Lecture 07-08 Fixed Income Valulation

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Fixed income valuation
Professor Sang Byung Seo
[email protected]

Last class
• Other investment criteria
• Internal rate of return (IRR)
• Payback period
• Profitability index
• How do CFOs make capital budgeting decision?

Today
• Fixed income valuation
• Problem set #2
• Due by Feb 10th (Wednesday) 11:59 PM CST

• Problem set #3
• Will be posted at 7 PM today on Blackboard

• Due by Feb 15th (Monday) 11:59 PM CST

First midterm
• Midterm #1
• Feb 18th (Thursday) in class
• Feb 16th (Tuesday) – exam review

Bond
• What is a bond?
• A bond is a certificate that shows that a borrower owes a specified
sum and the dates on which the interest and principal must be paid.

• For instance, you might have a bond with
• 5 years to maturity
• Face value 𝐹 = $1,000
• Coupon rate 5%
• Interest (coupon) paid annually

• Bonds are also called fixed Income Securities.

Coupon?
• Why are interest payments called coupons?
• Bonds were historically
issued in the form of bearer
bonds. (i.e. physical
possession of the certificate
was proof of ownership.)
• At the date the coupon was
due, the owner would detach
the coupon and present it for
payment.

Zero-coupon bonds

Zero coupon bond
• Zero coupon bond
• Promise single payment at maturity
• Called Face value or Par value

• Sometimes called “pure discount bond” or “zero”

• Example: ZCB
• Face value = $1,000
• r = 7%
• 5-year maturity  price 𝑃 =

$1,000
1+0.07 5

= $712.99

• 3-year maturity  price 𝑃 =

$1,000
1+0.07 3

= $816.30

Some remarks
• We valued this bond directly applying tools we used
earlier.
• However, bonds are not valued by us deciding the
price.
• Instead, buyers and sellers determine the price.
• Given this price, we inversely calculate the “discount rate.”

Some remarks
• Suppose that interest rates do not change
• Purchase a five-year zero in 2016 (matures in 2021)
• Price P = $712.99

• Two years later, in 2018, this bond will become a three-year bond.
• Price P = $816.30

• Three years later, in 2021, the bond will mature.
• Price P = F = $1,000

• As the maturity gets shorter, the price converges to the
face value. (pulled to par)

YTM of a ZCB
• We can ask: given a price (say $712.99)
• What 𝒓 makes this price correct?
• YTM = discount rate such that the sum of the bond’s discounted
payments equals the price

• Find 𝒓 such that
1 + 𝑌𝑇𝑀
𝐹
𝑌𝑇𝑀 =
𝑃

𝑇

𝐹
𝑃 =
1 + 𝑌𝑇𝑀

𝑇

Price and YTM are
inversely related!

𝐹
=
𝑃

1
𝑇

−1

Because the face value is known with certainty, YTM
is known with certainty today. P is the price “today.”

Example
• In the previous example,
• For the 5-year bond,
𝑌𝑇𝑀 =

1000
712.99

1
5

− 1 = 0.07

• For the 3-year bond,
𝑌𝑇𝑀 =

1000
816.30

1
3

− 1 = 0.07

Interpretation of YTM
• Bond investors frequently use the YTM as a way of
evaluating the investment.
• Is this practice okay?
• In other words, what is the rate of return we actually earn on a
bond, and how does it related to YTM?

What is HPR?
• Return on investment (or holding period return)
• Equivalently, the annual return on an investment
• If 𝑉0 = value at 0 and 𝑉𝑇 = value at T,
𝑉0 1 + 𝐻𝑃𝑅

𝑇

= 𝑉𝑇

Equivalently
𝑉𝑇
𝐻𝑃𝑅 =
𝑉0

1
𝑇

−1

• It measures how fast your initial investment grows.
• Compare the initial value vs. the final value

YTM vs HPR
• If a zero-coupon bond is held to maturity,
• 𝑉0 = 𝑃
• 𝑉𝑇 = 𝐹

𝐹
𝐻𝑃𝑅 =
𝑃

1
𝑇

− 1 = 𝑌𝑇𝑀

• That is, in case of a zero-coupon bond, YTM is the
same as HPR if the bond is held to maturity.
• If we sell the bond prior to maturity, they might be
different!
• Consider the case where future interest rates are different from
today!

Example
• Assume you have a 10-year ZCB.
• F = $1,000 and today’s price is $450.11.

• What is the YTM?

1000
𝑌𝑇𝑀 =
450.11

1
10

− 1 = 0.0831

• If the bond is held until maturity (for 10 years)
𝐻𝑃𝑅 = 𝑌𝑇𝑀 = 0.0831

Example (cont’d)
• Suppose you decide to sell the bond after one year!
• Interest rates may change by then!
• Investors may be more or less willing to buy bonds next year.

• Suppose first that the YTM falls to 8% next year.
• Investors are more willing to buy bonds.

• Higher price  low interest rate

• What is the holding period return?

Example (cont’d)
• Holding period return
𝑉1
𝐻𝑃𝑅 =
−1
𝑉0

• What is 𝑉1 ?
• After a year, the ten-year bond becomes a nine-year bond.

• YTM = 8%
$1,000
𝑉1 =
1 + 0.08

𝟗

= $500.25

500.25
𝐻𝑃𝑅 =
− 1 = 11.14%
450.11

• HPR = 11.14% > 8.31% (YTM at time 0)

Example (cont’d)
• What is the yield had risen to 8.6%?
• Investors are less willing to buy bonds.
• Higher price  low interest rate
$1,000
𝑉1 =
1 + 0.086

𝟗

= $475.92

475.92
𝐻𝑃𝑅 =
− 1 = 5.7%
450.11

• HPR = 5.7% < 8.31% (YTM at time 0)
• You earned less than the YTM because yields were higher when
you sold the bond than when you bought it.

Example (cont’d)
• What is the yield stayed the same?
$1,000
𝑉1 =
1 + 0.0831

𝟗

= $487.51

487.51
𝐻𝑃𝑅 =
− 1 = 8.31%
450.11

• HPR = 8.31% = 8.31% (YTM at time 0)
• When yields stay the same, you actually earn the YTM.

What can we conclude?
• Unlike YTM
• HPR is uncertain. (unless you hold it to maturity)
• When we buy a bond, we do not know what price we will be
able to sell it for next period.
• Government bonds are indeed risk-less?
• No! (unless it matures next period.)
• It is subject to interest risk! (no default risk)
• Bond cash flows are known with certainty but the HPR are not.

Coupon Bonds

Coupon bonds
• Coupon bond cash flows

T-1

𝑪
𝑪
𝑷=
+
𝟏+𝒓
𝟏+𝒓

𝑪
𝟐 +⋯+ 𝟏+𝒓

𝑪+𝑭
𝑻−𝟏 + 𝟏 + 𝒓 𝑻

• It looks like an annuity plus a lump sum payment!
• Every year -- a coupon payment ($C)
• At maturity -- face value ($F) in addition to a coupon ($C)

• Coupon rate = C/F

T

Example
• A coupon bond with
• $1,000 face value
• 5% annual coupon (coupon rate = 5% )
• 10-year maturity

• Each CPN payment: $1000 × 5% = $50
0

$50

$50

0

1

2

$50


3 …

$50

$1050

9

10

YTM of a coupon bond
• The same definition!
• YTM = discount rate such that the sum of the bond’s discounted
payments equals the price

• Suppose that the bond price (𝑃) is given. Then YTM is defined as:
𝐶
𝐶
𝑃=
+
1 + 𝒀𝑻𝑴
1 + 𝒀𝑻𝑴

𝐶
2 + ⋯ + 1 + 𝒀𝑻𝑴

𝐶+𝐹
𝑇−1 + 1 + 𝒀𝑻𝑴

• YTM is very similar to IRR!
• IRR: discount rate such that NPV = 0
• YTM: discount rate such that

Sum of discounted bond payoffs = P (price of the bond)

𝑇

YTM and bond price
• Result 1
• Bond sells at par (𝑃 = 𝐹) if the YTM is equal to the coupon
rate. In other words,
𝑷=𝑭

if

𝑪
𝒀𝑻𝑴 =
𝑭

Example
• Example
• Calculate the price of a 10-year maturity bond that pays annual
5% coupons. The face value is $1,000 and the YTM is 5%.
50
50
50
50
1000
𝑃=
+
+ ⋯+
+
+
2
9
10
1.05 1.05
1.05
1.05
1.0510

50 × 𝐴𝐹(10𝑌, 5%)
1
1
100
𝑃 = 50 ×

+
= 1000
10
10
0.05 0.05 1 + 0.05
1.05

386.09

613.91

What if YTM ≠ C/F
𝑌𝑇𝑀 < 𝐶/𝐹

𝑌𝑇𝑀 > 𝐶/𝐹

𝑃 > $1000

$1000

𝑃 < $1000

𝑌𝑇𝑀 = 𝐶/𝐹

YTM and bond price
• Result 2
• Bond sells at discount (𝑃 < 𝐹) if the YTM is larger than the
coupon rate. In other words,
𝑷<𝑭

if

𝑪
𝒀𝑻𝑴 >
𝑭

• Result 3
• Bond sells at premium (𝑃 > 𝐹) if the YTM is smaller than the
coupon rate. In other words,
𝑷>𝑭

if

𝑪
𝒀𝑻𝑴 <
𝑭

Example (revisited)
• Example
• Calculate the price of a 10-year maturity bond that pays annual
5% coupons. The face value is $1,000.
• YTM = 3%

50
50
50
50
1000
𝑃=
+
+ ⋯+
+
+
= $1,170.60
2
9
10
10
1.03 1.03
1.03
1.03
1.03
• YTM = 7%
50
50
50
50
1000
𝑃=
+
+ ⋯+
+
+
= $859.53
2
9
10
10
1.07 1.07
1.07
1.07
1.07

Intuition
• Bond payments are fixed.
• Interest rate (or YTM) > coupon rate
• You are not willing to pay $1,000 for these payments.
• Note that the price of a bond with the higher coupon rate (=YTM) is
$1,000. So this bond should be cheaper than $1,000.
• The bond sells at discount

• Interest rate (or YTM) < coupon rate
• You are willing to pay more than $1,000 for these payments.
• You can buy a bond with the higher coupon rate at par.
• Bonds sell at premium.

Etc.
• Why are zero-coupon bonds called pure discount
bonds?
• Coupon rate = 0% < YTM  𝑃 < 𝐹
• Thus, zero-coupon bonds always sell at discount:

• When do coupon bonds sell at par?
• When they are first issued!

• But as time goes by, market interest rates change!
• Higher interest rates  bonds sell at discount
• Lower interest rates  bonds sell at premium

HPR on a coupon bond
• Recall that for a ZCB that is held until maturity,
• The HPR is always equal to the YTM.

• Is this true for a coupon bond?
• If it is held until maturity, is the HPR always the same as the
YTM?
• NO!

• This is true only if intermediate coupon payments are reinvested at the YTM.

Example
• Consider the following bond:
• F = $1,000
• C = $80

• T=4
• YTM = 8%

• The price of this bond is $1,000.
• Why? No calculation is needed.

• How do we compute the holding period of return?

Example (cont’d)
• Note that
𝑉4
𝐻𝑃𝑅 =
𝑉0

1
4

−1

• 𝑉0 = 𝑃 = $1,000
• The value in 4 years (𝑉4 ) depends on the re-investment rate for
coupons.

𝑉4 = 1080 + ? ? ? ? ?
• For example, if the reinvestment rate is 0%,
𝑉4 = 1080 + 80 + 80 + 80

Example (cont’d)
• Suppose that we can re-invest the coupons at the YTM
(8%).

𝑉4
𝐻𝑃𝑅 =
𝑉0

1
4

$1360.49
−1=
$1000

1
4

− 1 = 8% = 𝑌𝑇𝑀

Example (cont’d)
• Suppose that we can coupons are re-invested at 6%.

𝑉4
𝐻𝑃𝑅 =
𝑉0

1
4

$1350
−1=
$1000

1
4

− 1 = 7.8% < 𝑌𝑇𝑀

HPR vs YTM: summary
• In general, HPR = YTM if
• We can re-invest coupons at the YTM.

• Even if the bond is held until maturity,
• The HPR can be different from the YTM if intermediate
coupons are not re-invested at the YTM.

• In case of a ZCB that is held until maturity,
• No intermediate coupons -- reinvestment rates are irrelevant so
always we obtain HPR = YTM

Selling bonds before maturity
• Suppose that we sold the coupon bond before maturity
(say, year 3). Assume we can re-invest at the YTM (8%).
• Time 1 coupon: $80



Time 3: $80 1.08

2

= $93.31

• Time 2 coupon: $80



Time 3: $80 1.08

1

= $86.40

• Time 3 coupon: $80



Time 3: $80

• Time 3 selling bond (at 8% YTM)
• 𝑃𝑟𝑖𝑐𝑒 𝑎𝑡 𝑦𝑒𝑎𝑟 3 =

1080
1.08

HPR = YTM if
(1) intermediate coupons are
reinvested at YTM
(2) YTM remains the same
when selling the bond

= $1000

93.31 + 86.40 + 80 + 1000
𝐻𝑃𝑅 =
1000

1
3

− 1 = 8% = 𝑌𝑇𝑀

Selling bonds before maturity
• Assume a 6% re-investment rate.
• Time 1 coupon: $80



Time 3: $80 1.06

2

= $89.89

• Time 2 coupon: $80



Time 3: $80 1.06

1

= $84.80

• Time 3 coupon: $80



Time 3: $80

• Time 3 selling bond (at 6% YTM)
• 𝑃𝑟𝑖𝑐𝑒 𝑎𝑡 𝑦𝑒𝑎𝑟 3 =

1080
1.06

= $1018.87

89.89 + 84.80 + 80 + 1018.87
𝐻𝑃𝑅 =
1000

1
3

− 1 = 8.4%

Semi-annual bond
• In the real world, bonds usually pay semi-annually or
quarterly.
• The US treasury (and often corporations) issue bonds that pay semiannually.

T

𝑪/𝟐
𝑪/𝟐
𝑷=
𝒓𝒂 +
𝒓𝒂
𝟏+
𝟏
+
𝟐
𝟐

𝑪/𝟐
𝑪/𝟐 + 𝑭
+⋯+
+
𝟐
𝟐𝑻−𝟏
𝒓𝒂
𝒓𝒂 𝟐𝑻
𝟏+
𝟏+
𝟐
𝟐

• Note that YTM (𝒓𝒂 ) is not an EAR. It is a SAIR (or APR).

Example
• A semi-annual bond with
• Coupon rate = 8%
• YTM = 10%
• T = 10
• F = $1000
40
40
40
40
1000
𝑃=
+
+ ⋯+
+
+
2
19
20
1.05 1.05
1.05
1.05
1.0520

Yield Curve

Treasury securities
• Treasury Bills
• Short-term bonds with maturities of no more than 1 year
• Zero-coupon bonds

• Treasury Notes
• Intermediate-term to long-term bonds
• Typically maturities of 2,3,5,7 or 10 years

• Semi-annual coupon bonds

• Treasury Bonds
• Longer than 10 years
• Typically issued with maturity of 30 years
• Semi-annual coupon bonds

The yield curve
• Different maturities of government bonds
 different YTMs for different maturities

• The yield curve is not always
upward sloping!
• Flat
• Downward sloping
• Hump-shaped
• Inverted hump-shaped

Zero curve
• Zero-coupon yield curve (in short zero curve)
• Special type of yield curve
• For each maturity, the curve represents the yield extracted from
the zero-coupon bond with that maturity.

• Notation
• YTM for a 1-year ZCB  𝑃1 =

• YTM for a 2-year ZCB  𝑃2 =
• YTM for a 3-year ZCB  𝑃3 =

𝐹
(1+𝑟1 )
𝐹
1+𝑟2 2
𝐹
1+𝑟3 3

or 𝑟1 =

or 𝑟2 =
or 𝑟3 =

𝐹
𝑃1
𝐹
𝑃2
𝐹
𝑃3

−1
1
2
1
3

−1
−1

Example
• Suppose that the ZCB prices are as follows:
• 1-year bond price 𝑃1 = $943.40
• 2-year bond price 𝑃2 = $873.44

• 3-year bond price 𝑃3 = $793.83

• Zero-coupon curve?
• 1-year zero-coupon bond yield 𝑟1 =

• 2-year zero-coupon bond yield 𝑟2 =
• 3-year zero-coupon bond yield 𝑟3 =

1000
943.40
1000
873.44
1000
793.83

1
1
1
2
1
3

− 1 = 6%

− 1 = 7%
− 1 = 8%

Single discount rate?
• When considering the time value of money,
• We assumed that there was a single interest rate that we could
use to discount cash flows across different periods.
𝑪𝟏
𝑪𝟐
𝑷=
+
𝟏+𝒓
𝟏+𝒓

𝑪𝑻−𝟏
𝑪𝑻
𝟐 + ⋯ + 𝟏 + 𝒓 𝑻−𝟏 + 𝟏 + 𝒓

𝑻

• Now that we are talking about bonds and the yield curve, it is a
good time to show how we can relax this assumption.

𝑪𝟏
𝑪𝟐
𝑷=
+
𝟏 + 𝒓𝟏
𝟏 + 𝒓𝟐

𝑪𝑻−𝟏
𝟐 +⋯+ 𝟏 +𝒓
𝑻−𝟏

𝑪𝑻
𝑻−𝟏 + 𝟏 + 𝒓
𝑻

where 𝒓𝟏 , 𝒓𝟏 , … , 𝒓𝑻−𝟏 , 𝒓𝑻 are ZCB yields.

𝑻

ZCB yields as discount rates
• Why?
• 𝑛-year ZCB yield is used for discounting a CF at year 𝑛.
𝐹
n−year ZCB price 𝑃𝑛 =
1 + 𝒓𝒏

𝑛

• Note that $𝐹 after 𝑛 year is equivalent to $𝑃𝑛 today.

• Discount factor
𝑪𝟏
𝑪𝟐
𝑷=
+
𝟏 + 𝒓𝟏
𝟏 + 𝒓𝟐

1
𝐷𝐹𝑛 =
1 + 𝒓𝒏
𝑪𝑻
+ ⋯+
𝟐
𝟏 + 𝒓𝑻

𝑻

𝑛

= 𝑪𝟏 𝑫𝑭𝟏 + 𝑪𝟐 𝑫𝑭𝟐 + ⋯ + 𝑪𝑻 𝑫𝑭𝑻

Example (revisted)
• We calculated the ZCB yields as the following:
• 𝑟1 = 6%, 𝑟2 = 7%, 𝑟3 = 8%
• Discount factors:
1
𝐷𝐹1 =
1 + 0.06

1

= 0.9434

1
𝐷𝐹2 =
1 + 0.07

2

= 0.8734

1
1 + 0.08

3

= 0.7938

𝐷𝐹3 =

• We use these discount factors to calculate the present
value of a cash flow stream.

Example (cont’d)
• What is the price of a 3-year coupon bond with $1000
face value and 5% coupon rate?
50
𝑃=
1 + 0.06

50
+
1
1 + 0.07

1050
+
2
1 + 0.08

3

= $924.37

• Equivalently,
𝑃 = 50 × 𝐷𝐹1 + 50 × 𝐷𝐹2 + 1050 × 𝐷𝐹3
= 50 0.9434 + 50 0.8734 + 1050 0.7938
= $924.37

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