Quantitative Methods

Question 1

1.1 nominal risk-free rate =

Answer 1

nominal risk-free rate = real risk-free rate + expected inflation rate

Question 2

1.1 nominal rate of interest =

Answer 2

nominal rate of interest =

nominal risk-free rate

+ default risk premium

+ liquidity premium

+ maturity risk premium

Question 3

1.1 The real risk-free rate can be thought of as:

A) approximately the nominal risk-free rate plus the expected inflation rate.
B) approximately the nominal risk-free rate reduced by the expected inflation rate.
C) exactly the nominal risk-free rate reduced by the expected inflation rate.

Answer 3

B) approximately the nominal risk-free rate reduced by the expected inflation rate.

Question 4

1.2 What is the present value of four $100 end-of-year payments if the first payment is to be received three years from today and the appropriate rate of return is 9%?

Answer 4

Step 1: N = 4; I/Y = 9; PMT = −100; FV = 0; CPT → PV = PV2 = $323.97

Step 2: N = 2; I/Y = 9; PMT = 0; FV = −323.97; CPT → PV = PV0 = $272.68

Question 5

1.2 Annuity Due: What is the future value of an annuity that pays $200 per year at the beginning of each of the next three years, commencing today, if the cash flows can be invested at an annual rate of 10%?

Answer 5

To solve this problem, put your calculator in the BGN mode ([2nd] [BGN] [2nd] [SET] [2nd] [QUIT] on the TI or [g] [BEG] on the HP), then input the relevant data and compute FV.

N = 3; I/Y = 10; PMT = –200; PV = 0; CPT → FV = $728.20

Question 6

1.2 Annuity Due: Given a discount rate of 10%, what is the present value of an annuity that makes $200 payments at the beginning of each of the next three years, starting today?

Answer 6

First, let’s solve this problem using the calculator’s BGN mode. Set your calculator to the BGN mode ([2nd] [BGN] [2nd] [SET] [2nd] [QUIT] on the TI or [g] [BEG] on the HP), enter the relevant data, and compute PV.

N = 3; I/Y = 10; PMT = −200; FV = 0; CPT → PV = $547.11

Question 7

1.2 Perpetuity: Kodon Corporation issues preferred stock that will pay $4.50 per year in annual dividends beginning next year and plans to follow this dividend policy forever. Given an 8% rate of return, what is the value of Kodon’s preferred stock today?

Answer 7

PV perpetuity = 4.50 / 0.08 = $56.25

Question 8

1.2 Deferred Perpetuity: Kodon Corporation issues preferred stock that will pay $4.50 per year in annual dividends beginning next year and plans to follow this dividend policy forever. Given an 8% rate of return, what is the value of Kodon’s preferred stock today?

Assume the Kodon preferred stock in the preceding examples is scheduled to pay its first dividend in four years, and is non-cumulative (i.e.,does not pay any dividends for the first three years). Given an 8% required rate of return, what is the value of Kodon’s preferred stock today?

Answer 8

PV perpetuity = 4.50 / 0.08 = $56.25

FV = -56.25; N = 3; I/Y = 8; PMT = 0; CPT → PV = $44.65

Question 9

1.3
0yr = +0; 1yr = +300; 2yr = +600; 3yr = +200

Using a rate of return of 10%, compute the future value of the 3-year uneven cash flow stream described above at the end of the third year.

Answer 9

On calc:
CF; CF0 = 0; C01 = 300; F01 = 1; C02 = 600; F02 = 1; C03 = 200; F03 = 1
NPV; I = 10; NPV = [CPT] = $918.86

N = 3; I/Y = 10; PV = 918.86; PMT = 0; CPT → FV = $1,223

Question 10

1.3
0yr = +0; 1yr = +300; 2yr = +600; 3yr = +200

Compute the present value of this 3-year uneven cash flow stream described previously using a 10% rate of return.

Answer 10

On calc:
CF; CF0 = 0; C01 = 300; F01 = 1; C02 = 600; F02 = 1; C03 = 200; F03 = 1
NPV; I = 10; NPV = [CPT] = $918.86

Question 11

1.3 At an expected rate of return of 7%, how much must be deposited at the end of each year for the next 15 years to accumulate $3,000?

Answer 11

N = 15; I/Y = 7; FV = +$3,000; CPT → PMT = −$119.38 (ignore sign)

Question 12

1.3 Suppose you are considering applying for a $2,000 loan that will be repaid with equal end-of-year payments over the next 13 years. If the annual interest rate for the loan is 6%, how much will your payments be?

Answer 12

N = 13; I/Y = 6; PV = −2,000; CPT → PMT = $225.92

Question 13

1.3 How many $100 end-of-year payments are required to accumulate $920 if the discount rate is 9%?

Answer 13

I/Y = 9%; FV = $920; PMT = −$100; CPT → N = 7 years

Question 14

1.3 Suppose you have a $1,000 ordinary annuity earning an 8% return. How many annual end-of-year $150 withdrawals can be made?

Answer 14

I/Y = 8; PMT = 150; PV = −1,000; CPT → N = 9.9 years

Question 15

1.3 Suppose you have the opportunity to invest $100 at the end of each of the next five years in exchange for $600 at the end of the fifth year. What is the annual rate of return on this investment?

Answer 15

N = 5; FV = $600; PMT = −100; CPT → I/Y = 9.13%

Question 16

1.3 What rate of return will you earn on an ordinary annuity that requires a $700 deposit today and promises to pay $100 per year at the end of each of the next 10 years?

Answer 16

N = 10; PV = −700; PMT = 100; CPT → I/Y = 7.07%

Question 17

1.3 Suppose you must make five annual $1,000 payments, the first one starting at the beginning of Year 4 (end of Year 3). To accumulate the money to make these payments, you want to make three equal payments into an investment account, the first to be made one year from today. Assuming a 10% rate of return, what is the amount of these three payments?

Answer 17

calculator to the BGN mode
N = 5; I/Y = 10; PMT = −1,000; CPT → PV = PV3 = $4,169.87

calculator is in the END mode
N = 3; I/Y = 10; FV = −4,169.87; CPT → PMT = $1,259.78

Question 18

1.3 A security will make the following payments at the end of the next four years: $100, $100, $400, and $100. Calculate the present value of these cash flows using the concept of the present value of an annuity when the appropriate discount rate is 10%.

Answer 18

On calc:
CF; CF0 = 0; C01 = 100; F01 = 1; C02 = 100; F02 = 1; C03 = 400; F03 = 1; C04 = 100; F04 = 1
NPV; I = 10; NPV = [CPT] = $542.38

Question 19

1.4 John plans to invest $2,500 in an account that will earn 8% per year with quarterly compounding. How much will be in the account at the end of two years?

Answer 19

N = 2 * 4 = 8; I/Y = 8% / 4 = 2%; PV = -$2,500; PMT = 0; CPT → FV = 2,929.15

Question 20

1.4 Alice would like to have $5,000 saved in an account at the end of three years. If the return on the account is 9% per year with monthly compounding, how much must Alice deposit today in order to reach her savings goal in three years?

Answer 20

N = 3 * 12 = 36; I/Y = 9% / 12 = 0.75%; PMT = 0; FV = -$5,000; CPT → PV = 3,820.74

Question 21

1.4 Using a stated rate of 6%, compute Effective Annual Rates for semiannual, quarterly, monthly, and daily compounding.

Answer 21

On calc
Semi: [ICONV]; NOM = 6; C/Y = 2; EFF → CPT = 6.09%
Quarterly: [ICONV]; NOM = 6; C/Y = 4; EFF → CPT = 6.14%
Monthly: [ICONV]; NOM = 6; C/Y = 12; EFF → CPT = 6.17%
Daily: [ICONV]; NOM = 6; C/Y = 365; EFF → CPT = 6.18%

Question 22

2.1 Compare Numerical / Quantitative Data:
Discrete -
Continuous -

Answer 22

Discrete - Data is countable units.
Continuous - Data can take on fractional value.

Question 23

2.1 Compare Categorical / Qualitative Data:
Nominal -
Ordinal -

Answer 23

Nominal - Data are labels with no logical order.
Ordinal - Data are labels that can be ordered or ranked.

Question 24

2.2 The vertical axis of a histogram shows:

A) the frequency with which observations occur.
B) the range of observations within each interval.
C)the intervals into which the observations are arranged.

Answer 24

A
In a histogram, the intervals are on the horizontal axis and the frequency is on the vertical axis.

Question 25

2.2 In which type of bar chart does the height or length of a bar represent the cumulative frequency for its category?

A)Stacked bar chart.
B)Grouped bar chart.
C)Clustered bar chart.

Answer 25

A
In a stacked bar chart, the height or length of a bar represents the cumulative frequency of a category. In a grouped or clustered bar chart, each category is displayed with bars side by side that together represent the cumulative frequency.

Question 26

2.2 An analyst who wants to illustrate the relationships among three variables should most appropriately construct:

A)a bubble line chart.
B)a scatter plot matrix.
C)a frequency polygon.

Answer 26

B
With a scatter plot matrix, an analyst can visualize the relationships among three variables by organizing scatter plots of the relationships between each pair of variables.

Question 27

2.3 A portfolio consists of 50% common stocks, 40% bonds, and 10% cash. If the return on common stocks is 12%, the return on bonds is 7%, and the return on cash is 3%, what is the portfolio return?

Answer 27

XW = (0.50 × 0.12) + (0.40 × 0.07) + (0.10 × 0.03) = 0.091, or 9.1%

Question 28

2.3 Geometric Mean
For the last three years, the returns for Acme Corporation common stock have been –9.34%, 23.45%, and 8.92%. Compute the compound annual rate of return over the three-year period.

Answer 28

1+RG= ^3√(1−0.0934)×(1+0.2345)×(1+0.0892)
=^3√1.21903
=(1.21903)^1/3 = 1.06825
RG = 1.06825 - 1 = 6.825%

Question 29

2.3 Harmonic Mean
An investor purchases $1,000 of mutual fund shares each month, and over the last three months, the prices paid per share were $8, $9, and $10. What is the average cost per share?

Answer 29

XH = (3) / ((1/8)+(1/9)+(1/10)) = $8.926 per share

Question 30

2.3
Calculate the arithmetic, geometric, and harmonic means of 8, 9, 10

Answer 30

Arithmetic: (8+9+10) / 3 = 9
Geometric: ^3√8910 or (8910)^(1/3) = 8.963
Harmonic: 3 / ((1/8)+(1/9)+(1/10)) = 8.926

Question 31

2.3 Understand when to use to use the following:
Arithmetic mean
Geometric mean
Trimmed mean
Winsorized mean
Harmonic mean

Answer 31

Arithmetic mean. Estimate the next observation, expected value of a distribution.
Geometric mean. Compound rate of returns over multiple periods.
Trimmed mean. Estimate the mean without the effects of a given percentage of outliers.
Winsorized mean. Decrease the effect of outliers on the mean.
Harmonic mean. Calculate the average share cost from periodic purchases in a fixed dollar amount.

Question 32

2.4 What is the third quartile for the following distribution of returns?

8%, 10%, 12%, 13%, 15%, 17%, 17%, 18%, 19%, 23%

Answer 32

The third quartile is the point below which 75% of the observations lie. Recognizing that there are 10 observations in the data set, the third quartile can be identified as:
Ly=(10+1)×75/100=8.25
When the data are arranged in ascending order, the third quartile is a fourth (0.25) of the way from the eighth data point (18%) to the ninth data point (19%), or 18.25%. This means that 75% of all observations lie below 18.25%.

Question 33

2.4 What is the mean absolute deviation (MAD) of the investment returns for the five investment managers if the managers’ individual returns were 30%, 12%, 25%, 20%, and 23%?

Answer 33

X=[30+12+25+20+23] / 5=22%
MAD=[|30−22|+|12−22|+|25−22|+|20−22|+|23−22|] / (5)
MAD=[8+10+3+2+1] / (5) =4.8%

Question 34

2.4 Assume that the 5-year annualized total returns for the five investment managers used in the preceding examples represent only a sample of the managers at a large investment firm. What is the sample variance of these returns?
annualized returns: [30%, 12%, 25%, 20%, 23%]

Answer 34

X=[30+12+25+20+23] / 5=22%
s^2=[(30−22)^2+(12−22)^2+(25−22)^2+(20−22)^2+(23−22)^2] / (5−1) =44.5(%^2)
or
on calc
[2nd] [Data] [“enter returns for x’s”] [2nd] [Stat] [“2nd enter until 1-V]: Sx = 0.0667
0.0667^2 = 0.00445

Question 35

2.4 Compute the sample standard deviation based on the result of the preceding example.
annualized returns: [30%, 12%, 25%, 20%, 23%]
sample variance: 44.5(%^2)

Answer 35

s = [44.5(%2)]1/2 = 6.67%
or
√0.00445 = 0.0667
or
on calc
[2nd] [Data] [“enter returns for x’s”] [2nd] [Stat] [“2nd enter until 1-V]: Sx = 0.0667

Question 36

2.4 You have just been presented with a report that indicates that the mean monthly return on T-bills is 0.25% with a standard deviation of 0.36%, and the mean monthly return for the S&P 500 is 1.09% with a standard deviation of 7.30%. Your unit manager has asked you to compute the CV for these two investments and to interpret your results.

Answer 36

CV T-bills = 0.36 / 0.25=1.44
CV S&P 500 = 7.30 / 1.09=6.70

These results indicate that there is less dispersion (risk) per unit of monthly return for T-bills than for the S&P 500

Question 37

2.4 XYZ Corp. Annual Stock Returns

20x1, 20x2, 20x3, 20x4, 20x5, 20x6
22%, 5%, –7%, 11%, 2%, 11%

Assume an investor has a target return of 11% for XYZ stock. What is the stock’s target downside deviation?

Answer 37

Deviations from the target return:

22% – 11% = 11%

5% – 11% = –6%

–7% – 11% = –18%

11% – 11% = 0%

2% – 11% = –9%

11% – 11% = 0%

Target downside deviation =
(√(–6)^2+(–18)^2+(–9)^2) / (6–1) = √88.2 = 9.39%

Question 38

2.5 The variance of returns on stock A is 0.0028, the variance of returns on stock B is 0.0124, and their covariance of returns is 0.0058. Calculate and interpret the correlation of the returns for stocks A and B.

Answer 38

First, it is necessary to convert the variances to standard deviations:

sA = (0.0028)½ = 0.0529

sB = (0.0124)½ = 0.1114

Now, the correlation between the returns of stock A and stock B can be computed as follows:

ρAB = 0.0058 / (0.0529)(0.1114) = 0.9842

Question 39

3.1 Probability that a horse will win a race is 12.5%.
Odds for:
Odds against:

Answer 39

Odds for: 0.125 / (1 - 0.125) = 1/7
Odds against: (1-0.125) / 0.125 = 7/1

Question 40

3.1 Consider the following information:

P(I) = 0.4, the probability of the monetary authority increasing interest rates (I) is 40%.
P(R | I) = 0.7, the probability of a recession (R) given an increase in interest rates is 70%.
What is P(RI), the joint probability of a recession and an increase in interest rates?

Answer 40

Applying the multiplication rule, we get the following result:

P(RI) = P(R | I) × P(I)

P(RI) = 0.7 × 0.4

P(RI) = 0.28

Question 41

3.1 Using the information in our previous interest rate and recession example and the fact that the unconditional probability of a recession, P(R), is 34%, determine the probability that either interest rates will increase or a recession will occur.

Answer 41

P(R or I) = P(R) + P(I) – P(RI)

P(R or I) = 0.34 + 0.40 – 0.28

P(R or I) = 0.46

Question 42

3.1 What is the probability of rolling three 4s in one simultaneous toss of three dice?

Answer 42

Since the probability of rolling a 4 for each die is 1/6, the probability of rolling three 4s is:

P(three 4s on the roll of three dice) = 1/6 × 1/6 × 1/6 = 1/216 = 0.00463

Question 43

3.2The probability distribution of EPS for Ron’s Stores is given in the figure below. Calculate the expected earnings per share.

(Probability, Earnings Per Share)
10%, £1.80
20%, £1.60
40%, £1.20
30%, £1.00

100%

Answer 43

on calc
[2nd] [Data] [“enter earnings for x’s”] [“enter probability for y’s”] [2nd] [Stat] [“2nd enter until 1-V]: x = 1.28

Question 44

3.3 Portfolio Standard Deviation
A portfolio is 30% invested in stockes, σ = 20%, with the remainder in bonds, σ = 12%. The correlation of bond returns with stock returns is 0.60. What is the standard deviation of portfolio returns?

Answer 44

√(0.3^20.2^2)+(0.7^20.12^2)+(2(0.3)(0.7)(0.6)(0.2)(0.12)) = 12.9%

Question 45

3.3 Consider a portfolio consisting of eight stocks. Your goal is to designate four of the stocks as “long-term holds,” three of the stocks as “short-term holds,” and one stock as “sell.” How many ways can these eight stocks be labeled?

Answer 45

8! / (4!×3!×1!) = 40,320 / (24×6×1) = 280

Question 46

3.3 How many ways to choose 3 from 8, if order doesn’t matter?

Answer 46

on calc
[8] [2nd] [nCr] [3] = 56

Question 47

3.3 How many ways to choose 3 from 8, if order does matter?

Answer 47

on calc
[8] [2nd] [nPr] [3] = 336

Question 48

3.3 Given the conditional probabilities in the table below and the unconditional probabilities P(Y = 1) = 0.3 and P(Y = 2) = 0.7, what is the expected value of X?

xi P(xi | Y = 1) P(xi | Y = 2)
0 0.2 0.1
5 0.4 0.8
10 0.4 0.1

Answer 48

E(X | Y = 1) = (0.2)(0) + (0.4)(5) + (0.4)(10) = 6

E(X | Y = 2) = (0.1)(0) + (0.8)(5) + (0.1)(10) = 5

E(X) = (0.3)(6) + (0.7)(5) = 5.30

Question 49

3.3 A discrete uniform distribution (each event has an equal probability of occurrence) has the following possible outcomes for X: [1, 2, 3, 4]. The variance of this distribution is closest to:

Answer 49

Expected value = (1/4)(1 + 2 + 3 + 4) = 2.5

Variance = (1/4)[(1 – 2.5)2 + (2 – 2.5)2 + (3 – 2.5)2 + (4 – 2.5)2] = 1.25

Note that since each observation is equally likely, each has 25% (1/4) chance of occurrence.

Question 50

4.1 Return on equity for a firm is defined as a continuous distribution over the range from –20% to +30% and has a cumulative distribution function of F(x) = (x + 20) / 50. Calculate the probability that ROE will be between 0% and 15%.

Answer 50

To determine the probability that ROE will be between 0% and 15%, we can first calculate the probability that ROE will be less than or equal to 15%, or F(15), and then subtract the probability that ROE will be less than zero, or F(0).

P(0 ≤ x ≤ 15) = F(15) – F(0)

F(15) = (15 + 20) / 50 = 0.70

F(0) = (0 + 20) / 50 = 0.40

F(15) – F(0) = 0.70 – 0.40 = 0.30 = 30%

Question 51

4.2 A key property of a normal distribution is that it:

A)
has zero skewness.
B)
is asymmetrical.
C)
has zero kurtosis.

Answer 51

A

Question 52

4.2 For a standard normal distribution, F(0) is:

A)
0.0.
B)
0.1.
C)
0.5.

Answer 52

C

Question 53

4.2
Portfolio Portfolio A Portfolio B Portfolio C
E(Rp) 5% 11% 18%
σp 8% 21% 40%

Given a threshold level of return of 4%, use Roy’s safety-first criterion to choose the optimal portfolio. Portfolio:

A)
A.
B)
B.
Incorrect Answer
C)
C.
Correct Answer

Question 54

4.3 If a stock’s initial price is $20 and its year-end price is $23, then its continuously compounded annual (stated) rate of return is:

Answer 54

ln(23 / 20) = 0.1398

Question 55

4.3 A stock doubled in value last year. Its continuously compounded return over the period was closest to:

Answer 55

ln(2) = 0.6931

Question 56

6.1