RPA2 Module 3

Question 1

Calculate the expected return of security A given the following information.
(Retirement Plan Investing Essentials, pp. 72-74)
Return (%) Probability
4 0.10
5 0.20
6 0.40
7 0.20
8 0.10

Answer 1

The expected return is simply the sum of the products of the returns multiplied by
the probabilities:
4 3 0.10 5 0.40
5 3 0.20 5 1.00
6 3 0.40 5 2.40
7 3 0.20 5 1.40
8 3 0.10 5 0.80
Total 5 6.00
The expected return is 6%

Question 2

What is the risk, that is, the standard deviation, of the returns in Learning Objective
1.1? (Retirement Plan Investing Essentials, pp. 77-78)
Return (%) Difference* Difference Squared Difference Squared 3 Probability
4 -2 4 0.40
5 -1 1 0.20
6 0 0 0.00
7 1 1 0.20
8 2 4 0.40
Total 1.20

Answer 2

The standard deviation is the square root of 1.2, or 1.095%.

Question 3

What is the expected return on a pension plan portfolio that has the following
securities? (Retirement Plan Investing Essentials, pp. 76-77)
Investment Amount (in Millions) Expected Return (%)
A 12 6
B 24 8
C 46 9
D 38 10
The pension portfolio has a total of $120 million invested, and the percentages
(weights) invested in assets A-D are:
A 0.10
B 0.20
C 0.38
D 0.32

Answer 3

The expected return on the portfolio is simply a weighted average of the expected
returns on the individual securities, or:
A 0.10 3 6 5 0.60
B 0.20 3 8 5 1.60
C 0.38 3 9 5 3.42
D 0.32 3 10 5 3.20
Total 5 8.82%

Question 4

Since the expected return of a portfolio is a weighted average of the expected
returns of the individual securities comprising the portfolio, does it follow that
a portfolio’s risk is the weighted average of the risk of the individual securities
comprising the portfolio?

Answer 4

No, it does not follow. Although the expected return of a portfolio is a weighted
average of its securities’ expected returns, portfolio risk is not a weighted average of
the risk of the individual securities contained in the portfolio. In fact, portfolio risk
is always less than a weighted average of the risks of the securities in the portfolio
unless the securities have outcomes that vary together exactly, which is an almost
impossible occurrence

Question 5

Assuming the risk of each security is 0.20 and the risk of all securities is
independent, what is the portfolio risk when a portfolio consists of (a) 16
securities or (b) 49 securities?

Answer 5

(a) 0.20/161/2 5 0.05

(b) 0.20/491/2 5 0.0286

Question 6

To what principle can risk reduction be attributed?

Answer 6

The Law of Large Numbers. - The larger the sample size, the more likely it is that the sample mean will be close to the population expected value.

Risk reduction in the case of independent risk sources can be thought of as the insurance principle, named for the idea that an insurance company reduces its risk by writing many policies against many independent sources of risk.

Question 7

Describe the relationship between the returns on two securities if the correlation
coefficient is (a) 11.0, (b) 0 or (c) 21.0.

Answer 7

(a) A correlation coefficient of +1.0 indicates a perfect direct linear relationship.
Combining securities with perfect positive correlation with each other provides
no reduction in portfolio risk.
(b) A correlation coefficient of 0 shows no relationship between the returns on the
securities. Combining two securities with zero correlation with each other
reduces the risk of the portfolio. However, portfolio risk is not eliminated in the
case of zero correlation. While a zero correlation between two security returns is
better than a positive correlation, it does not produce the risk reduction benefits
of a negative correlation coefficient.
(c) When the correlation coefficient is 21.0, there is a perfect inverse relationship.
Combining two securities with perfect negative correlation with each other
could eliminate risk altogether.

Question 8

Assume that a portfolio consists of only two stocks, A and B, and that 50% of the
funds are invested in each stock. Further assume that the standard deviation of
returns is 25% for each stock. What are the implications for portfolio risk if the
correlation coefficient between stock A and stock B is (a) 11.0, (b) 21.0 or (c) 0?

Answer 8

(a) Portfolio risk will not be reduced at all. The risk (as measured by standard
deviation) will be 25% for Stock A, Stock B and the combined portfolio.
(b) Portfolio risk will be eliminated. The risk for Stocks A and B will be 25%, but
portfolio risk will be 0%. There will be no fluctuations in returns.
(c) With a correlation coefficient of 0, the portfolio risk will be reduced but not
eliminated.

Question 9

Distinguish the difference between covariance and the correlation coefficient.

Answer 9

Covariance is absolute.

Correlation Coefficient is a relative measure

Both between 2 securities

Question 10

What are the three factors that determine portfolio risk?

Answer 10

The three factors that determine portfolio risk are:

(1) The variance of each security
(2) The covariances between securities
(3) The portfolio weights for each security.

Question 11

Assume that a portfolio consists of only two securities, X and Y, and that 50%
of the portfolio is invested in each. The standard deviation of X is 35%, and the
standard deviation of Y is 25%. Using equation 3-12 on page 84 of the text, what is
the portfolio risk if the correlation coefficient is (a) 11, (b) 21, (c) 0 or (d) 10.55?

Answer 11

(a) (0.5)2
(0.35)2 1 (0.5)2
(0.25)2 1 (2)(0.5)(0.5)(1)(.035)(0.25) 5 0.09
and the square root of .09 5 30%.

(b) (0.5)2
(0.35)2 1 (0.5)2
(0.25)2 1 (2)(0.5)(0.5)(-1)(0.35)(0.25) 5 0.0025
and the square root of .0025 5 5%.

(c) (0.5)2
(0.35)2 1 (0.5)2
(0.25)2 1 (2)(0.5)(0.5)(0)(0.35)(0.25) 5 0.05
and the square root of .05 5 22%.

(d) (0.5)2
(0.35)2 1 (0.5)2
(0.25)2 1 (2)(0.5)(0.5)(0.55)(0.35)(0.25) 5 0.07
and the square root of .07 5 26%.

Question 12

For a portfolio with only two securities, explain (in words, not numbers) the
relationship between the correlation coefficient and the level of portfolio risk.

Answer 12

All other things being equal, the lower the correlation coefficient, the lower the portfolio risk.

In other words, risk can be reduced when securities are added to a
portfolio, and the risk reduction effect is greatest when the returns have a low correlation.

Question 13

Describe how the number of securities held in a portfolio determines how each individual security’s risk or covariance impacts the overall risk associated with the portfolio generally.

Answer 13

As the number of securities held in a portfolio increases, the importance of each individual security’s risk (variance) decreases, and the importance of the covariance relationship increases

Question 14

What is the major practical problem in using the Markowitz portfolio selection
model?

Answer 14

It requires a full set of covariances between the returns of all securities being considered in order to calculate portfolio variance.

There are [n (n 2 1)]/2 unique covariances for a set of n securities. For example, with 200 securities, there are 19,900 unique covariances.

Question 15

According to the Markowitz model framework, what constitutes an efficient
portfolio?

Answer 15

One that has the smallest portfolio risk for a given level of expected return or the largest expected return for a given level of risk.

Question 16

Explain the concept of dominance.

Answer 16

Portfolios on the efficient frontier (efficient set) dominate other portfolios because
they offer the best return-risk combinations available to investors. Portfolios with
inferior return-risk relationships are “dominated.”

Question 17

Explain why, in the Markowitz analysis, the portfolio weights, or percentages of
investable funds to be invested in each security, are the only variable to solve for
in determining efficient portfolios.

Answer 17

The portfolio weights are the only variable to solve for because all of the analysis
of other components—the expected returns, the standard deviations and the
correlation coefficients for the securities being considered—are inputs

Question 18

With financial markets across countries becoming more integrated, should
investors ignore international diversification?

Answer 18

The correlation-based argument of risk diversification is very simplistic and dated and that investors must take a global approach in all aspects of their investment management.

Question 19

Describe the difference between diversifiable and nondiversifiable risk.

Answer 19

Diversifiable risk, or nonsystematic risk, is a unique risk related to a particular
security (company) and can be eliminated (though not completely) through a well diversified portfolio.

Nondiversifiable risk, or systematic risk, is the variability in a security’s total returns that is directly associated with overall movements in the general market or economy and cannot be avoided regardless of how well a portfolio is diversified.

Virtually all securities have some systematic risk, whether bonds or stocks, because systematic risk directly encompasses interest rate risk, recession, inflation and so on.

Question 20

How many securities are enough to diversify properly?

Answer 20

Traditionally, as few as 20 stocks were believed to be adequate. Recent studies have
indicated that 50-60 stocks appear to be needed to ensure adequate diversification.
As the number of stocks increases, the probability of underperforming the market
decreases.

Question 21

What is the definition of beta as conceptualized in the CAPM?

Answer 21

Beta is a relative measure of risk—the risk of an individual stock relative to the
market portfolio of all stocks. Thus, beta is a measure of the systematic risk of a
security that cannot be avoided through diversification.

Question 22

Suppose the risk-free rate is 5%, the expected rate of return on the market portfolio
is 20% and the beta coefficient is 1.2. Using the CAPM, what is the required rate of
return on the asset?

Answer 22

The required rate of return is:

0.05 1 1.2(0.20 2 0.05) 5 23%

Question 23

What is the graphic depiction of how risk and required rate of return are related within the CAPM framework?

Answer 23

The Security market line (SML).

Required rate of return is on the vertical axis, and beta, the measure of risk, is on the horizontal axis. The slope of the line is the difference between the required rate of return on the market index and RF, the risk-free rate.

Question 24

Does the SML hold constant over time?

Answer 24

The SML shows the required return and risk at a particular point in time. The SML can, and does, change over time as a result of (1) changes in the risk-free rate and (2) changes in the risk premium, which reflects investor beliefs.