console.log("==============================================================================")
console.log("file: PCM20180415_MinimumVarianceTwoAssetPortfolio  *** PCM *** 2018/04/24 ** ")
console.log("      the numerical example was taken from:                                   ")
console.log("          Cottin, Claudia & Döhler, Sebastian, Risikoanalyse: Modellierung,   ")
console.log("                  Beurteilung und Management von Risiken mit Praxisbeispielen,")
console.log("                  Kap.4.2.2.3, S.181f., Wiesbaden: Vieweg + Teubner, 2009,    ")
console.log("                  ISBN 978-3-8348-0594-2                                      ")
console.log("==============================================================================")
var startTime = Date.now()
var sigmaA = 0.2
console.log("sigmaA = ", sigmaA)
var sigmaB = 0.4
console.log("sigmaB = ", sigmaB)
var muA = 0.1
console.log("muA = ", muA)
var muB = 0.2
console.log("muB = ", muB)
var rhoAB =[1.0, 0.5, 0.0,-0.5,-1.0]
console.log("vector of hypothetical asset correlations"); print(rhoAB)
console.log("==============================================================================")
//-------------------------------------------------------------------------------------------
var runTime = function(startTime) {
  var stopTime = Date.now()
  var runTimeInMsec = (stopTime - startTime)
  var runTimeInSec = runTimeInMsec/1000.
  var runTimeInMin = runTimeInSec/60.
  console.log("runTimeInMsec ---> ", runTimeInMsec)
  console.log("runTimeInSec  ---> ", runTimeInSec)
  console.log("runTimeInMin  ---> ", runTimeInMin)
}
//-------------------------------------------------------------------------------------------
var xMVP_funct = function(rAB) {
  (Math.pow(sigmaB,2)
   - sigmaA*sigmaB*rAB)/(Math.pow(sigmaA,2) - 2*sigmaA*sigmaB*rAB + Math.pow(sigmaB,2))
}
var OneMinus_xMVP_funct = function(rAB) {
  (Math.pow(sigmaA,2)
   - sigmaA*sigmaB*rAB)/(Math.pow(sigmaA,2) - 2*sigmaA*sigmaB*rAB + Math.pow(sigmaB,2))
}
//-------------------------------------------------------------------------------------------
var muMVP = function(xMVP) {
  return (muB - (muB - muA) * xMVP)
}
//-------------------------------------------------------------------------------------------
var varSigmaPSQ = function(x, rho){
  var sigmaASQ = Math.pow(sigmaA, 2)
  var sigmaBSQ = Math.pow(sigmaB, 2)
  var sigmaA_sigmaB_rhoAB = sigmaA*sigmaB*rho
  var sigmaPSQ = Math.pow(x,2)*(sigmaASQ - 2*sigmaA_sigmaB_rhoAB + sigmaBSQ) +
      2*x*(sigmaA_sigmaB_rhoAB - sigmaBSQ) + sigmaBSQ
  return sigmaPSQ
}
//-------------------------------------------------------------------------------------------
var muP = function(x) {
  var mu = x*muA + (1 - x) * muB
  return mu
}
//-------------------------------------------------------------------------------------------
console.log("==============================================================================")
var xs = mapN(function(n){n/100.0 - 1.0}, 301)
var muPs = map(muP, xs)
viz.line(xs, muPs, {xLabel:"Proportion x(A) of asset A", yLabel:"expected return mu(A)"})
console.log("==============================================================================")
map(function(rho){
  console.log("============================================================================")
  console.log("rho_AB = ", rho)
  console.log("==============")
  var xMVP = xMVP_funct(rho)
  console.log("xMVP = ", xMVP)
  var OneMinus_xMVP = OneMinus_xMVP_funct(rho)
  console.log("1-xMVP = ", OneMinus_xMVP)
  var xMVP_check = xMVP + OneMinus_xMVP
  console.log("check: |xMVP+(1-xMVP) - 1| < 10E6 = ", Math.abs(xMVP_check - 1) < 10E6)
  console.log("muMVP = ", muMVP(xMVP))
  var sigmaPSQ_MVP = varSigmaPSQ(xMVP, rho)
  console.log("minimum variance sigma^2_P_MVP = ", sigmaPSQ_MVP)
  console.log("minimum variance sigma_P_MVP = ", Math.sqrt(sigmaPSQ_MVP))
  var sigmaPSQs = map(function(x){varSigmaPSQ(x,rho)}, xs)
  viz.line(xs, sigmaPSQs, {xLabel:"Proportion x(A) of asset A", yLabel:"Risk(P) = sigmaP^2"})
  var sigmaPs = map(function(sigmaPSQ){Math.sqrt(sigmaPSQ)}, sigmaPSQs)
  viz.line(xs,sigmaPs,{xLabel:"Proportion x(A) of asset A", yLabel:"sqrt(Risk(P)) = sigmaP"})
  viz.line(muPs, sigmaPs, {xLabel:"muP", yLabel:"sigmaP"})
  viz.line(sigmaPs, muPs, {xLabel:"sigmaP", yLabel:"muP"})
}, rhoAB)
  console.log("============================================================================")
console.log("runTime = "); display(runTime(startTime))
console.log("==============================================================================")

The problem is described e.g. in Cottin, 2009, ch.4.2.2. From ch. 4.2.2.3 we used their numerical example with two assets with μ(A) = 0.1, μ(B) = 0.2, σ(A) = 0.2, and σ(B) = 0.4. Thus asset B has both a higher return and risk.

For the correlation ρ(A,B) Cottin & Döhler assumed values +1.0, 0.0,-1.0. To that set we added two correlations +0.5 and -0.5.

Output of our WebPPL program is x_MVP (= proportion of asset A in the minimum variance portfolio (MVP)), 1 - x_MVP (= proportion of asset B in the minimum variance portfolio), variance and standard deviation of the MVP, and the expected return μ_MVP of the MVP.

Furthermore, we display the σ-μ and μ-σ diagrams. These diagrams demonstrate the efficient border of portfolio; that is what kind of portfolios can be generated by mixing A and B.

In contrast to Cottin & Döhler and others we use only one constraint. Weights have to sum to one. They need be constrained like 0 <= x_MVP, (1-x_MVP) <= 1.0. In the case of negative weights we have a short position.

This is quite natural when we already possess a portfolio and we only want to readjust weights. Then a negative weight means "sell" and and a weight greater than 1.0 means "buy more".

This special case occured in our analysis when rho(A,B) = +1.0. Cottin & Döhler do not explicitly offer weights. They only write that the minimum variance portfolio should contain only asset A with 100%. Our WebPPL program computes weights x_MVP(A) = x_MVP = 2.0 and x_MVP(B) = (1-x_MVP) = -1.0. That is a recommendation for a portfolio owner to sell asset B and reinvest this budget into asset A.