or outlier time period.

We can make this test more explicit by using this subset resampling technique to bootstrap a distribution of annualized returns for a top-minus-bottom quintile long/short portfolio. Specifically, we randomly select a subset of assets and generate our 3-month momentum signals. We construct a dollar-neutral long/short portfolio by going long assets falling in the top quintile and short assets falling in the bottom quintile. We then select a random sub-period and calculate the annualized return.

Only 207 of the 10,000 samples fall below 0%, indicating a high statistical likelihood that the outperformance of recent winners over recent losers is not an effect dominated by a specific subset of assets or time-periods.

While this commentary provides a first step towards analyzing quantitative style signals across fixed income exposures, more tests need to be run to develop greater confidence in their efficacy.

## Quantifying Timing Luck

By Corey Hoffstein

On January 22, 2018

In Craftsmanship, Risk Management, Weekly Commentary

This blog post is available as a PDF download here.SummaryAs a pre-emptive warning, this week’s commentary is a math derivation. We think it is a very

relevantderivation – one which we have not seen before – but a derivation nonetheless. If math is not your thing, this might be one to skip.If math

isyour thing: consider this a request for comments. The derivation here will be rather informal sketch, and we think there are other improvements still lingering.What is “Timing Luck?”The basic concept of timing luck is that

whenwe choose to rebalance can have a profound impact on our performance results. For example, if we rebalance an investment strategy once a month, the choice to rebalance at the end of the month will lead to different performance than had we elected to rebalance mid-month.We call this performance differential “timing luck,” and we believe it is an overlooked, non-negligible portfolio construction risk.

As an example, consider a simple stock/cash timing model that rebalances monthly, investing in a broad U.S. equity index when its 12-1 month return is positive, and a constant maturity 1-year U.S. Treasury index otherwise. Depending on which day of the month you choose to rebalance (we will assume 21 variations to represent 21 trading days), your results may be dramatically different.

Source: Kenneth French Data Library, Federal Reserve of St. Louis. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.The best performing strategy had an annualized return of 11.1%, while the worst returned just 9.6%. Compounded over 55 years, and that 150 basis point (“bps”) differential leads to an astounding difference in final wealth. With a standard deviation between 50-year annualized returns of 0.42%, the 1-year annualized estimate of performance variation due to timing luck is 314bps!

Again, an identical process is employed: the only difference between these results is the choice of what day of the month to rebalance.

That small choice, and the good luck or misfortune it realizes, can easily be the difference between “hired” and “fired.”

Is There a Solution to Timing Luck?In the past, we have argued that

overlapping portfolioscan be utilized to minimize the impact of timing luck. The idea of overlapping portfolios is as follows: given an investment process and a holding period, we can invest across multiple managers that invest utilizing the same process but have offset holding periods.[1]For example, below each manager has a four time-step holding period, and we utilize four managers to minimize timing luck from a single implementation.

The proof that this approach minimizes timing luck is as follows.

Assume that we have N managers, all following an identical investment process with identical holding period, but whose rebalance points are offset from one another by one period.

Consider that at any point in time, we can define the portfolio of Manager #2 to be the portfolio of Manager #1 plus a dollar-neutral long/short portfolio that captures the differences in holdings between them. Similarly, Manager #3’s portfolio can be thought of as Manager #2’s portfolio plus a dollar-neutral long/short portfolio. This continues in a circular manner, where Manager #1’s portfolio can be thought of as Manager #N’s portfolio plus a dollar-neutral long/short.

Given that the managers all follow an identical process, we would expect them to have the same long-term expected return. Thus, the expected return of the dollar-neutral long/short portfolios is zero.

However, the

varianceof the dollar-neutral long/short portfolios captures the risk of timing luck.In allocating capital between the N portfolios, our goal is to minimize timing luck. Put another way, we want to find the allocation that results in the minimum variance portfolio of the long/short portfolios. Fortunately, there is a simple, closed form solution for calculating the minimum variance portfolio:

Here,

wis our solution (an Nx1 vector of weights), Sigma is the covariance matrix and is an Nx1 vector of 1s. To solve this equation, we need the covariance matrix between the long/short portfolios. Since each portfolio is employing an identical process, we can assume that each of the long/short portfolios should have equal variance. Without loss of generality, we can assume variances are equal to 1 and replace our covariance matrix, Sigma, with a correlation matrix, C.The correlations between long/short portfolios will largely depend on the process in question and the amount of overlap between portfolios. That said, because each manager runs an identical process, we would expect that the long-term correlation between Portfolio #2’s long/short and Portfolio #1’s long/short to be identical to the correlation between Portfolio #3’s long/short and Portfolio #2’s. Similarly, the correlation between Portfolio #3’s and Portfolio #1’s long/shorts should be the same as the correlation between Portfolio #N’s and Portfolio #2’s.

Following this logic (and remembering the circular nature of the rebalances), we can ignore exact numbers and fill in a correlation matrix using variables:

This correlation matrix has two special properties. First, being a correlation matrix, it is symmetric. Second, it is circulant: each row is rotated one element to the right of the preceding row. A special property of a symmetric circulant matrix is that its inverse – in this case

C– is also symmetric circulant. This property guarantees that^{-1}Cis equal to^{-1}1k1for some constantk.Which means we can re-write our minimum variance solution as:

Since the constant will cancel out, we are left with:

Thus, our optimal solution is an equal-weight allocation to all N portfolios.

Highlighted in gold below, we can see the result of this approach using the same stock/cash example as before. Specifically, the gold portfolio uses each of the 21 variations as a different sub-portfolio.

Source: Kenneth French Data Library. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.While we have a solution for timing luck, a question that lingers is: “how much will timing luck affect my particular strategy?”

The SetupWe assume an active investment strategy with constant portfolio of variance (

S), constant and continuous annualized turnover (^{2}T; e.g. 0.5 for 50% annual turnover), and consistent rebalances at discrete frequency (f; e.g. 1/12 for monthly).We will also assume that the portfolio contains no static components. This allows us to interpret 100% turnover as meaning that the entire portfolio was turned over, rather than that 50% of the portfolio was turn over twice.

To quantify the magnitude of timing luck, we will calculate the variance of a dollar-neutral, long/short portfolio that is long a discrete implementation (i.e. rebalancing at a fixed interval) of this strategy (D) and short the theoretically optimal infinite overlapping portfolio implementation (M – for “meta”).

As before, the expected return of this long/short is zero, but its variance captures the return differences created by timing luck.

Differences between the Discrete and Continuous PortfoliosThe long/short portfolio is defined as (D – M). However, we would expect the holdings of D to overlap with the holdings of M. How much overlap will depend on both portfolio turnover and rebalance frequency.

Assume, for a moment, that M does not have infinite overlapping portfolios, but a finite number N, each uniformly spaced across the holding period.

If we assume 100% turnover that is continuous, we would expect that the first overlapping portfolio, implemented at t=1/N, to have (1 – 1/N) percent of its holdings identical to D (i.e. not “turned over”). On the other hand, the portfolio implemented at t = (N-1)/N will have just 1/N percent of its holdings identical to D.

Thus, we can say that if M contains N discrete overlapping portfolios, we can expect M and D to overlap by:

Which we can reduce,

If we take the limit as N goes to infinity – i.e. we have infinite overlapping portfolios – then we are simply left with:

Thus, the overlap we expect between our discretely implemented portfolio, D, and the portfolio with infinite overlapping portfolios, M, is a simple function of the expected turnover during the holding period.

We can then define our long/short portfolio:

Where Q is the portfolio of holdings in M that are

notin D.We should pause here, for a moment, as this is where our assumption of “no static portfolio elements” becomes relevant. We defined (1) to be the amount M and D overlap. Technically, if we allow securities to be sold and then repurchased, (1) represents a lower limit to how much M and D overlap. As an absurd example, consider a portfolio that creates 100% turnover by buying and selling the same 1% of the portfolio 100 times. Thus, Q in (6) need not necessarily be unique from D; part of D could be contained in Q.

By assuming that no part of the portfolio is static, we are assuming that over the (very) long run, the average turnover experience over a holding period does not include repurchase of sold securities, and thus (1) is the amount of overlap and D and Q are independent holdings.

This assumption is likely fairer for traditionally active portfolios that focus on security selection, but potentially less realistic for tactical strategies that often sell and re-purchase the same exposure. More on this later.

Defining,

We can re-write,

Solving for Timing LuckWe can then solve for the variance of the long/short portfolio,

Expanding:

As D and Q both represent viable allocation schemes for the portfolio, we will assume that they share the same long-term portfolio variance, S

^{2}. This assumption may be fair, over the long run, for traditional stock-selection portfolios, but likely less fair for highly tactical portfolios that can meaningfully shift their portfolio risk exposures.Thus,

Replacing back our definition for

a, we are left with:Or, that the annualized volatility due to timing luck (L) is:

What is Corr(D,Q)?The least easily interpreted – or calculated – term in our equation is the correlation between our discrete portfolio, D, and the non-overlapping securities found in the infinite overlapping portfolios implementation, Q.

The intuitive interpretation here is that when the securities held in our discrete portfolio are highly correlated to those that are not held but the optimal strategy recommends we hold, then we would expect the difference to have less impact. On the other hand, if those securities are negatively correlated, then the discrete rebalance choice could lead to significant additional volatility.

Estimating this value, however, may be difficult to do empirically.

One potential answer is to use the intra-portfolio correlation (“IPC”) of an equal-weight portfolio of representative assets or securities. The intuition here is that we expect each asset to experience, on average, an equivalent amount of turnover due to our assumption that there are no static positions in the portfolio.

Thus, taking the IPC of an equal-weight portfolio of representative securities allows us to express the view that while we do not know which securities will be different at any given point in time, we expect over the long-run that all securities will be “missing” with equal frequency and magnitude, and therefore the IPC is representative of the long-term correlation between D and Q.

Estimating Timing Luck in our Stock/Cash Tactical StrategyThe assumptions required for our estimate of timing luck may work well with traditional security selection portfolios (or, at least, quantitative implementations of factors like value, momentum, defensive etc.), but will it work with tactical portfolios?

Using our prior stock/cash example, let’s estimate the expected magnitude of timing luck. Using one of the discrete implementations, we estimate that turnover is 67% per year. Our rebalance frequency is monthly (1/12) and the intra-portfolio correlation between stocks and bonds is assumed to be 0%. Finally, the long-term volatility of the strategy is about 12.2%.

Using these figures, we estimate:

This is a somewhat disappointing result, as we had calculated prior that the actual timing luck was 314bps. Our estimate is less than 1/6

^{th}of the actual figure!Part of the problem may be that many of the assumptions we outlined are violated with our example tactical strategy. We think the bigger problem is that our

estimatesfor these variables, when using a highly tactical strategy, are simply wrong.In our equation, we assumed that turnover would be continuous. This is because we are using turnover as a proxy for the decay speed of our alpha signal.

What does this mean? As an example, value strategies rely on value signals that tend to decay slowly. When a stock is identified as being a value stock, it tends to stay that way for some time. Therefore, if you build a portfolio off of these signals, you would expect low turnover. Momentum signals, on the other hand, tend to decay more quickly. A stock that is labeled as high momentum this month may no longer be high momentum in three months’ time. Thus, momentum strategies tend to be high turnover.

This relationship does not necessarily hold for tactical strategies.

In our tactical example, we rebalance monthly because we believe the time-series momentum has a short forecast horizon. However, with only two assets, the strategy can go years without turnover. Worse, the same strategy might

missa signal because it is only sampling in a discrete manner and therefore understate true turnover in a continuous framework.If we were to look at the turnover of a tactical strategy implemented with the same rules but rebalanced

daily, we would see a turnover rate over 300%. This would increase our estimate up to 215bps. Still well below the realized 314bps, but certainly high enough to raise eyebrows about the impact of timing luck in tactical portfolios not implemented using overlapping portfolios.We should also remember that timing luck is determined by the difference in holdings between the discrete strategy and the meta strategy. We had assumed that the portfolios D and Q would have the same volatility, but in a strategy that shifts between stocks and bonds, this most certainly is not the case. This means that long-run volatility in such a tactical strategy can actually be misleadingly low.

Consider the situation when the tactical strategy goes to cash based upon a short-lived signal; i.e. the meta strategy will not build a significant cash position. The realized volatility of the strategy will dampen the perceived timing luck, when in reality the volatility difference between the two portfolios is quite large.

In our specific tactical example, we know that when D is stocks, Q is bonds and vice versa. With this insight, we can re-write equation (10):

Which we can simplify as:

Which is simply just a constant times the variance of a portfolio that is 100% long stocks and -100% short bonds (or vice versa; the variance will be the same).

If we use this equation and the variance of a long/short stock/bond portfolio and our prior estimate of 300% turnover, we get an estimate of timing luck volatility of 191bps.

Note that using this concept, there may be a more generic solution that is possible using some measure of active variance (likely scaled by active share).ConclusionIn this piece we have demonstrated the potentially massive impact of timing luck, addressed how to solve for it, and derived a model that can be used to estimate the magnitude of timing luck risk in strategies that do not employ an overlapping portfolios technique.

While our derived approach is not perfect – as we saw in its application with our tactical example – we believe it is an important step forward in being able to quantify the potential risk that timing luck creates.

[1] In reality, we probably wouldn’t hire a different manager to implement the same strategy with different rebalance timing even if we could find such managers. A more feasible solution would be for a single manager to run different sleeves implementing each rebalance iteration.