Tổng hợp kiến thức môn đầu tư tài chính

THEORIES OF CHAPTER 2
1. Money market
- Financial Market include Money Market (short term) and Capital Market (long term)
- The money market = cash equivalent = cash for short: because it’s highly liquid, and
relatively low – risk debt instruments
- The money market is very short – term debt securities and highly marketable
2. Treasury bills
- Treasury bill (T – bill) issued at a discount from face value and returning the face
amount at maturity
- Maturities: 4, 13, 26, 52 weeks (1, 3, 6, 12 months)
- Characteristics:
o Can purchase from Treasury or on the secondary market
o Low transaction cost
o Little price risk
o The income earned on T – bills is taxable at the federal level, exempt (miễn)
from all state and local taxes
- Bid price: lower price, if you wanted to sell a bill to a dealer: giá bán
- Asked price: you have to pay to buy a bill: giá mua
Calculate: Bid price, Asked price, Yield to Maturity, the price yesterday by method:
bank – discount
1: Calculate discount rate of bid price and asked price
𝑑𝑎𝑦𝑠𝑡𝑜𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦
𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡𝑟𝑎𝑡𝑒 = 𝑎𝑠𝑘 ×
360
2: Calculate asked price and bid price (***Bid < Asked***)
𝐴𝑠𝑘𝑒𝑑𝑝𝑟𝑖𝑐𝑒 = 𝐹 × (1 − 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡𝑟𝑎𝑡𝑒)
3: Calculate asked yield
𝐹
365
𝐴𝑠𝑘𝑒𝑑𝑦𝑖𝑒𝑙𝑑 = >
− 1? ×

𝐴𝑠𝑘𝑒𝑑𝑝𝑟𝑖𝑐𝑒
𝑑𝑎𝑦𝑠𝑡𝑜𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦
4: Calculate the price yesterday:
𝑃F − 𝑃G
𝐶𝐻𝐺% =
× 100%
𝑃G
3. Certificates of Deposit: higher interest saving, cannot withdrawal at any time
4. Commercial paper:
- Commercial paper is short – term unsecured debt issued by large corporation
- CP is backed by a bank line of credit, which gives the borrower access to cash that can
be used if needed to pay off the paper at maturity
5. Bankers’ Acceptances:
- Bankers’ Acceptances is an order to a bank by a customer to pay a sum of money at a
future date
- It is very safe (low risk) because the bank accept to pay it if the customer cannot
6. Repos and reserves:
Repos
Reserves
Short – term
Short – term
The dealer sell and buy back with a higher The dealer buy from the investor and resell
price
with a higher price
7. LIBOR market: Premier short -term interest, reference rate for a wide range of
transactions
8. The bond market: Longer – term borrowing or debt instruments, fixed – income
capital market. These fomulas can result in fixed income (coupon rate)
9. Treasury Notes and Bonds:

- T – bills < 1 year
- T – notes from 1 year to 10 years
- T – bonds > 10 years
10. Federal Agency Debt: dự án này được quốc hội hỗ trợ do vốn tư nhân không đủ
11. Municipal bonds:
- Issued by state and local governments
- It is exempt from federal income taxation
Comparing bonds:
𝑟 × (1 − 𝑡) = 𝑟H
r : taxable bonds
rm: municipal bonds/ free – tax paid bonds
t: tax rate
Choose the higher to invest
12. Corporate bonds:
- Long – term debt issued by private corporations paying semiannual coupons and
returning the face value of the bond at maturity
- Callable bond: công ty được mua lại bất kỳ lúc nào với giá do người phát hành ấn định
- Convertible bonds: đối trái phiếu sang cổ phiếu với giá do người phát hành ấn định
13. Common stock:
- Represent owership shares in a corporation
- Have voting rights and may receive dividends
- 2 characteristics of common stock: residual claim and limited liability
o Residual claim: Khi phá sản công ty trả nợ theo thứ tự như sau:
Thuế à Lương à Nhà cung cấp à Trái phiếu à Chủ nợ à Common stock
o Limited liability: khi phá sản các cổ đông chỉ mất phần vốn góp
14. Preferred Stock:
- Fixed stream of income each year (dividend: preferred stock à common stock)
- Similar perpetuity: Trái phiếu không kỳ hạn
- Not give the holder voting power

EXERCISE
1. What are the key differences between common stock, preferred stock, and corporate
bonds?
- Common stock: have voting rights and may receive dividends if the company make a
profit. When the company goes bankrupt, stockholder are the last in line of all those who
have a claim on the assets and income of the corporation
- Preferred stock: not give the holder voting power and receive a fixed stream of income
each year
- Corporate bond: not give the holder voting power and recerive semiannual coupon and
the face value of the bond at maturity. When the company goes bankrupt, corporate bond
have a claim on the assets and income of the corporation among three of this
2. Why do most professionals consider the Wilshire 5000 a better index of the
performance of the broad stock market than the Dow Jones Industrial Average?
Because the Wilshire 5000 Index includes more than 5000 stocks, while the Dow Jones
Industrial Average includes 30 large companies. The Wilshire 5000 Index has more
company than the Dow Jones, so the Wilshire 5000 Index can be shown a trend of the market
exactly.
3. What features of money market securities distinguish them from other fixed – income
securities?
Money markets include short – term, highly liquid and relatively low – risk debt instruments


The bond market is composed of long-term borrowing or debt. Most of them promise a fixed
stream of income that is determined by a specified formula.
4. What are the major components of the money market?
The major components of the money market are Treasury bills, certificates of deposit,
commercial paper, bankers' acceptances, Eurodollars, repos, reserves, federal funds, and
brokers' calls.
5. Describe alternative ways that an investor may add positions in international equity to
his or her portfolio

American Depository Receipts, or ADRs, are certificates traded in U.S. markets that
represent ownership in shares of a foreign company. Investors may also purchase shares of
foreign companies on foreign exchanges. Lastly, investors may use international mutual
funds to own shares indirectly.
6. Why are high – tax – bracket investors more inclined to invest in municipal bonds than
are low – bracket investors?
Because municopal bonds except their interest income is exempt from federal income
taxation.
7. What is meant by the LIBOR rate? The Federal funds rate?
The LIBOR is the rate at which large banks in London are willing to lend money among
themselves.
The Fed funds rate is the rate of interest on very short-term loans among financial
institutions in the U.S.
8. How does a municipal revenue bond differ from a general obligation bond? Which
would you expect to have a lower yield to maturity?
General obligation bonds are backed by the “full faith and credit” of the issuer, while
revenue bonds are issued to finance particular projects and are backed either by the revenues
from that project or by the municipal agency operating the project
9. Why are corporations more apt to hold preferred stock than are other potential
investors?
Preferred stocks may be moẻ apt for cổprations because of its tax treatment. Because they
are treated as dividends rather than interest on debt, they are not tax – deductible expenses
for the firm. They make desirable fixed – income investments for some corporations
10. What is meant by limited liability?
Limited liability means that the most shareholders can lose in event of the failure of the
corporation is their original investment.
11. Which of the following correctly describes a repurchase agreement?
a. The sale of a security with a commitment to repurchase the same security at a specified
future date and a designated price
b. The sale of a security with a commitment to repurchase the same security at a future date

left unspecified, at a designated price
c. The purchase of a security with a commitment to purchase more of the same security at a
specified future date
12. Why are money market securities sometimes referred to as “cash equivalent”?
Money markets include short – term, highly liquid and relatively low – risk debt instruments
13. A municipal bond carries a coupon rate of 6¾% and is trading at par. What would be
the equivalent taxable yield of this bond to a taxpayer in a 35% tax bracket?
0% because municipal bond except their interest income is exempt from federal income
taxation
14. Suppose that short – term municipal bonds currently offer yields of 4%, while
comparable taxable bonds pay 5%. Which gives you the higher after – tax yield if your
tax bracket is:


a.
b.
c.
d.

Zero
10%
20%
30%

SOLVE
(
)
𝑟 1 − 𝑡 = 𝑟H ó 5% . (1 – t) = 4% ó t = 20%
15. An investor is in a 30% combined federal plus state tax bracket. If corporate bonds
offer 9% yields, what must municipals offer for the investor to prefer them to

corporate bonds?
𝑟(1 − 𝑡) < 𝑟H ó 9% (1 – 30%) < rm ó rm > 6.3%
So municipal bonds have the coupon rate more than 6.3% for the investor to prefer them to
corporate bonds
16. Find the equivalent taxable yield of the municipal bond in Problem 14 for tax brackets
of zero, 10%, 20% and 30%
t = 0%: 𝑟(1 − 𝑡) = 𝑟H ó 5% . (1 – 0%) = rm ó rm = 5%
t = 10%: 𝑟(1 − 𝑡) = 𝑟H ó 5% . (1 – 10%) = rm ó rm = 4.5%
t = 20%: 𝑟(1 − 𝑡) = 𝑟H ó 5% . (1 – 20%) = rm ó rm = 4%
t = 30%: 𝑟(1 − 𝑡) = 𝑟H ó 5% . (1 – 30%) = rm ó rm = 3.5%
17. Look at the Treasury bond maturing in November 2040

a. How much would you have to pay to purchase one of these bonds?
Asked price = 98.0000 x 1,000 = 980,000
b. What is its coupon rate?
4.25%
c. What is the current yield (i.e, coupon income as a fraction of bond price) of the bond?
4.371%
18. Look at the listing for General Dynamics

a. What was the firm’s closing price yesterday?
75.60
J,LLL
b. How many shares could you buy for $5,000?
= 66
MJ.OL.
c. What would be your annual dividend income from those shares?


The 1.88 value in DIV column means that the last quarterly dividend payment was $0.47

per share, which is consistent with annual dividend payments of $0.47 x 4 = $1.88
ð 66 x $1.88 = $124.08
d. What must be General Dynamics’ earnings per share?
P
P
MJ.OL
P/E = QPR à EPS = P/Q = TL.UV = 6.92
19. Consider the three stocks in the following table. Pt represents price at time t, and Qt
represents shares outstanding at time t. Stock C splits two – for – one in the last period

a. Calculate the rate of return on a price – weighted index of the three stocks for the first
period (t = 0 to t = 1)
95 + 45 + 110
− 1 = 4.17%
90 + 50 + 100
b. What must happen to the divisor for the price – weighted index in year 2?
UJ[\J[TTL
The index value before the stock plit was
= 83.33
]
We must find a new divisor, d, that leaves the index unchanged after C splits and its
price falls to $55. Therefore we solve for d in the following equation:
95 + 45 + 55
= 83.3 → 𝑑 = 2.34
𝑑
c. Calculate the rate of return of the price – weighted index for the second period (t = 1 to
t = 2)
95 × 100 + 45 × 200 + 110 × 200
− 1 = 0%
95 × 100 + 45 × 200 + 55 × 400

20. Using the data in the previous problem, calculate the first – period rates of return on
the following indexes of the three stocks:
a. A market value – weighted index
Price 0
Share 0
Price 1
Share 1
Inital
Final
Value
Value
A
$90
100
$95
100
$9,000
$9,500
B
$50
200
$45
200
$10,000
$9,000
C
$100
200
$110
200

$20,000
$22,000
$39,000
$40,500
\L,JLL
A market value weight index = 100 × ]U,LLL = 104
b. An equally weighted index
Price 0
Share 0

Price 1

Share 1

Return

Return/inital
value
5.56%
-10%
10%

A
$90
100
$95
100
$500
B
$50

200
$45
200
-$1,000
C
$100
200
$110
200
$2,000
J.JO%aTL%[TL%
A equally weighted index = 100 ×
= 185
]
21. What problems would confront a mutual fund trying to create an index fund tied to
an equally weighted index of a broad stock market?


In an equally – weighted index (EWI) fund, each stock is given equal weight regardless of
its matket capitalization. So smaller cap stocks will have the same weight as larger cap
stocks.
Contrast this with a market – weight index fund in which stocks are held in proportion to
market weight
- Sine more volatile small – cap stocks have the same weight as large – cap stocks, EWI
tend to be more volatile than market – capitalization indices
- Turnover rates tend to be higher than market – weight fund becasue as stocks that have
positive returns will increase weight in the fund. Stocks with negative returns will
decrease weight in the fund. Therefore an EWI fund must be rebalanced back to return
to its original target weights. Many of the transactions would be among the smaller, less
– liquid stocks

22. What would happen to the divisor of the Dow Jones Industrial Average if FedEx, with
a current price of around $95 per share, replaced AT&T (with a current value of about
$31 per share)?
The Dow Jones Industrial Average is calculated by adding the price of these 30 stocks and
dividing the sum by 30. The price of AT&T is higher than the price of FedEx, so if the
FedEX is replaced by AT&T, the divisor of DJ will increase because the divisor is weight
sensitive and is consists of the sum of all the 30 stocks.
23. A T – bill with face value $10,000 and 87 days to maturity is selling at a bank discount
ask yield of 3.4%. What is the price of the bill? What is its bond equivalent yield?
bM
Discount rateasked price = 3.4% x ]OL = 0.82%
Asked price = 10,000 x (1 – 0.82%) = 9,918
TL,LLL
]OJ
Equivalent yieldasked price = c U,UTb − 1d × bM = 3.48

24. Which security should sell at a greater price?
a. A 10 – year Treasury bond with a 9% coupon rate or a 10 – year T – bond with a 10%
coupon
𝐶↑
1
𝐹
𝑃 ↑=
>1 −
?+
f
(1 + 𝑟 )
(1 + 𝑟 )f
𝑟
So a 10 – year T – bond with a 10% coupon have payment higher than a 10 – year

Treasury bond with a 9% coupon. So a 10 – year T – bond with a 10% coupon sell at a
greater price
b. A three – month expiration call option with an exercise price of $40 or a three -month
call on the same stock with an exercise price of $35
Call option: The higher exercise price, the less cost of the contract. So a three month
call option with an exercise price of $35 sells at a greater price
c. A put option on a stock selling at $50 or a put option on another stock selling at $60
(All other relevant features of the stocks and options are assumed to be identical)
Put option: The higher exercise price, the higher cost of the contract. So a put option
with an exercise price of $60 sells at a greater price
25. Look at the furtures listings for corn:


a. Suppose you can buy one contract for December 2011 delivery. If the contract closes
in December at a price of $6.43 per bushel, what will be your profit or loss? (Each
contract calls for delivery of 5,000 bushels)
The exercise price is $6.37, while the market price is $6.43. So the investor will make
a profit = 5,000 x ($6.43 - $6.37) = $300
b. How many December 2011 maturity contracts are outstanding?
Open interest is the number of outstanding contracts: 487465
26. Look at the Apple options. Suppose you buy an August expiration call option with
exercise price $355

a. If the stock price in August is $367, will you exercise your call? What are the profit and
rate of return on your position?
The profit = $367 - $355 = $12 but the cost paid is $13.7 so it is loss, so I won’t exercise
my call and I lose the fee of this contract
If the fee is higher than ($13.7 – $12 = $1.7) $1.7, I will exercise my call
b. What if you had bought the August call with exercise price $360?
The profit = $367 - $360 = $7 but the cost paid is $11.15 so it is loss, so I won’t exercise

my call and I lose the fee of this contract
If the fee is higher than ($11.15 – $7 = $4.15) $4.15, I will exercise my call
c. What if you had bought an August put with exercise price $355?
The loss = $367 - $355 = $12, the market price is $367 while the exercise price of put
option $355. So I cancel this contract and pay a fee
If the fee is higher than ($12 + $11.1 = $23.1) $23.1, I will exercise my put
27. What options position is associated with:


a. The right to buy an asset at a specified price?
Call option
b. The right to sell an asset at a specified price?
Put option
c. The obligation to buy an asset at a specified price? Long postition (lệnh mua)
d. The obligation to sell an asset at a specified price? Short position (lệnh bán)
28. Why do call options with exercise prices higher than the price of the underlying stock
sell for positive prices?
Because the investor can make a profit from the difference between the stock price and the
exercise price
29. Both a call and a put currently are traded on stock XYZ; both have strike prices of
$50 and maturities of six months. What will be the profit to an investor who buys the
call for $4 in the following scenarios for stock prices in six months? (a) $40; (b) $45;
(c) $50; (d) $55; (e) $60. What will be the profit in each scenario to an investor who
buys the put for $6?
Market
Call option (Mua)
Put option (Bán)
price
(cost = $4, exercise price = $50)
(cost = $6 exercise price = $50)

$40
Loss = fee = $4
Profit = ($50 - $40 - $6) = $4

$45

Loss = fee = $4

Exercise put option: ($50 - $45 - $6)
= -1 à loss = $1
Not exercise put option: loss = fee =
$6
ð Exercise put option: loss = $1

$50

Loss = fee = $4

Loss = fee = $6

$55

Profit = ($55 - $50 - $4) = $1

Loss = fee = $6

$60

Profit = ($60 - $50 - $4) = $16

Loss = fee = $6

30. What would you expect to happen to the spread between yields on commercial paper
and treasury bills if the economy were to enter a steep recession?
The spread will widen because commercial paper is riskier, so the rate of return will
increase. Deterioration of the economy increases credit risk, that is, the likelihood of default.
Investors will demand a greater premium on debt securities subject to default risk
31. Examine the stocks listed in Figure 2.8. For how many of these stocks is the 52 – week
high price at least 50% greater than the 52 – week low price? What do you conclude
about the volatility of prices on individual stocks?

32. Find the after – tax return to a corporation that buys a share of preferred stock at $40,
sells it at year – end at $40, and receives a $4 year – end dividend. The firm is in the
30% tax bracket
The after – tax return = ($40 + $4 - $40) x 30% = $1.2
33. Explain the difference between a put option and a short position in a futures contract


A put option gives the holder the right (but not the obligation) to sell the underlying asset at
the exercise price
A short position in a futures contract carries an obligation to sell the underlying asset at the
futures price
34. Explain the difference between a call option and a long position in a futures contract
A call option gives the holder the right (but not the obligation) to buy the underlying asset
at the exercise price
A long position in a futures contract carries an obligation to buy the underlying asset at the
futures price
Concept check 2.1: What were the bid price, asked price, and yield to maturity of the
3.5% Febuary 2018 Treasury bond? What was its asked price the previous day?

Bid price = 107.2969 % x 1,000 = 1,073
Asked price = 107.3594 % x 1,000 = 1,073.6
Yield to maturity based on asked price = 2.294
P aP
T,LM].OaPh
CHG = g P h ó -0.547% =
à Po = 1,079.5
P
h

h

Concept check 2.2: Suppose your tax bracket is 28%. Would you prefer to earn a 6%
taxable return or a 4% tax – free paid? What is the equivalent taxable yield of the 4%
tax – free yield?
r x (1 – t) = 6% x (1 – 28%) = 4.32% > 4%. So I prefer to earn a 6% taxable.
𝑟 × (1 − 𝑡) = 𝑟H ⟺ 𝑟 × (1 − 28%) = 4% à r = 5.56%
Concept check 2.3:
a. If you buy 100 shares of IBM common stock, to what are you entitled? Have voting
rights and may receive dividends
b. What is the most money you can make over the next year? Your potential gain is
unlimited because IBM’s dtock price has no upper bound
c. If you pay $95 per share, what is the most money you could lose over the year? If
the company goes bankrupt, you will lose $95 per share.
Concept check 2.4: Suppose XYZ’s final price increases to $110, while ABC falls to $20.
Find the percentage change in the price – weighter average of these two stocks. Compare
that to the percentage return of a portfolio that holds one share in each company

Index:

Inital index value =

VJ[TLL
V

= 62.5


Final value =
Portfolio:

VL[TTL
V

= 65
OJ.aOV.J

Percentage change in index = OV.J = 4%
Inital value = ($25 + $100) = $125
Final value = ($20 + $110) = $130
T]LaTVJ
Percentage change in portfolio value = TVJ = 4%

à It is equal
Concept check 2.5: Reconsider companies XYZ and ABC from Concept Check Question
2.4. Calculate the percentage change in the market value – weighted index. Compare that
to the rate of return of a portfolio that holds $500 of ABC stock for every $100 of XYZ
(i.e., an index portfolio)
OUL
The percentage change in market value – weighted index = OLL = 115%

Concept check 2.6: What would be the profit or loss per share of stock to an investor who
bought the July 2011 expiration Apple call option with exercise price $355, if the stock
price at the expiration of the option is $365? What about a purchaser of the put option
with the same exercise price and expiration?

Investor who bought call option: The profit = $365 - $355 - $5.6 = $4.4 per share
The purchaser of the put option: they cancel the contract because the market price is higher
than the exercise price so they make a loss equally a fee = $0.9.
THEORIES OF CHAPTER 10
1. Bond characteristics:
- Bond: a security that obligates the issuer to make specified payments to the holder
over a period of time
- Characteristics: Face value, coupon rate, duration (time to maturity)
- Payment of the bond: principal (face value, par value), interest (coupon payment)
2. Callable bond:
- Bonds that may be repurchased by the issuer at a specified call price during the call
period
- When the market interest rate â à the company will call the bond (khi lãi suất thị
trường giảm à giá trái phiếu sẽ tăng à khi giá tăng = giá thu hồi đã ấn định trước
à thu hồi)
- Tại sao lại như vậy?


3.
4.
5.
6.

Ví dụ khi cơng ty vay ngân hàng với lãi suất thị trường là 8% trong khi đó phát
hành trái phiếu với lãi suất coupon là 10%. Khi lãi suất thị trường cịn 5%, thì cơng

ty sẽ thu hồi lại do phải trả lãi tận 10% trong khi đó trên thị trường chỉ có 5%
Convertible bond: a bond with an option allowing the bondholder to exchange the
bond for a specified number of shares of common stock in the firm
Put bond: a bond that the holder may choose either to exchange for par value at some
date or to extend for a given number of years
Premium > < discount bond:
- Premium bond: bonds selling above par value
- Discount bond: bonds selling below par value
Formula:
6.1.Bond pricing: (giá lúc phát hành/giá tại lúc trả coupon)
f

𝐵𝑜𝑛𝑑𝑣𝑎𝑙𝑢𝑒 = l
FmT

𝐶𝑜𝑢𝑝𝑜𝑛 𝑃𝑎𝑟𝑣𝑎𝑙𝑢𝑒 𝐶
1
𝑃𝑎𝑟𝑣𝑎𝑙𝑢𝑒
+
= >1 −
?+
F
f
f
(1 + 𝑟)
(1 + 𝑟)
𝑟
(1 + 𝑟)
(1 + 𝑟)f

= C x Annuity factor (r, T) + Par value x PV factor (r, T)
Nếu trái phiếu giữ đến ngày đáo hạn thì r = YTM
Nếu bond value = par value à coupon rate = r
- ráàPâ
- T càng lơn à P càng thấp
6.2.Accrued interest and quoted bond price: (giá mua tại 1 ngày bất kỳ)
𝐴𝑐𝑐𝑟𝑢𝑒𝑑𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 𝑠𝑒𝑚𝑖𝑎𝑛𝑛𝑢𝑎𝑙𝑐𝑜𝑢𝑝𝑜𝑛𝑝𝑎𝑦𝑚𝑒𝑛𝑡 ×

𝑑𝑎𝑦𝑠𝑠𝑖𝑛𝑐𝑒𝑙𝑎𝑠𝑡𝑐𝑜𝑢𝑝𝑜𝑛𝑝𝑎𝑦𝑚𝑒𝑛𝑡
𝑑𝑎𝑦𝑠𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑖𝑛𝑔𝑐𝑜𝑢𝑝𝑜𝑛𝑝𝑎𝑦𝑚𝑒𝑛𝑡

Quoted price = cơng thức ở trên
à Invoice price = Accrued interest + Quoted price
15/7 20/8
15/2
15/7

15/2

Accrued interest
Quoted price (t = 3) à Invoice = quoted + accrued
(Phần này trả cho người nắm giữ lúc trước)
- Par value chỉ được nhận lại cho người nắm giữ cuối cùng
- Khác với cổ phiếu, dividend của cổ phiếu không cần chia cho người trước như
accrued interest mà dividend người nắm giữ đến khi phát dividend sẽ lấy hết
6.3.The rate of return:
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 + 𝑝𝑟𝑖𝑐𝑒𝑎𝑝𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛
𝑁𝑜𝑚𝑖𝑛𝑎𝑙𝑟𝑒𝑡𝑢𝑟𝑛 =
𝐼𝑛𝑖𝑡𝑎𝑙𝑝𝑟𝑖𝑐𝑒
1 + 𝑁𝑜𝑚𝑖𝑛𝑎𝑙𝑟𝑒𝑡𝑢𝑟𝑛

𝑅𝑒𝑎𝑙𝑟𝑒𝑡𝑢𝑟𝑛 =
−1
1 + 𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛
Price appreciation: sự thay đổi giá (P1 – Po): imputed interest income
Inital price = Po
6.4.Current yield:
𝑎𝑛𝑛𝑢𝑎𝑙𝑐𝑜𝑢𝑝𝑜𝑛 𝑠𝑒𝑚𝑖𝑐𝑜𝑢𝑝𝑜𝑛 × 2
𝐶𝑢𝑟𝑟𝑒𝑛𝑡𝑦𝑖𝑒𝑙𝑑 =
=
𝑏𝑜𝑛𝑑𝑝𝑟𝑖𝑐𝑒
𝑏𝑜𝑛𝑑𝑝𝑟𝑖𝑐𝑒
6.5.Yield to call:
Giá thị trường tại thời điểm thu hồi
Giá thu hồi
𝑃=

𝐶
1
𝐶𝑎𝑙𝑙𝑝𝑟𝑖𝑐𝑒
>1 −
?+
f
𝑟
(1 + 𝑟)
(1 + 𝑟)f

Thời gian nắm giữ đến lúc thu hồi

Yield to call

6.6.Realized compound return: compound rate of return on a bond with all coupons
reinvestment until maturity
(Lấy tiền coupon đi đầu tư thêm với lãi suất = lãi suất thị trường)
Giá trị ban đầu (present value)
Realized compound return
𝑉G (1 + 𝑟)V = 𝑉V
V2 = reinvestment rate x coupon payment + the total value of year 2
6.7.Holding – period return: khoản thu nhập nhận được khi nắm giữ trái phiếu
𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 + (𝑃T − 𝑃G )
𝐻𝑃𝑅 =
𝑃G
6.8.Stated YTM and Expected YTM:
Công ty sắp phá sản
Bình thường
Khả năng nhận được khi đáo hạn
Expected YTM
Stated YTM
Coupon payment
Xx
Xx
Number of semiannual periods
Yy
Yy
Final payment
Change
Price
Zz
Zz
- Sự khác biệt nằm ở chỗ final payment (do công ty phá sản nên không có khả

năng trả nợ nỗi = par value mà chỉ nhận được …% trên par value)
6.9.Forward rate:
(1 + 𝑦u )u = (1 + 𝑦uaT )uaT × (1 + 𝑓)
EXERCISE
1. Define the following types of bonds:
a. Catastrophe bond: A bond issued by Oriental Land Corporation, which
manages Tokyo Disneyland, with a final payment that depends on whether
there has been an earthquake near the park
b. Eurobond: Are denominated in one currency, usually that if the issuer, but sold
in other national markets
c. Zero – coupon bond: A bond paying no coupons that sells at a discount and
provides only a payment of par value at maturity
d. Samurai bond: Yen – denominated bonds sold in Japan by non – Japanese
issuers
e. Junk bond: A bond rated BB or lower by Standard & Poor’s, or Ba or lower by
Moody’s, or an unrated bond
f. Convertible bond: A bond with an option allowing the bondholder to exchange
the bond for a specified number of shares of common stock in the firm
g. Serial bond: Bonds issued with staggered maturity dates. As bonds mature
sequentially, the principal repayment burden for the firm is spread over time
h. Equipment obligation bond: A collateralized bond in which the collateral is
equipment owned by the firm. If the firm defaults on the bond, the bondholders
would receive the equipment
i. Original – issued discount bond: A bond issued at a discount to the face value
j. Indexed bond: A bond make a payments that are tied to a general price index
or the price of a particular commondity
2. What is the option embedded in a callable bond? A puttable bond?


-

Callable bond: repurchased by the issuer at a specified call price during the call
period
- Puttable bond: to exchange for par value at some date or to extend for a given
number of years
3. What would be the likely effect on the yield to maturity of a bond resulting from:
a. An increase in the issuing firm’s times – interest – earned ratio? â default risk,
á bond price, â YTM
b. An increase in the issuing firm’s debt – equity ratio? á default risk, â bond
price, á YTM
c. An increase in the issuing firm’s quick ratio? â default risk, á bond price, â
YTM
4. A coupon bond paying semiannual interest is reported as having an ask price of
117% of its $1,000 par value. If the last interest payment was made one month
ago and the coupon rate is 6%, what is the invoice price of the bond?
vwxyyzu{|}wyF{G~ãGuãwxH|uF
Accrued interest = Semiannual coupon payment ì vwxyy|ã|wFzuã{G~ãGuãwxH|uFy
]L

= 6% × $1,000 × TbV = $9.89
Invoice price = quoted price + accrued interest = 117% x $1,000 + $9.89 = $1179.89
5. A zero – coupon bond with face value $1,000 and maturity of five years sells for
$746.22. What is its yield to maturity? What will happen to its yield to maturity
if its price falls immeadiately to $730?
𝐹𝑉
1000
𝑃=
⟺ 746.22 =
⟺ 𝑌𝑇𝑀 = 6.03%
F

(1 + 𝑌𝑇𝑀)J
(1 + 𝑌𝑇𝑀)
𝐹𝑉
1000
𝑃† =
⟺ 730 =
⟺ 𝑌𝑇𝑀† = 6.50%
†
F
(1 + 𝑌𝑇𝑀 )
(1 + 𝑌𝑇𝑀† )J
à So the price falls, the YTM increase
6. Why do bond prices go down when interest rates go up? Don’t investors like high
interest rates?
‡ˆ
𝑃 = (T[‰fŠ)g : The inverse relationship between price and interest rates of bonds is the
primary reason behind fall in prices of bonds due to rising interest rates
7. Two bonds have identical times to maturity and coupon rates. One is callable at
105, the other at 110. Which should have the higher yield to maturity? Why?
The bond callable at 105 should sell at a lower price because the call provision is more
valuable to the issuing firm. Therefore, its yield to maturity should be higher
8. Consider a bond with a 10% coupon and with yield to maturity = 8%. If the
bond’s YTM remains constant, then in one year will the bond price be higher,
lower, or unchanged? Why?
Lower. The value of the bond will decrease in value when yield to maturity is lower
than the coupon rate
9. A bond with an annual coupon rate of 4.8% sells for $970. What is the bond’s
current yield?
𝑎𝑛𝑛𝑢𝑎𝑙𝑐𝑜𝑢𝑝𝑜𝑛 4.8% × 1,000
𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝑦𝑖𝑒𝑙𝑑 =

=
= 4.95%
𝑏𝑜𝑛𝑑𝑝𝑟𝑖𝑐𝑒
970
10. An investor believes that a bond may temporarily increase in credit risk. Which
of the following would be the most liquid method of exploiting this?
a. The purchase of a credit default swap
b. The sale of a credit default swap
c. The short sale of the bond


11. Which is the following most accurately describes the behavior of credit default
swaps?
a. When credit risk increases, swap premiums increase
b. When credit and interest rate risk increases, swap premiums increase
c. When credit risk increases, swap premiums increase, but when interest rate risk
increases, swap premiums decrease
12. You buy an eight – year bond that has 6% current yield and a 6% coupon (paid
annually). In one year, promised yields to maturity have risen to 7%. What is
your holding – period return?
PV = $60 x Annuity factor (6%,8) + $1,000 x PV factor (6%,8) = $1,000
In 1 year: P = $60 x Annuity factor (7%,7) + $1,000 x PV factor (7%,7) = $946
The loss in holding the bond = $946 - $1,000 = - $54
But you receive the coupon payment, so the return = - $54 + $60 = $6
$O
The holding – period return = $T,LLL = 0.6%
13. The started yield to maturity and realized compound yield to maturity of a
(default – fee) zero – coupon bond will always be equal. Why?
The realized compound yield is the yield to maturity on the bond supposing that
coupon payments are reinvested at the going interest rate when they are received. If a

zero-coupon bond does not provide any coupons that may be reinvested, then the
realized compound yield to maturity of zero coupon bonds will always be equal to the
stated yield to maturity on the bond. When the stated yield to maturity is unchanged
over the period, the rate of return on therealized compounded yield to maturity of a
zero-coupon bond will equal that yield. The bond must offer a rate of return
competitive with those available on others.
14. Which security has a higher effective annual interest rate?
a. A three – month T – bill with face value of $100,000 currently selling at
$97,645
3 months = ¼ year
100,000 \
>
? − 1 = 10%
97,645
b. A coupon bond selling at par and paying a 10% coupon semiannually
TL%
Semiannual coupon payment = V × 100,000 = 5,000
100,000 + 5,000 × 2 V
>
? − 1 = 21%
100,000
à The coupon bond has the higher effective annual interest rate
15. Tresury bonds paying an 8% coupon rate with semiannual payments currently
sell at par value. What coupon rate would they have to pay in order to sell at par
if they paid their coupons annually?
Treasury bonds have 8% on coupon rate with semiannual so the interest rate = 8% (sell
at par value)
8% coupon rate because they sell at par so the coupon rate = the interest rate = 8%
16. Consider a bond paying a coupon rate of 10% per year semiannually when the
market interest rate is only 4% per haft – year. The bond has three years until

maturity
a. Find the bond’s price today and six months from now after the next coupon is
paid
𝑃L =

𝐶
1
𝐹𝑉
5% × 1000
1
1000
>1 −
?+
=
>1 −
?+
= 1052.42
(1 + 𝑟)F
(1 + 𝑟)F
𝑟
4%
(1 + 4%)O
(1 + 4%)O


𝑃T =

𝐶
1
𝐹𝑉

5% × 1000
1
1000
>1 −
?+
=
>1 −
?+
= 1044.52
F
F
J
(1 + 𝑟)
(1 + 𝑟)
𝑟
4%
(1 + 4%)
(1 + 4%)J

b. What is the total rate of return on the bond
50 + (1044.52 − 1052.42)
𝑅𝑎𝑡𝑒𝑜𝑓𝑟𝑒𝑡𝑢𝑟𝑛 =
= 4%
1052.42
17. A 20 – year maturity bond with par value $1,000 makes semiannual coupon
payments at a coupon rate of 8%. Find the bond equivalent and effective annual
yield to maturity of the bond if the bond price is:
a. $950
𝑃=

𝐶
1
𝐹𝑉
4% × 1000
1
1000
>1 −
?+
⟺ 950 =
>1 −
?+
→ 𝑟 = 4.27%
(1 + 𝑟)F
(1 + 𝑟)F
(1 + 𝑟)\L
(1 + 𝑟)\L
𝑟
𝑟

So the bond equivalent = 4.27% x 2 = 8.54%
b. $1,000
Sell at the price = par value à the bond equivalent = 8%
c. $1,050
𝑃=

𝐶
1
𝐹𝑉
4% × 1000
1

1000
>1 −
?+
⟺ 1050 =
>1 −
?+
→ 𝑟 = 3.76%
(1 + 𝑟)F
(1 + 𝑟)F
(1 + 𝑟)\L
(1 + 𝑟)\L
𝑟
𝑟

So the bond equivalent = 4.27% x 2 = 8.54%
18. Redo the previous problem using the same data, but now assume that the bond
makes its coupon payment annually. Why are the yields you compute lower in
this case? The same but change semiannually to annually
19. Calculate both the real and nominal rates of return on the TIPS bond in the
second and third years

The second year:

𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 + 𝑃𝑟𝑖𝑐𝑒𝑎𝑝𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 42.02 + (1050.6 − 1020)
=
= 7.12%
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝑝𝑟𝑖𝑐𝑒
1020
1 + 𝑁𝑜𝑚𝑖𝑛𝑎𝑙𝑟𝑒𝑡𝑢𝑟𝑛
1 + 7.12%

𝑅𝑒𝑎𝑙𝑟𝑒𝑡𝑢𝑟𝑛 =
−1=
− 1 = 4%
1 + 𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛
1 + 3%

𝑁𝑜𝑚𝑖𝑛𝑎𝑙𝑟𝑒𝑡𝑢𝑟𝑛 =

The third year:

𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 + 𝑃𝑟𝑖𝑐𝑒𝑎𝑝𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 42.44 + (1061.11 − 1050.6)
=
= 5.04%
𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝑝𝑟𝑖𝑐𝑒
1050.6
1 + 𝑁𝑜𝑚𝑖𝑛𝑎𝑙𝑟𝑒𝑡𝑢𝑟𝑛
1 + 5.04%
𝑅𝑒𝑎𝑙𝑟𝑒𝑡𝑢𝑟𝑛 =
−1=
− 1 = 4%
1 + 𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛
1 + 1%

𝑁𝑜𝑚𝑖𝑛𝑎𝑙𝑟𝑒𝑡𝑢𝑟𝑛 =

20. Fill in the table below for the following zero – coupon bonds, all of which have
par values opf $1,000


𝐹𝑉

(1 + 𝑟 )F
Price
Maturity
YTM
400
20
4.69%
500
20
3.53%
500
10
7.18%
385.54
10
10%
463.19
10
8%
400
11.91
8%
21. A bond has a par value of $1,000, a time to maturity of 10 years, and a coupon
rate of 8% with interest paid annually. If the current market price is $800, what
will be the approximate capital gain yield of this bond over the next year if its
yield to maturity remains unchanged?
In the first year:
𝑃=

𝑃=

𝐶
1
𝐹𝑉
8% × 1000
1
1000
>1 −
?+
>1 −
?+
⟺ 800 =
→ 𝑟 = 11.46%
F
F
10
(1 + 𝑟)
(1 + 𝑟)
(1 + 𝑟)
(1 + 𝑟)10
𝑟
𝑟

In the next year:
𝑃=

𝐶
1
𝐹𝑉
8% × 1000

1
1000
>1 −
?+
>1 −
?+
=
= 811.70
(1 + 𝑟)F
(1 + 𝑟)F
(1 + 11.46%)9
(1 + 11.46%)9
𝑟
11.46%

8% × 1000 + (811.70 − 800)
= 11.46%
800
22. A bond with a coupon rate of 7% makes semiannual coupon payments on
January 15 and July 15 of each year. The Wall Street Journal reports the ask
price for the bond on January 30 at 100:02. What is the invoice price of the bond?
The coupon period has 182 days
100:02 means 100 2/32 percent of par = 100.0625 = $1,000.625
vwxyyzu{|}wyF{G~•Gu•wxH|uF
Accrued interest = Semiannual coupon payment ì vwxyy|ã|wFzuã{G~ãGuãwxH|uFy
=

TJ

= 3.5% x $1,000 × TbV = $2.885

à Invoice price = Quoted price + Accrued interest = $1,000.625 + $2.885 = $1,003.51
23. A bond has a current yield of 9% and a yield to maturity of 10%. Is the bond
selling above or below par value? Explain
If the yield to maturity is greater than the current yield, then the bond offers the
prospect of price appreciation as it approaches its maturity date. Therefore, the bond
must be selling below par value
24. Is the coupon rate of the bond in the previous problem more or less than 9%
The coupon rate is less than 9%. If coupon divided by price equals 9%, and price is
less than par, then price divided by par is less than 9%
25. Consider a bond with a settlement date of February 22, 2012, and a maturity date
of March 15, 2020. The coupon rate is 5.5%. If the yield to maturity of the bond
is 5.34% (bond equivalent yield, semiannual compouding), what is the list price
of the bond on the settlement date? What is the accrued interest on the bond?
What is the invoice price of the bond?
Price on settlement date:
𝑃=

𝐶
1
𝐹𝑉
5.5% × 1000
1
1000
>1 −
?+
>1 −
?+
=
= 1,016.93
F

F
16
(1 + 𝑟)
(1 + 𝑟)
(1 + 5.34%)
(1 + 5.34%)16
𝑟
5.34%

The last coupon payment is September 15, 2011; so days since last coupon payment to
the settlement date is 160
vwxyyzu{|}wyF{G~•Gu•wxH|uF
Accrued interest = Semiannual coupon payment × vwxyy|•|€wFzu•{G~•Gu•wxH|uFy


TOL

= 5.5% x 1000 × TbV = 48.35
Invoice price = P + Accrued inmterest = 1016.93 + 48.35 = $1,065.28
26. Now suppose the bond in the previous question is selling for 102. What is the
bond’s yield to maturity? What would the yield to maturity be at a price of 102 if
the bond paid its coupons only once per year?
The YTM for semiannually:
𝑃=

𝐶
1
𝐹𝑉
5.5% × 100
1

100
>1 −
?+
>1 −
?+
⟺ 102 =
→ 𝑟 = 5.31%
(1 + 𝑟)F
(1 + 𝑟)F
(1 + 𝑟)16
(1 + 𝑟)16
𝑟
𝑟

The YTM for annually:
𝑃=

𝐶
1
𝐹𝑉
5.5% × 2 × 100
1
100
>1 −
?+
>1 −
?+
⟺ 102 =
→ 𝑟 = 10.62%
F

F
8
(1 + 𝑟)
(1 + 𝑟)
(1 + 𝑟)
(1 + 𝑟)8
𝑟
𝑟

27. A 10 – year bond of a firm in severe financial distress has a coupon rate of 14%
and sells for $900. The firm is currently renegotiating the debt, and it appears
that the lenders will allow the firm to reduce coupon payments on the bond to one
– half the originally contracted amount. The firm can handle these lower
payments. What are the stated and expected yields to maturity of the bonds? The
bond makes its coupon payments annually
Expected YTM
Stated YTM
Coupon payment
$140
$140
Number of annually periods
10
10
Final payment
$500
$1,000
Price
$900
$900
YTM

13.16%
16.07%
28. A two-year bond with par value $1,000 making annual coupon payments of $100
is priced at $1,000. What is the yield to maturity of the bond? What will be the
realized compound yield to maturity if the one-year interest rate next year turns
out to be (a) 8%, (b) 10%, (c) 12%?
The bond sell at the price = the par value à YTM = the coupon rate = 10%
Năm 1 đem đi tái đầu tư với lãi suất 8%
(a) 8%: Vo (1 + r)2 = V1 ó 1000 (1 + r)2 = 1000 + 100 + 100 + 8% x 100 à r = 9.91%
Tiền coupon năm 2
(b) 10%: Vo (1 + r)2 = V1 ó 1000(1 + r)2 = 1000 + 100 + 100 + 10% x 100 à r = 10%
(c) 12%: Vo (1 + r)2 = V1 ó 1000(1 + r)2 = 1000 + 100 + 100 + 12% x 100 à r =
10.09%
29. Suppose that today’s date is April 15. A bond with a 10% coupon paid
semiannually every January 15 and July 15 is listed in The Wall Street Journal
as selling at an ask price of 101:04. If you buy the bond from a dealer today, what
price will you pay for it?
Quoted price = 101 4/32 percent of par = 101.125 = $1,011.25
vwxyyzu{|}wyF{G~•Gu•wxH|uF
Accrued interest = Semiannual coupon payment ì vwxyy|ã|wFzuã{G~ãGuãwxH|uFy
UT

= 5% x 1000 ì TbV = $25
Invoice price = Quoted price + Accrued interest = 1,011.25 + 25 = $1,036.25
30. Assume that two firms issued bonds with the following characteristics. Both
bonds are issued at par


Ignoring credit quality, identify four features of these issues that might account
for the lower coupon on the ABC debt. Explain

Factors that might make the ABC debt more attractive to investors, therefore justifying
a lower coupon rate and yield to maturity, are:
- The ABC debt is a larger issue and therefore may sell with greater liquidity.
- An option to extend the term from 10 years to 20 years is favorable if interest rates
ten years from now are lower than today’s interest rates. In contrast, if interest rates
increase, the investor can present the bond for payment and reinvest the money for
a higher return.
- In the event of trouble, the ABC debt is a more senior claim. It has more underlying
security in the form of a first claim against real property.
- The call feature on the XYZ bonds makes the ABC bonds relatively more attractive
since ABC bonds cannot be called from the investor.
- The XYZ bond has a sinking fund requiring XYZ to retire part of the issue each
year. Since most sinking funds give the firm the option to retire this amount at the
lower of par or market value, the sinking fund can be detrimental for bondholders.
31. A large corporation issued both fixed and floating – rate notes five years ago, with
terms given in the following table:

a. Why is the price range greater for the 9% coupon bond than the floating-rate note?
The floating rate note pays a coupon that adjusts to market levels. Therefore, it will
not experience dramatic price changes as market yields fluctuate. The fixed rate note
will therefore have a greater price range
b. What factors could explain why the floating-rate note is not always sold at par value?
- The yield spread between one-year Treasury bills and other money market
instruments of comparable maturity could be wider (or narrower) than when the
bond was issued.


-

The credit standing of the firm may have eroded (or improved) relative to Treasury

securities, which have no credit risk. Therefore, the 2% premium would become
insufficient to sustain the issue at par.
- The coupon increases are implemented with a lag, i.e., once every year. During a
period of changing interest rates, even this brief lag will be reflected in the price of
the security.
c. Why is the call price for the floating-rate note not of great importance to investors?
The risk of call is low. Because the bond will almost surely not sell for much above
par value (given its adjustable coupon rate), it is unlikely that the bond will ever be
called.
d. Is the probability of call for the fixed-rate note high or low?
The fixed-rate note currently sells at only 88% of the call price, so that yield to maturity
is greater than the coupon rate. Call risk is currently low, since yields would need to
fall substantially for the firm to use its option to call the bond.
e. If the firm were to issue a fixed-rate note with a 15-year maturity, callable after five
years at 106, what coupon rate would it need to offer to issue the bond at par value?
The 9% coupon notes currently have a remaining maturity of fifteen years and sell at
a yield to maturity of 9.9%. This is the coupon rate that would be needed for a newlyissued fifteen-year maturity bond to sell at par.
f. Why is an entry for yield to maturity for the floating-rate note not appropriate?
Because the floating rate note pays a variable stream of interest payments to maturity,
the effective maturity for comparative purposes with other debt securities is closer to
the next coupon reset date than the final maturity date. Therefore, yield-to-maturity is
an indeterminable calculation for a floating rate note, with “yield-to-recoupon date” a
more meaningful measure of return.
32. A 30-year maturity, 8% coupon bond paying coupons semiannually is callable in
five years at a call price of $1,100. The bond currently sells at a yield to maturity
of 7% (3.5% per half-year).
a. What is the yield to call?
The present value of this bond:
𝑃=

𝐶
1
𝐹𝑉
4% × 1000
1
1000
>1 −
?+
=
>1 −
?+
= 1,124.72
(1 + 𝑟)F
(1 + 𝑟)F
(1 + 3.5%)OL
(1 + 3.5%)OL
𝑟
3.5%

Yield to call:
𝑃=

𝐶
1
𝐶𝑎𝑙𝑙𝑝𝑟𝑖𝑐𝑒
4% × 1000
1
1100
>1 −
?+

⟺ 1,124.72 =
>1 −
?+
(1 + 𝑟)F
(1 + 𝑟)F
(1 + 𝑟)TL
(1 + 𝑟)TL
𝑟
𝑟
→ 𝑟 = 3.37%

b. What is the yield to call if the call price is only $1,050?
𝑃=

𝐶
1
𝐶𝑎𝑙𝑙𝑝𝑟𝑖𝑐𝑒
4% × 1000
1
1050
>1 −
?+
⟺ 1,124.72 =
>1 −
?+
F
F
TL
(1 + 𝑟)
(1 + 𝑟)

(1 + 𝑟)
(1 + 𝑟)TL
𝑟
𝑟
→ 𝑟 = 2.98%

c. What is the yield to call if the call price is $1,100 but the bond can be called in two
years instead of five years?
𝑃=

𝐶
1
𝐶𝑎𝑙𝑙𝑝𝑟𝑖𝑐𝑒
4% × 1000
1
1100
>1 −
?+
⟺ 1,124.72 =
>1 −
?+
F
F
\
(1 + 𝑟)
(1 + 𝑟)
(1 + 𝑟)
(1 + 𝑟)\
𝑟
𝑟

→ 𝑟 = 3.03%

33. A newly issued 20-year maturity, zero-coupon bond is issued with a yield to
maturity of 8% and face value $1,000. Find the imputed interest income in the
first, second, and last year of the bond’s life
The price when it’s issued:
𝐹𝑉
1000
𝑃=
=
= 214.55
F
(1 + 𝑟 )
(1 + 8%)VL
The price in the first year:


𝐹𝑉
1000
=
= 231.71
F
(1 + 𝑟 )
(1 + 8%)TU
à Imputed interest income in the first year = 231.71 – 214.55 = 17.16
The price in the second year:
𝐹𝑉
1000
𝑃=
=

= 250.25
(1 + 𝑟)F (1 + 8%)Tb
à Imputed interest income in the second year = 250.25 – 231.71 =18.54
The price in the 19th year:
𝐹𝑉
1000
𝑃=
=
= 925.93
F
(1 + 𝑟 )
(1 + 8%)T
à Imputed interest income in the last year = 1000 – 925.93 = 74.07
34. A newly issued 10-year maturity, 4% coupon bond making annual coupon
payments is sold to the public at a price of $800. What will be an investor’s taxable
income from the bond over the coming year? The bond will not be sold at the end
of the year. The bond is treated as an original-issue discount bond.
𝑃=

𝐶
1
𝐹𝑉
4% × 1000
1
1000
>1 −
?+
⟺ 800 =
>1 −
?+

→ 𝑟 = 6.82%
(1 + 𝑟)F
(1 + 𝑟)F
(1 + 𝑟)TL
(1 + 𝑟)TL
𝑟
𝑟
𝐶
1
𝐹𝑉
4% × 1000
1
1000
𝑃T = >1 −
?+
=
>1 −
?+
= 814.86
(1 + 𝑟)F
(1 + 𝑟)F
(1 + 6.82%)U
(1 + 6.82%)U
𝑟
6.82%

𝑃=

à taxable income = 4% x 1000 + (814.86 – 800) = $54.86
35. Masters Corp. issues two bonds with 20-year maturities. Both bonds are callable

at $1,050. The first bond is issued at a deep discount with a coupon rate of 4%
and a price of $580 to yield 8.4%. The second bond is issued at par value with a
coupon rate of 8.75%.
a. What is the yield to maturity of the par bond? Why is it higher than the yield of the
discount bond?
The YTM of the par bond: YTM = coupon rate = 8.75%
Because the coupon rate of the par bond is higher than the discount bond
b. If you expect rates to fall substantially in the next two years, which bond would you
prefer to hold?
The 4% bond offers a greater expected return
c. In what sense does the discount bond offer “implicit call protection”?
Implicit call protection is offered in the sense that any likely fall in yields would
not be nearly enough to make the firm consider calling the bond. In this case, the
call feature is almost irrelevant
36. Under the expectations hypothesis, if the yield curve is upward - sloping, the
market must expect an increase in short - term interest rates. True/ false/
uncertain? Why?
True. Becasue this analysis to infer the market’s expectation of future short – term
rates
37. The yield curve is upward - sloping. Can you conclude that investors expect short
- term interest rates to rise? Why or why not?
If the yield curve is upward sloping, you cannot conclude that investors expect short –
term interest rates to rise because the rising slope could be due to either exepectations
of future increases in rates or the demand of investors for a risk premium on long –
term bonds. In fact the yield curve can be upward sloping even in the absense of
expectations of future increases in rates
38. Assume you have a one-year investment horizon and are trying to choose among
three bonds. All have the same degree of default risk and mature in 10 years. The
first is a zero-coupon bond that pays $1,000 at maturity. The second has an 8%

coupon rate and pays the $80 coupon once per year. The third has a 10% coupon
rate and pays the $100 coupon once per year.
a. If all three bonds are now priced to yield 8% to maturity, what are their prices?
The zero – coupon bond:
𝑃=

𝐹𝑉
1000
=
= 463.19
(1 + 𝑟)F (1 + 8%)TL

The 8% coupon rate: P = par value = 1000
The 10% coupon rate:
𝑃=

𝐶
1
𝐹𝑉
100
1
1000
>1 −
?+
=
>1 −
?+
= 1134.2
(1 + 𝑟)F

(1 + 𝑟)F 8%
(1 + 8%)TL
(1 + 8%)TL
𝑟

b. If you expect their yields to maturity to be 8% at the beginning of next year,
what will their prices be then? What is your rate of return on each bond during
the one-year holding period?
The zero – coupon bond in the next year:
𝑃=

𝐹𝑉
1000
=
= 500.25
(1 + 𝑟)F (1 + 8%)U

JLL.VJa\O].TU

à Rate of return =
= 8%
\O].TU
The 8% coupon rate in the next year: P = par value = 1000
bL
à Rate of return = TLLL = 8%
The 10% coupon rate in the next year:
𝑃=

𝐶
1

𝐹𝑉
100
1
1000
>1 −
?+
=
>1 −
?+
= 1124.94
F
F
U
(1 + 𝑟)
(1 + 𝑟)
(1 + 8%)
(1 + 8%)U
𝑟
8%
TLL[(TTV\.U\aTT]\.V)

à Rate of return =
= 8%
TT]\.V
39. Under the liquidity preference theory, if inflation is expected to be falling over
the next few years, long-term interest rates will be higher than short-term rates.
True/ false/ uncertain? Why?
40. The current yield curve for default – free zero – coupon bonds is as follows:

a. What are the implied one-year forward rates?

(1 + 𝑦u )u = (1 + 𝑦uaT )uaT × (1 + 𝑓)
Maturity
YTM
Forward rate
1
10%
2
11%
12.01%
3
12%
14.03%
b. Assume that the pure expectations hypothesis of the term structure is correct. If
market expectations are accurate, what will the pure yield curve (that is, the
yields to maturity on one- and two-year zero-coupon bonds) be next year?
Maturity
Expected future Price
YTM
1000
1
12.01%
= 892.78
2

1 + 12.01%
1000
= 782.93
(1 + 12.01%) × (1 + 14.03%)

13.02%

c. If you purchase a two-year zero-coupon bond now, what is the expected total rate
of return over the next year? What if you purchase a three-year zero-coupon
bond? (Hint: Compute the current and expected future prices.) Ignore taxes.
Maturity
YTM
Price


1
2
3
bUV.Mb
2 year: bTT.OV − 1 = 10%

10%
11%
12%

909.09
811.62
711.78

MbV.U]

3 year: MTT.Mb − 1 = 10%
41. The yield to maturity on one-year zero-coupon bonds is 8%. The yield to maturity
on two-year zero-coupon bonds is 9%.
a. What is the forward rate of interest for the second year?
Total proceeds dollar invested 2 years will be: = $1+ (1 + 9%)2= $1.1881

Total proceeds dollar invested 1 year will be: = $1 x (1 + 8%) x (1 + f) = $1.2321
à f = 10%
b. If you believe in the expectations hypothesis, what is your best guess as to the
expected value of the short-term interest rate next year?
𝑃=

1000
= 909.09
1 + 10%

c. If you believe in the liquidity preference theory, is your best guess as to next
year’s short-term interest rate higher or lower than in (b)?
42. The following table contains spot rates and forward rates for three years.
However, the labels got mixed up. Can you identify which row of the interest rates
represents spot rates and which one the forward rates?

- Line 1: Spot rates
- Line 2: Forward rates
43. Consider the following $1,000 par value zero-coupon bonds:

According to the expectations hypothesis, what is the market’s expectation of the
one-year interest rate three years from now?
(1 + 𝑦u )u = (1 + 𝑦uaT )uaT × (1 + 𝑓)
Maturity
YTM
Forward rate
1
5%
2
6%

7.01%
3
6.5%
7.51%
4
7%
8.51%
44. A newly issued bond pays its coupons once a year. Its coupon rate is 5%, its
maturity is 20 years, and its yield to maturity is 8%.
a. Find the holding-period return for a one-year investment period if the bond is
selling at a yield to maturity of 7% by the end of the year.
The present value:
𝑃=

𝐶
1
𝐹𝑉
50
1
1000
>1 −
?+
=
>1 −
?+
= 705.46
(1 + 𝑟)F
(1 + 𝑟)F 8%
(1 + 8%)VL
(1 + 8%)VL

𝑟


The value after 1 year
𝑃=

𝐶
1
𝐹𝑉
50
1
1000
>1 −
?+
=
>1 −
?+
= 793.29
(1 + 𝑟)F
(1 + 𝑟)F 7%
(1 + 7%)TU
(1 + 7%)TU
𝑟

→ 𝐻𝑃𝑅 =

50 + (793.29 − 705.46)
= 19.54%
705.46

b. If you sell the bond after one year when its yield is 7%, what taxes will you owe
if the tax rate on interest income is 40% and the tax rate on capital gains income
is 30%? The bond is subject to original-issue discount (OID) tax treatment.
Using OID tax rules so the constant yield method are obtained à YTM = 8%:
Po = 705.46
P1 = 711.89
Interest in 1 year: = $50 + ($711.89 - $705.46) = $56.43
à Tax on interest income = 40% x 56.43 = $22.57
Capital gain in first year = Actual price at 7% YTM – constant yield price
= 793.29 – 711.89 = $81.40
à Tax on capital gain = 30% x 81.40 = $24.42
Total taxes = 22.57 + 24.42 = $46.99
c. What is the after-tax holding-period return on the bond?
𝐴𝑓𝑡𝑒𝑟𝑡𝑎𝑥𝐻𝑃𝑅 =

50 + (793.29 − 705.46) − 46.99
= 12.88%
705.46

d. Find the realized compound yield before taxes for a two-year holding period,
assuming that (i) you sell the bond after two years, (ii) the bond yield is 7% at the
end of the second year, and (iii) the coupon can be reinvested for one year at a
3% interest rate.
The value after 2 year
𝑃=

𝐶
1
𝐹𝑉
50

1
1000
>1 −
?+
=
>1 −
?+
= 798.82
(1 + 𝑟)F
(1 + 𝑟)F 7%
(1 + 7%)Tb
(1 + 7%)Tb
𝑟

The coupon can be reinvested for one year at a 3% interest rate: $50 x 3% = $1.5
Total income after 2 year = Value after 2 year + 2 x coupon payment + reinvested
= 798.82 + 2 x 50 + 1.5 = 900.32
Realized compound yield before taxes:
900.32 = 705.46 × (1 + 𝑟)V
à r = 12.97%
e. Use the tax rates in part (b) to compute the after-tax two-year realized compound
yield. Remember to take account of OID tax rules.
Using OID tax rules so the constant yield method are obtained à YTM = 8%:
Coupon interest received in first year:
$50.00
Less: tax on coupon interest 40%:
– 20.00
Less: tax on imputed interest (0.40*$6.43): – 2.57
Net cash flow in first year:
$27.43

The year-1 cash flow can be invested at an after-tax rate of: 3%x(1 - 40%) = 1.8%
By year 2, this investment will grow to: $27.43 × (1+ 1.8%) = $27.92
In two years, sell the bond for:
$798.82
Less: tax on imputed interest in second year:
– 2.78 [0.40 × $6.95]
Add: after-tax coupon interest received in year 2:
+ 30.00 [$50 × (1 – 0.40)]
Less: Capital gains tax on
– 23.99
(sales price – constant yield value):
[0.30 × (798.82 – 718.84)]
Add: CF from first year's coupon (reinvested):
+ 27.92 [from above]
Total
$829.97
THEORIES OF CHAPTER 13: EQUITY VALUATION


1. Book value: The net worth of common equity according to a firm’s balance sheet (giá trị
sổ sách à lúc phát hành à giá trị đầu tiên ghi vào sổ sách kế toán) (Giá trị giao dịch
market value à giá đang giao dịch)
2. Limitations of Book Value:
- Liquidation value: net amount that can be realized by selling the assets of a firm and
paying off the debt
- Replacement cost (chi phí bán tài sản của công ty): cost to replace a firm’s assets
Hw€Ž|F•w}~|
- Tobin’s q = €|•}w{|H|uF{GyF
3. Analyse a financial highlight:

Số lượng CP đang lưu hành
Giá trị vốn hoá thị trường = price per share x common shares outstanding
(giá trị có được do phát hành cổ phiếu ra thị trường)

𝐸𝑃𝑆 =

𝑁𝑒𝑡𝑖𝑛𝑐𝑜𝑚𝑒
𝐶𝑜𝑚𝑚𝑜𝑛𝑠ℎ𝑎𝑟𝑒𝑠𝑜𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔

Hầu như các tỷ lệ của Microsoft đều nhỏ
hơn ngành à CP đang được định giá thấp

4. Intrinsic value (giá trị thực/giá trị nội tại):
- CAPM:
Risk – return
Tỷ suất sinh lời của GTCG free – risk (T – bill)

Risk

𝑘 = 𝑟 = 𝑟• + 𝛽[𝐸(𝑟H ) − 𝑟• ]
Exected rate of return on the market portfolio
Market risk premium
(Lợi nhuận kỳ vọng của cổ phiếu này)
Required return: lợi nhuận yêu cầu: k, r
Expected return: E(rm)
- Tỷ suất sinh lời kỳ vọng:
Dividend kỳ vọng
Capital gain = P1 – Po
𝐸 (𝐷T ) + [𝐸(𝑃T ) − 𝑃G ]
𝑃G

𝐸 (𝐷T ) 𝐸 (𝑃T ) − 𝑃G
=
+

𝑃G
𝑃G

𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑𝐻𝑃𝑅 = 𝐸 (𝑟) =

Expected dividend yield

Expected rate of price appreciation


ð If a stock is priced “correctly”, it will offer investors a “fair” return, that is, its
expected return will equal its required return. An underpriced stock will privide an
expected return greater than the required return (nhà đầu tư nên đầu tư vào
underpriced stock)
- Intrinsic value: the present value / of a firms expected future net cash flows / discounted
/ by the required rate of return (Giá trị hiện tại của dòng tiền kỳ vọng tương lai chiết
khẩu bởi tỷ suất sinh lời yêu cầu)
Không cố định (khác với Trái phiếu do coupon payment là cố định cịn cổ phiếu thì biến động)
𝐼𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐𝑣𝑎𝑙𝑢𝑒 = 𝑉G =

𝐸 (𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 ) + 𝐸(𝑚𝑎𝑟𝑘𝑒𝑡𝑝𝑟𝑖𝑐𝑒)
1+𝑘

K: market capitalization rate (trường hợp 1 năm)
o Nếu Vo > Po: cổ phiếu đang được định giá thấp à NĐT nên mua
o Nếu Vo < Po: cổ phiếu đang được định giá cao à NĐT nên mua ít đi

5. Dividend discount models:
- 1 năm:
𝐷T + 𝑃T
𝑉G =
1+𝑘
- 2 năm:
𝐷T
𝐷V + 𝑃V
𝑉G =
+
1 + 𝑘 (1 + 𝑘 )V
- H năm:
𝐷T
𝐷V
𝐷— + 𝑃—
𝑉G =
+
+⋯+
V
(1 + 𝑘 )—
1 + 𝑘 (1 + 𝑘 )
- Dividend discount models: a formila for the intrinsic value of a firm euqal to the present
value of all expected future dividends (Giá trị nội tại = Toàn bộ các dòng tiền cổ tức kỳ
vọng trong tương lai chiết khấu về hiện tại)
𝐷T
𝐷V
𝐷]
𝑉G =
+
+

+⋯
1 + 𝑘 (1 + 𝑘 )V (1 + 𝑘 )]
6. The Constant – Growth DDM:
𝐷T = 𝐷G (1 + 𝑔)
𝐷V = 𝐷G (1 + 𝑔)V
𝐷] = 𝐷G (1 + 𝑔)]
𝐷G (1 + 𝑔) 𝐷G (1 + 𝑔)V 𝐷G (1 + 𝑔)]
𝐷T
𝑉G =
+
+
+⋯ =
V
]
(1 + 𝑘 )
(1 + 𝑘 )
1+𝑘
𝑘−𝑔
The constant – growth rate DDM implies that a stock’s value will be greater:
- The larger its expected dividend per share (D á à công ty làm dự án tốt à V á)
- The lower the market capitalization rate (K â à rủi ro cổ phiếu giảm à V á)
- The higher the expected growth rate of dividends (g á à dividend á à lợi nhuận tạo
ra nhiều à V á)
𝐷T
𝑃G =
𝑘−𝑔
Nếu cổ tức tăng trường là g thì P cũng tăng theo tốc độ tăng trưởng là g:
𝑃T = 𝑃G (1 + 𝑔)
Vậy:
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑𝐻𝑃𝑅 = 𝐸 (𝑟) = 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑦𝑖𝑒𝑙𝑑 + 𝐶𝑎𝑝𝑖𝑡𝑎𝑙𝑔𝑎𝑖𝑛

𝐷T 𝑃T − 𝑃G 𝐷T
=
+
=
+𝑔
𝑃G
𝑃G
𝑃G