Quantitative Methods

Question 1

Assume an investor makes the following investments:
Today, she purchases a share of stock in Redwood Alternatives for $50.00.
After one year, she purchases an additional share for $75.00.
After one more year, she sells both shares for $100.00 each.
There are no transaction costs or taxes. The investor’s required return is 35.0%.
During year one, the stock paid a $5.00 per share dividend. In year two, the stock paid a$7.50 per share dividend.
The time-weighted return is:

A) 51.4%.
B) 51.7%.
C) 23.2%.

Answer 1

A is accurate.

To calculate the time-weighted return:

Step 1: Separate the time periods into holding periods and calculate the return over that period:

Holding period 1:
P0 = $50.00
D1 = $5.00
P1 = $75.00 (from information on second stock purchase)
HPR 1 = (75 – 50 + 5) / 50 = 0.60, or 60%
Holding period 2:
P1 = $75.00
D2 = $7.50
P2 = $100.00
HPR 2 = (100 – 75 + 7.50) / 75 = 0.433, or 43.3%.

Step 2: Use the geometric mean to calculate the return over both periods
Return = [(1 + HPR1) × (1 + HPR2)]^1/2 – 1 = [(1.60) × (1.433)]^1/2 – 1 = 0.5142, or 51.4%.

Question 2

If a stock decreases from $90 to $80, the continuously compounded rate of return for the period is:

A) -0.1250.
B) -0.1000.
C) -0.1178.

Answer 2

C is correct.

This is given by the natural logarithm of the new price divided by the old price; ln(80 /90) = -0.1178.

Question 3

An investor expects a stock currently selling for $20 per share to increase to $25 by year-end. The dividend last year was $1 but he expects this year’s dividend to be $1.25. What is the expected holding period return on this stock?

A) 24.00%.
B) 28.50%.
C) 31.25%.

Answer 3

C is correct.

Return = [dividend + (ending value - beginning value)] / beginning price value
= [1.25 + (25 – 20)] / 20 = 6.25 / 20 = 0.3125

Question 4

Vega research has been conducting investor polls for Third State Bank. They have found the most investors are not willing to tie up their money in a 1-year (2-year) CD unless they receive at least 1.0% (1.5%) more than they would on an ordinary savings account. If the savings account rate is 3%, and the bank wants to raise funds with 2-year CDs, the yield must be at least:

A) 4.0%, and this represents a required rate of return.
B) 4.5%, and this represents a discount rate.
C) 4.5%, and this represents a required rate of return.

Answer 4

C is correct.

Since we are taking the view of the minimum amount required to induce investors to lend funds to the bank, this is best described as a required rate of return. Based upon the numerical information, the rate must be 4.5% (= 3.0 + 1.5).

Question 5

Wei Zhang has funds on deposit with Iron Range bank. The funds are currently earning 6%interest. If he withdraws $15,000 to purchase an automobile, the 6% interest rate can be best thought of as a(n):

A) discount rate.
B) financing cost.
C) opportunity cost.

Answer 5

C is correct.

Since Wei will be foregoing interest on the withdrawn funds, the 6% interest can be best characterized as an opportunity cost — the return he could earn by postponing his auto purchase until the future.

Question 6

Selmer Jones has just inherited some money and wants to set some of it aside for a vacation in Hawaii one year from today. His bank will pay him 5% interest on any funds he deposits. In order to determine how much of the money must be set aside and held for the trip, he should use the 5% as a:

A) discount rate.
B) opportunity cost.
C) required rate of return.

Answer 6

A is accurate.

He needs to figure out how much the trip will cost in one year, and use the 5% as a discount rate to convert the future cost to a present value. Thus, in this context the rate is best viewed as a discount rate.

Question 7

An investor makes the following investments:
She purchases a share of stock for $50.00.
After one year, she purchases an additional share for $75.00.
After one more year, she sells both shares for $100.00 each.
There are no transaction costs or taxes.
During year one, the stock paid a $5.00 per share dividend. In year 2, the stock paid a $7.50 per share dividend. The investor’s required return is 35%. Her money-weighted return is closest to:

A) 48.9%.
B) 16.1%.
C) -7.5%.

Answer 7

A is accurate.

To determine the money weighted rate of return, use your calculator’s cash flow and IRR functions. The cash flows are as follows:

CF0: initial cash outflow for purchase = $50
CF1: dividend inflow of $5 - cash outflow for additional purchase of $75 = net cash outflow of -$70
CF2: dividend inflow (2 × $7.50 = $15) + cash inflow from sale (2 × $100 = $200) = net cash inflow of $215
Enter the cash flows and compute IRR:
CF0 = -50; CF1 = -70; CF2 = +215; CPT IRR = 48.8607

Question 8

A 10% coupon bond was purchased for $1,000. One year later the bond was sold for $915 to yield 11%. The investor’s holding period yield on this bond is
closest to:

A) 1.5%.
B) 9.0%.
C) 18.5%.

Answer 8

A is accurate.

HPY = [(interest + ending value) / beginning value] – 1
= [(100 + 915) / 1,000] – 1
= 1.015 – 1 = 1.5%

Question 9

An investor buys a non-dividend paying stock for $100 at the beginning of the year with 50% initial margin. At the end of the year, the stock price is $95. Deflation of 2% occurred during the year. Which of the following return measures for this investment will be greatest?

A) Leveraged return.
B) Real return.
C) Nominal return.

Answer 9

B is correct.

No calculations are needed. The real return is greater than the nominal return because the inflation rate is negative. The leveraged return is more negative than the nominal return because the investment lost value and leverage magnifies the loss.

Question 10

The continuously compounded rate of return that will generate a one-year holding period return of -6.5% is closest to:

A) -5.7%.
B) -6.3%.
C) -6.7%.

Answer 10

C is correct.

Continuously compounded rate of return = ln(1 – 0.065) = -6.72%.

Question 11

Time-weighted returns are used by the investment management industry because they:

A) take all cash inflows and outflows into account using the internal rate of return.
B) result in higher returns versus the money-weighted return calculation.
C) are not affected by the timing of cash flows.

Answer 11

C is correct.

Time-weighted returns are not affected by the timing of cash flows. Money-weighted returns, by contrast, will be higher when funds are added at a favorable investment period or will be lower when funds are added during an unfavorable period. Thus, time-weighted returns offer a better performance measure because they are not affected by the timing of flows into and out of the account.

Question 12

Which of the following is most accurate
with respect to the relationship of the money-weighted return to the time-weighted return? If funds are contributed to a portfolio just prior to a period of favorable performance, the:

A) money-weighted rate of return will tend to be depressed.
B) money-weighted rate of return will tend to be elevated.
C) time-weighted rate of return will tend to be elevated.

Answer 12

B is correct.

The time-weighted returns are what they are and will not be affected by cash inflows or outflows. The money-weighted return is susceptible to distortions resulting from cash inflows and outflows. The money-weighted return will be biased upward if the funds are invested just prior to a period of favorable performance and will be biased downward if funds are invested just prior to a period of relatively unfavorable performance. The opposite will be true for cash outflows.

Question 13

Computing the internal rate of return of the inflows and outflows of a portfolio would give the:

A) money-weighted return.
B) net present value.
C) time-weighted return.

Answer 13

A is accurate.

The money-weighted return is the internal rate of return on a portfolio that equates the present value of inflows and outflows over a period of time.

Question 14

A stock that pays no dividend is currently priced at €42.00. One year ago the stock was €44.23. The continuously compounded rate of return is closest to:

A) –5.17%.
B) –5.04%.
C) +5.17%.

Answer 14

A is accurate.

ln(S1/S0 ) = ln(42.00/44.23 ) = ln (0.9496) = − 0.0517 = − 5.17%

Question 15

Over a period of one year, an investor’s portfolio has declined in value from 127,350 to 108,427. What is the continuously compounded rate of return?

A) -14.86%.
B) -13.84%.
C) -16.09%.

Answer 15

C is correct.

The continuously compounded rate of return = ln(S1/ S0) = ln(108,427 / 127,350) = –16.09%.

Question 16

An investor buys a stock on March 24 for $63.25. The stock pays quarterly dividends of $0.54on May 1 and August 1. On September 27, the investor sells the stock for $62.80. The investor’s holding period return is closest to:

A) 2.5%.
B) 1.0%.
C) 2.0%.

Answer 16

B is correct.

[(62.80 + 0.54 + 0.54)/63.25] - 1 = 0.01 = 1%.

Because we are asked for the HPR, the beginning and ending dates are irrelevant. If we had been asked to annualize the return, we would need to know the length of the holding period.

Question 17

For a given stated annual rate of return, compared to the effective rate of return with discrete compounding, the effective rate of return with continuous compounding will be:

A) the same.
B) higher.
C) lower.

Answer 17

B is correct.

A higher frequency of compounding leads to a higher effective rate of return. The effective rate of return with continuous compounding will, therefore, be greater than any effective rate of return with discrete compounding

Question 18

Stock XYZ is purchased on January 2 at a price of $12 per share. The investor receives a quarterly dividend of $0.60 per share on April 1, and the stock closes on June 30 at $13 per share. The holding period return is closest to:

A) 13.33%.
B) 8.33%.
C) 18.33%.

Answer 18

A is accurate.

The holding period return is equal to the change in value from the beginning to the end of the holding period, which will include not only the change in price but also any dividends received over the period. For each share, the price increased by $1, and the dividend received was $0.60. The calculation is equal to:

Pt − P0 + Divt/P0

[(13 - 12 + 0.60)/12] = 13.33%

Ignoring the dividend produces an 8.33% return, and doubling the dividend produces an18.33% return. It is important to note that only one dividend was received in the six-month period, and that was on April 1.

Question 19

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 19

B is correct.

The approximate relationship between nominal rates, real rates and expected inflation rates can be written as:

Nominal risk-free rate = real risk-free rate + expected inflation rate.

Therefore we can rewrite this equation in terms of the real risk-free rate as:

Real risk-free rate = Nominal risk-free rate – expected inflation rate

The exact relation is: (1 + real)(1 + expected inflation) = (1 + nominal)

Question 20

An investor begins with a $100,000 portfolio. At the end of the first period, it generates$5,000 of income, which he does not reinvest. At the end of the second period, he contributes $25,000 to the portfolio. At the end of the third period, the portfolio is valued at$123,000. The portfolio’s money-weighted return per period is closest to:

A) 1.20%.
B) –0.50%.
C) 0.94%.

Answer 20

C is correct.

Using the financial calculator, the initial investment (CF0) is –100,000. The income is +5,000 (CF1), and the contribution is –25,000 (CF2). Finally, the ending value is +123,000 (CF3) available to the investor. Compute IRR = 0.94

Question 21

An investor buys one share of stock for $100. At the end of year one she buys three more shares at $89 per share. At the end of year two she sells all four shares for $98 each. The stock paid a dividend of $1.00 per share at the end of year one and year two. What is the investor’s money-weighted rate of return?

A) 0.06%.
B) 5.29%.
C) 6.35%.

Answer 21

C is correct.

T = 0: Purchase of first share = -$100.00
T = 1: Dividend from first share = +$1.00
Purchase of 3 more shares = -$267.00
T = 2: Dividend from four shares = +4.00
Proceeds from selling shares = +$392.00
The money-weighted return is the rate that solves the equation:
$100.00 = -$266.00 / (1 + r) + 396.00 / (1 + r)
2
.
CFO = -100; CF1 = -266; CF2 = 396; CPT → IRR = 6.35%.

Question 22

A stated interest rate of 9% compounded continuously results in an effective annual rate closest to:

A) 9.42%.
B) 9.20%.
C) 9.67%.

Answer 22

A is correct.

The effective annual rate with continuous compounding = (e^r) – 1 = e^0.09 – 1 = 0.09417, or 9.42%.

Question 23

A security portfolio earns a gross return of 7.0% and a net return of 6.5%. The difference of 0.5% most likely results from:

A) inflation.
B) fees.
C) taxes.

Answer 23

B is correct.

The net return on a portfolio is its gross return minus management and administrative fees. A return adjusted for taxes is called an after-tax return. A return adjusted for inflation is called a real return.

Question 24

A stock increased in value last year. Which will be greater, its continuously compounded or its holding period return?

A) Its continuously compounded return.
B) Its holding period return.
C) Neither, they will be equal

Answer 24

B is correct.

When a stock increases in value, the holding period return is always greater than the continuously compounded return that would be required to generate that holding period return. For example, if a stock increases from $1 to $1.10 in a year, the holding period return is 10%. The continuously compounded rate needed to increase a stock’s value by10% is Ln(1.10) = 9.53%.

Question 25

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 expected inflation 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 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.

Answer 25

A is accurate. The required interest rate on a security is made up of the nominal rate which is in turn made up of the real risk-free rate plus the expected inflation rate. It should also contain a liquidity premium as well as a premium related to the maturity of the security

Question 26

An investor buys one share of stock for $100. At the end of year one she buys three more shares at $89 per share. At the end of year two she sells all four shares for $98 each. The stock paid a dividend of $1.00 per share at the end of year one and year two. What is the investor's time-weighted rate of return? A) 0.06%. B) 11.24%. C) 6.35%.

Answer 26

A is accurate. The holding period return in year one is ($89.00 – $100.00 + $1.00) / $100.00 = -10.00%. The holding period return in year two is ($98.00 – $89.00 + $1.00) / $89 = 11.24%. The time-weighted return is [{1 + (-0.1000)}{1 + 0.1124}] 1/2 – 1 = 0.06%.

Question 27

Based on the advice of his financial advisor regarding dollar cost averaging, a client invests $2,000 each month into a blue-chip stock. The stock price on the date of purchase each month over a four-month stretch was $12, $14, $11, and $9. Using the harmonic mean, the average cost per share of the stock is closest to: A) $11.50. B) $11.75. C) $11.20.

Answer 27

C is correct. The formula to calculate the harmonic mean is equal to: X = 4/(1/12 + 1/14 + 1/11 + 1/9) = 11.2113 Note that the arithmetic mean stock price is $11.50, and because the harmonic mean will always be less than the arithmetic mean for any dataset with unequal values, $11.75 would never be possible.

Question 28

Over the last four years, an investor's portfolio has the following returns: 5.26%, –2.10%,`3.86%, and 8.18%. The arithmetic mean return is closest to: A) 3.73%. B) 3.80%. C) 3.76%.

Answer 28

B is accurate. The arithmetic mean is equal to the average of the four data points, calculated by summing all four returns and dividing by the number of returns: (0.0526 + 0.021 + 0.0386 + 0.0818)/4 = 0.0380 = 3.8%

Question 29

A dataset contains six values, none of which are equal. The arithmetic mean of the data is 13.25, and the geometric mean of the data is 12.75. The harmonic mean will be: A) less than 12.75. B) between 12.75 and 13.25. C) greater than 13.25.

Answer 29

A is accurate. For any dataset where the values are not equal, the harmonic mean will be less than the geometric mean (which, in turn, will be less than the arithmetic mean). Here, the arithmetic mean is 13.25, and the geometric mean is 12.75—so the harmonic mean must be less than 12.75. It is worth noting that all three means are equal if every value in the dataset is the same.

Question 30

The product of the arithmetic mean and the harmonic mean is the: A) square root of the geometric mean. B) square of the geometric mean. C) geometric mean.

Answer 30

B is accurate. The mathematical relationship among arithmetic, geometric, and harmonic means is as follows: arithmetic mean × harmonic mean = (geometric mean)^2.

Question 31

Which of the following return measures is best described as purely representing time preference? A) Real risk-free interest rate. B) Total rate of return. C) Nominal risk-free interest rate

Answer 31

A is accurate. The real risk-free interest rate represents time preference, or the degree to which consumers prefer consumption in the present to an equal amount of consumption in the future. Other measures of return include time preference, but it also reflect other factors, such as risk or expected inflation.

Question 32

bond was purchased exactly one year ago for $910 and was sold today for $1,020. During the year, the bond made two semi-annual coupon payments of $30. What is the holding period return? A) 12.1%. B) 18.7%. C) 6.0%.

Answer 32

B is accurate. HPY = (1,020 + 30 + 30 – 910) / 910 = 0.1868 or 18.7%.

Question 33

An investor sold a 30-year bond at a price of $850 after he purchased it at $800 a year ago. He received $50 of interest at the time of the sale. The annualized holding period return is: A) 12.5%. B) 15.0%. C) 6.25%.

Answer 33

A is accurate. The holding period return (HPR) is calculated as follows: HPR = (Pt – Pt-1+ Dt) / Pt where: Pt = price per share at the end of time period t Dt = cash distributions received during time period t. Here, HPR = (850 – 800 + 50) / 800 = 0.1250, or 12.50%.

Question 34

A stock is currently worth $75. If the stock was purchased one year ago for $60, and the stock paid a $1.50 dividend during the year, what is the holding period return? A) 24.0%. B) 22.0%. C) 27.5%.

Answer 34

C is correct. HPR = [ending value – beginning value] / beginning value = (75 + 1.50 – 60) / 60 = 27.5%.

Question 35

T-bill yields can be thought of as: A) nominal risk-free rates because they contain an inflation premium. B) nominal risk-free rates because they do not contain an inflation premium. C) real risk-free rates because they contain an inflation premium

Answer 35

A is accurate. T-bills are government issued securities and are therefore considered to be default risk free. More precisely, they are nominal risk-free rates rather than real risk-free rates since they contain a premium for expected inflation.

Question 36

If an investor bought a stock for $32 and sold it nine months later for $37.50 after receiving $2 in dividends, what was the holding period return on this investment? A) 23.44%. B)17.19%. C) 32.42%.

Answer 36

A is accurate. HPR = [ending value – beginning value] / beginning value HPR = [(2 + 37.50) – 32] / 32 = 0.2344 = 23.44%

Question 37

Assuming at least some variations in a set of data, the: A) arithmetic mean is greater than geometric mean, which is greater than the harmonic mean. B) geometric mean is greater than the arithmetic mean, which is greater than the harmonic mean. C) harmonic mean is greater than the geometric mean, which is greater than the arithmetic mean.

Answer 37

A is accurate. As long as there is variability in the data, the arithmetic mean is greater than geometric mean, which is greater than the harmonic mean.

Question 38

An investor buys a share of stock for $200.00 at time t = 0. At time t = 1, the investor buys an additional share for $225.00. At time t = 2 the investor sells both shares for $235.00. During both years, the stock paid a per share dividend of $5.00. What are the approximate time-weighted and money-weighted returns respectively? A) 10.8%; 9.4%. B) 7.7%; 7.7%. C) 9.0%; 15.0%.

Answer 38

A is accurate. Time-weighted return = (225 + 5 – 200) / 200 = 15%; (470 + 10 – 450) / 450 = 6.67%; [(1.15)(1.0667)]^1/2 – 1 = 10.8% Money-weighted return: 200 + [225 / (1 + return)] = [5 / (1 + return)] + [480 / (1 + return)^2];money return = approximately 9.4% Note that the easiest way to solve for the money-weighted return is to set up the equation and plug in the answer choices to find the discount rate that makes outflows equal to inflows.

Question 39

An analyst evaluates a dataset with eight values. From the dataset, she calculates the geometric mean to be 8.50. If the arithmetic mean is equal to 8.90, the harmonic mean is closest to: A) 8.63. B) 8.12. C) 9.30.

Answer 39

B is accurate. The relationship between the arithmetic, harmonic, and geometric mean is equal to: arithmetic mean × harmonic mean = (geometric mean)^2 8.90 × harmonic mean = (geometric mean)^2 = (8.50)^2 harmonic mean = 8.12 Note: This could also be answered without performing calculations, knowing that harmonic < geometric < arithmetic, where values are not equal.

Question 40

On January 1, Jonathan Wood invests $50,000. At the end of March, his investment is worth $51,000. On April 1, Wood deposits $10,000 into his account, and by the end of June, his account is worth $60,000. Wood withdraws $30,000 on July 1 and makes no additional deposits or withdrawals the rest of the year. By the end of the year, his account is worth$33,000. The time-weighted return for the year is closest to: A) 7.0%. B) 5.5%. C) 10.4%.

Answer 40

C is accurate. January – March return = 51,000 / 50,000 − 1 = 2.00% April – June return = 60,000 / (51,000 + 10,000) − 1 = –1.64% July – December return = 33,000 / (60,000 − 30,000) − 1 = 10.00% Time-weighted return = [(1 + 0.02)(1 − 0.0164)(1 + 0.10)] − 1 = 0.1036 or 10.36%

Question 41

An investor with a buy-and-hold strategy who makes quarterly deposits into an account should most appropriately evaluate portfolio performance using the portfolio's: A) arithmetic mean return. B) geometric mean return. C) money-weighted return.

Answer 41

B is correct. Geometric mean return (time-weighted return) is the most appropriate method for performance measurement as it does not consider additions to or withdrawals from the account.

Question 42

The most appropriate measure of the increase in the purchasing power of a portfolio's value over a given span of time is a(n): A) after-tax return. B) real return. C) holding period return

Answer 42

B is accurate. A real return is adjusted for the effects of inflation and is used to measure the increase in purchasing power over time.

Question 43

An annuity will pay eight annual payments of $100, with the first payment to be received oneyear from now. If the interest rate is 12% per year, what is the present value of this annuity? A) $496.76. B) $1,229.97. C) $556.38.

Answer 43

A is accurate. N = 8; I/Y = 12%; PMT = -$100; FV = 0; CPT → PV = $496.76.

Question 44

A 15-year zero-coupon German government bond has an annualized yield of –1.5%.Assuming annual compounding, the price of the bond per €100 of principal is closest to: A) €125. B) €115. C) €105.

Answer 44

A is accurate. N = 15; I/Y = –1.5; FV = 100; PMT = 0; CPT PV = –125.45.

Question 45

An investor purchases a stock on January 1. The annual dividend payments for a stock investment for the next four years, beginning on December 31, are $50, $75, $100, and$125. Based on the cash flow additivity principle, the present value of this series of cashflows will be equivalent to the present value of a $50 annuity and the present value of what series of cash flows? A) $0, $0, $125, and $125. B) $75, $50, $25, and $0. C) $0, $25, $50, and $75.

Answer 45

C is correct. The cash flow additivity principle states that the PV of any stream of cash flows is equal to the sum of the PVs of all of the cash flows. The cash flows are $50, $75, $100, and $125.So, if one stream of cash flows is equal to $50 each year, subtract $50 from each original cash flow to get the second stream of cash flows. The PV of 50, 75, 100, and 125 = PV of 50, 50, 50, and 50 + PV of 0, 25, 50, and 75. The order matters, as the PV will be different (and higher) if the higher cash flows comes before the lower ones.

Question 46

Wortel Industries has preferred stock outstanding that paying an annual dividend of $3.75 per share. If an investor wants to earn a rate of return of 8.5%, how much should he be willing to pay for a share of Wortel preferred stock? A) $31.88. B) $44.12. C) $42.10.

Answer 46

B is accurate. To calculate the price, we need to discount the future dividend stream at the investor's required return. The stream of dividends is a perpetuity (a fixed dividend each year forever). Given the PV of a perpetuity = cash flow / discount rate Then price = $3.75 / 0.085 = $44.12

Question 47

A bond pays annual coupon interest of £40 and returns its face value of £1,000 in five years. The bond's yield to maturity is 4.5%. Its price today is closest to: A) £946. B) £978. C) £957.

Answer 47

B is correct. N = 5; I/Y = 4.5; PMT = 40; FV = 1,000; CPT PV = –978.05.

Question 48

An investor purchases a 10-year, $1,000 par value bond that pays annual coupons of $100. If the market rate of interest is 12%, what is the current market value of the bond? A) $950. B) $887. C) $1,124.

Answer 48

B is correct. Note that bond problems are just mixed annuity problems. You can solve bond problems directly with your financial calculator using all five of the main TVM keys at once. For bond-types of problems the bond's price (PV) will be negative, while the coupon payment (PMT)and par value (FV) will be positive. N = 10; I/Y = 12; FV = 1,000; PMT = 100; CPT → PV = –886.99.

Question 49

Assuming a constant rate of growth in dividends, we can estimate an equity share's: A) dividend yield as the sum of its required rate of return and its growth rate. B) growth rate as the sum of its dividend yield and its required rate of return. C) required rate of return as the sum of its dividend yield and growth rate.

Answer 49

C is correct. Starting with the Gordon growth model, we can solve for the estimated required rate of return, constant growth rate, or dividend yield as follows: ke = + gc gc = ke − = ke − gc D1 V0 D1 V0 D1 V0 see Module 2.2, LOS 2.b.

Question 50

A financial advisor recommends to her client that he buy a 6-year, $1,000 face value bond that pays annual interest of 5%. The yield to maturity is 4.5%, and the client intends to hold the bond as an investment until it matures. The value of the bond today is closest to: A) $1,000. B) $975. C) $1,025.

Answer 50

C is correct. With a fixed-coupon, annual-pay bond, the annual interest payment and the principal payment are discounted at the yield to maturity. The calculator solution is to solve for present value while setting the number of periods (N) to 6, the annual payment (PMT) to50 (which is 1,000 × 5%), the future value (FV) to 1,000, and the yield (I/Y) to 4.5%: 50/1.045 + 50/1.045^2+...+ 1,050/1.045^6 = 1, 025.79 This can also be answered using the calculator: N = 6; I/Y = 4.5; PMT = 50; FV = 1,000. CPTPV = –1,025.79. It is also worth noting that because the yield to maturity (4.5%) is below the coupon rate(5%), the bond's current price must be above the par value of $1,000. $975 would only be possible if the yield was above the coupon rate.

Question 51

An investor makes 48 monthly payments of $500 each beginning today into an account that will have a value of $29,000 at the end of four years. The stated annual interest rate is closest to: A) 10.00%. B) 9.00%. C) 9.50%.

Answer 51

B is accurate. Because this is an annuity due (payments at the start of each period) the calculator must first be set to BGN mode. N = 48; PMT = 500; FV = –29,000; PV = 0; CPT I/Y = 0.7532 This percentage is a monthly rate because the time periods were entered as 48 months. It must be converted to a stated annual percentage rate (APR) by multiplying by the number of compounding periods per year: 0.7532 × 12 = 9.04%.

Question 52

An investor pays $726.27 for a zero-coupon bond with a face value of $1,000 and maturing in 10 years. Bonds with similar risk profiles and with similar terms yield 3.00%. The yield to maturity for this bond is closest to: A) 3.25%. B) 2.75%. C) 3.00%.

Answer 52

A is accurate. A zero-coupon bond pays no interest, but it is most often purchased at a price heavily discounted from par value. The equation that shows the relationship between the present value (the purchase price), the future value, time, and yield to maturity is shown as follows: $726.27 = $1,000/(1+r)^10 1 + r = √1,000/726.27 r = 1.0325 − 1 = 0.0325, or 3.25 This can also be answered using the calculator: N = 10; PV = –726.27; PMT = 0; FV = 1,000.CPT I/Y = 3.25. 2.75% is the yield to maturity if the present value is incorrectly input as $762.27 instead of$726.27. The yield on similar bonds does not reflect the yield on a specific bond.

Question 53

A perpetual bond with a face value of $100,000 pays annual interest of 5%. The bond is quoted at a yield of 7%. The bond's price is closest to: A) $140,000. B) $71,500. C) $98,100.

Answer 53

B is accurate. $100,000(0.05)/0.07 = $71, 428.57.

Question 54

An analyst is using the constant growth dividend discount model (DDM) to evaluate XYZ stock. The stock is currently trading at $20 per share and recently paid an annual dividend of$1.50. Assuming a constant growth rate of 4.5%, the implied required rate of return on the stock is closest to: A) 12.00%. B) 12.34%. C) 11.68%.

Answer 54

B is accurate. The Gordon growth model, also known as the DDM, takes the next period's dividend and divides it by the difference between the required return and the growth rate. The formula can be algebraically manipulated to isolate the required rate of return. The calculation to determine the required rate of return is shown: ke = D1/V0 + gc = 1.50(1.045)/20 + 0.045 = 0.1234, or 12.34 The 11.68% answer option is the output if the current dividend is discounted by the growth rate rather than increased by the growth rate to get to the next period's dividend. The 12.00% answer option is the output if the current dividend is used in the calculation without adjusting for the growth rate.

Question 55

An investor spends $365,000 purchasing zero-coupon bonds with a total face value of $500,000 and maturing in 10 years. For the annualized rate of return to be above 3.20%, the bond's price will have to be: A) equivalent to $365,000. B) lower than $365,000. C) higher than $365,000.

Answer 55

B is accurate. With a future value of $500,000, a present value of $365,000, and a maturity of 10 years,the annualized rate of return is calculated as shown: = $365, 000 (1 + r)10 = = 1.36986 r = 1.369861/10 − 1 = 0.0320 $500,000 (1+r)10 $500,000 $365,000 On the calculator, N = 10; PV = –365,000; PMT = 0; FV = 500,000; CPT I/Y = 3.2. Because the annualized return is 3.20% and the question asks about what the bond's pricemust be to be above 3.20%, the price of the bond must be below the purchase price of$365,000. The relationship between the price and rate of return is inverse; for the rate ofreturn to be above 3.20%, the price must fall.

Question 56

A bond with a 10-year maturity has a face value of $10,000 and pays annual interest of $600.The bond is issued at a price of $9,500. The bond's yield to maturity will be: A) greater than 6%. B) equal to 6%. C) less than 6%.

Answer 56

A is accurate. No calculations are needed to answer this question. This bond was issued at a price of $9,500, which is below face value of $10,000. The bond is considered a discount bond, and this results from a situation where the bond's coupon rate is below the yield to maturity. With annual interest of $600 on a face value of $10,000, the coupon rate is equal to 6% (600 / 10,000). The yield to maturity must be greater than 6% for the bond to be issued at a discounted price.

Question 57

A pure discount instrument with a face value of ¥500 million matures nine years from today and has a current price of ¥350 million. The instrument's annualized yield is closest to: A) 3.3%. B) 4.7%. C) 4.0%.

Answer 57

C is correct. (1 + r)^9 = 500/350 r = 500/350^(1/9) − 1 = 4.04%.

Question 58

An investor looks at her monthly brokerage statement and notices that the yield to maturity on her 5-year corporate bond with a 4% annual coupon rate has gone from 4.2% last month to 3.8% this month. The statement will reflect a bond price that, over the last month, has: A) decreased. B) remained flat. C) increased.

Answer 58

C is accuarte. Bond prices and yields move in opposite directions, such that if the yield has dropped from 4.2% to 3.8%, it must be a case that the price of the bond has increased. A decrease in price would align with an increase in yield to maturity. If the price had remained flat, the yield would be unchanged.

Question 59

Given a 5% discount rate, the present value of $500 to be received three years from today is: A) $400. B) $432. C) $578.

Answer 59

B is accurate. N = 3; I/Y = 5; FV = 500; PMT = 0; CPT → PV = 431.92. or: 500/1.05^3 = 431.92.

Question 60

An investor is deciding whether to buy a 1-year bond two years in a row or lock in the rate on a 2-year bond today. The 1-year spot interest rate is 5.25%, and the 2-year spot interest rate is 6.50%. Which of the following statements is most accurate regarding implied forward rates and the investor's options? A) The expected rate on a 1-year bond one year from today is equal to 7.76%. B) The forward rate will be between 5.25% and 6.50%. C) The investor is better off locking in the 2-year rate at 6.50%.

Answer 60

A is accurate. Implied forward rates can be derived based on observable spot rates in the fixed income market. The result is that the implied 1-year forward rate one year in the future can be derived based on this formula: The forward rate (1y1y) is equal to 7.76%. The forward rate will be higher than both spot rates, which means it cannot be between5.25% and 6.50%. The investor should be indifferent between the 2-year bond paying6.50% and 1-year bonds at 5.25% and 7.76%.

Question 61

Assume that one- and two-year risk-free rates are 1.80% and 2.50%, respectively. Using the cash flow additivity principle, the one-year reinvestment rate, one year from now is closest to:' A) 2.8%. B) 3.2%. C) 3.5%.

Answer 61

B is accurate. F 1,1 = (1+r2)^2/(1+r1)^1 - 1 = 1.025^2/ 1.018^1 − 1 = 3.2%

Question 62

An equity investor has a required return of 7% and purchases preferred stock with a $50 per share par value and an annual dividend of $3.20. The value of the preferred stock is closest to: A) $46. B) $50. C) $43.

Answer 62

A is accurate. The value of preferred stock, based on the assumption that the annual dividend will be paid in perpetuity, is equal to: Dp/kP = 3.20/0.07 = 45.71 The correct answer is 45.71, which is closest to $46 per share.

Question 63

A stock is expected to pay a dividend next year of $2.40. An analyst expects the dividend to grow at a constant annual rate of 4% and believes investors' required rate of return on the stock is 7%. The analyst will estimate a value for this stock that is closest to: A) $85.60. B) $80.00. C) $83.20.

Answer 63

B is accurate. Applying the Gordon growth model, $2.40/(0.07-0.04) = $80

Question 64

Given investors require an annual return of 12.5%, a perpetual bond (i.e., a bond with no maturity/due date) that pays $87.50 a year in interest should be valued at: A) $70. B) $700. C) $1,093.

Answer 64

B is correct. 87.50 ÷ 0.125 = $700.

Question 65

To determine whether the current price of a common stock is aligned with its intrinsic value, an analyst wants to use the Gordon growth model. To appropriately apply the model, the analyst will need to estimate: A) the dividend to be received next year. B) a fluctuating growth rate assigned to dividends. C) a growth rate that is above the required return.

Answer 65

A is accurate. The Gordon growth model, also known as the constant growth dividend discount model(DDM), takes the next period's dividend and divides it by the difference between the required return and the growth rate. The growth rate is assumed to be constant, and it must be below the required return—or else the denominator of the calculation will be negative, making it invalid.

Question 66

Bill Jones is creating a charitable trust to provide six annual payments of $20,000 each, beginning next year. How much must Jones set aside now at 10% interest compounded annually to meet the required disbursements? A) $154,312.20. B) $87,105.21. C) $95,815.74.

Answer 66

B is correct. N = 6, PMT = -$20,000, I/Y = 10%, FV = 0, Compute PV → $87,105.21.

Question 67

Given the following cash flow stream: Annual Cash Flow at the End of Year 1 = $4,000 Annual Cash Flow at the End of Year 2 = $2,000 Annual Cash Flow at the End of Year 3 = 0 Annual Cash Flow at the End of Year 4 = -$1,000 Using a 10% discount rate, the present value of this cash flow stream is: A) $3,636.00. B) $4,606.00. C) $3,415.00.

Answer 67

B is correct. PV(1): N = 1; I/Y = 10; FV = -4,000; PMT = 0; CPT → PV = 3,636 PV(2): N = 2; I/Y = 10; FV = -2,000; PMT = 0; CPT → PV = 1,653 PV(3): 0 PV(4): N = 4; I/Y = 10; FV = 1,000; PMT = 0; CPT → PV = -683 Total PV = 3,636 + 1,653 + 0 – 683 = 4,606

Question 68

Assuming the 1-year riskless interest rates on the U.S. dollar and British pound are 3.5% and4.0% respectively, the forward exchange rate between the two currencies will be different than the spot rate by approximately: A) 0.50%. B) 3.75%. C) 7.50%.

Answer 68

A is accurate. The percentage difference between forward and spot exchange rates is approximately equal to the difference between the interest rates in the two countries. Although there is amore refined calculation, the difference between the forward and spot rates will be approximately equal to 4.0% – 3.5% = 0.50%. 3.75% is just the average of the two rates, and 7.50% adds them together instead of taking the difference.

Question 69

Compute the present value of a perpetuity with $100 payments beginning four years fromnow. Assume the appropriate annual interest rate is 10%. A) $683. B) $751. C) $1,000.

Answer 69

B is accurate. Compute the present value of the perpetuity at (t = 3). Recall, the present value of a perpetuity or annuity is valued one period before the first payment. So, the present value at t = 3 is 100 / 0.10 = 1,000. Now it is necessary to discount this lump sum to t = 0.Therefore, present value at t = 0 is 1,000 / (1.10)^3 = 751.

Question 70

A 5-year, 8% coupon bond with a par value of $1,000 pays interest annually. The price is$942.50, and the yield to maturity is 9.50%. If the price of the bond moves to $963.75, the yield to maturity will be closest to: A) 10.07%. B) 8.55%. C) 8.93%.

Answer 70

C is accurate. Because the price of the bond increases, the yield to maturity will fall from its current level. The current level is 9.50%, which means the yield cannot be 10.07%. The calculation for the yield can be derived using a financial calculator: PV = -963.75, FV = 1000, N = 5 years, PMT = 80 (8% of par), Solve for I/Y = 8.93

Question 71

A share of George Co. preferred stock is selling for $65. It pays a dividend of $4.50 per year and has a perpetual life. The rate of return it is offering its investors is closest to: A) 6.9%. B) 4.5%. C) 14.4%.

Answer 71

A is correct. 4.5 / 65 = 0.0692, or 6.92%.

Question 72

An investor with USD1,000,000 is undecided between two mutually exclusive opportunities with the following cash flows: Opportunity 1 Time 0 = –1,000,000 Time 1 = 500,000 Time 2 = 500,000 Time 3 = 500,000 Opportunity 2 Time 0 = –1,000,000 Time 1 = 400,000 Time 2 = 500,000 Time 3 = 600,000 The investor's required return is 11% per year. Which opportunity should the investor choose? A) The investor should be indifferent between the two opportunities. B) The investor should choose Opportunity 1. C) The investor should choose Opportunity 2.

Answer 72

B is accurate. Although this problem may be solved by calculating the individual NPVs of each opportunity (Opportunity 1: 221.86 and Opportunity 2: 204.89), another approach would be to use the cash flow additivity principle as follows: Opportunity 1 Time 1 = 500,000 Time 2 = 500,000 Time 3 = 500,000 Opportunity 2 Time 1 = 400,000 Time 2 = 500,000 Time 3 = 600,000 Cash flow difference Time 1 = +100,000 Time 2 = 0 Time 3 = –100,000 Because the present value of the cash flow difference arising at Time 1 (in favor of Opportunity 1) must exceed the present value of the negative cash flow difference arising at Time 3 (in favor of Opportunity 2) at any positive discount rate, Opportunity 1 is preferred.

Question 73

A pure discount instrument with a face value of ¥100 million matures 12 years from today. If its yield to maturity is 3%, its price today is closest to: A) ¥71 million. B) ¥70 million. C) ¥72 million.

Answer 73

B is accurate. ¥100,000,000(1.03)^–12 = ¥70,137,988.

Question 74

Using a constant growth dividend discount model (DDM), an analyst assumes a required return on equity of 9.75%. The current stock price is $30 per share, and the next period's dividend is $2.40 per share. The constant growth rate implied in the model is closest to: A) 1.75%. B) 1.89%. C) 1.83%.

Answer 74

A is accurate. The Gordon growth model, also known as the DDM, takes the next period's dividend and divides it by the difference between the required return and the growth rate. The formula can be algebraically manipulated to isolate the implied growth rate. The calculation to determine the growth rate is shown: gc = ke − D1/V0= 0.0975 − 2.40/30= 0.0975 − 0.08 = 0.0175, or 1.75 The 1.83% answer option takes the correct answer of 1.75% and adds the dividend yield of8%: 1.75% + 0.08% = 1.83%. The 1.89% answer option takes the correct answer of 1.75%and grows it by multiplying it by the dividend yield of 8%: 1.75% × (1.08) = 1.89%.

Question 75

An investment product promises to pay a lump sum of $25,458 at the end of 9 years. If an investor feels this investment should produce a rate of return of 14%, compounded annually, the present value is closest to: A) $9,426.00. B) $7,618.00. C) $7,829.00.

Answer 75

C is accurate. 25,458 / 1.14 9 = 7,828.54 Alternatively, N = 9; I/Y = 14; FV = -25,458; PMT = 0; CPT → PV = $7,828.54.

Question 76

An investor is choosing between two possible investments. Both have identical future cashflows in all situations, but the investor notices a slight discrepancy in price between the two. What action will this investor take based on the no-arbitrage principle? A) Wait for the prices to further diverge, then sell the higher-priced investment. B) Do nothing, as there cannot be a price divergence based on the rule. C) Act quickly by buying the lower-priced investment, as the prices will quickly converge.

Answer 76

C is accurate. The no-arbitrage principle (law of one price) states that the price for an investment will be the same if two sets of future cash flows are identical under all conditions. Although there should not be a discrepancy in theory, there may be one for a short time period. If there isa slight price discrepancy between these investments, it will not last long, so the investor should act quickly and buy the lower-priced investment. The prices would not further diverge.

Question 77

Abeta's stock is trading at $47. Abeta just paid a dividend of $1.50, and markets assume a constant growth rate in dividends of 4%. Abeta's required return on equity is closest to: A) 8.1%. B) 6.5%. C) 7.3%.

Answer 77

C is correct. To calculate the implied cost of equity, we rearrange the constant growth formula as follows: r = D0×(1+g)/P0 + g = 1.50×1.04/47.00 + 0.04 = 7.32%

Question 78

A bond pays annual coupon interest of £60 and returns its face value of £1,000 in seven years. The bond's price today is £1,045. Its yield to maturity is closest to: A) 5.2%. B) 6.8%. C) 6.0%.

Answer 78

A is accurate. N = 7; PMT = 60; FV = 1,000; PV = –1,045; CPT I/Y = 5.2162.

Question 79

A loan of $15,000 is to be paid off in monthly payments over 5 years at 12% annual interest. What is the amount of each payment? A) $334. B) $1,802. C) $4,161.

Answer 79

A is accurate. I = 12 / 12 = 1; N = 5 × 12 = 60; PV = 15,000; CPT → PMT = 333.67.

Question 80

A pure discount instrument with a face value of €1 million matures eight years from today. If its yield to maturity is –1.5%, its price today is closest to: A) €1.13 million. B) €0.98 million. C) €0.89 million.

Answer 80

A is accurate. Given these three answer choices, you can choose the correct answer without performing the calculation. With a negative yield, the price of a single future cash flow must be greater than the amount of the cash flow. In this case, €1,000,000(1 0.015)^–8 = €1,128,522.