Question 1-5

# Load libraries
library(quantmod)
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library(tidyquant)
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## Loading required package: lubridate
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library(lubridate)
library(timetk)
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library(purrr)
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library(tibble)
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library(PerformanceAnalytics)
library(magrittr)
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library(readr)
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library(xts)
library(dplyr)
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## # work, which breaks lag(my_xts). Calls to lag(my_xts) that you type or       #
## # source() into this session won't work correctly.                            #
## #                                                                             #
## # Use stats::lag() to make sure you're not using dplyr::lag(), or you can add #
## # conflictRules('dplyr', exclude = 'lag') to your .Rprofile to stop           #
## # dplyr from breaking base R's lag() function.                                #
## #                                                                             #
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library(ggplot2)
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library(tidyverse)
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## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
# Define the ticker names and the time frame
tickers <- c("SPY", "QQQ", "EEM", "IWM", "EFA", "TLT", "IYR", "GLD")
data = new.env()
getSymbols(tickers, src = 'yahoo', from = '2010-01-01', to = Sys.Date(), auto.assign = TRUE)
## [1] "SPY" "QQQ" "EEM" "IWM" "EFA" "TLT" "IYR" "GLD"
ETFList <- merge(Ad(SPY), Ad(QQQ),Ad(EEM), Ad(IWM),Ad(EFA),Ad(TLT),Ad(IYR),Ad(GLD))
colnames(ETFList) <- c("SPY", "QQQ", "EEM", "IWM", "EFA", "TLT", "IYR", "GLD")
head(ETFList)
##                 SPY      QQQ      EEM      IWM      EFA      TLT      IYR
## 2010-01-04 86.86006 40.73327 31.82711 52.51540 37.52378 60.71097 28.10298
## 2010-01-05 87.09000 40.73327 32.05812 52.33483 37.55685 61.10308 28.17047
## 2010-01-06 87.15129 40.48757 32.12519 52.28559 37.71561 60.28508 28.15819
## 2010-01-07 87.51921 40.51388 31.93890 52.67136 37.57008 60.38649 28.40971
## 2010-01-08 87.81044 40.84736 32.19226 52.95863 37.86774 60.35949 28.21954
## 2010-01-11 87.93310 40.68062 32.12519 52.74523 38.17862 60.02823 28.35450
##               GLD
## 2010-01-04 109.80
## 2010-01-05 109.70
## 2010-01-06 111.51
## 2010-01-07 110.82
## 2010-01-08 111.37
## 2010-01-11 112.85
tail(ETFList)
##               SPY    QQQ   EEM    IWM   EFA    TLT   IYR    GLD
## 2024-05-31 527.37 450.71 41.79 205.77 81.18 90.142 86.67 215.30
## 2024-06-03 527.80 453.13 42.23 204.61 81.41 91.600 86.43 217.22
## 2024-06-04 528.39 454.37 41.64 201.97 81.31 92.670 87.20 215.27
## 2024-06-05 534.67 463.53 42.31 205.06 81.88 93.350 86.99 217.82
## 2024-06-06 534.66 463.37 42.52 203.59 82.16 93.210 87.13 219.43
## 2024-06-07 534.01 462.96 42.04 201.20 81.27 91.500 86.43 211.60
ETFList.xts <- xts(ETFList)
head(ETFList.xts)
##                 SPY      QQQ      EEM      IWM      EFA      TLT      IYR
## 2010-01-04 86.86006 40.73327 31.82711 52.51540 37.52378 60.71097 28.10298
## 2010-01-05 87.09000 40.73327 32.05812 52.33483 37.55685 61.10308 28.17047
## 2010-01-06 87.15129 40.48757 32.12519 52.28559 37.71561 60.28508 28.15819
## 2010-01-07 87.51921 40.51388 31.93890 52.67136 37.57008 60.38649 28.40971
## 2010-01-08 87.81044 40.84736 32.19226 52.95863 37.86774 60.35949 28.21954
## 2010-01-11 87.93310 40.68062 32.12519 52.74523 38.17862 60.02823 28.35450
##               GLD
## 2010-01-04 109.80
## 2010-01-05 109.70
## 2010-01-06 111.51
## 2010-01-07 110.82
## 2010-01-08 111.37
## 2010-01-11 112.85
#weekly returns
weekly.returns <- to.weekly(ETFList.xts, indexAt = "last", OHLC = FALSE)
ETF.weekly.returns <- na.omit(Return.calculate(weekly.returns, method = "log"))
head(ETF.weekly.returns)
##                     SPY          QQQ         EEM         IWM          EFA
## 2010-01-15 -0.008150601 -0.015151625 -0.02936203 -0.01310478 -0.003499690
## 2010-01-22 -0.039762693 -0.037556138 -0.05739696 -0.03110020 -0.057354180
## 2010-01-29 -0.016805488 -0.031515088 -0.03415371 -0.02659375 -0.026141646
## 2010-02-05 -0.006821007  0.004430798 -0.02861898 -0.01407309 -0.019238775
## 2010-02-12  0.012855578  0.017985138  0.03278985  0.02909825  0.005231234
## 2010-02-19  0.028288930  0.024157310  0.02415974  0.03288524  0.022734791
##                      TLT          IYR          GLD
## 2010-01-15  1.984813e-02 -0.006324314 -0.004589867
## 2010-01-22  1.005034e-02 -0.042683138 -0.033851808
## 2010-01-29  3.363696e-03 -0.008483257 -0.011354686
## 2010-02-05 -5.404219e-05  0.003218624 -0.012153575
## 2010-02-12 -1.965233e-02 -0.007603475  0.022294524
## 2010-02-19 -8.238809e-03  0.048966649  0.022447945
tail(ETF.weekly.returns)
##                      SPY         QQQ           EEM          IWM          EFA
## 2024-05-03  5.943814e-03  0.01034081  0.0310881818  0.017688930  0.012238795
## 2024-05-10  1.850598e-02  0.01499672  0.0007061029  0.011865941  0.017832925
## 2024-05-17  1.639581e-02  0.02170547  0.0299014258  0.018284194  0.016051747
## 2024-05-24 -1.890615e-05  0.01360895 -0.0177403672 -0.012768598 -0.007253045
## 2024-05-31 -3.917469e-03 -0.01593594 -0.0290080361  0.001605029  0.001602629
## 2024-06-07  1.251220e-02  0.02681654  0.0059644694 -0.022459638  0.001107988
##                      TLT          IYR          GLD
## 2024-05-03  0.0214673251  0.014550063 -0.017040258
## 2024-05-10  0.0031118742  0.020856044  0.026642296
## 2024-05-17  0.0139938967  0.024058882  0.022380376
## 2024-05-24 -0.0001094056 -0.037483488 -0.035219098
## 2024-05-31 -0.0102294348  0.018398305 -0.002875542
## 2024-06-07  0.0149527865 -0.002772941 -0.017334690
#monthly returns 
monthly.returns <- to.monthly(ETFList.xts, indexAt = "last", OHLC = FALSE)
ETF.monthly.returns <- na.omit(Return.calculate(monthly.returns, method = "log"))
head(ETF.monthly.returns)
##                    SPY         QQQ          EEM         IWM          EFA
## 2010-02-26  0.03071797  0.04501048  0.017607814  0.04377884  0.002664225
## 2010-03-31  0.05909833  0.07428091  0.077987050  0.07909456  0.061898157
## 2010-04-30  0.01535164  0.02217755 -0.001662958  0.05523135 -0.028446480
## 2010-05-28 -0.08278894 -0.07679883 -0.098645386 -0.07835796 -0.118702247
## 2010-06-30 -0.05312751 -0.06161676 -0.014085325 -0.08059623 -0.020834952
## 2010-07-30  0.06606928  0.07006953  0.103751723  0.06514069  0.109843801
##                     TLT         IYR          GLD
## 2010-02-26 -0.003430556  0.05313332  0.032223422
## 2010-03-31 -0.020787431  0.09302110 -0.004396045
## 2010-04-30  0.032678413  0.06192367  0.057168645
## 2010-05-28  0.049822159 -0.05851417  0.030056879
## 2010-06-30  0.056359320 -0.04782677  0.023280093
## 2010-07-30 -0.009508752  0.08988441 -0.052210723
tail(ETF.monthly.returns)
##                    SPY         QQQ          EEM         IWM          EFA
## 2024-01-31  0.01580099  0.01802859 -0.046318705 -0.03979283 -0.004522438
## 2024-02-29  0.05087068  0.05148515  0.040832472  0.05472731  0.029425449
## 2024-03-28  0.03217864  0.01266909  0.026891965  0.03427554  0.033228193
## 2024-04-30 -0.04115501 -0.04472312 -0.002193254 -0.07093080 -0.032969371
## 2024-05-31  0.04934208  0.05970011  0.019328923  0.04915477  0.049363177
## 2024-06-07  0.01251220  0.02681654  0.005964469 -0.02245964  0.001107988
##                     TLT          IYR          GLD
## 2024-01-31 -0.022707353 -0.052324523 -0.014330371
## 2024-02-29 -0.022779484  0.021101450  0.004553167
## 2024-03-28  0.007798144  0.018368747  0.083130157
## 2024-04-30 -0.066732274 -0.084688296  0.029456829
## 2024-05-31  0.028460902  0.048098120  0.016059563
## 2024-06-07  0.014952787 -0.002772941 -0.017334690
#tibble format
ETF.monthly.returns.tibble <- as_tibble(ETF.monthly.returns)
ETF.monthly.returns.tibble
## # A tibble: 173 × 8
##        SPY      QQQ      EEM     IWM      EFA      TLT     IYR      GLD
##      <dbl>    <dbl>    <dbl>   <dbl>    <dbl>    <dbl>   <dbl>    <dbl>
##  1  0.0307  0.0450   0.0176   0.0438  0.00266 -0.00343  0.0531  0.0322 
##  2  0.0591  0.0743   0.0780   0.0791  0.0619  -0.0208   0.0930 -0.00440
##  3  0.0154  0.0222  -0.00166  0.0552 -0.0284   0.0327   0.0619  0.0572 
##  4 -0.0828 -0.0768  -0.0986  -0.0784 -0.119    0.0498  -0.0585  0.0301 
##  5 -0.0531 -0.0616  -0.0141  -0.0806 -0.0208   0.0564  -0.0478  0.0233 
##  6  0.0661  0.0701   0.104    0.0651  0.110   -0.00951  0.0899 -0.0522 
##  7 -0.0460 -0.0527  -0.0329  -0.0774 -0.0387   0.0806  -0.0131  0.0555 
##  8  0.0858  0.124    0.111    0.117   0.0951  -0.0255   0.0453  0.0467 
##  9  0.0375  0.0615   0.0297   0.0406  0.0373  -0.0457   0.0386  0.0362 
## 10  0      -0.00173 -0.0295   0.0343 -0.0494  -0.0170  -0.0160  0.0209 
## # ℹ 163 more rows
setwd("D:/School Folder/HW/Investment")
ff3 <- read.csv("F-F_Research_Data_Factors.csv",  header = TRUE)
str(ff3)
## 'data.frame':    1174 obs. of  5 variables:
##  $ X     : int  192607 192608 192609 192610 192611 192612 192701 192702 192703 192704 ...
##  $ Mkt.RF: num  2.96 2.64 0.36 -3.24 2.53 2.62 -0.06 4.18 0.13 0.46 ...
##  $ SMB   : num  -2.56 -1.17 -1.4 -0.09 -0.1 -0.03 -0.37 0.04 -1.65 0.3 ...
##  $ HML   : num  -2.43 3.82 0.13 0.7 -0.51 -0.05 4.54 2.94 -2.61 0.81 ...
##  $ RF    : num  0.22 0.25 0.23 0.32 0.31 0.28 0.25 0.26 0.3 0.25 ...
head(ff3)
##        X Mkt.RF   SMB   HML   RF
## 1 192607   2.96 -2.56 -2.43 0.22
## 2 192608   2.64 -1.17  3.82 0.25
## 3 192609   0.36 -1.40  0.13 0.23
## 4 192610  -3.24 -0.09  0.70 0.32
## 5 192611   2.53 -0.10 -0.51 0.31
## 6 192612   2.62 -0.03 -0.05 0.28
tail(ff3)
##           X Mkt.RF   SMB   HML   RF
## 1169 202311   8.84 -0.02  1.64 0.44
## 1170 202312   4.87  6.34  4.93 0.43
## 1171 202401   0.70 -5.09 -2.38 0.47
## 1172 202402   5.06 -0.24 -3.49 0.42
## 1173 202403   2.83 -2.49  4.19 0.43
## 1174 202404  -4.67 -2.39 -0.51 0.47
glimpse(ff3)
## Rows: 1,174
## Columns: 5
## $ X      <int> 192607, 192608, 192609, 192610, 192611, 192612, 192701, 192702,…
## $ Mkt.RF <dbl> 2.96, 2.64, 0.36, -3.24, 2.53, 2.62, -0.06, 4.18, 0.13, 0.46, 5…
## $ SMB    <dbl> -2.56, -1.17, -1.40, -0.09, -0.10, -0.03, -0.37, 0.04, -1.65, 0…
## $ HML    <dbl> -2.43, 3.82, 0.13, 0.70, -0.51, -0.05, 4.54, 2.94, -2.61, 0.81,…
## $ RF     <dbl> 0.22, 0.25, 0.23, 0.32, 0.31, 0.28, 0.25, 0.26, 0.30, 0.25, 0.3…
colnames(ff3) <- paste(c("date","Mkt-RF","SMB","HML","RF"))
ff3.digit <- ff3 %>% mutate(date = as.character(date))%>% 
  mutate(date=ymd(parse_date(date,format="%Y%m"))) %>%
  mutate(date = rollback(date))
head(ff3.digit)
##         date Mkt-RF   SMB   HML   RF
## 1 1926-06-30   2.96 -2.56 -2.43 0.22
## 2 1926-07-31   2.64 -1.17  3.82 0.25
## 3 1926-08-31   0.36 -1.40  0.13 0.23
## 4 1926-09-30  -3.24 -0.09  0.70 0.32
## 5 1926-10-31   2.53 -0.10 -0.51 0.31
## 6 1926-11-30   2.62 -0.03 -0.05 0.28
ff3.digit.xts <- xts(ff3.digit[,-1],order.by=as.Date(ff3.digit$date))
head(ff3.digit.xts)
##            Mkt-RF   SMB   HML   RF
## 1926-06-30   2.96 -2.56 -2.43 0.22
## 1926-07-31   2.64 -1.17  3.82 0.25
## 1926-08-31   0.36 -1.40  0.13 0.23
## 1926-09-30  -3.24 -0.09  0.70 0.32
## 1926-10-31   2.53 -0.10 -0.51 0.31
## 1926-11-30   2.62 -0.03 -0.05 0.28
#merge of monthly returns and fama french
final.data <- merge(ff3.digit,ETF.monthly.returns)
tail(final.data)
##              date Mkt-RF   SMB   HML   RF       SPY        QQQ         EEM
## 203097 2023-10-31   8.84 -0.02  1.64 0.44 0.0125122 0.02681654 0.005964469
## 203098 2023-11-30   4.87  6.34  4.93 0.43 0.0125122 0.02681654 0.005964469
## 203099 2023-12-31   0.70 -5.09 -2.38 0.47 0.0125122 0.02681654 0.005964469
## 203100 2024-01-31   5.06 -0.24 -3.49 0.42 0.0125122 0.02681654 0.005964469
## 203101 2024-02-29   2.83 -2.49  4.19 0.43 0.0125122 0.02681654 0.005964469
## 203102 2024-03-31  -4.67 -2.39 -0.51 0.47 0.0125122 0.02681654 0.005964469
##                IWM         EFA        TLT          IYR         GLD
## 203097 -0.02245964 0.001107988 0.01495279 -0.002772941 -0.01733469
## 203098 -0.02245964 0.001107988 0.01495279 -0.002772941 -0.01733469
## 203099 -0.02245964 0.001107988 0.01495279 -0.002772941 -0.01733469
## 203100 -0.02245964 0.001107988 0.01495279 -0.002772941 -0.01733469
## 203101 -0.02245964 0.001107988 0.01495279 -0.002772941 -0.01733469
## 203102 -0.02245964 0.001107988 0.01495279 -0.002772941 -0.01733469
final.data.tibble <- as_tibble(final.data)
head(final.data.tibble)
## # A tibble: 6 × 13
##   date       `Mkt-RF`   SMB   HML    RF    SPY    QQQ    EEM    IWM     EFA
##   <date>        <dbl> <dbl> <dbl> <dbl>  <dbl>  <dbl>  <dbl>  <dbl>   <dbl>
## 1 1926-06-30     2.96 -2.56 -2.43  0.22 0.0307 0.0450 0.0176 0.0438 0.00266
## 2 1926-07-31     2.64 -1.17  3.82  0.25 0.0307 0.0450 0.0176 0.0438 0.00266
## 3 1926-08-31     0.36 -1.4   0.13  0.23 0.0307 0.0450 0.0176 0.0438 0.00266
## 4 1926-09-30    -3.24 -0.09  0.7   0.32 0.0307 0.0450 0.0176 0.0438 0.00266
## 5 1926-10-31     2.53 -0.1  -0.51  0.31 0.0307 0.0450 0.0176 0.0438 0.00266
## 6 1926-11-30     2.62 -0.03 -0.05  0.28 0.0307 0.0450 0.0176 0.0438 0.00266
## # ℹ 3 more variables: TLT <dbl>, IYR <dbl>, GLD <dbl>

Question 6 & 7

## # A tibble: 6 × 13
##   date       `Mkt-RF`   SMB   HML    RF    SPY    QQQ    EEM    IWM     EFA
##   <date>        <dbl> <dbl> <dbl> <dbl>  <dbl>  <dbl>  <dbl>  <dbl>   <dbl>
## 1 2010-02-28     6.31  1.48  2.21  0.01 0.0307 0.0450 0.0176 0.0438 0.00266
## 2 2010-03-31     2     4.87  2.89  0.01 0.0307 0.0450 0.0176 0.0438 0.00266
## 3 2010-04-30    -7.89  0.09 -2.44  0.01 0.0307 0.0450 0.0176 0.0438 0.00266
## 4 2010-05-31    -5.57 -1.82 -4.7   0.01 0.0307 0.0450 0.0176 0.0438 0.00266
## 5 2010-06-30     6.93  0.2  -0.31  0.01 0.0307 0.0450 0.0176 0.0438 0.00266
## 6 2010-07-31    -4.77 -3    -1.9   0.01 0.0307 0.0450 0.0176 0.0438 0.00266
## # ℹ 3 more variables: TLT <dbl>, IYR <dbl>, GLD <dbl>
## # A tibble: 6 × 13
##   date       `Mkt-RF`   SMB   HML    RF    SPY    QQQ     EEM     IWM     EFA
##   <date>        <dbl> <dbl> <dbl> <dbl>  <dbl>  <dbl>   <dbl>   <dbl>   <dbl>
## 1 2014-08-31    -1.97 -3.71 -1.34     0 0.0125 0.0268 0.00596 -0.0225 0.00111
## 2 2014-09-30     2.52  4.21 -1.81     0 0.0125 0.0268 0.00596 -0.0225 0.00111
## 3 2014-10-31     2.55 -2.06 -3.09     0 0.0125 0.0268 0.00596 -0.0225 0.00111
## 4 2014-11-30    -0.06  2.49  2.27     0 0.0125 0.0268 0.00596 -0.0225 0.00111
## 5 2014-12-31    -3.11 -0.56 -3.58     0 0.0125 0.0268 0.00596 -0.0225 0.00111
## 6 2015-01-31     6.13  0.63 -1.86     0 0.0125 0.0268 0.00596 -0.0225 0.00111
## # ℹ 3 more variables: TLT <dbl>, IYR <dbl>, GLD <dbl>
##               spy_rf       qqq_rf       eem_rf       iwm_rf       efa_rf
## one.vec -0.259568091 -0.255856713 -0.268135401 -0.262252287 -0.265459883
##          0.003088849  0.003088849  0.003088849  0.003088849  0.003088849
##               tlt_rf       iyr_rf       gld_rf
## one.vec -0.268164590 -0.263574692 -0.266379172
##          0.003088849  0.003088849  0.003088849
##            [,1]       [,2]       [,3]       [,4]       [,5]       [,6]
## [1,] 0.06420763 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
## [2,] 0.00000000 0.06493947 0.00000000 0.00000000 0.00000000 0.00000000
## [3,] 0.00000000 0.00000000 0.06524272 0.00000000 0.00000000 0.00000000
## [4,] 0.00000000 0.00000000 0.00000000 0.06567052 0.00000000 0.00000000
## [5,] 0.00000000 0.00000000 0.00000000 0.00000000 0.06454722 0.00000000
## [6,] 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.06400105
## [7,] 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
## [8,] 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
##            [,7]       [,8]
## [1,] 0.00000000 0.00000000
## [2,] 0.00000000 0.00000000
## [3,] 0.00000000 0.00000000
## [4,] 0.00000000 0.00000000
## [5,] 0.00000000 0.00000000
## [6,] 0.00000000 0.00000000
## [7,] 0.06486568 0.00000000
## [8,] 0.00000000 0.06445405
##          spy_rf   qqq_rf   eem_rf   iwm_rf   efa_rf   tlt_rf   iyr_rf   gld_rf
## spy_rf 1.983866 1.892215 1.983010 1.939507 1.963226 1.983226 1.949286 1.970023
## qqq_rf 1.892215 1.930103 1.954660 1.911779 1.935159 1.954873 1.921418 1.941859
## eem_rf 1.983010 1.954660 2.113695 2.003513 2.028015 2.048675 2.013615 2.035037
## iwm_rf 1.939507 1.911779 2.003513 2.025231 1.983525 2.003731 1.969440 1.990393
## efa_rf 1.963226 1.935159 2.028015 1.983525 2.072329 2.028236 1.993525 2.014734
## tlt_rf 1.983226 1.954873 2.048675 2.003731 2.028236 2.112899 2.013834 2.035259
## iyr_rf 1.949286 1.921418 2.013615 1.969440 1.993525 2.013834 2.044236 2.000428
## gld_rf 1.970023 1.941859 2.035037 1.990393 2.014734 2.035259 2.000428 2.086164
##               [,1]
## spy_rf  0.58088051
## qqq_rf  0.98030183
## eem_rf -0.36111138
## iwm_rf  0.27759947
## efa_rf -0.07056341
## tlt_rf -0.37135684
## iyr_rf  0.13622854
## gld_rf -0.17197872
## # A tibble: 6 × 8
##      SPY    QQQ    EEM    IWM     EFA      TLT    IYR    GLD
##    <dbl>  <dbl>  <dbl>  <dbl>   <dbl>    <dbl>  <dbl>  <dbl>
## 1 0.0307 0.0450 0.0176 0.0438 0.00266 -0.00343 0.0531 0.0322
## 2 0.0307 0.0450 0.0176 0.0438 0.00266 -0.00343 0.0531 0.0322
## 3 0.0307 0.0450 0.0176 0.0438 0.00266 -0.00343 0.0531 0.0322
## 4 0.0307 0.0450 0.0176 0.0438 0.00266 -0.00343 0.0531 0.0322
## 5 0.0307 0.0450 0.0176 0.0438 0.00266 -0.00343 0.0531 0.0322
## 6 0.0307 0.0450 0.0176 0.0438 0.00266 -0.00343 0.0531 0.0322
##         SPY         QQQ         EEM         IWM         EFA         TLT 
## 0.001782446 0.002514705 0.002818131 0.003246166 0.002122232 0.001575751 
##         IYR         GLD 
## 0.002440871 0.002029005
##               SPY           QQQ           EEM           IWM           EFA
## SPY  0.0017815872  1.949330e-03  0.0016256263  0.0021343254  0.0017034184
## QQQ  0.0019493298  2.513494e-03  0.0017485820  0.0022184458  0.0018077286
## EEM  0.0016256263  1.748582e-03  0.0028167730  0.0020930648  0.0020600505
## IWM  0.0021343254  2.218446e-03  0.0020930648  0.0032446026  0.0020655945
## EFA  0.0017034184  1.807729e-03  0.0020600505  0.0020655945  0.0021212099
## TLT -0.0001804641 -8.928701e-07 -0.0002112621 -0.0004222711 -0.0002268046
## IYR  0.0016158804  1.655386e-03  0.0016142757  0.0020528427  0.0016273203
## GLD  0.0001932109  2.768823e-04  0.0007213619  0.0001103875  0.0003290724
##               TLT          IYR          GLD
## SPY -1.804641e-04 0.0016158804 0.0001932109
## QQQ -8.928701e-07 0.0016553865 0.0002768823
## EEM -2.112621e-04 0.0016142757 0.0007213619
## IWM -4.222711e-04 0.0020528427 0.0001103875
## EFA -2.268046e-04 0.0016273203 0.0003290724
## TLT  1.574992e-03 0.0003986696 0.0004710134
## IYR  3.986696e-04 0.0024396953 0.0004255513
## GLD  4.710134e-04 0.0004255513 0.0020280272
##               SPY           QQQ           EEM           IWM           EFA
## SPY  1.000000e+00  1.444855e-04  1.757362e-05  7.791546e-05  4.654652e-05
## QQQ  1.444855e-04  1.000000e+00  2.137781e-05  9.159013e-05  5.586450e-05
## EEM  1.757362e-05  2.137781e-05  1.000000e+00  1.260329e-05  9.285003e-06
## IWM  7.791546e-05  9.159013e-05  1.260329e-05  1.000000e+00  3.143922e-05
## EFA  4.654652e-05  5.586450e-05  9.285003e-06  3.143922e-05  1.000000e+00
## TLT -2.574992e-06 -1.440822e-08 -4.972160e-07 -3.356119e-06 -1.349276e-06
## IYR  5.695586e-05  6.598808e-05  9.385237e-06  4.030378e-05  2.391476e-05
## GLD  4.390047e-06  7.114914e-06  2.703517e-06  1.397071e-06  3.117402e-06
##               TLT          IYR          GLD
## SPY -2.574992e-06 5.695586e-05 4.390047e-06
## QQQ -1.440822e-08 6.598808e-05 7.114914e-06
## EEM -4.972160e-07 9.385237e-06 2.703517e-06
## IWM -3.356119e-06 4.030378e-05 1.397071e-06
## EFA -1.349276e-06 2.391476e-05 3.117402e-06
## TLT  1.000000e+00 3.059323e-06 2.329989e-06
## IYR  3.059323e-06 1.000000e+00 5.200160e-06
## GLD  2.329989e-06 5.200160e-06 1.000000e+00
##           [,1]
## SPY 0.15452805
## QQQ 0.10950721
## EEM 0.09776889
## IWM 0.08485290
## EFA 0.12981531
## TLT 0.17486769
## IYR 0.11286057
## GLD 0.13579938

Question 8

library(quadprog)
## Warning: package 'quadprog' was built under R version 4.3.1
# Function to compute GMV portfolio weights
gmv_weights <- function(cov_matrix) {
  n <- ncol(cov_matrix)
  dvec <- rep(0, n)
  Amat <- cbind(rep(1, n), diag(1, n))
  bvec <- c(1, rep(0, n))
  solve.QP(Dmat = cov_matrix, dvec = dvec, Amat = Amat, bvec = bvec, meq = 1)$solution
}

# Compute GMV weights based on CAPM covariance matrix
gmv_weights_capm <- gmv_weights(cov.mat)
gmv_weights_capm
## [1]  2.924817e-01  7.075183e-01  7.314936e-17  0.000000e+00  5.570595e-18
## [6]  4.011246e-17  1.419811e-17 -4.627908e-17
# Compute GMV weights based on FF 3-factor model covariance matrix
gmv_weights_ff <- gmv_weights(cov.matff)
gmv_weights_ff
## [1]  3.479815e-01 -7.695800e-18 -3.916171e-18  0.000000e+00  4.859879e-02
## [6]  4.120942e-01 -5.665377e-18  1.913254e-01
# Prepare data for SIT
returns <- final.data.tibble %>% select(date, starts_with("SPY"), starts_with("QQQ"), starts_with("EEM"), starts_with("IWM"), starts_with("EFA"), starts_with("TLT"), starts_with("IYR"), starts_with("GLD"))
returns_xts <- xts(returns[,-1], order.by = returns$date)

# GMV portfolio weights function
gmv_weights <- function(cov_matrix) {
  n <- ncol(cov_matrix)
  dvec <- rep(0, n)
  Amat <- cbind(rep(1, n), diag(1, n))
  bvec <- c(1, rep(0, n))
  solve.QP(Dmat = cov_matrix, dvec = dvec, Amat = Amat, bvec = bvec, meq = 1)$solution
}

Question 9

# Calculate the average monthly returns for the past year (2010 - May 2024)
final.data4 <- final.data.tibble[final.data.tibble$date>="2010-01-01"&final.data.tibble$date<="2024-05-31",]
average_returns <- colMeans(final.data4[final.data4$date >= "2010-01-01" & final.data4$date <= "2024-05-31", (ncol(final.data4)-7):ncol(final.data4)])

# Estimate the portfolio return for May 2024 using the weights from question 5 (CAPM Model)
estimated_portfolio_return_may_2024_CAPM <- sum(mvp_capm * average_returns)

# Estimate the portfolio return for May 2024 using the weights from question 6 (FF 3 Factor Model)
estimated_portfolio_return_may_2024_FF3 <- sum(MVP_FF3 * average_returns)

# Print or store the realized portfolio returns
print(paste("Realized portfolio return in May 2024 (CAPM model):", estimated_portfolio_return_may_2024_CAPM))
## [1] "Realized portfolio return in May 2024 (CAPM model): 0.0210295299964941"