Bond Valuation

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Lesson Plan•What’s A Bond?•Types of Bonds•Bond Valuation Techniques•The Bangladeshi Bond Market•Problem Set

•A bond is a long-term debt instrument issued by a corporation or a government. It is a fixed-income security. •A bond is a formal contract to repay borrowed money with interest at fixed intervals. •A bond provides the borrower with external funds to:

– finance long term investments (for corporations)– finance current expenditures (for municipal, state or national governments).

What’s a bond?

Difference between stocks and bonds

Shares of stock BondEquity stake in the firmCreditor stake in the firm[i.e. owners] [.i.e. lenders]

Undefined term Defined term (maturity)(outstanding indefinitely)Types of bonds

• A non-zero coupon-paying bond is a coupon-paying bond with a finite life. •A zero-coupon bond is a bond that pays no interest but sells at a deep discount from face value. • A perpetual bond is a bond that never matures. It has an infinite life.

Types of bonds• A fixed rate bond is a bond whose coupon rate remains constant throughout the life of the bond. •A floating rate bond is a bond with a variable coupon that is linked to a reference rate of interest, such as the LIBOR. [LIBOR + 20 b.p.]• An inflation-linked bond is a bond whose coupon payments and principal are indexed to inflation.

– Gilts in the UK– TIPS in the US

• A Municipal bond (or muni) is a bond issued by a municipality, city, state or their agencies and is usually tax exempt. •A lottery bond is a bond with coupon payments like fixed-rate bonds, but the issuer will redeem randomly selected individual bonds for a higher value than the bond’s face value. • An asset-backed security (e.g. CDOs, CMOs, MBSs) is a bond whose interest and principal payments are backed by underlying cash flows from other assets.

Coupon rate =Annual coupon payments (CP)

Maturity value (MV)

For example:Annual coupon payments = $80Face value = $1,000Coupon rate =

$80/$1,000 = 0.08 = 8%

The maturity value•The maturity value (MV) [or face value] of a bond is the stated value.

– In the case of a U.S. bond, the MV is usually $1,000. – In the case of a Bangladeshi prize bond, the MV is usually 100 Taka. Coupon rate

•The coupon rate [or coupon yield] of a bond is the stated rate of interest.

Sample problem #1

•If the annual coupon payments are $70 and the face value of a bond is $1,000, what is its coupon rate?

Annual coupon payments = $70Face value = $1,000Coupon rate = $70/$1,000

= 0.07 = 7%

Coupon rate =

Annual coupon payments (CP)

Maturity value (MV)

Sample problem #2

•If the annual coupon payments are $100 and the face value of a bond is $1,000, what is its coupon rate?Annual coupon

payments (CP)

Maturity value (MV)

Coupon rate=

Annual coupon payments = $100Face value = $1,000Coupon rate = $100/$1,000

= 0.10 = 10%

The discount rate•The discount rate [or capitalization rate] of a bond is dependent on the risk of the bond.

•The discount rate (kd) is composed of the risk-free

rate plus a premium for risk. Bond valuation•Bond value = PV of coupons + PV of MV•Bond value = PV annuity + PV of lump sum

Remember: as interest rates increase, the PVs decrease.So, as interest rates increase, bond prices decrease.

V = CP (PVIFA kd, nn) + MV (PVIF kd, nn)

A bond has a $1,000 face value and provides an 8% annual coupon for 3030 years. The appropriate discount rate is 10%. What is the value of the bond?

Bond valuationA bond has a $ 1,000 face value and provides an 8% annual coupon for 3030 years. The appropriate discount rate is 10%. What is the value of the bond?VV = CP (PVIFA 10%, 3030) +

MV (PVIF 10%, 3030) = $80 (9.427) + $1,000 (0.057) = $754.16 + 57.00 = $811.16$811.16

Sample problem #1A bond has a $ 1,000 face value and provides a 6% annual coupon for 2020 years. The appropriate discount rate is 6%. What is the value of the bond?VV = CP (PVIFA 6%, 2020) +

MV (PVIF 6%, 2020) = $60 (11.4699) + $1,000 (0.3118) = $688.19 + 311.80 = $999.99$999.99

A bond has a $ 1,000 face value and provides a 6% annual coupon for 1010 years. The appropriate discount rate is 10%. What is the value of the bond?VV = CP (PVIFA 10%, 1010) + MV (PVIF 10%, 1010)

= $60 (6.1446) + $1,000 (0.3855) = $368.68 + 385.50= $754.18$754.18

Sample problem #2

Semi-annual compounding

•Most bonds pay coupon payments twice a year. •They pay ½ of the annual coupon payments.

Adjustments needed for semi-annual compounding:

(1) Divide kkdd by 22

(2) Multiply nn by 22

(3) Divide CPCP by 22

Semi-annual compoundingA non-zero coupon bond adjusted for semi-annual compounding:

V = CP/2 (PVIFA kd/2, n*2n*2) + MV (PVIF kd/2, n*2n*2)

Sample problem #1

A bond has a $ 1,000 face value and provides an 8% semi-annual coupon for 1515 years. The appropriate discount rate is 10% (annual rate). What is the value of the coupon bond?VV = CP/2 (PVIFA 10%/2,

15*215*2) + MV (PVIF 10%/2, 15*215*2) = $40 (15.3725) + $1,000 (0.2314) = $614.90 + 231.40= $846.30$846.30

Sample problem #2A bond has a $ 1,000 face

value and provides a 6% semi-annual coupon for 1515 years. The appropriate discount rate is 6% (annual rate). What is the value of the coupon bond?VV = CP/2 (PVIFA 6%/2,

15*215*2) + MV (PVIF 6%/2, 15*215*2) = $30 (19.6004) + $1,000 (0.4120) = $588.01 + 412.00= $1,000.01$1,000.01

Zero-coupon bond valuation•A zero-coupon bond is a bond that pays no interest but sells at a deep discount from its face value.

V = MV (PVIF kd, nn)

Zero-coupon bond example

A bond has a $1,000 face value and a 3030-year life. The appropriate discount rate is 10%. What is the value of the zero-coupon bond? VV = MV (PVIF 10%,

3030) = $1,000

(0.057)= $57.00$57.00

A bond has a $1,000 face value and a 3030-year life. The appropriate discount rate is 6%. What is the value of the zero-coupon bond?

VV = MV (PVIF 6%, 3030) = $1,000 (0.1741)= $174.10$174.10

Zero-coupon bond example #2

Zero-coupon bond example #3A bond has a $1,000 face value and a 1010-year life. The appropriate discount rate is 6%. What is the value of the zero-coupon bond? VV = MV (PVIF 6%, 1010)

= $1,000 (0.5584)= $558.40$558.40Perpetual

bond valuation

•A perpetual bond is a bond that never matures. It has an infinite life.

V = CP / kd

Perpetual bond example

A bond has a $1,000 face value and provides an 8% coupon. The appropriate discount rate is 10%. What is the value of the perpetual bond?CP = $1,000 * (8%) = $80kd = 10% = 0.10VV = CP / kd

= $80 / 0.10 = $800$800

Perpetual bond example #2

A bond has a $1,000 face value and provides a 6% coupon. The appropriate discount rate is 20%. What is the value of the perpetual bond?

CP = $1,000 * (6%) = $60kd = 20% = 0.20VV = CP / kd

= $60 / 0.20 = $300$300

Perpetual bond example #3

A bond has a $1,000 face value and provides a 5% coupon. The appropriate discount rate is 8%. What is the value of the perpetual bond?

CP = $1,000 * (5%) = $50kd = 8% = 0.08VV = CP / kd

= $50 / 0.08 = $625$625

Yield to maturity (YTM)

•The yield to maturity (YTM) of a bond is the discount rate which returns the market price of the bond.

– YTM is often used to price a bond.

– Bond prices are often quoted in terms of YTM.

YTM calculations

•You want to determine the YTM for an issue of outstanding bonds at your firm.

•Your firm has an issue of 10% annual coupon bonds with 15 years 15 years left to maturity.

•The bonds have a current market value of $1,250.

What is the YTM?

YTM solution (9%)

$1,250 = $100 (PVIFA 9%, 15) +

$1,000 (PVIF 9%, 15)

$1,250 = $100 (8.061) + $1,000 (.275)

$1,250 = $806.10 + $275.00

= $1,081.10

[Rate is too high]

YTM solution (7%)

$1,250 = $100 (PVIFA 7%, 15) +

$1,000 (PVIF 7%, 15)

$1,250 = $100 (9.108) + $1,000 (.362)

$1,250 = $910.80 + $362.00

= $1,272.80

[Rate is too low]

YTM solution (interpolate)

.07 $1,273.02 IRR $1,250

$192.09 $1,081

X $23.02 $192

.07 $1,273.02 IRR $1,250

$192.09 $1,081

X $23.02 $192

.07 $1,273.02 YTMYTM $1,250$1,250

$192.09 $1,081

($23)(0.02) $192

X = X = .0024

YTMYTM = .07 + .0024 = .0724 or 7.24%7.24%

YTM problem #1

What is the YTM?

YTM problem #2

What is the YTM?

Premium vs. discount bond

•A bond selling for more than its face value is called a premium bond.

•A bond selling for less than its face value is called a discount bond.

Current vs. coupon yield

The current yield is the coupon payment (CP) as a percentage of the current bond price (V).

Current yield = CP / V

•The coupon yield of a bond is the stated rate of interest.

Coupon yield =Annual coupon payments (CP)

Maturity value (MV)

YTM, current & coupon yield

•When a bond sells at a discountYTM > current yield > coupon yield

•When a bond sells at a premiumCoupon yield > current yield > YTM

•When a bond sells at parYTM = current yield = coupon yield

The Bangladeshi bond market

•Bangladesh’s bond market is TINY.

•It accounts for only 12% of the GDP.

•In South Asian countries, bonds account for an average of 34% of the GDP.

•In USA, the bond market accounts for 21% of the GDP, in Korea 29%, in Japan 16% and for select OECD countries 16%.

•The bond market is dominated by short and long term government securities.

•The Bangladesh Bank sells 28-day, 91-day, 182-day and 364-day treasury bills and five, ten, fifteen and twenty year treasury bonds.

•The central bank holds auctions every week and any investor can bid for bonds through any bank.

•An auction committee fixes the coupon rate which is applicable throughout the instrument’s life.

•These government bonds pay interest or coupon payments every six months.

•Generally, these bonds are of 100,000 (or 1 Lakh) Taka denomination or their multiples.

•There is a secondary bond market; however, the trading activity is very limited.

•Government paper has been traded since 2005.

•Corporate bonds are far and few in between.

•The corporate bond market is highly under-developed and has only ten bonds.

•All corporate bonds are issued through private placement.

•Notable corporate bonds include the Islamic Bank Bangladesh Limited (IBBL) Mudaraba Perpetual bond. <- Perpetual bond

•GrameenPhone issued 540-day and 720-day bonds worth US $62 million via Citi. (coupon: 14.5%) <- Fixed rate bond.

Bond Valuation

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