Chiến lược làm bài tổng quát
-
Phân tích đề thuộc dạng nào
-
Áp dụng công thức chính xác
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Bấm máy tính nhanh và chính xác
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Lưu ý: Viết công thức ra nháp. Làm bài chỉ cần bước thế số và ra đáp án. Dùng dấu “,”
cho phần ngàn, dấu “.” cho phần thập phân. Làm tròn 2 số sau phần thập phân
-
Target: Làm 1 câu trong vòng 20 GIÂYYYYYY
CHAPTER 2 – HOW TO CALCULATE PRESENT VALUE?
A. LÝ THUYẾT:
1. Present Value is defined as future cash flows discounted to the present at an appropriate
discount rate
2. If the present value of the cash flow X is $240, and the present value cash flow Y $160, then
the present value of the combined cash flow is:
 $400
3. The rate of return is also called: I) discount rate; II) hurdle rate; III) opportunity cost of capital.
4. The present value formula for one period cash flow is: PV = C1/(1 + r)
5. An annuity is defined as equal cash flows at equal intervals of time for a specified period of
time
6. Example of annuity: If you receive $1,000 payment at the end each year for the next five years
7. You are considering investing in a retirement fund that requires you to deposit $5,000 per year,
and you want to know how much the fund will be worth when you retire. What financial
technique should you use to calculate this value?
 Future value of an annuity
8. According to the net present value rule, an investment in a project should be made if the Net
present value is positive.
9. NPV rule and the rate of return rule: Accept a project if its NPV > 0, Reject a project if the
NPV < 0, Accept a project if its rate of return > opportunity cost of capital
10. The net present value formula for one period is: NPV = C0 + [C1/(1 + r)]; NPV = PV required
investment.
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11. The managers of a firm can maximize stockholder wealth by:
 Taking all projects with positive NPVs
12. A perpetuity is defined as Equal cash flows at equal intervals of time forever. Example: consols
13. The concept of compound interest is most appropriately described as: Interest earned on
interest
14. The opportunity cost of capital for a risky project is the expected rate of return on a portfolio
of securities of similar risks as the project
B. Dạng bài “Single payment cashflow” – Chỉ có 𝑪𝟎 và 𝑪𝒏
Chến lược làm bài:
-
Nhận dạng câu hỏi có cụm “The present value + of + FUTURE VALUE”
-
Dùng công thức:
𝑃𝑉 =
𝑃𝑉 =
𝐶𝑛
(1 + 𝑟)𝑛
𝐶𝑛
𝑟 𝑛×𝑚
(1 + )
𝑚
1. The present value of $100 expected in two years from today at a discount rate of 6% is:
 $89.00
2. Present value of $121,000 expected to be received one year from today at an interest rate
(discount rate) of 10% per year is:
 $110,000
3. Present Value of $100,000 that is, expected, to be received at the end of one year at a discount
rate of 25% per year is:
 $80,000
4. Find the future value of an investment of $100,000 made today for five years and paying 8.75
percent for the following compounding periods:
a) Quarterly ($154,154.24)
b) Monthly ($154,637.37)
c) Daily ($154,874.91)
5. Jennifer is looking to invest some money, so that she has $5,500 at the end of three years.
Which investment should she make given the following choices:
a) 4.2% compounded daily ($4,848.92)
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b) 4.9% compounded monthly ($4,749.54)
c) 5.2% compounded quarterly ($4,710.31)
6. Alibaba is looking to invest some money, so that she has $5,500 at the end of three years.
Which investment should she make given the following choices:
a) 4.2% compounded daily ($4,848.92)
b) 4.9% compounded monthly ($4,749.54)
7. Find the future value of an investment of $2,500 made today for the following rates and
periods:
a) 6.25 percent compounded semiannually for 12 years ($5,232.09)
b) 7.63 percent compounded quarterly for 6 years ($3,934.48)
c) 8.9 percent compounded monthly for 10 years ($6,067.86)
d) 10 percent compounded daily for 3 years ($3,374.51)
8. Caroline Weslin needs to decide whether to accept a bonus of $1,820 today or wait two years
and receive $2,100 then. She can invest at 6 percent. What should she do?
 $2,044.95 < $2,100
9. Samantha is looking to invest some money, so that she has $5,500 at the end of three years.
Which investment should she make given the following choices:
a) 4.2% compounded daily ($4,848.92)
b) 4.9% compounded monthly ($4,749.54)
c) 5.2% compounded quarterly ($4,710.31)
d) 5.4% compounded annually ($4,697.22)
10. Anna saves $7,250 and she needs some money at the end of five years. Which investment
should she make given the following choices:
a) 6.25% compounded quarterly ($9,410.20)
b) 6.5% compounded semi-annually ($9,394.41)
11. You have won a prize of 52 million VNĐ to be paid exactly after 3 years. You was offered 45
million VNĐ today as a consideration for his agreement to sell the right to receive the prize.
The market interest rate is 12% and the interest is compounded on quarterly basis. Whether the
offer should be accepted or not?
 VNĐ 36,471,754 < VNĐ 45,000,000
C. Dạng bài “Multiple payment cashflows” – Có 𝑪𝟎, 𝑪𝟏, 𝑪𝟐, …., 𝑪𝒏
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Chiến lược làm bài:
-
Nhận dạng câu hỏi có following cash flows (dòng tiền cho từng năm)
-
Dùng công thức:
𝑃𝑉 =
𝐶1
𝐶𝑛
𝐶2
+ ⋯+
+
(1 + 𝑟 )𝑛
(1 + 𝑟)1 (1 + 𝑟)2
𝐹𝑉𝑛 = 𝐶0 × (1 + 𝑟)𝑛 + 𝐶1 × (1 + 𝑟)𝑛−1 + ⋯ + 𝐶𝑛−1 × (1 + 𝑟)
1. What is the present value of the following cash flow at a discount rate of 9%?
Year -1
Year - 2
Year - 3
$100,000
$150,000
$200,000
 $372,431.81
2. At an interest rate of 10%, which of the following cash flows should you prefer?
Option
Year 1
Year 2
Year 3
A
500
300
100
B
100
300
500
C
300
300
300
D
Any of the above as they all add up to $900
 Option A
3. What is the present value of the following cash flow at a discount rate of 16% APR?
Year
1
2
CF ($)
-100,000
300,000
 $136,741.97
4. A project is expected to produce cash flows of $5,000, $8,000, and $16,000 over the next three
years, respectively. After three years, the project will be discontinued. What is this project
worth today at a discount rate of 15 percent?
 $20,917.24
5. What is the future value of the following cash flow at the end of five years with the discount
rate of 9%?
Year
0
CF
$250,000
1
2
3
4
5
$120,000
$80,000
$50,000
$25,000
 $714,607
D. Dạng bài “Net present value”
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Chiến lược làm bài:
-
Nhận dạng câu hỏi có kêu tính NPV của dự án.
-
Vận dụng NPV rule: chấp nhận dự án khi NPV > 0, từ chối dự án khi NPV âm.
-
Dùng công thức:
𝑁𝑃𝑉 = 𝐶0 +
𝐶𝑛
𝐶2
𝐶1
+ ⋯+
+
2
1
(1 + 𝑟)
(1 + 𝑟)
(1 + 𝑟 )𝑛
(𝐶𝑜 < 0)
1. An initial investment of $400,000 will produce an end of year cash flow of $480,000. What is
the NPV of the project at a discount rate of 20%?
 $0
2. If the present value of a cash flow generated by an initial investment of $200,000 is $250,000.
What is the NPV of the project?
 $50,000
3. What is the net present value of the following cash flow at a discount rate of 11%?
Year
0
1
2
CF ($)
-120,000
300,000
-100,000
 $69,108.03
4. A project is expected to produce cash flows of $5,000, $8,000, and $16,000 over the next three
years, respectively. After three years, the project will be discontinued. What is the Net Present
Value of project at a discount rate of 15 percent if initial investment cost is $25,000?
 – $4,082.76
5. A project is expected to produce cash flows of $5,000, $8,000, and $16,000 over the next three
years, respectively. After three years, the project will be discontinued. What is the Net Present
Value of project at a discount rate of 15 percent if initial investment cost is $15,000?
 $5,917.24
6. FFF Company wants to purchase an equipment for $320,000. It will produce the following net
cash flows:
Year
Net Cash Flows
1
$130,000
2
280,000
3
110,000
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Assume that the cost of capital is 10%. What is the net present value?
 $112,231
7. What is the net present value (NPV) of the following cash flows at a discount rate of 9%?
Year
0
1
2
3
CF ($)
-250,000
100,000
150,000
200,000
 $122,431.81
8. A project requires an initial investment of $21,600 and will produce cash inflows of $4,900,
$14,200, and $8,700 over the next three years, respectively. What is the project's NPV at a
required return of 14 percent? The project will be accepted or not? Why?
 – $503.06
9. What is the net present value of the following cash flow at a discount rate of 12%?
Year
0
1
2
3
4
5
CF
($150,000)
$50,000
$30,000
$75,000
$10,000
($75,000)
 – 64,259.64
10. FFF Company wants to purchase an equipment for $420,000. It will produce the following net
cash flows:
Year
Net Cash Flows ($)
1
130,000
2
-60,000
3
280,000
4
210,000
Assume that the cost of capital is 10%. What is the net present value? Should FFF Company
accept this project? Why?
 $2,396
11. What is the net present value of the following cash flow at a discount rate of 12%?
Year
0
1
2
3
4
5
CF
-120,000
35,000
35,000
35,000
35,000
-10,000
 – $19,367.04
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E. Dạng bài “Annuity”
Chiến lược làm bài:
-
Nhận dạng câu hỏi có từ “annuity”
-
1 số ví dụ khác về annuity:
-
•
Sau khi về hưu, mỗi năm sẽ nhận 75,000$ trong vòng 25 năm.
•
Mỗi năm trả nợ với 1 khoản tiền bằng nhau, trong vòng 20 năm.
•
Mỗi năm đóng tiền học là 12,500$ trong vòng 6 năm.
•
Cho thuê xe trong vòng 5 năm, mỗi tháng nhận được $250.
•
Nhà máy hoạt động tạo ra dòng tiền đều $150,000 mỗi năm, trong vòng 8 năm.
Lưu ý: Dòng tiền đều bắt đầu phát sinh từ lúc nào. Nếu là cuối năm 1 thì dùng công thức
tính C0 bình thường. Nếu là đầu năm 1 (today) thì dùng công thức là đang tính C(-1),
rồi nhân lại cho (1+r) để ra C0.
-
Dùng công thức:
𝐶 0 = 𝐶1 × [
1 − (1 + 𝑟)−𝑛
]
𝑟
𝐶 𝑛 = 𝐶0 × [
(1 + 𝑟)𝑛 − 1
𝑟
]
1. What is the present value of $1000 per year annuity for five years at an interest rate of 12%?
 $3,604.78
2. What is the present value of $5000 per year annuity at a discount rate of 10% for 6 years?
 $16,760.78
3. What is the present value of $1000 per year annuity at a discount rate of 12% for 8 years?
 $ 4,967.64
4. After retirement, you expect to live for 25 years. You would like to have $75,000 income each
year. How much should you have saved in the retirement to receive this income, if the interest
is 9% per year (assume that the payments start one year after the retirement)?
 $736,693.47
5. After retirement, you expect to live for 25 years. You would like to have $75,000 income each
year. How much should you have saved in the retirement to receive this income, if the interest
is 9% per year (assume that the payments start on the day of retirement)?
 $802,995.88
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6. John House has taken a $250,000 mortgage on his house at an interest rate of 6% per year. If
the mortgage calls for twenty equal annual payments, what is the amount of each payment?
 $21,796.14
7. Mr. Hopper is expected to retire in 25 years and he wishes accumulate $750,000 in his
retirement fund by that time. If the interest rate is 10% per year, how much should Mr. Hopper
put into the retirement fund each year in order to achieve this goal? [Assume that the payments
are made at the end of each year]
 $7,626.05
8. Mr. Hopper is expected to retire in 30 years and he wishes accumulate $1,000,000 in his
retirement fund by that time. If the interest rate is 12% per year, how much should Mr. Hopper
put into the retirement fund each year in order to achieve this goal?
 $4,143.66
9. Ms. Colonial has just taken out a $150,000 mortgage at an interest rate of 6% per year. If the
mortgage calls for equal monthly payments for twenty years, what is the amount of each
payment? (Assume monthly compounding or discounting.)
 $1254.70
10. Mr. William expects to retire in 30 years and would like to accumulate $1 million in the pension
fund. If the annual interest rate is 12% per year, how much should Mr. Williams put into the
pension fund each month in order to achieve his goal? Assume that Mr. Williams will deposit
the same amount each month into his pension fund and also use monthly compounding.
 $286.13
11. You have to pay $12,500 a year in school fees at the end of each of the next six years. If the
interest rate is 8%, how much do you need to set aside today to cover these bills? You have
invested $70,476 at 8%. After paying the above school fees, how much would remain at the
end of the six years?
 $20,137.44
12. After retirement, you expect to live for 20 years. You would like to have $7,000 income each
year. How much should you have saved in the retirement to receive this income, if the interest
is 10% per year (assume that the payments end on the day of retirement)?
 $59,595
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13. You borrow $150,000 to buy a house. The mortgage rate is 8.5 percent and the loan period is
30 years. Payments are made monthly. If you pay for the house according to the loan
agreement, how much total interest will you pay?
 $265,213.28
14. The discount rate is 10 percent for five years, compounded quarterly. What is the difference in
the present value of these two sets of the following payments?
Case 1: you receive $200 on the first of each quarter.
Case 2: you receive $200 on the last day of each quarter.
 $77.95
15. You agree to lease a car for 5 years at $250 per month. You are not required to pay any money
up front or at the end of your agreement. If your opportunity cost of capital is 0.3% per month,
what is the cost of the lease?
 $13,708.72
16. A factory costs $750,000. You reckon that it will produce an inflow after operating costs of
$150,000 a year for 8 years. If the opportunity cost of capital is 14%, what is the net present
value of the factory? What will the factory be worth at the end of five years?
 NPV = $54,170.42
 PV(5) = $514,962.15
17. I order to save money for travelling to Europe, I have to deposit 30 million VNĐ at the end of
each year, during 3 years. Help me to calculate the total money I get in my account at the end
of year 3, if the interest rate is 8%, compounding semi-annually?
 198.99 million VNĐ
18. Khanh is planning for retirement. He expects to retire 28 years from now, at which time he
wishes to have accumulated $2,500,000 in his retirement fund (money at that time). If the
interest rate is 3% per year, how much should Khanh put into his retirement fund at the end of
each year in order to achieve his goal?
 $58,233.08
19. After retirement, you would like to have $80,000 income each year for 20 years. How much
should you have saved in the retirement to receive this income, if the interest is 3.0% per year
(Assume that the payments start one year after the retirement)
 $1,190,197.99
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F. Dạng bài “Perpetuity”
Chiến lược làm bài:
-
Nhận dạng câu hỏi có từ “perpetuity”
-
1 số ví dụ khác về perpetuity:
•
Sau khi về hưu, mỗi năm sẽ nhận 80,000$ cho đến vĩnh viễn.
•
Trong thực tế, trái phiếu consols của Anh, là có trả trái tức (coupon) đều cho
đến vĩnh viễn.
-
Lưu ý: Dòng tiền đều vĩnh viên bắt đầu phát sinh từ lúc nào. Nếu là cuối năm 1 thì dùng
công thức tính C0 bình thường. Nếu là đầu năm 1 (today) thì dùng công thức là đang
tính C(-1), rồi nhân lại cho (1+r) để ra C0. N ếu là cuối năm 4 thì dung công thức để tính
C3, rồi chiết khấu 3 năm về để tính C0.
-
Dùng công thức:
𝐶0 =
𝐶1
𝑟
1. What is the present value of $10,000 per year perpetuity at an interest rate of 10%?
 $100,000
2. You would like to have enough money saved to receive $80,000 per year perpetuity after
retirement so that you and your family can lead a good life. How much would you need to save
in your retirement fund to achieve this goal (assume that the perpetuity payments start one year
from the date of your retirement. The interest rate is 8%)?
 $1,000,000
3. You would like to have enough money saved to receive $100,000 per year perpetuity after
retirement so that you and your family can lead a good life. How much would you need to save
in your retirement fund to achieve this goal (assume that the perpetuity payments start one year
from the date of your retirement. The interest rate is 12.5%)?
 $800,000
4. You would like to have enough money saved to receive a $50,000 per year perpetuity after
retirement so that you and your family can lead a good life. How much would you need to save
in your retirement fund to achieve this goal (assume that the perpetuity payments start on the
day of retirement. The interest rate is 8%)?
 $675,000
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5. You would like to have enough money saved to receive an $80,000 per year perpetuity after
retirement so that you and your family can lead a good life. How much would you need to save
in your retirement fund to achieve this goal (assume that the perpetuity payments start on the
day of retirement. The interest rate is 10%)?
 $880,000
6. What is the present value of cash flows of $5,000 start at the end of year 4 in perpetuity?
Assuming a rate of return of 10%?
 $37,565
7. Calculate the present value of cash flows of $9,500 start at the end of year 5 in perpetuity.
Assuming a rate of return of 8.25%?
 $83,860.61
G. Dạng bài “Growth Annuity”
Chiến lược làm bài:
-
Nhận dạng câu hỏi có từ “growing annuity”
-
1 số ví dụ khác về growing annuity:
•
Sau khi về hưu, năm đầu nhận đc 60,000$, các năm sau mỗi năm số tiền nhận
được tăng trưởng với tốc độ g = 4%, nhận tiền như vậy trong vòng 25 năm.
-
Dùng công thức:
𝑛
𝐶
1+𝑔
) ]
𝐶 0 = 1 × [1 − (
1+𝑟
𝑟−𝑔
1. You would like to have enough money saved to receive a growing annuity for 25 years,
growing at a rate of 4% per year, the first payment being $60,000 after retirement, so that you
and your family can lead a good life. How much would you need to save in your retirement
fund to achieve this goal? (assume that the growing perpetuity payments start one year from
the date of your retirement. The interest rate is 12%)?
 $632,390
2. You would like to have enough money saved to receive a growing annuity for 20 years,
growing at a rate of 5% per year, the first payment being $50,000 after retirement. That way,
you hope that you and your family can lead a good life after retirement. How much would you
need to save in your retirement fund to achieve this goal. (assume that the growing annuity
payments start one year from the date of your retirement. The interest rate is 10%)?
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 $605,604.20
H. Dạng bài “Growth Perpetuity”
Chiến lược làm bài:
-
Nhận dạng câu hỏi có từ “perpetuity” và có cho growth rate (g%)
-
1 số ví dụ khác về growing perpetuity:
•
Cuối năm 1 nhận đc 1,000$, cuối các năm kế tiếp nhận được số tiền tang trưởng
với tốc độ tăng trưởng g%, cho đến vĩnh viễn.
•
Cuối năm 4 nhận được 5,000$, cuối năm thứ 5 cho đến vĩnh viễn, mỗi năm số
tiền nhận được tăng trưởng với tốc độ g%.
-
Lưu ý: Phải xác định được dòng tiền đầu tiên phát sinh tại thời điểm nào. Nếu tại thời
điểm cuối năm 1 thì dùng công thức tính C0 bình thường. Nếu là đầu năm 1 (today) thì
dùng công thức là đang tính C(-1), rồi nhân lại cho (1+r) để ra C0. Nếu là cuối năm 4 thì
dùng công thức để tính C3, rồi chiết khấu 3 năm về để tính C0.
-
Dùng công thức:
𝐶0 =
𝐶1
𝑟−𝑔
1. What is the present value of cash flows of $5,000 start at the end of year 4 in perpetuity,
assuming a rate of return of 10% and a constant growth rate of 4%?
 $62,609.57
2. Calculate the present value of cash flows of $9,500 start at the end of year 5 in perpetuity.
Assuming a rate of return of 8.25% and a constant growth rate of 3.5%?
 $145,652.64
I. Dạng bài “Discount Factor”
Chiến lược làm bài:
-
Nhận dạng câu hỏi kêu tính cái gì?
𝑛 − 𝑦𝑒𝑎𝑟 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 =
1
(1 + 𝑟 )𝑛
𝑃𝑉 = 𝐹𝑉 × (𝑛 − 𝑦𝑒𝑎𝑟 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑓𝑎𝑐𝑡𝑜𝑟)
1 − (1 + 𝑟)−𝑛
𝑟
(1 + 𝑟)𝑛 − 1
𝐹𝑢𝑡𝑢𝑟𝑒 𝑉𝑎𝑙𝑢𝑒 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 𝐹𝑎𝑐𝑡𝑜𝑟 =
𝑟
𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 𝐹𝑎𝑐𝑡𝑜𝑟 =
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1. If the interest rate is 12%, what is the 2-year discount factor?
 0.7972
2. One-year discount factor at a discount rate of 25% per year is:
 0.8
3. The one-year discount factor at an interest rate of 100% per year is:
 0.5
4. If the one-year discount factor is 0.8333, what is the discount rate (interest rate) per year?
 20%
5. If the present value of $480 to be paid at the end of one year is $400, what is the one-year
discount factor?
 0.8333
6. If the present value of $250 expected to be received one year from today is $200, what is the
discount rate?
 25%
7. If the one-year discount factor is 0.90, what is the present value of $120 to be received one
year from today?
 $108
8. If the present value of $600 expected to be received one year from today is $400, what is the
one-year discount rate?
 50%
9. What is the present value annuity factor at a discount rate of 11% for 8 years?
 5.1461
10. What is the present value annuity factor at an interest rate of 9% for 6 years?
 4.4859
11. If the present value annuity factor at 8% APR for 10 years is 6.71, what is the equivalent future
value annuity factor?
 14.487
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J. Dạng APR (Annual Percentage Rate)
Chiến lược làm bài:
-
Nhận dạng bài cho APR
-
Xem kêu tính FV bằng tính lãi suất đơn hay lãi suất kép (lãi chồng lại)
-
Nếu apply lãi suất đơn thì:
-
Nếu apply lãi suất kép thì:
𝐹𝑉𝑛 = 𝑃𝑉 × (1 + 𝐴𝑃𝑅 × 𝑛)
𝐹𝑉𝑛 = 𝑃𝑉 × (1 + 𝐴𝑃𝑅 )𝑛
1. If you invest $100 at 12% APR for three years, how much would you have at the end of 3 years
using simple interest?
 FV = 100 + (100 * 0.12 * 3) = $136
2. If you invest $100 at 12% APR for three years, how much would you have at the end of 3 years
using compound interest?
 FV = 100 * (1.12^3) = $140.49
K. Dạng EAR
Chiến lược làm bài:
-
Nhận dạng bài:
•
Cho EAR, kêu tính Nominal rate:
1
𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑟𝑎𝑡𝑒 = [(1 + 𝐸𝐴𝑅)𝑚 − 1] × 𝑚
•
Cho Nominal rate compounded m, kêu tính EAR:
𝐸𝐴𝑅 = (1 +
𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑟𝑎𝑡𝑒 𝑚
) −1
𝑚
•
Cho Nominal rate compounded continuously, kêu tính EAR:
•
Cho Nominal rate per m (VD: 1% / tháng), kêu tính APR: Nominal rate × m
𝐸𝐴𝑅 = 𝑒 𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑟𝑎𝑡𝑒 − 1
1. An investment at 10.47% effective rate compounded monthly is equal to a nominal (annual)
rate of:
 Nominal rate = [(1.1047)^(1/12) - 1] * 12 = 0.1 = 10%
2. An investment at 12% nominal rate compounded monthly is equal to an annual rate of:
 12.68%
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3. An investment at 10% nominal rate compounded continuously is equal to an equivalent annual
rate of:
 (e^(0.1)) - 1 = 0.10517 = 10.517%
4. Given a monthly rate of 1%, what is the Effective Annual Rate (EAR)? What is the Annual
Percentage Rate (APR)?
 EAR = (1+1%)12 – 1 = 12.68%
 APR = 1%*12 = 12%
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