Derivatives: Statistics and Analysis

DERIVATIVES
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STATISTICS
DERIVATIVES
Student Name:
Name of Institution:

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> #Importing the dataset
> Data_set<-read.csv("F:/Data.csv")
> #Question 1:
> #Excess daily returns has been calculated by getting the difference between the returns and the
risk- free interest rate.The risk- free rate has been taken to be 2% (0.02). The caluculations have
been done in excel. The values of excess daily returns can be extracted from data set using the
code below.
> Daily_Excess_Returns<-Data_set$Excess_Return #Extracting the daily returns from the data
set.
> Daily_Excess_Returns
[1] -0.45 -2.23 0.38 1.02 0.76 0.66 0.67 -0.23 0.11 0.01 1.07 0.34
[13] -0.55 -0.44 -0.20 -0.22 -0.29 0.05 0.05 -0.45 1.24 1.34 0.52 1.12
[25] 0.47 -0.04 -0.22 0.47 0.45 -0.77 1.12 0.15 0.24 1.25 0.92 -0.53
[37] 0.35 1.41 -0.75 -0.81 0.95 0.29 -2.75 -4.00 2.12 -1.19 -4.24 2.42
[49] 1.17 0.21 2.06 1.78 -0.24 0.19 -0.17 -0.11 2.00 1.05 -1.43 -0.50
[61] -1.43 1.08 1.25 0.14 1.27 0.44 1.88 0.14 -1.61 0.23 0.00 -0.19
[73] -2.39 -0.67 -0.21 -2.83 -2.54 3.67 -3.83 -0.46 2.25 -2.44 0.73 1.49
[85] 0.52 -2.11 0.58 2.22 -0.46 1.00 -0.51 0.86 2.37 -0.22 -0.05 -1.15
[97] -0.14 -2.53 -0.40 2.51 -0.21 -0.83 1.04 -0.82 0.10 1.53 0.82 0.11
[109] 1.67 0.92 -0.21 -0.36 -0.91 0.12 -0.28 -0.12 0.74 -0.62 1.10 -0.19
[121] -0.15 -0.74 1.09 0.49 1.84 0.82 0.39 0.25 -1.37 0.49 0.29 0.71
[133] -0.18 0.80 -0.32 0.87 -0.48 0.18 -0.97 -0.60 -2.18 0.40 -1.74 1.12
[145] 0.03 1.33 -1.15 1.40 1.45 0.69 -0.22 0.14 2.23 0.23 -0.40 0.97
[157] -0.21 -0.60 0.29 0.75 0.38 1.75 -2.64 -2.18 -2.47 0.29 0.21 1.20
[169] 0.23 0.93 0.41 0.29 0.08 -0.57 -0.60 0.75 -1.37 -0.25 -0.21 0.04
[181] -0.01 0.74 0.17 1.42 0.88 -0.07 1.01 -0.50 -0.37 -0.65 -2.09 -0.44
[193] 0.01 0.55 1.07 -0.27 0.60 0.28 -1.57 0.55 -0.44 1.23 -0.54 0.46
[205] 0.43 -0.05 0.57 -0.25 -0.11 -0.87 0.38 -2.19 -0.78 -1.37 -0.10 -4.82
[217] -0.41 2.70 -1.17 3.32 -0.47 -2.39 -0.69 0.85 -0.46 -4.80 3.89 -2.16
[229] -2.31 1.37 2.90 0.16 -0.57 -0.32 0.54 3.04 -0.69 -1.87 -2.62 0.06
[241] -0.61 1.91 -0.27 -4.43 -1.11 1.56 -0.58 2.58 -0.28 3.55 -0.05 0.99
> #Question 1 a: The mean and standard deviation of the excess daily retuns
> Mean_of_excess_dailyretuns<-mean(Daily_Excess_Returns) #The mean of excess daily retuns
> Mean_of_excess_dailyretuns
[1] 0.03130952
#Report
#The mean of daily returns is 0.031. This mean is equivalent to 3.1%. The mean value of the
daily returns is more than the risk- free market return which is 2%. Therefore, it is clear that the
software industry is profitable (Masayuki, 2010)
> Std_deviation_excess_dailyreturns<-sd(Daily_Excess_Returns)# The stdanrd deviation
> Std_deviation_excess_dailyreturns
[1] 1.383512
#Report
#The standard deviation is the measure of the spread of risk (Natalia, et al., 2014). The standard
deviation is also equivalent to the returns.

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> #Question 1 b:
> #Estimating alpha, Beta, t-statistics and R^2 for gold and software
> #i. Estimating alpha, Beta,t-statistics and R^2 for Software
> software_portfolio_returns<-Data_set$Software_PortforlioReturn
> Software_excessReturn<-Data_set$Software.Excess_Return
> CAMPEstimates1<-lm(software_portfolio_returns~Software_excessReturn)
> summary(CAMPEstimates1)
Call:
lm(formula = software_portfolio_returns ~ Software_excessReturn)
Residuals:
Min 1Q Median 3Q Max
-2.933e-15 -2.701e-17 9.900e-18 4.429e-17 9.691e-16
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.000e-02 1.354e-17 1.477e+15 <2e-16 ***
Software_excessReturn 1.000e+00 9.806e-18 1.020e+17 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 2.149e-16 on 250 degrees of freedom
Multiple R-squared: 1, Adjusted R-squared: 1
F-statistic: 1.04e+34 on 1 and 250 DF, p-value: < 2.2e-16
> #ii. Estimating alpha, Beta, t-statistics and R^2 in gold
> gold_portfolioRetuns<-Data_set$Gold_PortflioRetuen
> gold_excessRetuns<-Data_set$Gold_ExcessReturn
> CAMPEstimates2<-lm(gold_portfolioRetuns~gold_excessRetuns)
> summary(CAMPEstimates2)
Call:
lm(formula = gold_portfolioRetuns ~ gold_excessRetuns)
Residuals:
Min 1Q Median 3Q Max
-9.584e-15 9.000e-19 3.760e-17 7.620e-17 1.052e-15
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.000e-02 3.919e-17 5.103e+14 <2e-16 ***
gold_excessRetuns 1.000e+00 2.437e-17 4.103e+16 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

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Residual standard error: 6.215e-16 on 250 degrees of freedom
Multiple R-squared: 1, Adjusted R-squared: 1
F-statistic: 1.684e+33 on 1 and 250 DF, p-value: < 2.2e-16
#Report
#The value of alpha is 2.0, the value of Beta is 3.919, the t- value is 5.103 and the R squared
value is 1. The value of alpha represents the return on software industry, the value of Beta
represents the market return. The value of t-statistics is more than the significant level hence
there is a significant difference in the returns of the individual daily returns. The value of R
squared is 1 indicating that the sample explain 100% of the population (Masayuki, 2010).
#Question 1 c : Reason for Hedging portfolio
The reason for hedging portfolio is to spread the risks associated with investments (Itkin &
Andrey, 2013). The spreading of risk through hedging is achieved by portfolio diversification
(De, et al., 2010)
> #Question 2
> #2a. Estimating CAMP. The codes below will give the parameters of CAPM. Therefore, to get
the actual CAPM, the estimated parameters are used to write the model.
> #i. CAPM for softawre
> software_portfolio_returns<-Data_set$Software_PortforlioReturn
> Software_Return<-Data_set$MKT_Ret
> CAMPEstimates3<-lm(software_portfolio_returns~Software_Return)
> summary(CAMPEstimates3)
Call:
lm(formula = software_portfolio_returns ~ Software_Return)
Residuals:
Min 1Q Median 3Q Max
-2.3862 -0.3413 0.0023 0.3325 2.1322
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.01580 0.03870 0.408 0.683
Software_Return 1.30021 0.04063 32.002 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.6141 on 250 degrees of freedom
Multiple R-squared: 0.8038, Adjusted R-squared: 0.803
F-statistic: 1024 on 1 and 250 DF, p-value: < 2.2e-16
> #ii. CAMP for gold

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> gold_portfolioRetuns<-Data_set$Gold_PortflioRetuen
> gold_Return<-Data_set$MKT_Ret
> CAPMEstimates4<-lm(gold_portfolioRetuns~gold_Return)
> summary(CAPMEstimates4)
Call:
lm(formula = gold_portfolioRetuns ~ gold_Return)
Residuals:
Min 1Q Median 3Q Max
-7.1970 -0.9230 -0.0616 0.7486 7.6693
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.06359 0.09922 -0.641 0.522181
gold_Return 0.36667 0.10417 3.520 0.000513 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.574 on 250 degrees of freedom
Multiple R-squared: 0.04722, Adjusted R-squared: 0.04341
F-statistic: 12.39 on 1 and 250 DF, p-value: 0.0005125
>
#Report
#The value of alpha is 0.06559 indicating that the return on gold industry is 6.6% while the
return on the software industry is 1.6%. The market return in relation to the software industry is
3.9% while the market return in relation to the gold industry is 10%. Both industries demonstrate
that there is a significant difference in the daily returns. The R squared value in relation to the
software industry is 80.38% while in R squared value in relation to the gold industry is 47.22%.
#Question 2b
#Based on the above results, the best hedge is the gold industry. Gold industry is the best hedge
since it has higher returns compared to the software industry.
> #Question 3
> #3a i. Plotting S$P 500 index level as a function of date
> S_and_P500_index.level<-Data_set$SP500_Idx #Extracting the values of S$P500 index level
> Date_Values<-Data_set$Date #Extracting the data values
> plot(S_and_P500_index.level,Date_Values)

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> #3a ii. Plot of Nasdaq 100 index level as a function of the date
> Nasdaq100_index_leve<-Data_set$Nasdaq100_Idx #Extrating the values of Nasdaq 100 index
level
> plot(Nasdaq100_index_leve,Date_Values)