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Harvey and Liu - Backtesting
require(MASS) # for mvrnorm()
### Generate empirical p-value distributions based on Harvey, Liu and Zhu (2014) ------ Harvey and Liu
### (2014): "Backtesting", Duke University
sample_random_multests <- function(rho, m_tot, p_0, lambda, M_simu){
###Parameter input from Harvey, Liu and Zhu (2014) ############
###Default: para_vec = [0.2, 1377, 4.4589*0.1, 5.5508*0.001,M_simu]###########
p_0 <- p_0 ; # probability for a random factor to have a zero mean
lambda <- lambda; # average of monthly mean returns for true strategies
m_tot <- m_tot; # total number of trials
rho <- rho; # average correlation among returns
M_simu <- M_simu; # number of rows (simulations)
sigma <- 0.15/sqrt(12); # assumed level of monthly vol
N <- 240; #number of time-series
sig_vec <- c(1, rho*rep(1, m_tot-1))
SIGMA <- toeplitz(sig_vec)
MU <- rep(0,m_tot)
shock_mat <- mvrnorm(MU, SIGMA*(sigma^2/N), n = M_simu)
prob_vec <- replicate(M_simu, runif(m_tot,0,1))
mean_vec <- t(replicate(M_simu, rexp(m_tot, rate=1/lambda)))
m_indi <- prob_vec > p_0
mu_nul <- as.numeric(m_indi)*mean_vec #Null-hypothesis
tstat_mat <- abs(mu_nul + shock_mat)/(sigma/sqrt(N))
sample_random_multests <- tstat_mat
}
### Sharpe ratio adjustment due to testing multiplicity ------ Harvey and Liu
### (2014): "Backtesting", Duke University
Haircut_SR <- function(sm_fre, num_obs, SR, ind_an, ind_aut, rho, num_test, RHO){
###############################
####### Parameter inputs ######
### 'sm_fre': Sampling frequency; [1,2,3,4,5] = [Daily, Weekly, Monthly, Quarterly, Annual];
### 'num_obs': No. of observations in the frequency specified in the previous step;
### 'SR': Sharpe ratio; either annualized or in the frequency specified in the previous step;
### 'ind_an': Indicator; if annulized, 'ind_an' = 1; otherwise = 0;
### 'ind_aut': Indicator; if adjusted for autocorrelations, 'ind_aut' = 0; otherwise = 1;
### 'rho': Autocorrelation coefficient at the specified frequency ;
### 'num_test': Number of tests allowed, Harvey, Liu and Zhu (2014) find 315 factors;
### 'RHO': Average correlation among contemporaneous strategy returns.
### Calculating the equivalent annualized Sharpe ratio 'sr_annual', after
### taking autocorrelation into account
if(sm_fre == 1){
fre_out <- 'Daily'
} else if(sm_fre == 2){
fre_out <- 'Weekly'
} else if(sm_fre == 3){
fre_out <- 'Monthly'
} else if(sm_fre == 4){
fre_out <- 'Quarterly'
} else {
fre_out <- 'Annual'
}
if(ind_an == 1){
sr_out <- 'Yes'
} else {
sr_out <- 'No'
}
if(ind_an == 1 & ind_aut == 0){
sr_annual <- SR
} else if(ind_an ==1 & ind_aut == 1){
if(sm_fre ==1){
sr_annual <- SR*(1 + (2*rho/(1-rho))*(1- ((1-rho^(360))/(360*(1-rho)))))^(-0.5)
} else if(sm_fre ==2){
sr_annual <- SR*(1 + (2*rho/(1-rho))*(1- ((1-rho^(52))/(52*(1-rho)))))^(-0.5)
} else if(sm_fre ==3){
sr_annual <- SR*(1 + (2*rho/(1-rho))*(1- ((1-rho^(12))/(12*(1-rho)))))^(-0.5)
} else if(sm_fre ==4){
sr_annual <- SR*(1 + (2*rho/(1-rho))*(1- ((1-rho^(4))/(4*(1-rho)))))^(-0.5)
} else if(sm_fre ==5){
sr_annual <- SR
}
} else if(ind_an == 0 & ind_aut == 0){
if(sm_fre ==1){
sr_annual <- SR*sqrt(360)
} else if(sm_fre ==2){
sr_annual <- SR*sqrt(52)
} else if(sm_fre ==3){
sr_annual <- SR*sqrt(12)
} else if(sm_fre ==4){
sr_annual <- SR*sqrt(4)
} else if(sm_fre ==5){
sr_annual = SR
}
} else if(ind_an == 0 & ind_aut == 1){
if(sm_fre ==1){
sr_annual <- sqrt(360)*sr*(1 + (2*rho/(1-rho))*(1- ((1-rho^(360))/(360*(1-rho)))))^(-0.5)
} else if(sm_fre ==2){
sr_annual <- sqrt(52)*sr*(1 + (2*rho/(1-rho))*(1- ((1-rho^(52))/(52*(1-rho)))))^(-0.5)
} else if(sm_fre ==3){
sr_annual <- sqrt(12)*sr*(1 + (2*rho/(1-rho))*(1- ((1-rho^(12))/(12*(1-rho)))))^(-0.5);
} else if(sm_fre ==4){
sr_annual <- sqrt(4)*sr*(1 + (2*rho/(1-rho))*(1- ((1-rho^(4))/(4*(1-rho)))))^(-0.5)
} else if(sm_fre ==5){
sr_annual <- SR
}
}
### Number of monthly observations 'N' ###
if(sm_fre ==1){
N <- floor(num_obs*12/360)
} else if(sm_fre ==2){
N <- floor(num_obs*12/52)
} else if(sm_fre == 3){
N <- floor(num_obs*12/12)
} else if(sm_fre == 4){
N <- floor(num_obs*12/4)
} else if(sm_fre == 5){
N <- floor(num_obs*12/1)
}
### Number of tests allowed ###
M <- num_test;
###########################################
########### Intermediate outputs ##########
print('Inputs:')
print(paste('Frequency =', fre_out))
print(paste('Number of Observations = ', num_obs))
print(paste('Initial Sharpe Ratio = ', SR))
print(paste('Sharpe Ratio Annualized = ', sr_out))
print(paste('Autocorrelation = ', rho))
print(paste('A/C Corrected Annualized Sharpe Ratio = ', sr_annual))
print(paste('Assumed Number of Tests = ', M))
print(paste('Assumed Average Correlation = ', RHO))
############################################
########## Sharpe ratio adjustment #########
m_vec <- 1:(M+1);
c_const <- sum(1./m_vec);
########## Input for Holm & BHY ##########
### Parameter input from Harvey, Liu and Zhu (2014) %%%%%%%
para0 <- matrix(c(0, 1295, 3.9660*0.1, 5.4995*0.001,
0.2, 1377, 4.4589*0.1, 5.5508*0.001,
0.4, 1476, 4.8604*0.1, 5.5413*0.001,
0.6, 1773, 5.9902*0.1, 5.5512*0.001,
0.8, 3109, 8.3901*0.1, 5.5956*0.001),
nrow = 5, ncol = 4, byrow = TRUE)
### Interpolated parameter values based on user specified level of correlation RHO %%%%%%%%%%
if (RHO >= 0 & RHO < 0.2){
para_inter <- ((0.2 - RHO)/0.2)*para0[1,] + ((RHO - 0)/0.2)*para0[2,]
} else if (RHO >= 0.2 & RHO < 0.4) {
para_inter <- ((0.4 - RHO)/0.2)*para0[2,] + ((RHO - 0.2)/0.2)*para0[3,]
} else if (RHO >= 0.4 & RHO < 0.6){
para_inter <- ((0.6 - RHO)/0.2)*para0[3,] + ((RHO - 0.4)/0.2)*para0[4,]
} else if (RHO >= 0.6 & RHO < 0.8){
para_inter <- ((0.8 - RHO)/0.2)*para0[4,] + ((RHO - 0.6)/0.2)*para0[5,]
} else if (RHO >= 0.8 & RHO < 1.0){
para_inter <- ((0.8 - RHO)/0.2)*para0[4,] + ((RHO - 0.6)/0.2)*para0[5,]
} else {
### Default: para_vec = [0.2, 1377, 4.4589*0.1, 5.5508*0.001,M_simu]
para_inter <- para0[2,] ### Set at the preferred level if RHO is misspecified
}
WW <- 2000; ### Number of repetitions
### Generate a panel of t-ratios (WW*Nsim_tests) ###
Nsim_tests <- (floor(M/para_inter[2]) + 1)*floor(para_inter[2]+1); # make sure Nsim_test >= M
t_sample <- sample_random_multests(para_inter[1], Nsim_tests, para_inter[3], para_inter[4], WW)
# Sharpe Ratio, monthly
sr <- sr_annual / (12 ^(1/2))
T <- sr * 120^(1/2)
p_val <- 2 * (1 - pt(T, N - 1))
# Drawing observations from the underlying p-value distribution; simulate a
# large number (WW) of p-value samples
p_holm <- rep(1, WW)
p_bhy <- rep(1, WW)
for(ww in 1:WW){
yy <- t_sample[ww, 1:M]
t_value <- t(yy)
p_val_sub <- 2*(1- pnorm(t_value,0,1));
### Holm
p_val_all <- append(t(p_val_sub), p_val)
p_val_order <- sort(p_val_all)
p_holm_vec <- NULL
for(i in 1:(M+1)){
p_new <- NULL
for(j in 1:i){
p_new <- append(p_new, (M+1-j+1)*p_val_order[j])
}
p_holm_vec <- append(p_holm_vec, min(max(p_new),1))
}
p_sub_holm <- p_holm_vec[p_val_order == p_val]
p_holm[ww] <- p_sub_holm[1];
### BHY
p_bhy_vec <- NULL
for(i in 1:(M+1)){
kk <- (M+1) - (i-1)
if(kk == (M+1)){
p_new <- p_val_order[M+1]
} else {
p_new <- min( (M+1)*(c_const/kk)*p_val_order[kk], p_0)
}
p_bhy_vec <- append(p_new, p_bhy_vec)
p_0 <- p_new
}
p_sub_bhy <- p_bhy_vec[p_val_order == p_val]
p_bhy[ww] <- p_sub_bhy[1]
}
### Bonferroni ###
p_BON <- min(M*p_val,1)
### Holm ###
p_HOL <- median(p_holm)
### BHY ###
p_BHY <- median(p_bhy)
### Average ###
p_avg <- (p_BON + p_HOL + p_BHY)/3
# Invert to get z-score
z_BON <- qt(1 - p_BON/2, N - 1)
z_HOL <- qt(1 - p_HOL/2, N - 1)
z_BHY <- qt(1 - p_BHY/2, N - 1)
z_avg <- qt(1- p_avg/2, N - 1)
# Annualized Sharpe ratio
sr_BON <- (z_BON/sqrt(N)) * sqrt(12)
sr_HOL <- (z_HOL/sqrt(N)) * sqrt(12)
sr_BHY <- (z_BHY/sqrt(N)) * sqrt(12)
sr_avg <- (z_avg/sqrt(N))*sqrt(12)
# Calculate haircut
hc_BON <- (sr_annual - sr_BON)/sr_annual
hc_HOL <- (sr_annual - sr_HOL)/sr_annual
hc_BHY <- (sr_annual - sr_BHY)/sr_annual
hc_avg <- (sr_annual - sr_avg)/sr_annual
##################################
######### Final Output ###########
print("Bonferroni Adjustment:")
print(paste("Adjusted P-value =", p_BON))
print(paste("Haircut Sharpe Ratio =", sr_BON))
print(paste("Percentage Haircut =", hc_BON))
print("Holm Adjustment:")
print(paste("Adjusted P-value =", p_HOL))
print(paste("Haircut Sharpe Ratio =", sr_HOL))
print(paste("Percentage Haircut =", hc_HOL))
print("BHY Adjustment:")
print(paste("Adjusted P-value =", p_BHY))
print(paste("Haircut Sharpe Ratio =", sr_BHY))
print(paste("Percentage Haircut =", hc_BHY))
print("Average Adjustment:")
print(paste("Adjusted P-value =", p_avg))
print(paste("Haircut Sharpe Ratio =", sr_avg))
print(paste("Percentage Haircut =", hc_avg))
}
Haircut_SR(3,120,1,1,1,0.1,100,0.4)
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