Count the periods the manager beat the index, then put the size of the wins beside the frequency, or the number misleads.
A fund's batting average is the number of periods in which it beat its benchmark, divided by the number of periods measured. An illustrative equity fund that beat its index in 5 of 12 quarters has a batting average of 41.7 per cent, and still outperformed by 1.14 points a year, because its average winning quarter (1.16 points) was 3.01 times its average losing quarter (0.39 points). The batting average only means something next to that win/loss ratio.
Worked in full in Asset Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Consultants put the batting average on every manager scorecard because it is easy to read: above 50 per cent, the manager wins more often than it loses. Read alone, it is also easy to misread. A manager can bat below 50 per cent and add value, or bat 60 per cent and lose money. The calculation takes five minutes; the interpretation needs two more numbers.
| Quarter | Fund | Benchmark | Excess | Result |
|---|---|---|---|---|
| Q1 | 4.5 | 3.1 | +1.4 | Win |
| Q2 | −1.5 | −2.4 | +0.9 | Win |
| Q3 | 4.2 | 4.5 | −0.3 | Loss |
| Q4 | 0.8 | 1.2 | −0.4 | Loss |
| Q5 | −4.2 | −5.8 | +1.6 | Win |
| Q6 | 2.7 | 2.9 | −0.2 | Loss |
| Q7 | 5.5 | 6.1 | −0.6 | Loss |
| Q8 | −0.9 | −1.7 | +0.8 | Win |
| Q9 | 0.5 | 0.8 | −0.3 | Loss |
| Q10 | 3.1 | 3.6 | −0.5 | Loss |
| Q11 | −2.1 | −3.2 | +1.1 | Win |
| Q12 | 2.0 | 2.4 | −0.4 | Loss |
Batting average = periods with positive excess return ÷ total periods = 5 ÷ 12 = 41.7%
Excel, with excess returns in C2:C13: =COUNTIF(C2:C13,">0")/COUNT(C2:C13)
Up-market batting average = wins in quarters when the benchmark rose ÷ up quarters = 1 ÷ 8 = 12.5%
Down-market batting average = wins in quarters when the benchmark fell ÷ down quarters = 4 ÷ 4 = 100.0%
Excel: =COUNTIFS(B2:B13,">0",C2:C13,">0")/COUNTIF(B2:B13,">0")
Average win = (1.4 + 0.9 + 1.6 + 0.8 + 1.1) ÷ 5 = 5.8 ÷ 5 = 1.16 points
Average loss = (0.3 + 0.4 + 0.2 + 0.6 + 0.3 + 0.5 + 0.4) ÷ 7 = 2.7 ÷ 7 = 0.386 points
Win/loss ratio = 1.16 ÷ 0.386 = 3.01
Mean excess per period = batting average × average win − (1 − batting average) × average loss
= 41.7% × 1.16 − 58.3% × 0.386 = 0.258 points a quarter
Step 4 is the identity that makes the batting average useful. The mean excess return is exactly the batting average times the average win, less the losing frequency times the average loss. Here it checks to the sum of the excess column: 3.1 points over 12 quarters is 0.258 a quarter.
Compounded, the fund returned 15.1 per cent over the three years against 11.4 per cent for the benchmark: 4.79 per cent a year against 3.65 per cent, an annualised excess of 1.14 points. A scorecard that showed only the 41.7 per cent batting average would have flagged this manager for review.
The split by market direction explains the pattern. The fund lost to a rising index in 7 of 8 up quarters, by small amounts, and beat a falling index in all 4 down quarters, by larger ones. That is the signature of a low-beta or quality-tilted portfolio: it gives up a little in every rally and protects a lot in every sell-off. The same story appears in upside and downside capture ratios, which measure the size of the participation rather than its frequency.
Rearranging the identity, a manager breaks even when the win/loss ratio equals (1 − batting average) ÷ batting average.
| Batting average | Break-even win/loss ratio |
|---|---|
| 30.0% | 2.33 |
| 40.0% | 1.50 |
| 41.7% | 1.40 |
| 50.0% | 1.00 |
| 60.0% | 0.67 |
| 70.0% | 0.43 |
At 41.7 per cent the fund needs its average win to be at least 1.40 times its average loss, an average win of 0.54 points against the 0.386 loss. It delivers 1.16. If its winning quarters shrank to 0.80 points the mean excess would fall to 0.108 points a quarter, and at 0.40 points it would turn negative, at −0.058, with the batting average unchanged.
A batting average is a proportion, so it carries sampling noise. If a manager had no skill and each period were a coin toss, the standard error of the batting average would be the square root of 0.25 divided by the number of periods.
| Periods | Standard error, points | Low | High |
|---|---|---|---|
| 12 | 14.4 | 21.1% | 78.9% |
| 36 | 8.3 | 33.3% | 66.7% |
| 60 | 6.5 | 37.1% | 62.9% |
| 120 | 4.6 | 40.9% | 59.1% |
Twelve quarters cannot distinguish 41.7 per cent from 50 per cent. Even 36 monthly observations leave a band from 33.3 to 66.7 per cent. This is why most consultants compute the batting average on monthly data over at least five years, and why the win/loss ratio and the information ratio, covered in how to calculate tracking error and the information ratio, carry more weight in a review than the batting average itself.
Count the winning periods, divide by the total, then put the win/loss ratio and the up and down split beside it. Here a 41.7 per cent batting average sits on a 3.01 win/loss ratio and a 100.0 per cent down-market record, which together produce 1.14 points of annual outperformance. The mandate returns, fees and benchmark comparisons behind the book are in the free workbooks for Asset Management.
Above 50 per cent means the manager beats the benchmark more often than not, but a good figure depends on the win/loss ratio beside it. A manager batting 41.7 per cent only needs wins 1.40 times the size of its losses to add value; one batting 60.0 per cent can afford wins of 0.67 times its losses. Judge the two together, over at least five years of monthly data.
The batting average measures how often the fund beats its benchmark; the win/loss ratio measures how much it wins by relative to how much it loses by. In the illustrative case the fund wins 41.7 per cent of quarters, with an average win of 1.16 points against an average loss of 0.386, a ratio of 3.01. Mean excess equals batting average times average win less the losing frequency times average loss.
More than most scorecards use. Under a coin-toss null, the standard error is the square root of 0.25 divided by the number of periods: 14.4 points for 12 periods, 8.3 for 36 and 6.5 for 60. With 36 monthly returns, any batting average between 33.3 and 66.7 per cent is consistent with no skill at all.
This article is one calculation from Asset Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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