Performance evaluation

We're going to evaluate portfolio performance. The common way to do this is to estimate a portfolio's return adjusted for risk using a factor model with tradeable risk factors.

What's a risk factor? These risk factors are portfolios and the idea is that the expected excess return on these risk factors is compensation to investors for bearing the risk inherent in holding those portfolios. For the return variation in these factors to be "risky", it should be something that investors cannot easily diversify. If it was easy to diversify, then investors could put a small bit of the asset in their portfolio and capture the return without affecting portfolio volatility. That would imply being able to increase return without adding risk. Hence the requirement that a factor constitute return variation that is hard to diversify away.

The greater the riskiness of the factor, the greater the factor's expected return (i.e., the risk-return tradeoff). For example, most people feel that stocks are riskier than bonds and indeed stocks have historically had higher returns than bonds.

The risk adjustment involves estimating a portfolio's \(\beta\)'s on different risk factors. These \(\beta\)'s constitute the exposure of the portfolio to the risk factor. If the factor return goes up by 1%, then the portfolio's return goes up by \(\beta \times 1\%\).

We can estimate these \(\beta\)'s by OLS regressions of the portfolio's returns on contemporaneous returns of the risk factors. The slope coefficients on the risk factors are the portfolio's betas on the risk factors. The regression intercept is known as \(\alpha\). It represents the average return of the portfolio that is not explained by the portfolio's \(\beta\)'s on the risk factors. This alpha is the risk-adjusted return.

Intuitively, \(\alpha\) is the average return on a portfolio long the investment you are evaluating and short a portfolio with the same factor risk as that portfolio. If the factors and factor betas accurately measure the portfolio's risk, then the alpha is the portfolio's return that is unrelated to the portfolio's risk. Investors like positive alphas because that implies that the portfolio's return is higher than what investors require for bearing the portfolio's risk.

One thing to keep in mind is that throughout this discussion, we have discussed things from the perspective of arbitrage. That is, like a trader. We have not made any assumptions about utility functions or return distributions. This is the Arbitrage Pricing Theory (APT) of Stephen Ross (1976). He was motivated by the observation that

"... on theoretical grounds it is difficult to justify either the assumption [in mean-variance anlysis and CAPM] of normality in returns...or of quadratic preferences...and on empirical grounds the conclusions as well as the assumptions of the theory have also come under attack."

The APT way of thinking is less restrictive than economically motivated equilibrium asset pricing models. Which is nice. But it has the cost that it does not tell us as much. With the APT we cannot say precisely what a security's return should be. We can only say that if we go long a portfolio and short the portfolio that replicates its factor exposure, then the alpha shouldn't be too big. But if we're thinking like a trader, that's perhaps most of what we care about anyway.

#r "nuget: FSharp.Stats, 0.5.0"
#r "nuget: FSharp.Data, 5.0.2"
#r "nuget: Plotly.NET, 3.*"
#r "nuget: Plotly.NET.Interactive, 3.*"
#r "nuget: NovaSBE.Finance, 0.5.0"

#r "nuget: Quotes.YahooFinance, 0.0.5"

open System
open FSharp.Data
open Quotes.YahooFinance

open FSharp.Stats
open Plotly.NET
open NovaSBE.Finance
open NovaSBE.Finance.Ols

Environment.CurrentDirectory <- __SOURCE_DIRECTORY__

Data

We get the Fama-French 3-Factor asset pricing model data.

let ff3 = French.getFF3 French.Frequency.Monthly

Let's get a portfolio to analyze.

VBR is the Vanguard Small Cap Value ETF. It invests in small-cap value stocks.

type Return = { YearMonth: DateTime; Return: float }

let getReturns (ticker: string) =
    YahooFinance.History(
        ticker,
        startDate = DateTime(2010, 1, 1),
        endDate = DateTime(DateTime.Now.Year - 1, 12, 31),
        interval = Monthly
    )
    |> List.sortBy (fun x -> x.Date)
    |> List.pairwise
    |> List.map (fun (yesterday, today) ->
        { YearMonth = DateTime(today.Date.Year, today.Date.Month, 1)
          Return = today.AdjustedClose / yesterday.AdjustedClose - 1.0 })
    |> List.toArray

let vbr = getReturns "VBR"

We'll also use VTI as a proxy for the market later.

let vti = getReturns "VTI"

A function to accumulate simple returns.

let cumulativeReturn (xs: seq<DateTime * float>) =
    let mutable cr = 1.0

    [ for (dt, r) in xs do
          cr <- cr * (1.0 + r)
          dt, cr - 1.0 ]

Plot of vbr cumulative return.

let vbrChart =
    vbr
    |> Array.map (fun x -> x.YearMonth, x.Return)
    |> cumulativeReturn
    |> Chart.Line

You want to work with excess returns. If you have a zero-cost, long-short portfolio then it is already an excess return. But if you have a long portfolio such as we have with VBR, we need to convert it to an excess return.

type ExcessReturn =
    { YearMonth: DateTime
      ExcessReturn: float }

// ff3 indexed by month
let ff3ByMonth =
    ff3 |> Array.map (fun x -> DateTime(x.Date.Year, x.Date.Month, 1), x) |> Map

let vbrExcess =
    vbr
    |> Array.map (fun x ->
        { YearMonth = x.YearMonth
          ExcessReturn = x.Return - ff3ByMonth[x.YearMonth].Rf })

let vtiExcess =
    vti
    |> Array.map (fun x ->
        { YearMonth = x.YearMonth
          ExcessReturn = x.Return - ff3ByMonth[x.YearMonth].Rf })

Factor Models

For regression, it is helpful to have the portfolio return data merged into our factor model data.

type RegData =
    {
        Date: DateTime
        /// Make sure Portfolio is an Excess Return
        Portfolio: float
        MktRf: float
        Hml: float
        Smb: float
    }

let regData =
    vbrExcess
    |> Array.map (fun x ->
        let xff3 = ff3ByMonth[x.YearMonth]

        { Date = x.YearMonth
          Portfolio = x.ExcessReturn
          MktRf = xff3.MktRf
          Hml = xff3.Hml
          Smb = xff3.Smb })

One way to evaluate performance is Sharpe ratios.

/// Calculates sharpe ratio of a sequence of excess returns
let sharpe (xs: float seq) = (Seq.mean xs) / (Seq.stDev xs)

let annualizeMonthlySharpe monthlySharpe = sqrt (12.0) * monthlySharpe

Our portfolio.

regData |> Array.map (fun x -> x.Portfolio) |> sharpe |> annualizeMonthlySharpe

The market.

regData |> Array.map (fun x -> x.MktRf) |> sharpe |> annualizeMonthlySharpe

The HML factor.

regData |> Array.map (fun x -> x.Hml) |> sharpe |> annualizeMonthlySharpe

Now we can estimate our factor models using OLS.

let capmEstimate = Ols("Portfolio ~ MktRf", regData).fit()
let ff3Estimate = Ols("Portfolio ~ MktRf + Hml + Smb", regData).fit()

CAPM results.

capmEstimate.summary ()

What's the interpretation of the alpha?
What's the interpretation of the beta?
If you want to replicate VBR with the MKT factor and risk-free bonds, what are the weights that you would use?

Fama-French 3-Factor model results

ff3Estimate.summary ()

What's the interpretation of the alpha?
What's the interpretation of the factor betas?
If you want to replicate VBR with risk free bonds and the MKT, HML, and SMB factors, what are the weights that you would use?

Practice: What is the expected annual return of VBR? When answering this,

Assume that your alpha and beta estimates for the 3 factor model explaining VBR returns are accurate.

Use the average annual premia on the Fama and French factors from 1926 until end of 2022 as your estimate of the factors' expected returns.

// Answer here

You will probably see that the CAPM \(R^2\) is lower than the Fama-French \(R^2\). This means that you can explain more of the portfolio's returns with the Fama-French model. Or in trader terms, you can hedge the portfolio better with the multi-factor model.

Let's turn things around and see if we can explain the HML factor.

let hmlRegData =
    let vbrByMonth = vbrExcess |> Array.map (fun x -> x.YearMonth, x) |> Map

    [| for vti in vtiExcess do
           {| YearMonth = vti.YearMonth
              Hml = ff3ByMonth[vti.YearMonth].Hml
              Vti = vti.ExcessReturn
              Vbr = vbrByMonth[vti.YearMonth].ExcessReturn |} |]

hmlRegData[..3]

Explain the HML factor with VTI and VBR

let hmlModel = Ols("Hml ~ Vti + Vbr", hmlRegData).fit()
hmlModel.summary()

What's the interpretation of the alpha?
What's the interpretation of the betas?
If you want to replicate HML using VTI, VBR, and risk-free bonds, what are the weights you would use?

Note:

HML average return.

hmlRegData |> Array.averageBy (fun x -> 12.0 * x.Hml)

VTI average return

hmlRegData |> Array.averageBy (fun x -> 12.0 * x.Vti)

VBR average return.

hmlRegData |> Array.averageBy (fun x -> 12.0 * x.Vbr)

Information Ratios

We want residuals so that we can estimate information ratios.

let capmResiduals = capmEstimate.resid
let ff3Residuals = ff3Estimate.resid

In general I would write a function to do this. Function makes it a bit simpler to follow. It's hard for me to read the next few lines and understand what everything is. Too much going on.

let capmAlpha = 12.0 * capmEstimate.coefs["Intercept"]
let capmStDevResiduals = sqrt (12.0) * (Seq.stDev capmResiduals)
let capmInformationRatio = capmAlpha / capmStDevResiduals

val capmAlpha: float = -0.03043829182
val capmStDevResiduals: float = 0.07553316546
val capmInformationRatio: float = -0.4029791634

let ff3Alpha = 12.0 * ff3Estimate.coefs["Intercept"]
let ff3StDevResiduals = sqrt (12.0) * (Seq.stDev ff3Residuals)
let ff3InformationRatio = ff3Alpha / ff3StDevResiduals

val ff3Alpha: float = -0.008830715608
val ff3StDevResiduals: float = 0.03868132804
val ff3InformationRatio: float = -0.228294013

Here is the function version.

let informationRatio monthlyAlpha (monthlyResiduals: float array) =
    let annualAlpha = 12.0 * monthlyAlpha
    let annualStDev = sqrt (12.0) * (Seq.stDev monthlyResiduals)
    annualAlpha / annualStDev

informationRatio capmEstimate.coefs["Intercept"] capmResiduals

val informationRatio:
  monthlyAlpha: float -> monthlyResiduals: float array -> float
val it: float = -0.4029791634

informationRatio ff3Estimate.coefs["Intercept"] ff3Residuals

val it: float = -0.228294013

namespace System

Multiple items
namespace FSharp

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

Multiple items
namespace FSharp.Data

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

namespace Quotes

namespace Quotes.YahooFinance

namespace FSharp.Stats

namespace Plotly

namespace Plotly.NET

namespace NovaSBE

namespace NovaSBE.Finance

module Ols from NovaSBE.Finance

type Environment = static member Exit: exitCode: int -> unit static member ExpandEnvironmentVariables: name: string -> string static member FailFast: message: string -> unit + 1 overload static member GetCommandLineArgs: unit -> string array static member GetEnvironmentVariable: variable: string -> string + 1 overload static member GetEnvironmentVariables: unit -> IDictionary + 1 overload static member GetFolderPath: folder: SpecialFolder -> string + 1 overload static member GetLogicalDrives: unit -> string array static member SetEnvironmentVariable: variable: string * value: string -> unit + 1 overload static member CommandLine: string ...
<summary>Provides information about, and means to manipulate, the current environment and platform. This class cannot be inherited.</summary>

property Environment.CurrentDirectory: string with get, set
<summary>Gets or sets the fully qualified path of the current working directory.</summary>
<exception cref="T:System.ArgumentException">Attempted to set to an empty string ("").</exception>
<exception cref="T:System.ArgumentNullException">Attempted to set to <see langword="null" />.</exception>
<exception cref="T:System.IO.IOException">An I/O error occurred.</exception>
<exception cref="T:System.IO.DirectoryNotFoundException">Attempted to set a local path that cannot be found.</exception>
<exception cref="T:System.Security.SecurityException">The caller does not have the appropriate permission.</exception>
<returns>The directory path.</returns>

val ff3: French.FF3Obs array

module French from NovaSBE.Finance

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

type Frequency = | Daily | Monthly

union case French.Frequency.Monthly: French.Frequency

type Return = { YearMonth: DateTime 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 getReturns: ticker: string -> Return array

val ticker: string

Multiple items
val string: value: 'T -> string

--------------------
type string = String

type YahooFinance = static member Dividends: symbols: string list * ?startDate: DateTime * ?endDate: DateTime * ?interval: Interval * ?displayLogs: bool -> Dividend list static member History: symbol: string * ?startDate: DateTime * ?endDate: DateTime * ?interval: Interval * ?displayLogs: bool -> Quote list + 1 overload static member Meta: symbols: string list * ?startDate: DateTime * ?endDate: DateTime * ?interval: Interval * ?displayLogs: bool -> Meta list

static member YahooFinance.History: symbols: string seq * ?startDate: DateTime * ?endDate: DateTime * ?interval: Interval * ?displayLogs: bool -> Quote list
static member YahooFinance.History: symbol: string * ?startDate: DateTime * ?endDate: DateTime * ?interval: Interval * ?displayLogs: bool -> Quote list

property DateTime.Now: DateTime with get
<summary>Gets a <see cref="T:System.DateTime" /> object that is set to the current date and time on this computer, expressed as the local time.</summary>
<returns>An object whose value is the current local date and time.</returns>

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>

union case Interval.Monthly: Interval

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

val x: Quote

Quote.Date: DateTime

val pairwise: list: 'T list -> ('T * 'T) list

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

val yesterday: Quote

val today: Quote

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>

Quote.AdjustedClose: float
<summary> Split and dividend adjusted Close price </summary>

val toArray: list: 'T list -> 'T array

val vbr: Return array

val vti: Return array

val cumulativeReturn: xs: (DateTime * float) seq -> (DateTime * float) list

val xs: (DateTime * float) seq

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

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

val mutable cr: float

val dt: DateTime

val r: float

val vbrChart: GenericChart.GenericChart

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

val x: Return

Return.YearMonth: DateTime

Return.Return: 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)

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>

type ExcessReturn = { YearMonth: DateTime ExcessReturn: float }

val ff3ByMonth: Map<DateTime,French.FF3Obs>

val x: French.FF3Obs

French.FF3Obs.Date: DateTime

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

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

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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 vbrExcess: ExcessReturn array

val vtiExcess: ExcessReturn array

type RegData = { Date: DateTime Portfolio: float MktRf: float Hml: float Smb: float }

module Portfolio from NovaSBE.Finance

val regData: RegData array

val x: ExcessReturn

val xff3: French.FF3Obs

ExcessReturn.YearMonth: DateTime

ExcessReturn.ExcessReturn: float

French.FF3Obs.MktRf: float

French.FF3Obs.Hml: float

French.FF3Obs.Smb: float

val sharpe: xs: float seq -> float
Calculates sharpe ratio of a sequence of excess returns

val xs: float seq

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 mean: items: 'T seq -> 'U (requires member (+) and member Zero and member DivideByInt and member (/))
<summary> Computes the population mean (Normalized by N) </summary>
<param name="items">The input sequence.</param>
<remarks>Returns default value if data is empty or if any entry is NaN.</remarks>
<returns>population mean (Normalized by N)</returns>

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 annualizeMonthlySharpe: monthlySharpe: float -> float

val monthlySharpe: float

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

val x: RegData

RegData.Portfolio: float
Make sure Portfolio is an Excess Return

RegData.MktRf: float

RegData.Hml: float

val capmEstimate: RegressionResults

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>

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

val ff3Estimate: RegressionResults

member RegressionResults.summary: ?yname: string * ?xname: string seq * ?title: string * ?alpha: float * ?slim: bool -> string

val hmlRegData: {| Hml: float; Vbr: float; Vti: float; YearMonth: DateTime |} array

val vbrByMonth: Map<DateTime,ExcessReturn>

val vti: ExcessReturn

val hmlModel: RegressionResults

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

val x: {| Hml: float; Vbr: float; Vti: float; YearMonth: DateTime |}

anonymous record field Hml: float

anonymous record field Vti: float

anonymous record field Vbr: float

val capmResiduals: float array

property RegressionResults.resid: float array with get
<summary> The model residuals </summary>

val ff3Residuals: float array

val capmAlpha: float

property RegressionResults.coefs: Map<string,float> with get
<summary> The estimated model coefficients </summary>

val capmStDevResiduals: float

val capmInformationRatio: float

val ff3Alpha: float

val ff3StDevResiduals: float

val ff3InformationRatio: float

val informationRatio: monthlyAlpha: float -> monthlyResiduals: float array -> float

val monthlyAlpha: float

val monthlyResiduals: float array

type 'T array = 'T array

val annualAlpha: float

val annualStDev: float