(**
[](https://mybinder.org/v2/gh/nhirschey/teaching/gh-pages?filepath=stockslongrun.ipynb)
[](/Teaching//stockslongrun.fsx)
[](/Teaching//stockslongrun.ipynb)
# Stocks for the long run
To explore working with returns, we're going to simulate return data and look at how
our portfolio would grow.
We will do analysis using relatively simple syntax common
to many programming languages to make learning easier.
*)
#r "nuget: FSharp.Stats, 0.5.0"
#r "nuget: Plotly.NET, 3.*"
#r "nuget: Plotly.NET.Interactive, 3.*"
open FSharp.Stats
open Plotly.NET
(**
From 1/1871-1/2023, the US market return was annualized
7.5% with a 14% standard deviation (Robert Schiller data).
*)
let rnorm = Distributions.Continuous.Normal.Init 0.075 0.14
for i = 0 to 5 do
printfn $"{rnorm.Sample()}"(* output:
-0.04713500363917163
-0.10631365578089781
0.05753517879871741
-0.07849860985076322
-0.1481297214692095
-0.24806071516942002*)
(**
Put sampled values in a list.
*)
let returns = [ for i = 1 to 1000 do rnorm.Sample() ]
(**
Plot the distribution of returns.
*)
let hist = Chart.Histogram(returns)
hist(* output:
*)
let percentiles =
[ for p in [0.0; 0.01; 0.05; 0.5; 0.95; 0.99; 1.0] do
let pctl = Quantile.compute p returns
p, pctl ](* output:
val percentiles: (float * float) list =
[(0.0, -0.3829766046); (0.01, -0.2493394921); (0.05, -0.1579542475);
(0.5, 0.07277495078); (0.95, 0.2959389059); (0.99, 0.388052132);
(1.0, 0.5194030419)]*)
for (p, pctl) in percentiles do
printfn $"percentile %.2f{p} is {pctl}"(* output:
percentile 0.00 is -0.38297660460592675
percentile 0.01 is -0.24933949213419362
percentile 0.05 is -0.15795424750990175
percentile 0.50 is 0.07277495077614787
percentile 0.95 is 0.2959389059143977
percentile 0.99 is 0.38805213202301714
percentile 1.00 is 0.519403041929646*)
(**
A draw of 10-years of returns.
*)
let draw10y = [ for i = 1 to 10 do rnorm.Sample() ](* output:
val draw10y: float list =
[0.4727131934; 0.1613510637; 0.1437149688; 0.08306833005; -0.2342341369;
0.05660844679; -0.03931746301; 0.02412983575; 0.08168102008; -0.06920637651]*)
(**
Compute the cumulative product of the returns.
*)
let mutable cumprod = 1.0
for r in draw10y do
cumprod <- cumprod * (1.0 + r)
printfn $"r={round 3 r}"
printfn $" cumprod={round 3 cumprod}"(* output:
r=0.473
cumprod=1.473
r=0.161
cumprod=1.71
r=0.144
cumprod=1.956
r=0.083
cumprod=2.119
r=-0.234
cumprod=1.622
r=0.057
cumprod=1.714
r=-0.039
cumprod=1.647
r=0.024
cumprod=1.687
r=0.082
cumprod=1.824
r=-0.069
cumprod=1.698*)
(**
Let's plot the cumulative return.
*)
let draw10yCR =
let mutable year = 0
let mutable cumprod = 1.0
[ for r in draw10y do
cumprod <- cumprod * (1.0 + r)
year <- year + 1
year, cumprod ](* output:
val draw10yCR: (int * float) list =
[(1, 1.472713193); (2, 1.710337034); (3, 1.956138067); (4, 2.11863119);
(5, 1.622375442); (6, 1.714215595); (7, 1.646816987); (8, 1.686554411);
(9, 1.824313895); (10, 1.698059741)]*)
(**
Functional version of the cumulative return calculation.
*)
((0, 1.0), draw10y)
||> List.scan (fun (year, cumprod) r ->
(year + 1, cumprod * (1.0 + r)))
|> List.tail(* output:
val it: (int * float) list =
[(1, 1.472713193); (2, 1.710337034); (3, 1.956138067); (4, 2.11863119);
(5, 1.622375442); (6, 1.714215595); (7, 1.646816987); (8, 1.686554411);
(9, 1.824313895); (10, 1.698059741)]*)
(**
Plot the cumulative return.
*)
let draw10yCrPlot = Chart.Line(draw10yCR)
draw10yCrPlot(* output:
*)
(**
> **Practice**: Sample 20 years of returns and plot the cumulative return.
>
*)
// Answer here
(**
Let's simulate 1000 draws of 30 years of returns.
*)
let draw1k =
[ for i = 1 to 1_000 do
[ for y = 1 to 30 do rnorm.Sample() ] ]
let draw1kCR =
[ for life in draw1k do
let mutable accRet = 1.0
let mutable year = 0
[ for r in life do
accRet <- accRet * (1.0 + r)
year <- year+1
year, accRet ] ]
(**
Look at a few observtions from the first draw.
*)
draw1kCR[0][0..5](* output:
val it: (int * float) list =
[(1, 0.9762929061); (2, 0.9619054462); (3, 1.039451857); (4, 1.0884101);
(5, 1.149589257); (6, 1.585591653)]*)
(**
Plot the first draw
*)
Chart.Line(draw1kCR[0])(* output:
*)
(**
Plot the second draw
*)
Chart.Line(draw1kCR[1])(* output:
*)
(**
Plot the first and second together
*)
[ Chart.Line(draw1kCR[0])
Chart.Line(draw1kCR[1]) ]
|> Chart.combine (* output:
*)
(**
Plot all 1000 draws together
*)
[ for x in draw1kCR do Chart.Line(x) ]
|> Chart.combine
|> Chart.withLayoutStyle(ShowLegend=false)(* output:
*)
(**
A more useful version may be to plot the values at the end of the last year.
*)
let terminalValues =
[ for x in draw1kCR do
let (yr, ret) = x[x.Length-1]
ret ]
terminalValues |> Chart.Histogram(* output:
*)
(**
What is the chance we lose money?
*)
let nLoseMoney =
terminalValues
|> List.filter (fun x -> x < 1.0)
|> List.length
|> float
let chanceLose = nLoseMoney / (float terminalValues.Length)
printfn $"chance lose money=%.3f{chanceLose}"(* output:
chance lose money=0.003*)
(**
What is the chance we double our money?
*)
let nDoubleMoney =
terminalValues
|> List.filter (fun x -> x >= 2.0)
|> List.length
|> float
let chanceDouble = nDoubleMoney / (float terminalValues.Length)
printfn $"chance double money=%.3f{chanceDouble}"(* output:
chance double money=0.954*)
(**
> **Practice**: What is the chance we earn at least a 10% compound rate of return per year?
>
*)
// Answer here
(**
Now let's say we have 1m EUR and we want to live off of 50k EUR per year.
What is the chance that we can do that?
*)
let expenses = 50_000.0
let initialWealth = 1_000_000.0
let wealthEvolution =
[ for life in draw1k do
let mutable wealth = initialWealth
[ for r in life do
// We'll take expenses out at the start of the year.
if wealth > expenses then
wealth <- (wealth - expenses) * (1.0 + r)
else
wealth <- 0.0
wealth ] ]
let terminalWealth = [ for x in wealthEvolution do x[x.Length-1] ]
(**
What's the chance that we don't have enough money?
*)
let nBroke =
terminalWealth
|> List.filter (fun x -> x <= 0.0)
|> List.length
|> float
let chanceBroke = nBroke / (float terminalWealth.Length)
printfn $"chance broke=%.3f{chanceBroke}"(* output:
chance broke=0.130*)
(**
> **Practice**: What is our chance of going broke if the market's standard deviation is 20% per year?
>
*)
// Answer here
(**
> **Practice**: What is our chance of going broke if the market's
return is 5% per year?
>
*)
// Answer here