Predictability

We'll revisit some old topics but also touch on new ones. Objectives:

Simple vol managed
Mean-variance efficient version
- known mean and known variance
- known mean and predicted variance
- predicted mean and known variance
- predicted mean and predicted variance
Adding leverage constraints
Effect of different volatility predictions.

#r "nuget: NovaSBE.Finance, 0.5.0"
#r "nuget: FSharp.Stats, 0.5.0"
#r "nuget: ExcelProvider, 2.0.0"
#r "nuget: Plotly.NET, 3.*"
#r "nuget: Plotly.NET.Interactive, 3.*"

open System
open NovaSBE.Finance
open NovaSBE.Finance.Utils
open FSharp.Stats
open Plotly.NET
open Plotly.NET.Interactive

let ff3 = French.getFF3 French.Monthly
let ff3Daily = French.getFF3 French.Daily

Vol Managed Portfolios

Simple vol managed

Following Moreia and Muir (2017) this strategy is \(w_t = \frac{c}{\hat{\sigma}^2_t}\) where \(c\) is a constant that sets the strategie's standard deviation to that of the buy-hold portfolio.

The variance prediction is based on realized variance the prior month.

type ManagedObs =
    { Month: DateTime 
      RiskyWeight: float 
      Return: float }

/// Realized variance from month t-1 to predict month t. The key is month t, the value is predicted variance
let mmVarPred =
    ff3Daily
    |> Array.groupBy (fun x -> DateTime(x.Date.Year, x.Date.Month, 1))
    |> Array.map (fun (month, dayObs) ->
        let mu = dayObs |> Array.averageBy (fun x -> x.MktRf)
        let realizedVar = 
            dayObs 
            |> Array.map (fun x -> (x.MktRf - mu) ** 2.0) 
            |> Array.sum
        month.AddMonths(1), realizedVar)
    |> Map

let minMonth = mmVarPred.Keys |> Seq.min
let maxMonth = mmVarPred.Keys |> Seq.max
    
let simpleManaged =
    ff3
    |> Array.filter (fun x -> x.Date >= minMonth && x.Date <= maxMonth)
    |> Array.map (fun x ->
        let var = mmVarPred[x.Date]
        let w = 1.0 / var
        { Month = x.Date
          RiskyWeight = w
          Return = w * x.MktRf })


let buyHold =
    ff3
    |> Array.filter (fun x -> x.Date >= minMonth && x.Date <= maxMonth)
    |> Array.map (fun x ->
        { Month = x.Date
          RiskyWeight = 1.0
          Return = x.MktRf })

Let's compare the unnormalized managed vol port and the buy and hold.

let annualizedStDev (xs: seq<ManagedObs>) =
    xs
    |> Seq.map (fun x -> x.Return)
    |> stDev
    |> (*) (sqrt 12.0)

unnormalized managed vol.

let unnormalizedStDev =
    simpleManaged
    |> annualizedStDev
unnormalizedStDev

buy and hold.

let buyHoldStDev =
    buyHold
    |> annualizedStDev

Normalized managed vol

let mmConstant = buyHoldStDev / unnormalizedStDev
mmConstant

Normalized the strategy.

let simpleManaged2 =
    simpleManaged
    |> Array.map (fun x -> 
        { x with 
            RiskyWeight = mmConstant * x.RiskyWeight
            Return = mmConstant * x.Return })

simpleManaged2
|> annualizedStDev

Plot all portfolios.

let accumulateFromOne (xs: seq<ManagedObs>) =
    let sortedXs = xs |> Seq.sortBy (fun x -> x.Month)
    let mutable cr = 1.0
    [| for x in sortedXs do 
        cr <- cr * (1.0 + x.Return )
        x.Month, cr |]

[ "Buy and Hold", buyHold
  "Unnormalized Managed Vol", simpleManaged
  "Normalized Managed Vol", simpleManaged2 ]
|> List.map (fun (name, port) ->
    port
    |> accumulateFromOne
    |> Chart.Line
    |> Chart.withTraceInfo(Name=name))
|> Chart.combine
|> Chart.withYAxisStyle(AxisType=StyleParam.AxisType.Log)

Unnormalized doesn't make any sense.

[ "Buy and Hold", buyHold
  "Managed Vol", simpleManaged2 ]
|> List.map (fun (name, port) ->
    port
    |> accumulateFromOne
    |> Chart.Line
    |> Chart.withTraceInfo(Name=name))
|> Chart.combine
|> Chart.withYAxisStyle(AxisType=StyleParam.AxisType.Log)

Look at sharpe ratios.

let sharpe (xs: seq<ManagedObs>) =
    let sd = xs |> annualizedStDev
    let mean = xs |> Seq.averageBy (fun x -> 12.0 * x.Return)
    mean / sd

$"Buy and Hold: {buyHold |> sharpe}"
$"Unnormalized Managed Vol: {simpleManaged |> sharpe}"
$"Managed Vol: {simpleManaged2 |> sharpe}"

By decade.

let sharpeByDecade (xs: seq<ManagedObs>) =
    xs
    |> Seq.groupBy (fun x -> 10 * (x.Month.Year / 10))
    |> Seq.map (fun (decade, decadeObs) ->
        {| Decade = decade; Sharpe = sharpe decadeObs|})
    |> Array.ofSeq

let buyHoldDecades = buyHold |> sharpeByDecade
let simpleManaged2Decades = simpleManaged2 |> sharpeByDecade

for i = 0 to buyHoldDecades.Length-1 do 
    printfn $"Decade: {buyHoldDecades[i].Decade}"
    printfn $"  Buy-Hold/Managed: %.2f{buyHoldDecades[i].Sharpe} / %.2f{simpleManaged2Decades[i].Sharpe}"

Check alphas

let ff3Map = ff3 |> Array.map (fun x -> x.Date, x) |> Map

let regData =
    simpleManaged2
    |> Array.filter (fun x -> x.Month <= DateTime(2015,12,31) && x.Month >= DateTime(1927,1,1))
    |> Array.map (fun x ->
        {| Month = x.Month
           Managed = x.Return * 12.0 
           MktRf = ff3Map[x.Month].MktRf * 12.0 |})

open NovaSBE.Finance.Ols

Ols("Managed ~ MktRf", regData).fit().summary()

Mean-variance optimal managed vols

Mean-variance utility

let mvUtility gamma mu sigma =
    mu - 0.5 * gamma * sigma ** 2.0

/// Annualized buy-and-hold sample return.
let buyHoldAvg = buyHold |> Array.averageBy (fun x -> 12.0 * x.Return)

printfn $"Mean: {buyHoldAvg}"
printfn $"StDev: {buyHoldStDev}"

mvUtility 3.0 buyHoldAvg buyHoldStDev

Function to to it for our managed portfolios.

let managedMVUtility gamma (xs: seq<ManagedObs>) =
    let avg = xs |> Seq.averageBy (fun x -> 12.0 * x.Return)
    let stDev = xs |> annualizedStDev
    mvUtility gamma avg stDev

let portSummary name port =
    printfn $"_________\n{name}\nNumber of months: {Seq.length port}"
    printfn $"Sharpe: {port |> sharpe}"
    printfn $"MV Utility: {managedMVUtility 3.0 port}"
    printfn $"annaluzed stdev: {port |> annualizedStDev}"

portSummary "Buy and Hold" buyHold
portSummary "Unnormalized Managed Vol" simpleManaged
portSummary "Normalized Managed Vol" simpleManaged2

Mean-variance optimal weight.

let mvWeight gamma mu sigma =
    mu / (gamma * sigma ** 2.0)

mvWeight 3.0 buyHoldAvg buyHoldStDev

Managed portfolio assuming we knew the full-sample mean and standard deviation ahead of time. We cannot use this to trade, because it uses future information. But it's a good benchmark.

let w_staticMuSigma = mvWeight 3.0 buyHoldAvg buyHoldStDev
let managedStaticMuSigma =
    buyHold
    |> Array.map (fun x -> 
        { x with 
            RiskyWeight = w_staticMuSigma
            Return = w_staticMuSigma * x.Return })

portSummary "Managed, known mu and sigma" managedStaticMuSigma
portSummary "Managed, base case" simpleManaged2

Let's now try with a static mean return estimate, but we'll use the rolling variance estimate.

/// Calculates a mean-variance optimal port given gamma and estimates of mu's and sigma's.
let managedMVPort gamma muEstimates varEstimates =
    ff3
    |> Array.choose (fun x ->
        let sd = varEstimates |> Map.tryFind x.Date |> Option.map sqrt
        let mu = muEstimates |> Map.tryFind x.Date
        match sd, mu with
        | Some sd, Some mu ->
            let w = mvWeight 3.0 mu sd
            let result = 
                { Month = x.Date
                  RiskyWeight = w
                  Return = w * x.MktRf }
            Some result
        | _ -> None)

Full sample 'estimates'

let muEstFull = [ for x in ff3 do x.Date, buyHoldAvg / 12.0 ] |> Map
let sigmaEstFull = [ for x in ff3 do x.Date, buyHoldStDev / sqrt 12.0 ] |> Map

portSummary "Managed, static mu and static sigma" managedStaticMuSigma
portSummary "Managed, static mu and rolling sigma" (managedMVPort 3.0 muEstFull sigmaEstFull)

Now actual static mu but rolling sigma estimates.

let managedStaticMuRollingSigma = managedMVPort 3.0 muEstFull mmVarPred

portSummary "Managd, static mu and rolling sigma" managedStaticMuRollingSigma
portSummary "Managed, base case" simpleManaged2

How volatile is that portfolio?

let sigmaStaticMuRollingSigma = managedStaticMuRollingSigma |> annualizedStDev
sigmaStaticMuRollingSigma

Let's try rescaling our static mu rolling sigma.

let managedStaticMuRollingSigma2 = 
    managedStaticMuRollingSigma
    |> Array.map (fun x -> 
        let c = buyHoldStDev / sigmaStaticMuRollingSigma
        { x with 
            RiskyWeight = x.RiskyWeight * c
            Return = x.Return * c })

portSummary "Managed, static mu and rolling sigma" managedStaticMuRollingSigma2
portSummary "Managed base case" simpleManaged2

Why is the ex-post rescaling so critical to the utility calculation?

let managedHiVolDays =
    managedStaticMuRollingSigma
    |> Array.sortByDescending (fun x -> abs x.Return)
    |> Array.take 10
managedHiVolDays

Let's compare predicted and realized vols on those bad miss days.

[ for x in managedHiVolDays do
    let sd = sqrt (mmVarPred[x.Month]*12.0)
    let realized = (sqrt 12.0) * abs ff3Map[x.Month].MktRf
    {| Month = x.Month 
       RiskyWeight = x.RiskyWeight
       Predicted = sd
       Realized = realized |} ]

We might be able to control this ex-ante by using reasonable leverage limits.

let managedMVPortLimited gamma muEstimates varEstimates =
    ff3
    |> Array.choose (fun x ->
        let sd = varEstimates |> Map.tryFind x.Date |> Option.map sqrt
        let mu = muEstimates |> Map.tryFind x.Date
        match sd, mu with
        | Some sd, Some mu ->
            let w = mvWeight 3.0 mu sd
            let w2 = 
                if w > 2.0 then 2.0 
                elif w < 0.0 then 0.0 
                else w
            let result = 
                { Month = x.Date
                  RiskyWeight = w2
                  Return = w2 * x.MktRf }
            Some result
        | _ -> None)

let managedStaticMuRollingSigma3 = managedMVPortLimited 3.0 muEstFull mmVarPred


portSummary "Managed, static mu and rolling sigma" managedStaticMuRollingSigma3
portSummary "Managed base case" simpleManaged2
portSummary "Buy-and-hold" buyHold

Rolling 'mu estimates', because our portfolios still have a "look-ahead" bias becuase they know what future returns will be.

let muExpandingEstimate =
    let mutable acc = 0.0
    [| for i = 0 to ff3.Length-1 do
        acc <- acc + ff3[i].MktRf
        ff3[i].Date, acc / (float i + 1.0) |]

muExpandingEstimate
|> Chart.Line

How big of a burn-in period to use? Let's just use post-war period.

let muExpandingMap = muExpandingEstimate |> Map
let managedEstAll = 
    managedMVPortLimited 3.0 muExpandingMap mmVarPred
    |> Array.filter (fun x -> x.Month >= DateTime(1945,1,1))

let buyHold1945 = buyHold |> Array.filter (fun x -> x.Month >= DateTime(1945,1,1))
let simpleManaged21945 = 
    let since1945 = simpleManaged2 |> Array.filter (fun x -> x.Month >= DateTime(1945,1,1))
    let sd = since1945 |> annualizedStDev
    let sdBuyHold = buyHold1945 |> annualizedStDev
    let c = sdBuyHold / sd
    since1945
    |> Array.map (fun x -> 
        { x with 
            RiskyWeight = x.RiskyWeight * c 
            Return = x.Return * c })

portSummary "Managed, rolling mu and rolling sigma" managedEstAll
portSummary "Managed base case" simpleManaged21945
portSummary "Buy-and-hold" buyHold1945

Comparin decades.

let buyHold1945Decades = buyHold1945 |> sharpeByDecade
let managedEstAllDecades = managedEstAll |> sharpeByDecade

for i = 0 to buyHold1945Decades.Length-1 do 
    printfn $"Decade: {buyHold1945Decades[i].Decade}"
    printfn $"  Buy-Hold/Managed: %.2f{buyHold1945Decades[i].Sharpe} / %.2f{managedEstAllDecades[i].Sharpe}"

Comparing Vol forecasts

type ReturnObs = { Date: DateTime; Return: float}

type VolatilityPrediction = 
    { /// First date the prediction is valid for
      Date: DateTime
      /// The volatility prediction
      PredictedVol: float }

/// <summary>Calculates realized volatity of a return observation using an exponential weight.</summary>
/// <param name="width">The window width for calculating realized vol.</param>
/// <param name="lambda">The exponential weight</param>
/// <param name="data">The input data</param>
/// <returns>A prediction and date the prediction is valid for.</returns>
let expRealizedVol (width: int) (lambda: float) (data: array<ReturnObs>) =
    data
    |> Array.sortByDescending (fun x -> x.Date)
    |> Array.windowed (width + 1)
    |> Array.Parallel.map (fun window ->
        let mu = window[1..] |> Array.averageBy (fun x -> x.Return)
        let mutable acc = 0.0
        for t = 1 to width do 
            let w = (1.0 - lambda)*lambda**(float t - 1.0)
            acc <- acc + w * (window[t].Return - mu)**2.0
        { VolatilityPrediction.Date = window[0].Date; PredictedVol = sqrt acc })
    |> Array.rev

let expVarPred =
    ff3Daily
    |> Array.map (fun x -> { Date = x.Date; Return = x.MktRf })
    |> expRealizedVol 500 0.94 
    |> Array.groupBy (fun x -> DateTime(x.Date.Year, x.Date.Month, 1))
    |> Array.map (fun (month, xs) ->
        let last = xs |> Array.sortBy (fun x -> x.Date) |> Array.last
        month.AddMonths(1), 22.0 * last.PredictedVol ** 2.0)
    |> Map

Compare vol accuracies

let checkVols =
    [| for dt in expVarPred.Keys |> Seq.filter (fun x -> x >= DateTime(1945,1,1) ) do
        if expVarPred.ContainsKey (dt.AddMonths(1)) then
           {| Date = dt
              ExpVar = expVarPred[dt]
              Var22d = mmVarPred[dt]
              Actual = mmVarPred[dt.AddMonths(1)]|}|]

Exponential vol

Ols("Actual~ExpVar", checkVols).fit().summary()

Rolling vol

Ols("Actual~Var22d", checkVols).fit().summary()

Compare ports

let managedExpVar = 
    managedMVPortLimited 3.0 muExpandingMap expVarPred
    |> Array.filter (fun x -> x.Month >= DateTime(1945,1,1))

portSummary "Managed, rolling mu and exp. vol" managedExpVar
portSummary "Managed, rolling mu and rolling vol" managedEstAll
portSummary "Buy and hold" buyHold1945

namespace System

namespace NovaSBE

namespace NovaSBE.Finance

module Utils from NovaSBE.Finance

Multiple items
namespace FSharp

--------------------
namespace Microsoft.FSharp

namespace FSharp.Stats

namespace Plotly

namespace Plotly.NET

namespace Plotly.NET.Interactive

val ff3: French.FF3Obs array

module French from NovaSBE.Finance

val getFF3: frequency: French.Frequency -> French.FF3Obs array

union case French.Frequency.Monthly: French.Frequency

val ff3Daily: French.FF3Obs array

union case French.Frequency.Daily: French.Frequency

type ManagedObs = { Month: DateTime RiskyWeight: float Return: float }

Multiple items
[<Struct>] type DateTime = new: year: int * month: int * day: int -> unit + 16 overloads member Add: value: TimeSpan -> DateTime member AddDays: value: float -> DateTime member AddHours: value: float -> DateTime member AddMicroseconds: value: float -> DateTime member AddMilliseconds: value: float -> DateTime member AddMinutes: value: float -> DateTime member AddMonths: months: int -> DateTime member AddSeconds: value: float -> DateTime member AddTicks: value: int64 -> DateTime ...
<summary>Represents an instant in time, typically expressed as a date and time of day.</summary>

--------------------
DateTime ()
   (+0 other overloads)
DateTime(ticks: int64) : DateTime
   (+0 other overloads)
DateTime(ticks: int64, kind: DateTimeKind) : DateTime
   (+0 other overloads)
DateTime(date: DateOnly, time: TimeOnly) : DateTime
   (+0 other overloads)
DateTime(year: int, month: int, day: int) : DateTime
   (+0 other overloads)
DateTime(date: DateOnly, time: TimeOnly, kind: DateTimeKind) : DateTime
   (+0 other overloads)
DateTime(year: int, month: int, day: int, calendar: Globalization.Calendar) : DateTime
   (+0 other overloads)
DateTime(year: int, month: int, day: int, hour: int, minute: int, second: int) : DateTime
   (+0 other overloads)
DateTime(year: int, month: int, day: int, hour: int, minute: int, second: int, kind: DateTimeKind) : DateTime
   (+0 other overloads)
DateTime(year: int, month: int, day: int, hour: int, minute: int, second: int, calendar: Globalization.Calendar) : DateTime
   (+0 other overloads)

Multiple items
val float: value: 'T -> float (requires member op_Explicit)

--------------------
type float = Double

--------------------
type float<'Measure> = float

val mmVarPred: Map<DateTime,float>
Realized variance from month t-1 to predict month t. The key is month t, the value is predicted variance

Multiple items
type Array = new: unit -> Array static member geomspace: start: float * stop: float * num: int * ?IncludeEndpoint: bool -> float array static member linspace: start: float * stop: float * num: int * ?IncludeEndpoint: bool -> float array

--------------------
new: unit -> Array

val groupBy: projection: ('T -> 'Key) -> array: 'T array -> ('Key * 'T array) array (requires equality)

val x: French.FF3Obs

French.FF3Obs.Date: DateTime

property DateTime.Year: int with get
<summary>Gets the year component of the date represented by this instance.</summary>
<returns>The year, between 1 and 9999.</returns>

property DateTime.Month: int with get
<summary>Gets the month component of the date represented by this instance.</summary>
<returns>The month component, expressed as a value between 1 and 12.</returns>

val map: mapping: ('T -> 'U) -> array: 'T array -> 'U array

val month: DateTime

val dayObs: French.FF3Obs array

val mu: float

val averageBy: projection: ('T -> 'U) -> array: 'T array -> 'U (requires member (+) and member DivideByInt and member Zero)

French.FF3Obs.MktRf: float

val realizedVar: float

val sum: array: 'T array -> 'T (requires member (+) and member Zero)

DateTime.AddMonths(months: int) : DateTime

Multiple items
module Map from FSharp.Stats
<summary> Module to strore specialised computations on maps </summary>

--------------------
module Map from Microsoft.FSharp.Collections

--------------------
type Map<'Key,'Value (requires comparison)> = interface IReadOnlyDictionary<'Key,'Value> interface IReadOnlyCollection<KeyValuePair<'Key,'Value>> interface IEnumerable interface IStructuralEquatable interface IComparable interface IEnumerable<KeyValuePair<'Key,'Value>> interface ICollection<KeyValuePair<'Key,'Value>> interface IDictionary<'Key,'Value> new: elements: ('Key * 'Value) seq -> Map<'Key,'Value> member Add: key: 'Key * value: 'Value -> Map<'Key,'Value> ...

--------------------
new: elements: ('Key * 'Value) seq -> Map<'Key,'Value>

val minMonth: DateTime

property Map.Keys: Collections.Generic.ICollection<DateTime> with get

Multiple items
module Seq from FSharp.Stats
<summary> Module to compute common statistical measure </summary>

--------------------
module Seq from Microsoft.FSharp.Collections

--------------------
type Seq = new: unit -> Seq static member geomspace: start: float * stop: float * num: int * ?IncludeEndpoint: bool -> float seq static member linspace: start: float * stop: float * num: int * ?IncludeEndpoint: bool -> float seq

--------------------
new: unit -> Seq

val min: source: 'T seq -> 'T (requires comparison)

val maxMonth: DateTime

val max: source: 'T seq -> 'T (requires comparison)

val simpleManaged: ManagedObs array

val filter: predicate: ('T -> bool) -> array: 'T array -> 'T array

val var: float

val w: float

val buyHold: ManagedObs array

val annualizedStDev: xs: ManagedObs seq -> float

val xs: ManagedObs seq

Multiple items
val seq: sequence: 'T seq -> 'T seq

--------------------
type 'T seq = Collections.Generic.IEnumerable<'T>

val map: mapping: ('T -> 'U) -> source: 'T seq -> 'U seq

val x: ManagedObs

ManagedObs.Return: float

val stDev: items: 'T seq -> 'U (requires member (-) and member Zero and member DivideByInt and member (+) and member ( * ) and member (+) and member (/) and member Sqrt)
<summary> Computes the sample standard deviation </summary>
<param name="items">The input sequence.</param>
<remarks>Returns NaN if data is empty or if any entry is NaN.</remarks>
<returns>standard deviation of a sample (Bessel's correction by N-1)</returns>

val sqrt: value: 'T -> 'U (requires member Sqrt)

val unnormalizedStDev: float

val buyHoldStDev: float

val mmConstant: float

val simpleManaged2: ManagedObs array

ManagedObs.RiskyWeight: float

val accumulateFromOne: xs: ManagedObs seq -> (DateTime * float) array

val sortedXs: ManagedObs seq

val sortBy: projection: ('T -> 'Key) -> source: 'T seq -> 'T seq (requires comparison)

ManagedObs.Month: DateTime

val mutable cr: float

Multiple items
module List from FSharp.Stats
<summary> Module to compute common statistical measure on list </summary>

--------------------
module List from Microsoft.FSharp.Collections

--------------------
type List = new: unit -> List static member geomspace: start: float * stop: float * num: int * ?IncludeEndpoint: bool -> float list static member linspace: start: float * stop: float * num: int * ?IncludeEndpoint: bool -> float list

--------------------
type List<'T> = | op_Nil | op_ColonColon of Head: 'T * Tail: 'T list interface IReadOnlyList<'T> interface IReadOnlyCollection<'T> interface IEnumerable interface IEnumerable<'T> member GetReverseIndex: rank: int * offset: int -> int member GetSlice: startIndex: int option * endIndex: int option -> 'T list static member Cons: head: 'T * tail: 'T list -> 'T list member Head: 'T member IsEmpty: bool member Item: index: int -> 'T with get ...

--------------------
new: unit -> List

val map: mapping: ('T -> 'U) -> list: 'T list -> 'U list

val name: string

val port: ManagedObs array

type Chart = static member AnnotatedHeatmap: zData: #('a1 seq) seq * annotationText: #(string seq) seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?X: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?XGap: int * [<Optional; DefaultParameterValue ((null :> obj))>] ?Y: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?YGap: int * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a5 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a5 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?ColorBar: ColorBar * [<Optional; DefaultParameterValue ((null :> obj))>] ?ColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowScale: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?ReverseScale: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?ZSmooth: SmoothAlg * [<Optional; DefaultParameterValue ((null :> obj))>] ?Transpose: bool * [<Optional; DefaultParameterValue ((false :> obj))>] ?UseWebGL: bool * [<Optional; DefaultParameterValue ((false :> obj))>] ?ReverseYAxis: bool * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a1 :> IConvertible and 'a5 :> IConvertible) + 1 overload static member Area: x: #IConvertible seq * y: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowMarkers: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiOpacity: float seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?TextPosition: TextPosition * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiTextPosition: TextPosition seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerOutline: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerSymbol: MarkerSymbol * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiMarkerSymbol: MarkerSymbol seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Marker: Marker * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineWidth: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineDash: DrawingStyle * [<Optional; DefaultParameterValue ((null :> obj))>] ?Line: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?StackGroup: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?Orientation: Orientation * [<Optional; DefaultParameterValue ((null :> obj))>] ?GroupNorm: GroupNorm * [<Optional; DefaultParameterValue ((null :> obj))>] ?FillColor: Color * [<Optional; DefaultParameterValue ((false :> obj))>] ?UseWebGL: bool * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a2 :> IConvertible) + 1 overload static member Bar: values: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Keys: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiOpacity: float seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerOutline: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerPatternShape: PatternShape * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiMarkerPatternShape: PatternShape seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerPattern: Pattern * [<Optional; DefaultParameterValue ((null :> obj))>] ?Marker: Marker * [<Optional; DefaultParameterValue ((null :> obj))>] ?Base: #IConvertible * [<Optional; DefaultParameterValue ((null :> obj))>] ?Width: 'a4 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiWidth: 'a4 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?TextPosition: TextPosition * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiTextPosition: TextPosition seq * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a2 :> IConvertible and 'a4 :> IConvertible) + 1 overload static member BoxPlot: [<Optional; DefaultParameterValue ((null :> obj))>] ?X: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Y: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?FillColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?Marker: Marker * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?WhiskerWidth: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?BoxPoints: BoxPoints * [<Optional; DefaultParameterValue ((null :> obj))>] ?BoxMean: BoxMean * [<Optional; DefaultParameterValue ((null :> obj))>] ?Jitter: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?PointPos: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?Orientation: Orientation * [<Optional; DefaultParameterValue ((null :> obj))>] ?OutlineColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?OutlineWidth: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?Outline: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?AlignmentGroup: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?OffsetGroup: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?Notched: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?NotchWidth: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?QuartileMethod: QuartileMethod * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a2 :> IConvertible) + 2 overloads static member Bubble: x: #IConvertible seq * y: #IConvertible seq * sizes: int seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiOpacity: float seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?TextPosition: TextPosition * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiTextPosition: TextPosition seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerOutline: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerSymbol: MarkerSymbol * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiMarkerSymbol: MarkerSymbol seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Marker: Marker * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineWidth: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?LineDash: DrawingStyle * [<Optional; DefaultParameterValue ((null :> obj))>] ?Line: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?StackGroup: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?Orientation: Orientation * [<Optional; DefaultParameterValue ((null :> obj))>] ?GroupNorm: GroupNorm * [<Optional; DefaultParameterValue ((false :> obj))>] ?UseWebGL: bool * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a2 :> IConvertible) + 1 overload static member Candlestick: ``open`` : #IConvertible seq * high: #IConvertible seq * low: #IConvertible seq * close: #IConvertible seq * x: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a5 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a5 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Line: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?IncreasingColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?Increasing: FinanceMarker * [<Optional; DefaultParameterValue ((null :> obj))>] ?DecreasingColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?Decreasing: FinanceMarker * [<Optional; DefaultParameterValue ((null :> obj))>] ?WhiskerWidth: float * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a5 :> IConvertible) + 1 overload static member Column: values: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Keys: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiOpacity: float seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerOutline: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerPatternShape: PatternShape * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiMarkerPatternShape: PatternShape seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerPattern: Pattern * [<Optional; DefaultParameterValue ((null :> obj))>] ?Marker: Marker * [<Optional; DefaultParameterValue ((null :> obj))>] ?Base: #IConvertible * [<Optional; DefaultParameterValue ((null :> obj))>] ?Width: 'a4 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiWidth: 'a4 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?TextPosition: TextPosition * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiTextPosition: TextPosition seq * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a2 :> IConvertible and 'a4 :> IConvertible) + 1 overload static member Contour: zData: #('a1 seq) seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?X: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Y: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a4 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a4 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?ColorBar: ColorBar * [<Optional; DefaultParameterValue ((null :> obj))>] ?ColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowScale: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?ReverseScale: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Transpose: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContourLineColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContourLineDash: DrawingStyle * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContourLineSmoothing: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContourLine: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContoursColoring: ContourColoring * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContoursOperation: ConstraintOperation * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContoursType: ContourType * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowContourLabels: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?ContourLabelFont: Font * [<Optional; DefaultParameterValue ((null :> obj))>] ?Contours: Contours * [<Optional; DefaultParameterValue ((null :> obj))>] ?FillColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?NContours: int * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a1 :> IConvertible and 'a4 :> IConvertible) static member Funnel: x: #IConvertible seq * y: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?Width: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?Offset: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?TextPosition: TextPosition * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiTextPosition: TextPosition seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Orientation: Orientation * [<Optional; DefaultParameterValue ((null :> obj))>] ?AlignmentGroup: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?OffsetGroup: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?MarkerOutline: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?Marker: Marker * [<Optional; DefaultParameterValue ((null :> obj))>] ?TextInfo: TextInfo * [<Optional; DefaultParameterValue ((null :> obj))>] ?ConnectorLineColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?ConnectorLineStyle: DrawingStyle * [<Optional; DefaultParameterValue ((null :> obj))>] ?ConnectorFillColor: Color * [<Optional; DefaultParameterValue ((null :> obj))>] ?ConnectorLine: Line * [<Optional; DefaultParameterValue ((null :> obj))>] ?Connector: FunnelConnector * [<Optional; DefaultParameterValue ((null :> obj))>] ?InsideTextFont: Font * [<Optional; DefaultParameterValue ((null :> obj))>] ?OutsideTextFont: Font * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a2 :> IConvertible) static member Heatmap: zData: #('a1 seq) seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?Name: string * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Optional; DefaultParameterValue ((null :> obj))>] ?X: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?XGap: int * [<Optional; DefaultParameterValue ((null :> obj))>] ?Y: #IConvertible seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?YGap: int * [<Optional; DefaultParameterValue ((null :> obj))>] ?Text: 'a4 * [<Optional; DefaultParameterValue ((null :> obj))>] ?MultiText: 'a4 seq * [<Optional; DefaultParameterValue ((null :> obj))>] ?ColorBar: ColorBar * [<Optional; DefaultParameterValue ((null :> obj))>] ?ColorScale: Colorscale * [<Optional; DefaultParameterValue ((null :> obj))>] ?ShowScale: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?ReverseScale: bool * [<Optional; DefaultParameterValue ((null :> obj))>] ?ZSmooth: SmoothAlg * [<Optional; DefaultParameterValue ((null :> obj))>] ?Transpose: bool * [<Optional; DefaultParameterValue ((false :> obj))>] ?UseWebGL: bool * [<Optional; DefaultParameterValue ((false :> obj))>] ?ReverseYAxis: bool * [<Optional; DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart (requires 'a1 :> IConvertible and 'a4 :> IConvertible) + 1 overload ...

static member Chart.Line: xy: (#IConvertible * #IConvertible) seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowMarkers: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Name: string * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiOpacity: float seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?TextPosition: StyleParam.TextPosition * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiTextPosition: StyleParam.TextPosition seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerColorScale: StyleParam.Colorscale * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerOutline: Line * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerSymbol: StyleParam.MarkerSymbol * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiMarkerSymbol: StyleParam.MarkerSymbol seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Marker: TraceObjects.Marker * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineColorScale: StyleParam.Colorscale * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineWidth: float * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineDash: StyleParam.DrawingStyle * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Line: Line * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?StackGroup: string * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Orientation: StyleParam.Orientation * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?GroupNorm: StyleParam.GroupNorm * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Fill: StyleParam.Fill * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?FillColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((false :> obj))>] ?UseWebGL: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart.GenericChart (requires 'a2 :> IConvertible)
static member Chart.Line: x: #IConvertible seq * y: #IConvertible seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowMarkers: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Name: string * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Opacity: float * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiOpacity: float seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Text: 'a2 * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiText: 'a2 seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?TextPosition: StyleParam.TextPosition * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiTextPosition: StyleParam.TextPosition seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerColorScale: StyleParam.Colorscale * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerOutline: Line * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MarkerSymbol: StyleParam.MarkerSymbol * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiMarkerSymbol: StyleParam.MarkerSymbol seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Marker: TraceObjects.Marker * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineColorScale: StyleParam.Colorscale * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineWidth: float * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineDash: StyleParam.DrawingStyle * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Line: Line * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?StackGroup: string * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Orientation: StyleParam.Orientation * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?GroupNorm: StyleParam.GroupNorm * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Fill: StyleParam.Fill * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?FillColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((false :> obj))>] ?UseWebGL: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((true :> obj))>] ?UseDefaults: bool -> GenericChart.GenericChart (requires 'a2 :> IConvertible)

static member Chart.withTraceInfo: [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Name: string * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Visible: StyleParam.Visible * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowLegend: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LegendRank: int * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LegendGroup: string * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LegendGroupTitle: Title -> (GenericChart.GenericChart -> GenericChart.GenericChart)

static member Chart.combine: gCharts: GenericChart.GenericChart seq -> GenericChart.GenericChart

static member Chart.withYAxisStyle: [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?TitleText: string * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?TitleFont: Font * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?TitleStandoff: int * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Title: Title * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Color: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?AxisType: StyleParam.AxisType * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MinMax: (#IConvertible * #IConvertible) * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Mirror: StyleParam.Mirror * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowSpikes: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?SpikeColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?SpikeThickness: int * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowLine: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?LineColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowGrid: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?GridColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ZeroLine: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ZeroLineColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Anchor: StyleParam.LinearAxisId * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Side: StyleParam.Side * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Overlaying: StyleParam.LinearAxisId * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Domain: (float * float) * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Position: float * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?CategoryOrder: StyleParam.CategoryOrder * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?CategoryArray: #IConvertible seq * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?RangeSlider: LayoutObjects.RangeSlider * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?RangeSelector: LayoutObjects.RangeSelector * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?BackgroundColor: Color * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?ShowBackground: bool * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?Id: StyleParam.SubPlotId -> (GenericChart.GenericChart -> GenericChart.GenericChart)

module StyleParam from Plotly.NET

type AxisType = | Auto | Linear | Log | Date | Category | MultiCategory member Convert: unit -> obj override ToString: unit -> string static member convert: (AxisType -> obj) static member toString: (AxisType -> string)
<summary> Sets the axis type. By default (Auto), plotly attempts to determined the axis type by looking into the data of the traces that referenced the axis in question. </summary>

union case StyleParam.AxisType.Log: StyleParam.AxisType

val sharpe: xs: ManagedObs seq -> float

val sd: float

val mean: float

val averageBy: projection: ('T -> 'U) -> source: 'T seq -> 'U (requires member (+) and member DivideByInt and member Zero)

val sharpeByDecade: xs: ManagedObs seq -> {| Decade: int; Sharpe: float |} array

val groupBy: projection: ('T -> 'Key) -> source: 'T seq -> ('Key * 'T seq) seq (requires equality)

val decade: int

val decadeObs: ManagedObs seq

val ofSeq: source: 'T seq -> 'T array

val buyHoldDecades: {| Decade: int; Sharpe: float |} array

val simpleManaged2Decades: {| Decade: int; Sharpe: float |} array

val i: int

property Array.Length: int with get
<summary>Gets the total number of elements in all the dimensions of the <see cref="T:System.Array" />.</summary>
<exception cref="T:System.OverflowException">The array is multidimensional and contains more than <see cref="F:System.Int32.MaxValue">Int32.MaxValue</see> elements.</exception>
<returns>The total number of elements in all the dimensions of the <see cref="T:System.Array" />; zero if there are no elements in the array.</returns>

val printfn: format: Printf.TextWriterFormat<'T> -> 'T

val ff3Map: Map<DateTime,French.FF3Obs>

val regData: {| Managed: float; MktRf: float; Month: DateTime |} array

module Ols from NovaSBE.Finance

Multiple items
module Ols from NovaSBE.Finance

--------------------
type Ols<'Record> = new: formula: string * data: 'Record seq -> Ols<'Record> member fit: unit -> RegressionResults member df_model: int member df_resid: int member endog_names: string member exog_names: string array member k_constant: int
<summary>Create a model from a formula and collection of records</summary>

--------------------
new: formula: string * data: 'Record seq -> Ols<'Record>

val mvUtility: gamma: float -> mu: float -> sigma: float -> float

val gamma: float

val sigma: float

val buyHoldAvg: float
Annualized buy-and-hold sample return.

val managedMVUtility: gamma: float -> xs: ManagedObs seq -> float

val avg: float

val stDev: float

val portSummary: name: 'a -> port: ManagedObs seq -> unit

val name: 'a

val port: ManagedObs seq

val mvWeight: gamma: float -> mu: float -> sigma: float -> float

val w_staticMuSigma: float

val managedStaticMuSigma: ManagedObs array

val managedMVPort: gamma: 'a -> muEstimates: Map<DateTime,float> -> varEstimates: Map<DateTime,float> -> ManagedObs array
Calculates a mean-variance optimal port given gamma and estimates of mu's and sigma's.

val gamma: 'a

val muEstimates: Map<DateTime,float>

val varEstimates: Map<DateTime,float>

val choose: chooser: ('T -> 'U option) -> array: 'T array -> 'U array

val sd: float option

val tryFind: key: 'Key -> table: Map<'Key,'T> -> 'T option (requires comparison)

module Option from Microsoft.FSharp.Core

val map: mapping: ('T -> 'U) -> option: 'T option -> 'U option

val mu: float option

union case Option.Some: Value: 'T -> Option<'T>

val result: ManagedObs

union case Option.None: Option<'T>

val muEstFull: Map<DateTime,float>

val sigmaEstFull: Map<DateTime,float>

val managedStaticMuRollingSigma: ManagedObs array

val sigmaStaticMuRollingSigma: float

val managedStaticMuRollingSigma2: ManagedObs array

val c: float

val managedHiVolDays: ManagedObs array

val sortByDescending: projection: ('T -> 'Key) -> array: 'T array -> 'T array (requires comparison)

val abs: value: 'T -> 'T (requires member Abs)

val take: count: int -> array: 'T array -> 'T array

val realized: float

val managedMVPortLimited: gamma: 'a -> muEstimates: Map<DateTime,float> -> varEstimates: Map<DateTime,float> -> ManagedObs array

val w2: float

val managedStaticMuRollingSigma3: ManagedObs array

val muExpandingEstimate: (DateTime * float) array

val mutable acc: float

val muExpandingMap: Map<DateTime,float>

val managedEstAll: ManagedObs array

val buyHold1945: ManagedObs array

val simpleManaged21945: ManagedObs array

val since1945: ManagedObs array

val sdBuyHold: float

val buyHold1945Decades: {| Decade: int; Sharpe: float |} array

val managedEstAllDecades: {| Decade: int; Sharpe: float |} array

type ReturnObs = { Date: DateTime Return: float }

type VolatilityPrediction = { Date: DateTime PredictedVol: float }

val expRealizedVol: width: int -> lambda: float -> data: ReturnObs array -> VolatilityPrediction array
<summary>Calculates realized volatity of a return observation using an exponential weight.</summary>
<param name="width">The window width for calculating realized vol.</param>
<param name="lambda">The exponential weight</param>
<param name="data">The input data</param>
<returns>A prediction and date the prediction is valid for.</returns>

val width: int

Multiple items
val int: value: 'T -> int (requires member op_Explicit)

--------------------
type int = int32

--------------------
type int<'Measure> = int

val lambda: float

val data: ReturnObs array

type 'T array = 'T array

val x: ReturnObs

ReturnObs.Date: DateTime

val windowed: windowSize: int -> array: 'T array -> 'T array array

module Parallel from Microsoft.FSharp.Collections.ArrayModule

val window: ReturnObs array

ReturnObs.Return: float

val t: int

val rev: array: 'T array -> 'T array

val expVarPred: Map<DateTime,float>

val x: VolatilityPrediction

VolatilityPrediction.Date: DateTime
First date the prediction is valid for

val xs: VolatilityPrediction array

val last: VolatilityPrediction

val sortBy: projection: ('T -> 'Key) -> array: 'T array -> 'T array (requires comparison)

val last: array: 'T array -> 'T

VolatilityPrediction.PredictedVol: float
The volatility prediction

val checkVols: {| Actual: float; Date: DateTime; ExpVar: float; Var22d: float |} array

val dt: DateTime

val filter: predicate: ('T -> bool) -> source: 'T seq -> 'T seq

val x: DateTime

member Map.ContainsKey: key: 'Key -> bool

val managedExpVar: ManagedObs array