Short version of MPT

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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