The source files can be found in examples/.
Observed factors, from raw prices to a cross-validated portfolio
A cross-sectional factor model estimates the return of each factor from the cross-section of asset returns. Some factor returns need no estimate, because you can read them off the market. The return of a currency is one: an asset priced in euros earns the euro's move against your base currency, and the exchange rate states that move. The price of oil is another. This page puts both kinds of factor into one CrossSectionalFactorPrior: factors the fit estimates, and observed factors whose returns come in as data.
We draw the data from a model we wrote, so every number below has a known answer.
- Build asset prices in three currencies, the exchange rates, the cash indices and an oil price, and convert them to returns.
- Fit a prior with a market factor, a size factor, two currency factors and an oil factor.
- Measure what the observed factors change in
mu, insigma, in the summary and in the attribution. - Run the prior inside a
NestedClustered, aSubsetResamplingand a walk-forward, where every sub-problem fits its own factors. - Compare the log and the simple return rules.
Reach for an observed factor when the return of the factor is a price series you hold and the exposure of each asset to it is either known, such as the currency an asset is priced in, or measurable, such as its sensitivity to oil. When you hold every factor return as a series and no asset trait drives the exposures, FactorPrior is the simpler tool.
using PortfolioOptimisers, StableRNGs, Statistics, LinearAlgebra, Dates, PrettyTables, DataFrames, Clarabel, TimeSeriesnumfmt = (v, i, j) -> begin return isa(v, AbstractFloat) ? round(v; sigdigits = 4) : vend;1. Prices in three currencies
The universe holds 60 assets. A third of them are priced in US dollars, the base currency, a third in euros and a third in yen. The local log return of each asset is the sum of four parts:
- a market return times the asset's beta,
- a size return times the asset's standardised log market capitalisation of the day before,
- the log return of oil times the asset's oil sensitivity,
- an idiosyncratic return with a daily volatility of 1.2 %.
The base-currency price of an asset is its local price times the exchange rate of its currency, in dollars per unit. Each currency also has a cash total-return index, which accumulates the cash rate of each day: 4 % a year for the dollar, 2 % for the euro and 0 % for the yen.
The function returns the four price tables as TimeArrays, a panel with the market capitalisation and the currency of each asset, and the true parameters.
function synthetic_world(; T = 750, N = 60, seed = 1_370_001) rng = StableRNG(seed) days = filter(d -> Dates.dayofweek(d) <= 5, Date(2018, 1, 1):Day(1):(Date(2018, 1, 1) + Day(2 * T + 10)))[1:(T + 1)] currencies = ["USD", "EUR", "JPY"] code = [mod(i - 1, 3) + 1 for i in 1:N] beta = 1.0 .+ 0.3 .* randn(rng, N) b_oil = 0.5 .* randn(rng, N) logcap = 22.0 .+ 1.2 .* randn(rng, 1, N) .+ 0.02 .* cumsum(randn(rng, T + 1, N); dims = 1) zsize = (logcap .- mean(logcap; dims = 2)) ./ std(logcap; dims = 2) market = 0.0003 .+ 0.009 .* randn(rng, T) size_ret = -0.0002 .+ 0.003 .* randn(rng, T) oil = 0.02 .* randn(rng, T) local_lr = transpose(beta) .* market .+ view(zsize, 1:T, :) .* size_ret .+ transpose(b_oil) .* oil .+ 0.012 .* randn(rng, T, N) local_px = 100 .* exp.(vcat(zeros(1, N), cumsum(local_lr; dims = 1))) fx_lr = hcat(0.006 .* randn(rng, T), 0.007 .* randn(rng, T)) fx = vcat([1.1 0.009], [1.1 0.009] .* exp.(cumsum(fx_lr; dims = 1))) rates = [0.04 0.02 0.0] ./ 252 cash = cumprod(vcat(ones(1, 3), 1 .+ rates .+ 0.00002 .* randn(rng, T, 3)); dims = 1) base_px = local_px .* hcat(ones(T + 1), fx)[:, code] oil_px = 70 .* exp.(vcat(0.0, cumsum(oil))) pnl = asset_panel([NumericPanelInput(; name = "market_cap", vals = exp.(logcap)), CategoricalPanelInput(; name = "currency", vals = repeat(permutedims(currencies[code]), T + 1), levels = currencies)]; amsk = trues(T + 1, N), emsk = trues(T + 1, N)) return (; X = TimeArray(days, base_px, ["A" * lpad(i, 2, '0') for i in 1:N]), fx = TimeArray(days, fx, ["EUR", "JPY"]), cash = TimeArray(days, cash[:, 2:3], ["EUR", "JPY"]), cash_usd = TimeArray(days, cash[:, 1:1], ["USD"]), oil = TimeArray(days, oil_px, ["OIL"]), pnl, truth = (; code, beta, b_oil, zsize, local_lr, fx_lr))endworld = synthetic_world()truth = world.truth;A currency factor reads the currency excess return: what a dollar investor earns by holding a deposit in the currency, funded in dollars. currency_excess_index builds a price level for it from the exchange rate and the two cash indices, so that its return is the exchange-rate return plus the euro or yen cash return, less the dollar cash return. The dollar takes no index, because its excess return over itself is zero.
The currency indices and the oil price belong to no asset, so they are not panel fields. They go to price_ingestion as its E argument, the exogenous series, which the conversion turns into returns beside the asset returns.
E = merge(currency_excess_index(world.fx, world.cash, world.cash_usd), world.oil)pr = price_ingestion(PriceIngestion(), world.X; E = E, pnl = world.pnl)rd = prices_to_returns(pr; ret_method = :log)ReturnsResult
nx ┼ 60-element Vector{String}
X ┼ 750×60 Matrix{Float64}
nf ┼ nothing
F ┼ nothing
nb ┼ nothing
B ┼ nothing
ne ┼ Vector{String}: ["EUR", "JPY", "OIL"]
E ┼ 750×3 Matrix{Float64}
ts ┼ 750-element Vector{Date}
iv ┼ nothing
ivpa ┼ nothing
pnl ┼ AssetPanel
│ pf ┼ 2-element Vector{AbstractPanelField}
│ │ NumericPanelField ⋯
│ │ CategoricalPanelField ⋯
│ amsk ┼ 750×60 AllTrueMask
│ emsk ┴ 750×60 AllTrueMask
We convert with the log rule. Section 5 shows why. The table compares rd.E with the log returns of the three price levels we passed in. Its largest difference is zero, so rd.E holds the returns of the indices and of oil, under the same rule as the asset returns.
pretty_table(DataFrame("Series" => rd.ne, "Largest abs(rd.E - log return of the level)" => vec(maximum(abs, rd.E - diff(log.(values(E)); dims = 1); dims = 1))); formatters = [numfmt], title = "The exogenous series after the conversion") The exogenous series after the conversion
┌────────┬─────────────────────────────────────────────┐
│ Series │ Largest abs(rd.E - log return of the level) │
│ String │ Float64 │
├────────┼─────────────────────────────────────────────┤
│ EUR │ 5.161e-17 │
│ JPY │ 8.396e-16 │
│ OIL │ 8.431e-16 │
└────────┴─────────────────────────────────────────────┘2. The prior
The factor list holds four members.
"market"is aConstantExposure, an intercept: every asset has an exposure of one, and the fit estimates its return."size"is a style factor onLogMarketCap. The fit estimates its return too."currency"is aCurrencyExposure. It reads thecurrencyfield of the panel and gives one factor per currency, with an exposure of one on the asset's own currency.base = "USD"gives the dollar no factor. Each factor return is the column ofrd.Ethat the currency names."oil"is anObservedExposure. The exposure of each asset is itsEWMacroSensitivityto the"OIL"column ofrd.E, a beta on oil after the market is removed. The factor return is the same"OIL"column.
The fit estimates the market and size returns from the returns that are left after it subtracts the currency and oil factors.
style(d) = CompositeExposure(; descriptors = [d], family = "style")oil_sensitivity = CompositeExposure(; descriptors = [EWMacroSensitivity(; series = "OIL")], outlier = nothing, scoring = nothing, family = "macro")factors = ["market" => ConstantExposure(), "size" => style(LogMarketCap()), "currency" => CurrencyExposure(; base = "USD"), "oil" => ObservedExposure(; xe = oil_sensitivity, series = "OIL")]pe = CrossSectionalFactorPrior(; factors = factors)CrossSectionalFactorPrior
factors ┼ Vector{Pair{String}}: Pair{String}["market" => ConstantExposure
family ┴ String: "market"
, "size" => CompositeExposure
descriptors ┼ 1-element Vector{PanelFieldLog}
│ PanelFieldLog ⋯
weights ┼ nothing
min_coverage ┼ Float64: 0.0
outlier ┼ CrossSectionalWinsoriser
│ low ┼ Float64: 0.01
│ high ┴ Float64: 0.99
scoring ┼ CrossSectionalStandardiser
│ min_group_size ┼ Int64: 8
│ atol ┴ Float64: 1.0e-12
group ┼ nothing
bw ┼ String: "benchmark_weights"
family ┴ String: "style"
, "currency" => CurrencyExposure
field ┼ String: "currency"
family ┼ String: "currency"
base ┴ String: "USD"
, "oil" => ObservedExposure
xe ┼ CompositeExposure
│ descriptors ┼ 1-element Vector{EWMacroSensitivity}
│ │ EWMacroSensitivity ⋯
│ weights ┼ nothing
│ min_coverage ┼ Float64: 0.0
│ outlier ┼ nothing
│ scoring ┼ nothing
│ group ┼ nothing
│ bw ┼ String: "benchmark_weights"
│ family ┴ String: "macro"
series ┼ String: "OIL"
family ┴ String: "macro"
]
neutralise ┼ nothing
families ┼ nothing
unseen ┼ ZeroUnseenMember()
cre ┼ CrossSectionalLinearRegression
│ alg ┼ PseudoInverseFallback()
│ intercept ┼ Bool: false
│ ex ┴ Transducers.ThreadedEx{@NamedTuple{}}: Transducers.ThreadedEx()
wa ┼ MarketCapWeights
│ p ┴ Float64: 0.5
pe ┼ EmpiricalPrior
│ ce ┼ RegimeAdjustedExpWeightedCovariance
│ │ decay ┼ Float64: 0.9828205985452511
│ │ cor_decay ┼ nothing
│ │ min_obs ┼ Int64: 40
│ │ hac_lags ┼ nothing
│ │ hac_floor ┼ NoHacFloor()
│ │ hac_vol_before ┼ VolatilityBeforeUpdate()
│ │ cor_vol ┼ BiasCorrectedVolatility()
│ │ regime_method ┼ FirstMomentRegimeAdjusted
│ │ │ x ┴ Float64: 0.7978845608028654
│ │ regime_decay ┼ Float64: 0.9659363289248455
│ │ regime_min_obs ┼ Int64: 20
│ │ regime_target ┼ PortfolioTarget
│ │ │ w ┼ nothing
│ │ │ indefinite ┴ RefuseIndefiniteWarmUp()
│ │ regime_lohi_mult ┼ Tuple{Float64, Float64}: (0.7, 1.6)
│ │ min_val ┼ Float64: 1.0e-12
│ │ centring ┼ PreCentred()
│ │ debias ┼ ExactDebias()
│ │ cache ┴ nothing
│ me ┼ ExpWeightedExpectedReturns
│ │ decay ┼ Float64: 0.9828205985452511
│ │ min_obs ┼ Int64: 40
│ │ cache ┴ nothing
│ horizon ┼ nothing
│ fill_limit ┼ nothing
│ x_src ┴ nothing
ve ┼ RegimeAdjustedExpWeightedVariance
│ decay ┼ Float64: 0.9828205985452511
│ min_obs ┼ Int64: 40
│ hac_lags ┼ nothing
│ hac_floor ┼ NoHacFloor()
│ regime_method ┼ FirstMomentRegimeAdjusted
│ │ x ┴ Float64: 0.7978845608028654
│ regime_decay ┼ Float64: 0.9659363289248455
│ regime_min_obs ┼ Int64: 20
│ regime_lohi_mult ┼ Tuple{Float64, Float64}: (0.7, 1.6)
│ min_val ┼ Float64: 1.0e-12
│ centring ┼ PreCentred()
│ debias ┼ ExactDebias()
│ cache ┴ nothing
ce ┼ ExpWeightedCovariance
│ decay ┼ Float64: 0.9828205985452511
│ min_obs ┼ Int64: 40
│ centring ┼ PreCentred()
│ cache ┴ nothing
pdm ┼ Posdef
│ alg ┼ ClippedNearestCorrelation
│ │ tau ┼ Float64: 1.0e-13
│ │ higham ┼ Bool: false
│ │ iter ┴ Int64: 100
│ kwargs ┴ @NamedTuple{}: NamedTuple()
f_mp ┼ MatrixProcessing
│ pdm ┼ Posdef
│ │ alg ┼ UnionAll: NearestCorrelationMatrix.Newton
│ │ kwargs ┴ @NamedTuple{}: NamedTuple()
│ dn ┼ nothing
│ dt ┼ nothing
│ alg ┼ nothing
│ order ┴ NTuple{4, Symbol}: (:pdm, :dn, :dt, :alg)
mp ┼ MatrixProcessing
│ pdm ┼ Posdef
│ │ alg ┼ UnionAll: NearestCorrelationMatrix.Newton
│ │ kwargs ┴ @NamedTuple{}: NamedTuple()
│ dn ┼ nothing
│ dt ┼ nothing
│ alg ┼ nothing
│ order ┴ NTuple{4, Symbol}: (:pdm, :dn, :dt, :alg)
srep ┼ NoSystematicRepair()
th ┼ Float64: 0.0
bp ┼ Float64: 1.0
mcap ┼ String: "market_cap"
bw ┼ String: "benchmark_weights"
lag ┼ Int64: 1
minra ┼ nothing
rfe ┼ nothing
lambda ┼ PrecisionBlend
│ err ┴ CurrentForecastError()
c ┼ Float64: 1.0
ofit ┼ ScoreNeutralisation()
l1mask ┼ MaskedLeverageOne()
lx ┼ nothing
mtx_sqrt ┼ EigenFallbackSquareRoot()
ex ┼ Transducers.ThreadedEx{@NamedTuple{}}: Transducers.ThreadedEx()
choice ┼ BatchChoice()
carry ┼ FoldOrRefit()
x_src ┴ nothing
cross_sectional_factor_axis states the factor axis before any fit. You can write a constraint against these names, or check that a currency you expect has a factor.
axis = cross_sectional_factor_axis(pe, rd)pretty_table(DataFrame("Factor" => axis.nf, "Family" => axis.fam); title = "The factor axis before the fit")The factor axis before the fit
┌──────────────┬──────────┐
│ Factor │ Family │
│ String │ String │
├──────────────┼──────────┤
│ market │ market │
│ size │ style │
│ currency=EUR │ currency │
│ currency=JPY │ currency │
│ oil │ macro │
└──────────────┴──────────┘Now the fit. rr is the fitted factor model. rr.nf and rr.fam repeat the axis, and rr.fx holds the returns of the observed factors over the observations of the fit. The estimated factors come first on the axis, and the observed ones after them.
prior_full = prior(pe, rd)rr = prior_full.rrn_obs = size(rr.fx, 2)kind = [k > length(rr.nf) - n_obs ? "observed" : "estimated" for k in eachindex(rr.nf)]fit_rows = (size(rd.X, 1) - size(rr.fx, 1) + 1):size(rd.X, 1)61:750The prior fits the last 690 of the 750 observations. The oil sensitivity needs a warm-up before its first value, and the fit starts after it. The table prints, for each factor, how closely the last row of fitted exposures follows the true exposure, and the largest difference between rr.fx and the rows of rd.E the fit covers. The currency exposures are the one-hot currency of each asset, so we compare them with that.
true_exposure = hcat(ones(60), truth.zsize[end, :], truth.code .== 2, truth.code .== 3, truth.b_oil)pretty_table(DataFrame("Factor" => rr.nf, "Family" => rr.fam, "Kind" => kind, "corr(fitted, true) exposure" => [k == 1 ? NaN : cor(rr.M[:, k], true_exposure[:, k]) for k in eachindex(rr.nf)], "Largest abs(rr.fx - rd.E)" => [if k > length(rr.nf) - n_obs maximum(abs, rr.fx[:, k - length(rr.nf) + n_obs] - rd.E[fit_rows, k - length(rr.nf) + n_obs]) else NaN end for k in eachindex(rr.nf)]); formatters = [numfmt], title = "What the fit recovered") What the fit recovered
┌──────────────┬──────────┬───────────┬─────────────────────────────┬───────────────────────────┐
│ Factor │ Family │ Kind │ corr(fitted, true) exposure │ Largest abs(rr.fx - rd.E) │
│ String │ String │ String │ Float64 │ Float64 │
├──────────────┼──────────┼───────────┼─────────────────────────────┼───────────────────────────┤
│ market │ market │ estimated │ NaN │ NaN │
│ size │ style │ estimated │ 0.9975 │ NaN │
│ currency=EUR │ currency │ observed │ 1.0 │ 0.0 │
│ currency=JPY │ currency │ observed │ 1.0 │ 0.0 │
│ oil │ macro │ observed │ 0.9958 │ 0.0 │
└──────────────┴──────────┴───────────┴─────────────────────────────┴───────────────────────────┘The market exposure is one for every asset, so it has no correlation to print. The currency exposures are exactly the currency of each asset, the size and oil exposures follow the truth with a correlation above 0.99, and the observed factor returns are the columns of rd.E with no change.
3. What the observed factors change
The covariance of the prior is M * fpr.sigma * M' plus the idiosyncratic variance of each asset, where M holds the exposures of the last row and fpr.sigma is the factor covariance. Split the factor covariance into its estimated block, its observed block and the cross terms between them, and the variance of each asset splits into four parts. The first table checks that the four parts add up to diag(sigma). The second averages the share of each part over the assets of each currency.
est = findall(==("estimated"), kind)obs = findall(==("observed"), kind)M = rr.MS = prior_full.fpr.sigmapart(a, b) = vec(sum((M[:, a] * S[a, b]) .* M[:, b]; dims = 2))var_est = part(est, est)var_obs = part(obs, obs)var_cross = 2 .* part(est, obs)var_idio = rr.vs[end, :]var_total = diag(prior_full.sigma)pretty_table(DataFrame("Largest abs(sum of the parts - diag(sigma))" => maximum(abs, var_est + var_obs + var_cross + var_idio - var_total)); formatters = [numfmt], title = "The four parts of the variance of each asset")currency_names = ["USD", "EUR", "JPY"]share(v) = [mean((v ./ var_total)[truth.code .== c]) for c in 1:3]pretty_table(DataFrame("Currency" => currency_names, "Estimated factors" => share(var_est), "Observed factors" => share(var_obs), "Cross terms" => share(var_cross), "Idiosyncratic" => share(var_idio)); formatters = [numfmt], title = "Mean share of the variance of an asset") The four parts of the variance of each asset
┌─────────────────────────────────────────────┐
│ Largest abs(sum of the parts - diag(sigma)) │
│ Float64 │
├─────────────────────────────────────────────┤
│ 2.168e-19 │
└─────────────────────────────────────────────┘
Mean share of the variance of an asset
┌──────────┬───────────────────┬──────────────────┬─────────────┬───────────────┐
│ Currency │ Estimated factors │ Observed factors │ Cross terms │ Idiosyncratic │
│ String │ Float64 │ Float64 │ Float64 │ Float64 │
├──────────┼───────────────────┼──────────────────┼─────────────┼───────────────┤
│ USD │ 0.2897 │ 0.1543 │ 0.001338 │ 0.5547 │
│ EUR │ 0.2657 │ 0.24 │ -0.02047 │ 0.5148 │
│ JPY │ 0.2486 │ 0.332 │ -0.06358 │ 0.4829 │
└──────────┴───────────────────┴──────────────────┴─────────────┴───────────────┘A dollar asset has no currency exposure, so its observed share is the oil part alone. A yen asset carries the most volatile exchange rate, and its observed share is the largest.
mu splits the same way, with no cross term: it is M * fpr.mu. The table prints the mean of each part over the assets of each currency, annualised.
mu_est = M[:, est] * prior_full.fpr.mu[est]mu_obs = M[:, obs] * prior_full.fpr.mu[obs]group_mean(v) = [252 * mean(v[truth.code .== c]) for c in 1:3]pretty_table(DataFrame("Currency" => currency_names, "From estimated factors" => group_mean(mu_est), "From observed factors" => group_mean(mu_obs), "mu" => group_mean(prior_full.mu), "Largest abs(sum - mu)" => maximum(abs, mu_est + mu_obs - prior_full.mu)); formatters = [numfmt], title = "Annualised mean of mu by currency") Annualised mean of mu by currency
┌──────────┬────────────────────────┬───────────────────────┬─────────┬───────────────────────┐
│ Currency │ From estimated factors │ From observed factors │ mu │ Largest abs(sum - mu) │
│ String │ Float64 │ Float64 │ Float64 │ Float64 │
├──────────┼────────────────────────┼───────────────────────┼─────────┼───────────────────────┤
│ USD │ 0.2938 │ 0.007177 │ 0.301 │ 4.337e-19 │
│ EUR │ 0.2665 │ -0.2531 │ 0.01339 │ 4.337e-19 │
│ JPY │ 0.2706 │ -0.01184 │ 0.2587 │ 4.337e-19 │
└──────────┴────────────────────────┴───────────────────────┴─────────┴───────────────────────┘The same prior without the observed factors
We fit the prior again with the market and size factors alone. The currency and oil moves then have nowhere to go but the idiosyncratic return of each asset. The data were drawn with an idiosyncratic volatility of 1.2 % a day, 19.05 % a year, for every asset. The table prints the mean annualised idiosyncratic volatility of the last row under each prior.
pe_estimated = CrossSectionalFactorPrior(; factors = factors[1:2])prior_estimated = prior(pe_estimated, rd)idio_vol(p) = [mean(sqrt.(252 .* p.rr.vs[end, truth.code .== c])) for c in 1:3]pretty_table(DataFrame("Currency" => currency_names, "With observed factors" => idio_vol(prior_full), "Without observed factors" => idio_vol(prior_estimated), "True" => fill(0.012 * sqrt(252), 3)); formatters = [numfmt], title = "Mean annualised idiosyncratic volatility") Mean annualised idiosyncratic volatility
┌──────────┬───────────────────────┬──────────────────────────┬─────────┐
│ Currency │ With observed factors │ Without observed factors │ True │
│ String │ Float64 │ Float64 │ Float64 │
├──────────┼───────────────────────┼──────────────────────────┼─────────┤
│ USD │ 0.1944 │ 0.2362 │ 0.1905 │
│ EUR │ 0.1995 │ 0.2521 │ 0.1905 │
│ JPY │ 0.1942 │ 0.2755 │ 0.1905 │
└──────────┴───────────────────────┴──────────────────────────┴─────────┘With the observed factors the idiosyncratic volatility is close to the true value for every currency. Without them it is too high by the oil move for dollar assets, and by the oil move and the exchange rate for euro and yen assets.
The summary
factor_model_summary reports the return statistics of each factor and the regression statistics of the fit. An observed factor has return statistics, read off its observed series. The fit did not estimate its return, so it has no t-statistic, and its regression columns are NaN.
summary_full = factor_model_summary(rr; ppy = 252)pretty_table(DataFrame("Factor" => rr.nf, "Kind" => kind, "Annual return" => summary_full.ann_return, "Annual volatility" => summary_full.ann_volatility, "Mean abs(t)" => summary_full.mean_abs_t, "Share of abs(t) > 2" => summary_full.t_rate); formatters = [numfmt], title = "Factor model summary") Factor model summary
┌──────────────┬───────────┬───────────────┬───────────────────┬─────────────┬─────────────────────┐
│ Factor │ Kind │ Annual return │ Annual volatility │ Mean abs(t) │ Share of abs(t) > 2 │
│ String │ String │ Float64 │ Float64 │ Float64 │ Float64 │
├──────────────┼───────────┼───────────────┼───────────────────┼─────────────┼─────────────────────┤
│ market │ estimated │ -0.07586 │ 0.1683 │ 4.872 │ 0.7518 │
│ size │ estimated │ -0.11 │ 0.08218 │ 1.841 │ 0.3774 │
│ currency=EUR │ observed │ -0.0002504 │ 0.09734 │ NaN │ NaN │
│ currency=JPY │ observed │ -0.01385 │ 0.1067 │ NaN │ NaN │
│ oil │ observed │ 0.0958 │ 0.33 │ NaN │ NaN │
└──────────────┴───────────┴───────────────┴───────────────────┴─────────────┴─────────────────────┘The attribution
factor_attribution splits the risk and the return of a book over the factors. We take an equal-weighted book. The predicted attribution reads the moments of the prior. The realised one reads the returns, and under se = true it also states the standard error of each factor's return contribution. That error comes from the estimate of the factor return, so an observed factor, which the fit reports as its observed return, has none, and its error is NaN.
The standard error reads the idiosyncratic variance of every observation, and the first rows of the fit are the warm-up of that variance, where it is unknown. A realised call over every row of the fit therefore gives NaN for every factor. The rolling form takes a window and a step, and its windows end at window, window + step and so on. The attribution pairs the exposures of one observation with the returns of the next, so it reads rr.lag rows fewer than the fit. We count the rows of the warm-up, and choose the window and the step so that the last window starts after the warm-up and ends at the last row.
w_eq = fill(1 / 60, 60)predicted = factor_attribution(w_eq, prior_full; ppy = 252)warmup = count(r -> any(isnan, r), eachrow(rr.vs))aligned = size(rr.fx, 1) - rr.lagrealised = last(factor_attribution(w_eq, prior_full, rd.X, aligned - warmup; step = warmup, se = true, ppy = 252))window_rows = fit_rows[(warmup + rr.lag + 1):end]101:750The table prints, for each factor, the predicted share of the variance of the book, the realised mean return of the factor and its standard error. The last column is the annualised mean of the observed series over the rows of the window, the return that the attribution reports for an observed factor.
pretty_table(DataFrame("Factor" => rr.nf, "Kind" => kind, "Predicted % variance" => predicted.fbd.pct_var, "Realised mu contribution" => realised.fbd.mu_contrib, "Standard error" => realised.fbd.mu_se, "Realised factor mean" => realised.fbd.mu, "252 * mean of the series" => [if k in obs 252 * mean(rd.E[window_rows, k - first(obs) + 1]) else NaN end for k in eachindex(rr.nf)]); formatters = [numfmt], title = "Attribution of the equal-weighted book") Attribution of the equal-weighted book
┌──────────────┬───────────┬──────────────────────┬──────────────────────────┬────────────────┬──────────────────────┬──────────────────────────┐
│ Factor │ Kind │ Predicted % variance │ Realised mu contribution │ Standard error │ Realised factor mean │ 252 * mean of the series │
│ String │ String │ Float64 │ Float64 │ Float64 │ Float64 │ Float64 │
├──────────────┼───────────┼──────────────────────┼──────────────────────────┼────────────────┼──────────────────────┼──────────────────────────┤
│ market │ estimated │ 0.9652 │ -0.1035 │ 0.02148 │ -0.1035 │ NaN │
│ size │ estimated │ -0.01406 │ 0.07857 │ 0.0169 │ -0.1155 │ NaN │
│ currency=EUR │ observed │ 0.02076 │ 0.001372 │ NaN │ 0.004117 │ 0.004117 │
│ currency=JPY │ observed │ -0.01002 │ -0.01004 │ NaN │ -0.03013 │ -0.03013 │
│ oil │ observed │ 0.0003551 │ 0.001086 │ NaN │ 0.01798 │ 0.01798 │
└──────────────┴───────────┴──────────────────────┴──────────────────────────┴────────────────┴──────────────────────┴──────────────────────────┘4. Where a single fit cannot reach
A meta-optimiser splits the universe and solves a sub-problem on each part. Each sub-problem fits the prior again on its own assets, so it estimates its own market and size returns. The observed factor returns are the same columns of rd.E in every sub-problem.
A cluster holds fewer assets than the universe, and the prior refuses an observation with fewer than minra assets, 30 by default. We lower it, and ask for three clusters, so that the smallest cluster still has enough assets to estimate two factors.
#! A cluster smaller than `minra` stops the whole optimisation with an `ArgumentError`.pe_small = CrossSectionalFactorPrior(; factors = factors, minra = 10)solver = Solver(; name = :clarabel, solver = Clarabel.Optimizer, check_sol = (; allow_local = true, allow_almost = true), settings = Dict("verbose" => false))inner = MeanRisk(; obj = MinimumRisk(), opt = JuMPOptimiser(; pe = pe_small, slv = solver, wb = WeightBounds(; lb = 0.0, ub = 0.2)))clustered = optimise(NestedClustered(; pe = pe_small, cle = ClustersEstimator(; onc = OptimalNumberClusters(; alg = 3)), opti = inner, opto = EqualWeighted()), rd)clusters = [findall(==(k), assignments(clustered.clr)) for k in 1:(clustered.clr.k)]3-element Vector{Vector{Int64}}:
[1, 5, 8, 10, 12, 13, 16, 19, 21, 23 … 43, 44, 48, 50, 52, 54, 55, 57, 58, 60]
[2, 3, 4, 6, 7, 14, 15, 18, 22, 25, 26, 31, 33, 40, 45, 47, 49, 51, 56]
[9, 11, 17, 20, 35, 37, 39, 41, 46, 53, 59]The table prints, for each cluster, how many assets of each currency it holds, the largest difference between its observed factor returns and rd.E, and the largest difference between its market factor return and the one of the full fit. A cluster with no yen asset still carries the yen factor, with an exposure of zero for every asset.
pretty_table(DataFrame("Cluster" => eachindex(clusters), "USD" => [count(==(1), truth.code[c]) for c in clusters], "EUR" => [count(==(2), truth.code[c]) for c in clusters], "JPY" => [count(==(3), truth.code[c]) for c in clusters], "Largest abs(rr.fx - rd.E)" => [maximum(abs, r.pr.rr.fx - rd.E[fit_rows, :]) for r in clustered.resi], "Largest abs(market return - full fit)" => [maximum(abs, r.pr.fpr.X[:, 1] - prior_full.fpr.X[:, 1]) for r in clustered.resi]); formatters = [numfmt], title = "Each cluster fits its own market return") Each cluster fits its own market return
┌─────────┬───────┬───────┬───────┬───────────────────────────┬───────────────────────────────────────┐
│ Cluster │ USD │ EUR │ JPY │ Largest abs(rr.fx - rd.E) │ Largest abs(market return - full fit) │
│ Int64 │ Int64 │ Int64 │ Int64 │ Float64 │ Float64 │
├─────────┼───────┼───────┼───────┼───────────────────────────┼───────────────────────────────────────┤
│ 1 │ 11 │ 8 │ 11 │ 0.0 │ 0.02774 │
│ 2 │ 7 │ 5 │ 7 │ 0.0 │ 0.01543 │
│ 3 │ 2 │ 7 │ 2 │ 0.0 │ 0.05187 │
└─────────┴───────┴───────┴───────┴───────────────────────────┴───────────────────────────────────────┘A SubsetResampling solves the same problem on random subsets of the assets and averages the weights. Each subset fits its own factors too.
resampled = optimise(SubsetResampling(; subset_size = 0.9, n_subsets = 8, rng = StableRNG(7), pe = pe_small, opt = inner), rd)pretty_table(DataFrame("Subset" => eachindex(resampled.ress), "Assets" => [length(r.w) for r in resampled.ress], "Largest abs(rr.fx - rd.E)" => [maximum(abs, r.pr.rr.fx - rd.E[fit_rows, :]) for r in resampled.ress], "Largest abs(market return - full fit)" => [maximum(abs, r.pr.fpr.X[:, 1] - prior_full.fpr.X[:, 1]) for r in resampled.ress]); formatters = [numfmt], title = "Each subset fits its own market return") Each subset fits its own market return
┌────────┬────────┬───────────────────────────┬───────────────────────────────────────┐
│ Subset │ Assets │ Largest abs(rr.fx - rd.E) │ Largest abs(market return - full fit) │
│ Int64 │ Int64 │ Float64 │ Float64 │
├────────┼────────┼───────────────────────────┼───────────────────────────────────────┤
│ 1 │ 54 │ 0.0 │ 0.002602 │
│ 2 │ 54 │ 0.0 │ 0.003833 │
│ 3 │ 54 │ 0.0 │ 0.003573 │
│ 4 │ 54 │ 0.0 │ 0.002339 │
│ 5 │ 54 │ 0.0 │ 0.003568 │
│ 6 │ 54 │ 0.0 │ 0.005898 │
│ 7 │ 54 │ 0.0 │ 0.002024 │
│ 8 │ 54 │ 0.0 │ 0.002164 │
└────────┴────────┴───────────────────────────┴───────────────────────────────────────┘A walk-forward fits the prior on each training window and holds the book over the next one. A fold takes its rows of rd, and rd.E with them, so the observed factors of a fold are the series of its own training rows. The table prints, for each fold, the rows it trained on, the rows its prior fitted after the warm-up, and the largest difference between its observed factor returns and the rows of rd.E it fitted.
walk = IndexWalkForward(250, 125)folds = cross_val_predict(inner, rd, walk)train_start = [125 * (k - 1) + 1 for k in eachindex(folds.pred)]fold_fx = [p.res.pr.rr.fx for p in folds.pred]pretty_table(DataFrame("Fold" => eachindex(folds.pred), "Training rows" => ["$(s) to $(s + 249)" for s in train_start], "Fitted rows" => [size(f, 1) for f in fold_fx], "Largest abs(rr.fx - rd.E)" => [maximum(abs, f - rd.E[(s + 250 - size(f, 1)):(s + 249), :]) for (f, s) in zip(fold_fx, train_start)]); formatters = [numfmt], title = "Each fold reads the exogenous series of its own rows") Each fold reads the exogenous series of its own rows
┌───────┬───────────────┬─────────────┬───────────────────────────┐
│ Fold │ Training rows │ Fitted rows │ Largest abs(rr.fx - rd.E) │
│ Int64 │ String │ Int64 │ Float64 │
├───────┼───────────────┼─────────────┼───────────────────────────┤
│ 1 │ 1 to 250 │ 190 │ 0.0 │
│ 2 │ 126 to 375 │ 190 │ 0.0 │
│ 3 │ 251 to 500 │ 190 │ 0.0 │
│ 4 │ 376 to 625 │ 190 │ 0.0 │
└───────┴───────────────┴─────────────┴───────────────────────────┘The predicted returns of the walk-forward carry the exogenous series of the test rows, so a step after the walk-forward, such as an attribution of the held book, reads the series of the observations it describes. The cell prints whether folds.mrd.E equals the rows of rd.E that the folds tested.
folds.mrd.E == rd.E[251:end, :]true5. The two return rules
An asset priced in euros has, in dollars, the return (1 + local) * (1 + fx) - 1. The prior subtracts the currency excess return of the asset's currency from that, and the regression then reads what is left as the local return. Under the log rule the base-currency return is the sum of the local log return and the log return of the exchange rate, so the subtraction leaves the local log return plus a term that is the same for every asset of one currency: the difference of the two cash rates. The currency factor absorbs that term. Under the simple rule, the default of prices_to_returns, the subtraction also leaves the cross term local * fx, which differs from asset to asset and no factor can absorb.
We repeat the subtraction by hand on both conversions, and compare what is left with the true local return of each asset.
rd_simple = prices_to_returns(pr; ret_method = :simple)foreign = findall(>(1), truth.code)function local_returns(r) L = copy(r.X) for i in foreign L[:, i] .-= r.E[:, truth.code[i] - 1] end return Lendlocal_simple = expm1.(truth.local_lr)left_log = local_returns(rd) - truth.local_lrleft_simple = local_returns(rd_simple) - local_simplefunction spread(left) return maximum(maximum(std(left[t, truth.code .== c]) for c in 2:3) for t in axes(left, 1))endcross_term = hcat([local_simple[:, i] .* expm1.(truth.fx_lr[:, truth.code[i] - 1]) for i in foreign]...)750×40 Matrix{Float64}:
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-4.94047e-6 3.96025e-6 -5.96452e-6 -3.64917e-6 -4.7594e-7 1.70204e-5 2.73101e-6 -1.74005e-7 4.44364e-6 -4.96725e-6 4.7187e-6 1.04546e-5 2.32652e-6 -9.97725e-6 -4.46667e-6 -1.02134e-5 -1.40022e-6 -2.2943e-5 -1.74371e-6 -4.80286e-6 -4.28574e-6 -1.347e-6 5.45862e-6 -9.78473e-6 -2.46637e-6 1.03399e-5 1.82971e-6 -9.70933e-6 -7.33089e-6 7.1268e-7 -5.44158e-6 -2.54921e-5 1.00361e-6 -4.42429e-6 1.945e-6 -1.65161e-5 2.25578e-6 -1.68699e-5 5.07254e-6 -4.3279e-6
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6.37424e-5 0.000107198 7.96032e-6 7.54271e-5 3.85775e-5 5.67768e-5 2.57415e-6 4.7862e-5 4.27394e-5 2.1124e-5 0.000131385 3.83397e-5 0.000203894 -4.70941e-5 2.19969e-5 4.66896e-6 3.16553e-5 7.03782e-5 0.000164897 1.79643e-5 7.18555e-5 2.39763e-5 0.000218166 8.8766e-6 1.3075e-5 7.58768e-5 0.000148186 -1.16408e-5 5.90511e-6 4.63852e-5 9.82215e-5 -4.82847e-5 -4.72487e-6 3.36463e-5 5.35653e-5 5.64262e-5 9.38608e-5 2.71822e-6 3.09829e-5 1.50903e-5
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0.000141507 -1.86595e-5 6.6324e-5 -1.1322e-5 -0.000101703 1.55554e-5 -8.8652e-6 7.58799e-6 5.02627e-5 -5.47241e-6 4.89861e-5 1.12876e-5 4.97136e-5 2.37378e-5 4.80603e-5 2.28631e-5 0.000149254 1.01467e-5 3.63966e-6 1.78938e-5 1.98743e-5 1.45819e-6 0.000129547 -1.90412e-5 0.000116774 3.11184e-5 -8.5389e-6 5.41898e-6 0.000371378 -8.58518e-6 0.000140393 6.66254e-6 -7.96024e-5 -6.78164e-6 5.02572e-5 -1.09202e-5 6.37776e-5 3.72836e-5 1.25446e-5 -3.29707e-6
5.81978e-5 -5.88051e-5 -5.74635e-5 -5.56028e-5 -4.77488e-5 -7.11443e-5 0.000231727 3.25849e-5 -5.50131e-5 3.48026e-6 0.000102153 -3.77799e-5 0.000112351 0.000106508 6.44943e-5 -5.14458e-5 4.30337e-5 2.03999e-5 -0.000120608 -4.75026e-6 7.68367e-5 3.04751e-5 0.000124687 -9.47032e-7 -0.000206667 -1.96661e-5 0.000328385 -3.01791e-5 6.64206e-5 5.35507e-6 0.000289502 5.47377e-5 -5.76059e-5 -7.29773e-5 0.000195973 1.8835e-6 8.2608e-5 2.24676e-6 0.000267329 2.89957e-6
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6.42354e-5 -1.8102e-6 2.13544e-5 -6.05757e-6 -0.00018416 -9.57549e-6 0.000105273 9.0891e-6 -0.000113639 9.92905e-6 9.79086e-5 -2.08921e-5 1.26417e-5 3.07731e-5 -0.000120879 2.59517e-6 6.06856e-5 1.56904e-5 -4.97446e-5 -1.31972e-6 -0.000186271 -1.06745e-5 3.88801e-5 2.88272e-5 -3.72128e-5 -4.26025e-6 -3.10233e-5 4.73947e-6 3.73078e-5 -1.13505e-5 0.000205968 4.30089e-5 -3.64921e-6 -7.71406e-6 0.000141897 2.27543e-5 0.000103815 2.4183e-5 6.66465e-5 7.40215e-6
-2.81103e-6 7.77099e-6 3.27605e-5 -2.6266e-7 5.51057e-5 -6.30686e-8 -2.12437e-5 1.09454e-5 -1.065e-6 1.03033e-5 6.06549e-5 3.22713e-6 2.66532e-6 2.7332e-6 2.3447e-6 4.64993e-6 -1.10549e-5 1.18716e-5 3.69792e-5 -1.93114e-6 6.45243e-6 5.54135e-6 8.6823e-6 6.72297e-6 -5.4588e-5 2.78946e-6 -1.2059e-5 -5.89535e-6 1.1837e-5 1.26869e-5 8.56724e-6 4.04879e-6 6.7022e-5 6.65777e-7 8.59787e-5 -1.44359e-6 -4.47668e-6 5.77353e-6 -3.75337e-6 3.41023e-6
-9.11585e-5 0.000147039 -0.000121982 -0.000103442 9.14995e-6 0.000122789 1.49538e-5 -0.000198034 6.25826e-6 6.07094e-5 0.000134493 -3.54085e-5 1.24918e-5 -0.000153608 -9.07612e-5 -1.72134e-5 -4.37492e-5 -0.000198553 -9.69447e-5 -0.000138819 -3.46124e-5 -8.74604e-5 3.75706e-5 -0.000288126 -0.000107521 0.000183379 0.000107989 -0.000299606 -0.000135186 -5.11904e-5 -3.91452e-5 -0.00042502 -8.72941e-5 3.37053e-5 0.000157531 -0.000254937 -9.27574e-5 -0.000234874 5.89864e-5 -0.00013156
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0.000198233 8.50238e-6 -1.34008e-5 -1.73671e-5 0.000119612 -1.01667e-5 0.000248784 9.61076e-7 0.000111721 -1.80745e-5 3.99024e-5 -1.78372e-6 0.000168157 5.39858e-7 -5.8995e-5 -7.39547e-6 7.74289e-5 7.10213e-7 -0.000103228 -1.48988e-5 -4.48335e-5 -9.28745e-6 0.000153961 1.20819e-6 -6.38108e-5 -1.45651e-5 9.39877e-5 3.03648e-6 4.30147e-5 -6.34321e-6 0.000130755 -2.93171e-6 2.86497e-5 -7.07692e-7 3.96066e-5 -5.01049e-6 0.000136114 -4.80923e-6 0.000108228 -1.66381e-5
0.000166534 7.7139e-5 -5.18451e-5 5.06813e-5 -1.42107e-5 4.08515e-5 7.95323e-5 7.20739e-5 -6.99069e-5 2.8446e-5 -0.00017146 0.000101227 -8.47754e-5 -9.45126e-5 -5.05829e-5 -2.01162e-5 -0.000136622 -2.62083e-5 -0.000101993 1.68322e-5 -7.45102e-5 1.15533e-5 -0.000147512 7.57378e-6 7.37719e-5 3.4583e-5 -5.26638e-5 -4.40084e-5 2.644e-5 0.000103179 -0.00018515 -2.70732e-5 -1.81019e-5 2.83602e-5 -0.000103412 4.22716e-5 -2.14541e-5 -3.85226e-5 -0.000162278 2.64426e-5
-6.61528e-5 0.000113011 -1.2635e-5 9.93142e-5 1.99512e-5 9.00454e-5 6.20703e-5 -0.000221352 -5.30115e-5 -0.000168744 4.48965e-5 -1.13073e-5 -8.77496e-6 -8.43055e-5 9.62492e-5 0.00010736 -4.88144e-5 -6.19757e-5 2.27158e-5 -9.82892e-5 3.54756e-5 -1.78936e-5 4.86766e-5 8.05953e-5 -2.95064e-5 -5.22222e-5 8.64144e-5 0.000216793 -2.31997e-5 0.000226674 6.11928e-5 -0.000130449 -6.54417e-5 0.000171121 -3.87652e-5 -7.30255e-5 3.35744e-6 1.0236e-5 -1.28558e-5 2.65587e-5
-2.99579e-5 -0.000127716 2.54275e-5 -6.41862e-5 -2.68777e-5 -4.40039e-5 9.26064e-5 -5.987e-5 1.55852e-5 -3.39151e-5 5.88647e-5 -6.83058e-5 8.32422e-5 0.000101047 5.7402e-5 5.49126e-6 1.57856e-5 4.20346e-6 5.60108e-5 -4.91794e-5 4.53751e-6 -5.45172e-5 3.90523e-5 2.51577e-6 7.23107e-6 -8.48121e-5 9.01036e-5 2.23846e-5 3.03179e-5 7.17014e-6 4.55852e-5 0.000113915 -1.19772e-5 -5.31876e-5 5.81564e-5 -2.61515e-5 3.937e-5 7.82467e-5 5.27827e-5 -4.37565e-5
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4.45849e-5 5.89977e-8 8.555e-5 6.82054e-7 -5.0447e-6 4.46421e-6 -1.02369e-5 -2.8535e-6 0.000136279 5.89061e-7 7.60449e-5 -2.84179e-6 -1.56918e-5 -4.70686e-6 7.08236e-5 5.7343e-7 -1.46373e-5 -3.71061e-6 4.57928e-5 -2.23064e-6 0.00010361 -2.08931e-6 -4.47076e-5 -6.27981e-6 4.48353e-5 4.41144e-6 -8.91981e-6 -2.31233e-6 2.7122e-5 -5.48254e-6 5.30575e-5 -1.9892e-6 6.38304e-5 9.68886e-7 5.45155e-6 -3.40313e-6 -5.71564e-5 -7.16026e-6 -6.37645e-5 -4.84215e-6
-8.78165e-5 0.000182924 -0.000136926 -5.00794e-5 4.93763e-5 7.20127e-5 1.8876e-5 0.000229681 -5.21308e-5 3.18573e-5 1.72468e-5 1.70533e-5 5.57213e-5 0.000130356 -9.70809e-5 4.9404e-5 -4.88395e-5 0.000107497 -6.66821e-5 -3.84845e-5 1.29411e-5 9.28541e-5 1.20339e-5 5.59073e-5 -4.53787e-5 3.05941e-6 6.73856e-5 2.05479e-5 5.53898e-5 0.000132456 -2.45967e-5 0.000273038 -0.000180549 0.000123226 -4.36266e-5 3.06963e-5 1.66175e-5 3.1228e-5 -2.57171e-5 -0.000108027
-0.000136998 -3.10985e-5 0.000114338 -1.68758e-5 -0.000150043 -3.62709e-5 -0.000164099 1.45281e-5 -0.000143634 -6.76827e-6 -6.43275e-5 -2.22922e-5 -0.000174892 1.24677e-5 1.56045e-5 -2.46452e-7 -0.000137227 -1.8958e-6 -0.000113199 2.92555e-6 -7.14048e-5 -2.56378e-5 -0.000231696 1.16793e-6 -0.000185019 -2.12237e-5 -5.86307e-5 1.10057e-5 7.18014e-5 -3.19497e-5 -0.000131354 1.99621e-5 -7.66248e-5 -2.47564e-5 -0.000114683 1.15029e-6 0.000102075 -1.47177e-6 -4.83194e-5 -5.52585e-6
5.57118e-7 -8.95388e-5 9.22117e-8 6.04762e-5 -3.16718e-7 -6.46914e-6 -1.0222e-6 -1.49035e-5 -2.18287e-7 -0.000150534 -5.40814e-7 -4.83869e-5 -1.0776e-7 7.85067e-5 -1.22546e-6 9.59902e-5 3.20013e-7 2.88203e-5 7.51392e-7 0.000179723 -6.97521e-7 -4.40623e-5 2.222e-8 6.97393e-5 -1.19519e-7 -0.000207676 -8.37235e-7 -1.9172e-5 1.28498e-8 -3.85209e-5 -1.17268e-6 0.000213489 5.77813e-7 3.12679e-5 2.07144e-6 0.000181663 -1.10778e-6 -8.25786e-6 -3.25629e-7 -3.6007e-5
1.80142e-5 -4.98327e-5 -4.1751e-5 -1.49179e-5 -3.4011e-5 1.47288e-5 -1.73465e-5 -0.000229175 -1.6065e-5 5.26089e-5 8.14349e-5 -8.9229e-5 -1.66433e-5 -0.000256258 -4.50757e-5 -0.000177032 -6.58587e-5 -0.000252808 -8.25121e-5 -0.00017213 -8.18995e-5 5.15842e-5 -2.77155e-5 -7.87325e-5 4.30202e-6 0.00024288 -6.68111e-5 -6.95081e-5 -4.89677e-5 0.000118932 -1.73264e-5 -0.000455155 -3.31322e-5 -0.000140811 -5.91715e-5 -0.000314448 -1.11104e-5 -0.000228435 3.01093e-5 -0.000213715
7.33221e-5 -4.28013e-5 0.000178167 7.60491e-5 0.000166835 0.00018403 0.000193706 -4.84643e-5 0.000201855 7.25683e-5 5.98308e-5 -0.000107079 -0.000245389 -0.000162642 0.00023976 9.09504e-6 -0.000190793 -6.41162e-5 0.000414976 -5.61936e-5 0.000143676 3.01039e-5 4.15522e-5 -2.58335e-5 0.000132043 3.15399e-6 -0.000159816 -0.000144344 3.20909e-5 -3.3005e-5 4.62888e-5 -0.000142958 0.000139512 3.99826e-6 -0.000180255 -3.47414e-5 0.000248502 -0.000151975 -0.000151292 1.18485e-6
-2.45615e-5 -0.000113014 -1.04651e-5 -0.000183355 9.40566e-5 -0.000199415 -0.000247864 -7.98296e-5 -0.00014403 -9.6692e-5 -0.000132537 -0.000140911 -0.000476959 7.69257e-5 2.85757e-5 -6.59979e-5 -0.00019383 -0.000115314 -3.6628e-5 -2.39465e-5 -8.20875e-5 -0.000162451 -0.000187332 1.49353e-5 0.000232568 -0.000233416 -4.96378e-5 5.18801e-5 0.000144399 -3.05233e-5 -7.12832e-5 0.000152704 -8.4651e-5 -0.000183736 -0.000366174 -1.96948e-5 -0.000144033 -3.29829e-5 -0.000164963 7.27972e-5
2.66691e-6 -0.000126613 -5.50449e-6 -0.000347519 -3.5434e-6 -5.47357e-5 5.83213e-6 2.30872e-5 4.02182e-6 -0.000168492 -2.2596e-6 -0.000272905 5.23126e-6 0.000174732 1.28102e-5 -0.000158785 6.99541e-6 -0.000231225 1.15786e-5 -0.000187934 6.61135e-6 -0.000258392 5.61282e-6 -2.6605e-5 -3.32893e-6 -0.000352301 7.78598e-6 -0.0002235 -5.88347e-7 -0.000174245 9.80471e-6 -0.000297173 1.05018e-5 4.02801e-5 3.44275e-6 -0.000232472 1.39352e-5 -0.000236706 -2.42274e-6 -2.39668e-5
4.50321e-6 1.94327e-5 -7.83373e-6 -3.589e-5 -4.27614e-6 -0.000157819 3.25469e-6 -6.75454e-5 -1.35648e-5 -9.66417e-5 -1.22394e-6 1.50024e-5 -1.44772e-5 -3.36364e-5 -1.83172e-5 8.67495e-5 -7.22191e-6 -3.66066e-5 -1.45756e-5 5.69843e-6 -2.50739e-6 -5.95312e-5 6.66504e-6 3.71549e-5 -1.25365e-5 -6.52409e-5 -5.30943e-6 6.32776e-5 -1.8443e-6 -5.87406e-6 -1.15292e-5 -3.7385e-6 -9.3317e-6 -3.42158e-5 -7.66573e-6 -6.34874e-5 -8.95965e-6 8.10715e-5 -8.67834e-6 -4.5555e-5
4.51808e-5 -1.19336e-5 -1.29187e-6 -1.89976e-5 -9.34591e-5 -4.60746e-5 -4.66931e-6 -5.39937e-6 3.02524e-5 -4.71285e-6 -1.45933e-5 -2.5476e-5 1.12089e-5 -5.52714e-5 -6.23295e-5 -5.12539e-6 -8.05721e-5 -9.89138e-6 8.00405e-5 3.75563e-5 1.44906e-5 6.14031e-5 -4.70843e-5 -2.92185e-5 2.28289e-5 -1.80566e-5 3.73114e-6 -1.27483e-5 -1.08313e-5 1.62646e-5 -3.69083e-6 -1.09077e-5 2.49785e-5 -1.61397e-5 1.5128e-5 -1.54712e-5 4.27542e-5 -3.69266e-5 -1.22403e-6 -1.45614e-5The first column prints, for each rule, the largest standard deviation across the assets of one currency at one observation of what is left after the subtraction, less the true local return. A term that the currency factor can absorb has a spread of zero. The other columns print the size of the cross term against the size of the local returns.
pretty_table(DataFrame("Rule" => [":log", ":simple"], "Largest spread within a currency" => [spread(left_log), spread(left_simple)], "Mean abs(cross term)" => [0.0, mean(abs, cross_term)], "Largest abs(cross term)" => [0.0, maximum(abs, cross_term)], "Mean abs(local return)" => [mean(abs, truth.local_lr[:, foreign]), mean(abs, local_simple[:, foreign])]); formatters = [numfmt], title = "What the subtraction leaves in the local returns") What the subtraction leaves in the local returns
┌─────────┬──────────────────────────────────┬──────────────────────┬─────────────────────────┬────────────────────────┐
│ Rule │ Largest spread within a currency │ Mean abs(cross term) │ Largest abs(cross term) │ Mean abs(local return) │
│ String │ Float64 │ Float64 │ Float64 │ Float64 │
├─────────┼──────────────────────────────────┼──────────────────────┼─────────────────────────┼────────────────────────┤
│ :log │ 1.825e-16 │ 0.0 │ 0.0 │ 0.01506 │
│ :simple │ 0.0007153 │ 7.81e-5 │ 0.001536 │ 0.01507 │
└─────────┴──────────────────────────────────┴──────────────────────┴─────────────────────────┴────────────────────────┘Under the log rule the spread is at the level of rounding. Under the simple rule the mean cross term is about half a percent of the mean local return, the largest is about a tenth of it, and the regression reads the cross term as idiosyncratic return. Convert with :log when the universe holds more than one currency.
What to take away
rd.Ecarries the currency excess indices and the oil price as returns, under the rule of the asset returns, and the prior reads its observed factor returns from it with no change.- With two currency factors and an oil factor, the idiosyncratic volatility of every currency group is within 0.01 of the true 0.19 a year. Without them it is 0.24 to 0.28.
- An observed factor has return statistics but no t-statistic and no standard error, because the fit did not estimate its return.
- Each cluster, each subset and each fold fits its own market and size returns, and reads the same observed returns for the rows it covers.
- Under the simple rule the subtraction of the currency leaves a cross term in each local return, and under the log rule it does not.
Where to go next
- Cross-sectional factor model, end to end covers the estimated factors, the uncertainty sets and the constrained book of a cross-sectional prior.
- Cross-sectional factor model through a Pipeline reaches the same book with each estimator as a named step.
- Meta-optimisers covers the nested clustered optimisation in more depth.
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