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R-view of Profit Hurdle in Harvey & Liu (2014)
### Required returns due to testing multiplicity ------ Harvey and Liu
### (2014): "Backtesting", Duke University
Profit_Hurdle <- function (num_tests, num_obs, alpha_sig, vol_annual, RHO){
###############################
####### Parameter inputs ######
### 'num_tests': No. of tests one allows for in multiple tests;
### 'num_obs': No. of monthly observations for a strategy;
### 'alpha_sig': Significance level (e.g., 5#);
### 'vol_annual': Annual return volatility (e.g., 0.05 or 5#).
NN <- num_tests
Obs <- num_obs
alpha0 <- alpha_sig
vol_anu <- vol_annual
###Independent test ####
#B_ind = norminv( (1- alpha0/2),0,1);
B_ind <- qnorm( (1- alpha0/2),0,1)
###Bonferroni ####
p0_mat <- alpha0/NN
t0_mat <- qnorm( (1-p0_mat/2),0,1)
BF <- t0_mat
###Input for Holm and 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(NN/para_inter[2]) + 1)*floor(para_inter[2]+1); # make sure Nsim_test >= num_tests
t_sample <- sample_random_multests(para_inter[1], Nsim_tests, para_inter[3], para_inter[4], WW)
### Holm #####
HL_mat <- NULL
for(ww in 1:WW){
yy <- t_sample[ww, 1:NN] ### Use the ww'th row of t-sample ###
p_sub <- 2*(1-pnorm(yy))
p_new <- sort(p_sub)
KK <- length(p_new)
comp_vec <- NULL
for(kk in 1:KK){
comp_vec[kk] <- alpha0/(KK + 1-kk)
}
comp_res <- p_new > comp_vec
comp_new <- cumsum(as.numeric(comp_res))
if(sum(comp_new) == 0){
HL <- 1.96
} else {
p0 <- p_new[comp_new == 1]
HL <- qnorm((1 - p0/2),0,1)
}
HL_mat <- append(HL_mat, HL)
}
### BHY ####
BHY_mat <- NULL
for(ww in 1:WW){
yy <- t_sample[ww, 1:NN] ### Use the ww'th row of t-sample ###
p_sub <- 2*(1-pnorm(yy))
if(length(p_new) <= 1){
BH00 <- 1.96
} else {
p_new11 <- sort(p_sub, decreasing = TRUE)
KK <- length(p_new11)
comp_vec0 <- NULL
cons_vec <- 1:KK
cons_norm <- sum(1/cons_vec)
for(kk in 1:KK){
comp_vec0[kk] <- (alpha0*kk)/(KK*cons_norm)
}
comp_vec <- sort(comp_vec0, decreasing = TRUE)
comp_res11 <- as.numeric(p_new11 <= comp_vec)
if(sum(comp_res11) == 0){
BH00 <- 1.96;
} else {
p0 <- p_new11[comp_res11 ==1]
b0 <- which(abs(p_new11 - p0[1]) == min(abs(p_new11 - p0[1])))
if(b0 == 1){
p1 <- p0[1]
} else {
p1 <- p_new11[(b0-1)]
}
BH00 <- qnorm((1 - (p0[1]+p1)/4),0,1)
}
}
BHY_mat <- append(BHY_mat,BH00)
}
tcut_vec <- c(B_ind, BF, median(HL_mat), median(BHY_mat))
ret_hur <- ((vol_anu/sqrt(12))/sqrt(Obs))*tcut_vec
print('Inputs:')
print(paste('Significance Level = ',alpha0*100))
print(paste('Number of Observations = ', num_obs))
print(paste('Annualized Return Volatility = ', vol_anu*100))
print(paste('Assumed Number of Tests = ', NN))
print(paste('Assumed Average Correlation = ', RHO))
print('Outputs:')
print('Minimum Average Monthly Return:')
print(paste('Independent = ', ret_hur[1]*100))
print(paste('Bonferroni = ', ret_hur[2]*100))
print(paste('Holm = ', ret_hur[3]*100))
print(paste('BHY = ', ret_hur[4]*100))
print(paste('Average for Multiple Tests = ', mean(ret_hur[-1])*100))
}
Profit_Hurdle(300, 240, 0.05, 0.1, 0.4)
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