Optimal Risky Portfolios

Portfolio of Two Risky Assets

Suppose you hold a proportion w in asset A and (1-w) in asset B

The portfolio expected return and risk is given by E(RP ) = wE(RA ) + (1 − w) E (RB )

σ p 2 = w2σ A 2 + (1 − w) 2 σ B 2 + 2w(1 − w)σ A ⋅ σ B ⋅ ρ AB

An Example

Suppose you hold two assets in your portfolio, GE and IBM.

Let the portfolio weight of GE be w, and then the portfolio weight of IBM be (1-w)

If w = 1, you hold only GE,

If w = 0, you hold only IBM,

If w = 0.5, you have an equally weighted or naively diversified portfolio.

When choosing the optimal allocation between a risk-free asset and a risky portfolio, we have assumed that we have already selected the optimal risky portfolio.

In this section, we learn how to determine the optimal risky portfolio.

We start from two risky assets. Most of the intuition carries to the case of more than two risky assets.

Return and Risk of A Portfolio

Given the expected returns of A and B, variances of A and B, and the covariance

(correlation) between A and B, we compute the expected return and variance of the portfolio for a series of portfolio weights.

Then we plot the expected returns against variances. An Example

Based on data for 1982-2001, we find that

The average monthly return is 1.68% for GE, and 1.22% for IBM

The standard deviation is 6.49% for GE and 8.10% for IBM

The correlation between GE and IBM is

0.377

1

Equally weighted portfolio

Equally weighted portfolio σ p 2 = w2σ A 2 + (1 − w) 2 σ B 2 + 2w(1 − w)σ A ⋅ σ B ⋅ ρ AB

The expected return of the equally weighted portfolio is:

0.5*1.68%+0.5*1.22%=1.45%

The standard deviation of this portfolio is more complicated

= 0.52 × 6.49% 2 + 0.52 × 8.10% 2 +

2 × 0.5 × 0.5 × 6.49% × 8.10% × 0.377

= 0.00368

⇒ σ p = 6.07%

Another Portfolio

Diversification Benefit

Calculate the expected return and standard deviation of the portfolio consisting of 80%

GE and 20% IBM

0.018

Variance

0.017

$1,000 GE, $0 IBM

0.016

Standard Deviation

The portfolio risk is lower than either of the individual stocks

This is called the benefit of diversification

I repeat the above steps for other portfolio weights, w=0, 0.1, 0.2, …, 0.9, 1.0

I plot the expected return against standard deviation Diversification and Risk

Portfolios of GE and IBM

$800 GE, $200 IBM

0.015

NonSystematic

$600 GE, $400 IBM

$400 GE, $600 IBM

0.014

$200 GE, $800 IBM

0.013

0.012

This portfolio is less risky than either of GE and IBM!!!

$0 GE, $1000 IBM

Investment Opportunity Set

Systematic

0.011

49

47

45

43

41

39

37

35

33

31

29

23

27

25

17

0.095

21

0.09

15

0.085

19

0.08

11

0.075

13

0.07

9

0.065

7

0.06

5

0.055

3

1

0.01

0.05

Number of Assets

Expected Return

2

Feasible Portfolios

With N Risky Assets

Efficient Frontier

Investment Opportunity Set or Feasible Set

Expected

Return E(R)

You can construct the efficient frontier using one of two equivalent approaches.

B

C

A

Efficient frontier Std dev σ

You can do this using the Solver in Excel.

What if a risk-free asset is available? Utility Maximization

Higher Utility

Indifference Curves

C

Expected

Return E(R)

For given expected return, find the minimum variance

For given variance, find the maximum expected return

.A

B

We have covered the capital allocation problem between a risk-free asset and a risky asset. Recall that the capital allocation line is the straight line through the risk-free asset and the risky asset.

A is the Utility maximizing risky-asset portfolio

Std dev σ

The Optimal Risky Portfolio has the Highest Sharpe Ratio

Capital Market Line

Expected

Return E(R)

Optimal Risky

Portfolio

D

•

C

•

E

Riskless

Asset •

•

•

CALB

B

What if a risk-free asset is available?

CALA

•A

The feasible set of portfolios becomes more attractive We can identify an optimal risky portfolio which dominates all other risky…