指数ETF基金的组合分析方法初探

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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

 # Load package
 library(tidyquant)
 library(broom)

 # Load data for portfolios
 symbols <- c("SPY", "SHY", "GLD")
 symbols_low <- tolower(symbols)

 prices <- getSymbols(symbols, src = "yahoo",
                      from = "1990-01-01",
                      auto.assign = TRUE) %>%
   map(~Ad(get(.))) %>%
   reduce(merge) %>%
   `colnames<-`(symbols_low)

 prices_monthly <- to.monthly(prices, indexAt = "last", OHLC = FALSE)
 ret <- ROC(prices_monthly)["2005/2019"]

 # Load benchmark data
 bench_sym <- c("VTI", "VXUS", "BND", "BNDX")
 bench <- getSymbols(bench_sym, src = "yahoo",
                     from = "1990-01-01",
                     auto.assign = TRUE) %>%
   map(~Ad(get(.))) %>%
   reduce(merge) %>%
   `colnames<-`(tolower(bench_sym))
 bench <- to.monthly(bench, indexAt = "last", OHLC = FALSE)
 bench_ret <- ROC(bench)["2014/2019"]

 # Create different weights and portflios
 # Equal weigthed
 wt1 <- rep(1/(ncol(ret)), ncol(ret))
 port1 <- Return.portfolio(ret, wt1) %>%
   `colnames<-`("ret")

 # Risk portfolio
 wt2 <- c(0.9, 0.1, 0)
 port2 <- Return.portfolio(ret, weights = wt2) %>%
   `colnames<-`("ret")

 # Naive portfolio
 wtn <- c(0.5, 0.5, 0)
 portn <- Return.portfolio(ret, wtn)

 # Data frame of portfolios
 port_comp <- data.frame(date = index(port1), equal = as.numeric(port1),
                         risky = as.numeric(port2),
                         naive = as.numeric(portn)) 

 # Benchmark portfolio
 wtb <- c(0.24, 0.21, 0.22, 0.33)
 portb <- Return.portfolio(bench_ret, wtb, rebalance_on = "quarters") %>%
   `colnames<-`("bench")

 # Graph of portfolios vs. benchmark
 port_comp %>%
   filter(date >= "2014-01-01") %>%
   mutate(bench = portb) %>%
   gather(key,value, -date) %>%
   group_by(key) %>%
   mutate(value = cumprod(value+1)) %>%
   ggplot(aes(date, value*100, color = key)) +
   geom_line() +
   scale_color_manual("", labels = c("Bench", "Equal", "Naive", "Risky"),
                      values = c("purple", "blue", "black", "red")) +
   labs(x = "",
        y = "Index",
        title = "The three portfolios with a benchmark",
        caption = "Source: Yahoo, OSM estimates") +
   theme(legend.position = "top",
         plot.caption = element_text(hjust = 0))

 # summary
 port_comp %>%
   filter(date >= "2014-01-01") %>%
   mutate(bench = as.numeric(portb)) %>%
   rename("Equal" = equal,
          "Naive" = naive,
          "Risky" = risky,
          "Bench" = bench) %>%
   gather(Asset, value, -date) %>%
   group_by(Asset) %>%
   summarise(`Mean (%)` = round(mean(value, na.rm = TRUE),3)*1200,
             `Volatility (%)` = round(sd(value, na.rm = TRUE)*sqrt(12),3)*100,
             `Sharpe` = round(mean(value, na.rm = TRUE)/sd(value, na.rm=TRUE)*sqrt(12),2),
             `Cumulative (%)` = round(prod(1+value, na.rm = TRUE),3)*100) %>%
   knitr::kable(caption = "Annualized performance metrics") 

 # Portfolio
 mean_ret <- apply(ret[,c("spy", "shy", "gld")],2,mean)
 cov_port <- cov(ret[,c("spy", "shy", "gld")])

 port_exam <- data.frame(ports = colnames(port_comp)[-1],
                         ret = as.numeric(apply(port_comp[,-1],2, mean)),
                         vol = as.numeric(apply(port_comp[,-1], 2, sd)))

 bench_exam <- data.frame(ports = "bench",
                          ret = mean(bench_ret),
                          vol = sd(bench_ret))

 bench_spy <- data.frame(ports = "sp",
                         ret = mean(ret$spy),
                         vol = sd(ret$spy))

 bench_spy_14 <- data.frame(ports = "sp",
                         ret = mean(ret$spy["2014/2019"]),
                         vol = sd(ret$spy["2014/2019"]))

 mean_ret_14 <- apply(ret[,c("spy", "shy", "gld")]["2014/2019"],2,mean)

 cov_port_14 <- cov(ret[,c("spy", "shy", "gld")]["2014/2019"])

 port_exam_14 <- port_comp %>%
   filter(date >= "2014-01-01") %>%
   select(-date) %>%
   gather(ports, value) %>%
   group_by(ports) %>%
   summarise_all(list(ret = mean, vol = sd)) %>%
   data.frame()

 ### Random weighting
 # wts for full period
 wts <- matrix(nrow = 1000, ncol = 3)
 set.seed(123)
 for(i in 1:1000){
   a <- runif(1,0,1)
   b <- c()
   for(j in 1:2){
     b[j] <- runif(1,0,1-sum(a,b))
   }
   if(sum(a,b) < 1){
     inc <- (1-sum(a,b))/3
     vec <- c(a+inc, b+inc)
   }else{
     vec <- c(a,b)
   }
   wts[i,] <- sample(vec,replace = FALSE)
 }

 # wts for 2014
 wts1 <- matrix(nrow = 1000, ncol = 3)
 set.seed(123)
 for(i in 1:1000){
   a <- runif(1,0,1)
   b <- c()
   for(j in 1:2){
     if(j == 2){
       b[j] <- 1 - sum(a,b)
     }
     else {
       b[j] <- runif(1,0,1-sum(a,b))
     }
   vec <- c(a,b)
   }
   wts1[i,] <- sample(vec,replace = FALSE)
 }

 # Calculate random portfolios
 # Weighting: wts
 port <- matrix(nrow = 1000, ncol = 2)
 for(i in 1:1000){
   port[i,1] <- as.numeric(sum(wts[i,] * mean_ret))
   port[i,2] <- as.numeric(sqrt(t(wts[i,] %*% cov_port %*% wts[i,])))
 }

 colnames(port) <- c("returns", "risk")
 port <- as.data.frame(port)
 port <- port %>%
   mutate(sharpe = returns/risk)

 # Calculate random portfolios since 2014
 # Weighting: wts1
 port_14 <- matrix(nrow = 1000, ncol = 2)
 for(i in 1:1000){
   port_14[i,1] <- as.numeric(sum(wts1[i,] * mean_ret_14))
   port_14[i,2] <- as.numeric(sqrt(t(wts1[i,] %*% cov_port_14 %*% wts1[i,])))
 }

 colnames(port_14) <- c("returns", "risk")
 port_14 <- as.data.frame(port_14)
 port_14 <- port_14 %>%
   mutate(sharpe = returns/risk)

 # Grraph with Sharpe ratio
 port %>%
   ggplot(aes(risk*sqrt(12)*100, returns*1200, color = sharpe)) +
   geom_point(size = 1.2, alpha = 0.4) +
   geom_point(data = port_exam, aes(port_exam[1,3]*sqrt(12)*100,
                                    port_exam[1,2]*1200),
              color = "red", size = 6) +
   geom_point(data = port_exam, aes(port_exam[2,3]*sqrt(12)*100,
                                    port_exam[2,2]*1200),
              color = "purple", size = 7) +
   geom_point(data = port_exam, aes(port_exam[3,3]*sqrt(12)*100,
                                    port_exam[3,2]*1200),
              color = "black", size = 5) +
   scale_x_continuous(limits = c(0,14)) +
   labs(x = "Risk (%)",
        y = "Return (%)",
        title = "Simulated portfolios",
        color = "Sharpe ratio") +
   scale_color_gradient(low = "red", high = "green") +
   theme(legend.position = c(0.075,.8),
         legend.key.size = unit(.5, "cm"),
         legend.background = element_rect(fill = NA))

 # Graph since 2014
 port_14 %>%
   ggplot(aes(risk*sqrt(12)*100, returns*1200, color = sharpe)) +
   geom_point(size = 1.2, alpha = 0.4) +
   geom_point(data = port_exam_14, aes(port_exam_14[1,3]*sqrt(12)*100,
                                    port_exam_14[1,2]*1200),
              color = "blue", size = 6) +
   geom_point(data = port_exam_14, aes(port_exam_14[3,3]*sqrt(12)*100,
                                    port_exam_14[3, 2]*1200),
              color = "purple", size = 7) +
   geom_point(data = port_exam_14, aes(port_exam_14[2,3]*sqrt(12)*100,
                                    port_exam_14[2,2]*1200),
              color = "black", size = 5) +
   scale_x_continuous(limits = c(0,14)) +
   labs(x = "Risk (%)",
        y = "Return (%)",
        title = "Simulated portfolios since 2014",
        color = "Sharpe ratio") +
   scale_color_gradient(low = "red", high = "green") +
   theme(legend.position = c(0.075,0.8),
         legend.background = element_rect(fill = NA),
         legend.key.size = unit(.5, "cm"))

 # Portfolios benchmarked vs Vanguard
 port_14 %>%
   mutate(Bench = returns - bench_exam$ret) %>%
   # mutate(Bench = ifelse(Bench > 0, 1, 0)) %>%
   ggplot(aes(risk*sqrt(12)*100, returns*1200, color = Bench)) +
   geom_point(size = 1.2, alpha = 0.4) +
   scale_color_gradient(low = "red", high = "green") +
   geom_point(data = port_exam_14, aes(port_exam_14[1,3]*sqrt(12)*100,
                                    port_exam_14[1,2]*1200),
              color = "blue", size = 6) +
   geom_point(data = port_exam_14, aes(port_exam_14[3,3]*sqrt(12)*100,
                                    port_exam_14[3,2]*1200),
              color = "purple", size = 7) +
   geom_point(data = port_exam_14, aes(port_exam_14[2,3]*sqrt(12)*100,
                                    port_exam_14[2,2]*1200),
              color = "black", size = 5) +
   labs(x = "Risk (%)",
        y = "Return (%)",
        title = "Simulated portfolios since 2014") +
   theme(legend.position = c(0.06,0.8),
         legend.background = element_rect(fill = NA),
         legend.key.size = unit(.5, "cm"))

 # Portfolios benchmarked vs Vanguard
 port_14 %>%
   mutate(Bench = returns - bench_exam$ret) %>%
   mutate(Bench = ifelse(Bench > 0, 1, 0)) %>%
   ggplot(aes(risk*sqrt(12)*100, returns*1200, color = Bench)) +
   geom_point(size = 1.2, alpha = 0.4) +
   scale_color_gradient(low = "red", high = "green") +
   geom_point(data = port_exam_14, aes(port_exam_14[1,3]*sqrt(12)*100,
                                    port_exam_14[1,2]*1200),
              color = "blue", size = 6) +
   geom_point(data = port_exam_14, aes(port_exam_14[3,3]*sqrt(12)*100,
                                    port_exam_14[3,2]*1200),
              color = "purple", size = 7) +
   geom_point(data = port_exam_14, aes(port_exam_14[2,3]*sqrt(12)*100,
                                    port_exam_14[2,2]*1200),
              color = "black", size = 5) +
   labs(x = "Risk (%)",
        y = "Return (%)",
        title = "Simulated portfolios") +
   theme(legend.position = c(0.05,0.8),
         legend.background = element_rect(fill = NA),
         legend.key.size = unit(.5, "cm"))

 # Count how many portfolios are negative
 pos_b <- port_14 %>%
   mutate(Bench = returns - bench_exam$ret) %>%
   mutate(Bench = ifelse(Bench > 0, 1, 0)) %>%
   summarise(bench = round(mean(Bench),2)*100) %>%
   as.numeric()

 port_list_14 <- list()
 for(i in 1:1000){
   port_list_14[[i]] <- Return.portfolio(ret["2014/2019"], wts[i,]) %>%
     data.frame() %>%
     summarise(returns = mean(portfolio.returns),
               excess_ret = mean(portfolio.returns) - mean(portb$bench),
               track_err = sd(portfolio.returns - portb$bench),
               risk = sd(portfolio.returns))
 }

 port_info <- port_list_14 %>% bind_rows
 rfr <- mean(ret$shy)

 # Graph info
 port_info %>%
   mutate(info_ratio = excess_ret/track_err) %>%
   ggplot(aes(risk*sqrt(12)*100, returns*1200, color = info_ratio)) +
   geom_point(size = 1.2, alpha = 0.4) +
   geom_point(data = port_exam_14, aes(port_exam_14[1,3]*sqrt(12)*100,
                                       port_exam_14[1,2]*1200),
              color = "blue", size = 6) +
   geom_point(data = port_exam_14, aes(port_exam_14[3,3]*sqrt(12)*100,
                                       port_exam_14[3,2]*1200),
              color = "purple", size = 7) +
   geom_point(data = port_exam_14, aes(port_exam_14[2,3]*sqrt(12)*100,
                                       port_exam_14[2,2]*1200),
              color = "black", size = 5) +
   labs(x = "Risk (%)",
        y = "Return (%)",
        title = "Simulated portfolios") +
   theme(legend.position = c(0.075,0.8),
         legend.background = element_rect(fill = NA),
         legend.key.size = unit(.5, "cm")) +
   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。