Accounting financial

Annual savings amount with annuity due to fund fixed perpetuity

1. Hope wants to establish a charitable foundation that will make annual scholarship payments of $23,000 per year forever. Hope wants the foundation to make the first annual $23,000 scholarship payment in 7 years from today and she wants scholarship payments of $23,000 per year to continue every year after that first payment. To fund the foundation, Hope plans to make equal annual donations to the foundation for 6 years. How much does Hope need to donate to the foundation each year for 6 years to have exactly enough in the foundation to make the planned annual scholarship payments if she makes her first annual donation to the foundation today, all annual donations to the foundation are equal, and funds held by the foundation are expected to earn 7.4 percent per year?

A. An amount equal to or greater than $30,000 but less than $42,000

B. An amount equal to or greater than $42,000 but less than $54,000

C. An amount equal to or greater than $54,000 but less than $62,000

D. An amount equal to or greater than $62,000 but less than $70,000

E. An amount less than $30,000 or an amount equal to or greater than $70,000

Step 1: Determine how much savings needs to be accumulated to pay for the annual scholarship

Step 2: Determine how much needs to be saved each year (?B) to accumulate the amount identified in step 1

Step 1: Determine how much savings needs to be accumulated to pay for the annual scholarship

The scholarship payments reflect a fixed perpetuity with payments of $23,000 and a discount rate of 7.4%. Therefore, the present value of this fixed perpetuity as of 6 years from now, which is 1 year before the first payment, can be found as C/r. The present value is the amount needed one year before the scholarship payments start to pay the annual scholarships of $23,000 forever.

As of 6 years from today, PV6 = C/r = 23,000 / .074 = $310,811

Hope needs to accumulate $310,811 as of 6 years from today

Step 2: Determine how much needs to be saved each year (?B) to accumulate the amount identified in step 1

If Hope donates a fixed amount of money for 6 years with her first donation today and her last in 5 years from today, then the amount that she needs to save each year to accumulate $310,811 in 6 years is the annual payment associated with a 6-period annuity due with a future value of 310,811.

BEGIN mode

Enter 6 7.4 0 310,811

N I% PV PMT FV

Solve for -40,050

Answer: A. An amount equal to or greater than $30,000 but less than $42,000

1. Hope wants to establish a charitable foundation that will make annual scholarship payments of $29,000 per year forever. Hope wants the foundation to make the first annual $29,000 scholarship payment in 7 years from today and she wants scholarship payments of $29,000 per year to continue every year after that first payment. To fund the foundation, Hope plans to make equal annual donations to the foundation for 6 years. How much does Hope need to donate to the foundation each year for 6 years to have exactly enough in the foundation to make the planned annual scholarship payments if she makes her first annual donation to the foundation today, all annual donations to the foundation are equal, and funds held by the foundation are expected to earn 7.4 percent per year?

A. An amount equal to or greater than $30,000 but less than $42,000

B. An amount equal to or greater than $42,000 but less than $54,000

C. An amount equal to or greater than $54,000 but less than $62,000

D. An amount equal to or greater than $62,000 but less than $70,000

E. An amount less than $30,000 or an amount equal to or greater than $70,000

Step 1: Determine how much savings needs to be accumulated to pay for the annual scholarship

Step 2: Determine how much needs to be saved each year (?B) to accumulate the amount identified in step 1

Step 1: Determine how much savings needs to be accumulated to pay for the annual scholarship

The scholarship payments reflect a fixed perpetuity with payments of $29,000 and a discount rate of 7.4%. Therefore, the present value of this fixed perpetuity as of 6 years from now, which is 1 year before the first payment, can be found as C/r. The present value is the amount needed one year before the scholarship payments start to pay the annual scholarships of $29,000 forever.

As of 6 years from today, PV6 = C/r = 29,000 / .074 = $391,892

Hope needs to accumulate $391,892 as of 6 years from today

Step 2: Determine how much needs to be saved each year (?B) to accumulate the amount identified in step 1

If Hope donates a fixed amount of money for 6 years with her first donation today and her last in 5 years from today, then the amount that she needs to save each year to accumulate $391,892 in 6 years is the annual payment associated with a 6-period annuity due with a future value of 391,892.

BEGIN mode

Enter 6 7.4 0 391,892

N I% PV PMT FV

Solve for -50,498

Answer: B. An amount equal to or greater than $42,000 but less than $54,000

1. Hope wants to establish a charitable foundation that will make annual scholarship payments of $23,000 per year forever. Hope wants the foundation to make the first annual $23,000 scholarship payment in 7 years from today and she wants scholarship payments of $23,000 per year to continue every year after that first payment. To fund the foundation, Hope plans to make equal annual donations to the foundation for 6 years. How much does Hope need to donate to the foundation each year for 6 years to have exactly enough in the foundation to make the planned annual scholarship payments if she makes her first annual donation to the foundation today, all annual donations to the foundation are equal, and funds held by the foundation are expected to earn 4.7 percent per year?

A. An amount equal to or greater than $30,000 but less than $42,000

B. An amount equal to or greater than $42,000 but less than $54,000

C. An amount equal to or greater than $54,000 but less than $62,000

D. An amount equal to or greater than $62,000 but less than $70,000

E. An amount less than $30,000 or an amount equal to or greater than $70,000

Step 1: Determine how much savings needs to be accumulated to pay for the annual scholarship

Step 2: Determine how much needs to be saved each year (?B) to accumulate the amount identified in step 1

Step 1: Determine how much savings needs to be accumulated to pay for the annual scholarship

The scholarship payments reflect a fixed perpetuity with payments of $23,000 and a discount rate of 4.7%. Therefore, the present value of this fixed perpetuity as of 6 years from now, which is 1 year before the first payment, can be found as C/r. The present value is the amount needed one year before the scholarship payments start to pay the annual scholarships of $23,000 forever.

As of 6 years from today, PV6 = C/r = 23,000 / .047 = $489,362

Hope needs to accumulate $489,362 as of 6 years from today

Step 2: Determine how much needs to be saved each year (?B) to accumulate the amount identified in step 1

If Hope donates a fixed amount of money for 6 years with her first donation today and her last in 5 years from today, then the amount that she needs to save each year to accumulate $489,362 in 6 years is the annual payment associated with a 6-period annuity due with a future value of 489,362.

BEGIN mode

Enter 6 4.7 0 489,362

N I% PV PMT FV

Solve for -69,236

Answer: D. An amount equal to or greater than $62,000 but less than $70,000

1. Hope wants to establish a charitable foundation that will make annual scholarship payments of $20,000 per year forever. Hope wants the foundation to make the first annual $20,000 scholarship payment in 7 years from today and she wants scholarship payments of $20,000 per year to continue every year after that first payment. To fund the foundation, Hope plans to make equal annual donations to the foundation for 6 years. How much does Hope need to donate to the foundation each year for 6 years to have exactly enough in the foundation to make the planned annual scholarship payments if she makes her first annual donation to the foundation today, all annual donations to the foundation are equal, and funds held by the foundation are expected to earn 4.7 percent per year?

A. An amount equal to or greater than $30,000 but less than $42,000

B. An amount equal to or greater than $42,000 but less than $54,000

C. An amount equal to or greater than $54,000 but less than $62,000

D. An amount equal to or greater than $62,000 but less than $70,000

E. An amount less than $30,000 or an amount equal to or greater than $70,000

Step 1: Determine how much savings needs to be accumulated to pay for the annual scholarship

Step 2: Determine how much needs to be saved each year (?B) to accumulate the amount identified in step 1

Step 1: Determine how much savings needs to be accumulated to pay for the annual scholarship

The scholarship payments reflect a fixed perpetuity with payments of $20,000 and a discount rate of 4.7%. Therefore, the present value of this fixed perpetuity as of 6 years from now, which is 1 year before the first payment, can be found as C/r. The present value is the amount needed one year before the scholarship payments start to pay the annual scholarships of $20,000 forever.

As of 6 years from today, PV6 = C/r = 20,000 / .047 = $425,532

Hope needs to accumulate $425,532 as of 6 years from today

Step 2: Determine how much needs to be saved each year (?B) to accumulate the amount identified in step 1

If Hope donates a fixed amount of money for 6 years with her first donation today and her last in 5 years from today, then the amount that she needs to save each year to accumulate $425,532 in 6 years is the annual payment associated with a 6-period annuity due with a future value of 425,532.

BEGIN mode

Enter 6 4.7 0 425,532

N I% PV PMT FV

Solve for -60,205

Answer: C. An amount equal to or greater than $54,000 but less than $62,000

Price stock with non-constant growth

2. If 1) the expected return for Belmont Books stock is 9.5 percent; 2) the dividend is expected to be $4.38 in one year, $6.33 in two years, $0 in three years, and $2.54 in four years; and 3) after the dividend is paid in four years, the dividend is expected to begin growing by 4.5 percent a year forever, then what is the current price of one share of the stock?

A. An amount less than $45.20

B. An amount between $45.20 and just less than $46.20

C. An amount between $46.20 and just less than $47.20

D. An amount between $47.20 and just less than $48.20

E. An amount equal to or greater than $48.20

Belmont Books dividends are expected to grow at a variable rate before settling into a long-term growth rate after the dividend is paid in 4 years.

The stock of a company that pays a dividend that does not grow at its long-term constant rate during the next N+1 years, before growing forever at that constant rate after N+1 years can be valued as

P0 = D1/(1+R) + D2/(1+R)2 + ... + DN/(1+R)N + PN/(1+R)N where PN = DN+1/(R − g)

Starting after the dividend is paid in 4 years, Belmont Books dividends will be growing at a constant rate of 4.5 percent, so N + 1 = 4 and N = 3

P0 = D1/(1+R) + D2/(1+R)2 + D3/(1+R)3 + P3/(1+R)3 where P3 = D4/(R − g)

P3 = D4/(R − g) = 2.54/ (.095 – .045) = 2.54 / .050 = 50.80

P0 = 4.38/(1.095) + 6.33/(1.095)2 + 0/(1.095)3 + 50.80/(1.095)3

= 4.00 + 5.28 + 0 + 38.69 = $47.97

(answers may differ slightly due to rounding)

2. If 1) the expected return for Belmont Books stock is 9.5 percent; 2) the dividend is expected to be $4.38 in one year, $4.62 in two years, $0 in three years, and $2.54 in four years; and 3) after the dividend is paid in four years, the dividend is expected to begin growing by 4.5 percent a year forever, then what is the current price of one share of the stock?

A. An amount less than $45.30

B. An amount between $45.30 and just less than $46.30

C. An amount between $46.30 and just less than $47.30

D. An amount between $47.30 and just less than $48.30

E. An amount equal to or greater than $48.30

Belmont Books dividends are expected to grow at a variable rate before settling into a long-term growth rate after the dividend is paid in 4 years.

The stock of a company that pays a dividend that does not grow at its long-term constant rate during the next N+1 years, before growing forever at that constant rate after N+1 years can be valued as

P0 = D1/(1+R) + D2/(1+R)2 + ... + DN/(1+R)N + PN/(1+R)N where PN = DN+1/(R − g)

Starting after the dividend is paid in 4 years, Belmont Books dividends will be growing at a constant rate of 4.5 percent, so N + 1 = 4 and N = 3

P0 = D1/(1+R) + D2/(1+R)2 + D3/(1+R)3 + P3/(1+R)3 where P3 = D4/(R − g)

P3 = D4/(R − g) = 2.54/ (.095 – .045) = 2.54 / .050 = 50.80

P0 = 4.38/(1.095) + 4.62/(1.095)2 + 0/(1.095)3 + 50.80/(1.095)3

= 4.00 + 3.85 + 0 + 38.69 = $46.54

(answers may differ slightly due to rounding)

2. If 1) the expected return for Belmont Books stock is 9.5 percent; 2) the dividend is expected to be $4.38 in one year, $6.33 in two years, $0 in three years, and $3.81 in four years; and 3) after the dividend is paid in four years, the dividend is expected to begin growing by 4.5 percent a year forever, then what is the current price of one share of the stock?

A. An amount less than $67.10

B. An amount between $67.10 and just less than $68.10

C. An amount between $68.10 and just less than $69.10

D. An amount between $69.10 and just less than $70.10

E. An amount equal to or greater than $70.10

Belmont Books dividends are expected to grow at a variable rate before settling into a long-term growth rate after the dividend is paid in 4 years.

The stock of a company that pays a dividend that does not grow at its long-term constant rate during the next N+1 years, before growing forever at that constant rate after N+1 years can be valued as

P0 = D1/(1+R) + D2/(1+R)2 + ... + DN/(1+R)N + PN/(1+R)N where PN = DN+1/(R − g)

Starting after the dividend is paid in 4 years, Belmont Books dividends will be growing at a constant rate of 4.5 percent, so N + 1 = 4 and N = 3

P0 = D1/(1+R) + D2/(1+R)2 + D3/(1+R)3 + P3/(1+R)3 where P3 = D4/(R − g)

P3 = D4/(R − g) = 3.81/ (.095 – .045) = 3.81 / .050 = 76.20

P0 = 4.38/(1.095) + 6.33/(1.095)2 + 0/(1.095)3 + 76.20/(1.095)3

= 4.00 + 5.28 + 0 + 58.04 = $67.32

(answers may differ slightly due to rounding)

2. If 1) the expected return for Belmont Books stock is 9.5 percent; 2) the dividend is expected to be $4.38 in one year, $4.62 in two years, $0 in three years, and $3.81 in four years; and 3) after the dividend is paid in four years, the dividend is expected to begin growing by 4.5 percent a year forever, then what is the current price of one share of the stock?

A. An amount less than $64.40

B. An amount between $64.40 and just less than $65.40

C. An amount between $65.40 and just less than $66.40

D. An amount between $66.40 and just less than $67.40

E. An amount equal to or greater than $67.40

Belmont Books dividends are expected to grow at a variable rate before settling into a long-term growth rate after the dividend is paid in 4 years.

The stock of a company that pays a dividend that does not grow at its long-term constant rate during the next N+1 years, before growing forever at that constant rate after N+1 years can be valued as

P0 = D1/(1+R) + D2/(1+R)2 + ... + DN/(1+R)N + PN/(1+R)N where PN = DN+1/(R − g)

Starting after the dividend is paid in 4 years, Belmont Books dividends will be growing at a constant rate of 4.5 percent, so N + 1 = 4 and N = 3

P0 = D1/(1+R) + D2/(1+R)2 + D3/(1+R)3 + P3/(1+R)3 where P3 = D4/(R − g)

P3 = D4/(R − g) = 3.81/ (.095 – .045) = 3.81 / .050 = 76.20

P0 = 4.38/(1.095) + 4.62/(1.095)2 + 0/(1.095)3 + 76.20/(1.095)3

= 4.00 + 3.85 + 0 + 58.04 = $65.89

(answers may differ slightly due to rounding)

Compute and compare 3 loan EARs

3. Bianca is deciding among 3 loans that would each involve her receiving $8,000 today and then paying back the original principal and all accrued interest in 1 year from today. Loan A has an APR of 14.40%, compounded annually. Loan B has an APR of 13.60%, compounded quarterly. Loan C has an APR of 13.60%, compounded continuously. Which of the following assertions is true if Bianca prefers loans with lower costs more than she prefers loans with higher costs?

A. Bianca would prefer loan A to loan B and Bianca would prefer loan A to loan C

B. Bianca would prefer loan A to loan B and Bianca would prefer loan C to loan A

C. Bianca would prefer loan B to loan A and Bianca would prefer loan A to loan C

D. Bianca would prefer loan B to loan A and Bianca would prefer loan C to loan A

To answer this question, find and compare the EARs of the loans. Loans with lower EAR have lower costs, all else equal, because EAR reflects the true cost of a loan. Therefore, a loan with a lower EAR would be preferred to a comparable loan with a higher EAR.

When compounding is not continuous:

EAR = [(1 + periodic rate)# of periods in a year] – 1

= [(1 + (APR/# periods in a year))# of periods in a year] – 1

When compounding is continuous:

EAR = eAPR – 1

EAR(A) = [(1 + (.1440 / 1))1] – 1 = [(1 + .1440)1] – 1 = [(1.1440)1] – 1 = .1440 = 14.40%

EAR(B) = [(1 + (.1360 / 4))4] – 1 = [(1 + .0340)4] – 1 = [(1.0340)4] – 1 = .1431 = 14.31%

EAR(C) = e.1360 – 1 = 1.1457 – 1 = .1457 = 14.57%

Since EAR(A) > EAR(B), Bianca would prefer loan B to loan A

Since EAR(C) > EAR(A), Bianca would prefer loan A to loan C

Answer:

C. Bianca would prefer loan B to loan A and Bianca would prefer loan A to loan C

3. Bianca is deciding among 3 loans that would each involve her receiving $8,000 today and then paying back the original principal and all accrued interest in 1 year from today. Loan A has an APR of 15.50%, compounded annually. Loan B has an APR of 14.80%, compounded quarterly. Loan C has an APR of 14.80%, compounded continuously. Which of the following assertions is true if Bianca prefers loans with lower costs more than she prefers loans with higher costs?

A. Bianca would prefer loan A to loan B and Bianca would prefer loan A to loan C

B. Bianca would prefer loan A to loan B and Bianca would prefer loan C to loan A

C. Bianca would prefer loan B to loan A and Bianca would prefer loan A to loan C

D. Bianca would prefer loan B to loan A and Bianca would prefer loan C to loan A

To answer this question, find and compare the EARs of the loans. Loans with lower EAR have lower costs, all else equal, because EAR reflects the true cost of a loan. Therefore, a loan with a lower EAR would be preferred to a comparable loan with a higher EAR.

When compounding is not continuous:

EAR = [(1 + periodic rate)# of periods in a year] – 1

= [(1 + (APR/# periods in a year))# of periods in a year] – 1

When compounding is continuous:

EAR = eAPR – 1

EAR(A) = [(1 + (.1550 / 1))1] – 1 = [(1 + .1550)1] – 1 = [(1.1550)1] – 1 = .1550 = 15.50%

EAR(B) = [(1 + (.1480 / 4))4] – 1 = [(1 + .0370)4] – 1 = [(1.0370)4] – 1 = .1564 = 15.64%

EAR(C) = e.1480 – 1 = 1.1595 – 1 = .1595 = 15.95%

Since EAR(B) > EAR(A), Bianca would prefer loan A to loan B

Since EAR(C) > EAR(A), Bianca would prefer loan A to loan C

Answer:

A. Bianca would prefer loan A to loan B and Bianca would prefer loan A to loan C

3. Bianca is deciding among 3 loans that would each involve her receiving $8,000 today and then paying back the original principal and all accrued interest in 1 year from today. Loan A has an APR of 15.30%, compounded annually. Loan B has an APR of 14.40%, compounded continuously. Loan C has an APR of 14.40%, compounded quarterly. Which of the following assertions is true if Bianca prefers loans with lower costs more than she prefers loans with higher costs?

A. Bianca would prefer loan A to loan B and Bianca would prefer loan A to loan C

B. Bianca would prefer loan A to loan B and Bianca would prefer loan C to loan A

C. Bianca would prefer loan B to loan A and Bianca would prefer loan A to loan C

D. Bianca would prefer loan B to loan A and Bianca would prefer loan C to loan A

To answer this question, find and compare the EARs of the loans. Loans with lower EAR have lower costs, all else equal, because EAR reflects the true cost of a loan. Therefore, a loan with a lower EAR would be preferred to a comparable loan with a higher EAR.

When compounding is not continuous:

EAR = [(1 + periodic rate)# of periods in a year] – 1

= [(1 + (APR/# periods in a year))# of periods in a year] – 1

When compounding is continuous:

EAR = eAPR – 1

EAR(A) = [(1 + (.1530 / 1))1] – 1 = [(1 + .1530)1] – 1 = [(1.1530)1] – 1 = .1530 = 15.30%

EAR(B) = e.1440 – 1 = 1.1549 – 1 = .1549 = 15.49%

EAR(C) = [(1 + (.1440 / 4))4] – 1 = [(1 + .0360)4] – 1 = [(1.0360)4] – 1 = .1520 = 15.20%

Since EAR(A) < EAR(B), Bianca would prefer loan A to loan B

Since EAR(C) < EAR(A), Bianca would prefer loan C to loan A

Answer:

B. Bianca would prefer loan A to loan B and Bianca would prefer loan C to loan A

3. Bianca is deciding among 3 loans that would each involve her receiving $8,000 today and then paying back the original principal and all accrued interest in 1 year from today. Loan A has an APR of 16.40%, compounded annually. Loan B has an APR of 15.60%, compounded continuously. Loan C has an APR of 15.60%, compounded quarterly. Which of the following assertions is true if Bianca prefers loans with lower costs more than she prefers loans with higher costs?

A. Bianca would prefer loan A to loan B and Bianca would prefer loan A to loan C

B. Bianca would prefer loan A to loan B and Bianca would prefer loan C to loan A

C. Bianca would prefer loan B to loan A and Bianca would prefer loan A to loan C

D. Bianca would prefer loan B to loan A and Bianca would prefer loan C to loan A

To answer this question, find and compare the EARs of the loans. Loans with lower EAR have lower costs, all else equal, because EAR reflects the true cost of a loan. Therefore, a loan with a lower EAR would be preferred to a comparable loan with a higher EAR.

When compounding is not continuous:

EAR = [(1 + periodic rate)# of periods in a year] – 1

= [(1 + (APR/# periods in a year))# of periods in a year] – 1

When compounding is continuous:

EAR = eAPR – 1

EAR(A) = [(1 + (.1640 / 1))1] – 1 = [(1 + .1640)1] – 1 = [(1.1640)1] – 1 = .1640 = 16.40%

EAR(B) = e.1560 – 1 = 1.1688 – 1 = .1688 = 16.88%

EAR(C) = [(1 + (.1560 / 4))4] – 1 = [(1 + .0390)4] – 1 = [(1.0390)4] – 1 = .1654 = 16.54%

Since EAR(A) < EAR(B), Bianca would prefer loan A to loan B

Since EAR(A) < EAR(C), Bianca would prefer loan A to loan C

Answer:

A. Bianca would prefer loan A to loan B and Bianca would prefer loan A to loan C

Quantitative: find YTM of one bond and use it to find price of a second bond

4. Bond A and bond B both pay annual coupons, mature in 8 years, have a face value of $1000, pay their next coupon in 12 months, and have the same yield-to-maturity. Bond A has a coupon rate of 6.5 percent and is priced at $1,050.27. Bond B has a coupon rate of 7.4 percent. What is the price of bond B?

A. $1,106.83 (plus or minus $4)

B. $995.63 (plus or minus $4)

C. $1,050.27 (plus or minus $4)

D. $1,000.00 (plus or minus $4)

E. None of the above is within $4 of the correct answer

Approach: Find YTM of bond A and use it to compute price of bond B, since they have the same YTM

YTM of bond A:

N = 8 years × 1 coupon per year = 8

PV = -1,050.27

PMT = par × coupon rate ÷ # coupons per year = 1000 × 6.5% ÷ 1 = 65

FV = 1000

END mode

Enter 8 -1,050.27 65 1000

N I% PV PMT FV

Solve for 5.70

I% = YTM ÷ # coupons per year = YTM ÷ 1

So YTM = I% × # coupons per year = 5.70% × 1 = 5.70%

YTM for bond A = 5.70 percent, so for bond B

N = 8 years × 1 coupon per year = 8

I% = YTM ÷ # coupons per year = 5.70 ÷ 1 = 5.70

PMT = par × coupon rate ÷ # coupons per year = 1000 × 7.4% ÷ 1 = 74.00

FV = 1000

END mode

Enter 8 5.70 74 1000

N I% PV PMT FV

Solve for -1,106.83

The value of bond B is $1,106.83 (answer may differ slightly due to rounding I%) 4. Bond A and bond B both pay annual coupons, mature in 9 years, have a face value of $1000, pay their next coupon in 12 months, and have the same yield-to-maturity. Bond A has a coupon rate of 6.5 percent and is priced at $1,055.13. Bond B has a coupon rate of 7.4 percent. What is the price of bond B?

A. $1,117.15 (plus or minus $4)

B. $995.40 (plus or minus $4)

C. $1,055.13 (plus or minus $4)

D. $1,000.00 (plus or minus $4)

E. None of the above is within $4 of the correct answer

Approach: Find YTM of bond A and use it to compute price of bond B, since they have the same YTM

YTM of bond A:

N = 9 years × 1 coupon per year = 9

PV = -1,055.13

PMT = par × coupon rate ÷ # coupons per year = 1000 × 6.5% ÷ 1 = 65

FV = 1000

END mode

Enter 9 -1055.13 65 1000

N I% PV PMT FV

Solve for 5.70

I% = YTM ÷ # coupons per year = YTM ÷ 1

So YTM = I% × # coupons per year = 5.70% × 1 = 5.70%

YTM for bond A = 5.70 percent, so for bond B

N = 9 years × 1 coupon per year = 9

I% = YTM ÷ # coupons per year = 5.70 ÷ 1 = 5.70

PMT = par × coupon rate ÷ # coupons per year = 1000 × 7.4% ÷ 1 = 74.00

FV = 1000

END mode

Enter 9 5.70 74 1000

N I% PV PMT FV

Solve for -1,117.15

The value of bond B is $1,117.15 (answer may differ slightly due to rounding I%)

4. Bond A and bond B both pay annual coupons, mature in 8 years, have a face value of $1000, pay their next coupon in 12 months, and have the same yield-to-maturity. Bond A has a coupon rate of 6.5 percent and is priced at $1,056.78. Bond B has a coupon rate of 7.4 percent. What is the price of bond B?

A. $1,113.56 (plus or minus $4)

B. $1,001.91 (plus or minus $4)

C. $1,056.78 (plus or minus $4)

D. $1,000.00 (plus or minus $4)

E. None of the above is within $4 of the correct answer

Approach: Find YTM of bond A and use it to compute price of bond B, since they have the same YTM

YTM of bond A:

N = 8 years × 1 coupon per year = 8

PV = -1,056.78

PMT = par × coupon rate ÷ # coupons per year = 1000 × 6.5% ÷ 1 = 65

FV = 1000

END mode

Enter 8 -1,056.78 65 1000

N I% PV PMT FV

Solve for 5.60

I% = YTM ÷ # coupons per year = YTM ÷ 1

So YTM = I% × # coupons per year = 5.60% × 1 = 5.60%

YTM for bond A = 5.60 percent, so for bond B

N = 8 years × 1 coupon per year = 8

I% = YTM ÷ # coupons per year = 5.60 ÷ 1 = 5.60

PMT = par × coupon rate ÷ # coupons per year = 1000 × 7.4% ÷ 1 = 74.00

FV = 1000

END mode

Enter 8 5.60 74 1000

N I% PV PMT FV

Solve for -1,113.57

The value of bond B is $1,113.57 (answer may differ slightly due to rounding I%)

4. Bond A and bond B both pay annual coupons, mature in 9 years, have a face value of $1000, pay their next coupon in 12 months, and have the same yield-to-maturity. Bond A has a coupon rate of 6.5 percent and is priced at $1,055.13. Bond B has a coupon rate of 7.4 percent. What is the price of bond B?

A. $1,124.60 (plus or minus $4)

B. $1,002.31 (plus or minus $4)

C. $1,062.30 (plus or minus $4)

D. $1,000.00 (plus or minus $4)

E. None of the above is within $4 of the correct answer

Approach: Find YTM of bond A and use it to compute price of bond B, since they have the same YTM

YTM of bond A:

N = 9 years × 1 coupon per year = 9

PV = -1,055.13

PMT = par × coupon rate ÷ # coupons per year = 1000 × 6.5% ÷ 1 = 65

FV = 1000

END mode

Enter 9 -1055.13 65 1000

N I% PV PMT FV

Solve for 5.70

I% = YTM ÷ # coupons per year = YTM ÷ 1

So YTM = I% × # coupons per year = 5.70% × 1 = 5.70%

YTM for bond A = 5.70 percent, so for bond B

N = 9 years × 1 coupon per year = 9

I% = YTM ÷ # coupons per year = 5.70 ÷ 1 = 5.70

PMT = par × coupon rate ÷ # coupons per year = 1000 × 7.4% ÷ 1 = 74.00

FV = 1000

END mode

Enter 9 5.70 74 1000

N I% PV PMT FV

Solve for -1,117.15

The value of bond B is $1,117.15 (answer may differ slightly due to rounding I%)

None of the possible answers is within $4 of $1,117.15

Find P0 from D1, D2, D3, & P2

5. The next three annual dividends paid by Mindy’s Mending stock are expected to be $2.79 in one year, $7.43 in two years, and $3.05 in three years. The price of the stock is expected to be $54.78 in two years. The expected annual return for the stock is 15.20 percent. What is the current price of one share of Mindy’s Mending stock?

A. $49.30 (plus or minus $0.05)

B. $51.29 (plus or minus $0.05)

C. $45.85 (plus or minus $0.05)

D. $49.83 (plus or minus $0.05)

E. None of the above is within $0.05 of the correct answer

P0 = [D1 / (1 + R)] + [(D2 + P2) / (1 + R)2]

= [2.79 / 1.1520] + [(7.43 + 54.78) / (1.1520)2]

= $49.30

The expected dividend in 3 years is irrelevant

5. The next three annual dividends paid by Mindy’s Mending stock are expected to be $2.79 in one year, $7.43 in two years, and $3.05 in three years. The price of the stock is expected to be $45.78 in two years. The expected annual return for the stock is 15.20 percent. What is the current price of one share of Mindy’s Mending stock?

A. $42.52 (plus or minus $0.05)

B. $44.51 (plus or minus $0.05)

C. $39.96 (plus or minus $0.05)

D. $43.05 (plus or minus $0.05)

E. None of the above is within $0.05 of the correct answer

P0 = [D1 / (1 + R)] + [(D2 + P2) / (1 + R)2]

= [2.79 / 1.1520] + [(7.43 + 45.78) / (1.1520)2]

= $42.52

The expected dividend in 3 years is irrelevant

5. The next three annual dividends paid by Mindy’s Mending stock are expected to be $7.43 in one year, $2.79 in two years, and $3.05 in three years. The price of the stock is expected to be $54.78 in two years. The expected annual return for the stock is 15.20 percent. What is the current price of one share of Mindy’s Mending stock?

A. $49.83 (plus or minus $0.05)

B. $51.82 (plus or minus $0.05)

C. $46.38 (plus or minus $0.05)

D. $49.30 (plus or minus $0.05)

E. None of the above is within $0.05 of the correct answer

P0 = [D1 / (1 + R)] + [(D2 + P2) / (1 + R)2]

= [7.43 / 1.1520] + [(2.79 + 54.78) / (1.1520)2]

= $49.83 The expected dividend in 3 years is irrelevant

5. The next three annual dividends paid by Mindy’s Mending stock are expected to be $7.43 in one year, $2.79 in two years, and $3.05 in three years. The price of the stock is expected to be $45.78 in two years. The expected annual return for the stock is 15.20 percent. What is the current price of one share of Mindy’s Mending stock?

A. $43.05 (plus or minus $0.05)

B. $45.01 (plus or minus $0.05)

C. $40.46 (plus or minus $0.05)

D. $42.52 (plus or minus $0.05)

E. None of the above is within $0.05 of the correct answer

P0 = [D1 / (1 + R)] + [(D2 + P2) / (1 + R)2]

= [7.43 / 1.1520] + [(2.79 + 45.78) / (1.1520)2]

= $43.05

The expected dividend in 3 years is irrelevant

Current yield today from previous P, today’s P, and return

6. Bonds issued by Mindy’s Mending have a par value of $1000, were priced at $1,220.00 six months ago, and are priced at $1,140.00 today. The bonds pay semi-annual coupons and just made a coupon payment. If the bonds had a percentage return over the past 6 months (from 6 months ago to today) of -2.10%, then what is the current yield of the bonds today?

A. 9.54% (plus or minus 0.05 percentage points)

B. 8.91% (plus or minus 0.05 percentage points)

C. 6.26% (plus or minus 0.05 percentage points)

D. 5.73% (plus or minus 0.05 percentage points)

E. None of the above is within 0.05 percentage points of the correct answer

Current yield = annual coupons / bond value

Current yield today = annual coupons / bond value today

Bond value today = $1,140.00

Annual coupons = 2 × semi-annual coupons

The semi-annual coupons on the bond can be found from the percentage return on the bonds

Percentage return = (cash flow from investment + ending value – initial value) / initial value

= (semi-annual coupon + ending value – initial value) / initial value

Percentage return = -2.10% = -.0210

Initial value = price of bond 6 months ago = $1,220.00

Ending value = price of bond today = $1,140.00

Percentage return = (semi-annual coupon + ending value – initial value) / initial value

So -.0210 = (semi-annual coupon + 1,140.00 – 1,220.00) / 1,220.00

= (semi-annual coupon + (-80.00)) / 1,220.00

= (semi-annual coupon – 80.00) / 1,220.00

So -.0210 × 1,220.00 = -25.62 = (semi-annual coupon – 80.00)

So semi-annual coupon = -25.62 + 80.00 = 54.38

Annual coupons = 2 × semi-annual coupons

= 2 × 54.38

= $108.76

Current yield today = annual coupons / bond value today

= 108.76 / 1,140.00

= .0954 = 9.54%

6. Bonds issued by Mindy’s Mending have a par value of $1000, were priced at $1,140.00 six months ago, and are priced at $1,060.00 today. The bonds pay semi-annual coupons and just made a coupon payment. If the bonds had a percentage return over the past 6 months (from 6 months ago to today) of -2.10%, then what is the current yield of the bonds today?

A. 10.58% (plus or minus 0.05 percentage points)

B. 9.84% (plus or minus 0.05 percentage points)

C. 11.47% (plus or minus 0.05 percentage points)

D. 12.51% (plus or minus 0.05 percentage points)

E. None of the above is within 0.05 percentage points of the correct answer

Current yield = annual coupons / bond value

Current yield today = annual coupons / bond value today

Bond value today = $1,060.00

Annual coupons = 2 × semi-annual coupons

The semi-annual coupons on the bond can be found from the percentage return on the bonds

Percentage return = (cash flow from investment + ending value – initial value) / initial value

= (semi-annual coupon + ending value – initial value) / initial value

Percentage return = -2.10% = -.0210

Initial value = price of bond 6 months ago = $1,140.00

Ending value = price of bond today = $1,060.00

Percentage return = (semi-annual coupon + ending value – initial value) / initial value

So -.0210 = (semi-annual coupon + 1,060.00 – 1,140.00) / 1,140.00

= (semi-annual coupon + (-80.00)) / 1,140.00

= (semi-annual coupon – 80.00) / 1,140.00

So -.0210 × 1,140.00 = -23.94 = (semi-annual coupon – 80.00)

So semi-annual coupon = -23.94 + 80.00 = 56.06

Annual coupons = 2 × semi-annual coupons

= 2 × 56.06

= $112.12

Current yield today = annual coupons / bond value today

= 112.12 / 1,060.00

= .1058 = 10.58%

6. Bonds issued by Mindy’s Mending have a par value of $1000, were priced at $1,220.00 six months ago, and are priced at $1,140.00 today. The bonds pay semi-annual coupons and just made a coupon payment. If the bonds had a percentage return over the past 6 months (from 6 months ago to today) of -1.20%, then what is the current yield of the bonds today?

A. 11.47% (plus or minus 0.05 percentage points)

B. 10.71% (plus or minus 0.05 percentage points)

C. 12.51% (plus or minus 0.05 percentage points)

D. 9.54% (plus or minus 0.05 percentage points)

E. None of the above is within 0.05 percentage points of the correct answer

Current yield = annual coupons / bond value

Current yield today = annual coupons / bond value today

Bond value today = $1,140.00

Annual coupons = 2 × semi-annual coupons

The semi-annual coupons on the bond can be found from the percentage return on the bonds

Percentage return = (cash flow from investment + ending value – initial value) / initial value

= (semi-annual coupon + ending value – initial value) / initial value

Percentage return = -1.20% = -.0120

Initial value = price of bond 6 months ago = $1,220.00

Ending value = price of bond today = $1,140.00

Percentage return = (semi-annual coupon + ending value – initial value) / initial value

So -.0120 = (semi-annual coupon + 1,140.00 – 1,220.00) / 1,220.00

= (semi-annual coupon + (-80.00)) / 1,220.00

= (semi-annual coupon – 80.00) / 1,220.00

So -.0120 × 1,220.00 = -14.64 = (semi-annual coupon – 80.00)

So semi-annual coupon = -14.64 + 80.00 = 65.36

Annual coupons = 2 × semi-annual coupons

= 2 × 65.36

= $130.72

Current yield today = annual coupons / bond value today

= 130.72 / 1,140.00

= .1147 = 11.47%

6. Bonds issued by Mindy’s Mending have a par value of $1000, were priced at $1,140.00 six months ago, and are priced at $1,060.00 today. The bonds pay semi-annual coupons and just made a coupon payment. If the bonds had a percentage return over the past 6 months (from 6 months ago to today) of -1.20%, then what is the current yield of the bonds today?

A. 12.51% (plus or minus 0.05 percentage points)

B. 11.64% (plus or minus 0.05 percentage points)

C. 9.54% (plus or minus 0.05 percentage points)

D. 10.58% (plus or minus 0.05 percentage points)

E. None of the above is within 0.05 percentage points of the correct answer

Current yield = annual coupons / bond value

Current yield today = annual coupons / bond value today

Bond value today = $1,060.00

Annual coupons = 2 × semi-annual coupons

The semi-annual coupons on the bond can be found from the percentage return on the bonds

Percentage return = (cash flow from investment + ending value – initial value) / initial value

= (semi-annual coupon + ending value – initial value) / initial value

Percentage return = -1.20% = -.0120

Initial value = price of bond 6 months ago = $1,140.00

Ending value = price of bond today = $1,060.00

Percentage return = (semi-annual coupon + ending value – initial value) / initial value

So -.0120 = (semi-annual coupon + 1,060.00 – 1,140.00) / 1,140.00

= (semi-annual coupon + (-80.00)) / 1,140.00

= (semi-annual coupon – 80.00) / 1,140.00

So -.0120 × 1,140.00 = -13.68 = (semi-annual coupon – 80.00)

So semi-annual coupon = -13.68 + 80.00 = 66.32

Annual coupons = 2 × semi-annual coupons

= 2 × 66.32

= $132.64

Current yield today = annual coupons / bond value today

= 132.64 / 1,060.00

= .1251 = 12.51%