Description
Question: Estimate the copula correlation coefficient using Maximum Likelihood Estimation
Your goal for this question is to estimate the copula correlation coefficient () between continuous compounded daily returns of two stock indexes, S&P 500 and NASDAQ, using Maximum Likelihood Estimation. “Copula_HW_Data.xlsx” is the daily data of the stock indexes from Dec 01, 2015 to Dec 31, 2018 (for 1000 consecutive trading days). Below are some suggested steps:
- Calculate continuous compounded daily returns of two stock indexes;
- Transform daily returns of each stock index return into a new variable that has a standard normal distribution on a “percentile to percentile” basis;
- Write out the likelihood function, which is a function of the copula correlation coefficient ().
- Find the copula correlation coefficient () that maximizes the likelihood function.
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Explanation & Answer
Please view explanation and answer below.
5. An investor has a $1000 portfolio of three stocks, each of which has a normal distribution of
returns. We are told the annual variance on the return of his portfolio are 1.5, -0.75, and 0.25
respectively. His portfolio weight in the first stock is ½.
a) Find his annual Var (0.05). You can use the fact that z (0.05) = -1.65
VaR = [Rp- (z)(s)] V
= [0.5 - (-1.65*1.2247] * 1000 + [0.5 - (-1.65*-0.8660)] * 1000 + [0.5 - (-1.65*0.5)] * 1000
= $2,916.89
b) Find the contribution of the 3rd stocks to his overall VaR. You may use the facts that the portfolio
weights sum to 1 and that
VaR = [Rp- (z)(s)] V
= [0.5 - (-1.65*0.5)] * 1000
= $1,325
c) Using his marginal VaR, find the approximate increase in his VaR if he increases his position in the
2nd stock by $20
The annual variance of the second portfolio = -0.75
Stock = $1,020
[0.5 - (-1.65*-0.8660)] * 1020
=-$928.94
Increase = -947.52 -928.94
The approximate increase 18.58 due increase in stock by $20 in 2nd portfolio
Please view explanation and answer below.
Date
S&P 500
Nasdaq
S&P 500 returns
Nasdaq returns
12/31/2018
2506.85
6635.28
0.8492%
0.7709%
12/28/2018
2485.74
6584.52
-0.1242%
0.0764%
12/27/2018
2488.83
6579.49
0.8563%
0.3834%
12/26/2018
2467.70
6554.36
4.9594%
5.8363%
12/24/2018
2351.10
6192.92
-2.7112%
-2.2118%
12/21/2018
2416.62
6332.99
-2.0588%
-2.9934%
12/20/2018
2467.42
6528.41
-1.5772%
-1.6336%
12/19/2018
2506.96
6636.83
-1.5396%
-2.1681%
12/18/2018
2546.16
6783.91
0.0086%
0.4469%
12/17/2018
2545.94
6753.73
-2.0773%
-2.2708%
12/14/2018
2599.95
6910.66
-1.9087%
-2.2583%
12/13/2018
2650.54
7070.33
-0.0200%
-0.3942%
12/12/2018
2651.07
7098.31
0.5420%
0.9454%
12/11/2018
2636.78
7031.83
-0.0356%
0.1611%
12/10/2018
2637.72
7020.52
0.1762%
0.7357%
12/7/2018
2633.08
6969.25
-2.3320%
-3.0468%
12/6/2018
2695.95
7188.26
-0.1522%
0.4167%
12/4/2018
2700.06
7158.43
-3.2365%
-3.8041%
12/3/2018
2790.37
7441.51
1.0941%
1.5138%
11/30/2018
2760.17
7330.54
0.8171%
0.7900%
11/29/2018
2737.80
7273.08
-0.2183%
-0.2539%
11/28/2018
2743.79
7291.59
2.2974%
2.9493%
11/27/2018
2682.17
7082.70
0.3262%
0.0120%
11/26/2018
2673.45
7081.85
1.5532%
2.0589%
11/23/2018
2632.56
6938.98
-0.6555%
-0.4772%
11/21/2018
2649.93
6972.25
0.3043%
0.9181%
11/20/2018
2641.89
6908.82
-1.8151%
-1.7025%
11/19/2018
2690.73
7028.48
-1.6643%
-3.0270%
11/16/2018
2736.27
7247.87
0.2223%
-0.1537%
11/15/2018
2730.20
7259.03
1.0594%
1.7185%
11/14/2018
2701.58
7136.39
-0.7567%
-0.8954%
11/13/2018
2722.18
7200.87
-0.1482%
0.0000%
11/12/2018
2726.22
7200.87
-1.9701%
-2.7816%
11/9/2018
2781.01
7406.90
-0.9199%
-1.6463%
11/8/2018
2806.83
7530.88
-0.2509%
-0.5266%
11/7/2018
2813.89
7570.75
2.1209%
2.6409%
11/6/2018
2755.45
7375.96
0.6259%
0.6428%
11/5/2018
2738.31
7328.85
0.5600%
-0.3825%
11/2/2018
2723.06
7356.99
-0.6317%
-1.0367%
11/1/2018
2740.37
7434.06
1.0558%
1.7542%
10/31/2018
2711.74
7305.90
1.0851%
2.0142%
10/30/2018
2682.63
7161.65
1.5667%
1.5795%
10/29/2018
2641.25
7050.29
-0.6560%
-1.6313%
10/26/2018
2658.69
7167.21
-1.7327%
-2.0651%
10/25/2018
2705.57
7318.34
1.8625%
2.9534%
10/24/2018
2656.10
7108.40
-3.0864%
-4.4254%
10/23/2018
10/22/2018
10/19/2018
10/18/2018
10/17/2018
10/16/2018
10/15/2018
10/12/2018
10/11/2018
10/10/2018
10/9/2018
10/8/2018
10/5/2018
10/4/2018
10/3/2018
10/2/2018
10/1/2018
9/28/2018
9/27/2018
9/26/2018
9/25/2018
9/24/2018
9/21/2018
9/20/2018
9/19/2018
9/18/2018
9/17/2018
9/14/2018
9/13/2018
9/12/2018
9/11/2018
9/10/2018
9/7/2018
9/6/2018
9/5/2018
9/4/2018
8/31/2018
8/30/2018
8/29/2018
8/28/2018
8/27/2018
8/24/2018
8/23/2018
8/22/2018
8/21/2018
8/20/2018
8/17/2018
2740.69
2755.88
2767.78
2768.78
2809.21
2809.92
2750.79
2767.13
2728.37
2785.68
2880.34
2884.43
2885.57
2901.61
2925.51
2923.43
2924.59
2913.98
2914.00
2905.97
2915.56
2919.37
2929.67
2930.75
2907.95
2904.31
2888.80
2904.98
2904.18
2888.92
2887.89
2877.13
2871.68
2878.05
2888.60
2896.72
2901.52
2901.13
2914.04
2897.52
2896.74
2874.69
2856.98
2861.82
2862.96
2857.05
2850.13
7437.54
7468.63
7449.03
7485.14
7642.70
7645.49
7430.74
7496.89
7329.06
7422.05
7738.02
7735.95
7788.45
7879.51
8025.09
7999.55
8037.30
8046.35
8041.97
7990.37
8007.47
7993.25
7986.96
8028.23
7950.04
7956.11
7895.79
8010.04
8013.71
7954.23
7972.47
7924.16
7902.54
7922.73
7995.17
8091.25
8109.54
8088.36
8109.69
8030.04
8017.90
7945.98
7878.46
7889.10
7859.17
7821.01
7816.33
-0.5512%
-0.4300%
-0.0361%
-1.4392%
-0.0253%
2.1496%
-0.5905%
1.4206%
-2.0573%
-3.2864%
-0.1418%
-0.0395%
-0.5528%
-0.8169%
0.0712%
-0.0397%
0.3641%
-0.0007%
0.2763%
-0.3289%
-0.1305%
-0.3516%
-0.0369%
0.7841%
0.1253%
0.5369%
-0.5570%
0.0275%
0.5282%
0.0357%
0.3740%
0.1898%
-0.2213%
-0.3652%
-0.2803%
-0.1654%
0.0134%
-0.4430%
0.5701%
0.0269%
0.7670%
0.6199%
-0.1691%
-0.0398%
0.2069%
0.2428%
0.3323%
-0.4163%
0.2631%
-0.4824%
-2.0616%
-0.0365%
2.8900%
-0.8824%
2.2899%
-1.2529%
-4.0833%
0.0268%
-0.6741%
-1.1557%
-1.8141%
0.3193%
-0.4697%
-0.1125%
0.0545%
0.6458%
-0.2136%
0.1779%
0.0788%
-0.5141%
0.9835%
-0.0763%
0.7639%
-1.4263%
-0.0458%
0.7478%
-0.2288%
0.6097%
0.2736%
-0.2548%
-0.9060%
-1.1875%
-0.2255%
0.2619%
-0.2630%
0.9919%
0.1514%
0.9051%
0.8570%
-0.1349%
0.3808%
0.4879%
0.0599%
0.1257%
8/16/2018
8/15/2018
8/14/2018
8/13/2018
8/10/2018
8/9/2018
8/8/2018
8/7/2018
8/6/2018
8/3/2018
8/2/2018
8/1/2018
7/31/2018
7/30/2018
7/27/2018
7/26/2018
7/25/2018
7/24/2018
7/23/2018
7/20/2018
7/19/2018
7/18/2018
7/17/2018
7/16/2018
7/13/2018
7/12/2018
7/11/2018
7/10/2018
7/9/2018
7/6/2018
7/5/2018
7/3/2018
7/2/2018
6/29/2018
6/28/2018
6/27/2018
6/26/2018
6/25/2018
6/22/2018
6/21/2018
6/20/2018
6/19/2018
6/18/2018
6/15/2018
6/14/2018
6/13/2018
6/12/2018
2840.69
2818.37
2839.96
2821.93
2833.28
2853.58
2857.70
2858.45
2850.40
2840.35
2827.22
2813.36
2816.29
2802.60
2818.82
2837.44
2846.07
2820.40
2806.98
2801.83
2804.49
2815.62
2809.55
2798.43
2801.31
2798.29
2774.02
2793.84
2784.17
2759.82
2736.61
2713.22
2726.71
2718.37
2716.31
2699.63
2723.06
2717.07
2754.88
2749.76
2767.32
2762.59
2773.75
2779.66
2782.49
2775.63
2786.85
7806.52
7774.12
7870.89
7819.71
7839.11
7891.78
7888.33
7883.66
7859.68
7812.01
7802.69
7707.29
7671.79
7630.00
7737.42
7852.18
7932.24
7840.77
7841.87
7820.20
7825.30
7854.44
7855.12
7805.72
7825.98
7823.92
7716.61
7759.20
7756.20
7688.39
7586.43
7502.67
7567.69
7510.30
7503.68
7445.08
7561.63
7532.01
7692.82
7712.95
7781.51
7725.59
7747.03
7746.38
7761.04
7695.70
7703.79
0.7919%
-0.7602%
0.6389%
-0.4006%
-0.7114%
-0.1442%
-0.0262%
0.2824%
0.3538%
0.4644%
0.4926%
-0.1040%
0.4885%
-0.5754%
-0.6562%
-0.3032%
0.9102%
0.4781%
0.1838%
-0.0948%
-0.3953%
0.2161%
0.3974%
-0.1028%
0.1079%
0.8749%
-0.7094%
0.3473%
0.8823%
0.8481%
0.8621%
-0.4947%
0.3068%
0.0758%
0.6179%
-0.8604%
0.2205%
-1.3725%
0.1862%
-0.6346%
0.1712%
-0.4023%
-0.2126%
-0.1017%
0.2472%
-0.4026%
0.1743%
0.4168%
-1.2295%
0.6545%
-0.2475%
-0.6674%
0.0437%
0.0592%
0.3051%
0.6102%
0.1194%
1.2378%
0.4627%
0.5477%
-1.3883%
-1.4615%
-1.0093%
1.1666%
-0.0140%
0.2771%
-0.0652%
-0.3710%
-0.0087%
0.6329%
-0.2589%
0.0263%
1.3906%
-0.5489%
0.0387%
0.8820%
1.3440%
1.1164%
-0.8592%
0.7642%
0.0882%
0.7871%
-1.5413%
0.3933%
-2.0904%
-0.2610%
-0.8811%
0.7238%
-0.2768%
0.0084%
-0.1889%
0.8490%
-0.1050%
0.5726%
6/11/2018
6/8/2018
6/7/2018
6/6/2018
6/5/2018
6/4/2018
6/1/2018
5/31/2018
5/30/2018
5/29/2018
5/25/2018
5/24/2018
5/23/2018
5/22/2018
5/21/2018
5/18/2018
5/17/2018
5/16/2018
5/15/2018
5/14/2018
5/11/2018
5/10/2018
5/9/2018
5/8/2018
5/7/2018
5/4/2018
5/3/2018
5/2/2018
5/1/2018
4/30/2018
4/27/2018
4/26/2018
4/25/2018
4/24/2018
4/23/2018
4/20/2018
4/19/2018
4/18/2018
4/17/2018
4/16/2018
4/13/2018
4/12/2018
4/11/2018
4/10/2018
4/9/2018
4/6/2018
4/5/2018
2782.00
2779.03
2770.37
2772.35
2748.80
2746.87
2734.62
2705.27
2724.01
2689.86
2721.33
2727.76
2733.29
2724.44
2733.01
2712.97
2720.13
2722.46
2711.45
2730.13
2727.72
2723.07
2697.79
2671.92
2672.63
2663.42
2629.73
2635.67
2654.80
2648.05
2669.91
2666.94
2639.40
2634.56
2670.29
2670.14
2693.13
2708.64
2706.39
2677.84
2656.30
2663.99
2642.19
2656.87
2613.16
2604.47
2662.84
7659.93
7645.51
7635.07
7689.24
7637.86
7606.46
7554.33
7442.12
7462.45
7396.59
7433.85
7424.43
7425.96
7378.46
7394.04
7354.34
7382.47
7398.30
7351.63
7411.32
7402.88
7404.97
7339.91
7266.90
7265.21
7209.62
7088.15
7100.90
7130.70
7066.27
7119.80
7118.68
7003.74
7007.35
7128.60
7146.13
7238.06
7295.24
7281.10
7156.28
7106.65
7140.25
7069.03
7094.30
6950.34
6915.11
7076.55
0.1069%
0.3126%
-0.0714%
0.8567%
0.0703%
0.4480%
1.0849%
-0.6880%
1.2696%
-1.1564%
-0.2357%
-0.2023%
0.3248%
-0.3136%
0.7387%
-0.2632%
-0.0856%
0.4061%
-0.6842%
0.0883%
0.1708%
0.9371%
0.9682%
-0.0266%
0.3458%
1.2811%
-0.2254%
-0.7206%
0.2549%
-0.8187%
0.1114%
1.0434%
0.1837%
-1.3381%
0.0056%
-0.8537%
-0.5726%
0.0831%
1.0662%
0.8109%
-0.2887%
0.8251%
-0.5525%
1.6727%
0.3337%
-2.1920%
0.6863%
0.1886%
0.1367%
-0.7045%
0.6727%
0.4128%
0.6901%
1.5078%
-0.2724%
0.8904%
-0.5012%
0.1269%
-0.0206%
0.6438%
-0.2107%
0.5398%
-0.3810%
-0.2140%
0.6348%
-0.8054%
0.1140%
-0.0282%
0.8864%
1.0047%
0.0233%
0.7711%
1.7137%
-0.1796%
-0.4179%
0.9118%
-0.7518%
0.0157%
1.6411%
-0.0515%
-1.7009%
-0.2453%
-1.2701%
-0.7838%
0.1942%
1.7442%
0.6984%
-0.4706%
1.0075%
-0.3562%
2.0713%
0.5095%
-2.2813%
0.4891%
4/4/2018
4/3/2018
4/2/2018
3/29/2018
3/28/2018
3/27/2018
3/26/2018
3/23/2018
3/22/2018
3/21/2018
3/20/2018
3/19/2018
3/16/2018
3/15/2018
3/14/2018
3/13/2018
3/12/2018
3/9/2018
3/8/2018
3/7/2018
3/6/2018
3/5/2018
3/2/2018
3/1/2018
2/28/2018
2/27/2018
2/26/2018
2/23/2018
2/22/2018
2/21/2018
2/20/2018
2/16/2018
2/15/2018
2/14/2018
2/13/2018
2/12/2018
2/9/2018
2/8/2018
2/7/2018
2/6/2018
2/5/2018
2/2/2018
2/1/2018
1/31/2018
1/30/2018
1/29/2018
1/26/2018
2644.69
2614.45
2581.88
2640.87
2605.00
2612.62
2658.55
2588.26
2643.69
2711.93
2716.94
2712.92
2752.01
2747.33
2749.48
2765.31
2783.02
2786.57
2738.97
2726.80
2728.12
2720.94
2691.25
2677.67
2713.83
2744.28
2779.60
2747.30
2703.96
2701.33
2716.26
2732.22
2731.20
2698.63
2662.94
2656.00
2619.55
2581.00
2681.66
2695.14
2648.94
2762.13
2821.98
2823.81
2822.43
2853.53
2872.87
7042.11
6941.28
6870.12
7063.45
6949.23
7008.81
7220.54
6992.67
7166.68
7345.29
7364.30
7344.24
7481.99
7481.74
7496.81
7511.01
7588.32
7560.81
7427.95
7396.65
7372.01
7330.71
7257.87
7180.56
7273.01
7330.35
7421.46
7337.39
7210.09
7218.23
7234.31
7239.47
7256.43
7143.62
7013.51
6981.96
6874.49
6777.16
7051.98
7115.88
6967.53
7240.95
7385.86
7411.48
7402.48
7466.51
7505.77
1.1566%
1.2615%
-2.2337%
1.3770%
-0.2917%
-1.7276%
2.7157%
-2.0967%
-2.5163%
-0.1844%
0.1482%
-1.4204%
0.1703%
-0.0782%
-0.5725%
-0.6364%
-0.1274%
1.7379%
0.4463%
-0.0484%
0.2639%
1.1032%
0.5072%
-1.3324%
-1.1096%
-1.2707%
1.1757%
1.6028%
0.0974%
-0.5497%
-0.5841%
0.0373%
1.2069%
1.3402%
0.2613%
1.3915%
1.4936%
-3.7536%
-0.5002%
1.7441%
-4.0979%
-2.1209%
-0.0648%
0.0489%
-1.0899%
-0.6732%
1.1841%
1.4526%
1.0358%
-2.7370%
1.6436%
-0.8501%
-2.9323%
3.2587%
-2.4280%
-2.4316%
-0.2581%
0.2731%
-1.8411%
0.0033%
-0.2010%
-0.1891%
-1.0188%
0.3638%
1.7886%
0.4232%
0.3342%
0.5634%
1.0036%
1.0767%
-1.2711%
-0.7822%
-1.2277%
1.1458%
1.7656%
-0.1128%
-0.2223%
-0.0713%
-0.2337%
1.5792%
1.8551%
0.4519%
1.5633%
1.4361%
-3.8971%
-0.8980%
2.1292%
-3.7760%
-1.9620%
-0.3457%
0.1216%
-0.8576%
-0.5231%
1.2766%
1/25/2018
1/24/2018
1/23/2018
1/22/2018
1/19/2018
1/18/2018
1/17/2018
1/16/2018
1/12/2018
1/11/2018
1/10/2018
1/9/2018
1/8/2018
1/5/2018
1/4/2018
1/3/2018
1/2/2018
12/29/2017
12/28/2017
12/27/2017
12/26/2017
12/22/2017
12/21/2017
12/20/2017
12/19/2017
12/18/2017
12/15/2017
12/14/2017
12/13/2017
12/12/2017
12/11/2017
12/8/2017
12/7/2017
12/6/2017
12/5/2017
12/4/2017
12/1/2017
11/30/2017
11/29/2017
11/28/2017
11/27/2017
11/24/2017
11/22/2017
11/21/2017
11/20/2017
11/17/2017
11/16/2017
2839.25
2837.54
2839.13
2832.97
2810.30
2798.03
2802.56
2776.42
2786.24
2767.56
2748.23
2751.29
2747.71
2743.15
2723.99
2713.06
2695.81
2673.61
2687.54
2682.62
2680.50
2683.34
2684.57
2679.25
2681.47
2690.16
2675.81
2652.01
2662.85
2664.11
2659.99
2651.50
2636.98
2629.27
2629.57
2639.44
2642.22
2647.58
2626.07
2627.04
2601.42
2602.42
2597.08
2599.03
2582.14
2578.85
2585.64
7411.16
7415.06
7460.29
7408.03
7336.38
7296.05
7298.28
7223.69
7261.06
7211.78
7153.57
7163.58
7157.39
7136.56
7077.91
7065.53
7006.90
6903.39
6950.16
6939.34
6936.25
6959.96
6965.36
6960.96
6963.85
6994.76
6936.58
6856.53
6875.80
6862.32
6875.08
6840.08
6812.84
6776.38
6762.21
6775.37
6847.59
6873.97
6824.39
6912.36
6878.52
6889.16
6867.36
6862.48
6790.71
6782.79
6793.29
0.0603%
-0.0560%
0.2174%
0.8067%
0.4385%
-0.1616%
0.9415%
-0.3524%
0.6750%
0.7034%
-0.1112%
0.1303%
0.1662%
0.7034%
0.4029%
0.6399%
0.8303%
-0.5183%
0.1834%
0.0791%
-0.1058%
-0.0458%
0.1986%
-0.0828%
-0.3230%
0.5363%
0.8974%
-0.4071%
-0.0473%
0.1549%
0.3202%
0.5506%
0.2932%
-0.0114%
-0.3739%
-0.1052%
-0.2025%
0.8191%
-0.0369%
0.9849%
-0.0384%
0.2056%
-0.0750%
0.6541%
0.1276%
-0.2626%
0.8196%
-0.0526%
-0.6063%
0.7055%
0.9766%
0.5528%
-0.0306%
1.0326%
-0.5147%
0.6833%
0.8137%
-0.1397%
0.0865%
0.2919%
0.8286%
0.1752%
0.8367%
1.4994%
-0.6729%
0.1559%
0.0445%
-0.3407%
-0.0775%
0.0632%
-0.0415%
-0.4419%
0.8387%
1.1675%
-0.2803%
0.1964%
-0.1856%
0.5117%
0.3998%
0.5380%
0.2095%
-0.1942%
-1.0547%
-0.3838%
0.7265%
-1.2726%
0.4920%
-0.1544%
0.3174%
0.0711%
1.0569%
0.1168%
-0.1546%
1.2985%
11/15/2017
11/14/2017
11/13/2017
11/10/2017
11/9/2017
11/8/2017
11/7/2017
11/6/2017
11/3/2017
11/2/2017
11/1/2017
10/31/2017
10/30/2017
10/27/2017
10/26/2017
10/25/2017
10/24/2017
10/23/2017
10/20/2017
10/19/2017
10/18/2017
10/17/2017
10/16/2017
10/13/2017
10/12/2017
10/11/2017
10/10/2017
10/9/2017
10/6/2017
10/5/2017
10/4/2017
10/3/2017
10/2/2017
9/29/2017
9/28/2017
9/27/2017
9/26/2017
9/25/2017
9/22/2017
9/21/2017
9/20/2017
9/19/2017
9/18/2017
9/15/2017
9/14/2017
9/13/2017
9/12/2017
2564.62
2578.87
2584.84
2582.30
2584.62
2594.38
2590.64
2591.13
2587.84
2579.85
2579.36
2575.26
2572.83
2581.07
2560.40
2557.15
2569.1...
