Naive portfolio selection rule

Question

I have a xts file with monthly returns for 17 industry portfolios. The data looks as follows:

             Cars Chems Clths Cnstr Cnsum Durbl FabPr Finan  Food Machn Mines   Oil Other Rtail Steel Trans
1926-07-31   4.77  1.20 -0.09  4.20  2.05  1.33  0.61  0.44  0.46  2.06  2.65 -2.57  1.99  1.46  3.05 -0.69
1926-08-31  -1.11  3.55  3.57  0.85  4.10  0.75 -0.49  8.84  4.72  5.57  1.16  3.85  4.81  0.63 -0.58  4.96
1926-09-30  -3.39  1.85 -4.89 -1.06  2.50  1.27 -3.10 -2.55  1.66  0.52  1.44 -4.93 -2.09 -1.20  2.28  0.06
1926-10-31 -10.66 -9.15  0.49 -6.49 -1.41 -5.02 -3.92 -4.40 -4.79 -4.52  5.73  0.23 -3.50 -2.44 -4.98 -2.79
1926-11-30  -0.73  4.98  2.66  2.91  8.35  0.12  1.36 -0.27  7.04 -0.75  1.13  2.92 -0.47  1.72  1.81  1.38
1926-12-31   5.14  2.59  2.30  3.37  1.96  4.23  2.22  2.40 -1.39  2.93 -1.38  6.39  2.59  3.06  2.17  2.18
           Utils
1926-07-31  4.85
1926-08-31 -2.00
1926-09-30  2.06
1926-10-31 -2.98
1926-11-30  5.71
1926-12-31  1.72

My aim is to perform a backtest with a naive portfolio selection rule. Instead of holding an equal-weighted portfolio, i want to assing the weights according to the following naive rule:

• Assign weights 2/N to each asset that has above-median historical returns
• assign weight 0 if below-median historical returns

Instead of the equal-weighted vector:

w <- c(rep(1/17,17))

This weighting vector works well for getting the portfolio returns. To do so I used this function:

portfolio_returns_tq_rebl <- 
  returns %>% 
  tq_portfolio(assets_col = symbol,
               returns_col = return,
               weights = w, # here i want to have a weighting function?!
               col_rename = "returns",
               rebalance_on = "months")

I stuck in incorporate a weighting function to a standard backtest script (tidyquant, PerformanceAnalytics, quantmod). In most of these, only optimization-problems can be solved but not simple naive rules.

Does somebody have an idea in how to pursue such a backtest with a simple portfolio selection rule?

Thanks for your help!

Answer 1

If an alternative package were acceptable, too: here is a sketch how to do it with PMwR , which I maintain. I start with an example dataset: 17 industry portfolios from Kenneth French's website (probably the same dataset that you use).

library("PMwR")
library("NMOF")

P <- French(tempdir(),
            "17_Industry_Portfolios_daily_CSV.zip",
            frequency = "daily",
            price.series = TRUE)


str(P)
## 'data.frame':    24935 obs. of  17 variables:
##  $ Food : num  1 1 1 1 1.01 ...
##  $ Mines: num  1 1 1.01 1 1 ...
##  $ Oil  : num  1 1.01 1.01 1.02 1.01 ...
##  $ Clths: num  1 1 1 1 1.01 ...
##  $ Durbl: num  1 0.989 0.983 0.965 0.964 ...
##  $ Chems: num  1 1.01 1.02 1.02 1.03 ...
##  $ Cnsum: num  1 1 1.01 1.01 1.01 ...
##  $ Cnstr: num  1 1 1 1.01 1.01 ...
##  $ Steel: num  1 0.994 1.006 1.007 1.007 ...
##  $ FabPr: num  1 0.992 1.002 1.008 1.032 ...
##  $ Machn: num  1 0.999 1.003 1.008 1.006 ...
##  $ Cars : num  1 0.999 1.009 1.018 1.019 ...
##  $ Trans: num  1 1 1 1 1 ...
##  $ Utils: num  1 1.01 1.01 1.02 1.02 ...
##  $ Rtail: num  1 1 1 0.998 0.992 ...
##  $ Finan: num  1 1.01 1.01 1.01 1 ...
##  $ Other: num  1 1 1 1.01 1.01 ...

Backtests can be run with function btest . The main "ingredient" of a backtest is a "signal" function that is called at any instant of time and returns the desired portfolio. An example: the function here looks back 250 days, computes the returns of the assets and then keeps those assets that have an above-median return.

above_median <- function() {
    ## get the most recent 250 days
    H <- Close(n = 250)

    ## compute total return of industries
    R <- H[nrow(H), ] / H [1L, ]

    ## include only those with an above-median return
    include <- R > median(R)
    w <- numeric(ncol(H))
    w[include] <- 1/sum(include)
    w
}

This function is passed to btest with the instruction to call it every quarter.

bt <- btest(prices = list(as.matrix(P)),
            timestamp = as.Date(row.names(P)),
            signal = above_median,
            do.signal = "lastofquarter",
            b = 250, ## burnin
            initial.cash = 100,
            convert.weights = TRUE)

You can analyse the results.

summary(NAVseries(bt))
journal(bt)

---

Update, following the comment: btest does not impose restrictions on data frequency. Here is the example with monthly data, starting with monthly returns.

P <- French(tempdir(),
            "17_Industry_Portfolios_CSV.zip",
            price.series = FALSE)
head(P)  ## returns
##               Food   Mines     Oil  Clths   Durbl   Chems  Cnsum   Cnstr
## 1926-07-31  0.0048  0.0378 -0.0141 0.0602 -0.0162  0.0846 0.0142  0.0231
## 1926-08-31  0.0291  0.0069  0.0360 0.0015 -0.0196  0.0570 0.0584  0.0433
## ....

Transform the returns into total-return series:

P <- apply(P + 1, 2, cumprod)
head(P)  ## returns => 'prices'
##                Food    Mines       Oil    Clths     Durbl    Chems    Cnsum
## 1926-07-31 1.004800 1.037800 0.9859000 1.060200 0.9838000 1.084600 1.014200
## 1926-08-31 1.034040 1.044961 1.0213924 1.061790 0.9645175 1.146422 1.073429

Adjust the signal function for monthly data:

above_median <- function() {
    ## get the most recent 12 months
    H <- Close(n = 12)

    ## compute total return of industries
    R <- H[nrow(H), ] / H [1L, ]

    ## include only those with an above-median return
    include <- R > median(R)
    w <- numeric(ncol(H))
    w[include] <- 1/sum(include)
    w
}

Run the backtest, with an appropriate burnin b .

bt <- btest(prices = list(as.matrix(P)),
            timestamp = as.Date(row.names(P)),
            signal = above_median,
            do.signal = "lastofquarter",
            b = 12, ## burnin
            initial.cash = 100,
            convert.weights = TRUE)

unique(journal(bt)$timestamp)  ## timestamps of trades 
## [1] "1927-09-30" "1927-12-31" "1928-03-31" "1928-06-30" "1928-09-30"
## [6] "1928-12-31" "1929-03-31" "1929-06-30" "1929-09-30" "1929-12-31"
## ....