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试想一下，大多数基金“推荐”的配置策略都假设某种股票/债券组合。如果我们想寻求成本最小收益最高的组合（以yahoo finance上的数据来分析，因为美国股市数据更容易获得）。那么什么才是一个好的组合成为了我们的问题？指数基金包括几乎所有的股票和债券。几乎包含了美国股票及债券市场的组成的四种ETF是VTI、VXUS、BND、BNDX。让我们从这些开始数据分析。使用R语言来完成分析程序
1 # Load package 2 library(tidyquant) 3 library(broom) 4 5 # Load data for portfolios 6 symbols <- c("SPY", "SHY", "GLD") 7 symbols_low <- tolower(symbols) 8 9 prices <- getSymbols(symbols, src = "yahoo", 10 from = "1990-01-01", 11 auto.assign = TRUE) %>% 12 map(~Ad(get(.))) %>% 13 reduce(merge) %>% 14 `colnames<-`(symbols_low) 15 16 prices_monthly <- to.monthly(prices, indexAt = "last", OHLC = FALSE) 17 ret <- ROC(prices_monthly)["2005/2019"] 18 19 # Load benchmark data 20 bench_sym <- c("VTI", "VXUS", "BND", "BNDX") 21 bench <- getSymbols(bench_sym, src = "yahoo", 22 from = "1990-01-01", 23 auto.assign = TRUE) %>% 24 map(~Ad(get(.))) %>% 25 reduce(merge) %>% 26 `colnames<-`(tolower(bench_sym)) 27 bench <- to.monthly(bench, indexAt = "last", OHLC = FALSE) 28 bench_ret <- ROC(bench)["2014/2019"] 29 30 # Create different weights and portflios 31 # Equal weigthed 32 wt1 <- rep(1/(ncol(ret)), ncol(ret)) 33 port1 <- Return.portfolio(ret, wt1) %>% 34 `colnames<-`("ret") 35 36 # Risk portfolio 37 wt2 <- c(0.9, 0.1, 0) 38 port2 <- Return.portfolio(ret, weights = wt2) %>% 39 `colnames<-`("ret") 40 41 # Naive portfolio 42 wtn <- c(0.5, 0.5, 0) 43 portn <- Return.portfolio(ret, wtn) 44 45 # Data frame of portfolios 46 port_comp <- data.frame(date = index(port1), equal = as.numeric(port1), 47 risky = as.numeric(port2), 48 naive = as.numeric(portn)) 49 50 # Benchmark portfolio 51 wtb <- c(0.24, 0.21, 0.22, 0.33) 52 portb <- Return.portfolio(bench_ret, wtb, rebalance_on = "quarters") %>% 53 `colnames<-`("bench") 54 55 # Graph of portfolios vs. benchmark 56 port_comp %>% 57 filter(date >= "2014-01-01") %>% 58 mutate(bench = portb) %>% 59 gather(key,value, -date) %>% 60 group_by(key) %>% 61 mutate(value = cumprod(value+1)) %>% 62 ggplot(aes(date, value*100, color = key)) + 63 geom_line() + 64 scale_color_manual("", labels = c("Bench", "Equal", "Naive", "Risky"), 65 values = c("purple", "blue", "black", "red")) + 66 labs(x = "", 67 y = "Index", 68 title = "The three portfolios with a benchmark", 69 caption = "Source: Yahoo, OSM estimates") + 70 theme(legend.position = "top", 71 plot.caption = element_text(hjust = 0)) 72 73 # summary 74 port_comp %>% 75 filter(date >= "2014-01-01") %>% 76 mutate(bench = as.numeric(portb)) %>% 77 rename("Equal" = equal, 78 "Naive" = naive, 79 "Risky" = risky, 80 "Bench" = bench) %>% 81 gather(Asset, value, -date) %>% 82 group_by(Asset) %>% 83 summarise(`Mean (%)` = round(mean(value, na.rm = TRUE),3)*1200, 84 `Volatility (%)` = round(sd(value, na.rm = TRUE)*sqrt(12),3)*100, 85 `Sharpe` = round(mean(value, na.rm = TRUE)/sd(value, na.rm=TRUE)*sqrt(12),2), 86 `Cumulative (%)` = round(prod(1+value, na.rm = TRUE),3)*100) %>% 87 knitr::kable(caption = "Annualized performance metrics") 88 89 # Portfolio 90 mean_ret <- apply(ret[,c("spy", "shy", "gld")],2,mean) 91 cov_port <- cov(ret[,c("spy", "shy", "gld")]) 92 93 port_exam <- data.frame(ports = colnames(port_comp)[-1], 94 ret = as.numeric(apply(port_comp[,-1],2, mean)), 95 vol = as.numeric(apply(port_comp[,-1], 2, sd))) 96 97 bench_exam <- data.frame(ports = "bench", 98 ret = mean(bench_ret), 99 vol = sd(bench_ret)) 100 101 bench_spy <- data.frame(ports = "sp", 102 ret = mean(ret$spy), 103 vol = sd(ret$spy)) 104 105 bench_spy_14 <- data.frame(ports = "sp", 106 ret = mean(ret$spy["2014/2019"]), 107 vol = sd(ret$spy["2014/2019"])) 108 109 mean_ret_14 <- apply(ret[,c("spy", "shy", "gld")]["2014/2019"],2,mean) 110 111 cov_port_14 <- cov(ret[,c("spy", "shy", "gld")]["2014/2019"]) 112 113 port_exam_14 <- port_comp %>% 114 filter(date >= "2014-01-01") %>% 115 select(-date) %>% 116 gather(ports, value) %>% 117 group_by(ports) %>% 118 summarise_all(list(ret = mean, vol = sd)) %>% 119 data.frame() 120 121 122 ### Random weighting 123 # wts for full period 124 wts <- matrix(nrow = 1000, ncol = 3) 125 set.seed(123) 126 for(i in 1:1000){ 127 a <- runif(1,0,1) 128 b <- c() 129 for(j in 1:2){ 130 b[j] <- runif(1,0,1-sum(a,b)) 131 } 132 if(sum(a,b) < 1){ 133 inc <- (1-sum(a,b))/3 134 vec <- c(a+inc, b+inc) 135 }else{ 136 vec <- c(a,b) 137 } 138 wts[i,] <- sample(vec,replace = FALSE) 139 } 140 141 # wts for 2014 142 wts1 <- matrix(nrow = 1000, ncol = 3) 143 set.seed(123) 144 for(i in 1:1000){ 145 a <- runif(1,0,1) 146 b <- c() 147 for(j in 1:2){ 148 if(j == 2){ 149 b[j] <- 1 - sum(a,b) 150 } 151 else { 152 b[j] <- runif(1,0,1-sum(a,b)) 153 } 154 vec <- c(a,b) 155 } 156 wts1[i,] <- sample(vec,replace = FALSE) 157 } 158 159 # Calculate random portfolios 160 # Weighting: wts 161 port <- matrix(nrow = 1000, ncol = 2) 162 for(i in 1:1000){ 163 port[i,1] <- as.numeric(sum(wts[i,] * mean_ret)) 164 port[i,2] <- as.numeric(sqrt(t(wts[i,] %*% cov_port %*% wts[i,]))) 165 } 166 167 colnames(port) <- c("returns", "risk") 168 port <- as.data.frame(port) 169 port <- port %>% 170 mutate(sharpe = returns/risk) 171 172 # Calculate random portfolios since 2014 173 # Weighting: wts1 174 port_14 <- matrix(nrow = 1000, ncol = 2) 175 for(i in 1:1000){ 176 port_14[i,1] <- as.numeric(sum(wts1[i,] * mean_ret_14)) 177 port_14[i,2] <- as.numeric(sqrt(t(wts1[i,] %*% cov_port_14 %*% wts1[i,]))) 178 } 179 180 colnames(port_14) <- c("returns", "risk") 181 port_14 <- as.data.frame(port_14) 182 port_14 <- port_14 %>% 183 mutate(sharpe = returns/risk) 184 185 # Grraph with Sharpe ratio 186 port %>% 187 ggplot(aes(risk*sqrt(12)*100, returns*1200, color = sharpe)) + 188 geom_point(size = 1.2, alpha = 0.4) + 189 geom_point(data = port_exam, aes(port_exam[1,3]*sqrt(12)*100, 190 port_exam[1,2]*1200), 191 color = "red", size = 6) + 192 geom_point(data = port_exam, aes(port_exam[2,3]*sqrt(12)*100, 193 port_exam[2,2]*1200), 194 color = "purple", size = 7) + 195 geom_point(data = port_exam, aes(port_exam[3,3]*sqrt(12)*100, 196 port_exam[3,2]*1200), 197 color = "black", size = 5) + 198 scale_x_continuous(limits = c(0,14)) + 199 labs(x = "Risk (%)", 200 y = "Return (%)", 201 title = "Simulated portfolios", 202 color = "Sharpe ratio") + 203 scale_color_gradient(low = "red", high = "green") + 204 theme(legend.position = c(0.075,.8), 205 legend.key.size = unit(.5, "cm"), 206 legend.background = element_rect(fill = NA)) 207 208 # Graph since 2014 209 port_14 %>% 210 ggplot(aes(risk*sqrt(12)*100, returns*1200, color = sharpe)) + 211 geom_point(size = 1.2, alpha = 0.4) + 212 geom_point(data = port_exam_14, aes(port_exam_14[1,3]*sqrt(12)*100, 213 port_exam_14[1,2]*1200), 214 color = "blue", size = 6) + 215 geom_point(data = port_exam_14, aes(port_exam_14[3,3]*sqrt(12)*100, 216 port_exam_14[3, 2]*1200), 217 color = "purple", size = 7) + 218 geom_point(data = port_exam_14, aes(port_exam_14[2,3]*sqrt(12)*100, 219 port_exam_14[2,2]*1200), 220 color = "black", size = 5) + 221 scale_x_continuous(limits = c(0,14)) + 222 labs(x = "Risk (%)", 223 y = "Return (%)", 224 title = "Simulated portfolios since 2014", 225 color = "Sharpe ratio") + 226 scale_color_gradient(low = "red", high = "green") + 227 theme(legend.position = c(0.075,0.8), 228 legend.background = element_rect(fill = NA), 229 legend.key.size = unit(.5, "cm")) 230 231 # Portfolios benchmarked vs Vanguard 232 port_14 %>% 233 mutate(Bench = returns - bench_exam$ret) %>% 234 # mutate(Bench = ifelse(Bench > 0, 1, 0)) %>% 235 ggplot(aes(risk*sqrt(12)*100, returns*1200, color = Bench)) + 236 geom_point(size = 1.2, alpha = 0.4) + 237 scale_color_gradient(low = "red", high = "green") + 238 geom_point(data = port_exam_14, aes(port_exam_14[1,3]*sqrt(12)*100, 239 port_exam_14[1,2]*1200), 240 color = "blue", size = 6) + 241 geom_point(data = port_exam_14, aes(port_exam_14[3,3]*sqrt(12)*100, 242 port_exam_14[3,2]*1200), 243 color = "purple", size = 7) + 244 geom_point(data = port_exam_14, aes(port_exam_14[2,3]*sqrt(12)*100, 245 port_exam_14[2,2]*1200), 246 color = "black", size = 5) + 247 labs(x = "Risk (%)", 248 y = "Return (%)", 249 title = "Simulated portfolios since 2014") + 250 theme(legend.position = c(0.06,0.8), 251 legend.background = element_rect(fill = NA), 252 legend.key.size = unit(.5, "cm")) 253 254 # Portfolios benchmarked vs Vanguard 255 port_14 %>% 256 mutate(Bench = returns - bench_exam$ret) %>% 257 mutate(Bench = ifelse(Bench > 0, 1, 0)) %>% 258 ggplot(aes(risk*sqrt(12)*100, returns*1200, color = Bench)) + 259 geom_point(size = 1.2, alpha = 0.4) + 260 scale_color_gradient(low = "red", high = "green") + 261 geom_point(data = port_exam_14, aes(port_exam_14[1,3]*sqrt(12)*100, 262 port_exam_14[1,2]*1200), 263 color = "blue", size = 6) + 264 geom_point(data = port_exam_14, aes(port_exam_14[3,3]*sqrt(12)*100, 265 port_exam_14[3,2]*1200), 266 color = "purple", size = 7) + 267 geom_point(data = port_exam_14, aes(port_exam_14[2,3]*sqrt(12)*100, 268 port_exam_14[2,2]*1200), 269 color = "black", size = 5) + 270 labs(x = "Risk (%)", 271 y = "Return (%)", 272 title = "Simulated portfolios") + 273 theme(legend.position = c(0.05,0.8), 274 legend.background = element_rect(fill = NA), 275 legend.key.size = unit(.5, "cm")) 276 277 # Count how many portfolios are negative 278 pos_b <- port_14 %>% 279 mutate(Bench = returns - bench_exam$ret) %>% 280 mutate(Bench = ifelse(Bench > 0, 1, 0)) %>% 281 summarise(bench = round(mean(Bench),2)*100) %>% 282 as.numeric() 283 284 port_list_14 <- list() 285 for(i in 1:1000){ 286 port_list_14[[i]] <- Return.portfolio(ret["2014/2019"], wts[i,]) %>% 287 data.frame() %>% 288 summarise(returns = mean(portfolio.returns), 289 excess_ret = mean(portfolio.returns) - mean(portb$bench), 290 track_err = sd(portfolio.returns - portb$bench), 291 risk = sd(portfolio.returns)) 292 } 293 294 295 port_info <- port_list_14 %>% bind_rows 296 rfr <- mean(ret$shy) 297 298 # Graph info 299 port_info %>% 300 mutate(info_ratio = excess_ret/track_err) %>% 301 ggplot(aes(risk*sqrt(12)*100, returns*1200, color = info_ratio)) + 302 geom_point(size = 1.2, alpha = 0.4) + 303 geom_point(data = port_exam_14, aes(port_exam_14[1,3]*sqrt(12)*100, 304 port_exam_14[1,2]*1200), 305 color = "blue", size = 6) + 306 geom_point(data = port_exam_14, aes(port_exam_14[3,3]*sqrt(12)*100, 307 port_exam_14[3,2]*1200), 308 color = "purple", size = 7) + 309 geom_point(data = port_exam_14, aes(port_exam_14[2,3]*sqrt(12)*100, 310 port_exam_14[2,2]*1200), 311 color = "black", size = 5) + 312 labs(x = "Risk (%)", 313 y = "Return (%)", 314 title = "Simulated portfolios") + 315 theme(legend.position = c(0.075,0.8), 316 legend.background = element_rect(fill = NA), 317 legend.key.size = unit(.5, "cm")) + 318 scale_color_gradient("Information ratio", low = "red", high = "green")
总结一下结论？如果您有定义良好的约束条件，那么查看不同的投资组合分配以获得所需的风险/回报参数是非常好的。如果你没有，那么合并一个足够广泛的组合来包含尽可能多的可投资风险资产是有帮助的。使用调整后的Sharpe比率来观察组合的超额回报率是很有用的，这个投资组合比率揭示了一个重要的信息：即一个包含大部分相似资产的投资组合是否因偏离基准而得到收益上补偿。在这种情况下，我们的投资组合并不是，但那可能是由于gold exposure。因此，使用不关联资产的投资组合可以降低总投资金额，比如关注某个特定指数的成分股来指定投资组合，就能够最大限度的利用资金。