Russian mutual funds: skill versus luck

-0,051639

-0,069948

-33,48%

-68,69%

-0,000037

rajffajzen elektroenergetika

-0,000669

-0,000888

-0,046540

-0,062344

-43,35%

-72,53%

-0,000034

rgs elektroenergetika

-0,000552

-0,000725

-0,043421

-0,056826

-36,43%

-65,24%

0,000053

gorizont aktsii plyus khedzh

-0,000337

-0,000503

-0,060767

-0,082496

-4,68%

NA

0,000156

otkrytie agressivnyj

-0,000500

-0,000621

-0,082728

-0,112121

-15,89%

NA

0,000170

Some funds did not exist 3 years ago and thus have NA in column “3-year”. Full-length returns for these funds are: “sberbank globalnyj internet” - 60,2% 2,5 years, “gorizont aktsii plyus khedzh” - lost 20,4% during 2,5 years, “otkrytie agressivnyj” - lost 21% during 2 years. The last two funds have random returns at 5% significance level for three base models. Also, “gorizont aktsii plyus khedzh” performance is better than the other ones among negative skill funds and it is nominated only by 2209 observations model. So, “gorizont aktsii plyus khedzh” can be ignored. “otkrytie agressivnyj” is still assumed to have negative skills. Among funds with positive skills there is one with negative 3-year returns and low 1-year. The fund is “univer fond aktsij”. But it is among the skilled because it earns 272% during 8,5 years and has high mean returns. This is the reason two models treated the fund as it has positive nonrandom returns. The model with 2209 observations of pseudo residuals claims this fund has random returns. So, the results differs for models, also, recent fund`s performance is below average. Thus “univer fond aktsij” should be ignored.

The two skilled funds that are left are almost even. The bootstrap approach withstood the test - Sharpe and Sortino ratios are the top one (only one fund has higher Sharpe ratio - “tkb bnp pariba perspektivnye investitsii”, which was excluded because of low r-squared), same goes for mean values and real alphas, as for 1- and 3-year returns - the results differ: negative skill are among the worst, positive only one is among the best. The other two have great overall returns. So, the bootstrap approach can provide strong results, which with careful treatment can be trusted.

The final decision is to invest in two funds “sberbank globalnyj internet” and “interfin telekom”. The decision is made at the begging of 2014 year. The investment is oriented on 1-3 years. The results from such investment are measured ex-post. In the table below there are the prices of funds` shares and changes in Russian market - MSCI Russia (with local currency).



Table 17

Ex-post results for positive skill funds.

Instrument	09.01.2014	31.12.2014	Change,%	04.05.2015	Change,%
sberbank globalnyj internet	1593,78 rub.	1752,28 rub.	9,94%	2025,21 rub.	27,07%
interfin telekom	5948,81 rub.	5949,53 rub.	0,01%	6988,16 rub.	17,47%
MSCI Russia	768,79	662,24	-13,86%	817,55	6,34%

The year 2014 was harsh for Russian economy. USA and EU imposed sanctions because of Ukrainian matter. The benchmark dropped by more than 13,8% in national currency, in US dollars the fall was even deeper. But the picked funds has shown positive dynamics and outperform benchmark. If an investor has made a portfolio of these two funds, than in less than 1,5 year he earns 22,27% and in annual form it is 16,28%. This result is higher than mostly any deposit in Russia. There are funds that have higher returns during January 2014 - April 2015, but at least peaked funds are among 1^st quarter in terms of returns.

The skilled funds from bootstrap returns were tested ex-post, their results were not the best but at least acceptable. The funds with negative skill should be also tested. The funds with negative picking skills after filtering bootstrap results are: “alfakapital elektroenergetika”, “pallada energetika”, “interfin elektroenergetika”, “rajffajzen elektroenergetika”, “rgs elektroenergetika”. Their returns during January 2014 - April 2015 are presented in the table below.

Table 18

Ex-post results for negative skill funds

Instrument	09.01.2014	31.12.2014	Change,%	04.05.2015	Change,%	Liquidation
MSCI Russia	768,79	662,24	-13,86%	817,55	6,34%	-
alfakapital elektroenergetika	284,3 rub	313,88 rub	10,40%	340,39 rub	19,73%	-
pallada energetika	1 332,69 rub	1 299,53 rub	-2,49%	-	-	11.11.2014
interfin elektroenergetika	331,78 rub	262,15 rub	-20,99%	351,09 rub	5,82%	-
rajffajzen elektroenergetika	2 680,47 rub	2 316,28 rub	-13,59%	2 807,36 rub	4,73%	-
rgs elektroenergetika	324,93 rub	238,18 rub	-26,70%	305,85 rub	-5,87%	-
otkrytie agressivnyj	789,96 rub	737,49 rub	-6,64%	-	-	16.10.2014

According to the table 2 out of 6 funds have been liquidated during the period. Three more funds have negative one-year performance and underperform the market in 1 year and 4 months period. But one of the funds assumed to have negative skills outperform the benchmark and even one of the positive skill funds. “Alfakapital elektroenergetika” provides 19,73% for the period January 2014 - April 2015, which is 14,46% per annum. The reason can be this: the fund was at its minimum value (even below the bottom of 2008 crisis, when minimum value was 329,6 rubles) on the first days of testing period and this growth is a correction. But it does not change the fact that this investment provide good returns. Thus bootstrap approach can be used alongside other instruments in decision making process. With careful adjustment this approach can provide valuable information and show acceptable results.



Conclusion

Investors all over the world are looking for promising assets. Some of them want to earn high returns, others - to save and diversify their wealth. Mutual funds are one of the instruments of financial market that are managed by professionals. It is a good choice for investments for both: people that are not professional participants of financial market and are willing to give their money in hands of professionals to be managed skillfully, and also as promising assets for market participants and even companies and other funds. But as well as other type of assets there are funds that can show good performance and also funds that provide losses. There is asymmetry of information on the market of mutual funds: investors do not have full information of managers` skills, but managers are aware of their possibilities. In order to get more information about funds` abilities investors, analysts and economists are looking for methods to measure the performance of a fund and its managers` skills. Bootstrap approach and similar to it methods are built up to deal with this information asymmetry.

But before bootstrap the coefficient that is supposed to measure managers` picking skills is Jensen alpha. It represents the value of how much more or less returns the manager systematically provides in terms of a certain model. But what if the period that was used in the model was just a lucky time for that manager? What if the returns are due to luck and not skill? The alpha describes both luck and skill. And the investors probably do not want to invest in a lucky fund without goo skill as the fortune will not always be on the funds side. Thus there is a need to distinguish skill from luck. This goal can be achieved with approaches that simulate random portfolios with similar characteristics for a fund. These portfolios represents luck and their alpha distribution is nothing else but distribution of luck. One of approaches that can do this is bootstrap approach. It was already used by several economists for different markets all over the world including Russia.

The approach used in this research is similar to those in other papers about bootstrap approach. Aside from unique sample (time period, funds in the sample) and special models (no bootstrap approach in other papers is based on the models in this research) this paper also uncovers settings in bootstrap approach that strongly influence the result. These settings and assumptions are based on economic point of view and not only one set up can be used to investigate the question of skill presence for funds` managers. Moreover, in this paper to get final results is used a combination of set ups for bootstrap approach filtered with other famous statistics that measure fund`s performance. Since the research question is to develop and apply the method that can measure funds managers` skills, the resulted method involves not only bootstrap approach but also other statistics to support bootstrap`s results.

The result of combined method is acceptable, the funds that are assumed with skill have the direction of their returns vector in ex-post period predicted by the approach. According to this research method the market of mutual funds in Russia is inefficient, most funds have no evidence of skill, among the funds with skilled managers negative nonrandom returns prevail. Bootstrap approach provides evidence of negative picking skill for 6 funds and positive for only 2 funds out of 175. For these funds zero hypothesis about random returns is rejected, for all the others - accepted. The distribution of simulated alphas in comparison with real alphas suggests that most equity funds do not have sufficient skill to provide benchmark adjusted returns to cover the costs of management. Thus the zero hypothesis for the market is accepted. It claims random returns for the funds in general, only 4,6% of funds have nonrandom returns.

Further investigation is needed as the research has limitations. The main one is connected with models` specifications. One of possible enhancement of this approach is to create a well-suited model that can satisfy all funds` requirements. Equity funds invest have different investment strategies: aside from Russian blue chips, they invest in industries on Russian market, companies of low capitalization (others in high), commodities (including in foreign currencies), global market and in funds (Russian and foreign). For proper analysis the model`s specification for the whole market (one for all funds) should include many factors like industry indexes. But those alone will be not enough, factors like SMB, HML estimated for Russian market may also be of good use, as well as commodity indexes, currencies, global market indexes and etc. The models used in this research does not reflect each fund`s specification. The best model with Brent oil and USD/RUB can follow foreign assets and “Oil and gas” industry a bit (still very low r-squared and not enough explanatory factors), but it cannot describe some funds at all. Thus the results of this research are sufficient for some funds and inappropriate for others as r-squared is low for many funds. So, the result for whole market may be insignificant. But for single funds (if a model has r-squared sufficient enough) the bootstrap approach with cautious adjustment can provide stable results that are significant. The method provided by this research can still provide useful information about funds. Even with models that have low r-squared one can filter the results and make appropriate conclusions.

This paper and its methods can be useful for investors in their decision making process, as the approach can help to pick a skilled fund. What is more, it can provide useful information for possible stakeholders. This research`s methods can also help managers to test and track their skills or may be even make a decision to change their management policy.



References

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Internet resources:

1. Consultant plus. Informational portal, available at: http://www.consultant.ru

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3. Morgan Stanley Capital International. Indexes, available at: https://www.msci.com/end-of-day-data-search

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6. R-forge. R informational portal, available at: http://r-forge.r-project.org

Appendix

Script in R

#Preparing initial data

bench=bencmarks[2:2341,2]

equity1=mf[,-seq(1,NCOL(mf),3)]

equity2=equity1[,seq(1,NCOL(equity1),2)]

equity=equity2[1:2223,]

BrentRub<-Brent[,2]*UsdRub[,2]

BrentRub<-as.numeric(BrentRub)

Fund_names=colnames(equity)

#From prices to returns

equity_returns = (head(equity, -1) - tail(equity, -1)) / tail(equity, -1)

bench_returns = (head(bench, -1) - tail(bench, -1)) / tail(bench, -1)

Brent_returns = (head(Brent[,2], -1) - tail(Brent[,2], -1)) / tail(Brent[,2], -1)

UsdRub_returns = (head(UsdRub[,2], -1) - tail(UsdRub[,2], -1)) / tail(UsdRub[,2], -1)

Rf_returns<-(as.numeric(Risk_free_rate[,2])/100+1)^(1/247)-1

BrentRub_returns = (head(BrentRub, -1) - tail(BrentRub, -1)) / tail(BrentRub, -1)

summary(bench_returns)

sd(bench_returns)

n_obs = apply(equity_returns,2, function (x) NROW( x[!is.na(x)] ))

#First, adjust date in time series

b<-bencmarks[-c(1,43,146,169,170,174:176,214,257:262,303,407,431,432,475,485,518:523,564,668,693,698,737,746,778:783, 824,825,928,953,958,998,1007,1008,1039:1044,1086, 1189,1213,1219,1258,1268,1299:1305,1347, 1348,1449,1450,1474,1479,1480,1518,1528,1562:1568,1608,1617,1646,1712,1713,1736,1739,1741,1742, 1743,1776,1780,1782,1789,1813,1823:1828,1868,1874,1956,1973,1997,1998,2003,2041,2049,2050, 2083:2088,2129,2194,2233,2235,2257,2258,2264,2274,2296,2301,2310,2342:2623),]

equityfunds<-mf[-c(780,960,1242,1284,1388,1413,1493,1634,1660,1899,1948,2142,2224:2995),]

equityfunds[2211,1]

b[2211,]

Funds_adj<-equityfunds[-c(248),]

Benchmark_adj<-b[-c(248),]

#Now we can estimate returns

Funds1=Funds_adj[,-seq(1,NCOL(Funds_adj),3)]

Funds_adj_col=Funds1[,seq(1,NCOL(Funds1),2)]

Benchmark_adj_col=Benchmark_adj[,2]

##deleting funds operating less than an year

Funds_returns = (head(Funds_adj_col, -1) - tail(Funds_adj_col, -1)) / tail(Funds_adj_col, -1)

Funds_returns_adj=Funds_returns[,-c(76,77,79,84,88,92,103,115,177,178,180,181,182,189,191,192,193,196,200,201,202,204,213,214,215,217,218)]

Funds_returns_adj$oplot_metallurgiya<-NULL

Funds_returns_adj$denezhkin_kamen<-NULL

Funds_returns_adj$oplot___alternativnaya_energetika<-NULL

Funds_returns_adj$shag_pervyj<-NULL

Funds_returns_adj$rgs_perspektivnye_investitsii<-NULL

Funds_returns_adj$alfa_kapital_perspektiva<-NULL

Funds_returns_adj$oplot<-NULL

Funds_returns_adj$kanopus<-NULL

Funds_returns_adj$rskhb_fond_aktsij<-NULL

Funds_returns_adj$vysokie_tekhnologii<-NULL

Funds_returns_adj$almaz<-NULL

Funds_returns_adj$pallada_fond_aktsij_vtorogo_eshelon<-NULL

Funds_returns_adj$arsagera_a_6.4<-NULL

Funds_returns_adj$adekta_fond_aktsij_vtorogo_eshelona<-NULL

Funds_returns_adj$everest_aktsii<-NULL

Funds_returns_adj$energokapital_mirovye_rynki<-NULL

Funds_returns_adj$investitsionnyj_respublikanskij_fon<-NULL

benchmark_returns = (head(Benchmark_adj_col, -1) - tail(Benchmark_adj_col, -1)) / tail(Benchmark_adj_col, -1)

Micex_returns = (head(MICEX[,2], -1) - tail(MICEX[,2], -1)) / tail(MICEX[,2], -1)

####Brent, UsdRub, risk-free rate were already adjusted in excel

#Creating series with statistics

n_observations = apply(Funds_returns_adj,2, function (x) NROW( x[!is.na(x)] ))

Mean_returns<-data.frame(MEAN=numeric(175))

for (i in 1:175){

Mean_returns[i,]<-mean(Funds_returns_adj[,i],na.rm=TRUE)

}

Median_returns<-data.frame(MEDIAN=numeric(175))

for (i in 1:175){

Median_returns[i,]<-median(Funds_returns_adj[,i],na.rm=TRUE)

}

StDev_returns<-data.frame(StDev=numeric(175))

for (i in 1:175){

StDev_returns[i,]<-sd(Funds_returns_adj[,i],na.rm=TRUE)

}

Min_returns<-data.frame(Min=numeric(175))

for (i in 1:175){

Min_returns[i,]<-min(Funds_returns_adj[,i],na.rm=TRUE)

}

Max_returns<-data.frame(Max=numeric(175))

for (i in 1:175){

Max_returns[i,]<-max(Funds_returns_adj[,i],na.rm=TRUE)

}

#Distribution of statistics

plot(density(n_obs), type = "l", col = "black", main = "Distribution of n_obs", xlab = "observations", ylab = "density")

plot(density(n_observations), type = "l", col = "black", main = "Distribution of n_obs", xlab = "observations", ylab = "density")

plot(density(Mean_returns[,1]), type = "l", col = "black", main = "Distribution of mean", xlab = "Mean", ylab = "density")

abline(v =0, col="black")

abline(v=mean(benchmark_returns),col="red")

plot(density(Median_returns[,1]), type = "l", col = "black", main = "Distribution of median", xlab = "Median", ylab = "density")

abline(v =0, col="black")

abline(v=median(benchmark_returns),col="red")

plot(density(Max_returns[,1]), type = "l", col = "black", main = "Distribution of max", xlab = "Max", ylab = "density")

abline(v=max(benchmark_returns),col="red")

plot(density(Min_returns[,1]), type = "l", col = "black", main = "Distribution of min", xlab = "Min", ylab = "density")

abline(v=min(benchmark_returns),col="red")

plot(density(StDev_returns[,1]), type = "l", col = "black", main = "Distribution of standart deviation", xlab = "Standart Deviation", ylab = "density")

abline(v =0, col="black")

abline(v=sd(benchmark_returns),col="red")

#### Discriptive statistics for all funds

FR<-as.numeric(as.matrix(Funds_returns_adj))

summary(FR, na.rm=TRUE)[1:6]

sd(FR, na.rm=TRUE)

plot(density(na.omit(FR)), type = "l", xlim=c(-0.06,0.06), col = "black", main = "Distribution of returns", xlab = "observations", ylab = "density")

abline(v =0, col="red")

Stat_funds <- do.call(data.frame,

list(mean = apply(Funds_returns_adj, 2, mean, na.rm=TRUE),

min = apply(Funds_returns_adj, 2, min, na.rm=TRUE),

median = apply(Funds_returns_adj, 2, median, na.rm=TRUE),

max = apply(Funds_returns_adj, 2, max, na.rm=TRUE),

sd = apply(Funds_returns_adj, 2, sd, na.rm=TRUE),

n = apply(Funds_returns_adj, 2, function (x) NROW( x[!is.na(x)] ))))

####Normality test - Jarque Bera

JarqueBera<-data.frame(JarqueBera=numeric(175))

for (i in 1:175) {

JarqueBera[i,]<-jbTest(na.omit(Funds_returns_adj[,i]))@test$p.value[3]

}

#alphas estimations

##Cov and var for B

Covar<-data.frame(Covariance=numeric(175))

for (i in 1:175) {

Covar[i,]=cov(Funds_returns_adj[,i],benchmark_returns,use="complete.obs")

}

Var_benchmark<-var(benchmark_returns)

sd(benchmark_returns)^2

##Having rf,rm,r,var,cov we can estimate Jensen alphas

###Trying lm and equation to see the difference in alphas (without Brent oil in rubles)

alpha<-mean(Funds_returns_adj[,1],na.rm=TRUE)-mean(Rf_returns)-(cov(Funds_returns_adj[,1],benchmark_returns,use="complete.obs")/Var_benchmark)*(mean(benchmark_returns)-mean(Rf_returns))

RiskMarketPremium<-benchmark_returns-Rf_returns[1:2209]

RiskMarketPremium_Micex<-Micex_returns-Rf_returns[1:2209]

Regression<-lm(Funds_returns_adj[,1]-Rf_returns[1:2209]~RiskMarketPremium)

Beta<-cov(Funds_returns_adj[,1],benchmark_returns,use="complete.obs")/Var_benchmark

alpha

Beta

Regression

####The results are the same for this particular case, but for funds with NA they will differ, we should use regression

########################################

#Now we can get distribution of alphas

ReturnsaboveRf<-Funds_returns_adj-Rf_returns[1:2209]

## For Brent+UsdRub

Regression_Alpha_Brent_Rub<-data.frame(Alpha=numeric(175))

for (i in 1:175) {

Regression_Alpha_Brent_Rub[i,]<-summary(lm(ReturnsaboveRf[,i]~RiskMarketPremium+Brent_returns+UsdRub_returns))$coefficients[1,1]

}

summary(Regression_Alpha_Brent_Rub)

sd(Regression_Alpha_Brent_Rub[,1])

## For BrentRub

Regression_Alpha_BrentRub<-data.frame(Alpha=numeric(175))

for (i in 1:175) {

Regression_Alpha_BrentRub[i,]<-summary(lm(ReturnsaboveRf[,i]~RiskMarketPremium+BrentRub_returns))$coefficients[1,1]

}

summary(Regression_Alpha_BrentRub)

sd(Regression_Alpha_BrentRub[,1])

##For CAPM

Regression_Alpha_CAPM<-data.frame(Alpha=numeric(175))

for (i in 1:175) {

Regression_Alpha_CAPM[i,]<-summary(lm(ReturnsaboveRf[,i]~RiskMarketPremium))$coefficients[1,1]

}

summary(Regression_Alpha_CAPM)

sd(Regression_Alpha_CAPM[,1])

#R-squared

R_squared_Brent_Rub<-data.frame(R_squared=numeric(175))

for (i in 1:175) {

R_squared_Brent_Rub[i,]<-summary(lm(ReturnsaboveRf[,i]~RiskMarketPremium+Brent_returns+UsdRub_returns))$r.squared

}

summary(R_squared_Brent_Rub)

sd(R_squared_Brent_Rub[,1])

R_squared_BrentRub<-data.frame(R_squared=numeric(175))

for (i in 1:175) {

R_squared_BrentRub[i,]<-summary(lm(ReturnsaboveRf[,i]~RiskMarketPremium+BrentRub_returns))$r.squared

}

summary(R_squared_BrentRub)

sd(R_squared_BrentRub[,1])

R_squared_CAPM<-data.frame(R_squared=numeric(175))

for (i in 1:175) {

R_squared_CAPM[i,]<-summary(lm(ReturnsaboveRf[,i]~RiskMarketPremium))$r.squared

}

summary(R_squared_CAPM)

sd(R_squared_CAPM[,1])

R_squared<-matrix(nrow=175,ncol=3)

colnames(R_squared)<-c('CAPM','BrentRub','Brent+Rub')

R_squared[,1]<-R_squared_CAPM[,1]

R_squared[,2]<-R_squared_BrentRub[,1]

R_squared[,3]<-R_squared_Brent_Rub[,1]

plot(density(R_squared[,1]), lty=3, col = "blue", main = "R_squared", xlab = "R_squared", ylab = "density")

lines(density(R_squared[,2]), lty=2, col = "red", main = "R_squared", xlab = "R_squared", ylab = "density")

lines(density(R_squared[,3]), type = "l", col = "black", main = "R_squared", xlab = "R_squared", ylab = "density")

mean(R_squared)

####Testing R-squared for Micex

Regression_Alpha_Brent_Rub_Micex<-data.frame(Alpha=numeric(175))

for (i in 1:175) {

Regression_Alpha_Brent_Rub_Micex[i,]<-summary(lm(ReturnsaboveRf[,i]~RiskMarketPremium_Micex+Brent_returns+UsdRub_returns))$coefficients[1,1]

}

R_squared_Brent_Rub_Micex<-data.frame(Alpha=numeric(175))

for (i in 1:175) {

R_squared_Brent_Rub_Micex[i,]<-summary(lm(ReturnsaboveRf[,i]~RiskMarketPremium_Micex+Brent_returns+UsdRub_returns))$r.squared

}

summary(R_squared_Brent_Rub_Micex)

sd(R_squared_Brent_Rub_Micex[,1])

########################################

#Graph for Real Alpha

plot(density(Regression_Alpha_Brent_Rub[,]), type = "l", col = "black", main = "Real Alpha(Brent+Rub)", xlab = "Alpha", ylab = "density")

abline(v =0, col="red")

plot(density(Regression_Alpha_BrentRub[,]), type = "l", col = "black", main = "Real Alpha(BrentRub)", xlab = "Alpha", ylab = "density")

abline(v =0, col="red")

plot(density(Regression_Alpha_CAPM[,]), type = "l", col = "black", main = "Real Alpha(CAPM)", xlab = "Alpha", ylab = "density")

abline(v =0, col="red")

#all on one pic

plot(density(Regression_Alpha_Brent_Rub[,]), type = "l", ylim=c(0,1400), col = "black", main = "Real Alpha", xlab = "Alpha", ylab = "density")

lines(density(Regression_Alpha_BrentRub[,]), lty=2, col = "red", main = "Real Alpha(BrentRub)", xlab = "Alpha", ylab = "density")

lines(density(Regression_Alpha_CAPM[,]), lty=3, col = "blue", main = "Real Alpha(CAPM)", xlab = "Alpha", ylab = "density")

abline(v =0, col="green")

###Yearly

Regression_Alpha_perannum_Brent_Rub<-data.frame(Alpha=numeric(175))

Regression_Alpha_perannum_Brent_Rub<-(Regression_Alpha_Brent_Rub+1)^247-1

plot(density(Regression_Alpha_perannum_Brent_Rub), type = "l", col = "black", main = "Real Year Alpha(Brent+Rub)", xlab = "Alpha", ylab = "density")

abline(v =0, col="red")

Regression_Alpha_perannum_BrentRub<-data.frame(Alpha=numeric(175))

Regression_Alpha_perannum_BrentRub<-(Regression_Alpha_BrentRub+1)^247-1

plot(density(Regression_Alpha_perannum_BrentRub), type = "l", col = "black", main = "Real Year Alpha(BrentRub)", xlab = "Alpha", ylab = "density")

abline(v =0, col="red")

Regression_Alpha_perannum_CAPM<-data.frame(Alpha=numeric(175))

Regression_Alpha_perannum_CAPM<-(Regression_Alpha_CAPM+1)^247-1

plot(density(Regression_Alpha_perannum_CAPM), type = "l", col = "black", main = "Real Year Alpha(CAPM)", xlab = "Alpha", ylab = "density")

abline(v =0, col="red")

#all in one

plot(density(Regression_Alpha_perannum_Brent_Rub), type = "l", ylim = c(0,5.5), col = "black", main = "Real Year Alpha", xlab = "Alpha", ylab = "density")

abline(v =0, col="red")

lines(density(Regression_Alpha_perannum_BrentRub), lty = 2, col = "green")

lines(density(Regression_Alpha_perannum_CAPM), lty = 3, col = "blue")

legend (0.4,5,c("Brent and Rub","Brent in Rub","CAPM"),lty=c(1,2,3), col=c("black","green","blue"))

######################################

#Best and worst Alphas

BrentandRubAlpha<-data.frame(Alpha=numeric(175))

rownames(BrentandRubAlpha)<-colnames(Funds_returns_adj)

BrentandRubAlpha[,1]<-Regression_Alpha_Brent_Rub[,1]

Brent_Rub_sort_alphas<-BrentandRubAlpha[order(BrentandRubAlpha[,1]),, drop=FALSE]

Brent_Rub_top_alphas<-Brent_Rub_sort_alphas[171:175,,drop=FALSE]

Brent_Rub_worst_alphas<-Brent_Rub_sort_alphas[1:5,,drop=FALSE]

BrentinRubAlpha<-data.frame(Alpha=numeric(175))

rownames(BrentinRubAlpha)<-colnames(Funds_returns_adj)

BrentinRubAlpha[,1]<-Regression_Alpha_BrentRub[,1]

BrentRub_sort_alphas<-BrentinRubAlpha[order(BrentinRubAlpha[,1]),, drop=FALSE]

BrentRub_top_alphas<-BrentRub_sort_alphas[171:175,,drop=FALSE]

BrentRub_worst_alphas<-BrentRub_sort_alphas[1:5,,drop=FALSE]

CAPMAlpha<-data.frame(Alpha=numeric(175))

rownames(CAPMAlpha)<-colnames(Funds_returns_adj)

CAPMAlpha[,1]<-Regression_Alpha_CAPM[,1]

CAPM_sort_alphas<-CAPMAlpha[order(CAPMAlpha[,1]),, drop=FALSE]

CAPM_top_alphas<-CAPM_sort_alphas[171:175,,drop=FALSE]

CAPM_worst_alphas<-CAPM_sort_alphas[1:5,,drop=FALSE]

###################################

###Regressions with 500 observations

Short_Alpha_Brent_Rub<-data.frame(Alpha=numeric(175))

for (i in 1:175) {

Short_Alpha_Brent_Rub[i,]<-summary(lm(ReturnsaboveRf[1:500,i]~RiskMarketPremium[1:500]+Brent_returns[1:500]+UsdRub_returns[1:500]))$coefficients[1,1]

}

summary(Short_Alpha_Brent_Rub)

sd(Short_Alpha_Brent_Rub[,1])

Short_Alpha_BrentRub<-data.frame(Alpha=numeric(175))

for (i in 1:175) {

Short_Alpha_BrentRub[i,]<-summary(lm(ReturnsaboveRf[1:500,i]~RiskMarketPremium[1:500]+Brent_returns[1:500]))$coefficients[1,1]

}

summary(Short_Alpha_BrentRub)

sd(Short_Alpha_BrentRub[,1])

Short_Alpha_CAPM<-data.frame(Alpha=numeric(175))

for (i in 1:175) {

Short_Alpha_CAPM[i,]<-summary(lm(ReturnsaboveRf[1:500,i]~RiskMarketPremium[1:500]))$coefficients[1,1]

}

summary(Short_Alpha_CAPM)

sd(Short_Alpha_CAPM[,1])

###

Short_BrentandRubAlpha<-data.frame(Alpha=numeric(175))

rownames(Short_BrentandRubAlpha)<-colnames(Funds_returns_adj)

Short_BrentandRubAlpha[,1]<-Short_Alpha_Brent_Rub[,1]

Short_Brent_Rub_sort_alphas<-Short_BrentandRubAlpha[order(Short_BrentandRubAlpha[,1]),, drop=FALSE]

Short_Brent_Rub_top_alphas<-Short_Brent_Rub_sort_alphas[171:175,,drop=FALSE]

Short_Brent_Rub_worst_alphas<-Short_Brent_Rub_sort_alphas[1:5,,drop=FALSE]

Short_BrentinRubAlpha<-data.frame(Alpha=numeric(175))

rownames(Short_BrentinRubAlpha)<-colnames(Funds_returns_adj)

Short_BrentinRubAlpha[,1]<-Short_Alpha_BrentRub[,1]

Short_BrentRub_sort_alphas<-Short_BrentinRubAlpha[order(Short_BrentinRubAlpha[,1]),, drop=FALSE]

Short_BrentRub_top_alphas<-Short_BrentRub_sort_alphas[171:175,,drop=FALSE]

Short_BrentRub_worst_alphas<-Short_BrentRub_sort_alphas[1:5,,drop=FALSE]

Short_CAPMAlpha<-data.frame(Alpha=numeric(175))

rownames(Short_CAPMAlpha)<-colnames(Funds_returns_adj)

Short_CAPMAlpha[,1]<-Short_Alpha_CAPM[,1]

Short_CAPM_sort_alphas<-Short_CAPMAlpha[order(Short_CAPMAlpha[,1]),, drop=FALSE]

Short_CAPM_top_alphas<-Short_CAPM_sort_alphas[171:175,,drop=FALSE]

Short_CAPM_worst_alphas<-Short_CAPM_sort_alphas[1:5,,drop=FALSE]

#####################Bootstrap#####################

#Store Beta

Beta1_Brent_Rub<-data.frame(Beta1=numeric(175))

for (i in 1:175) {

Beta1_Brent_Rub[i,]=summary(lm(ReturnsaboveRf[,i]~RiskMarketPremium+Brent_returns+UsdRub_returns))$coefficients[2,1]

}

Beta2_Brent_Rub<-data.frame(Beta2=numeric(175))

for (i in 1:175) {

Beta2_Brent_Rub[i,]=summary(lm(ReturnsaboveRf[,i]~RiskMarketPremium+Brent_returns+UsdRub_returns))$coefficients[3,1]

}

Beta3_Brent_Rub<-data.frame(Beta3=numeric(175))

for (i in 1:175) {

Beta3_Brent_Rub[i,]=summary(lm(ReturnsaboveRf[,i]~RiskMarketPremium+Brent_returns+UsdRub_returns))$coefficients[4,1]

}

Beta1_BrentRub<-data.frame(Beta1=numeric(175))

for (i in 1:175) {

Beta1_BrentRub[i,]=summary(lm(ReturnsaboveRf[,i]~RiskMarketPremium+BrentRub_returns))$coefficients[2,1]

}

Beta2_BrentRub<-data.frame(Beta2=numeric(175))

for (i in 1:175) {

Beta2_BrentRub[i,]=summary(lm(ReturnsaboveRf[,i]~RiskMarketPremium+BrentRub_returns))$coefficients[3,1]

}

Beta1_CAPM<-data.frame(Beta1=numeric(175))

for (i in 1:175) {

Beta1_CAPM[i,]=summary(lm(ReturnsaboveRf[,i]~RiskMarketPremium))$coefficients[2,1]

}

#Store residuals

Residuals_Brent_Rub<-matrix(nrow=2209,ncol=175)

colnames(Residuals_Brent_Rub)=colnames(Funds_returns_adj)

for (i in 1:175) {

fit1<-lm(ReturnsaboveRf[,i]~RiskMarketPremium+Brent_returns+UsdRub_returns, na.action=na.exclude)

Residuals_Brent_Rub[,i]<-resid(fit1)

}

#Tests

JarqueBera_residuals_Brent_Rub<-data.frame(JarqueBera=numeric(175))

for (i in 1:175) {

JarqueBera_residuals_Brent_Rub[i,]<-jbTest(na.omit(Residuals_Brent_Rub[,i]))@test$p.value[3]

}

DurbinWatson_residuals_Brent_Rub<-data.frame(DurbinWatson=numeric(175))

for (i in 1:175) {

DurbinWatson_residuals_Brent_Rub[i,]<-dwtest(lm(ReturnsaboveRf[,i]~RiskMarketPremium+Brent_returns+UsdRub_returns), alt='two.sided')$p.value

}

autocor_Brent_Rub<-data.frame(Autocorrelation=numeric(175))

for (i in 1:175) {

autocor_Brent_Rub[i,]<- if (DurbinWatson_residuals_Brent_Rub[i,]<0.05) {1} else {0}

}

sum(autocor_Brent_Rub)

###

Residuals_BrentRub<-matrix(nrow=2209,ncol=175)

colnames(Residuals_BrentRub)=colnames(Funds_returns_adj)

for (i in 1:175) {

fit2<-lm(ReturnsaboveRf[,i]~RiskMarketPremium+Brent_returns, na.action=na.exclude)

Residuals_BrentRub[,i]<-resid(fit2)

}

#Tests

JarqueBera_residuals_BrentRub<-data.frame(JarqueBera=numeric(175))

for (i in 1:175) {

JarqueBera_residuals_BrentRub[i,]<-jbTest(na.omit(Residuals_BrentRub[,i]))@test$p.value[3]

}

DurbinWatson_residuals_BrentRub<-data.frame(DurbinWatson=numeric(175))

for (i in 1:175) {

DurbinWatson_residuals_BrentRub[i,]<-dwtest(lm(ReturnsaboveRf[,i]~RiskMarketPremium+BrentRub_returns), alt='two.sided')$p.value

}

autocor_BrentRub<-data.frame(Autocorrelation=numeric(175))

for (i in 1:175) {

autocor_BrentRub[i,]<- if (DurbinWatson_residuals_BrentRub[i,]<0.05) {1} else {0}

}

sum(autocor_BrentRub)

###

Residuals_CAPM<-matrix(nrow=2209,ncol=175)

colnames(Residuals_CAPM)=colnames(Funds_returns_adj)

for (i in 1:175) {

fit3<-lm(ReturnsaboveRf[,i]~RiskMarketPremium, na.action=na.exclude)

Residuals_CAPM[,i]<-resid(fit3)

}

#Tests

JarqueBera_residuals_CAPM<-data.frame(JarqueBera=numeric(175))

for (i in 1:175) {

JarqueBera_residuals_CAPM[i,]<-jbTest(na.omit(Residuals_CAPM[,i]))@test$p.value[3]

}

DurbinWatson_residuals_CAPM<-data.frame(DurbinWatson=numeric(175))

for (i in 1:175) {

DurbinWatson_residuals_CAPM[i,]<-dwtest(lm(ReturnsaboveRf[,i]~RiskMarketPremium), alt='two.sided')$p.value

}

autocor_CAPM<-data.frame(Autocorrelation=numeric(175))

for (i in 1:175) {

autocor_CAPM[i,]<- if (DurbinWatson_residuals_CAPM[i,]<0.05) {1} else {0}

}

sum(autocor_CAPM)

#how to make random residuals

randomRes_Brent_Rub<-sample(nrow(Residuals_Brent_Rub))

ResidualsRandom_Brent_Rub<-Residuals_Brent_Rub[randomRes_Brent_Rub,]

#how to restore returns from residuals with zero alpha to get pseudo returns

#checking we are doing it right

Regression_Alpha_Brent_Rub[1,1]

Beta1_Brent_Rub[1,1]

Beta2_Brent_Rub[1,1]

Beta3_Brent_Rub[1,1]

RiskMarketPremium[1]

Residuals_Brent_Rub[1,1]

Regression_Alpha_Brent_Rub[1,1]+Beta1_Brent_Rub[1,1]*RiskMarketPremium[1]+Beta2_Brent_Rub[1,1]*Brent_returns[1]+Beta3_Brent_Rub[1,1]*UsdRub_returns[1]+Residuals_Brent_Rub[1,1]

ReturnsaboveRf[1,1]

#Now back to pseudo returns

NewReturns_Brent_Rub<-data.frame(Returns=numeric(2209))

NewReturns_Brent_Rub[,1]=Beta1_Brent_Rub[1,1]*RiskMarketPremium+Beta2_Brent_Rub[1,1]*Brent_returns+Beta3_Brent_Rub[1,1]*UsdRub_returns+ResidualsRandom_Brent_Rub[,1]

#how to get pseudo alpha from pseudo returns

lm(NewReturns_Brent_Rub[,1]~RiskMarketPremium+Brent_returns+UsdRub_returns)

New_Alpha<-summary(lm(NewReturns_Brent_Rub[,1]~RiskMarketPremium+Brent_returns+UsdRub_returns))$coefficients[1,1]

########################################

#Now we can create a loop

##Brent_Rub loop

Boot_Alpha_Brent_Rub<-matrix(nrow=1000,ncol=175)

colnames(Boot_Alpha_Brent_Rub)=colnames(Funds_returns_adj)

for (j in 1:1000) {

Pseudo_RaRF_Brent_Rub<-matrix(nrow=2209,ncol=175)

colnames(Pseudo_RaRF_Brent_Rub)=colnames(Funds_returns_adj)

RR_Brent_Rub<-matrix(nrow=2209, ncol=175)

for (i in 1:175) {

RR_Brent_Rub[,i]<-cbind(rbind(sample(Residuals_Brent_Rub[1:n_observations[i],i],replace=TRUE))[1:2209])

}

for (i in 1:175) {

Pseudo_RaRF_Brent_Rub[,i]<-Beta1_Brent_Rub[i,1]*RiskMarketPremium+Beta2_Brent_Rub[i,1] *Brent_returns+Beta3_Brent_Rub[i,1]*UsdRub_returns+RR_Brent_Rub[,i]

}

for (n in 1:175) {

Boot_Alpha_Brent_Rub[j,n]<-summary(lm(Pseudo_RaRF_Brent_Rub[,n]~RiskMarketPremium+Brent_returns+UsdRub_returns, na.action=na.exclude))$coefficients[1,1]

}

}

##BrentRub loop

Boot_Alpha_BrentRub<-matrix(nrow=1000,ncol=175)

colnames(Boot_Alpha_BrentRub)=colnames(Funds_returns_adj)

for (j in 1:1000) {

Pseudo_RaRF_BrentRub<-matrix(nrow=2209,ncol=175)

colnames(Pseudo_RaRF_BrentRub)=colnames(Funds_returns_adj)

RR_BrentRub<-matrix(nrow=2209, ncol=175)

for (i in 1:175) {

RR_BrentRub[,i]<-cbind(rbind(sample(Residuals_BrentRub[1:n_observations[i],i],replace=TRUE))[1:2209])

}

for (i in 1:175) {

Pseudo_RaRF_BrentRub[,i]<-Beta1_BrentRub[i,1]*RiskMarketPremium+Beta2_BrentRub[i,1]*BrentRub_returns+ RR_BrentRub[,i]

}

for (n in 1:175) {

Boot_Alpha_BrentRub[j,n]<-summary(lm(Pseudo_RaRF_BrentRub[,n]~RiskMarketPremium+BrentRub_returns, na.action=na.exclude))$coefficients[1,1]

}

}

##CAPM loop

Boot_Alpha_CAPM<-matrix(nrow=1000,ncol=175)

colnames(Boot_Alpha_CAPM)=colnames(Funds_returns_adj)

for (j in 1:1000) {

Pseudo_RaRF_CAPM<-matrix(nrow=2209,ncol=175)

colnames(Pseudo_RaRF_CAPM)=colnames(Funds_returns_adj)

RR_CAPM<-matrix(nrow=2209, ncol=175)

for (i in 1:175) {

RR_CAPM[,i]<-cbind(rbind(sample(Residuals_CAPM[1:n_observations[i],i],replace=TRUE))[1:2209])

}

for (i in 1:175) {

Pseudo_RaRF_CAPM[,i]<-Beta1_CAPM[i,1]*RiskMarketPremium+RR_CAPM[,i]

}

for (n in 1:175) {

Boot_Alpha_CAPM[j,n]<-summary(lm(Pseudo_RaRF_CAPM[,n]~RiskMarketPremium, na.action=na.exclude))$coefficients[1,1]

}

}

########################################

#Real and bootstrap alphas qus

summary(as.vector(Boot_Alpha_Brent_Rub))

sd(as.vector(Boot_Alpha_Brent_Rub))

summary(Regression_Alpha_Brent_Rub[,])

summary(as.vector(Boot_Alpha_BrentRub))

sd(as.vector(Boot_Alpha_BrentRub))

summary(Regression_Alpha_BrentRub[,])

summary(as.vector(Boot_Alpha_CAPM))

sd(as.vector(Boot_Alpha_CAPM))

summary(Regression_Alpha_CAPM[,])

##These alphas represents luck

plot(density(Boot_Alpha_Brent_Rub), type = "l", col = "black", main = "Bootstrap Alpha for Brent and USDRub", xlab = "Alpha", ylab = "density", xlim=c(-0.0017,0.0017), ylim=c(0,1400))

lines(density(Regression_Alpha_BrentRub[,]), lty=2, col="red", )

abline(v =0, col="red")

plot(density(Boot_Alpha_BrentRub), type = "l", col = "black", main = "Bootstrap Alpha for Brent in Rub", xlab = "Alpha", ylab = "density", xlim=c(-0.001,0.001))

abline(v =0, col="red")

plot(density(Boot_Alpha_CAPM), type = "l", col = "black", main = "Bootstrap Alpha for CAPM", xlab = "Alpha", ylab = "density", xlim=c(-0.001,0.001))

abline(v =0, col="red")

#Now we should make 95% interval

Boot_Alpha_Sorted_Brent_Rub<-apply(Boot_Alpha_Brent_Rub, 2, sort)

Bad_skill_25_alphas_Brent_Rub<-(Boot_Alpha_Sorted_Brent_Rub[25,]+Boot_Alpha_Sorted_Brent_Rub[26,])/2

Good_skill_25_alphas_Brent_Rub<-(Boot_Alpha_Sorted_Brent_Rub[974,]+Boot_Alpha_Sorted_Brent_Rub[975,])/2

Bad_skill_50_alphas_Brent_Rub<-(Boot_Alpha_Sorted_Brent_Rub[50,]+Boot_Alpha_Sorted_Brent_Rub[51,])/2

Good_skill_50_alphas_Brent_Rub<-(Boot_Alpha_Sorted_Brent_Rub[949,]+Boot_Alpha_Sorted_Brent_Rub[950,])/2

Boot_Alpha_Sorted_BrentRub<-apply(Boot_Alpha_BrentRub, 2, sort)

Bad_skill_25_alphas_BrentRub<-(Boot_Alpha_Sorted_BrentRub[25,]+Boot_Alpha_Sorted_BrentRub[26,])/2

Good_skill_25_alphas_BrentRub<-(Boot_Alpha_Sorted_BrentRub[974,]+Boot_Alpha_Sorted_BrentRub[975,])/2

Bad_skill_50_alphas_BrentRub<-(Boot_Alpha_Sorted_BrentRub[50,]+Boot_Alpha_Sorted_BrentRub[51,])/2

Good_skill_50_alphas_BrentRub<-(Boot_Alpha_Sorted_BrentRub[949,]+Boot_Alpha_Sorted_BrentRub[950,])/2

Boot_Alpha_Sorted_CAPM<-apply(Boot_Alpha_CAPM, 2, sort)

Bad_skill_25_alphas_CAPM<-(Boot_Alpha_Sorted_CAPM[25,]+Boot_Alpha_Sorted_CAPM[26,])/2

Good_skill_25_alphas_CAPM<-(Boot_Alpha_Sorted_CAPM[974,]+Boot_Alpha_Sorted_CAPM[975,])/2

Bad_skill_50_alphas_CAPM<-(Boot_Alpha_Sorted_CAPM[50,]+Boot_Alpha_Sorted_CAPM[51,])/2

Good_skill_50_alphas_CAPM<-(Boot_Alpha_Sorted_CAPM[949,]+Boot_Alpha_Sorted_CAPM[950,])/2

#Compare funds alphas with bootstrap bad and good results

Res_25_Brent_Rub<-data.frame(Skill=numeric(175))

for (i in 1:175) {

Res_25_Brent_Rub[i,]<-if (Regression_Alpha_Brent_Rub[i,] > Good_skill_25_alphas_Brent_Rub[i]) {1}

else {if (Regression_Alpha_Brent_Rub[i,] < Bad_skill_25_alphas_Brent_Rub[i]) {-1} else {0}}

}

NumResSkill_25_Brent_Rub<-data.frame(Result_Brent_Rub=numeric(175))

for (i in 1:175) {

NumResSkill_25_Brent_Rub[i,]<-if (Res_25_Brent_Rub[i,]==1) {1} else {0}

}

sum(NumResSkill_25_Brent_Rub)

NumResBad_25_Brent_Rub<-data.frame(Result_Brent_Rub=numeric(175))

for (i in 1:175) {

NumResBad_25_Brent_Rub[i,]<-if (Res_25_Brent_Rub[i,]== -1) {1} else {0}

}

sum(NumResBad_25_Brent_Rub)

Res_50_Brent_Rub<-data.frame(Skill=numeric(175))

for (i in 1:175) {

Res_50_Brent_Rub[i,]<-if (Regression_Alpha_Brent_Rub[i,] > Good_skill_50_alphas_Brent_Rub[i]) {1}

else {if (Regression_Alpha_Brent_Rub[i,] < Bad_skill_50_alphas_Brent_Rub[i]) {-1} else {0}}

}

NumResSkill_50_Brent_Rub<-data.frame(Result_Brent_Rub=numeric(175))

for (i in 1:175) {

NumResSkill_50_Brent_Rub[i,]<-if (Res_50_Brent_Rub[i,]==1) {1} else {0}

}

sum(NumResSkill_50_Brent_Rub)

NumResBad_50_Brent_Rub<-data.frame(Result_Brent_Rub=numeric(175))

for (i in 1:175) {

NumResBad_50_Brent_Rub[i,]<-if (Res_50_Brent_Rub[i,]== -1) {1} else {0}

}

sum(NumResBad_50_Brent_Rub)

##

Res_25_BrentRub<-data.frame(Skill=numeric(175))

for (i in 1:175) {

Res_25_BrentRub[i,]<-if (Regression_Alpha_BrentRub[i,] > Good_skill_25_alphas_BrentRub[i]) {1}

else {if (Regression_Alpha_BrentRub[i,] < Bad_skill_25_alphas_BrentRub[i]) {-1} else {0}}

}

NumResSkill_25_BrentRub<-data.frame(Result_BrentRub=numeric(175))

for (i in 1:175) {

NumResSkill_25_BrentRub[i,]<-if (Res_25_BrentRub[i,]==1) {1} else {0}

}

sum(NumResSkill_25_BrentRub)

NumResBad_25_BrentRub<-data.frame(Result_BrentRub=numeric(175))

for (i in 1:175) {

NumResBad_25_BrentRub[i,]<-if (Res_25_BrentRub[i,]== -1) {1} else {0}

}

sum(NumResBad_25_BrentRub)

Res_50_BrentRub<-data.frame(Skill=numeric(175))

for (i in 1:175) {

Res_50_BrentRub[i,]<-if (Regression_Alpha_BrentRub[i,] > Good_skill_50_alphas_BrentRub[i]) {1}

else {if (Regression_Alpha_BrentRub[i,] < Bad_skill_50_alphas_BrentRub[i]) {-1} else {0}}

}

NumResSkill_50_BrentRub<-data.frame(Result_BrentRub=numeric(175))

for (i in 1:175) {

NumResSkill_50_BrentRub[i,]<-if (Res_50_BrentRub[i,]==1) {1} else {0}

}

sum(NumResSkill_50_BrentRub)

NumResBad_50_BrentRub<-data.frame(Result_BrentRub=numeric(175))

for (i in 1:175) {

NumResBad_50_BrentRub[i,]<-if (Res_50_BrentRub[i,]== -1) {1} else {0}

}

sum(NumResBad_50_BrentRub)

##

Res_25_CAPM<-data.frame(Skill=numeric(175))

for (i in 1:175) {

Res_25_CAPM[i,]<-if (Regression_Alpha_CAPM[i,] > Good_skill_25_alphas_CAPM[i]) {1}

else {if (Regression_Alpha_CAPM[i,] < Bad_skill_25_alphas_CAPM[i]) {-1} else {0}}

}

NumResSkill_25_CAPM<-data.frame(Result_CAPM=numeric(175))

for (i in 1:175) {

NumResSkill_25_CAPM[i,]<-if (Res_25_CAPM[i,]==1) {1} else {0}

}

sum(NumResSkill_25_CAPM)

NumResBad_25_CAPM<-data.frame(Result_CAPM=numeric(175))

for (i in 1:175) {

NumResBad_25_CAPM[i,]<-if (Res_25_CAPM[i,]== -1) {1} else {0}

}

sum(NumResBad_25_CAPM)

Res_50_CAPM<-data.frame(Skill=numeric(175))

for (i in 1:175) {

Res_50_CAPM[i,]<-if (Regression_Alpha_CAPM[i,] > Good_skill_50_alphas_CAPM[i]) {1}

else {if (Regression_Alpha_CAPM[i,] < Bad_skill_50_alphas_CAPM[i]) {-1} else {0}}

}

NumResSkill_50_CAPM<-data.frame(Result_CAPM=numeric(175))

for (i in 1:175) {

NumResSkill_50_CAPM[i,]<-if (Res_50_CAPM[i,]==1) {1} else {0}

}

sum(NumResSkill_50_CAPM)

NumResBad_50_CAPM<-data.frame(Result_CAPM=numeric(175))

for (i in 1:175) {

NumResBad_50_CAPM[i,]<-if (Res_50_CAPM[i,]== -1) {1} else {0}

}

sum(NumResBad_50_CAPM)

###############Funds Alphas and their skill###############

###For 2.5%

Compare_25<-matrix(nrow=175,ncol=12)

rownames(Compare_25)=colnames(Funds_returns_adj)

colnames(Compare_25)<-c("Real_Brent_Rub", "Good_Skill_Brent_Rub", "Bad_Skill_Brent_Rub", "Skill_Brent_Rub","Real_BrentRub", "Good_Skill_BrentRub", "Bad_Skill_BrentRub", "Skill_BrentRub","Real_CAPM", "Good_Skill_CAPM", "Bad_Skill_CAPM", "Skill_CAPM")

for (i in 1:175) {

Compare_25[i,1]<-Regression_Alpha_Brent_Rub[i,]

Compare_25[i,2]<-Good_skill_25_alphas_Brent_Rub[i]

Compare_25[i,3]<-Bad_skill_25_alphas_Brent_Rub[i]

Compare_25[i,4]<-Res_25_Brent_Rub[i,]

Compare_25[i,5]<-Regression_Alpha_BrentRub[i,]

Compare_25[i,6]<-Good_skill_25_alphas_BrentRub[i]

Compare_25[i,7]<-Bad_skill_25_alphas_BrentRub[i]

Compare_25[i,8]<-Res_25_BrentRub[i,]

Compare_25[i,9]<-Regression_Alpha_CAPM[i,]

Compare_25[i,10]<-Good_skill_25_alphas_CAPM[i]

Compare_25[i,11]<-Bad_skill_25_alphas_CAPM[i]

Compare_25[i,12]<-Res_25_CAPM[i,]

}

attach(mtcars)

par(mfrow=c(2,1))

plot(Regression_Alpha_Brent_Rub[,], main = "Brent and Rub Result", xlab="fund")

abline(h=0, col="blue")

lines(Good_skill_25_alphas_Brent_Rub, col="green")

lines(Bad_skill_25_alphas_Brent_Rub, col="red")

plot(Compare_25[,4], ylim= c(-1,1), ylab="skill", xlab="fund")

plot(Regression_Alpha_BrentRub[,], main = "Brent in Rub Result", xlab="fund")

abline(h=0, col="blue")

lines(Good_skill_25_alphas_BrentRub, col="green")

lines(Bad_skill_25_alphas_BrentRub, col="red")

plot(Compare_25[,8], ylim= c(-1,1), ylab="skill", xlab="fund")

plot(Regression_Alpha_CAPM[,], main = "CAPM Result", xlab="fund")

abline(h=0, col="blue")

lines(Good_skill_25_alphas_CAPM, col="green")

lines(Bad_skill_25_alphas_CAPM, col="red")

plot(Compare_25[,12], ylab="skill", xlab="fund")

###For 5%

Compare_50<-matrix(nrow=175,ncol=12)

rownames(Compare_50)=colnames(Funds_returns_adj)

colnames(Compare_50)<-c("Real_Brent_Rub", "Good_Skill_Brent_Rub", "Bad_Skill_Brent_Rub", "Skill_Brent_Rub","Real_BrentRub", "Good_Skill_BrentRub", "Bad_Skill_BrentRub", "Skill_BrentRub","Real_CAPM", "Good_Skill_CAPM", "Bad_Skill_CAPM", "Skill_CAPM")

for (i in 1:175) {

Compare_50[i,1]<-Regression_Alpha_Brent_Rub[i,]

Compare_50[i,2]<-Good_skill_50_alphas_Brent_Rub[i]

Compare_50[i,3]<-Bad_skill_50_alphas_Brent_Rub[i]

Compare_50[i,4]<-Res_50_Brent_Rub[i,]

Compare_50[i,5]<-Regression_Alpha_BrentRub[i,]

Compare_50[i,6]<-Good_skill_50_alphas_BrentRub[i]

Compare_50[i,7]<-Bad_skill_50_alphas_BrentRub[i]

Compare_50[i,8]<-Res_50_BrentRub[i,]

Compare_50[i,9]<-Regression_Alpha_CAPM[i,]

Compare_50[i,10]<-Good_skill_50_alphas_CAPM[i]

Compare_50[i,11]<-Bad_skill_50_alphas_CAPM[i]

Compare_50[i,12]<-Res_50_CAPM[i,]

}

attach(mtcars)

par(mfrow=c(2,1),ask=F)

plot(Regression_Alpha_Brent_Rub[,], main = "Brent and Rub Result", xlab="fund")

abline(h=0, col="blue")

lines(Good_skill_50_alphas_Brent_Rub, col="green")

lines(Bad_skill_50_alphas_Brent_Rub, col="red")

plot(Compare_50[,4], ylab="skill", xlab="fund")

plot(Regression_Alpha_BrentRub[,], main = "Brent in Rub Result", xlab="fund")

abline(h=0, col="blue")

lines(Good_skill_50_alphas_BrentRub, col="green")

lines(Bad_skill_50_alphas_BrentRub, col="red")

plot(Compare_50[,8], ylab="skill", xlab="fund")

plot(Regression_Alpha_CAPM[,], main = "CAPM Result", xlab="fund")

abline(h=0, col="blue")

lines(Good_skill_50_alphas_CAPM, col="green")

lines(Bad_skill_50_alphas_CAPM, col="red")

plot(Compare_50[,12], ylab="skill", xlab="fund")

#####With pseuode residuals with 2209 observations######

##Brent_Rub loop 2209

Boot_Alpha_Brent_Rub_2209<-matrix(nrow=1000,ncol=175)

colnames(Boot_Alpha_Brent_Rub_2209)=colnames(Funds_returns_adj)

for (j in 1:1000) {

RandomRes_Brent_Rub_2209<-matrix(nrow=2209,ncol=175)

colnames(RandomRes_Brent_Rub_2209)=colnames(Funds_returns_adj)

Pseudo_RaRF_Brent_Rub_2209<-matrix(nrow=2209,ncol=175)

colnames(Pseudo_RaRF_Brent_Rub_2209)=colnames(Funds_returns_adj)

for (k in 1:175) {

RandomRes_Brent_Rub_2209[,k]<-sample(na.omit(Residuals_Brent_Rub[,k]), size=2209, replace=T)

}

for (i in 1:175) {

Pseudo_RaRF_Brent_Rub_2209[,i]<-Beta1_Brent_Rub[i,1]*RiskMarketPremium+Beta2_Brent_Rub[i,1]*Brent_returns+Beta3_Brent_Rub[i,1]*UsdRub_returns+RandomRes_Brent_Rub_2209[,i]

}

for (n in 1:175) {

Boot_Alpha_Brent_Rub_2209[j,n]<-summary(lm(Pseudo_RaRF_Brent_Rub_2209[,n]~RiskMarketPremium+Brent_returns+UsdRub_returns))$coefficients[1,1]

}

}

summary(as.vector(Boot_Alpha_Brent_Rub_2209[,]))

sd(as.vector(Boot_Alpha_Brent_Rub_2209[,]))

summary(Regression_Alpha_Brent_Rub[,])

plot(density(Boot_Alpha_Brent_Rub_2209), type = "l", col = "black", main = "Bootstrap Alpha for Brent and USDRub", xlab = "Alpha", ylab = "density", xlim=c(-0.001,0.001))

abline(v =0, col="red")

Boot_Alpha_Sorted_Brent_Rub_2209<-apply(Boot_Alpha_Brent_Rub_2209, 2, sort)

Bad_skill_25_alphas_Brent_Rub_2209<-(Boot_Alpha_Sorted_Brent_Rub_2209[25,]+Boot_Alpha_Sorted_Brent_Rub_2209[26,])/2

Good_skill_25_alphas_Brent_Rub_2209<-(Boot_Alpha_Sorted_Brent_Rub_2209[974,]+Boot_Alpha_Sorted_Brent_Rub_2209[975,])/2

Bad_skill_50_alphas_Brent_Rub_2209<-(Boot_Alpha_Sorted_Brent_Rub_2209[50,]+Boot_Alpha_Sorted_Brent_Rub_2209[51,])/2

Good_skill_50_alphas_Brent_Rub_2209<-(Boot_Alpha_Sorted_Brent_Rub_2209[949,]+Boot_Alpha_Sorted_Brent_Rub_2209[950,])/2

Res_25_Brent_Rub_2209<-data.frame(Skill=numeric(175))

for (i in 1:175) {

Res_25_Brent_Rub_2209[i,]<-if (Regression_Alpha_Brent_Rub[i,] > Good_skill_25_alphas_Brent_Rub_2209[i]) {1}

else {if (Regression_Alpha_Brent_Rub[i,] < Bad_skill_25_alphas_Brent_Rub_2209[i]) {-1} else {0}}

}

NumResSkill_25_Brent_Rub_2209<-data.frame(Result_Brent_Rub_2209=numeric(175))

for (i in 1:175) {

NumResSkill_25_Brent_Rub_2209[i,]<-if (Res_25_Brent_Rub_2209[i,]==1) {1} else {0}

...