BUSI 223 - Advanced Quantitative Problems on
Time Value of Money
PROBLEM SET
1. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
2. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
3. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
4. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
5. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
6. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
7. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
8. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
9. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
10. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
11. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
12. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
13. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
14. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
15. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
16. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
17. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
18. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
19. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
20. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
21. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
22. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
23. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
24. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
25. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
26. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
27. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
28. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
29. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
30. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
31. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
32. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
33. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
34. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
35. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
36. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
37. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
38. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
39. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
40. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
41. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
42. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
43. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
44. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
45. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
46. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
47. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
48. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
49. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
50. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
51. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
52. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
53. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
54. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
55. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
56. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
57. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
58. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
59. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
60. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
61. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
62. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
63. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
64. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
65. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
66. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
67. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
68. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
69. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
70. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
71. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
72. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
73. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
74. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
75. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
76. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
77. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
78. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
79. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
80. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
81. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
82. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
83. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
84. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
85. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
86. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
87. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
88. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
89. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
90. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
91. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
92. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
93. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
94. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
95. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
96. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
97. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
98. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
99. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
100. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
101. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
102. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
103. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
104. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
105. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
106. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
107. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
108. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
109. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
110. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
111. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
112. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
113. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
114. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
115. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
116. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
117. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
118. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
119. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
120. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
121. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
122. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
123. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
124. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
125. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
126. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
127. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
128. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
129. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
130. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
131. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
132. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
133. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
134. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
135. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
136. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
137. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
138. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
139. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
140. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
141. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
142. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
143. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
144. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
145. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
146. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
147. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
148. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
149. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
150. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
151. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
152. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
153. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
154. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
155. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
156. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
157. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
158. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
159. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
160. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
161. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
162. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
163. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
164. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
165. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
166. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
167. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
168. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
169. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
170. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
171. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
172. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
173. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
174. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
175. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
176. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
177. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
178. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
179. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
180. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
181. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
182. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
183. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
184. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
185. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
186. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
187. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
188. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
189. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
190. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
191. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
192. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
193. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
194. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
195. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
196. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
197. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
198. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
199. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
200. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
201. A project requires an initial investment of $500,000 and is expected to generate cash
flows of $150,000 at the end of each year for the next 5 years. After that, the cash flows
are expected to grow at a rate of 3% per year indefinitely. If the required rate of return is
12%, what is the Net Present Value (NPV) of this project?
Solution:
– Step 1: Calculate the present value of the first 5 years of cash flows
𝑃𝑉1=150,000×1−(1+0.12)−5
0.12 = $541,231.22
– Step 2: Calculate the present value of the growing perpetuity starting at year 6
𝑃𝑉2=150,000×1.03
0.12−0.03 ×(1+0.12)−5 = $937,728.94
– Step 3: Calculate the total NPV
𝑁𝑃𝑉 = −500,000+541,231.22+937,728.94 = $978,960.16
202. An investor is considering two mutually exclusive projects. Project A requires an initial
investment of $200,000 and will generate annual cash flows of $50,000 for 6 years.
Project B requires an initial investment of $250,000 and will generate annual cash flows
of $65,000 for 6 years. The investor’s cost of capital is 10%. Which project should the
investor choose based on the Internal Rate of Return (IRR) and why?
Solution:
– Step 1: Calculate IRR for Project A
−200,000+50,000×1− (1+𝐼𝑅𝑅𝐴)−6
𝐼𝑅𝑅𝐴= 0
– Solving this equation (using a financial calculator or Excel) gives 𝐼𝑅𝑅𝐴=15.24%
– Step 2: Calculate IRR for Project B
−250,000+65,000×1−(1+𝐼𝑅𝑅𝐵)−6
𝐼𝑅𝑅𝐵= 0
– Solving this equation gives 𝐼𝑅𝑅𝐵=16.29%
– Step 3: Compare IRRs Project B has a higher IRR (16.29% > 15.24%), so based
solely on the IRR criterion, Project B should be chosen. However, it’s important
to note that IRR has limitations and NPV should also be considered for a
comprehensive decision.
203. A company is evaluating a project that requires an initial investment of $1,000,000. The
project is expected to generate cash flows of $300,000 per year for the first 3 years,
$400,000 per year for the next 3 years, and $500,000 per year for the final 3 years. If the
company’s cost of capital is 15%, what is the project’s profitability index?
Solution:
– Step 1: Calculate the present value of cash flows
𝑃𝑉𝐶𝐹
=300,000×1−(1+0.15)−3
0.15 +400,000
(1+0.15)3×1−(1+0.15)−3
0.15 +500,000
(1+0.15)6×1−(1+0.15)−3
0.15
𝑃𝑉𝐶𝐹 =661,499.61+603,778.26+517,912.45 = $1,783,190.32
– Step 2: Calculate the Profitability Index
𝑃𝐼 =𝑃𝑉𝐶𝐹
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 =1,783,190.32
1,000,000 = 1.78
204. An annuity pays $5,000 at the end of each year for 20 years. What is the present value
of this annuity if the interest rate is 8% for the first 10 years and 6% for the last 10
years?
Solution:
– Step 1: Calculate PV of the first 10 years
𝑃𝑉1= 5,000 ×1− (1+0.08)−10
0.08 = $33,927.87
– Step 2: Calculate PV of the last 10 years
𝑃𝑉2= 5,000×1−(1+0.06)−10
0.06 ×(1+0.08)−10 = $20,220.54
– Step 3: Sum the two present values
𝑃𝑉𝑡𝑜𝑡𝑎𝑙 =33,927.87+20,220.54 = $54,148.41
205. A bond with a face value of $1,000 pays a 5% coupon rate semi-annually and matures in
10 years. If the market interest rate is 6% (with semi-annual compounding), what is the
bond’s price? What is its duration?
Solution:
– Step 1: Calculate the bond price
𝑃𝑟𝑖𝑐𝑒 = 25×1−(1+0.03)−20
0.03 +1000
(1+0.03)20 =461.09+553.68 = $1,014.77
– Step 2: Calculate the bond’s duration
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 1
𝑃𝑟𝑖𝑐𝑒 ×∑ 𝑡 ×𝐶𝐹𝑡
(1+0.03)𝑡
20
𝑡=1
– where 𝐶𝐹𝑡 is the cash flow at time 𝑡 Calculating this (which is tedious by hand,
typically done with a financial calculator or spreadsheet) gives a duration of
approximately 8.11 years.
206. An investor wants to accumulate $1,000,000 for retirement in 30 years. They can earn
an 8% return on their investments. How much should they invest at the end of each year
to reach their goal?
Solution:
– Use the future value of an annuity formula and solve for the payment:
1,000,000 = 𝑃𝑀𝑇 ×(1+ 0.08)30 −1
0.08
– Solving for PMT:
𝑃𝑀𝑇 = 1,000,000×0.08
(1+0.08)30 −1 = $7,336.88
207. A company is considering replacing an old machine with a new one. The new machine
costs $500,000 and will save the company $100,000 per year in operating costs for 8
years. The old machine can be sold for $50,000 now. If the company’s cost of capital is
12%, should they replace the machine?
Solution:
– Step 1: Calculate the NPV of the replacement
𝑁𝑃𝑉 = −500,000+50,000+100,000×1−(1+0.12)−8
0.12
𝑁𝑃𝑉 = −450,000+457,289.55 = $7,289.55
– Step 2: Make a decision Since the NPV is positive, the company should replace
the machine.
208. An oil well is expected to produce 10,000 barrels of oil per year for the next 15 years. If
the current price of oil is $50 per barrel and is expected to grow at 3% per year, what is
the present value of this oil well if the discount rate is 10%?
Solution:
– Step 1: Calculate the present value of each year’s production
𝑃𝑉 = ∑10,000×50×(1.03)𝑡
(1.10)𝑡
15
𝑡=1
– Step 2: Use the growing annuity formula
𝑃𝑉 =500,000×1−(1.03
1.10)15
0.10−0.03 = $5,078,649.42
209. A lottery winner has the option of receiving $1,000,000 now or 20 annual payments of
$75,000. If the interest rate is 5%, which option should they choose?
Solution:
– Step 1: Calculate the present value of the annuity option
𝑃𝑉𝑎𝑛𝑛𝑢𝑖𝑡𝑦 =75,000×1−(1+ 0.05)−20
0.05 = $935,604.62
– Step 2: Compare with the lump sum option Since $1,000,000 > $935,604.62, the
winner should choose the lump sum option.
210. A perpetual bond pays $100 at the end of each year forever. If the market interest rate is
8%, what is the value of this bond? If interest rates suddenly drop to 6%, what is the
percentage change in the bond’s value?
Solution:
– Step 1: Calculate the initial bond value
𝑉1=100
0.08 = $1,250
– Step 2: Calculate the new bond value
𝑉2=100
0.06 = $1,666.67
– Step 3: Calculate the percentage change
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒𝐶ℎ𝑎𝑛𝑔𝑒 = 1,666.67−1,250
1,250 ×100 =33.33%
