Introduction toInvestmentPortfoliosD Keck

November 15, 2016

Mean and Variance

X: random variable with values x

MEAN(X) = =μX1n ∑n

i=1 xi

VAR(X) = = ( −σX2 1

n ∑ni=1 xi μX)2

VAR(X) = = ( − )( − )σX2 1

n ∑ni=1 xi μX xi μX

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Covariance

X: random variable with values xY: random variable with values y

COV(X, Y) = = = ( − )( − )σXY2 σXY

1n ∑n

i=1 xi μX yi μY

COV(X + Y) = + 2 ⋅ +σX2 σXY σY

2

COV( ⋅ X + ⋅ Y) = + 2 ⋅ ⋅ ⋅ +wX wY ⋅wX2 σX

2 wX wY σXY

⋅wY2 σY

2

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Matrix Algebra

2 random variables X and Y

covariance matrix

each element is

weight vector

C = [ ]σX2

σXY

σXY

σY2

σij

=ρijσij

⋅σi σj

w = [ ]wX

wY

COV( ⋅ X + ⋅ Y) = ⋅ C ⋅ wwX wY wT

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Matrix Algebra

μ = [ ]μX

μY

MEAN( ⋅ X + ⋅ Y) = ⋅ μwX wY wT

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Portfolio of Assets

P is a portfolio with m assets with weights and rates

Each asset has n return rates

[ , , . . .w1 w2 wm]T [ , , . . .r1 r2 rm]T

1 ≤ i ≤ m

= 1∑mi=1 wi

0 ≤ ≤ 1wi

= ⋅ C ⋅ wσP2 wT

= ⋅ rrP wT

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Portfolio Optimization

Minimize:

Subject to:

Plot the efficient frontier varying portfolio returnrate from that if the lowest return asset to thehighest return asset

= ⋅ C ⋅ wσP2 wT

= 1∑mi=1 wi

0 ≤ ≤ 1wi

= ⋅ rr⎯⎯⎯P wT

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Random Portfolios

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r = as.numeric(r[,2])m = length(r)n = 100000main = paste(as.character(n),' portfolios')mm = m + 1 s = as.numeric(s[,2])C = data.matrix(C[,2:mm])sdev=as.numeric(rep(0,n))mret=as.numeric(rep(0,n))for ( i in 1:n) { w=runif(6,0,100) w=w/sum(w) sdev[i]= sqrt(t(w) %*% C %*% w) mret[i] = t(w) %*% r}plot(sdev,mret, pch=19, col = "blue", cex = .6, xlim = c(0,.08points(s, r, pch=19, col = "red", cex = .8)grid()

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