Introduction to Portfolio Theory
Econ 424/Amath 462
Eric Zivot
Summer 2013
Updated: July 30, 2013

© Eric Zivot 2006

Portfolios of 2 Risky Assets# Set asset return distribution parameters> mu.A = 0.175> sig.A
sig A = 00.258
258> sig2.A = sig.A^2> mu.B = 0.055> sig.B = 0.115
0 11> sig2.B = sig.B^2> rho.AB = -0.164> sig.AB = rho.AB*sig.A*sig.B© Eric Zivot 2006

Example Portfolio 1: Long Only
# specify equally weighted portfolio> x.A = 0.5> x.B = 0.5# compute portfolio mean, variance and sd> mu.p1 = x.A*mu.A + x.B*mu.B> sig2.p1 = x.Ax.A^22 * sig2.A + x.B^2x.B 2 * sig2.B + 2*x.A*x.B*sig.AB
2 x.A x.B sig.AB> sig.p1 = sqrt(sig2.p1)> mu.p1[1] 0.115> sig2.p1[1] 0.01751> sig.p1[1] 0.1323© Eric Zivot 2006

Example Portfolio: Long-Short# specify long-short portfolio> x.A = 1.5> x.B = -0.5
# compute portfolio mean, variance and sd> mu.p2 = x.A*mu.A + x.B*mu.B> sig2.p2 = x.A^2 * sig2.A + x.B^2 * sig2.B + 2*x.A*x.B*sig.AB> sigsig.p2
p2 = sqrt(sig2.p2)
sqrt(sig2 p2)> mu.p2[1] 0.235> sig2.p2[1] 0.1604> sig
sig.p2p2[1] 0.4005© Eric Zivot 2006

Asset and Portfolio VaR> w0 = 100000# Individual asset VaR> VaR.AV R A = (mu.A(A + sig.A*qnorm(0.05))*w0i A*(0 05))* 0> VaR.A[1] -24937> VaR.B = (mu.B + sig.B*qnorm(0.05))*w0> VaR.B[1] -13416# Asset weighted VaR for equally weighted portfolio> x.A*VaR.A + x.B*VaR.B[1] -19177# VaR for equally weighted portfolio> VaR.p1 = (mu.p1 + sig.p1*qnorm(0.05))*w0> VaR
VaR.p1p1[1] -10268© Eric Zivot 2006

Create Portfolio Frontier> x.A = seq(from=-0.4, to=1.4, by=0.1)> x.B = 1 - x.A> mu.p = x.A*mu.A + x.B*mu.B> sig2.p = x.A^2 * sig2.A + x.B^2 * sig2.B ++2*x.A*x.B*sig.AB> sig.p = sqrt(sig2.p)> plot(sig.p, mu.p, type="b", pch=16,+
ylim=c(0, max(mu.p)), xlim=c(0, max(sig.p)),+xlab=expression(sigma[p]),ylab=expression(mu[p]),+col=c(rep("red", 6), rep("green", 13)))> text(x=sig.A, y=mu.A, labels="Asset A", pos=4)> text(x=sig.B, y=mu.B, labels="Asset B", pos=4)© Eric Zivot 2006

Portfolio Frontier

ExpectedGain

 AB  0.164

Risk© Eric Zivot 2006Portfolio Frontier as function of 

© Eric Zivot 2006

Portfolio Frontier as function of 

© Eric Zivot 2006

Expectedp
Gain

Efficient Portfolios

 AB  p0.164
Efficient
portfolios(g(green)) have thehighest expected return for a given
level of risk

Inefficient portfolios are in redRisk© Eric Zivot 2006

Computing the Minimum Variance Portfolio# minimum variance portfolio> xA.min = (sig2.B - sig.AB)/(sig2.A + sig2.B - 2*sig.AB)> xB.min = 1 - xA.min> xA.mini[1] 0.2021> xB.min[1] 0.7979# compute mean, variance and volatility> mu.p.min
mu p min = xAxA.min
min*mumu.A
A + xBxB.min
min*mumu.B
B> sig2.p.min = xA.min^2 * sig2.A + xB.min^2 * sig2.B ++2*xA.min*xB.min*sig.AB> sig.p.min = sqrt(sig2.p.min)> mu.p.min[1] 0.07925
g p> sig.p.min[1] 0.09782© Eric Zivot 2006

Global Minimum Variance Portfolio

min  p2  xA2 A2  xB2 B2  2 xA xB ABxA , xB

s.t. xA  xB  1© Eric Zivot 2006

Portfolios of T-Bills and 1 Risky Asset# Risk-free (T-Bill) rate
> r.f = 0.03
# T-bills + asset A> x.A = seq(from=0, to=1.4, by=0.1)
p= r.f + x.A*(mu.A(- r.f))> mu.p.A> sig.p.A = x.A*sig.A> sharpe.A = (mu.A - r.f)/sig.A> sharpe.A[1] 00.562562# T-bills + asset B> x.B = seq(from=0, to=1.4, by=0.1)> mu.p.B = r.f + x.B*(mu.B - r.f)
> sig.p.B = x.B*sig.B> sharpe.B = (mu.B - r.f)/sig.B> sharpe.B[1] 0.2174© Eric Zivot 2006

Portfolios of T-Bills and 1 Risky Asset
Portfolios of T-Billsand asset A areefficientffi i t relativel ti tto
portfolios of T-Billsand asset B

SRA = 0.562

SRB = 0.217© Eric Zivot 2006

Portfolios of T-Bills and 2 Risky Assets
Portfolios of asset A andasset B onlyyPortfolios of T-Bills andasset A

Portfolios of T-Bills and asset B© Eric Zivot 2006

Portfolios of T-Bills and 2 Risky Assets
Portfolios of T-Bills andtangency portfolio

Tangency portfolio has
the highest Sharpe slope© Eric Zivot 2006

Efficient Portfolios and Risk Preferences-100% T-Bills and200% tangency
portfolio

90% T-Bills and 10%tangency portfolio© Eric Zivot 2006

Interpreting Efficient Portfolios
e2: Efficient portfolio withsame mean as asset B. xt =0.31, xf = 0.69

e1: Efficient portfolio withsame volatility as asset B.
xt = 0.92, xf = 0.08

© Eric Zivot 2006