Equity Portfolio Management - Numerical Questions

Question 1

How is Effective Annual Interest Rate Calculated on the following bond?

• horizon, T: 0.5Y
• face value: $100
• price of the bond: $97.36

Answer 1

Step 1: Calculate the total return of the bond for the given horizon:
r_f(0.5) = 100 / 97.36-1 = 0.0271

Step 2: Calculate the EAR:
EAR = (1 + r_f) ^ 1/T –> EAR = (1 + 0.0271) ^ 1/0.5 = 1.0549 = 5.49%

Question 2

How is Effective Annual Interest Rate Calculated on the following bond?

• horizon, T: 25Y
• face value: $100
• price of the bond: $23.3

Answer 2

Step 1: Calculate the total return of the bond for the given horizon:
r_f(25) = 100 / 23.3 - 1 = 3.2918

Step 2: Calculate the EAR:
EAR = (1 + r_f) ^ 1/T –> EAR = (1 + 3.2918) ^ 1/25 = 1.060 = 6%

Question 3

How is the Annual Percentage Rate (APR) calculated for the following bond?

• horizon, T: 25Y
• face value: $100
• price of the bond: $23.3
• EAR: 1.0549

Answer 3

APR = ((1 + EAR)^T) - 1 / T
APR = ((1.0549)^0.5) - 1 /0.5 = 0.0542=5.42%

Question 4

Risk-free rate is the rate you would earn in risk-free assets. Which of the following options is NOT a risk-free asset?
A) T-bills
B) Money market funds
C) Certificates of Deposits (CD)
D) Yield obtained from bank depositing
E) All of the above options are considered risk-free

Answer 4

E) All of the above options are considered risk-free

Question 5

How is the Sharpe Ratio calculated? And how is it interpreted?

Answer 5

Sharpe Ratio = (Risk Premium)/(St.Dev.of Excess Return)
The importance of the trade-off between reward (the risk premium) and risk (as measured by the standard deviation or SD) suggests that we measure the attraction of a portfolio by the ratio of its risk premium to the SD of its excess returns. This reward-to-volatility measure is known as the Sharpe ratio. It is widely used to evaluate the performance of investment managers.
The higher the Sharpe Ratio, the better the performance.

Question 6

Determine the standard deviation of a random variable q with the following probability distribution:

Value Probability
0 0.25
1 0.25
2 0.50

Answer 6

Step 1: Compute Expected Return:
E(r)=(0.250)+(0.251)+(0.5*2)=1.25

Step 2: Determining the Variance:
Variance=σ^2 = SUM ( p(s) * [r(s) - E(r)] ^ 2 )
σ^2= [0.25 (0 - 1.25) ^ 2] + [0.25 (1 - 1.25) ^ 2] + [0.5(2 - 1.25) ^ 2] = 0.686875

Step 3: Determine the St. Dev.:
σ = √(0.686875) = 0.82878

Question 7

The continuously compounded annual return on a stock is normally distributed with a mean of 20% and standard deviation of 30%. With 95.44% confidence, we should expect its actual return in any particular year to be between which pair of values? (Hint: Look again at Figure 5.4.)

a. −40.0% and 80.0%
b. −30.0% and 80.0%
c. −20.6% and 60.6%
d. −10.4% and 50.4%

Answer 7

A) CORRECT

With the confidence level of 95.44%, we know that the value of a normally distributed variable will fall within two standard deviations of the mean. That is, with a mean of 20%, the interval will be:
Upper=20%+(230%)=80%
Lower=20%-(230%)=-40%

Question 8

According to the “68, 95, and 99.7 rule”, with a confidence level of 68, the expected actual return of a security will fall within _____ standard deviations from the mean.

A) 1
B) 2
C) 3

Answer 8

According to the “68, 95, and 99.7 rule”, with a confidence level of 68, the expected actual return of a security will fall within ONE standard deviation from the mean.

That is 68% of the data falls within one standard deviation.

Question 9

According to the “68, 95, and 99.7 rule”, with a confidence level of 99.7, the expected actual return of a security will fall within _____ standard deviations from the mean.

A) 1
B) 2
C) 3

Answer 9

According to the “68, 95, and 99.7 rule”, with a confidence level of 99.7, the expected actual return of a security will fall within THREE standard deviations from the mean.

That is 99.7% of the data falls within three standard deviations.

Question 10

According to the “68, 95, and 99.7 rule”, with a confidence level of 95, the expected actual return of a security will fall within _____ standard deviations from the mean.

A) 1
B) 2
C) 3

Answer 10

According to the “68, 95, and 99.7 rule”, with a confidence level of 95, the expected actual return of a security will fall within TWO standard deviations from the mean.

That is 95% of the data falls within two standard deviations.

Question 11

How is Expected Annual HPR calculated?

A) risk-free rate + average historical risk premium
B) risk-free rate + inflation rate
C) risk-free rate - average historical risk premium
D) risk-free rate - inflation rate

Answer 11

A) CORRECT

Expected Annual HPR = risk-free rate + average historical risk premium

Question 12

During a period of severe inflation, a bond offered a nominal HPR of 80% per year. The inflation rate was 70% per year.

A) What was the real HPR on the bond over the year?
B) Compare this real HPR to the approximation: r_real≈r_nom-i

Answer 12

A)
Real interest rate = [(1 + Nominal Rate) / (1+Inflation)] - 1
Real interest rate = [(1 + 0.8) / (1+0.7)] - 1 = 0.0588 = 5.88%

B)
Interest_Real(Approximation) ≈ nominal rate - inflation
Interest_Real (Approximation) ≈ 0.8 - 0.7 = 0.1 = 10%
There is a large difference between the actual real rate and the approximated real rate due to such a large inflation.

Question 13

Square foot rule (volatility across time horizons)

What is the daily volatility of a portfolio with a yearly volatility of 20%?
Assuming 250 days in a year (standard days of open markets).

Answer 13

We convert the yearly volatility to daily volatility as follows:

Yearly Volatility → Daily Volatility:
Yearly Volatility / √Days in a year

Assuming 250 standard trading days a year and yearly volatility of 20%:

Yearly Volatility → Daily Volatility = 0.2/√250 = 1.2%

Question 14

Compute daily VaR given probability of 2.5% and annual volatility of 20%. The value of an investment is equal to €15m.

Answer 14

Step 1: convert annual volatility to daily volatility using square-root rule:
assuming 250 trading days a year, we get the daily volatility of:
0.2/√250 = 1.2%

Step 2: calculate daily VaR by multiplying daily volatility with investment value and multiply with the probability level:
VaR = Initial Investmnet * Volatility * Probability level
VaR = 15m * 0.012 * 1.96

Probability levels:
• 1% VaR = 99% confidence level = 2.325
• 5% VaR = 95% confidence level = 1.645
• 2.5% VaR = 97.5% confidence level 1.96

Question 15

The probability multiple given 99% confidence level is ______

A) 1.645
B) 2.325
C) 1.96
D) 2.35

Answer 15

B) CORRECT
The probability multiple given 99% confidence level is 2.325

REMEMBER: the smaller the probability (i.e., the larger the confidence level), the bigger the multiple

Question 16

The probability multiple given 95% confidence level is ______

A) 1.645
B) 2.325
C) 1.96
D) 2.35

Answer 16

A) CORRECT
The probability multiple given 95% confidence level is 1.645

REMEMBER: the smaller the probability (i.e., the larger the confidence level), the bigger the multiple

Question 17

The probability multiple given 97.5% confidence level is ______

A) 1.645
B) 2.325
C) 1.96
D) 2.35

Answer 17

C) CORRECT
The probability multiple given 97.5% confidence level is 1.96

REMEMBER: the smaller the probability (i.e., the larger the confidence level), the bigger the multiple

Question 18

For an initial investment of EUR L=1,000, and an annual volatility equal to 20%, what is the two-year Value at Risk for a 5% probability?

Answer 18

Step 1: Convert annual volatility to two-year volatility using square root rule:

Yearly Volatility → TwoYear Volatility = Yearly Volatility/√(No.of two years every one year) → Yearly Volatility/√0.5
TwoYear Volatility = 0.2/√0.5 = 0.283

Step 2: Calculate two-year VaR:
VaR = volatility * probability multiple * initial investment
VaR = 0.283 * 1.645 * 1000 = 465

Question 19

Assume a portfolio with following securities:
-Stock A: βA=1.3 wA=0.8
-Stock B: βB=0.5 wB=0.2
What is the portfolio beta?

Answer 19

The portfolio beta is the weighted average beta of each security in the portfolio:
βportfolio = 0.8 (1.3) + 0.2 (0.5) = 1.14

Question 20

Which risk is non-diversifiable?

A) Systematic risk
B) Non-systematic risk

Answer 20

Systematic risk refers to macroeconomic risks. This is a common factor that affects all security returns. The market factor, m, measures unanticipated developments in the macroeconomy.

The systematic component of a portfolio variance, β_P^2 σ_M^2 depends on the average beta coefficient of the individual securities. This part of the risk depends on portfolio beta and σ_M^2 and will persist regardless of the extent of portfolio diversification. No matter how many stocks are held, their common exposure to the market will result in a positive portfolio beta and be reflected in portfolio systematic risk. I.e., systematic risk is non-diversifiable.

Question 21

Which of the following are examples of common economic factors? I.e., sources of systematic risk. Select 1-4
A) business cycles
B) stock repurchase
C) interest rates
D) cost of natural resources

Answer 21

Common economic factors/“Shocks” refers to unexpected changes to macroeconomic variables that cause, simultaneously, correlated shocks in the rates of return on stocks across the entire market.

ANSWER: all options are correct except B

Question 22

A scatter diagram plots returns of one security versus _____ .

A) its price
B) returns of another security or benchmark
C) excess return of the same security, relative to risk-free rate
D) excess return of the same security, relative to market excess return

Answer 22

CORRECT: B)
A scatter diagram plots returns of one security versus returns of another security (or benchmark such as market). Each point represents one PAIR of returns for a given holding period.

Question 23

A regression equation describes the average relationship between a dependent variable and one or more explanatory variables. In this context, residuals captures______

A) Parts of stock returns not explained by the explanatory variable. They measure the impact of firm-specific events during a particular period.
B) The risks that are non-diversifiable and therefore not captured by the explanatory variables.

Answer 23

A regression equation describes the average relationship between a dependent variable and one or more explanatory variables. In this context, residuals captures Parts of stock returns not explained by the explanatory variable. They measure the impact of firm-specific events during a particular period.

A IS CORRECT

Question 24

A security characteristic line (SCL) plots_____

A) the excess return on a security over the risk-free rate as a function of the excess return on the market.
B) the excess return on a security against its price
C) the returns of one security against the return of another security
D) the return of a security against the excess return of the same security relative to risk-free rate

Answer 24

A) IS CORRECT:

A security characteristic line (SCL) plots the excess return on a security over the risk-free rate as a function of the excess return on the market.

Question 25

According to the security risk in the index model, the total risk/ variance of any given security is______
A) systematic risk - firm-specific risk + residual risk
B) systematic risk + firm-specific risk
C) systematic risk + firm-specific risk + risk premium relative to the risk-free asset

Answer 25

CORRECT: B)

Security risk in the index model is the sum of the systematic risk that is non-diversifiable and the firm-specific risk that is diversifiable:

Total Risk = Systematic Risk + Firm Specific Risk

σ^2 = (β^2 * σ_M^2) + (σ^2 (e))

Question 26

How is the covariance between two securities calculated?

Answer 26

Covariance between two securities is calculated as the product of betas of the securities in the portfolio, multiplied with market index risk, σ_M^2:

Cov(r_i,r_j ) = Product of betas * Market Index Risk
Cov(r_i,r_j ) = (β_i * β_j) * σ_M^2

Question 27

Following is NOT true about the single-index model (with excess returns):

A) It can be written as the following regression equation:
R_i(t) = α_i + β_i * R_M(at time t) + e_i(at time t) : also known as the SCL
B) It drastically reduces the necessary inputs in the Markowitz portfolio selection procedure
C) The index model abstraction is crucial for the specialization of effort in security analysis. By decomposing uncertainty into these systemwide versus firm-specific sources, we vastly simplify the problem of estimating covariance and correlation.
D) Practitioners routinely estimate the index model using total rather than excess (deviations from T-bill rates) rates of return
E) All of the above are true

Answer 27

E) All of the above are true

Question 28

The following is true about the development of beta over time:
A) Betas show a tendency to be quite static over time - i.e., only negligible evolution occurs
B) Betas show a tendency to evolve toward 0 over time
C) Betas show a tendency to evolve toward 1 over time

Answer 28

C) Betas show a tendency to evolve toward 1 over time
• One explanation for this phenomenon is intuitive. A business is
established and a new firm may be more unconventional than an older
one in terms of technology, management style, etc. As it grows, it may
become more diversified; expanding to similar products and later to more
diverse operations. As the firm becomes more conventional, it starts to
resemble the rest of the economy more. Thus, its beta coefficient will tend
to change in the direction of 1.
• Second explanation is statistical. The average beta over all securities is 1.
Thus, before estimating the beta of a security, our best forecast would be
that it is 1.

Question 29

What is the beta of a risk-free asset?
A) 1
B) 0
C) depends

Answer 29

B) The risk-free asset beta is ALWAYS assumed to be 0 if not stated otherwise

Question 30

What are the standard deviations of stocks A and B?
STOCK A: 
 - Expected Return: 13%
 - Beta: 0.8
 - Firm-Specific Standard Deviation: 30%
STOCK B:
 - Expected Return: 18%
 - Beta: 1.2
 - Firm-Specific Standard Deviation: 40%
The market index has a standard deviation of 22% and the risk-free rate is 8%.

Answer 30

The standard deviation of each individual stock is given by:
σ_i = [(β_i^2 * σ_M^2) + σ^2 (e_i )] ^ (1/2)
σ_A = [(0.8^2 * 22^2) +30^2] ^ (1/2) = 34.78%
σ_B = [(1.2^2*22^2) +40^2 ] ^ (1/2) = 47.93%

Question 31

What is the portfolio beta given the following portfolio?
The market index has a standard deviation of 22% and the risk-free rate is 8%
STOCK A: weights: 30%
 - Expected Return: 13%
 - Beta: 0.8
 - Firm-Specific Standard Deviation: 30%
STOCK B: weights: 45%
 - Expected Return: 18%
 - Beta: 1.2
 - Firm-Specific Standard Deviation: 40%
T-BILL: weights 25%

Answer 31

The portfolio beta is the weighted average of the betas of the individual securities:
β_P = w_Aβ_A + w_Bβ_B + w_f β_f
β_P = 0.8(0.3) + 1.2(0.45) + 0(0.25) = 0.78

Question 32

A portfolio management organization analyzes 60 stocks and constructs a mean-variance efficient portfolio using only these 60 securities. WITHOUT using the index model, how many estimates of expected returns, variances, and covariances are needed to optimize this portfolio?

Answer 32

If we are not using an index model, we have to estimate:
- 60=n expected returns, since we need one expected return for each stock
- 60=n variances, since we need one variance for each stock
- The number of co-variances is calculated as:
covariance = n/2n-1
covariance = 60/260-1= 1770

ALTERNATIVE FASTER METHOD:
[(n^2+3n)/2)] –> [(3600+180)/2] = 1890

Question 33

Consider the following (excess return) index model results for STOCK A:
R_A = 1% + 1.2R_M
R squared_ A = 0.576
Residual standard deviation_A= 10.3%
Risk free rate = 6%

Using total return for the index regression, what is the alpha?

Answer 33

The regression equation under excess return:
r_A-r_f = α+β(r_M-r_f )
The regression equation under total return:
r_A=α + r_f (1-β) + (βr_M)

Using total returns, the intercept of the regression is equal to:
Intercept: α + r_f (1-β) - this is the part of the regression that is not affected by the market (i.e., it is not the systematic risk).

Plugging in our numbers and solve for alpha:
r_A = α + r_f (1-β) + (βr_M) –> 0.01 + 0.06 (1 - 1.2) = -0.002 = 0.2%

Question 34

Consider the two following (excess return) index model results for two stocks:
STOCK A:
R_A = 1% + 1.2R_M
R squared_ A = 0.576
Residual standard deviation_A= 10.3%
STOCK B:
R_B = -2% + 0.8R_M
R squared_B = 0.436
Residual standard deviation_B= 9.1%

Which stock has more firm-specific risk?

Answer 34

Firm-specific risk is measured by the residual standard deviation. Thus, stock A has more firm-specific risk: 10.3% > 9.1%.

Question 35

Consider the two following (excess return) index model results for two stocks:
STOCK A:
R_A = 1% + 1.2R_M
R squared_ A = 0.576
Residual standard deviation_A= 10.3%
STOCK B:
R_B = -2% + 0.8R_M
R squared_B = 0.436
Residual standard deviation_B= 9.1%

Which stock has greater market risk?

Answer 35

Market risk is measured by beta, the slop coefficient of the regression. A has a larger beta coefficient: 1.2>0.8, which means that stock A has a larger exposure to market risk

Question 36

Consider the two following (excess return) index model results for two stocks:
STOCK A:
R_A = 1% + 1.2R_M
R squared_ A = 0.576
Residual standard deviation_A= 10.3%
STOCK B:
R_B = -2% + 0.8R_M
R squared_B = 0.436
Residual standard deviation_B= 9.1%

For which stock does market movement explain a greater fraction of return variability?

Answer 36

R^2 measures the fraction of total variance of return explained by the market return. Since A’s R^2 is larger than B’s: 0.576>0.4356, a greater fraction of return variability is explained by market movement for stock A than for stock B

Question 37

What is the portfolio expected return given the following portfolio?
The market index has a standard deviation of 22% and the risk-free rate is 8%
STOCK A: weights: 30%
 - Expected Return: 13%
 - Beta: 0.8
 - Firm-Specific Standard Deviation: 30%
STOCK B: weights: 45%
 - Expected Return: 18%
 - Beta: 1.2
 - Firm-Specific Standard Deviation: 40%
T-BILL: weights 25%

Answer 37

he expected rate of return on a portfolio is the weighted average of the expected returns of the individual securities:
E(r_P ) = w_AE(r_A ) + w_BE(r_B ) + w_f*r_f
E(r_P ) = 0.13(0.3) + 0.18(0.45) + 0.08(0.25 = 0.14 –> 14%

Question 38

What is the formula for R^2 in the index model regression?

Answer 38

R^2 = (Explained Variance)/(Total Variance)
R_i^2 = (β_i^2 * σ_M^2) / (σ_i^2)

Question 39

A portfolio management organization analyzes 60 stocks and constructs a mean-variance efficient portfolio using only these 60 securities. USING THE INDEX MODEL, how many estimates of expected returns, variances, and covariances are needed to optimize this portfolio?

Answer 39

The number of parameter estimates needed in the index model for the portfolio of 60 stocks is:
- Estimates of the mean/ expected return of each security: E(r_i )=n=60
- Estimates of the sensitivity coefficient of each security: β_i=n=60
- Estimates of the firm-specific variance of each security: σ^2 (e_i )=n=60
- Estimates of the market mean: E(r_M )=1
- Estimates of the market variance: σ_M^2=1
No. estimates required: 60+60+60+1+1=182

ALTERNATIVE FASTER METHOD:
3n+2 : 3(60)+2=182

Question 40

A portfolio management organization analyzes 60 stocks and constructs a mean-variance efficient portfolio using only these 60 securities. WITHOUT USING THE INDEX MODEL, how many estimates of expected returns, variances, and covariances are needed to optimize this portfolio?