Efficient Frontier - Banking - Lecture Slides, Slides of Banking and Finance

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Efficient frontier

E(Rp )

σp

The general case – applied to two

assets

[ ]

( )

[ 2 ]

( )

( 2 )

( ) 2

( 1 ) 2 0

2 2 ( 1 ) 2 4 0

( 1 ) 2 ( 1 )

( 1 ) 2 ( 1 )

σ σ ρ ωσ σ

σ σ ρ σ ω

ω σ σ ρ ωσ σ σ ρ σσ

ω σ σ σ ρ ωσσ ρ σ σ

ωσ ω σ ρ σ σ ρ ωσσ

ωσ ω σ ρ σ σ ρ ωσσ ω

σ

σ ω σ ω σ ρ ω ω σ σ

σ ω σ ω σ ρ ω ω σσ

• −

−

⇒= + − = −

⇒= + − − =−

⇒= − − + − =

= − − + − =

= + − + −

= + − + −

d

d (^) p

p

p

Risk-free asset

• Lets introduce a risk-free asset that pays a rate

of interest R f

.

• The rate R (^) f is known with certainty and has

zero variance and therefore no covariance

with the portfolio.

• Such a rate could be a short-term government

bill or commercial bank deposit.

One bundle of risky assets

• Take one bundle of risky assets and allow the investor to lend

or borrow at the safe rate of interest. The investor can;

• Invest all his wealth in the risky bundle and undertake no

lending or borrowing.

• Invest less than his total wealth in the single risky bundle and

the rest in the risk-free asset.

• Invest more than his total wealth in the risky bundle by

borrowing at the risk-free rate and hold a levered portfolio.

• These choices are shown by the transformation line that

relates the return on the portfolio with one risk-free asset and

risk.

Linear Opportunity set

• Let the risk-free rate R (^) f = 10% and the return on the

bundle of assets R (^) N = 22.5%.

• The standard deviation of the returns on the bundle

σN = 24.87%.

• The weights on the risky bundle and the risk-free

asset can be varied to produce a range of new

portfolio returns.

Portfolio Risk and Return

State T-bill Equity E(Rp) σp

(1-φ) φ

1 1 0 10% 0%

2 0.5 0.5 16.25% 12.44%

3 0 1 22.5% 24.87%

4 -0.5 1.5 28.75% 37.31%

Transformation line

R f

No lending all

investment in

bundle

E(Rp )

σp

All lending

0.5 lending + 0.

in risky bundle

-0.5 borrowing + 1.

in risky bundle

A riskless asset and a risky portfolio

• An investor faces many bundles of risky assets (eg

from the London Stock Exchange).

• The efficient frontier defines the boundary of

efficient portfolios.

• The single risky asset is replaced by a risky portfolio.
• We can find a dominant portfolio with the riskless

asset that will be superior to all other combinations.

Borrowing and Lending

• The investor can lend or borrow at the risk-

free rate of interest rate.

• The risk-free rate of interest R f

represents the

rate on Treasury Bills or some other risk-free

asset.

• The efficiency boundary is redefined to

include borrowing.

Borrowing and lending frontier

E(R p )

σp

Rf

A

B

C

Combining borrowing and lending

E(R p )

σp

Rb

A

B

C

D

Rf

P

Q

Separation Principle

• Investor makes 2 separate decisions
• Given knowledge of expected returns, variances and

covariances the investor determines the efficient

frontier. The point M is located with reference to Rf.

• The investor determines the combination of the risky

portfolio and the safe asset (lending) or a leveraged

portfolio (borrowing).

Summary

• We have examine the theory of portfolio

diversification

• We have seen how the efficient frontier is

constructed.

• We have seen that portfolio diversification

reduces risk to the non-diversifiable

component.