F303 Intermediate Investments 1

Inside the Optimal Risky Portfolio

• New Terms:– Co-variance

– Correlation

– Diversification

• Diversification – the process of adding assets to a portfolio in order to reduce the risk of the overall portfolio

F303 Intermediate Investments 2

Types of Risk

• Systematic Risk – This risk is part of the economic system (it is systemic!). It is non-diversifiable and is a/k/a market risk

• Non-Systematic Risk is firm specific. It can be diversified away

• How can we tell if adding assets to a portfolio will reduce the overall risk of the portfolio?– Covariance

– Correlation

F303 Intermediate Investments 3

Diversification and Risk: An Example

• Two stock funds– Avers: A fund made up of Pizza Companies

– Zagrebs: A fund made up of beef producing companies

– What is the expected return on each fund?

Expected return:Scenario Probability Aver's Returns Zagreb ReturnsRecession 0.3333 0.0300 (0.0700) Normal 0.3333 0.0800 0.1500 Boom 0.3333 0.0500 0.3000

F303 Intermediate Investments 4

Diversification and Risk: An Example

• What is the individual deviation, variance and standard deviation for each fund?

Individual deviationsAvers Return Deviation Squared DeviationRecession 0.0300 - - Normal 0.0800 - - Boom 0.0500 - -

Zagreb's Return Deviation Squared DeviationRecession (0.0700) - - Normal 0.1500 - - Boom 0.3000 - -

F303 Intermediate Investments 5

Diversification and Risk: An Example

• What would happen if these two assets were combined in a single portfolio?

• What is the Variance?• What is the Standard Deviation?

50% 50%Combined Return Deviation Squared Deviation

Recession - - - Normal - - - Boom - - - Expected return -

F303 Intermediate Investments 6

Diversification and Risk: An Example

• How do we measure the Covariance and Correlation Coefficient?

• The Covariance = the product of the deviations:

Covariance Aver's Returns Deviations Zagreb Returns Deviation ProductScenarioRecession 0.0300 - (0.0700) (0.1967) - Normal 0.0800 - 0.1500 0.0233 - Boom 0.0500 - 0.3000 0.1733 -

F303 Intermediate Investments 7

Diversification and Risk: An Example

• Correlation Coefficient =

Covariance

SDA * SDZ

• If the Correlation Coefficient is < 1, the addition of the asset has diversification benefits, regardless of the other risk/return characteristics of the asset!

F303 Intermediate Investments 8

Three Rules for Portfolios Made Up of Two Risky Assets!

1. The rate of return on the portfolio is a weighted average of the returns on the component securities, with the investment proportions as weights

rp = wara + wzrz

2. The same holds true for the Expected rate of return

Rp = waE(ra) + wzE(rz)3. The variance of the rate of return on the two risky asset

portfolio is

V = (waSDa)2 + (wzSDz)2+2(waSDa)(WzSDz)Corraz