Investments - Math Practice

Question 1

Calculate the GMR of the following returns:
Year 1, 45% ; Year 2, -10% ; Year 3, -25% ; Year 4 30%

Answer 1

Answer: 6.21%

Question 2

Calculate the IRR of the following problem:

A company has a required rate of return of 5%, should they make a $10 million investment that is expected to bring in $1 million of profit each year for 15 years?

Answer 2

Answer: 5.56%

Question 3

Calculate Dollar-Weighted and Time-Weighted Returns for the following:

At the beginning of Year 1, an investor buys one share of Good Chips stock for $12. At the end of Year 1, the investor receives a dividend of $1 and buys another share of Good Chip stock for $16. At the end of Year 2, the investor sells both shares for $20 each.

Answer 3

Dollar-Weighted: 30.5%
CF0: –$12
CF1: $1 – $16 = –$15
CF2: $20 + $20 = $40

Time-Weighted: 33.3%
CF0: -$12
CF1: $1
CF2: $20

Question 4

What is the dollar-weighted return of an investment portfolio with the following cash flows over the past 3 years?

Beginning of Year 1 – $50,000 deposit
End of Year 1 – $5,000 deposit
End of Year 2 – $25,000 withdrawal
End of Year 3 – $50,000 withdrawal

Answer 4

Answer: 12.84%

P/YR = 1
50,000
[+/-]
[CFj]
5,000
[+/-]
[CFj]
25,000
[CFj]
50,000
[CFj]
[SHIFT]
[IRR/YR]

Question 5

Seth bought ABC stock for $21 per share 6 months ago. Seth sold ABC today for $29.40 per share. What is his annualized return?

Answer 5

Annualized return = ($29.4/$21)2 − 1 = (1.4)2 − 1 = 1.96 −1 = 0.96, or 96%

Question 6

A CD offers 3% per quarter. What is its annualized return?

EXACT Answer - DO THE MATH ON YOUR CALCULATOR

Answer 6

Annualized return = (1.03)4 − 1 = 1.1255 −1 = 0.1255, or 12.55%

Question 7

Calculate the standard deviation of the following returns by hand:
9%, -3%, 4%, 12%, 3%

Answer 7

Answer: 5.8%

1. Find mean of returns
2. Subtract each ret by mean
3. Square each “return-mean”
4. Add up Squared numbers
5. Divide by n-1

Question 8

Calculate the standard deviation of the following returns on your calculator:

5%, 9%, 1%, -2%, 7%

Answer 8

5, Σ+
9, Σ+
1,Σ+
2, +/-, Σ+
7, Σ+
SHIFT, 8 (Sx, Sy)

Question 9

Calculate the expected return of the following portfolio:

Answer 9

Answer: 33.5%

Question 10

According to the Capital Asset Pricing Model (CAPM), if the risk-free rate is 3% and the market portfolios expected return is 10%, what is the expected return of a stock with a beta of 0.7?

Answer 10

Answer: 7.9%

Expected return = rf + β (rm – rf)
Expected return = 3% + 0.7 (10% − 3%) = 3% + 0.7 (7%) = 3% + 4.9% = 7.9%

Question 11

Glen is considering two portfolios: 1) Portfolio A with a return of 14 percent and a standard deviation of 14 percent and 2) Portfolio B with a return of 4 percent and a standard deviation of 7 percent. Assuming the correlation between A and B is 0.0 and Glen invests 70 percent in A and 30 percent in B, what range of returns should this portfolio produce 95 percent of the time?

A) Between 1% and 21%
B) Between -9% and 31%
C) Between -1% and 21%
D) Between -19% and 41%

Answer 11

The correct answer is (B).
Standard deviation:

=SQRT[(0.7² x 14² ) + (0.3² x 7² ) + 0]
=SQRT[(0.49 x 196) + (.09 x 49)]
=SQRT[96.04 + 4.41]
=SQRT[100.45]
=10.02% or about 10%

Expected return = (0.7 × 14%) + (0.3 × 4%) = 11%
As noted in Chapter 2, the 95 percent confidence interval equals two standard deviations from the mean. Therefore the range -9 percent to 31 percent is the correct answer.

Question 12

Discount, Inc., expects to generate $100 million in operating cash flows during the next year. It estimates its long-term dividend growth rate to be 2%, and it has 50 million shares outstanding. What is the intrinsic value of Discount, Inc., if your required rate of return is 10%?

Answer 12

Answer: $25

Intrinsic value = Total market value / Outstanding Shares.

Total market value = Operating Cash Flow / (Required Rate of Return – Dividend Growth Rate)

Total market value = $100,000,000 / (0.10 − 0.02) = $100,000,000 / 0.08 = $1,250,000,000

Intrinsic value = $1,250,000,000 / 50,000,000 = $25

Question 13

The Nacho Equity Fund has a beta of 1.44 and a standard deviation of 20.8%. It has returned 12.9% during the past year when the return on one-year Treasury bills has been 3.2%. The Sharpe Ratio of the Nacho Equity Fund is closest to:

Answer 13

Answer: 0.466

Sharpe = (Portfolio Return – Risk-Free Rate) / (Standard Deviation)
Sharpe = (12.9 – 3.2) / 20.8 = 0.466

Question 14

The Southern Fund generated a return of 7% over the last year. The relevant market portfolio had a return of 10% and the risk-free return was 3%. Southern Fund’s beta is 0.6. The Jensen’s alpha for the fund is closest to:

Answer 14

Answer: -0.2

α = rp – [rf + βp × (rm – rf)]
α = 7 – [3 + 0.6 × (10 – 3)] = − 0.2

Question 15

Last year, the risk-free rate was 2% and a tech sector index returned –10%. What is Jensen’s alpha for a tech mutual fund with a beta of 1.2 that returned –11% over the same period?

Answer 15

Answer: 1.4

α = rp – [rf + βp × (rm – rf)]
α = -11 – [2 + 1.2 × (-10 – 2)]
α = -11 - [2 + (1.2 x –12)]
α = -11 - [2 - 14.4] = -11 + 12.4 = 1.4

Question 16

What is the Treynor ratio of a fund with a beta of 1.3 and a standard deviation of 15% if the fund returned -8% this year when the risk-free rate of return was 2%?

Answer 16

Answer: -7.69

Treynor ratio = (rp – rf) / β
Treynor ratio = (-8 – 2) / 1.3 = -7.7

Question 17

James buys a 6% coupon bond with a $1,000 par value on September 30th. The bond pays semiannual interest on Jan 1st and July 1st each year. What is the amount of accrued interest that James must pay to the seller of the bond?

Answer 17

Answer = $15

Semiannual coupon PMT x (Days since last coupon payment / Days in coupon period)
= ($1,000 x 6% / 2) x (90 / 180)
= $30 x 1 / 2
= $15

Question 18

All of the following regarding Treasury Inflation-Protected Securities (TIPS) are true EXCEPT:

A. TIPS are issued directly from the U.S. Treasury.
B. The interest rate for TIPS changes based on inflation.
C. TIPS can be purchased on a competitive or noncompetitive basis.
D. With deflation, the interest payments will decrease for TIPS.

Answer 18

B

TIPS are issued by the U.S. Treasury on both a competitive and noncompetitive basis. The interest rate does not change. Instead, the principal amount changes based on inflation or deflation.

Question 19

The Echo bond is a 6% coupon bond with semiannual coupon payments. It matures in 15 years with a $1,000 par value. If the YTM for this bond is 4%, what is the value of the bond?

Answer 19

Answer: $1,223.96

P/YR - 2
FV - 1000
N - 15x2 = 30
I - 4%
PMT - 30
PV = 1223.96

Question 20

Arthur is considering purchasing a 12-year bond that is selling for $1,300. What is the current yield for this bond if it has an 8% coupon, paid semiannually.

What is the current yield?

Answer 20

Answer: 6.15%

Current yield = ($1000*.08)/$1,300 = 6.15%
1000 = par value
.08 = interest rate
1300 = PV

Question 21

Carla is considering purchasing a 35-year bond that is selling for $500 and has a $1,000 par value. What is the YTM for this bond if it has a 2% coupon, paid semiannually?

Answer 21

Answer: 5.06%

P/YR = 2
PV = -500
N = 70 (35X2)
PMT = 10 (2% OF 1000 / 2)
FV = 1000
I = 5.06%

Question 22

A company with 500 million shares outstanding paid a dividend of $2 this year. If the company’s net earnings were $4 billion, what was its dividend payout ratio?

Answer 22

Answer: 25%

The payout ratio = dividend per share / EPS
The payout ratio = $2/($4,000,000,000/500,000,000) = 0.25 = 25%

Question 23

Monica purchases one share of stock on margin for $104. The initial margin is 50%. If the maintenance margin equals 35%, then Monica will receive a margin call if the stock falls below what number?.

Answer 23

Answer = $80

Question 24

Andre purchased shares of Latte Co. for $48 by using a margin account with a 50% initial margin rate and a 30% maintenance margin rate. At what price will Andre receive a margin call?

Answer 24

Answer = 34.29

The formula to determine the price at which a margin call will occur is as follows:
Debt / (1 − Maintenance Margin Rate). $24 / (1 − .30) = $34.29

Question 25

Coyote, Inc., has net earnings of $3 billion this year. It has 500 million shares of common stock outstanding, and it paid $0.25 per share per quarter this year as a dividend. Which of the following is correct?

A) The retention ratio equals 8.33%.
B) The payout ratio equals 8.33%.
C) The payout ratio equals 16.67%.
D) The retention ratio equals 16.67%

Answer 25

The correct answer is (C).
The dividend per share equals $1.00. The Earnings Per Share (EPS) equals $6.00, which is found by dividing net earnings by outstanding shares. The payout ratio = dividend per share / EPS.

Question 26

Darrel buys Hollandaise, Inc., stock for $112 using a margin account with a 50 percent initial margin and a 35 percent maintenance margin. Assuming the price of the stock drops to $56, how much would Darrel need to pay to restore the equity in his account to the maintenance margin?

Answer 26

The correct answer is (C).
Required equity: $56 × 35% = $19.60​
Current equity: $56 stock price − $56 loan = $0​
Margin call: $19.60 - $0 = $19.60

Question 27

A convertible bond has a par value of $1,000, but its current market price is $750. The current price of the issuing company’s stock is $17 and the conversion ratio is 30 shares. The bond’s conversion premium is closest to:

Answer 27

Answer: $240

The bond is convertible into 30 shares, which is worth $510 at $17 per share. The bond is trading at $750. Thus, the conversion premium equals the difference or $240. $750 – (30 × $17 = $510) = $240.

Question 28

A real estate investor bought a property for $150,000 and rented it out over 3 years before selling it for $160,000. His income and expenses over this period are shown in the table below. Based on this information, what was his dollar-weighted return?

Year Income Expenses
1 $7,000 -$10,000
2 $9,000 -$10,000
3 $10,000 -$1,000

Answer 28

First, calculate each time period’s total cash inflows and outflows:

Period Cash Flow
0 -150,000
1 -3,000
2 -1,000
3 169,000

Then, to solve this problem using an HP10bII+, use the following keystrokes:

150,000
[+/-]
[CFj]
3,000
[+/-]
[CFj]
1,000
[+/-]
[CFj]
169,000
[CFj]
[SHIFT]
[IRR/YR]
The correct answer is 3.1809, or about 3.18%.

Question 29

Cashflow, Inc. expects to generate $200 million in operating cash flows during the next year. It estimates its long-term dividend growth rate to be 3 percent, and it has 100 million shares outstanding. What is the intrinsic value of Cashflow, Inc., if your required rate of return is 9 percent?

Answer 29

Intrinsic value = Total market value ÷ Outstanding Shares
Total market value = Operating Cash Flow ÷ (Required Rate of Return – Dividend Growth Rate)

Total market value = $200,000,000 ÷ (0.09 − 0.03) = $200,000,000 ÷ 0.06 = $3,333,333,333
Intrinsic value = $3,333,333,333 ÷ 100,000,000 = $33.33

Question 30

The North Equity Fund has a beta of 1.67 and a standard deviation of 22.6 percent. It has returned 13.8 percent during the past year when the return on one-year treasury bills has been 3.6 percent. The Sharpe Ratio of the North Equity Fund is closest to

Answer 30

Sharpe Ratio = (Portfolio Return – Risk-Free Rate) ÷ (Standard Deviation)
Sharpe Ratio = (13.8 − 3.6) ÷ 22.6 = 0.45

Question 31

Identify the financial asset with the highest Treynor ratio if the risk-free rate of interest is 3%.

Answer 31

Fund 2

Question 32

Cody purchased 400 shares of NAY stock six years ago when it was trading at $65 per share. Unfortunately, NAY has been steadily declining. Cody sold his shares this year for $18 per share. This year Cody also sold 800 shares of a mutual fund that he purchased 6 months ago. His average cost per share was $15 and he sold the shares for $32. Assuming Cody had no other capital transactions this year, what is Cody’s tax consequence from these transactions?

A) $5,200 long-term loss this year
B) $18,800 long-term loss, $13,600 short-term gain taxed at ordinary income rates this year
C) $3,000 ordinary loss this year with a $15,800 ordinary loss carried forward up to 5 years
D) $3,000 ordinary loss this year with a $2,200 loss carried forward indefinitely

Answer 32

The correct answer is (D).
The NAY stock sale resulted in a long-term loss of $18,800 ($47 loss per share × 400 shares). The mutual fund sale resulted in a short-term gain of $13,600 ($17 gain × 800 shares). The entire $13,600 gain is offset by the loss, leaving a net long-term loss of $5,200. Cody can take $3,000 of the loss against ordinary income this year, and the remaining loss is carried forward indefinitely (until death).

Question 33

$100 of stock is purchased with a 50% initial margin and a 30% maintenance margin. At what price would a margin call occur?

Answer 33

The formula used to determine the price below when a margin call will occur is: debt / (1 − MMR). For example, if a 50% initial margin rate and a 30% maintenance margin rate apply, a stock that was purchased when the price was $100 would receive a margin call if the price falls below $50 / (1 − 0.30) = $71.42. The price drop is $28.58, or 1 − debt / (1 − MMR).= 1 − (0.50 / 0.70) = 28.58%.

Question 34

Byte, Inc., had net earnings of $2.5 billion last year. It has 500 million shares of common stock outstanding, which stayed the same since last year. It paid 50 cents per share per quarter this year as a dividend payment, which is the same as last year. This year’s earnings are 20% higher than last year. Which of the following is correct for this year?

A) The retention ratio equals 33.33%.
B) The retention ratio equals 50.00%.
C) The payout ratio equals 33.33%.
D) The payout ratio equals 40.00%.

Answer 34

The correct answer is (C).
The dividend per share equals $2.00. The EPS equals $5.00 last year, which is found by dividing net earnings by outstanding shares. This year’s EPS is 20% higher or $6.00.
The payout ratio = dividend / EPS $2.00 / $6.00 = 33.33%

Question 35

The Anderson bond is a 5 percent coupon bond with semiannual coupon payments that matures in 10 years. If the YTM for this bond is 4 percent, what is the value of the bond?

Answer 35

P/YR = 2
FV = $1,000
N = 20
i = 4.0%
Pmt = $25.00
Answer: PV = $1,081.76

Question 36

Sam has a $3-million fixed-income portfolio that consists of Bond A, Bond B, Bond C, and Bond D. The bonds have durations of 2, 3, 8, and 10, respectively. If Sam has 20 percent invested in Bond A, 30 percent in Bond B, and 25 percent invested in each of the other two bonds, what is the duration for the portfolio? Assume that the correlation among the bonds is 0.5.

Answer 36

(0.2 × 2) + (0.3 × 3) + (0.25 × 8) + (0.25 × 10) = 0.4 + 0.9 + 2.0 + 2.5 = 5.8

Question 37

Marleen, who lives in Louisiana, is in the 20 percent federal tax bracket and 5 percent state income tax bracket. Which of the following bonds that she is considering purchasing has the highest after-tax yield?

A) A treasury bond paying 4%
B) A corporate bond paying 4.5%
C) A Louisiana municipal bond paying 3.5%
D) A Texas municipal bond paying 3.6%

Answer 37

Corporate bonds are subject to federal and state income tax. Treasury bonds are subject to federal income tax only. Municipal bonds are not subject to federal income tax, but they are subject to state income tax if they are not issued by the taxpayer’s state of residence. The La. muni has the highest after-tax return.
Treasury = 3.200%
Corporate = 3.375%
La. Bond = 3.500%
Tex. Bond = 3.42%

Question 38

Ollie is considering two portfolios: 1) Portfolio A with a return of 10 percent and a standard deviation of 20 percent, and 2) Portfolio B with a return of 6 percent and a standard deviation of 8 percent. Assuming the correlation between A and B is zero and Ollie invests 40 percent in A and 60 percent in B, what is the portfolio standard deviation?

Answer 38

Portfolio standard deviation = sqrt[(w1^2 x SD1^2) + (w2^2 x SD2^2)]

Portfolio standard deviation = sqrt[(0.4^2 x 20^2) + (0.6^2 x 8^2)]

Portfolio standard deviation = sqrt[(0.16 x 400) + (0.36 x 64)]

Portfolio standard deviation = sqrt(64 + 23.04) = sqrt(87.04) = 9.33

Question 39

Sam’s retirement fund is expected to earn a nominal rate of 7 percent, and the inflation rate is estimated at 3 percent. What is Sam’s real rate of return?

Answer 39

Real return = (1.07 ÷ 1.03) − 1 = 3.8835%

Question 40

Austin invested in the Very Value mutual fund 5 years ago. His returns were 25%, -5%, 10%, 0%, and 50%, respectively. What is the difference between the arithmetic average and the geometric average return over the 5 years?

Answer 40

Arithmetic average = 16%

Geometric average = (1 + 25%) × (1 + -5%) × (1 + 10%) × (1 + 0%) × (1 + 50%) – 1 = 14.4%

Difference = 16% − 14.4% = 1.6%