为什么不解决.QP和Portfolio.optim产生相同的结果? [英] Why doesn't solve.QP and portfolio.optim generate identical results?

问题描述

Portfolio.optim {tseries}的文档说，solve.QP {quadprog}用于生成用于找到使Sharpe比率最大化的相切组合的解决方案.这意味着结果与任何一个函数都应该相同.我可能会忽略一些东西，但是在这个简单的示例中，我得到了相似但不完全相同的解决方案，它们使用Portfolio.optim和Solve.QP估算最佳投资组合权重.结果不应该一样吗?如果是这样，我在哪里出错?这是代码:

The documentation for portfolio.optim {tseries} says that solve.QP {quadprog} is used to generate the solution for finding the tangency portfolio that maximizes the Sharpe ratio. That implies that results should be identical with either function. I'm probably overlooking something, but in this simple example I get similar but not identical solutions for estimating optimal portfolio weights with portfolio.optim and solve.QP. Shouldn't the results be identical? If so, where am I going wrong? Here's the code:

library(tseries)
library(quadprog)


# 1. Generate solution with solve.QP via: comisef.wikidot.com/tutorial:tangencyportfolio

# create artifical data
set.seed(1)
nO     <- 100     # number of observations
nA     <- 10      # number of assets
mData  <- array(rnorm(nO * nA, mean = 0.001, sd = 0.01), dim = c(nO, nA))
rf     <- 0.0001     # riskfree rate (2.5% pa)
mu     <- apply(mData, 2, mean)    # means
mu2    <- mu - rf                  # excess means

# qp
aMat  <- as.matrix(mu2)
bVec  <- 1
zeros <- array(0, dim = c(nA,1))
solQP <- solve.QP(cov(mData), zeros, aMat, bVec, meq = 1)

# rescale variables to obtain weights
w <- as.matrix(solQP$solution/sum(solQP$solution))

# 2. Generate solution with portfolio.optim (using artificial data from above)
port.1 <-portfolio.optim(mData,riskless=rf)
port.1.w <-port.1$pw
port.1.w <-matrix(port.1.w)

# 3. Compare portfolio weights from the two methodologies:

compare <-cbind(w,port.1$pw)

compare

             [,1]        [,2]
 [1,]  0.337932967 0.181547633
 [2,]  0.073836572 0.055100415
 [3,]  0.160612441 0.095800361
 [4,]  0.164491490 0.102811562
 [5,]  0.005034532 0.003214622
 [6,]  0.147473396 0.088792283
 [7,] -0.122882875 0.000000000
 [8,]  0.127924865 0.067705050
 [9,]  0.026626940 0.012507530
[10,]  0.078949672 0.054834759

推荐答案

处理此类情况的唯一方法就是浏览源代码.在您的情况下，可以通过tseries:::portfolio.optim.default访问它.

The one and the only way to deal with such situations is to browse the source. In your case, it is accessible via tseries:::portfolio.optim.default.

现在，要找到这两个调用之间的区别，我们可以通过定义等效的辅助函数来缩小问题的范围:

Now, to find the difference between those two calls, we may narrow down the issue by defining an equivalent helper function:

foo <- function(x, pm = mean(x), covmat = cov(x), riskless = FALSE, rf = 0) 
{
  x <- mData
  pm <- mean(x)
  covmat <- cov(x)
  k <- dim(x)[2]
  Dmat <- covmat
  dvec <- rep.int(0, k)
  a1 <- colMeans(x) - rf
  a2 <- matrix(0, k, k)
  diag(a2) <- 1
  b2 <- rep.int(0, k)
  Amat <- t(rbind(a1, a2))
  b0 <- c(pm - rf, b2)
  solve.QP(Dmat, dvec, Amat, bvec = b0, meq = 1)$sol
}

identical(portfolio.optim(mData, riskless=TRUE, rf=rf)$pw,
          foo(mData, riskless=TRUE, rf=rf))
#[1] TRUE

这样，您可以看到1)riskless=rf不是预期的方式，riskless=TRUE, rf=rf是正确的方式； 2)Amat和bvec有一些区别.

With that, you can see that 1) riskless=rf is not the intended way, riskless=TRUE, rf=rf is the correct one; 2) there are several differences in Amat and bvec.

我不是投资组合优化方面的专家，所以我不知道这些额外约束背后的解释是什么，以及它们是否应该首先出现，但至少您可以看到是什么导致了差异.

I am not an expert in portfolio optimization, so I do not know what's the explanation behind these additional constraints and if they should be there in the first place, but at least you can see what exactly causes the difference.

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