- 25 Marks
Question
A lot of investors have been reading about something called the “new-fund effect”. That is the tendency of new funds to outperform their older peers because of any one of a number of
factors: better access to initial public offerings, more motivated managers, or better spreads on
trades. However, despite the potential growth benefits of new funds , their volatility makes many
investors uncomfortable. Consider a sample of 10 newly M- mutual funds and a sample of 10 newly
Q-mutual funds randomly selected from all mutuals funds in Ghana that are less than 18 months
old as follows:
| Annualized Performance of Newly M- Mutual Funds | Annualized Performance of Newly Q-Mutual Funds |
|---|---|
| 13.7 | 9.5 |
| 15.3 | 14.9 |
| 7.9 | 10.8 |
| 9.8 | 11.5 |
| 13.6 | 11.3 |
| 13.6 | 25.2 |
| 11.4 | 12.0 |
| 8.6 | 6.3 |
| 14.6 | 12.7 |
| 15.2 | 12.4 |
Using a hypothesis testing procedure, investigate whether there is sufficient evidence to conclude
that there is a significant difference in variance of newly created M and Q mutual funds.
Answer
To investigate whether there is a significant difference in the variances of the annualized performance of newly created M-mutual funds and Q-mutual funds, we use the F-test for equality of variances. This test is appropriate for comparing variances from two independent samples, assuming the data are normally distributed (a reasonable assumption for financial returns in hypothesis testing contexts). In the Ghanaian banking sector, such analysis is crucial for risk management, as higher variance indicates greater volatility in fund performance, which could affect investor confidence and compliance with Bank of Ghana’s operational risk standards under Basel II/III adaptations. For instance, during the 2017-2019 banking cleanup, volatile investment products contributed to instability in institutions like UT Bank, highlighting the need for variance assessments in portfolio decisions.
Step 1: State the Hypotheses
- Null Hypothesis (H₀): σ_M² = σ_Q² (The population variances of M and Q mutual funds are equal.)
- Alternative Hypothesis (H₁): σ_M² ≠ σ_Q² (The population variances differ.)
We use a two-tailed test at a significance level of α = 0.05 (common standard unless specified).
Step 2: Compute Sample Statistics
For M-mutual funds (n_M = 10):
- Mean (x̄_M) = 12.37
- Sample Variance (s_M²) = 7.57 (calculated as Σ(x_i – x̄)² / (n-1))
For Q-mutual funds (n_Q = 10):
- Mean (x̄_Q) = 12.66
- Sample Variance (s_Q²) = 24.43
The means are similar, but Q shows higher variance, suggesting potentially greater volatility.
Step 3: Calculate the F-Statistic
The F-statistic is the ratio of the larger variance to the smaller variance:
F = s_Q² / s_M² = 24.43 / 7.57 ≈ 3.23
Degrees of freedom: numerator df = n_Q – 1 = 9; denominator df = n_M – 1 = 9.
Step 4: Determine the Critical Values and p-Value
For a two-tailed F-test at α = 0.05 with df=9,9:
- Upper critical value F(0.975, 9,9) ≈ 4.03
- Lower critical value F(0.025, 9,9) ≈ 0.25 (or equivalently, 1/4.03 ≈ 0.25)
Since F = 3.23 < 4.03 and > 0.25, it falls within the acceptance region.
The p-value ≈ 0.096 (calculated using the F-distribution cumulative density function). Since p-value > 0.05, we fail to reject H₀.
Step 5: Conclusion
There is insufficient evidence at the 5% significance level to conclude that the variances differ significantly (F=3.23, p=0.096). The observed difference in sample variances could be due to random sampling variation. In practical terms, this suggests that the volatility (risk) of new M and Q mutual funds is statistically similar, aiding banks like Ecobank Ghana in diversifying investment portfolios without heightened variance concerns. However, ongoing monitoring is recommended, especially post-2022 DDEP, where market volatility increased, as per BoG’s Liquidity Risk Management Guidelines. If larger samples were available, the test power would improve, potentially detecting subtler differences.
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