Financial Engineering 3

References: Villalobos, Luenberger, Faerber

Lecture 10

Two Fund and One Fund Theorems

Lecture Topics • Two Fund Theorem • Minimum Variance Point • Portfolio Exercise • One Fund Theorem

Two Fund Theorem • Given two efficient portfolios (portfolios in E), all efficient

portfolios can be generated as a weighted combination of these two.

• In theory, two funds would be sufficient for all investors because they could achieve any point in E from a linear combination of these.

Two Funds and the Efficient Frontier

Standard Deviation

E xp

ec te

d R

et ur

n Portfolio 1 (P1)

Portfolio 2 (P2)

Minimum Variance Portfolio

Outline of the Proof • Consider the example where we compute the efficient portfolio

for two different desired returns; portfolios P1 and P2. • Now consider the convex combination of these two portfolios.

P = (1-α)P1 + α P2

• Notice that P is also a valid portfolio and that the last restriction of the Markowitz equations is met (sum of weights equals one).

• Notice the expected return of this new portfolio is equal to the convex combination of the returns of the original two portfolios.

• Also notice that each one of the portfolios makes the derivative of the L function (variance plus Lagrangian multipliers) equal to zero.

– Therefore, their convex combination should be also zero. – And therefore, is an optimal portfolio in the efficient frontier.

Minimum Variance Point • If we omit the targeted return constraints of the original

Markowitz model we end up with the following:

• Taking derivatives with respect to the w’s and λ wet get the system of linear equations:

• By solving these equations and standardizing the weights to one, we get the minimum variance point.

)1( 2 1

LMin 211, nijj n

ji i wwwww −−−−+= ∑ = λσ

λσσσ −+++= ∂ ∂

nnnnn n

www w L

2211

)1( 21 nwww L

−−−−= ∂ ∂

λ λ

( ) ( )1 2, 1 1 11Original: L 12 n n n

i j ij i i ii j i i w w w r r wσ λ λ

= = = = − − − −∑ ∑ ∑

Example • Consider the following information for six stocks:

YHOO GOOG JNJ KO UNP F Average Daily Return -5.01708E-05 0.002252 0.000163 0.000391 0.001014 -0.00072 Std. Dev. 0.022116796 0.021467 0.008106 0.008326 0.01513 0.020588 Variance 0.000489153 0.000461 6.57E-05 6.93E-05 0.000229 0.000424 Annual Return -0.012592864 0.56516 0.040795 0.098201 0.25456 -0.18083 Annual Std Dev 0.350395949 0.340096 0.128424 0.131914 0.23971 0.326176

Covariance Matrix YHOO GOOG JNJ KO UNP F

YHOO 0.000488586 0.000191 2.33E-05 5.36E-05 6.4E-05 0.000101 GOOG 0.000190765 0.00046 1.76E-05 3.62E-05 5.33E-05 5.51E-05 JNJ 2.32596E-05 1.76E-05 6.56E-05 2.4E-05 2.36E-05 2.51E-05 KO 5.35529E-05 3.62E-05 2.4E-05 6.92E-05 3.84E-05 4.75E-05 UNP 6.3952E-05 5.33E-05 2.36E-05 3.84E-05 0.000229 0.0001 F 0.000101058 5.51E-05 2.51E-05 4.75E-05 0.0001 0.000423

Example • If we solved the original Markowitz equations for returns of the

average daily return (0.000508163) and three times the average we get the following efficient portfolios:

P1 P2 YHOO -0.0216385 -0.1687143 GOOG 0.0748466 0.3641567 JNJ 0.4548215 0.2154583 KO 0.4159374 0.4790691 UNP 0.0903226 0.3187440 F -0.0142896 -0.2087139 λ1 0.0071678 0.0630134 λ2 0.0000413 0.0000202

Example • The corresponding minimum variance portfolio is:

Solving these equations we get:

This is also efficient.

YHOO GOOG JNJ KO UNP F λ2

0.000488586 0.000191 2.33E-05 5.36E-05 6.4E-05 0.000101 -1 0.000190765 0.00046 1.76E-05 3.62E-05 5.33E-05 5.51E-05 -1 2.32596E-05 1.76E-05 6.56E-05 2.4E-05 2.36E-05 2.51E-05 -1 5.35529E-05 3.62E-05 2.4E-05 6.92E-05 3.84E-05 4.75E-05 -1

6.3952E-05 5.33E-05 2.36E-05 3.84E-05 0.000229 0.0001 -1 0.000101058 5.51E-05 2.51E-05 4.75E-05 0.0001 0.000423 -1

1 1 1 1 1 1 0

0 0 0 0 0 0 1

=

w1 -0.00276 w2 0.037713 w3 0.485544 w4 0.407834 w5 0.061004 w6 0.010665 λ2 4.4E-05

Example • The three portfolios that we have are:

• Notice the min variance portfolio follows the rule: P = (1-α)P1 + α P2 with α = -0.128 α = (P- P1)/(P2 - P1)

– With two efficient portfolios we can replicate any other portfolio on the efficient frontier

– The two fund theorem applies.

P1 P2 PMin Var YHOO -0.0216385 -0.1687143 -0.0027611 GOOG 0.0748466 0.3641567 0.0377133 JNJ 0.4548215 0.2154583 0.4855440 KO 0.4159374 0.4790691 0.4078344 UNP 0.0903226 0.3187440 0.0610045 F -0.0142896 -0.2087139 0.0106650 λ1 0.0071678 0.0630134 λ2 0.0000413 0.0000202 0.0000440 ROR 0.0005082 0.0015245 0.0003777 Var 0.0000449 0.0001162 0.0000440 StDev 0.0067003 0.0107806 0.0066301

Example • The overall analysis is:

PMin Var α -0.2 -0.128350966 -0.1 0 0.1 0.2 0.3 YHOO 0.0077767 -0.0027611 -0.0069309 -0.0216385 -0.0363460 -0.0510536 -0.0657612 GOOG 0.0169846 0.0377133 0.0459156 0.0748466 0.1037776 0.1327086 0.1616396 JNJ 0.5026941 0.4855440 0.4787578 0.4548215 0.4308852 0.4069489 0.3830125 KO 0.4033110 0.4078344 0.4096242 0.4159374 0.4222506 0.4285637 0.4348769 UNP 0.0446383 0.0610045 0.0674804 0.0903226 0.1131647 0.1360069 0.1588490 F 0.0245953 0.0106650 0.0051529 -0.0142896 -0.0337320 -0.0531744 -0.0726169 ROR 0.000304898 0.000377717 0.00040653 0.000508163 0.000609796 0.000711428 0.000813061 Var 4.42501E-05 4.39587E-05 4.40044E-05 4.48938E-05 4.69183E-05 5.0078E-05 5.43728E-05 StDev 0.006652075 0.006630138 0.006633578 0.00670028 0.006849693 0.00707658 0.007373793

Inclusion of a Risk Free Asset • A risk-free asset is any asset with σ = 0.

• Let’s assume weight α for the risk-free asset and weight 1 − α for the “risky” asset. (rf = risk free return; r = mean of risky return)

( )mean 1fr rα α= + − ( )stdev 1 α σ= −

• As α varies, each portfolio on a straight line between the risk-free and risky assets in the mean and standard deviation space is generated.

– Take a look at the feasible region generated with shorting and without shorting.

Efficient Frontier

rf

Portfolio with return r

Risk-Free rate

One Fund Theorem • There is a single fund F of risky assets such that any efficient

portfolio can be constructed as a combination of the fund and the risk free asset.

One Fund Theorem

Original Efficient Portfolio

Risk-Free Rate

One-Fund, efficient Portfolio

rf

One Fund Theorem • There is a single fund F of risky assets such that any efficient

portfolio can be constructed as a combination of the fund and the risk free asset.

• The mean-variance analysis of Markowitz’ portfolio theory tells us that the portfolio we need to consider for an efficient investment must lie on an “efficient frontier” of the mean- variance graph.

• Because of the one-fund theorem, stating there is a single fund F of risky assets such that any efficient portfolio can be constructed as a combination of the fund and the risk free asset, everyone in the market will tend to hold the same portfolio and borrow or lend at the same rate.

• If there were only one portfolio that everyone seems to want to buy what is that portfolio?

The Market Portfolio • The fund everyone buys must be equal to the “market”

portfolio.

• The market portfolio should contain all the assets available.

• The market portfolio should contain each asset in the proportion that they are part of the overall market.

– It should contain shares of each asset in proportion of that stock’s representation in the entire market.

• The weight of an asset in the market portfolio should be equal to the proportion of that asset’s total market capitalization.

• What is the closest thing to this ideal market?

S&P 500!

Assignments • Read Luenberger Chapter 7. • Luenberger problems 6.3, 6.7, and 6.9.