Presenting The Keller Ratio

by: TrendXplorer
Summary

Many traditional return to risk measures are not apt for intuitive interpretation.

The Keller ratio is expressed as an adjusted return and therefore easy to interpret.

The Keller ratio allows for strategy selection optimally aligned with an investor’s risk appetite.

In our Vigilant Asset Allocation (VAA) paper (article), we introduced a new metric for assessing a portfolio’s equity line in terms of the reward to risk relationship: return adjusted for drawdown (RAD). We did choose RAD above the usual risk measures like the Sharpe and the MAR ratios (Sharpe: return divided by volatility; MAR: return divided by maximum drawdown), because most retail investors commonly identify true risk with maximum drawdown over volatility. Since RAD is an adjusted return, its interpretation is similar to any return (a simple percentage). For this reason we prefer RAD over MAR, which as such is just a numeric value with little context.

Frankly, albeit "return adjusted for drawdown" states exactly what RAD is all about, it is quite a mouthful. Therefore, and not only because RAD is his brainchild, but also to commemorate Wouter Keller’s contributions to the Tactical Asset Allocation (TAA) literature (FAA, MAA, CAA, EAA, PAA, and VAA; see SSRN) it only seems fitting to accredit the return adjusted for drawdown indicator with his name. So henceforward RAD is to be named the “Keller ratio.”

Every investor with skin in the game acknowledges a large portfolio drawdown as the ultimate investing risk. Large drawdowns are devastating to long term returns. For example, during the 2008 subprime crisis the S&P 500 Total Return index crashed over 50% in approximately 1.5 years from its late 2007 peak, needing 3 years for recovery to breakeven. This left Buy & Hold investors without any positive returns for over nearly five years, not to speak of the excruciating anxiety along the way.

The following table illustrates how severe drawdowns wreak havoc to portfolio performance. Total loss of principal is the biggest risk of all.

Drawdown Rovery Drawdown destroys investment capital, hence the required recovery percentage to get back to breakeven grows exponentially with drawdown. Living through the drawdown quagmire often causes anxiety and can possibly even become a confidence shattering experience. So keeping drawdowns as small as possible is key for ultimate profitability.

As such, maximum drawdown is a “left tail risk,” because it is located in the outer left part of the statistician’s normal distribution chart. In backtests covering only a limited number of years, maximum drawdown may be a single event occurrence. For any meaningful assessment of drawdown long-term backtests are required, preferably extending over multiple decades and comprised of several bull-bear market cycles with many peak-to-trough declines.

The Keller ratio adjusts return for drawdown such as to reflect the severity of the observed maximum drawdown. In case maximum drawdown is small, the return adjustment is only limited. But with large maximum drawdown, the impact of the return adjustment is amplified, in similar fashion to the exponentially growing recovery percentages shown in the above table.

To recall from our VAA-paper, the formula for the Keller ratio (K=(RAD)) is:

K = R * ( 1 - D / ( 1 - D ) ), if R >= 0% and D <= 50%, and K = 0% otherwise,

where R = CAGR and D = Maximum Drawdown of the portfolio equity line over the chosen backtest period, with D expressed as a positive value, and for our models measured at month’s ends. This K measure is based on the observation that a maximum drawdown of 50% often leads to the liquidation of a hedge fund. In this case the Keller ratio becomes 0%, independent of CAGR.

The observant reader recognizes the term D / ( 1 - D ) in the Keller formula, which is the algebraic expression of the increase in price necessary for recovery to breakeven at the previous top portfolio capital level after a drawdown of D. At D = 50%, this price gain equals 100%, so the ratio becomes 0%, reflecting the difficulty of getting back to the previous portfolio peak level after a severe drawdown.

Next, by generalizing the Keller formula, an adjustable threshold parameter can be implemented at which the Keller ratio becomes 0%. This allows for a tailored Keller measurement for better reflecting an investor’s risk preference:

K( Dmax ) = R * ( 1 - f * D / ( 1 - f * D ) ), where f = 0.5 / Dmax,
with Dmax being D at which K=0%

Accordingly, using 50%, 25%, 20%, and 10% as the respective threshold parameters, the formula for the Keller ratio becomes:

K(50%) = R * ( 1 - D / ( 1 - D ) ), if R >= 0% and D <= 50%, and
K(50%) = 0% otherwise.
K(25%) = R * ( 1 - 2D / ( 1 - 2D ) ), if R >= 0% and D <= 25%, and
K(25%) = 0% otherwise.
K(20%) = R * ( 1 - 2.5D / ( 1 - 2.5D ) ), if R >= 0% and D <= 20%, and K(20%) = 0% otherwise.
K(10%) = R * ( 1 - 5D / ( 1 - 5D ) ), if R >= 0% and D <= 10%, and
K(10%) = 0% otherwise.

The following table crystallizes the effect of the mentioned Keller thresholds for the VAA-G12 strategy covering 1970 - 2017 (mid) as described in our VAA-paper (see also end notes):

Keller Ratio Table NB! Results are derived from simulated monthly total return data. Furthermore, trading costs, slippage, and taxes are disregarded. Results are therefore purely hypothetical. Past performance is no guarantee of future results.

Using the Keller threshold as the allowed maximum portfolio drawdown, the Keller ratio allows the investor to select the strategy scenario optimally aligned with his risk appetite. An investor with an offensive risk profile might still feel comfortable with high drawdowns, therefore using K(50%) as scenario selector resulting in the T1/B4 setting. An investor with a moderate risk tolerance can select K(25%) as threshold value, leading to preference for the T5/B4 setting. In this particular instance the same diversified T5/B4 scenario happens to be selected too with the threshold lowered to K(20%) or K(10%) respectively. Other TAA strategies may offer broader ranges of choice for risk targeting through heightening or lowering the Keller threshold.

Point to note: With K(20%), by design the indicator value for the T1/B4 scenario becomes 0%, because the registered maximum portfolio drawdown of the T1/B4 scenario (D = 21.13%) exceeds the chosen K(20%) threshold (Dmax = 20%). The same is true for the T1/B4, T2/B4, and T3/B4 scenarios with the K(10%) threshold deployed.

To elaborate our preference for the Keller ratio over the usual ones like the Sharpe and MAR ratio, let’s revisit the reasoning in our VAA-paper. The Sharpe ratio is defined as the annual return R (often in excess over a target return like the risk-free rate) divided by the annual volatility V of the returns. The MAR ratio (similar to the Calmar ratio) is simply annual return R divided by maximum drawdown D (expressed as D >= 0%). Both measures assume that you can apply leverage to arrive at higher R, V and D combinations with the same Sharpe and MAR ratio. But as we know from leveraged ETFs, this only holds for constant growth (combined with a lending rate equal to the risk-free rate). However, in practice the resulting Sharpe ratio will be much less after leverage. Furthermore, not all investors - especially not retail investors - have access to cheap leverage at (near) risk-free rates. Therefore, when optimizing TAA-models using Sharpe or MAR ratios as target, one might get stuck at relative low returns R with low risk, especially when using a low (near) risk-free or zero target return for the Sharpe threshold.

As an alternative for the Keller ratio the Ulcer Performance Index (UPI) or “Martin Ratio” springs to mind. UPI is return divided by the Ulcer index (UI), where the Ulcer Index measures the depth and duration of percentage drawdowns in price from earlier highs. The greater a drawdown in value, and the longer it takes to recover to earlier highs, the higher the UI. Technically, UI is the square root of the mean of the squared percentage drawdowns in value. The squaring effect penalizes large drawdowns proportionately more than small drawdowns. (From Peter G. Martin’s explanation at tangotools.com). UPI takes the entire drawdown record into account, which is statistically preferable. However, using UPI as a target frequently results in lower optimal returns and broad top selections, because in general those diversified tops coincide with smaller drawdowns.

In our current research the Keller ratio is preferred, especially because of its risk targeting through lowering or heightening of the drawdown threshold. However, without the multi decade In-Sample optimization/Out-of-Sample (IS/OS) validation approach, as adhered to in our research, where IS/OS each cover several market cycles, optimizing for the Keller ratio bears the risk of data snooping because maximum drawdown is just a single data point, prone to overfitting.

End notes

  • The VAA-strategy is explained is this post: Breadth Momentum and Vigilant Asset Allocation.
  • AllocateSmartly will begin tracking VAA-G12 T2/B4 "in the near future."
  • Detailed views at the performance of VAA-G12 for the T2/B4, T3/B4, T4/B4, and T5/B4 scenarios along with signals for VAA-G12 with T2/B4 and T5/B4 are available through TrendXplorer.info where this contribution is simultaneously published.

Disclosure: I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.