# Buổi 1 - Quantitative - Reading 6: The time value of money & Reading 7: Statistical concepts and market returns

READING 6: THE TIME VALUE OF MONEY

Question 6.1: Which one of the following statements best describes the components of the required interest rate on a security?
A) The real risk-free rate, the default risk premium, a liquidity premium and a premium to reflect the risk associated with the maturity of the security.
B) The real risk-free rate, the expected inflation rate, the default risk premium, a liquidity premium and a premium to reflect the risk associated with the maturity of the security.
C) The nominal risk-free rate, the expected inflation rate, the default risk premium, a liquidity premium and a premium to reflect the risk associated with the maturity of the security
Explanation
B is correct. Required rate of return = nominal risk-free rate (real risk free rate + expected inflation) + default risk premium + liquidity risk premium + maturity risk premium

Question 6.2:
Sydney Burns, CFA, is considering the purchase of a bond issued by SubPrime Providers. The bond is highly liquid and has a maturity equal to that of a long-term Treasury bond. The SubPrime Providers bond carries a default risk premium of 5%. Burns notices that the difference in interest rates offered on long-term Treasury bonds and short-term Treasury bills currently equals 4%. The real risk-free rate equals 1% and the expected inflation rate equals 2%. Burns should expect the interest rate on the SubPrime Providers bond to:

A) be greater than or equal to 4%, and less than or equal to 8%.
B) be greater than or equal to 5%, and less than or equal to 9%.
C) be greater than or equal to 7%, and less than or equal to 12%.
Explanation
C is correct.The interest rate equals the sum of the real rate, the expected inflation rate, the total risk premium (which equals the sum of the maturity risk premium, the liquidity risk premium, and the default risk premium). The real rate equals 1%, the expected inflation rate equals 2%, the maturity risk premium equals 4%. Treasury bonds have no liquidity or default risk, so the interest rate on a long-term Treasury bond would be expected to equal 7%. Since the SubPrime Providers (long-term) bond incurs default risk, its interest rate must exceed that of the long-term Treasury bond (i.e., 7%). The SubPrime Providers bond is highly liquid, so it has no liquidity premium. The default risk premium for SubPrime Providers bond equals 5%. Taken together, this implies that the SubPrime Provider bond interest rate should exceed the Treasury bond interest by 5 percentage points (12%).

Question 6.3: Peter Wallace wants to deposit \$10,000 in a bank certificate of deposit (CD). Wallace is considering the following banks:
Bank A offers 5.85% annual interest compounded annually.
Bank B offers 5.75% annual interest rate compounded monthly.
Bank C offers 5.70% annual interest compounded daily.
Which bank offers the highest effective interest rate and how much?
A) Bank C, 5.87%.
B) Bank B, 5.90%.
C) Bank A, 5.85%
Explanation
Step 1: Calculate each EAR offered by each bank
Formula for calculating Effective Annual Return
EAR = (1+periodic rate)m -1
where:
periodic rate = stated annual rate/m
m = the number of compounding periods per year

Bank A’s EAR = 5.85%
Bank B, nominal = 5.75, m(C/Y) = 12 => EAR = 5.90%
Bank C, nominal = 5.70, m(C/Y) = 365 => EAR = 8.87%
Step 2: Chose the bank with the highest EAR
=> B is correct

Question 6.4: An investor would like to purchase a 20-year annuity that will pay \$45,000 each year beginning when she retires 15 years from now. The amount she would need to invest today to fund this annuity, assuming an annual return of 5%, in order fund this annuity, is closest to:

A) \$269,750.
B) \$274,340.
C) \$283,240

Explanation
C is corect. The amount needed to fund the annuity at t=15 is \$588,839. In BGN mode: N = 20; I/Y = 5; PMT = 45,000; FV = 0; CPT PV. The equivalent amount at t=0 is \$588,839 /(1.05^15) = \$283,242.

Question 6.5: Compute the present value of a perpetuity with \$100 payments beginning four years from now. Assume the appropriate annual interest rate is 10%.
A) \$1,000.
B) \$683.
C) \$751.
Explanation
Step 1: Calculated the present value of the perpetuity at t = 3 (because the present value of a perpetuity or annuity is valued one period before the first payment)
.
PVperpetuity = PMT/r
The present value at t = 3 is 100 / 0.10 = 1,000.
Step 2: Discount the result above to t=0
Present value at t = 0 is 1,000 / (1.10)^3 = 751 => C is correct

Question 6.6: Find the future value of the following uneven cash flow stream. Assume end of the year payments. The discount rate is 12%.
Year 1          -2,000
Year 2          -3,000
Year 3           6,000
Year 4         25,000
Year 5         30,000
A) \$58,164.58.
B) \$33,004.15.
C) \$65,144.33.
Explanation
Step 1: Calculate the FV of each year’s cashflow
N = 4; I/Y = 12; PMT = 0; PV = -2,000; CPT → FV = -3,147.04
N = 3; I/Y = 12; PMT = 0; PV = -3,000; CPT → FV = -4,214.78
N = 2; I/Y = 12; PMT = 0; PV = 6,000; CPT → FV = 7,526.40
N = 1; I/Y = 12; PMT = 0; PV = 25,000; CPT → FV = 28,000.00
N = 0; I/Y = 12; PMT = 0; PV = 30,000; CPT → FV = 30,000.00
Step 2: Sum the cash flows
=> Total cash flow’s future value = \$58,164.58 => A is correct

Question 6.7: Concerning an ordinary annuity and an annuity due with the same payments and positive interest rate, which of the following statements is most accurate?
A) There is no relationship.
B) The present value of the ordinary annuity is greater than an annuity due.
C) The present value of the ordinary annuity is less than an annuity due
Explanation
C is correct. With a positive interest rate, the present value of an ordinary annuity is less than the present value of an annuity due. The first cash flow in an annuity due is at the beginning of the period, while in an ordinary annuity, the first cash ow occurs at the end of the period. Therefore, each cash flow of the ordinary annuity is discounted one period more

Question 6.8: A client plans to retire in 15 years and will need to withdraw \$50,000 from his retirement account each year for 10 years, beginning on the day he retires. After that, he will need to withdraw \$20,000 per year for 25 years. The account returns 4% annually. The amount he needs to have in the account on the day he retires is closest to:
A) \$580,000.
B) \$640,000.
C) \$655,000.
Explanation
Step 1: Convert these cash flow into 2 cash flows and calculate based on additivity principle
Cash flow 1: A 35 year annuity due of \$20,000 per year
N = 35, PMT = 20,000, I/Y = 4, FV = 0; CPT PV = –388,224
Cash flow 2: A 10 year annuity due of \$30,000
N = 10, PMT = 30,000, I/Y = 4, FV = 0; CPT PV = –253,060
Step 2: Sum these two cash flows
Cash flow 1 + Cash flow 2 = \$641,284 => B is correct

READING 7: STATISTICAL CONCEPTS AND MARKET RETURNS

Question 7.1: Which of the following groups best illustrates a sample?
A) The set of all estimates for Exxon Mobil’s EPS for next financial year
B) The FTSE Eurotop 100 as a representation of the European stock market
C) UK shares traded on Wednesday of last week that also closed above £120/ share on the London Stock Exchange
Explanation
B is correct. The FTSE Eurotop 100 represents a sample of all European stocks. It is a subset of the population of all European stocks.

Question 7.2:
Which of the following measurement scales provides the least information?
A) Ratio.
B) Ordinal.
C) Nominal.
Explanation
C is correct. From least to most information, the ordering of measurements scales is nominal, ordinal, interval, and ratio.

Question 7.3: In descriptive statistics, an example of a parameter is the:
A) median of a population.
B) mean of a sample of observations.
C) standard deviation of a sample of observations.
Explanation
A is correct. Any descriptive measure of a population characteristic is referred to as a parameter.

Question 7.4: For the last four years, the returns for XYZ Corporation's stock have been 10.4%, 8.1%, 3.2%, and 15.0%. The equivalent compound annual rate is:
A) 8.9%.
B) 9.1%.
C) 9.2%
Explanation
Formula for calculating geometric mean of return: 1+ Rg = [(1+R1)(1+R2)...(1+Rn)]^(1/n)-1

=> The equivalent compound annual rate is (1.104 × 1.081 × 1.032 × 1.15) ^0.25 – 1 = 9.1%
=> B is correct

Question 7.5: A manager invests €5,000 annually in a security for four years at the prices
shown in the following table.
Purchase Price of Security (€)
Year 1      62.00
Year 2      76.00
Year 3      84.00
Year 4      90.00
The average price paid for the security is closest to:
A) €76.48.
B) €77.26.
C) €78.00.
Explanation Question 7.6: An analyst constructs a histogram and frequency polygon of monthly returns for aggressive equity funds over a 20-year period. Which of the following statements about these displays is most accurate?
A) The height of each bar in a frequency polygon represents the absolute frequency for each return interval.

B) Both a histogram and a frequency polygon provide a graphical display of data found in a frequency distribution.
C) To construct a histogram, the analyst would plot the midpoint of the return intervals on the x-axis and the absolute frequency for that interval on the y-axis, connecting neighboring points with a straight line.
Explanation
B is correct. Choice A describes a histogram, and Choice C describes a frequency polygon.

Question 7.7: Assume that the following returns are a sample of annual returns for firms in the clothing industry. Given the following sample of returns, what are the sample variance and standard deviation respectively?
Firm 1       Firm 2        Firm 3        Firm 4       Firm 5
15%           2%            5%            (7%)            0%
A) 32.4; 5.7.
B) 51.6; 7.2.
C) 64.5; 8.0.
Explanation Question 7.8: The following data points are observed returns.
4.2%, 6.8%, 7.0%, 10.9%, 11.6%, 14.4%, 17.0%, 19.0%, 22.5%
What return lies at the 70th percentile (70% of returns lie below this return)?
A) 17.0%.
B) 14.4%.
C) 19.0%.
Explanation
Formula for finding quantile: Ly = (n+1)
=> With 9 observations, the location of the 70th percentile is (9 + 1)(70/100) = 7. The seventh observation in ascending order is 17.0% => A is correct

Question 7.9: According to Chebyshev's inequality, the minimum proportion of observations falling within three standard deviations of the mean for a negatively skewed distribution is closest to:
A)68%.
B) 75%.
C) 89%.
Explanation
According to Chebyshev's inequality, the proportion of the observations within 3 standard deviations of the mean is at least 1 – (1/3^2) = 0.89 or 89%. This holds for any distribution, regardless of the shape.

Question 7.10: Which of the following statements about return distributions is most accurate?
A) With positive skewness, the median is greater than the mean.
B) If skewness is positive, the average magnitude of positive deviations from the mean is smaller than the average magnitude of negative deviations from the mean.
C) If a return distribution has positive excess kurtosis and the analyst uses statistical models that do not account for the fatter tails, the analyst will underestimate the likelihood of extreme outcomes.
Explanation
C is correct. If a return distribution has positive excess kurtosis, statistical models that do not account for the fatter tails will underestimate the likelihood of very bad or very good outcomes. A distribution with positive skewness will have a mean greater than the median and larger average positive deviations than average negative deviations.

Question 7.11: Which of the following most accurately describes a distribution that is more peaked than normal?
A) Mesokurtotic
B) Platykurtotic
C) Leptokurtotic
Explanation
C is correct. A distribution that is more peaked than normal is called leptokurtotic.