Choosing the risky asset weight in your portfolio

#r "nuget: FSharp.Stats, 0.5.0"
#r "nuget: Plotly.NET, 3.*"
#r "nuget: Plotly.NET.Interactive, 3.*"


open System

open FSharp.Stats

open Plotly.NET

Portfolios of a risky asset and the risk-free asset

Portfolio return

Suppose an investor has a portfolio comprised of a risky asset and the risk-free asset. The risky asset has return $r_i$ and the risk-free asset has return $r_f$. If $w$ is the weight that the investor puts on the risky asset, then the portfolio return is

\[r_p = w r_i + (1-w) r_f = w (r_i - r_f) + r_f\]

If we put this in terms of excess returns, meaning returns in excess of the risk-free rate, then the portfolio excess return is

\[r_p - r_f = w (r_i - r_f)\].

So if we work in excess returns the excess return of the portfolio is the risky weight times the risky asset's excess return.

When $w < 0$ we are short the risky asset and long the risk-free asset, when $0 < w < 1$ we are long both the risky and risk-free asset, and when $w > 1$ we are short the risk-free asset and long the risky asset.

Does this check out? Imagine that you have $1 and you borrow $1 at the risk-free rate for a net stake of $2. Assuming that you invest in an asset with a return of 15% and the risk-free rate is 4%, what are you left with?

let invest = 1.0m
let borrow = 1.0m 
let ret = 0.15m
let rf = 0.04m
let result = (invest + borrow)*(1.0m+ret)-borrow*(1.0m+rf)
printfn $"You are left with {result}"

You are left with 1.260

If we calculate this using weights ...

let w = (invest + borrow)/invest
let result2 = 1m + rf + w*(ret - rf)
printfn $"You are left with {result2}"

You are left with 1.26

... we get the same result.

result = result2

val it: bool = true

Portfolio variance

Recall the formula for variance of a portfolio consisting of stocks $x$ and $y$:

\[\sigma^2 = w_x^2 \sigma^2_x + w_y^2 \sigma^2_y + 2 w_x w_y cov(r_x,r_y),\]

where $w_x$ and $w_y$ are the weights in stocks $x$ and $y$, $r_x$ and $r_y$ are the stock returns, $\sigma^2$ is variance, and $cov(.,.)$ is covariance.

If one asset is the risk free asset (borrowing a risk-free bond), then this asset does not vary, so the risk free variance and covariance terms are zero. Thus we are left with the result that if we leverage risky asset $x$ by borrowing or lending the risk-free asset, then our leveraged portfolio's standard deviation ($\sigma$) is

\[\sigma^2 = w_x^2 \sigma^2_x \rightarrow \sigma = w_x \sigma_x.\]

Mean-Variance optimal weight

An investor with mean-variance preferences will try to maximize utility of the form

\[u = \mu - \gamma \frac{\sigma^2}{2}\]

where $\mu$ is the expected portfolio return, $\sigma$ is the standard deviation of the portfolio, and $\gamma$ is the investor's coefficient of relative risk aversion.

If our portfolio is comprised of a risky asset x and the risk-free asset, then this objective function can be written

\[u = w_x (\mu_x - r_f) + r_f - \gamma \frac{w_x^2 \sigma_x^2}{2}.\]

To maximize this with respect to $w_x$ we take the derivative and set it equal to zero:

\[\frac{\partial u}{\partial w_x} = (\mu_x - r_f) - \gamma w_x \sigma_x^2 = 0\]

This gives us the optimal weight

\[w_x = \frac{\mu_x - r_f}{\gamma \sigma_x^2}.\]

It is common to define $\mu$ as the expected excess return, so that $\mu_x = r_x - r_f$. Then the optimal weight is

\[w_x = \frac{\mu_x}{\gamma \sigma_x^2}.\]

Typical values for $\gamma$ range from two to five. Higher values for $\gamma$ indicate higher risk aversion.

Examples

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

[ for gamma in [1.0 .. 5.0 ] do 
    {| Gamma = gamma
       RiskyWeight = meanVarianceWeight 0.075 0.14 gamma |} ]

val meanVarianceWeight: mu: float -> sigma: float -> gamma: float -> float
val it: {| Gamma: float; RiskyWeight: float |} list =
  [{ Gamma = 1.0
     RiskyWeight = 3.826530612 }; { Gamma = 2.0
                                    RiskyWeight = 1.913265306 };
   { Gamma = 3.0
     RiskyWeight = 1.275510204 }; { Gamma = 4.0
                                    RiskyWeight = 0.9566326531 };
   { Gamma = 5.0
     RiskyWeight = 0.7653061224 }]

Kelly Criterion

The mean-variance optimal weight has a similar form to the optimal weight from the Kelly Criterion (Kelly, 1956). The Kelly Criterion is the weight that maximizes the expected geometric growth rate of an investor's wealth or, equivalently, the expected value of log wealth. It is often seen in industry as a formula for determining the optimal fraction of your wealth to bet on a risky asset. Originally developed for gambling, it can also be used for asset management.

The objective is to maximize wealth, which grows as

\[(1+w r_1)(1+w r_2)...(1+w r_T)=\prod_{i=1}^T (1 + w_i r_i)\]

It can be shown that the expected long-term growth rate is

\[\approx w \mu - \frac{1}{2} w^2 \sigma^2\]

This is maximized when

\[w = \frac{\mu}{\sigma^2}\]

where $\mu$ is the expected excess return of the risky asset and $\sigma^2$ is the variance of the risky asset.

let kellyWeight mu sigma =
    mu/(sigma**2.0)

kellyWeight 0.075 0.14

val kellyWeight: mu: float -> sigma: float -> float
val it: float = 3.826530612

Practice: Use a loop to calculate the Kelly Criterion weight for a range of $\mu$ and $\sigma$ values.

First, hold $\sigma$ fixed and vary $\mu$ from 0.05 to 0.10 by 0.01. What happens to the optimal weight as $\mu$ increases?

// Answer here

Second, hold $\mu$ fixed and vary $\sigma$ from 0.1 to 0.2 by 0.01. What happens to the optimal weight as $\sigma$ increases?

// Answer here

Simulating results from different portfolio rules

We can simulate the results of different portfolio rules, using the mean-variance weight and varying $\gamma$. The Kelly Criterion corresponds to $\gamma = 1$.

let seed = 99
Random.SetSampleGenerator(Random.RandThreadSafe(seed))

// Let's start with this sample of returns
let rnorm1 = Distributions.Continuous.Normal.Init 0.01 1.0

let careerLength = 30
let draws =
    [ for draw in [1 .. 100] do 
        [ for year in [1 .. 10_000] do
            rnorm1.Sample() ]]

Those are our returns. Let's calculate it for a particular gamma.

let gamma = 2.0
let ww = meanVarianceWeight 0.01 1.0 gamma

let investmentResult (riskyWeight: float) (returns: float list)  =
    let mutable wealth = 1.0
    for yearReturn in returns do 
        let newWealth = wealth*(1.0 + riskyWeight*yearReturn)
        // If we go bankrupt, we are done
        wealth <- max 0.0 newWealth
    // This is the last wealth value
    wealth

investmentResult  1.0 [0.1; 0.1; 0.1]

1.331

investmentResult  ww [0.1; 0.1; 0.1]

1.00150075

Now do it for all our draws.

let myResult =
    [ for draw in draws do investmentResult  ww draw]

myResult

[2.837887967; 1.023004546; 1.643739485; 2.12179225; 1.436309156; 2.346815953;
 0.8786401356; 3.792752596; 1.815664246; 2.972106488; 1.517589968; 2.72606676;
 1.155107973; 1.410694065; 1.632517331; 1.947730141; 2.279910946; 1.481477892;
 2.086684258; 0.538281238; 2.424849205; 1.468674191; 1.143783327; 1.833370522;
 1.6039682; 1.368936275; 0.616193267; 1.263363465; 1.947484509; 1.297083759;
 0.8644086517; 1.754023531; 1.036148661; 3.989536962; 0.9810447439; 3.340688473;
 1.148285313; 1.288333618; 1.734002934; 1.23840977; 2.793957289; 1.001289703;
 1.461894733; 3.625290137; 2.028797723; 2.550341752; 1.306375484; 1.828023863;
 1.044705207; 1.150132042; 1.927413813; 1.461602833; 1.065465273; 1.045489968;
 1.014470884; 1.671204029; 1.284125979; 2.096228342; 0.9096693827; 3.574390319;
 0.9957710939; 1.800322085; 2.254395349; 0.4030964431; 2.036395802; 1.689207898;
 1.345681325; 0.665578875; 0.440331489; 0.918066828; 1.896234599; 1.664944745;
 5.056735953; 0.8511369538; 3.299352623; 0.9708549067; 4.549881402; 2.913110159;
 0.5309154183; 1.944486473; 4.581272808; 3.034970632; 0.9535913349; 1.787086779;
 0.734172244; 1.464532329; 1.17019237; 0.8295362389; 1.195536513; 0.6561338853;
 1.353833407; 1.084669355; 0.666350047; 1.181805047; 1.374376343; 0.7532359753;
 1.322688859; 2.582040177; 2.217090877; 0.9919548743]

Now let's calculate some statistics.

type SimulationSummary =
    { Gamma: float 
      AvgLogWealth: float 
      AvgWealth: float
      AvgGeometricGrowth: float 
      MinWealth: float
      FractionBankrupt: float
      FractionLoseMoney: float }

let calcSummary (gamma: float) (nPeriods: int) (wealths: list<float>) =
    { Gamma = gamma
      AvgLogWealth = wealths |> List.map log |> List.average
      AvgWealth = wealths |> List.average
      AvgGeometricGrowth = 
        wealths 
        |> List.map (fun w -> w**(1.0/float nPeriods) - 1.0) 
        |> List.average
      MinWealth = wealths |> List.min
      FractionBankrupt = 
        let bankrupts = wealths |> List.filter (fun w -> w = 0.0)
        float bankrupts.Length / float wealths.Length
      FractionLoseMoney = 
        let lostMoney = wealths |> List.filter (fun w -> w < 1.0)
        float lostMoney.Length / float wealths.Length  
      }

calcSummary gamma careerLength myResult

val it: SimulationSummary = { Gamma = 2.0
                              AvgLogWealth = 0.3996341337
                              AvgWealth = 1.70989802
                              AvgGeometricGrowth = 0.01356404241
                              MinWealth = 0.4030964431
                              FractionBankrupt = 0.0
                              FractionLoseMoney = 0.21 }

Now a function to do it for a particular gamma

let gammaRecord (mu: float) (sigma: float) (draws: list<list<float>>) (gamma: float) =
    let w = meanVarianceWeight mu sigma gamma
    let myResult = draws |> List.map (investmentResult w)
    let careerLength = draws[0] |> List.length
    calcSummary gamma careerLength myResult

gammaRecord 0.01 1.0 draws 3.0

{ Gamma = 3.0
  AvgLogWealth = 0.2942149837
  AvgWealth = 1.426273684
  AvgGeometricGrowth = 2.942253804e-05
  MinWealth = 0.5613435247
  FractionBankrupt = 0.0
  FractionLoseMoney = 0.17 }

Now do it for many gammas.

let ruleResults =
    let gammas = [0.5 .. 0.1 .. 1.0] @ [1.25 .. 0.25 .. 2.0]
    [ for gamma in gammas do
        gammaRecord 0.01 1.0 draws gamma ]

let veryLongRunChart =
    ruleResults
    |> List.map (fun x -> 
        let kellyFraction = 1.0 / x.Gamma
        kellyFraction, x.AvgGeometricGrowth)
    |> Chart.Line
    |> Chart.withXAxisStyle("Kelly fraction")
    |> Chart.withYAxisStyle("Average geometric growth")

veryLongRunChart

A couple takeaways:

Even an investor who is not very risk averse should not bet more than 1x the Kelly bet. This is particularly true if we take into account uncertainty about expected returns and variances.
It takes a very long time to converge to the kelly result.

Now let's try with representative returns, monthly rebalancing.

let monthlyMu = 0.075/12.0
let monthlySigma = 0.14/sqrt 12.0
let rnorm2 = Distributions.Continuous.Normal.Init monthlyMu monthlySigma
let investmentCareer = 30
let drawsRealistic =
    [ for draw in [1 .. 1000] do 
        [ for year in [1 .. investmentCareer*12] do
            rnorm2.Sample() ]]

let realisticResults =
    let gammas = [0.75 .. 0.25 .. 3.0]
    [ for gamma in gammas do
        gammaRecord monthlyMu monthlySigma drawsRealistic gamma ]

The results:

realisticResults

val it: SimulationSummary list =
  [{ Gamma = 0.75
     AvgLogWealth = 3.450913017
     AvgWealth = 404528.9387
     AvgGeometricGrowth = 0.00969663175
     MinWealth = 0.0003612225864
     FractionBankrupt = 0.0
     FractionLoseMoney = 0.19 }; { Gamma = 1.0
                                   AvgLogWealth = 4.129875279
                                   AvgWealth = 17346.91124
                                   AvgGeometricGrowth = 0.01157315897
                                   MinWealth = 0.01479285394
                                   FractionBankrupt = 0.0
                                   FractionLoseMoney = 0.087 };
   { Gamma = 1.25
     AvgLogWealth = 4.011820663
     AvgWealth = 1924.313345
     AvgGeometricGrowth = 0.011228559
     MinWealth = 0.07365774793
     FractionBankrupt = 0.0
     FractionLoseMoney = 0.05 }; { Gamma = 1.5
                                   AvgLogWealth = 3.730401519
                                   AvgWealth = 446.1184822
                                   AvgGeometricGrowth = 0.01043149726
                                   MinWealth = 0.1711251698
                                   FractionBankrupt = 0.0
                                   FractionLoseMoney = 0.036 };
   { Gamma = 1.75
     AvgLogWealth = 3.433054336
     AvgWealth = 165.4387707
     AvgGeometricGrowth = 0.009593159598
     MinWealth = 0.2813529385
     FractionBankrupt = 0.0
     FractionLoseMoney = 0.024 }; { Gamma = 2.0
                                    AvgLogWealth = 3.158107278
                                    AvgWealth = 81.67908898
                                    AvgGeometricGrowth = 0.008819737755
                                    MinWealth = 0.3864483271
                                    FractionBankrupt = 0.0
                                    FractionLoseMoney = 0.016 };
   { Gamma = 2.25
     AvgLogWealth = 2.913705061
     AvgWealth = 48.21241209
     AvgGeometricGrowth = 0.008133277799
     MinWealth = 0.4789578792
     FractionBankrupt = 0.0
     FractionLoseMoney = 0.014 }; { Gamma = 2.5
                                    AvgLogWealth = 2.699006896
                                    AvgWealth = 31.99927166
                                    AvgGeometricGrowth = 0.00753093058
                                    MinWealth = 0.5573887274
                                    FractionBankrupt = 0.0
                                    FractionLoseMoney = 0.011 };
   { Gamma = 2.75
     AvgLogWealth = 2.510690679
     AvgWealth = 23.03190791
     AvgGeometricGrowth = 0.007003073346
     MinWealth = 0.6227931384
     FractionBankrupt = 0.0
     FractionLoseMoney = 0.008 }; { Gamma = 3.0
                                    AvgLogWealth = 2.345068506
                                    AvgWealth = 17.57720025
                                    AvgGeometricGrowth = 0.006539170532
                                    MinWealth = 0.6770275267
                                    FractionBankrupt = 0.0
                                    FractionLoseMoney = 0.006 }]

namespace System

Multiple items
namespace FSharp

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

namespace FSharp.Stats

namespace Plotly

namespace Plotly.NET

val invest: decimal

val borrow: decimal

val ret: decimal

val rf: decimal

val result: decimal

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

val w: decimal

val result2: decimal

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

val mu: float

val sigma: float

val gamma: float

val kellyWeight: mu: float -> sigma: float -> float

val seed: int

Multiple items
type Random = new: unit -> unit + 1 overload member GetItems<'T> : choices: ReadOnlySpan<'T> * length: int -> 'T array + 2 overloads member Next: unit -> int + 2 overloads member NextBytes: buffer: byte array -> unit + 1 overload member NextDouble: unit -> float member NextInt64: unit -> int64 + 2 overloads member NextSingle: unit -> float32 member Shuffle<'T> : values: Span<'T> -> unit + 1 overload static member Shared: Random
<summary>Represents a pseudo-random number generator, which is an algorithm that produces a sequence of numbers that meet certain statistical requirements for randomness.</summary>

--------------------
Random() : Random
Random(Seed: int) : Random

val SetSampleGenerator: rg: Random.IRandom -> unit
<summary> Sets the random number generator used for sampling. </summary>

Multiple items
type RandThreadSafe = interface IRandom new: unit -> RandThreadSafe + 1 overload val mutable rnd: ThreadLocal<Random>

--------------------
new: unit -> Random.RandThreadSafe
new: n: int -> Random.RandThreadSafe

val rnorm1: Distributions.ContinuousDistribution<float,float>

namespace FSharp.Stats.Distributions

namespace FSharp.Stats.Distributions.Continuous

type Normal = static member CDF: mu: float -> sigma: float -> x: float -> float static member CheckParam: mu: float -> sigma: float -> unit static member Estimate: samples: float seq -> ContinuousDistribution<float,float> static member Init: mu: float -> sigma: float -> ContinuousDistribution<float,float> static member InvCDF: mu: float -> sigma: float -> p: float -> float static member Mean: mu: float -> sigma: float -> float static member Mode: mu: float -> sigma: float -> float static member PDF: mu: float -> sigma: float -> x: float -> float static member Sample: mu: float -> sigma: float -> float static member SampleUnchecked: mu: float -> sigma: float -> float ...
<summary> Normal distribution. </summary>

static member Distributions.Continuous.Normal.Init: mu: float -> sigma: float -> Distributions.ContinuousDistribution<float,float>

val careerLength: int

val draws: float list list

val draw: int

val year: int

abstract Distributions.ContinuousDistribution.Sample: unit -> 'b

val ww: float

val investmentResult: riskyWeight: float -> returns: float list -> float

val riskyWeight: float

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

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

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

val returns: float list

type 'T list = List<'T>

val mutable wealth: float

val yearReturn: float

val newWealth: float

val max: e1: 'T -> e2: 'T -> 'T (requires comparison)

val myResult: float list

val draw: float list

type SimulationSummary = { Gamma: float AvgLogWealth: float AvgWealth: float AvgGeometricGrowth: float MinWealth: float FractionBankrupt: float FractionLoseMoney: float }

val calcSummary: gamma: float -> nPeriods: int -> wealths: float list -> SimulationSummary

val nPeriods: int

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

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

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

val wealths: float list

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 log: value: 'T -> 'T (requires member Log)

val average: list: 'T list -> 'T (requires member (+) and member DivideByInt and member Zero)

val w: float

val min: list: 'T list -> 'T (requires comparison)

val bankrupts: float list

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

property List.Length: int with get

val lostMoney: float list

val gammaRecord: mu: float -> sigma: float -> draws: float list list -> gamma: float -> SimulationSummary

val length: list: 'T list -> int

val ruleResults: SimulationSummary list

val gammas: float list

val veryLongRunChart: GenericChart.GenericChart

val x: SimulationSummary

val kellyFraction: float

SimulationSummary.Gamma: float

SimulationSummary.AvgGeometricGrowth: float

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: 'c * [<Runtime.InteropServices.Optional; Runtime.InteropServices.DefaultParameterValue ((null :> obj))>] ?MultiText: 'c 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 'c :> 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.withXAxisStyle: [<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)

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 GenericChart from Plotly.NET
<summary> Module to represent a GenericChart </summary>

val toChartHTML: gChart: GenericChart.GenericChart -> string
<summary> Converts a GenericChart to it HTML representation. The div layer has a default size of 600 if not specified otherwise. </summary>

val monthlyMu: float

val monthlySigma: float

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

val rnorm2: Distributions.ContinuousDistribution<float,float>

val investmentCareer: int

val drawsRealistic: float list list

val realisticResults: SimulationSummary list