The source files can be found in examples/.
Turnover and tracking
Turnover is the distance of the weights from a reference weight vector, such as your current weights when you rebalance. Tracking is the distance of the portfolio returns from the returns of a benchmark. Each comes in two forms. The constraint form, a keyword of the JuMPOptimiser, keeps the distance under a limit. The risk measure form makes the distance the quantity you minimise.
Turnover, through thetnkeyword, andTurnoverRiskMeasuremeasure the change of the weights from a reference weight vector.TrackingError, through thetrkeyword, andTrackingRiskMeasuremeasure the distance from a benchmark. The benchmark is either a weight vector,WeightsTracking, or a return series such as an index,ReturnsTracking.
Reach for turnover when trading is costly and you rebalance often, so the new book must stay close to the old one. Reach for tracking when your mandate judges you against a benchmark. To replicate an index, minimise the tracking error. To beat it within a limit, maximise the return with a tracking-error budget. Use the constraint form for a limit that a mandate sets, and the risk measure form when staying near the current book or the benchmark is the goal itself.
using PortfolioOptimisers, CSV, TimeSeries, DataFrames, PrettyTables, Clarabel, StatsPlots, GraphRecipesresfmt = (v, i, j) -> begin return if j == 1 v else isa(v, AbstractFloat) ? "$(round(v*100, digits=3)) %" : v endend;1. Data and two benchmarks
The data is one year of S&P 500 prices, with two benchmarks. One is the equal-weight book, which is a weight vector. The other is the S&P 500 index, which is a return series.
X = TimeArray(CSV.File(joinpath(@__DIR__, "..", "SP500.csv.gz")); timestamp = :Date)[(end - 252):end]rd = prices_to_returns(X)pr = prior(EmpiricalPrior(), rd)N = length(rd.nx)idx = TimeArray(CSV.File(joinpath(@__DIR__, "..", "SP500_idx.csv.gz")); timestamp = :Date)index_returns = vec(values(percentchange(idx)))[(end - 251):end]slv = Solver(; name = :clarabel, solver = Clarabel.Optimizer, settings = Dict("verbose" => false), check_sol = (; allow_local = true, allow_almost = true))rf = 4.2 / 100 / 252equal_weight = fill(1 / N, N)20-element Vector{Float64}:
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0.052. Turnover as a constraint
Turnover, through tn, limits how far each weight may move from a reference weight vector w, which is normally your current holdings. val is the largest change it allows for each asset. We maximise the ratio with the equal-weight book as the reference, for four values of val. For each result we print the sum of the absolute changes from the equal-weight book, and the largest weight.
turnover_vals = [0.005, 0.02, 0.1, 0.5]turnover_res = [optimise(MeanRisk(; obj = MaximumRatio(; rf = rf), opt = JuMPOptimiser(; pe = pr, slv = slv, tn = Turnover(; w = equal_weight, val = v)))) for v in turnover_vals]drift(w) = sum(abs, w .- equal_weight)pretty_table(DataFrame("Turnover budget" => turnover_vals, "Drift from start" => drift.(getproperty.(turnover_res, :w)), "Max weight" => maximum.(getproperty.(turnover_res, :w))); formatters = [resfmt], title = "Turnover budget and drift from the equal-weight book")Turnover budget and drift from the equal-weight book
┌─────────────────┬──────────────────┬────────────┐
│ Turnover budget │ Drift from start │ Max weight │
│ Float64 │ Float64 │ Float64 │
├─────────────────┼──────────────────┼────────────┤
│ 0.005 │ 10.0 % │ 5.5 % │
│ 0.02 │ 40.0 % │ 7.0 % │
│ 0.1 │ 125.917 % │ 15.0 % │
│ 0.5 │ 170.0 % │ 55.0 % │
└─────────────────┴──────────────────┴────────────┘3. Turnover as a risk measure
TurnoverRiskMeasure makes the turnover the quantity to minimise. We minimise it with the equal-weight book as the reference, and print the drift of the result from that book, the sum of the absolute changes of section 2. With no other term the turnover is smallest at the reference, and a drift near zero means that the result is the reference book. The measure is useful as one term of a problem with more than one risk measure. You can also compute it for a candidate portfolio, to find how far that portfolio is from the current weights.
res_min_turnover = optimise(MeanRisk(; r = TurnoverRiskMeasure(; w = equal_weight), obj = MinimumRisk(), opt = JuMPOptimiser(; pe = pr, slv = slv)))drift(res_min_turnover.w)1.6653345369377348e-154. Tracking a benchmark
With TrackingRiskMeasure we minimise the tracking error to a benchmark. A WeightsTracking benchmark is a portfolio of the same assets, so the minimum is that portfolio. A ReturnsTracking benchmark, here the return series of the S&P 500 index, gives the portfolio of our 20 assets whose returns have the smallest tracking error to the index.
res_replicate_ew = optimise(MeanRisk(; r = TrackingRiskMeasure(; tr = WeightsTracking(; w = equal_weight)), obj = MinimumRisk(), opt = JuMPOptimiser(; pe = pr, slv = slv)))res_replicate_idx = optimise(MeanRisk(; r = TrackingRiskMeasure(; tr = ReturnsTracking(; w = index_returns)), obj = MinimumRisk(), opt = JuMPOptimiser(; pe = pr, slv = slv)))pretty_table(DataFrame("Asset" => rd.nx, "Replicate EW" => res_replicate_ew.w, "Replicate index" => res_replicate_idx.w); formatters = [resfmt], title = "Minimum tracking-error weights for two benchmarks")Minimum tracking-error weights for two benchmarks
┌────────┬──────────────┬─────────────────┐
│ Asset │ Replicate EW │ Replicate index │
│ String │ Float64 │ Float64 │
├────────┼──────────────┼─────────────────┤
│ AAPL │ 5.0 % │ 13.692 % │
│ AMD │ 5.0 % │ 6.689 % │
│ BAC │ 5.0 % │ 3.613 % │
│ BBY │ 5.0 % │ 1.767 % │
│ CVX │ 5.0 % │ 3.767 % │
│ GE │ 5.0 % │ 7.456 % │
│ HD │ 5.0 % │ 8.681 % │
│ JNJ │ 5.0 % │ 0.001 % │
│ JPM │ 5.0 % │ 7.861 % │
│ KO │ 5.0 % │ 3.52 % │
│ LLY │ 5.0 % │ 2.906 % │
│ MRK │ 5.0 % │ 3.55 % │
│ MSFT │ 5.0 % │ 18.432 % │
│ PEP │ 5.0 % │ 5.272 % │
│ PFE │ 5.0 % │ 2.654 % │
│ PG │ 5.0 % │ 0.75 % │
│ RRC │ 5.0 % │ 1.049 % │
│ UNH │ 5.0 % │ 4.056 % │
│ WMT │ 5.0 % │ 3.014 % │
│ XOM │ 5.0 % │ 1.271 % │
└────────┴──────────────┴─────────────────┘5. Tracking as a constraint
To beat a benchmark within a limit, you maximise the return with the tracking error at or below a budget. TrackingError, through tr, keeps the tracking error at or below err. err is in the units of the tracking error, which is a norm of the return differences, not a distance between weights. We maximise the ratio against the equal-weight benchmark with four values of err. A small err keeps the portfolio near the benchmark, and a large one gives the optimiser more room to raise the return.
err_vals = [0.0005, 0.001, 0.005, 0.02]track_res = [optimise(MeanRisk(; obj = MaximumRatio(; rf = rf), opt = JuMPOptimiser(; pe = pr, slv = slv, tr = TrackingError(; tr = WeightsTracking(; w = equal_weight), err = e)))) for e in err_vals]pretty_table(DataFrame("Tracking-error budget" => err_vals, "Drift from benchmark" => drift.(getproperty.(track_res, :w)), "Max weight" => maximum.(getproperty.(track_res, :w))); formatters = [resfmt], title = "Drift from the equal-weight benchmark at four tracking-error budgets")Drift from the equal-weight benchmark at four tracking-error budgets
┌───────────────────────┬──────────────────────┬────────────┐
│ Tracking-error budget │ Drift from benchmark │ Max weight │
│ Float64 │ Float64 │ Float64 │
├───────────────────────┼──────────────────────┼────────────┤
│ 0.0005 │ 15.77 % │ 7.81 % │
│ 0.001 │ 31.247 % │ 10.606 % │
│ 0.005 │ 106.974 % │ 26.905 % │
│ 0.02 │ 180.0 % │ 66.03 % │
└───────────────────────┴──────────────────────┴────────────┘The alg field of TrackingError sets the norm of the tracking error, which acts on the series of differences between the portfolio returns and the benchmark returns. The default is L2Norm. L1Norm uses the sum of the absolute differences, LpNorm a p-norm of them, and LInfNorm the largest difference in one period. Each alg except LInfNorm divides its norm by a factor of the number of observations, and err is in the units of that result. With L1Norm the result is the mean absolute difference. LInfNorm does not divide, so its err bounds the largest difference in one period.
6. Comparing the approaches
We plot four of the portfolios: the equal-weight replica, two tracking-error budgets and one turnover limit.
results = [res_replicate_ew, track_res[1], track_res[3], turnover_res[2]]labels = ["Replicate EW", "Track err 5e-4", "Track err 5e-3", "Turnover 0.02"]plot_stacked_bar_composition(results, rd; xticks = (1:length(labels), labels))
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