# The Outperformance Probability of Mutual Funds

^{*}

## Abstract

**:**

## 1. Motivation

- The OP compares some strategy with a specified benchmark, which need not necessarily be the money-market account.
- It is a probability. Thus, it is easy to understand also for a nonacademic audience, more precisely, for people who are not educated in statistics or probability theory.
- The holding period of the investor is considered random. This enables us to compute the performance of an investment opportunity for arbitrary liquidity preferences.

- The first one is conceptual:
- (a)
- A simple thought experiment reveals that comparing two performance measures with one another, where each one compares an investment opportunity with the money-market account, can lead to completely different conclusions than evaluating only one performance measure that compares the given investment opportunities without taking the money-market account into consideration at all. The former comparison refers to the question of which of the two investment opportunities is better able to outperform the money-market account, whereas the latter comparison refers to the question of whether the first investment opportunity is able to outperform the second. In general, performance measures are not linear and so the former comparison does not imply the latter, i.e., an investment opportunity that is better able to produce excess returns than another investment opportunity need not be better than the other.
- (b)
- Most performance measures presume that the holding period of the investor is fixed. This assumption is clearly violated in real-life because investors usually do not know, in advance, when they will liquidate all their assets. We solve this problem by incorporating any holding-time distribution, which specifies the individual liquidity preference of the investor. It can be either discrete or continuous and it can have a finite or infinite right endpoint. Any fixed holding period can be considered a special case, which means that we can treat one-period models, too.

- The second one is empirical:
- (a)
- The natural logarithm of the assets under management of the mutual funds that are taken into consideration are highly correlated with their inverse coefficient of variation (ICV). The ICV is a return-to-risk measure, which is based on differential log-returns but not (necessarily) on excess log-returns, and in the Brownian-motion framework it is the main ingredient of the OP. This means that capital allocation and relative performance are strongly connected to one another, which suggests that market participants take differential (log-)returns implicitly into account when making their investment decisions.
- (b)
- We emphasize our results by comparing the p-values of the differences between the Sharpe ratios of all mutual funds and the Sharpe ratios of the given benchmarks. The p-values indicate that it is hard to distinguish between the former and the latter Sharpe ratios. For this reason, we cannot say that any fund is better than the benchmark by comparing two Sharpe ratios with one another. A completely different picture evolves when considering the p-values of the ICVs of all funds with respect to their benchmarks. Those p-values are much lower and economically significant, i.e., it turns out that most funds are able to beat their benchmarks.

## 2. The Outperformance Probability

#### 2.1. Basic Assumptions and Definition

**Definition**

**1**(Outperformance probability).

#### 2.2. Theoretical Properties

**Theorem**

**1**(Outperformance probability).

#### 2.3. Statistical Inference

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Definition**

**2**(ML-estimator).

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 2.4. Discussion

- It is based on a parametric model, namely the geometric Brownian motion. While this model is standard in financial mathematics, we may doubt that value processes follow a geometric Brownian motion in real life. The assumption that log-returns are independent and identically normally distributed contradicts the stylized facts of empirical finance.
- We assume that the time of liquidation does not depend on whether or not the strategy outperforms the benchmark. This assumption might be violated, for example, if investors suffer from the disposition effect, i.e., if they tend to sell winners too early and ride losers too long (Shefrin and Statman 1985).
- Also the holding-time distribution follows a parametric model, which can either be true or false, too. Indeed, we have to choose some model for T but need not necessarily know its parameters. However, in order to estimate the parameters of F, we would need appropriate data and then the statistical properties of the ML-estimator might change essentially.

## 3. Empirical Investigation

#### 3.1. General Observations

“The recent increase in the number and types of index funds that are available to individual investors makes this a matter of practical as well as theoretical significance. Numerous index funds, which track the Standard and Poor’s (S&P) 500 Index or various small-stock, bond, value, growth, or international indexes, are now widely available to individual investors. […] Given that there are sufficient index funds to span most investors’ risk choices, that the index funds are available at low cost, and that the low cost of index funds means that a combination of index funds is likely to outperform an active fund of similar risk, the question is, why select an actively managed fund?”

#### 3.2. Empirical Results

#### 3.2.1. Fixed Holding Period

#### 3.2.2. Uniformly Distributed Holding Period

#### 3.2.3. Exponentially Distributed Holding Period

#### 3.2.4. Weibull Distributed Holding Period

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | Each excess return is a differential return but not vice versa. |

2 | It is a matter of fact that some students do not understand even the distinction between parameter and estimator after attending a statistics course. |

3 | According to Lusardi and Mitchell (2013, pp. 15–16), we can expect that a large number of people cannot understand the question at all because they are unfamiliar with stocks, bonds, and mutual funds. |

4 | From now on, we will omit the subscript “$t\ge 0$” for notational convenience. |

5 | In the given example, the critical time point is, approximately, $T=5$ years. |

6 | The reason why we focus on ETFs is discussed in Section 3.1. |

7 | |

8 | $\mathrm{P}(X>Y)=0$ does not imply that $\mathrm{P}(X<Y)=1$ but $\mathrm{P}(X\le Y)=1$. |

9 | Thus, if a strategy has a higher OP than another strategy, given some holding-time distribution, the same holds true for any other holding-time distribution. |

10 | This term can be attributed to the estimation error regarding the standard deviation of X (Frahm 2018). |

11 | This is true whenever $\mathbf{E}\left({T}^{k}\right)<\infty $ for any $k\ge \frac{1}{2}$. |

12 | This is not a linear regression equation because $\mathbf{E}\left(\epsilon \right)>0$. |

13 | This kind of situation cannot happen in our Brownian-motion framework. |

14 | It holds that $\mathbf{E}\left({S}_{T}\right)={e}^{{\mu}_{S}T}$ and so the drift coefficient ${\mu}_{S}$ can be considered the internal rate of return on the fund. |

15 | |

16 | Note that this is precisely the case in which we have that $\mathrm{ICV}\le 0$. |

**Figure 3.**Outperformance probability (OP) depending on the inverse coefficient of variation (ICV) and the (maximal) holding period. On the

**left-hand**side, the holding period T is considered fixed, whereas on the

**right-hand**side, T is supposed to be uniformly distributed between 0 and M years.

**Figure 4.**Standard error of ${\widehat{\Pi}}_{n}$ depending on the ICV. On the

**left-hand**side, the holding period T is considered fixed and is equal to 5 years, whereas on the

**right-hand**side, T is supposed to be uniformly distributed between 0 and 10 years.

**Figure 5.**ICV against the natural logarithm of the assets under management of the mutual funds with respect to the S&P 500 (

**left**) and the Russell 1000 (

**right**).

**Table 1.**Basic characteristics of the benchmarks and funds that are taken into consideration, i.e., the estimated drift coefficients ($\widehat{\mu}$), diffusion coefficients ($\widehat{\sigma}$), growth rates ($\widehat{\gamma}$) as well as their expense ratios (ER) and assets under management (AUM).

Benchmark/Fund | Symbol | $\widehat{\mathit{\mu}}$ | $\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\gamma}}$ | ER (%) | AUM (bn $) |
---|---|---|---|---|---|---|

SPDR S&P 500 ETF | SPY | 0.0995 | 0.1815 | 0.0830 | 0.09 | 264.06 |

iShares Russell 1000 Growth ETF | IWF | 0.1064 | 0.1774 | 0.0906 | 0.20 | 44.44 |

iShares 7–10 Year Treasury Bond ETF | IEF | 0.0465 | 0.0656 | 0.0444 | 0.15 | 13.31 |

Money market | – | 0.0100 | 0.0007 | 0.0100 | – | – |

Cash | – | 0 | 0 | 0 | – | – |

Fidelity Blue Chip Growth | FBGRX | 0.1135 | 0.1919 | 0.0951 | 0.72 | 26.89 |

T. Rowe Price Blue Chip Growth | TRBCX | 0.1200 | 0.1974 | 0.1005 | 0.70 | 60.37 |

Franklin Growth A | FKGRX | 0.1064 | 0.1708 | 0.0918 | 0.84 | 14.96 |

JPMorgan Large Cap Growth A | OLGAX | 0.1105 | 0.1900 | 0.0924 | 0.94 | 15.15 |

Vanguard Morgan Growth | VMRGX | 0.1094 | 0.1889 | 0.0915 | 0.37 | 14.79 |

Morgan Stanley Institutional Growth A | MSEGX | 0.1299 | 0.2164 | 0.1064 | 0.89 | 6.93 |

BlackRock Capital Appreciation | MAFGX | 0.1060 | 0.1931 | 0.0873 | 0.76 | 3.13 |

AllianzGI Focused Growth Fund Admin | PGFAX | 0.1129 | 0.1942 | 0.0940 | 0.99 | 1.02 |

Goldman Sachs Large Cap Growth Insights | GCGIX | 0.1015 | 0.1829 | 0.0848 | 0.53 | 2.16 |

Pioneer Disciplined Growth A | PINDX | 0.0993 | 0.2077 | 0.0777 | 0.87 | 1.22 |

⌀ | 0.1110 | 0.1933 | 0.0922 | 0.76 | 14.66 |

Symbol | Ordinary | Generalized | ||
---|---|---|---|---|

SPY | IWF | IEF | ||

SPY | 0.4933 | – | – | – |

IWF | 0.5433 | – | – | – |

IEF | 0.5579 | – | – | – |

FBGRX | 0.5396 | 0.2480 | 0.1608 | 0.2970 |

TRBCX | 0.5576 | 0.3570 | 0.2854 | 0.3187 |

FKGRX | 0.5649 | 0.1433 | 0.0020 | 0.2914 |

OLGAX | 0.5292 | 0.1568 | 0.0788 | 0.2874 |

VMRGX | 0.5264 | 0.1954 | 0.0801 | 0.2824 |

MSEGX | 0.5544 | 0.3304 | 0.2903 | 0.3366 |

MAFGX | 0.4973 | 0.1014 | −0.0074 | 0.2628 |

PGFAX | 0.5304 | 0.2185 | 0.1338 | 0.2921 |

GCGIX | 0.5007 | 0.0436 | −0.1372 | 0.2541 |

PINDX | 0.4302 | −0.0028 | −0.1008 | 0.2198 |

⌀ | 0.5231 | 0.1792 | 0.0786 | 0.2842 |

**Table 3.**ICVs of the mutual funds with respect to the benchmarks. The standard errors can be approximated by $1/\sqrt{m}=0.2502$ (see Equation (2)).

Symbol | Benchmark | ||||
---|---|---|---|---|---|

SPY | IWF | IEF | LIBOR | Cash | |

FBGRX | 0.2133 | 0.1002 | 0.2245 | 0.4429 | 0.4947 |

TRBCX | 0.3038 | 0.2066 | 0.2432 | 0.4584 | 0.5088 |

FKGRX | 0.1825 | 0.0279 | 0.2307 | 0.4790 | 0.5373 |

OLGAX | 0.1339 | 0.0346 | 0.2156 | 0.4335 | 0.4858 |

VMRGX | 0.1677 | 0.0239 | 0.2115 | 0.4312 | 0.4839 |

MSEGX | 0.2539 | 0.1950 | 0.2503 | 0.4454 | 0.4913 |

MAFGX | 0.0669 | −0.0638 | 0.1896 | 0.4001 | 0.4516 |

PGFAX | 0.1790 | 0.0699 | 0.2182 | 0.4326 | 0.4838 |

GCGIX | 0.0378 | −0.1656 | 0.1865 | 0.4088 | 0.4632 |

PINDX | −0.0747 | −0.1845 | 0.1387 | 0.3259 | 0.3738 |

⌀ | 0.1464 | 0.0244 | 0.2109 | 0.4258 | 0.4774 |

**Table 4.**Test statistics ${\widehat{\mathrm{ICV}}}_{n}$ and $\Delta {\widehat{\mathrm{Sh}}}_{n}$ together with their corresponding p-values (in parentheses) for the null hypotheses ${H}_{0}\phantom{\rule{-0.166667em}{0ex}}:\mathrm{ICV}\le 0$ and ${H}_{0}\phantom{\rule{-0.166667em}{0ex}}:\Delta \mathrm{Sh}\le 0$, respectively.

Symbol | ${\widehat{\mathbf{ICV}}}_{\mathbf{n}}$ | $\Delta {\widehat{\mathbf{Sh}}}_{\mathbf{n}}$ | ||||
---|---|---|---|---|---|---|

SPY | IWF | IEF | SPY | IWF | IEF | |

FBGRX | 0.2133 | 0.1002 | 0.2245 | 0.0463 | −0.0183 | −0.0038 |

(0.1970) | (0.3444) | (0.1848) | (0.2908) | (0.5168) | (0.5233) | |

TRBCX | 0.3038 | 0.2066 | 0.2432 | 0.0643 | −0.0003 | 0.0142 |

(0.1123) | (0.2044) | (0.1654) | (0.2193) | (0.5003) | (0.4149) | |

FKGRX | 0.1825 | 0.0279 | 0.2307 | 0.0717 | 0.0071 | 0.0216 |

(0.2329) | (0.4556) | (0.1782) | (0.1756) | (0.4935) | (0.3822) | |

OLGAX | 0.1339 | 0.0346 | 0.2156 | 0.0360 | −0.0286 | −0.0141 |

(0.2963) | (0.4450) | (0.1944) | (0.3664) | (0.5264) | (0.5706) | |

VMRGX | 0.1677 | 0.0239 | 0.2115 | 0.0331 | −0.0315 | −0.0170 |

(0.2513) | (0.4619) | (0.1989) | (0.3315) | (0.5289) | (0.6200) | |

MSEGX | 0.2539 | 0.1950 | 0.2503 | 0.0611 | −0.0035 | 0.0110 |

(0.1550) | (0.2178) | (0.1585) | (0.3071) | (0.5032) | (0.4575) | |

MAFGX | 0.0669 | −0.0638 | 0.1896 | 0.0040 | −0.0606 | −0.0460 |

(0.3946) | (0.6006) | (0.2243) | (0.4829) | (0.5556) | (0.7294) | |

PGFAX | 0.1790 | 0.0699 | 0.2182 | 0.0371 | −0.0275 | −0.0130 |

(0.2371) | (0.3899) | (0.1915) | (0.3403) | (0.5252) | (0.5728) | |

GCGIX | 0.0378 | −0.1656 | 0.1865 | 0.0074 | −0.0571 | −0.0426 |

(0.4400) | (0.7460) | (0.2280) | (0.4587) | (0.5524) | (0.7818) | |

PINDX | −0.0747 | −0.1845 | 0.1387 | −0.0631 | −0.1277 | −0.1131 |

(0.6173) | (0.7696) | (0.2897) | (0.7481) | (0.6165) | (0.8875) |

Fund | SPY | IWF | IEF | LIBOR | Cash | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

OP | Std | p | OP | Std | p | OP | Std | p | OP | Std | p | OP | Std | p | |

FBGRX | 0.6833 | 0.1992 | 0.1788 | 0.5886 | 0.2176 | 0.3419 | 0.6921 | 0.1967 | 0.1644 | 0.8390 | 0.1367 | 0.0066 | 0.8657 | 0.1210 | 0.0013 |

TRBCX | 0.7515 | 0.1772 | 0.0779 | 0.6780 | 0.2006 | 0.1874 | 0.7068 | 0.1925 | 0.1414 | 0.8473 | 0.1320 | 0.0042 | 0.8724 | 0.1168 | 0.0007 |

FKGRX | 0.6584 | 0.2053 | 0.2203 | 0.5249 | 0.2227 | 0.4555 | 0.6970 | 0.1954 | 0.1566 | 0.8580 | 0.1257 | 0.0022 | 0.8852 | 0.1084 | 0.0002 |

OLGAX | 0.6176 | 0.2134 | 0.2907 | 0.5308 | 0.2225 | 0.4449 | 0.6851 | 0.1987 | 0.1758 | 0.8338 | 0.1395 | 0.0084 | 0.8613 | 0.1237 | 0.0017 |

VMRGX | 0.6462 | 0.2080 | 0.2411 | 0.5213 | 0.2228 | 0.4619 | 0.6819 | 0.1995 | 0.1810 | 0.8325 | 0.1402 | 0.0088 | 0.8604 | 0.1243 | 0.0019 |

MSEGX | 0.7149 | 0.1899 | 0.1289 | 0.6686 | 0.2029 | 0.2030 | 0.7122 | 0.1908 | 0.1331 | 0.8403 | 0.1359 | 0.0061 | 0.8640 | 0.1220 | 0.0014 |

MAFGX | 0.5595 | 0.2207 | 0.3938 | 0.4433 | 0.2209 | 0.6013 | 0.6642 | 0.2040 | 0.2105 | 0.8145 | 0.1496 | 0.0177 | 0.8437 | 0.1340 | 0.0052 |

PGFAX | 0.6556 | 0.2060 | 0.2251 | 0.5621 | 0.2204 | 0.3890 | 0.6872 | 0.1981 | 0.1723 | 0.8333 | 0.1398 | 0.0085 | 0.8603 | 0.1243 | 0.0019 |

GCGIX | 0.5336 | 0.2224 | 0.4399 | 0.3556 | 0.2084 | 0.7558 | 0.6617 | 0.2046 | 0.2147 | 0.8196 | 0.1470 | 0.0148 | 0.8498 | 0.1305 | 0.0037 |

PINDX | 0.4337 | 0.2201 | 0.6184 | 0.3400 | 0.2050 | 0.7825 | 0.6217 | 0.2127 | 0.2835 | 0.7669 | 0.1711 | 0.0594 | 0.7984 | 0.1574 | 0.0290 |

⌀ | 0.6254 | 0.2062 | 0.2815 | 0.5213 | 0.2144 | 0.4623 | 0.6810 | 0.1993 | 0.1833 | 0.8285 | 0.1417 | 0.0137 | 0.8561 | 0.1263 | 0.0047 |

Fund | SPY | IWF | IEF | LIBOR | Cash | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

OP | Std | p | OP | Std | p | OP | Std | p | OP | Std | p | OP | Std | p | |

FBGRX | 0.6716 | 0.1839 | 0.1754 | 0.5834 | 0.2042 | 0.3414 | 0.6798 | 0.1813 | 0.1607 | 0.8124 | 0.1208 | 0.0049 | 0.8359 | 0.1064 | 0.0008 |

TRBCX | 0.7341 | 0.1607 | 0.0726 | 0.6667 | 0.1854 | 0.1843 | 0.6932 | 0.1767 | 0.1371 | 0.8197 | 0.1164 | 0.0030 | 0.8418 | 0.1027 | 0.0004 |

FKGRX | 0.6485 | 0.1906 | 0.2179 | 0.5235 | 0.2099 | 0.4555 | 0.6842 | 0.1798 | 0.1528 | 0.8291 | 0.1107 | 0.0015 | 0.8531 | 0.0952 | 0.0001 |

OLGAX | 0.6106 | 0.1994 | 0.2896 | 0.5291 | 0.2096 | 0.4449 | 0.6733 | 0.1834 | 0.1723 | 0.8078 | 0.1235 | 0.0063 | 0.8321 | 0.1088 | 0.0011 |

VMRGX | 0.6372 | 0.1935 | 0.2392 | 0.5201 | 0.2100 | 0.4619 | 0.6703 | 0.1843 | 0.1777 | 0.8067 | 0.1241 | 0.0067 | 0.8312 | 0.1094 | 0.0012 |

MSEGX | 0.7007 | 0.1740 | 0.1244 | 0.6580 | 0.1879 | 0.2002 | 0.6982 | 0.1749 | 0.1286 | 0.8136 | 0.1201 | 0.0045 | 0.8345 | 0.1073 | 0.0009 |

MAFGX | 0.5560 | 0.2076 | 0.3937 | 0.4466 | 0.2078 | 0.6014 | 0.6539 | 0.1891 | 0.2078 | 0.7906 | 0.1331 | 0.0145 | 0.8165 | 0.1184 | 0.0037 |

PGFAX | 0.6459 | 0.1913 | 0.2228 | 0.5585 | 0.2073 | 0.3888 | 0.6752 | 0.1827 | 0.1688 | 0.8073 | 0.1237 | 0.0065 | 0.8312 | 0.1094 | 0.0012 |

GCGIX | 0.5317 | 0.2095 | 0.4399 | 0.3645 | 0.1939 | 0.7577 | 0.6516 | 0.1897 | 0.2122 | 0.7952 | 0.1306 | 0.0119 | 0.8219 | 0.1151 | 0.0026 |

PINDX | 0.4376 | 0.2069 | 0.6186 | 0.3499 | 0.1902 | 0.7850 | 0.6144 | 0.1987 | 0.2823 | 0.7480 | 0.1545 | 0.0542 | 0.7763 | 0.1407 | 0.0248 |

⌀ | 0.6174 | 0.1917 | 0.2794 | 0.5200 | 0.2006 | 0.4621 | 0.6694 | 0.1841 | 0.1800 | 0.8030 | 0.1258 | 0.0114 | 0.8274 | 0.1113 | 0.0037 |

Fund | SPY | IWF | IEF | LIBOR | Cash | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

OP | Std | p | OP | Std | p | OP | Std | p | OP | Std | p | OP | Std | p | |

FBGRX | 0.6598 | 0.1683 | 0.1712 | 0.5782 | 0.1905 | 0.3407 | 0.6672 | 0.1655 | 0.1562 | 0.7868 | 0.1087 | 0.0042 | 0.8081 | 0.0966 | 0.0007 |

TRBCX | 0.7165 | 0.1449 | 0.0675 | 0.6553 | 0.1699 | 0.1803 | 0.6795 | 0.1608 | 0.1322 | 0.7934 | 0.1050 | 0.0026 | 0.8134 | 0.0936 | 0.0004 |

FKGRX | 0.6386 | 0.1754 | 0.2147 | 0.5221 | 0.1972 | 0.4555 | 0.6713 | 0.1640 | 0.1481 | 0.8019 | 0.1002 | 0.0013 | 0.8237 | 0.0875 | 0.0001 |

OLGAX | 0.6035 | 0.1852 | 0.2881 | 0.5273 | 0.1969 | 0.4448 | 0.6613 | 0.1677 | 0.1681 | 0.7827 | 0.1110 | 0.0054 | 0.8046 | 0.0986 | 0.0010 |

VMRGX | 0.6281 | 0.1786 | 0.2365 | 0.5189 | 0.1973 | 0.4619 | 0.6586 | 0.1687 | 0.1735 | 0.7817 | 0.1115 | 0.0058 | 0.8038 | 0.0991 | 0.0011 |

MSEGX | 0.6863 | 0.1580 | 0.1193 | 0.6473 | 0.1726 | 0.1966 | 0.6840 | 0.1590 | 0.1236 | 0.7879 | 0.1081 | 0.0039 | 0.8067 | 0.0974 | 0.0008 |

MAFGX | 0.5526 | 0.1945 | 0.3934 | 0.4498 | 0.1948 | 0.6016 | 0.6436 | 0.1738 | 0.2044 | 0.7673 | 0.1194 | 0.0126 | 0.7905 | 0.1066 | 0.0032 |

PGFAX | 0.6362 | 0.1762 | 0.2197 | 0.5550 | 0.1942 | 0.3886 | 0.6631 | 0.1671 | 0.1645 | 0.7823 | 0.1112 | 0.0056 | 0.8038 | 0.0991 | 0.0011 |

GCGIX | 0.5298 | 0.1967 | 0.4398 | 0.3734 | 0.1790 | 0.7603 | 0.6414 | 0.1745 | 0.2089 | 0.7714 | 0.1172 | 0.0103 | 0.7954 | 0.1039 | 0.0022 |

PINDX | 0.4414 | 0.1937 | 0.6189 | 0.3600 | 0.1750 | 0.7882 | 0.6071 | 0.1843 | 0.2806 | 0.7290 | 0.1389 | 0.0496 | 0.7544 | 0.1262 | 0.0219 |

⌀ | 0.6093 | 0.1771 | 0.2769 | 0.5187 | 0.1867 | 0.4618 | 0.6577 | 0.1685 | 0.1760 | 0.7784 | 0.1131 | 0.0101 | 0.8004 | 0.1009 | 0.0033 |

Fund | SPY | IWF | IEF | LIBOR | Cash | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

OP | Std | p | OP | Std | p | OP | Std | p | OP | Std | p | OP | Std | p | |

FBGRX | 0.6756 | 0.1890 | 0.1764 | 0.5852 | 0.2088 | 0.3415 | 0.6840 | 0.1864 | 0.1619 | 0.8215 | 0.1266 | 0.0056 | 0.8462 | 0.1121 | 0.0010 |

TRBCX | 0.7400 | 0.1662 | 0.0744 | 0.6706 | 0.1905 | 0.1853 | 0.6978 | 0.1820 | 0.1385 | 0.8292 | 0.1222 | 0.0035 | 0.8524 | 0.1083 | 0.0006 |

FKGRX | 0.6519 | 0.1955 | 0.2186 | 0.5240 | 0.2144 | 0.4555 | 0.6886 | 0.1850 | 0.1540 | 0.8391 | 0.1165 | 0.0018 | 0.8643 | 0.1008 | 0.0001 |

OLGAX | 0.6130 | 0.2042 | 0.2899 | 0.5297 | 0.2141 | 0.4449 | 0.6773 | 0.1885 | 0.1734 | 0.8167 | 0.1293 | 0.0072 | 0.8422 | 0.1146 | 0.0014 |

VMRGX | 0.6403 | 0.1984 | 0.2398 | 0.5205 | 0.2145 | 0.4619 | 0.6743 | 0.1894 | 0.1787 | 0.8155 | 0.1300 | 0.0076 | 0.8413 | 0.1151 | 0.0015 |

MSEGX | 0.7055 | 0.1793 | 0.1259 | 0.6617 | 0.1930 | 0.2011 | 0.7029 | 0.1802 | 0.1301 | 0.8227 | 0.1259 | 0.0052 | 0.8447 | 0.1131 | 0.0012 |

MAFGX | 0.5572 | 0.2121 | 0.3937 | 0.4454 | 0.2124 | 0.6014 | 0.6574 | 0.1941 | 0.2086 | 0.7988 | 0.1389 | 0.0158 | 0.8258 | 0.1242 | 0.0043 |

PGFAX | 0.6492 | 0.1962 | 0.2235 | 0.5598 | 0.2119 | 0.3889 | 0.6793 | 0.1879 | 0.1699 | 0.8162 | 0.1296 | 0.0073 | 0.8413 | 0.1151 | 0.0015 |

GCGIX | 0.5324 | 0.2140 | 0.4399 | 0.3614 | 0.1988 | 0.7571 | 0.6551 | 0.1947 | 0.2129 | 0.8035 | 0.1364 | 0.0130 | 0.8315 | 0.1209 | 0.0031 |

PINDX | 0.4362 | 0.2115 | 0.6185 | 0.3465 | 0.1951 | 0.7842 | 0.6170 | 0.2034 | 0.2827 | 0.7544 | 0.1601 | 0.0560 | 0.7838 | 0.1465 | 0.0264 |

⌀ | 0.6201 | 0.1966 | 0.2801 | 0.5205 | 0.2053 | 0.4622 | 0.6734 | 0.1892 | 0.1811 | 0.8118 | 0.1316 | 0.0123 | 0.8374 | 0.1171 | 0.0041 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Frahm, G.; Huber, F.
The Outperformance Probability of Mutual Funds. *J. Risk Financial Manag.* **2019**, *12*, 108.
https://doi.org/10.3390/jrfm12030108

**AMA Style**

Frahm G, Huber F.
The Outperformance Probability of Mutual Funds. *Journal of Risk and Financial Management*. 2019; 12(3):108.
https://doi.org/10.3390/jrfm12030108

**Chicago/Turabian Style**

Frahm, Gabriel, and Ferdinand Huber.
2019. "The Outperformance Probability of Mutual Funds" *Journal of Risk and Financial Management* 12, no. 3: 108.
https://doi.org/10.3390/jrfm12030108