Fidelity Nasdaq-Composite-Index ETF (ONEQ) & Invesco (QQQ) vs. iShares Russell 2000 ETF (IWM), Vanguard Russell 2000 ETF (VTWO).

Do small-caps yield a higher average return compared to large-caps? What about the risk? What does the data reveal?

**Summary:**

**Returns of small-caps vs. large-caps should, theoretically, be different. This is not backed by data, however.****Data reveals the returns of Russell 2000 vs Nasdaq, Russell 2000 vs. Nasdaq 100 don't exhibit any difference, statistically.****The risk profile of Russell 2000, Nasdaq, and Nasdaq 100 is analyzed, and is statistically identical.**

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**Is it more profitable to invest in small-caps index, Russell 2000, compared to the Nasdaq Composite Index or Nasdaq 100 Index?**

The conventional viewpoint would say that the small firms should not be as profitable compared to the larger established players? Why? This is because the big guys should be able to achieve economies of scale and gain value through synergistic ventures that provide them a competitive advantage compared to the smaller guys. The bigger players, arguably, also have better access to capital and capital markets.

**What advantages do the smaller guys have, comparatively? **

Arguably, the smaller guys, most vitally, should, theoretically, be more ‘agile,’ and because of this, better able to navigate the volatile, uncertain, and ambiguous (VUCA) business environment.

For example, in a dynamic environment, with unanticipated changes constantly impacting business decision-making, the smaller firms, being less bureaucratic, with teams better integrated with the resilient but less obstructive and fluid organizational structure, should be able to react more rapidly and proactively, compared to larger firms with multiple complex entanglements that have to be cautiously weighed before making and implementing business decisions.

Furthermore, arguably, smaller firms are also newly created firms (less than five years old, for example) with novel creative approaches and products. They make an effort to do things differently.

Thus, we can consider two theoretical forces for large firms and small firms:

1. Large firms – the advantages that the larger firms have, due to economies of scale and value gained through synergistic ventures. Also, easier access to capital.

2. smaller firms – the advantages of agility, novel approaches and products, creativity and ability to react more rapidly due to less complex business entanglements.

The force more overpowering, thematically, should generate a higher return for investors; for example, if force 1. supersedes force 2., investment in the larger firms should be more profitable for investors, or vice versa. There is also the issue of risk profiles. Arguably, smaller firms should be riskier investments.

See also: How long does a market bubble last on average?

**So, what does the data reveal?**

These fundamental elements have been analyzed in this report, relying on data from the past two decades. As the two indices are compared in this work, ETFs based on these indices are also examined as the ETFs performance matches the performance of the index they follow.

The analysis of the returns for the last two decades of data reveals that the monthly mean return of Russell 2000 stands at 0.8%; the monthly mean returns of Nasdaq for the last two decades stand at 0.01%.; the monthly mean returns of Nasdaq 100 for the last two decades stand at 0.09%.

### Does this mean that the Russell 2000, and small- caps are a more profitable investment?

Hypothesis testing must be applied to assess statistical significance:

A paired comparison test has been conducted to evaluate statistical significance, as both the Russell 2000, Nasdaq 100, and Nasdaq are reliant on one primary economy, fundamentally.

Test is constructed as:

*H0: μd - *μ0* = 0 versus Ha: μd - *μ0*≠ 0.*

*, where *

*μd – the difference of the returns of the two tested indices. *

μ0 *– 0*

The null will hold if the difference in the mean returns of the tested, equals 0.

The t-critical value at the 98% significance level = 2.342.

The paired comparison test for the difference of means, Russell 2000 vs. Nasdaq 100, reveals a test value of 0.0243:

__Paired sample T-test, using T distribution (df=240) (two-tailed)____(validation)__** 1. H0 hypothesis**
Since p-value > α, H0 cannot be rejected.
The average of the difference of the sample is assumed to be

**equal to**the μ0. In other words, the difference between the average of test value and the μ0 is not big enough to be statistically significant.

**The p-value equals**

__2. P-value__**0.8079**, ( p(x≤T) = 0.5961 ). It means that the chance of type I error, rejecting a correct H0, is too high: 0.8079 (80.79%). The larger the p-value the more it supports H0.

**The test statistic T equals**

__3. The statistics__**0.2434**, which is in the 98% region of acceptance: [-2.342 : 2.342]. x=0.00065, is in the 98% region of acceptance: [-0.006225 : 0.006225]. The standard deviation of the difference, S' equals 0.00266, is used to calculate the statistic.

**The observed effect size d is**

__4. Effect size__**small**,

**0.016**. This indicates that the magnitude of the difference between the average and μ0 is small.

The paired comparison test for the difference of means, Russell 2000 vs. Nasdaq, reveals a test value of 0.082:

__Paired sample T-test, using T distribution (df=240) (two-tailed)____(validation)__** 1. H0 hypothesis**
Since p-value > α, H0 cannot be rejected.
The average of the difference of the sample is assumed to be

**equal to**the μ0. In other words, the difference between the average of test value and the μ0 is not big enough to be statistically significant.

__ 2. P-value__
The p-value equals

**0.9345**, ( p(x≤T) = 0.5327 ). It means that the chance of type I error, rejecting a correct H0, is too high: 0.9345 (93.45%). The larger the p-value the more it supports H0.

__The test statistic T equals__

**3. The statistics****0.08222**, which is in the 98% region of acceptance: [-2.342 : 2.342]. x=0.00016, is in the 98% region of acceptance: [-0.004656 : 0.004656]. The standard deviation of the difference, S' equals 0.00199, is used to calculate the statistic.

__The observed effect size d is__

**4. Effect size****small**,

**0.0053**. This indicates that the magnitude of the difference between the average and μ0 is small.

**These two nulls cannot be rejected, therefore. **

**So, what does this mean, simplistically?**

There is no value of statistical significance to support the premise that the returns of the tested indices exhibit a difference, i.e., Russell 2000 vs Nasdaq 100, and Russell 2000 vs Nasdaq, actually being different. This means that the difference of the returns over the last two decades equals 0. Russell 2000 doesn’t yield a higher capital growth than Nasdaq 100, Nasdaq, and vice versa.

**What about the Risk? Which is riskier, Russell 2000, Nasdaq Composite Index, or Nasdaq 100 Index?**

F test has been conducted to evaluate the difference in variance between Russell 2000 compared to Nasdaq, and Russell 2000 compared to Nasdaq. The Hypothesis is constructed as:

*H0: σ2Ru2 = σ2Nd100 versus Ha: σ2Ru2 ≠ σ2Nd100,*

*And*

*H0: σ2Ru2 = σ2Nd versus Ha: σ2Ru2 ≠ σ2Nd*

*, where*

*σ2Ru2 – the variance of the returns of Russell 2000;*

*σ2Nd100 – the variance of the returns of Nasdaq 100 (NDX), and*

*σ2Nd – the variance of the returns of Nasdaq (IXIC).*

The sample standard deviation of returns for Russell 2000 stands at 0.0577;

The sample standard deviation of returns for Nasdaq 100 stands at 0.0633, and

The sample standard deviation of returns for Nasdaq, identical to Russell 2000, stands at 0.0579.

The test reveals:

The f value of Russell 2000 compared to Nasdaq 100 stands at: 1.2:

__F test for variances, using F distribution (dfnum=240,dfdenom=240) (two-tailed)____(validation)__** 1. H0 hypothesis**
Since p-value > α, H0 is accepted.
The sample standard deviation (S) of

**NDX**population is considered to be

**equal to**the sample standard deviation (S) of

**Russell 2000's**population. In other words, the difference between the sample standard deviation (S) of the

**NDX**and

**Russell 2000**populations is not big enough to be statistically significant.

**The p-value equals**

__2. P-value__**0.152**, ( p(x≤F) = 0.924 ). It means that the chance of type I error, rejecting a correct H0, is too high: 0.152 (15.2%). The larger the p-value the more it supports H0.

**The test statistic F equals**

__3. The statistics__**1.2035**, which is in the 98% region of acceptance: [0.7399 : 1.3515]. S1/S2=1.1, is in the 98% region of acceptance: [0.8602 : 1.1625]. The 98% confidence interval of σ12/σ22 is: [0.8905 , 1.6266].

The f value of Russell 2000 compared to Nasdaq stands at: 1.003:

__F test for variances, using F distribution (dfnum=240,dfdenom=240) (two-tailed)____(validation)__** 1. H0 hypothesis**
Since p-value > α, H0 is accepted.
The sample standard deviation (S) of

**IXIC's**population is considered to be

**equal to**the sample standard deviation (S) of

**Russell 2000's**population. In other words, the difference between the sample standard deviation (S) of

**IXIC**and

**Russell 2000**populations is not big enough to be statistically significant.

**The p-value equals**

__2. P-value__**0.9573**, ( p(x≤F) = 0.5214 ). It means that the chance of type I error, rejecting a correct H0, is too high: 0.9573 (95.73%). The larger the p-value the more it supports H0.

**The test statistic F equals**

__3. The statistics__**1.0069**, which is in the 98% region of acceptance: [0.7399 : 1.3515]. S1/S2=1, is in the 98% region of acceptance: [0.8602 : 1.1625]. The 98% confidence interval of σ12/σ22 is: [0.7451 , 1.3609].

The critical f value for the test, at 98% confidence level =1.35

**The null cannot be rejected, therefore.**

**What does this mean, simplistically? Which index is riskier?**

The tests reveal no difference in the variance with any statistical significance. This means that the risk profiles of all three are identical. An investor, as per the data, should not favor one over the other, when choosing between the three.

There is no evidence that the small-caps in the last two decades have yielded a higher capital growth than the large-caps or Nasdaq; nor is there any evidence that the risk profile is any different. Picking either one, as per historical returns, should yield similar results, with a similar risk exposure.

It appears that the two forces mentioned do not supersede each other, and there is no evidence to favor one or the other. Cross-sectional data, nonetheless, may differ, i.e., data from a specific time; however, this difference should be short-term, and the long-term results should converge, as the data shows.

See also:

1. Why do all financial assets correlatively fall in a market crash

2. Which stock indices/index-based ETFs are the riskiest investments?

3. Which stock indices/index-based ETFs provide the best risk-adjusted returns

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