Econ 424
Introduction to Portfolio Theory

Eric Zivot

November 13, 2014

Introduction to Portfolio Theory
Investment in Two Risky Assets
 = simple return on asset A
 = simple return on asset B
0 = initial wealth
Assumptions•  and  are described by the CER model ∼ iid ( 2)  =  cov(  ) =   cor(  ) = • Investors like high [] = • Investors dislike high var() = 2• Investment horizon is one period (e.g., one month or one year)
Note: Traditionally in portfolio theory, returns are simple and not continuously
compoundedPortfolios$ in A
0$ in B
 = share of wealth in asset B =0
 = share of wealth in asset A =Long position   0
Short position  0 or   0Assumption: Allocate all wealth between assets A and B
 +  = 1
Portfolio Return =  +  

Portfolio Distribution
 = [] =  +  2 + 2  2 + 2  
2 = var() = 2   2 + 2  2 + 2  = 2     ∼ iid ( 2)
End of Period Wealth1 = 0(1 + ) = 0(1 +  +   )
1 ∼ (0(1 + ) 202)

Result: Portfolio SD is not a weighted average of asset SD unless  = 1 :³

´122222
 =  +   + 2  6=  +   for  6= 1If  = 1 then =   = and2 + 2  2 + 2   
2 = 2    = ( +   )2⇒  =  +  

Example Data = 0175  = 0055
2 = 0067  2 = 0013 = 0258  = 0115
 = −0004875 == −0164Note: Asset A has higher expected return and risk than asset B

Example: Long only two asset portfolio
Consider an equally weighted portfolio with  =  = 05 The expected
return, variance and volatility are = (05) · (0175) + (05) · (0055) = 0115

2 = (05)2 · (0067) + (05)2 · (0013)+ 2 · (05)(05)(−0004875) = 001751√ = 001751 = 01323

This portfolio has expected return half-way between the expected returns on assets A and B, but the portfolio standard deviation is less than half-way between
the asset standard deviations. This reflects risk reduction via diversification.

Example: Long-Short two asset portfolio
Next, consider a long-short portfolio with  = 15 and  = −05 In thisportfolio, asset B is sold short and the proceeds of the short sale are used toleverage the investment in asset A. The portfolio characteristics are = (15) · (0175) + (−05) · (0055) = 0235

2 = (15)2 · (0067) + (−05)2 · (0013)+ 2 · (15)(−05)(−0004875) = 01604√ = 001751 = 04005

This portfolio has both a higher expected return and standard deviation thanasset A

Portfolio Value-at-Risk• Assume an initial investment of $0 in the portfolio of assets A and B.
• Given that the simple return  ∼ ( 2), For  ∈ (0 1) the  ×
100% portfolio value-at-risk is VaR = 0)
= ( + 0 is the  quantile of the distribution of  and   =  quantilewhere 
of  ∼ (0 1).

Relationship between Portfolio VaR and Individual Asset VaR
Result: Portfolio VaR is not a weighted average of asset VaRVaR 6= VaR +  VaR
unless  = 1Asset VaRs for A and B are  = ( +    )VaR = 005
00
 

  = ( +    )VaR = 00500 Portfolio VaR is)
VaR = ( + 0∙´12 ¸
2 + 2  2 + 2   = ( +   ) + 2
0   ³

Portfolio weighted asset VaR is ) +  ( +    )VaR +  VaR = ( + 00  ]= [( +   ) + ( +   ) 0 )  = VaR
6= ( + 
0provided  6= 1
If  = 1 then  =   =  and2 + 2  2 + 2    = (  +   )22 = 2      ⇒  =  +  and soVaR +  VaR = VaRExample: Portfolio VaR and Individual Asset VaRConsider an initial investment of 0 =$100,000. The 5% VaRs on assets Aand B are  = (0175 + 0258(−1645)) · 100 000 = −24 937
VaR005 = 0050

  = (0055 + 0115(−1645)) · 100 000 = −13 416
VaR005 = 0050The 5% VaR on the equal weighted portfolio with  =  = 05 isVaR005 = 005
0 = (0115 + 01323(−1645)) · 100 000 = −10 268and the weighted average of the individual asset VaRs isVaR005 + VaR005 = 05(−24 937)+05(−13 416) = −19 177

Portfolio Frontier
Vary investment shares  and  and compute resulting values of  and2 Plot  against  as functions of  and • Shape of portfolio frontier depends on correlation between assets A and B• If  = −1 then there exists portfolio shares  and  such that2 = 0• If  = 1 then there is no benefit from diversification• Diversification is beneficial even if 0    1

Eﬃcient Portfolios
Definition: Portfolios with the highest expected return for a given level of risk,
as measured by portfolio standard deviation, are eﬃcient portfolios• If investors like portfolios with high expected returns and dislike portfolios
with high return standard deviations then they will want to hold eﬃcient
portfolios• Which eﬃcient portfolio an investor will hold depends on their risk preferences— Very risk averse investors dislike volatility and will hold portfolios near
the global minimum variance portfolio. They sacrifice expected returnfor the safety of low volatility— Risk tolerant investors don’t mind volatility and will hold portfolios thathave high expected returns. They gain expected return by taking on
more volatility.

Globabl Minimum Variance Portfolio• The portfolio with the smallest possible variance is called the global minimum variance portfolio.

• This portfolio is chosen by the most risk averse individuals• To find this portfolio, one has to solve the following constrained minimization problem
2 + 2  2 + 2  min 2 = 2   

  +  = 1

Review of Optimization Techniques: Constrained Optimization
Example: Finding the minimum of a bivariate function subject to a linear constraint =  ( ) = 2 +  2
min  =  ( )  +  = 1
Solution methods:• Substitution• Lagrange multipliers

Method of Substitution
Substitute  =  − 1 in  ( ) and solve univariate minimization =  (  − 1) = 2 + (1 − )2min  (  − 1)
First order conditions
 20=( + (1 − )) = 2 + 2(1 − )(−1)
= 4 − 2⇒  = 05Solving for  = 1 − 05 = 05

Method of Lagrange Multipliers
Idea: Augment function to be minimized with extra terms to impose constraints

1. Put constraints in homogeneous form
+ =1⇒+−1=0
2. Form Lagrangian function
(  ) = 2 +  2 + ( +  − 1)
 = Lagrange multiplier3. Minimize Lagrangian functionmin (  )First order conditions
(  )=2·+0=(  )
0==2·+(  )
0==+−1We have three linear equations in three unknowns. Solving gives
2 = 2 = − ⇒  = 2 − 1 = 0 ⇒  = 05  = 05

Example: Finding the Global Minimum Variance Portfolio
Two methods for solution• Analytic solution using Calculus• Numerical solution— use the Solver in Excel— use R function solve.QP() in package quadprog for quadratic optimization problems with equality and inequality constraints

Calculus Solution
Minimization problem
2 + 2  2 + 2  min 2 = 2   

  +  = 1
Use substitution method with
 = 1 − to give the univariate minimization
2 + (1 −  )2 2 + 2 (1 −  )min 2 = 2 First order conditions´ 2
 ³ 2 2220=
 = + (1 − )  + 2(1 − )


2 − 2(1 −  ) 2 + 2
= 2  (1 − 2)
2 −
min
min = 1 − min⇒  = 2
2 − 2
 + 

Excel Solver Solution
The Solver is an Excel add-in, that can be used to numerically solve generallinear and nonlinear optimization problems subject to equality or inequality
constraints• The solver is made by FrontLine Systems and is provided with Excel• The solver add-in may not be installed in a “default installation” of Excel— Tools/Add-Ins and check the Solver Add-In box— If Solver Add-In box is not available, the Solver Add-In must be installedfrom original Excel installation CD

Portfolios with a Risk Free Asset
Risk Free Asset• Asset with fixed and known rate of return over investment horizon• Usually use U.S. government T-Bill rate (horizons  1 year) or T-Noterate (horizon  1 yr)• T-Bill or T-Note rate is only nominally risk freeProperties of Risk-Free Asset = return on risk-free asset
[ ] =  = constantvar( ) = 0cov(  ) = 0  = return on any asset
Portfolios of Risky Asset and Risk Free Asset = share of wealth in T-Bills = share of wealth in asset B
 +  = 1
 = 1 − Portfolio return =   +  = (1 −  ) +  =  +  ( −  )Portfolio excess return
 −  =  ( −  )Portfolio Distribution
 = [] =  +  ( −  )2
2 = var() = 2  =   ∼ ( 2)

Risk Premium −  = excess expected return on asset B= expected return on risky asset over return on safe asset

For the portfolio of T-Bills and asset B
 −  =  ( −  )= expected portfolio return over T-Bill
The risk premia is an increasing function of the amount invested in asset B.

Leveraged Investment  0   1
Borrow at T-Bill rate to buy more of asset B
Result: Leverage increases portfolio expected return and risk
 =  +  ( −  ) =   ↑ ⇒  &  ↑

Determining Portfolio Frontier
Goal: Plot  vs.  =  +  ( −  )( −  )=  +Ã!
 − 
=  + =   ⇒  =whereÃ

 − !

= SR = Asset B Sharpe Ratio
= excess expected return per unit risk

Remarks• The Sharpe Ratio (SR) is commonly used to rank assets.
• Assets with high Sharpe Ratios are preferred to assets with low Sharpe
Ratios

Eﬃcient Portfolios with 2 Risky Assets and a Risk Free Asset
Investment in 2 Risky Assets and T-Bill
 = simple return on asset A
 = simple return on asset B
 =  = return on T-Bill
Assumptions•  and  are described by the CER model ∼  ( 2)  =  cov(  ) =   corr(  ) = Results:• The best portfolio of two risky assets and T-Bills is the one with the highest
Sharpe Ratio• Graphically, this portfolio occurs at the tangency point of a line drawn
from  to the risky asset only frontier.

• The maximum Sharpe Ratio portfolio is called the “tangency portfolio”

Mutual Fund Separation Theorem
Eﬃcient portfolios are combinations of two portfolios (mutual funds)• T-Bill portfolio• Tangency portfolio - portfolio of assets A and B that has the maximum
Shape ratio

Implication: All investors hold assets A and B according to their proportions in
the tangency portfolio regardless of their risk preferences.

Finding the tangency portfolio
max SR =  − 

subject to =  +  2 + 2  2 + 2  
2 = 2   

1 =  + 
Solution can be found analytically or numerically (e.g., using solver in Excel)

Using the substitution method it can be shown thattan
 =2 − ( −  )( −  )
 

2 + ( −  ) 2 − ( −  +  −  )( −  )  tantan
 = 1 − Portfolio characteristics³tantantan =   +  ´2³´2³´2tantan2tan
2 + 2tantan=  +   

Eﬃcient Portfolios: tangency portfolio plus T-Bills
tan = share of wealth in tangency portfolio
 = share of wealth in T-bills
tan +  = 1
 =  + tan(tan −  ) = tantan

Result: The weights tan and  are determined by an investor’s risk preferences• Risk averse investors hold mostly T-Bills• Risk tolerant investors hold mostly tangency portfolioExample
For the two asset example, the tangency portfolio is

³tan
tan
 = 46  = 054tan
 = (46)(175) + (54)(055) = 011

´2tan

= (46)2(067) + (54)2(013)+ 2(46)(54)(−005)= 0015√tan = 015 = 0124

Eﬃcient portfolios have the following characteristics
 =  + tan(tan −  )= 003 + tan(011 − 003)
 = tantan= tan(0124)

Problem: Find the eﬃcient portfolio that has the same risk (SD) as asset B?
That is, determine tan and  such that =  = 0114 = target risk
Note: The eﬃcient portfolio will have a higher expected return than asset B

Solution:
114 =  = tantan= tan(124)
114⇒ tan == 92
124
 = 1 − tan = 08
Eﬃcient portfolio with same risk as asset B has(92)(46) = 42 in asset A(92)(54) = 50 in asset B
08 in T-Bills
If  = 003 then expected Return on eﬃcient portfolio is = 03 + (92)(11 − 003) = 104

Problem: Assume that  = 003 Find the eﬃcient portfolio that has thesame expected return as asset B. That is, determine tan and  such that =  = 0055 = target expected return
Note: The eﬃcient portfolio will have a lower SD than asset B

Solution:
0055 =  = 003 + tan(11 − 03)
0055 − 003
tan == 31
11 − 03
 = 1 − tan = 69
Eﬃcient portfolio with same expected return as asset B has(31)(46) = 14 in asset A(31)(54) = 17 in asset B69 in T-Bills
The SD of the eﬃcient portfolio is = 31(124) = 038