BKM Chapter 7 - Optimal Risky Portfolios

Question 1

Two broad sources of uncertainty in the returns on risky assets

Answer 1

• those related to general economic conditions (systematic risk, such as business cycle, interest rate, exchange rate, can not be diversified) and
• those which are firm specific (research and development opportunities or personnel changes)

Question 2

Expected Return for a portfolio of two risky assets

Variance of the return (to reflect the portfolio risk)

Answer 2

Expected Return:

E„(r_p)… = w₁E(„r₁)… + ‚w₂(E„r₂)…

Variance:

σ_p² = ƒ w₁²*σ₁² + w₂²*σ₂²+2w₁w₂Cov(r₁,r₂)

since Cov(r1, r2) = σ₁σ₂ρ

σ_p² = ƒ w₁²*σ₁² + w₂²*σ₂²+2w₁w₂σ₁σ₂ρ

Portfolio of less than perfectly correlated assets always offer better risk-return opportunities and the lower the correlation, the greater the gain in efficiency.

Question 3

Risk and Return Trade-Offs (-1<ρ<1)

Answer 3

Question 4

Risk and Return Trade-Offs (ρ=1)

Answer 4

When the correlation is ρƒ=‚1, the portfolios all lie along a straight line;

Question 5

Risk and Return Trade-Offs (ρ= -1)

Answer 5

Question 6

Minimum Variance Portfolio

Answer 6

We know the portfolio variance is:

σ_p² = ƒ w₁²*σ₁² + w₂²*σ₂²+2w₁w₂σ₁σ₂ρ

If we want to minimize the variance, then we simply need to take the derivative of this with respect to w1, set the derivative equal to zero and solve for w₁. (w₂=1-w₁)

w₁ = [σ₂² - σ₁σ₂ρ]/[σ₁²+σ₂²-2σ₁σ₂ρ]

When ρ=-1, a perfectly hedged position can be obtained by choosing the portfolio proportions that solve the equation

w₁σ₁-w₂σ₂=0. The solution is w₁ = σ₂/σ₁+σ₂. These weights drive the standard deviation of the portfolio to zero. (This is also the optimal risky portfolio).

Benefits from diversification arise when correlation is less than perfectly positive:

i. the lower the correlation, the greater the potential benefit from diversification
ii. When perfect negative correlation exists, a perfect heaged opportunity exists and a zero-variance portfolio can be constructed.

Question 7

optimal risky portfolio

Answer 7

All we want to do is keep finding lines that go through one of the risky portfolios and the risk-free point along the y-axis and that is higher than the other lines.

Graphically, we simply want to find the tangent line to the portfolio opportunity set. The risky portfolio that produces this line is then the optimal risky portfolio and it is comprised of some optimal combination of the two risky assets.

Question 8

Determining the Optimal Portfolio of Risky Assets

Answer 8

Weight in Optimal Risky Portfolio can be obtained by maiximize the slope (sharp ratio). See attached image for formula.

Question 9

Markowitz Portfolio Selection Model:

Expected Return and Variance for n risky assets

Answer 9

see image

Question 10

Markowitz Portfolio Selection Model:

Minimum Variance Frontier & Global minimun variance portfolio

Answer 10

Minimum Variance Frontier: there will be an infinite number of points, but there will be an outer frontier that represents the minimum risk portfolio for any level of expected return.

There is a single portfolio combination that results in the Global Minimum Variance. Note also that all points along this frontier are possible, but only those above this global minimum variance portfolio are of interest because the points below are inefficient. They each have lower expected returns for any level of risk than some other point above the global minimum variance portfolio.

Question 11

Markowitz Portfolio Selection Model:

Steps of selection

Answer 11

Step 1: Identify the risk-return combinations available from the set of risky assets.

Step 2: Identify the optimal portfolio of risky assets by finding the portfolio weights that result in the steepest CAL. (trangent line of the efficient frontier)

Step 3: Choose an appropriate complete portfolio by mixing the risk free asset with the optimal risky portfolio.

(interception of the indifference curve and CAL)

Question 12

Seperation Property

Answer 12

Markowitz selection makes it clear that portfolio selection can be separated into two very distinct tasks.

1. The first is to select an optimal combination of risky assets. This is a purely mathematical exercise, given the assumptions of expected return, standard deviation and correlations for each of the risky assets, and is not affected by any individual’s risk preferences.
2. The second task is to allocate funds between the risk-free asset and the optimal risky portfolio. This is entirely based upon individual risk preferences.

Question 13

The Power of Diversification

Answer 13

When there are N assets each with a weight equal to wi = 1/N, the portfolio variance is:
σ_p²= (1/N) * Avg. Variance ‚+ (N-1)/N * Avg. Covariance

It shows that as N gets large (i.e. as we increase the number of
assets in our portfolio and continue to invest an equal amount in each asset) the first term becomes very small since the ratio 1/N approaches zero and the second term approaches the average covariance.

Question 14

Q. Why Distinguish between Asset allocation and Security selection?

Answer 14

1. Demand for sophisticated security selection has increased tremendously due to society’s need and ability to save for the future (e.g. for retirement, college, health care, etc).
2. Amateur investors are at a disadvantage in their ability select the proper securities to be held in their risky portfolio due to the widening spectrum of financial markets and financial instruments.
3. Strong economies of scale result when sophisticated investment analysis is conducted.

Since large investment companies are likely to invest in both domestic and international markets, management of each asset-class portfolio needs to be decentralized due to the expertise required. This requires that security selection of each asset-class portfolio be optimized independently.

Question 15

Risk Pooling and the Insurance Principle

& Risk Sharing

Answer 15

Risk pooling means merging uncorrelated risky assets to reduce risk.

Risk pooling with n assets:

• Sharp Ratio increase from R/σ to (n^0.5)R/σ
• Total risk of the pooling strategy will increase by the same multiple to σ(n^0.5). So risk pooling by itself does not reduce risk

For insurance, risk pooling entails selling many uncorrelated insurance policies (a.k.a. the insurance principle). The insurance principle tells us only that risk increases less than proportionally to the number of policies insured when the policies are uncorrelated; hence, the Sharpe ratio—increases.

Risk sharing: spreads or shares the risk of a fixed portfolio among many owners)

• Reaches same sharp ratio as risk pooling
• but lower volatility: σ/(n^0.5).

Question 16

Risk sharing in insurance context:

Answer 16

• Other investors buying shares in the insurer.
• Alternatively, insurers often “reinsure” their risk by selling off portions of the policies to other investors or insurance companies, thus explicitly sharing the risk.

Question 17

Why risk pooling might limit the potential economies of scale of an ever-growing portfolio of a large insurer

Answer 17

• Each policy written requires the insurer to set aside additional capital to cover potential losses.
• As the number of policies grows, the risk of the pool will grow - despite “diversification” across policies. Eventually, that growing risk will overwhelm the company’s available capital.

Question 18

With risk sharing, it appears one can set up an insurer of any size, building a large portfolio of policies and limiting total risk by selling shares among many investors.

Give two reasons why in reality it’s not possible.

Answer 18

• Burdens related to problems of managing very large firms will sooner or later eat into the increased gross margins.
• The possibility of error in assessing the risk of each policy or misestimating the correlations across losses on the pooled policies can cause an insurer to fail.