Financial Modeling Assignment: Portfolio Analysis and Option Pricing

This financial modeling assignment delves into various aspects of investment analysis and portfolio management. It begins with calculating the mean, variance, and standard deviation of monthly returns for several stocks, followed by computing their betas relative to the S&P 500. The assignment then proceeds to construct a covariance matrix and two envelope portfolios. Furthermore, it involves plotting the optimal portfolio on a graph and calculating its mean, variance, and beta. The solution also includes the computation of an optimum portfolio under short-sale constraints. The assignment extends to option pricing using both the binomial model and the Black-Scholes model, with a comparison of the results and a percentage error calculation. Finally, it presents a graph illustrating a put strategy, providing a comprehensive overview of financial modeling techniques.

Running head: FINANCIAL MODELLING
Financial Modelling
Name of the Student:
Name of the University:
Authors Note:

FINANCIAL MODELLING
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Table of Contents
1. Computing mean, variance and standard deviation of monthly returns:...............................2
2. Computing beta of stocks:......................................................................................................2
3. Computing covariance matrix of the stocks:..........................................................................5
4. Computing two envelope portfolios:......................................................................................5
5. Plotting optimal portfolio in graph:........................................................................................8
6. Computing mean, variance and beta of monthly returns of the portfolio:.............................8
7. Computing optimum portfolio with short sale constraint:.....................................................9
8a. Computing binomial model for the stock:............................................................................9
8b. Computing option using Black Scholes model:...................................................................9
8c. Computing percentage error:..............................................................................................10
9. Plotting graph with a pout strategy:.....................................................................................11
Bibliography:............................................................................................................................12

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1. Computing mean, variance and standard deviation of monthly returns:
Particulars Amazon.com
, Inc
American
Express Co
Apple
Inc
Macy's
Inc
United Health
Group Inc
Monthly Mean 2.620% 0.899% 1.382% -0.838% 2.435%
Monthly Variance 0.0
055 0.0035
0.0
048
0.
0093 0.0019
Monthly Standard
deviation
0.0
740 0.0590
0.0
693
0.
0965 0.0439
2. Computing beta of stocks:
Date Amazon.
com, Inc
America
n
Express
Co
Apple
Inc
Macy's
Inc
United
Health
Group Inc
S&P 500 Portfolio
Returns
01-01-13 5.67% 2.29% -15.56% 1.25% 1.77% 4.92% 7.74%
01-02-13 -0.46% 5.52% -3.14% 3.95% -3.24% 1.10% -12.61%
01-03-13 0.84% 8.20% 0.29% 1.78% 6.80% 3.54% 8.08%
01-04-13 -4.88% 1.40% 0.03% 6.39% 4.64% 1.79% 1.71%
01-05-13 5.89% 10.14% 1.56% 8.05% 4.41% 2.06% -0.46%
01-06-13 3.11% -1.26% -12.59% -0.71% 4.45% -1.51% 15.01%
01-07-13 8.13% -1.33% 13.21% 0.71% 10.67% 4.83% 23.61%
01-08-13 -6.96% -2.55% 7.39% -8.43% -1.54% -3.18% -0.62%
01-09-13 10.68% 4.90% -2.17% -2.65% -0.18% 2.93% 4.30%
01-10-13 15.22% 7.99% 9.20% 6.36% -4.79% 4.36% -14.13%

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01-11-13 7.81% 4.77% 6.19% 14.42% 8.72% 2.77% 7.18%
01-12-13 1.30% 5.59% 0.89% 0.26% 1.09% 2.33% -0.97%
01-01-14 -10.60% -6.50% -11.39% -0.38% -4.09% -3.62% -6.87%
01-02-14 0.95% 7.10% 5.00% 8.40% 6.68% 4.22% 2.68%
01-03-14 -7.37% -1.38% 1.98% 2.44% 5.93% 0.69% 7.32%
01-04-14 -10.08% -2.93% 9.48% -3.19% -8.86% 0.62% -21.50%
01-05-14 2.73% 4.55% 7.02% 4.19% 5.94% 2.08% 6.23%
01-06-14 3.84% 3.62% 2.73% -3.17% 2.63% 1.89% 6.81%
01-07-14 -3.70% -7.52% 2.83% -0.40% -0.86% -1.52% 0.72%
01-08-14 7.99% 1.75% 6.97% 7.50% 6.72% 3.70% 9.85%
01-09-14 -5.02% -2.27% -1.72% -6.83% -0.50% -1.56% 3.24%
01-10-14 -5.41% 2.72% 6.95% -0.62% 9.67% 2.29% 14.62%
01-11-14 10.31% 2.71% 9.64% 11.57% 3.74% 2.42% 0.63%
01-12-14 -8.72% 0.67% -7.46% 1.29% 2.46% -0.42% 1.13%
01-01-15 13.31% -14.24% 5.96% -2.89% 4.98% -3.15% 26.71%
01-02-15 6.98% 1.11% 9.21% -0.25% 6.72% 5.34% 14.87%
01-03-15 -2.14% -4.35% -3.19% 1.85% 4.02% -1.75% 9.42%
01-04-15 12.53% -0.86% 0.58% -0.43% -6.00% 0.85% -5.53%
01-05-15 1.75% 2.89% 4.02% 3.53% 7.61% 1.04% 11.44%
01-06-15 1.13% -2.54% -3.79% 0.77% 1.48% -2.12% 5.54%
01-07-15 21.12% -2.16% -3.35% 2.33% -0.49% 1.95% 9.95%
01-08-15 -4.44% 0.86% -7.16% -16.41% -4.81% -6.46% 0.79%
01-09-15 -0.20% -3.43% -2.35% -13.28% 0.27% -2.68% 12.82%
01-10-15 20.11% -1.18% 8.01% -0.66% 1.51% 7.97% 12.60%
01-11-15 6.03% -2.24% -1.01% -26.58% -4.40% 0.05% 14.82%
01-12-15 1.65% -2.96% -11.68% -11.08% 4.28% -1.77% 22.35%
01-01-16 -14.10% -26.24% -7.82% 14.43% -2.13% -5.21% -3.72%

FINANCIAL MODELLING
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01-02-16 -6.05% 3.81% -0.67% 6.70% 3.36% -0.41% -3.13%
01-03-16 7.18% 9.96% 11.97% 2.02% 7.91% 6.39% 9.61%
01-04-16 10.53% 6.36% -15.07% -10.77% 2.13% 0.27% 17.00%
01-05-16 9.15% 0.50% 6.32% -17.57% 1.50% 1.52% 18.73%
01-06-16 -1.00% -7.91% -4.36% 1.20% 5.48% 0.09% 16.02%
01-07-16 5.86% 5.91% 8.62% 6.40% 1.41% 3.50% -4.48%
01-08-16 1.35% 1.72% 1.80% 0.97% -5.12% -0.12% -12.20%
01-09-16 8.49% -2.38% 6.34% 2.38% 2.86% -0.12% 8.52%
01-10-16 -5.84% 3.65% 0.43% -1.52% 0.95% -1.96% -2.23%
01-11-16 -5.10% 8.12% -2.70% 14.54% 11.35% 3.36% 6.05%
01-12-16 2.38% 2.79% 4.68% -16.42% 1.08% 1.80% 12.57%
01-01-17 6.89% 3.06% 4.66% -19.25% 1.28% 1.77% 17.13%
01-02-17 2.58% 4.71% 12.12% 11.74% 2.00% 3.65% -8.80%
01-03-17 4.79% -1.19% 4.75% -11.40% -0.83% -0.04% 8.62%
01-04-17 4.25% 0.18% -0.01% -1.43% 6.42% 0.91% 16.43%
01-05-17 7.26% -2.96% 6.15% -21.79% 0.17% 1.15% 20.29%
01-06-17 -2.71% 9.06% -5.89% -1.11% 5.68% 0.48% 6.95%
01-07-17 2.02% 1.17% 3.22% 2.17% 3.39% 1.92% 5.03%
01-08-17 -0.73% 1.02% 9.77% -13.41% 3.63% 0.05% 14.09%
01-09-17 -1.98% 4.94% -6.21% 4.93% -1.55% 1.91% -9.49%
01-10-17 13.95% 5.44% 9.24% -15.11% 7.08% 2.19% 27.20%
01-11-17 6.27% 2.27% 1.65% 23.80% 8.19% 2.77% 1.09%
Particulars Amazon.co
m, Inc
American
Express Co
Apple
Inc
Macy's
Inc
United Health
Group Inc
Beta

FINANCIAL MODELLING
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1.45 1.28 1.18 1.00 0.65
3. Computing covariance matrix of the stocks:
Variance-Covariance Matrix
Amazon.co
m, Inc
American
Express Co
Apple
Inc
Macy's
Inc
United Health
Group Inc
Amazon.com, Inc 0.00548 0.00104 0.001
56
-
0.0007
0
0.00035
American
Express Co
0.00104 0.00348 0.000
77
0.0005
7
0.00071
Apple Inc 0.00156 0.00077 0.004
80
0.0002
8
0.00061
Macy's Inc -0.00070 0.00057 0.000
28
0.0093
2
0.00142
United Health
Group Inc
0.00035 0.00071 0.000
61
0.0014
2
0.00192
4. Computing two envelope portfolios:
Computing an envelope portfolio with constant 0%
z Envelope portfolio X
3.638279741 25.26%
-1.003380874 -6.97%

FINANCIAL MODELLING
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0.195186814 1.35%
-2.761517194 -19.17%
14.33689575 99.52%
100.00%
Computing an envelope portfolio with constant 1%
z Envelope portfolio Y
2.564877578 38.84%
-2.488044159 -37.68%
-0.789162338 -11.95%
-3.212129067 -48.64%
10.52777521 159.43%
100.00%
E (x) 0.4612
Var (x) 0.4612
Sigma (x) 67.91%
E (y) 0.3172
Var (y) 0.2511
Sigma (y) 50.11%
Cov(x,y) 0.3172
Corr(x,y) 0.9319
Portion of 0.3000

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E(rp) 36.04%
σp2 0.2978
σp 54.57%
Proportion of x Sigma Return
54.57% 36.04%
-2.2 58.16% 0.03%
-1.9 53.09% 4.35%
-1.6 48.95% 8.67%
-1.3 45.97% 12.99%
-1 44.40% 17.31%
-0.7 44.38% 21.63%
-0.4 45.91% 25.96%
-0.1 48.86% 30.28%
0.2 52.98% 34.60%
0.5 58.02% 38.92%
0.8 63.78% 43.24%
1.1 70.06% 47.56%
1.4 76.74% 51.89%
1.7 83.73% 56.21%
2 90.96% 60.53%
2.3 98.37% 64.85%

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40.00% 50.00% 60.00% 70.00% 80.00% 90.00% 100.00% 110.00%
0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
70.00%
17.31%
Efficient Frontier
Series2
5. Plotting optimal portfolio in graph:
0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00%
0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
70.00%
17.31%
Efficient Frontier
Series2
Portfolio Z
6. Computing mean, variance and beta of monthly returns of the portfolio:
Portfolio mean 6.11%
Portfolio sigma 10.30%
Portfolio beta 26.76%

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7. Computing optimum portfolio with short sale constraint:
Portfolio Z
0.2697
0.0000
0.0000
0.0000
0.7303
1.0000
Portfolio mean 2.48%
Portfolio sigma 3.95%
e=Theta 30.97%
8a. Computing binomial model for the stock:
175.61
174.19
172.78 172.78
171.38 171.38
170 170.00 170.00
168.63 168.63
167.26 167.26
165.91
164.57
8b. Computing option using Black Scholes model:
Input Data Value
Stock Price now (P) 171.5