Find the optimal trade to minimize variance given a constraint on the tracking error and some other simple constraints, including that the portfolio is long-only.
- vector of asset prices
- variance matrix for the assets
- information about the benchmark
- current portfolio (if it exists)
- Portfolio Probe
You need the prices at which assets trade and a variance matrix of the asset returns. You need either the benchmark to be an asset in the variance matrix or the weight vector of assets for the benchmark.
The holdings of the current portfolio need to be in a vector with names that are the asset identifiers.
You also need to have the Portfolio Probe package loaded into your R session:
If you don’t have Portfolio Probe, see “Demo or Buy”.
Doing the example
xaLWvar06variance matrix from “Returns to variance matrix” example
xaLWvar06EqWtvariance matrix from “Add benchmark to variance matrix” example
You need to have the package loaded into your R session:
The inputs we need in order to get our minimum variance portfolio are:
- vector of prices at which the assets may be traded
- variance matrix of the asset returns
- desired value of the new portfolio
- appropriate constraints
- current portfolio (optional)
- information about the benchmark
We start by naming the vector of prices that we want to use:
priceVector <- xassetPrices[251,]
These are the prices at the close of the last trading day of 2006. The first few values are:
> head(priceVector) XA101 XA103 XA105 XA107 XA108 XA111 33.56 72.25 74.39 192.06 5.91 15.98
The requirement for the prices is that it be a vector of positive numbers with names (that are the asset identifiers).
We create an object to serve as the current portfolio:
curPortfol <- (1:10) * 1000 names(curPortfol) <- colnames(xassetPrices)[1:10]
What is expected is a numeric vector of the number of units of each asset in the portfolio. The names of the vector are the identifiers of the assets that are used in the price vector and the variance matrix.
> curPortfol XA101 XA103 XA105 XA107 XA108 XA111 XA113 XA115 1000 2000 3000 4000 5000 6000 7000 8000 XA120 XA126 9000 10000
The value of the portfolio that we should specify is the current value of the existing portfolio adjusted by whatever cash flow is desired. Here we assume we want to take out $30,000 from the portfolio:
cashFlow <- -30000 grossVal <- as.numeric(valuation(curPortfol, priceVector, collapse=TRUE)) + cashFlow
We get the value of the current portfolio assuming the prices we are using and then add the cash flow. (The
as.numeric is merely for cosmetic reasons to make the result simpler.) We end up with:
> grossVal  2983430
The gross value of the portfolio that we want is slightly less than $3 million.
Optimization with simple variance matrix
For this option we need:
- a vector of the asset weights that comprise the benchmark
Further, we need the variance matrix to contain all of those assets. But the vector of benchmark weights need not contain all the assets in the variance nor need they be in any particular order.
create benchmark weight vector
For the example, we will create a weight vector that equally weights all of the assets in the universe:
benwt <- rep(1/350, 350) names(benwt) <- rownames(xaLWvar06)
The first few values are:
> head(benwt) XA101 XA103 XA105 XA107 0.002857143 0.002857143 0.002857143 0.002857143 XA108 XA111 0.002857143 0.002857143
The vector of benchmark weights need not have all of the assets that are in the variance matrix nor be in any particular order. It is mandatory, however, that all of the assets in the benchmark weight vector be in the variance.
We need to specify the gross value, that the portfolio is long-only, and the tracking error constraint of 5%.
We need to translate the 5% tracking error — the annualized standard deviation — to daily variance (which is the variance matrix that we have). We square the tracking error and divide by 252.
We’re now ready to do an optimization. We also demand that no more than 10 assets be in the portfolio.
opMinVarTEcon <- trade.optimizer(priceVector, variance=xaLWvar06, existing=curPortfol, gross=grossVal, long.only=TRUE, port.size=10, utility="minimum variance", bench.constraint=c(EqualWt=.05^2/252), bench.weights=list(EqualWt=benwt))
We specify the utility to be minimum variance. We still would have got the same thing without the specification, but there would have been a warning that it was guessing what utility we wanted.
The resulting object is printed like:
> opMinVarTEcon $new.portfolio XA336 XA448 XA461 XA486 XA530 XA577 XA683 XA709 6587 6260 2046 7877 6043 10336 4571 10886 XA870 XA966 53547 8461 $trade XA101 XA103 XA105 XA107 XA108 XA111 XA113 -1000 -2000 -3000 -4000 -5000 -6000 -7000 XA115 XA120 XA126 XA336 XA448 XA461 XA486 -8000 -9000 -10000 6587 6260 2046 7877 XA530 XA577 XA683 XA709 XA870 XA966 6043 10336 4571 10886 53547 8461 $results objective negutil cost 3.489308e-05 3.489308e-05 0.000000e+00 penalty 0.000000e+00 $converged  FALSE $objective.utility  "minimum variance" $alpha.values  NA $var.values V0 V0 -- EqualWt 3.489308e-05 9.920628e-06 $utility.values  3.489308e-05 $existing XA101 XA103 XA105 XA107 XA108 XA111 XA113 XA115 1000 2000 3000 4000 5000 6000 7000 8000 XA120 XA126 9000 10000 $violated NULL $timestamp  "Fri Sep 07 17:55:06 2012"  "Fri Sep 07 17:55:19 2012" $call trade.optimizer(prices = priceVector, variance = xaLWvar06, existing = curPortfol, gross = grossVal, long.only = TRUE, port.size = 10, utility = "minimum variance", bench.constraint = c(EqualWt = 0.05^2/252), bench.weights = list(EqualWt = benwt))
The first two components are the new (optimal) portfolio and the trade to achieve that. There are some additional components in the object that are not shown.
Optimization with benchmark in the variance matrix
For this option we need:
- a variance matrix that includes the benchmark
See “Add benchmark to variance matrix” to create a variance matrix that includes a benchmark.
The only difference between this version and the previous optimization is that the
bench.weights argument is not used here, we need a different variance matrix, and we need the name given in
bench.constraint to match whatever the benchmark is called in the variance.
trade.optimizer function does the optimization. Its first argument is the asset prices.
It is mandatory that the value of the resulting portfolio be specified. For long-only portfolios it is sufficient to state the desired gross value. The actual value of the portfolio will (usually) be slightly less than the specification:
> format(grossVal, nsmall=2, big.mark=",")  "2,983,430.00" > grossVal - as.numeric(valuation(opMinVarTEcon, collapse=TRUE))  287.42
Specifying the tracking error constraint
The constraint on the tracking error was imposed with:
bench.constraint = c(EqualWt = 0.05^2/252)
The object given to
bench.constraint looks like:
> c(EqualWt = 0.05^2/252) EqualWt 9.920635e-06
Alternatively we could have first created such an object:
benCon <- c(EqualWt = 0.05^2/252)
benCon <- 0.05^2/252 names(benCon) <- "EqualWt"
and then imposed the constraint as:
bench.constraint = benCon
If we just gave a number without it having names, then the optimizer wouldn’t know what was to be used as the benchmark.
Let’s look at the
utility.values components of the output:
$var.values V0 V0 -- EqualWt 3.489308e-05 9.920628e-06 $utility.values  3.489308e-05
var.values component has two elements. The first has a name that only involves the variance, the second’s name includes the benchmark as well. That means that the first is the portfolio variance and the second is the variance relative to the benchmark.
The first variance value is equal to the utility value. This indicates that it really is the portfolio variance that is being optimized.
Resulting volatility and tracking error
The volatility and tracking error for the optimal portfolio is a modification of the
> sqrt(opMinVarTEcon$var.values * 252) * 100 V0 V0 -- EqualWt 9.377129 4.999998
So the tracking error is right up against the 5% constraint (as should be expected). The portfolio volatility is about 9.5%.
Remember that this is predicted volatility. Since it was the quantity being optimized, it will be biased downward. The tracking error is also a predicted value. But since the tracking error is only being constrained and not optimized, we have no particular reason to believe it to be biased.
One component of the output to pay special attention to is ‘
violated‘ — this states which constraints, if any, are violated. You want this to be
It is probably not important whether ‘
FALSE. The optimization is likely to be good enough with or without convergence.
You can see more about the optimization with the summary of the object:
> summary(opMinVarTEcon) $results objective negutil cost 3.489308e-05 3.489308e-05 0.000000e+00 penalty 0.000000e+00 $objective.utility  "minimum variance" $alpha.values  NA $var.values V0 V0 -- EqualWt 3.489308e-05 9.920628e-06 $number.of.assets existing trade 10 20 new open 10 10 close universe.total 10 351 tradable select.universe 350 351 positions.notrade 0 $opening.positions  "XA336" "XA448" "XA461" "XA486" "XA530"  "XA577" "XA683" "XA709" "XA870" "XA966" $closing.positions  "XA101" "XA103" "XA105" "XA107" "XA108"  "XA111" "XA113" "XA115" "XA120" "XA126" $value.limits lower upper gross 2983132 2983430 net 2983132 2983430 long 2983132 2983430 short 0 0 $valuation.new.portfolio gross net long short 2983143 2983143 2983143 0 $valuation.trade gross net long short 5996572.58 -30287.42 2983142.58 3013430.00 $valuation.trade.fraction.of.gross gross net long short 2.01015286 -0.01015286 1.00000000 1.01015286
This has some pieces that are also in the
- The variance matrix needs to contain all of the assets that are in the price vector. It can have additional assets — these will be ignored (except for benchmarks). The order of the assets in the variance does not matter.
- All of the prices need to be in the same currency. You have to check that — the code has no way of knowing.
- It will still work if the object given as the prices is a one-column or one-row matrix. But it will complain about other matrices.