This document introduces concepts related to investment portfolios including mean, variance, covariance, and matrix algebra. It discusses how to calculate the mean, variance, and covariance of random variables and how these concepts apply to portfolio optimization. Specifically, it explains how to calculate the variance and expected return of a portfolio using covariance matrices and weight vectors. The document also provides examples of generating random portfolios and plotting the efficient frontier.

1. Introduction to
Investment
Portfolios
D Keck
November 15, 2016

2. Mean and Variance
X: random variable with values x
MEAN(X) = =μX
1
n ∑
n
i=1 xi
VAR(X) = = ( −σX
2 1
n ∑
n
i=1 xi μX)2
VAR(X) = = ( − )( − )σX
2 1
n ∑
n
i=1 xi μX xi μX
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3. Covariance
X: random variable with values x
Y: random variable with values y
COV(X, Y) = = = ( − )( − )σXY
2
σXY
1
n
∑n
i=1 xi μX yi μY
COV(X + Y) = + 2 ⋅ +σX
2
σXY σY
2
COV( ⋅ X + ⋅ Y) = + 2 ⋅ ⋅ ⋅ +wX wY ⋅wX
2
σX
2
wX wY σXY
⋅wY
2
σY
2
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4. Matrix Algebra
2 random variables X and Y
covariance matrix
each element is
weight vector
C =
[ ]
σX
2
σXY
σXY
σY
2
σij
=ρij
σij
⋅σi σj
w =
[ ]
wX
wY
COV( ⋅ X + ⋅ Y) = ⋅ C ⋅ wwX wY wT
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5. Matrix Algebra
μ =
[ ]
μX
μY
MEAN( ⋅ X + ⋅ Y) = ⋅ μwX wY wT
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6. 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∑m
i=1 wi
0 ≤ ≤ 1wi
= ⋅ C ⋅ wσP
2
wT
= ⋅ rrP wT
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7. Portfolio Optimization
Minimize:
Subject to:
Plot the efficient frontier varying portfolio return
rate from that if the lowest return asset to the
highest return asset
= ⋅ C ⋅ wσP
2
wT
= 1∑m
i=1 wi
0 ≤ ≤ 1wi
= ⋅ rr⎯⎯⎯
P wT
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8. Random Portfolios
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9. r = as.numeric(r[,2])
m = length(r)
n = 100000
main = 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,.
points(s, r, pch=19, col = "red", cex = .8)
grid()
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