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^-11 is equal to 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 (S²), 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, S².  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.

 

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[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.