Outline - New York University

Portfolio Selection with Two Risky Securities.

Professor Lasse H. Pedersen

Prof. Lasse H. Pedersen

1

Outline

Portfolio: expected return and SD Diversification Investment opportunity set Investor preference: risk-return tradeoff Optimal portfolio choice with 2 risky assets

Prof. Lasse H. Pedersen

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Portfolio Expected Return and Standard Deviation

The expected return on the portfolio is:

N

E(Rp ) = iE(Ri ) i =1

With 2 securities, the portfolio variance is:

2 p

=

12

2 1

+

22

2 2

+

2 1 2 12 1 2

The standard deviation is:

p =

2 p

Prof. Lasse H. Pedersen

3

Diversification with 2 assets:

Example

Suppose we have two assets, US and JP, with:

mean

volatility

US

13.6%

15.4%

JP

15.0%

23.0%

and with correlation 27%.

If an investor holds 60% in the US and 40% in JP what is the mean and volatility of the portfolio?

`volatility' is another word for `standard deviation'

Prof. Lasse H. Pedersen

4

Diversification with 2 assets: Example

Portfolio mean:

E(Rp) = 0.6*0.136 + 0.4*0.150 = 14.2% Portfolio variance:

var(Rp) = (0.6)2*(0.154)2 + (0.4)2*(0.230)2 +2*0.6*0.4*0.27*0.154*0.230

= 0.022

p = 14.7% This portfolio has higher expected return and lower risk than the US market alone!

Prof. Lasse H. Pedersen

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Risk and Return with Varying Weights

Let be the weight in the US, and 1- the weight in JP.

The expected return of the portfolio is:

E(rp) = *0.136 + (1-)*0.150 The variance of the portfolio return is:

var(rp) = 2*(0.154)2 + (1-)2*(0.230)2 +2**(1-)*0.27*0.154*0.230

What happens when we vary ?

Prof. Lasse H. Pedersen

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expected return Er

Varying the Portfolio Weights gives: The Investment Opportunity Set

0.152

0.15

JP

0.148

0.146

0.144

0.142

0.14

0.138

0.136

US

0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 s tandard deviation

w mean volatility 0.0 0.150 0.230 0.1 0.149 0.212 0.2 0.147 0.195 0.3 0.146 0.179 0.4 0.144 0.166 0.5 0.143 0.155 0.6 0.142 0.147 0.7 0.140 0.143 0.8 0.139 0.143 0.9 0.137 0.146 1.0 0.136 0.154

Portfolio Terminology

The investment opportunity set consists of all available risk-return combinations.

An efficient portfolio is a portfolio that has the highest possible expected return for a given standard deviation

The efficient frontier is the set of efficient portfolios. It is the upper portion of the minimum variance frontier starting at the minimum variance portfolio.

The minimum variance portfolio (mvp) is the portfolios that provides the lowest variance (standard deviation) among all possible portfolios of risky assets.

Prof. Lasse H. Pedersen

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Portfolio Terminology

Mean-s tandard deviation frontier for US and Japan 0.18

0.17

Efficient Frontier

0.16

short US

0.15

JP

0.14 US

0.13

Minimum

0.12

Variance short JP

Portfolio

0.11

0.1 0

0.05

0.1

0. 15

0.2

0.25

0. 3

0.35

0.4

Prof. Lasse H. Pedersen

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e xpec te d re turn

Investment Opportunity Set with Varying Correlations

0 .1 5 2 0.15

C a s e w it h = -1 C a s e w it h = 0 . 2 7 C a s e w it h = 1

0 .1 4 8

0 .1 4 6

0 .1 4 4

0 .1 4 2

0.14

0 .1 3 8

0 .1 3 6 0

0.05

US

0.1

0 .1 5

0.2

s t a n d a rd d e via t io n

Prof. Lasse H. Pedersen

JP

0 .2 5

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Optimal Portfolio Choice with 2 Risky Assets

Any (mean-variance) investor should choose an efficient portfolio to benefit from diversification.

The specific choice depends on the investor's risk aversion

A more risk-averse investor should choose a portfolio with ? lower risk ? lower expected return

Prof. Lasse H. Pedersen

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