On this page, I’m going to explain the information content in the charts and tables, and also mathematically define the terms used.
To begin, consider the following chart, which shows the historical performance of a 10 month moving average crossover strategy in local currency Swedish equities from 1920 to 2015:
The items in the charts are as follows:
- The blue line is the performance of the market timing strategy–in this case, the 10 month moving average crossover strategy. We abbreviate it as “MMA.” Note that the chart identifies the moving average period as “10” in the upper left corner.
- The gray line is a strategy, abbreviated “RISK”, that buys and holds the risk asset. The risk asset is shown to the right of RISK in the legend against a green background–in this case, it’s Sweden, Swedish Equities.
- The thin black line on top of the gray line is the 10 month moving average, shown for reference. Given the 95 year time scale of the chart, it’s difficult to see the individual crossover points. But for charts on shorter time scales, the crossover points will be more clear.
- The red line is the X/Y portfolio, abbreviated “X/Y.” It consists of a 70.9/29.1 RISK/SAFE asset ratio: a constant 70.9% exposure to the risk asset (Swedish Equities), and a constant 29.1% exposure to the safe asset (Swedish T-Bills). Those exposures are the same exposures that the MMA strategy has to each asset, except that rather than taking on the exposures at the same time, in a mixed portfolio, MMA takes the exposures on different times, in a homogeneous portfolio.
- The yellow line is a strategy, abbreviated SAFE, that buys and holds the safe asset. The safe asset is shown against a green background to the right of SAFE in the legend–in this case, it’s Swedish T-Bills.
- The purple line is a strategy, abbreviated “ANTI”, that does the exact opposite of the market timing strategy. In the current example, ANTI is a strategy that buys when MMA sells, and sells when MMA buys. If a given market timing strategy is outperforming the risk asset, then we should expect ANTI to underperform the safe asset, and vice-versa.
- The gray columns are U.S. recessionary periods.
- The dotted green line displays the market timing strategy’s “outperformance”, calculated as the ratio between the strategy’s cumulative trailing total return and the cumulative trailing total return of a buy and hold strategy that fully invests in the risk asset. Note that it takes its measurements off of the right y-axis, rather than the left y-axis. In the places where it’s rising, the market timing strategy is outperforming, seeing a gain relative to the fully invested buy and hold strategy, usually because it’s out of the market during a price drop. In the places where it’s falling, the market timing strategy is underperforming, seeing a loss relative to the fully invested buy and hold strategy, usually because it’s out of the market during a price gain. In the places where it’s flat, it’s tracking perfectly with a fully invested buy and hold strategy, usually because it’s fully invested.
- For each strategy (RISK, MMA, X/Y, ANTI, SAFE), the yellow-highlighted matrix to the right of the chart gives the numerical values of different performance measurements. “ANN TR” refers to annualized total return. “VOL” refers to realized volatility. “MAX DD” refers to maximum drawdown. “SORTINO” refers to the Sortino Ratio. “SHARPE” refers to the Sharpe ratio. “% TIME IN” gives the total share of the time that the strategies are in the market. “% MOS SWITCHED” gives the total share of all months in which the strategies conduct switches–from the safe asset into the risk asset, or from the risk asset into the safe asset.
- The risk-free rate for the period, and the slip losses applied to each round-trip transaction, are shown underneath the matrix against a gray background.
Now, consider the following table, which shows the market (risk asset) entry-exit dates for the strategy:
Each line contains the date, the action on that date (exit or entry), the value of the risk asset total return index on that date, and the value of the safe asset total return index on that date. The hypothetical actions are taken at the close of the last day of the stated month, because the quoted price and total return index values are end-of-the-month closing values.
Now, the combination of an exit and a subsequent reentry will cause the market timing strategy to register a gain or loss relative to the performance of the market, the risk asset. A relative gain, shown against a green background, occurs when the strategy exits the market and reenters at a lower price. The return produced by the safe asset in the interim adds to that gain. A relative loss, shown against a red background, occurs when the strategy exits the market and reenters at a higher price. The return produced by the safe asset in the interim reduces that loss.
The following formula is used to calculate each relative gain (and loss):
Relative Gain (or Loss) = (SAFE_re / SAFE_ex) / (RISK_re / RISK_ex) – 1
SAFE_re, SAFE_ex, RISK_re and RISK_ex represent the values of the total return indices of the safe asset and risk asset upon reentry and exit, doubly respectively. To clarify the math with an example, if a strategy exits into cash before the market gets cut in half, then, assuming no return in the safe asset in the interim, the relative gain for the strategy will be ((1/1) / (.5/1) – 1 = 100%. The strategy will have doubled its trailing return relative to the market (by keeping a constant value while the market was cut in half). If, in contrast, a strategy exits into cash before the market doubles, then, assuming no return in the safe asset in the interim, the loss will be ((1/1) / (2/1)) – 1 = -50%. The strategy will have seen its trailing return cut in half relative to the market (by keeping a constant value while the market doubles).
The mini-table in the far right of the chart restates the annualized total returns of the fully invested buy and hold strategy (RISK), the market timing strategy (MMA), and the X/Y portfolio (X/Y). The table below that table shows the following:
- Win Share: The number of exit-entry pairs that produce relative gains on the index as a percentage of the number of exit-entry pairs in total. In simple terms, the win share tells us the following: when the strategy trades out of and back into the risk asset, how often does the trade turn out to be a good one?
- Average Winner: the average relative gain of all exit-entry pairs that involve relative gains.
- Average Loser: the average relative loss of all exit-entry pairs that involve relative losses.
- Average Exit Duration: the average amount of continuous time that the strategy spends out of the market when it exits. Alternatively, the average amount of time that it takes for the strategy to reenter the market upon exiting.
The best way to efficiently process the information in the tables is to note (1) the frequency at which you see green trades occur, and (2) how much the strategy appears to gain on each green trade. What you will find, as you peruse the results, is that green trades are rare in the strategy, but when they happen, they tend to be large and impactful.
With respect to the math, the calculation of annualized total return and maximum drawdown are relatively sraightforward. We calculate annualized total return over a given number of years Y using the following basic equation:
Annualized Total Return = (Final Total Return Index Value / Initial Total Return Index Value) ^ (1 / Y) – 1
We calculate maximum drawdown as the largest peak to trough loss in the data set, measured from any point in time, to any future point in time. In the spirit of keeping all assets on an equal footing, the drawdowns are quoted as total return losses, where the positive offsets of reinvested dividends are included.
For annualized total return, all numbers in the backtests are nominal numbers. Because the numbers are nominal, certain local currency equity indices (e.g., Turkey, Brazil) end up exhibiting very high values. These high values do not represent superior corporate performance, but instead represent the consequences of extreme inflation problems. Because equities are a real asset, a claim on the output of real capital, their nominal sales, profits, market prices and returns tend to track with inflation.
With respect to reward-to-risk ratios, any ratios that we choose are unfortunately going to be arbitrary, since their mathematical arrangements will end up putting certain relative weightings on reward and risk respectively. The problem is that there are no clear criteria for what those weightings should be. The question ultimately comes down to personal preference: which is more important to you, as an investor–making money, or not temporarily losing money?
The most popular reward-to-risk ratio is the Sharpe ratio, which we will calculate on an annualized basis using the following formula:
Sharpe Ratio = (average of monthly excess returns / standard deviation of monthly excess returns) * sqrt (12).
The drawback to the Sharpe Ratio is that it punishes strategies for generating upside volatility, when there is no sound reason for an investor to be averse to upside volatility. What should matter to an investor is downside volatility.
The Sortino Ratio improves on the Sharpe Ratio by including only downside volatility in the denominator. It will be our preferred reward-to-risk ratio. We will calculate it on an annualized basis as follows:
Sortino Ratio = (average of monthly excess returns / standard deviation of negative monthly excess returns) * sqrt (12)
We should point out that though this definition of the Sortino Ratio is the most common definition given, it’s technically incorrect, or at least different from the definition put forth by the ratio’s original author. The difference, however, does not make a difference–comparisons and rankings that use the more common version end up virtually identical to those that use the more correct one. I prefer the incorrect version, for reasons that are beyond the scope of this piece.
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