Chapter 2: Diversification and risky asset allocation

Question 1

2.1 Expected returns and variances

Answer 1

• Expected return: Average return on a risky asset expected in the future.
• E(Rj)=Prob of Economy A * Return of Economy A+ Prob of Economy B * Return of Economy B.
• Projected (Expected) Risk Premium: Difference between expected return on a risky investment and the certain return on a risk-fee investment.
• Risk Premium= E(Rj) - Rf
• Look at Variance formula in page 41.

Question 2

2.2 Portfolios

Answer 2

• Portfolio: Group of assets such as stocks and bonds held by an investor.
• Portfolio Weight: Percentage of a portfolio’s total value invested in a particular asset. This is the best way to describe a portfolio. The weights sum up to 1.
• Portfolio Expected Return
  E(Rp) = x1 * E(R1)+x2 * E(R2)+…+xn * E(Rn)
• Portfolio Variance: Look at how to calculate it in page 45.

Question 3

2.3 Diversification and portfolio risk

Answer 3

Effect of Diversification:

• The standard deviation declines as the number of securities is increased. By the time there are 100 randomly chosen stocks (1% investment in each) the portfolio’s volatility has declined 64% (from 13.47 to 4.86%. 2/3 of the total risk associated with a random stock can be eliminated in an equally weighted portfolio of 222 stocks. Portfolio risk is reduced by 46% for a 10 stock portfolio and 53% for a 20 stock portfolio. A 200 stock portfolio provides 99.6% risk reduction.
• An important foundation of the diversification effect is the random selection of stocks. When the stocks are chosen randomly, the portfolio represents different sectors, market caps, etc (don’t have all your eggs in one basket)

Question 4

2.3 Diversification and portfolio risk

Answer 4

The principle of diversification

• Principle of diversification: Spreading an investment across a number of assets will eliminate some but not all the risk.
• Diversification: spreading an investment across assets.
• The benefit in terms of risk reduction from adding securities drops off as adding securities to our portfolio more and more. By there are 10 securities, most diversification effect is realized, by the time there are 30 or so there is little remaining benefit. In other words, the benefit of further diversification increases at a decreasing rate (Law of diminishing returns)
• Investors should think of 30 to 50 randomly chosen stocks when they are building a diversified portfolio.
• Some riskiness associated with individual assets can be eliminated by forming portfolios. There is a minimum level of risk that cannot be eliminated by diversifying. Diversification reduces risk, but only up to a point. Some risk is diversifiable and some not.

Question 5

2.3 Diversification and portfolio risk

Answer 5

The fallacy of time diversification:
* Time diversification fallacy: flawed logic that says that volatility cancels itself out with time. Volatility increases over time - it does not cancel out over time.

Question 6

2.4 Correlation and diversification

Answer 6

Why diversification works?

• Correlation: Tendency of returns on 2 assets to move together.
• Positive Correlation: If returns on 2 assets move up/down together. A correlation of +1 indicates the 2 assets have a perfect positive correlation. Perfect correlation does not necessarily mean the 2 assets move by the same amount.
• Negative Correlation: If returns on 2 assets move in opposite direction. A correlation of -1 indicates that they always move in opposite directions.
• Un-correlation: No particular relationship between 2 assets. A correlation of 0, if one asset is up/down there is no way to know what the other one is likely to do.
• Diversification works because security returns are generally not perfect correlated. If 2 assets are highly correlated they will offer limited diversification benefit. In contrast, if assets are negatively correlated, there will be a substancial diversification benefit because variation in the return on one asset tends to be offset by variation in the opposite direction from the other; in fact, if 2 assets have perfect negative correlation then it is possible to combine them such that all risk is eliminated.
• Covariance between two assets shows how returns of these two assets relate to each other and move with respect to each other.
• Negative covariance indicates 2 assets’ returns move opposite to each other, when the return of one asset increases the other one decreases.
• Positive covariance when the return of one asset increases the other return will also increase.
• See formula for covariance in page 53.

Question 7

2.4 Correlation and diversification

Answer 7

The importance of asset allocation (part 1)

• Asset allocation: How an investor spreads portfolio dollars among assets.
• Investment opportunity set: Collection of possible risk-return combinations available from portfolios of individual assets (combinations of risk and return available all fall on a smooth curve or “hyperbola”).
• Minimum variance portfolio: portfolio that has the smallest standard deviation (or variance) of all.
• Efficient portfolio: A portfolio that offers the highest return for its level of risk.

Question 8

2.5 The Markowitz efficient frontier

Answer 8

The importance of asset allocation (part 2)
* See page 60 for Rp formula for more than 3 stocks.

• Markowitz efficient frontier: The set of portfolios with the maximum return for a given standard deviation (only combinations that plot on the upper left-hand boundary; all the rest are inefficient). A primary reason that the Markowitz analysis is not usually extended to large collections of individual assets has to do with data requirements. The inputs into the analysis are: 1. Expected returns on all assets, 2. Standard deviations on all assets, 3. Correlations between every pair of assets. Moreover, these inputs have to be measured with precision.