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

**Summary**

- When two managers implement identical strategies, but merely choose to rebalance on different days, we call variance between their returns “timing luck.”
- Timing luck can easily be overcome by using a method of overlapping portfolios, but few firms do this in practice.
- We believe the magnitude of timing luck impact is much larger than most believe, particularly in tactical strategies.
- We derive a model to estimate the impact of timing luck, using only values that can be easily estimated from portfolios implemented without the overlapping portfolio technique.
- We find that timing luck looms large in many different types of strategies.

As a pre-emptive warning, this week’s commentary is a math derivation. We think it is a very *relevant *derivation – 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 *is* your 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 *when* we 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 portfolios* can 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 *variance *of 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, *w* is 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 ^{-1}* – is also symmetric circulant. This property guarantees that

*C*is equal to

^{-1}1*k1*for some constant

*k*.

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 Setup**

We assume an active investment strategy with constant portfolio of variance (*S ^{2}*), constant and continuous annualized turnover (

*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 Portfolios**

The 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 *not *in 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 Luck**

We 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 Strategy**

The 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 *estimates *for 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 *miss* a 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).*

**Conclusion**

In 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.

## 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

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 IdeaLet’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?, 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.[2]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.”

ThisMightNow be a Good IdeaYetthis 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 ValueFollowing 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

currentreal yield against atrailing averageof real yield will almost surely lead to anovervaluedconclusion, 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

differencefrom 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 excessof 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 MomentumFor 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.

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.(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.

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.Ironically, by doing

worsemid-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 CarryCarry 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 StrategiesWith 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.

StrategyAnnualized ReturnAnnualized VolatilityInformationRatio

MaxDrawdown

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. Performance measured from 6/1974 to 1/2018, representing the full overlapping investment period of the strategies.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/SMomentum L/SCarry L/SValue L/SMomentum L/SCarry L/STo 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.

ConclusionBond 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.