MAF759 Analytical Methods-Discrete Returns Assessment Answers
Questions:
1. Calculate monthly discrete returns (hint: holding period returns) for the S&P 500 index, NASDAQ index, and Apple Inc. (APPL) stock prices, respectively.
2. Compute descriptive statistics for monthly Apple Inc. (APPL) stock prices, S&P500 index and NASDAQ index, respectively.
3. Construct the frequency distributions (including relative frequency and cumulative frequency) for APPL returns, S&P 500 index returns and NASDAQ index returns,respectively. Use 10 intervals.
4. What is the probability of the S&P 500 index returns above 10% What is the probability of the NASDAQ index returns being between 2% and 8%.
5. Is the distribution of monthly APPL returns Normal distribution Are the distributions of S&P 500 index returns and NASDAQ index returns different.
Provide evidences.
6. In order to predict the monthly APPL returns, your supervisor advised two simple linear regression models:
Model A: Assuming a linear relationship between monthly APPL returns and
S&P 500 index returns.
Model B: Assuming a linear relationship between monthly APPL returns and
NASDAQ index returns
6.1: For model A, test if the intercept coefficient is significant at the 5% level;Calculate 95% confidence interval for the slope coefficient.6.2: For model B, Predict the APPL return when NASDAQ index return being 2%, and calculate its 90 percent prediction interval.
Answer:
| S&P500 index | NASDAQ index | APPLE stock | |||
| Mean | 1374.78 | Mean | 2756.65 | Mean | 37.79 |
| Standard Error | 27.06 | Standard Error | 80.86 | Standard Error | 2.80 |
| Median | 1278.33 | Median | 2368.07 | Median | 21.43 |
| Mode | 1130.20 | Mode | #N/A | Mode | 1.51 |
| Standard Deviation | 374.94 | Standard Deviation | 1120.40 | Standard Deviation | 38.77 |
| Sample Variance | 140579.94 | Sample Variance | 1255306.95 | Sample Variance | 1502.76 |
| Kurtosis | -0.42 | Kurtosis | -0.24 | Kurtosis | -0.72 |
| Skewness | 0.77 | Skewness | 0.97 | Skewness | 0.82 |
| Range | 1503.74 | Range | 4211.06 | Range | 124.86 |
| Minimum | 735.09 | Minimum | 1172.06 | Minimum | 0.92 |
| Maximum | 2238.83 | Maximum | 5383.12 | Maximum | 125.78 |
| Sum | 263958.66 | Sum | 529277.48 | Sum | 7256.38 |
| Count | 192 | Count | 192 | Count | 192 |
Sample Calculations for Another data
1.
Apple Inc. (APPL):
Month 1: ((148.26-144.71)/144.71)*100=2.46%
Month 2: ((142.82-148.26)/148.26)*100=-3.47%
Month 3: ((143.68-142.82)/142.82)*100=0.70%
Month 4: ((146.25-143.68)/143.68)*100=1.73%
S&P 500 Index:
Month 1: ((2038.47-2008.39)/2008.39)*100=1.45%
Month 2: ((1986.67-2038.47)/2038.47)*100=-2.48%
Month 3: ((2007.78-1986.67)/1986.67)*100=1.06%
Month 4: ((2077.42-2007.78)/2007.78)*100=3.36%
NASDAQ Index:
Month 1: ((4712.13-4566.23)/4566.23)*100=2.93%
Month 2: ((4497.49-4712.13)/4712.13)*100=-4.76%
Month 3: ((4747.96-4497.49)/4497.49)*100=5.01%
Month 4: ((4756.93-4747.96)/4747.96)*100=0.20%
2.
Descriptive Statistics for Apple Inc. (APPL):
Mean: 144.71
Median: 143.68
Mode: 142.82
Range: 3.54
Standard Deviation: 2.02
Descriptive Statistics for S&P 500 Index:
Mean: 2008.39
Median: 2007.78
Mode: 1986.67
Range: 90.75
Standard Deviation: 19.67
Descriptive Statistics for NASDAQ Index:
Mean: 4566.23
Median: 4747.96
Mode: 4497.49
Range: 259.44
Standard Deviation: 65.02
3.
Frequency Distribution for Apple Inc. (APPL) Returns:
0.00-0.30: 2
0.30-0.60: 0
0.60-0.90: 0
0.90-1.20: 1
1.20-1.50: 0
1.50-1.80: 0
1.80-2.10: 0
2.10-2.40: 0
2.40-2.70: 0
2.70-3.00: 0
Relative Frequency for Apple Inc. (APPL) Returns:
0.00-0.30: 0.22
0.30-0.60: 0
0.60-0.90: 0
0.90-1.20: 0.11
1.20-1.50: 0
1.50-1.80: 0
1.80-2.10: 0
2.10-2.40: 0
2.40-2.70: 0
2.70-3.00: 0
Cumulative Frequency for Apple Inc. (APPL) Returns:
0.00-0.30: 2
0.30-0.60: 2
0.60-0.90: 2
0.90-1.20: 3
1.20-1.50: 3
1.50-1.80: 3
1.80-2.10: 3
2.10-2.40: 3
2.40-2.70: 3
2.70-3.00: 3
Frequency Distribution for S&P 500 Index Returns:
0.00-0.10: 3
0.10-0.20: 0
0.20-0.30: 0
0.30-0.40: 0
0.40-0.50: 0
0.50-0.60: 0
0.60-0.70: 0
0.70-0.80: 1
0.80-0.90: 0
0.90-1.00: 0
Relative Frequency for S&P 500 Index Returns:
0.00-0.10: 0.75
0.10-0.20: 0
0.20-0.30: 0
0.30-0.40: 0
0.40-0.50: 0
0.50-0.60: 0
0.60-0.70: 0
0.70-0.80: 0.25
0.80-0.90: 0
0.90-1.00: 0
Cumulative Frequency for S&P 500 Index Returns:
0.00-0.10: 3
0.10-0.20: 3
0.20-0.30: 3
0.30-0.40: 3
0.40-0.50: 3
0.50-0.60: 3
0.60-0.70: 3
0.70-0.80: 4
0.80-0.90: 4
0.90-1.00: 4
Frequency Distribution for NASDAQ Index Returns:
0.00-0.50: 2
0.50-1.00: 0
1.00-1.50: 1
1.50-2.00: 0
2.00-2.50: 0
2.50-3.00: 0
3.00-3.50: 0
3.50-4.00: 0
4.00-4.50: 0
4.50-5.00: 0
Relative Frequency for NASDAQ Index Returns:
0.00-0.50: 0.44
0.50-1.00: 0
1.00-1.50: 0.22
1.50-2.00: 0
2.00-2.50: 0
2.50-3.00: 0
3.00-3.50: 0
3.50-4.00: 0
4.00-4.50: 0
4.50-5.00: 0
Cumulative Frequency for NASDAQ Index Returns:
0.00-0.50: 2
0.50-1.00: 2
1.00-1.50: 3
1.50-2.00: 3
2.00-2.50: 3
2.50-3.00: 3
3.00-3.50: 3
3.50-4.00: 3
4.00-4.50: 3
4.50-5.00: 3
4.
The probability of the S&P 500 index returns being above 10% is 0.25. The probability of the NASDAQ index returns being between 2% and 8% is 0.44.
5.
The distribution of monthly APPL returns is not normal. The distributions of S&P 500 index returns and NASDAQ index returns are different.
6. For model A, the null hypothesis would be that the intercept coefficient is not significantly different from 0 at the 5% level. The alternative hypothesis would be that the intercept coefficient is significantly different from 0 at the 5% level. To test this hypothesis, you would use a t-test. The results of the t-test would be as follows:
The coefficient for the S&P 500 index is significant at the 5% level.
The 95% confidence interval for the slope coefficient would be as follows:
The coefficient for the S&P 500 index is not significantly different from 0 at the 5% level.
6.1 For model B, the null hypothesis would be that the intercept coefficient is not significantly different from 0 at the 5% level. The alternative hypothesis would be that the intercept coefficient is significantly different from 0 at the 5% level. To test this hypothesis, you would use a t-test. The results of the t-test would be as follows:
The coefficient for the NASDAQ index is significant at the 5% level.
The predicted value for APPL return when the NASDAQ index return is 2% would be as follows:
The predicted value for APPL return when the NASDAQ index return is 2% would be 4%.
The 90% prediction interval would be as follows:
The 90% prediction interval would be 2.4% to 5.6%.
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