Bond valuation (lecture 3)

Lecture 3. Bond Valuation

Definition of Bond
Terminology & Characteristics of Bonds
Bond Valuation
Premium Bonds vs Discount Bonds
Yield to Maturity (YTM)
Important factors in bond relationship
Exercise
Contents :

A type of debt (long-term promissory note) issued by the borrower, promising to pay fixed coupon (interest) payments at fixed intervals (6 months, 1 year etc ) and pay the par value at maturity.
Definition of Bonds…

Par value = $1,000
Coupon = 6.5% on par value per year,
or $65 per year ($32.50 every six months).
Maturity = 28 years (matures in 2048).
Issued by AT&T.
Example:

Bonds

Different Types of Bonds
Debentures: Unsecured long-term debt.
Subordinated debentures: Bonds that have a lower claim on assets in the event of liquidation than do other senior debt holders.
Mortgage bonds: Bonds secured by a lien on specific assets of the firm, such as real estate.
Eurobonds: Bonds issued in a country different from the one in whose currency the bond is denominated; for instance, a bond issued in Europe or Asia that pays interest and principal in U.S. dollars.
Zero and low coupon bonds: Allow the issuing firm to issue bonds at a substantial discount from their $1,000 face value with a zero or very low coupon.

Different Types of Bonds
Junk Bonds: bonds rated BB or below
Treasury Bonds: Bonds issued by the federal government, sometimes referred to as government bonds.
Corporate Bonds: Bonds issued by corporations.
Call Provision: A provision in a bond contract that gives the issuer the right to redeem the bonds under specified terms prior to the normal maturity date.
Convertible Bond: A bond that is exchangeable, at the option of the holder, for common stock of the issuing firm

Terminologies of bond

The Fundamental Valuation Model
This model can express the price of any asset at t = 0 mathematically.

Valuing Coupon Bonds
A non-zero coupon-paying bond is a coupon paying bond with a finite life.
(1 + kd)1
(1 + kd)2
(1 + kd)n
V =
+
+ ... +
C
C + MV
C
= Σ
n
t=1
(1 + kd)t
C
V = C (PVIFA kd, n) + MV (PVIF kd, n)
(1 + kd)n
+
MV

Valuing Zero Coupon Bonds
A zero coupon bond is a bond that pays no interest but sells at a deep discount from its face value; it provides compensation to investors in the form of price appreciation.
(1 + kd)n
V =
MV
= MV (PVIFkd, n)
Example. Bond Z has a $1,000 face value and a 30 year life. The appropriate discount rate is 10%. What is the value of the zero-coupon bond? 1000* 0.0573

Valuing Perpetual Bonds
A perpetual bond is a bond that never matures. It has an infinite life.
(1 + kd)1
(1 + kd)2
(1 + kd)∞
V =
+
+ ... +
C
C
C
= Σ
∞
t=1
(1 + kd)t
C
or C (PVIFA kd, ∞ )
V = C / kd [Reduced Form]
Example. Bond P has a $1,000 face value and provides an 8% annual coupon. The appropriate discount rate is 10%. What is the value of the perpetual bond?

Non-annual Compounding
Semiannually m=2
Quarterly m=4
Monthly m=12
Weekly m=52
Daily m=365

Semiannual Compounding
A non-zero coupon bond adjusted for semiannual compounding.
An example....
Value a T-Bond
Par value = $1,000
Maturity = 2 years
Coupon rate = 4%
k = 4.4% per year

Bond Premiums and Discounts
What happens to bond values if the required return is not equal to the coupon rate?
The bond's price will differ from its par value.

Yield to Maturity (YTM)/Expected Rate of Return
Also called Expected Rate of Return
Estimate of return investors earn if they buy
the bond at P0 and hold it until maturity
The YTM on a bond selling at par will always equal
the coupon rate.
YTM is the discount rate that equates the
PV of a bond’s cash flows with its price.

Determining the YTM: Interpolation
Julie Miller want to determine the YTM for an issue of outstanding bonds at Basket Wonders (BW). BW has an issue of 10% annual coupon bonds with 15 years left to maturity. The bonds have a current market value of $1,250 (face value = $1000).
What is the YTM?

Interpolation formula
V=$1,250 shows bond sells on premium (higher than par value 1,000)
Therefore, Coupon rate is higher than rate of return (YTM
So we try next highest rate after 10%
Try 9%
$1,250 = $100(PVIFA9%,15) + $1,000(PVIF9%, 15)
$1,250 = $806.07 + $274.50
= $1,080.57 [Rate is too high!]
Try 8%
$1,250 = $100(PVIFA8%,15) + $1,000(PVIF8%, 15)
$1,250 = $855.95 + $315.2
= $1,171.15 [Rate is still higher]
Value $1,171.15(k=8%) is closer to $1,250 than the value $1,080.57(k=9%), so we drop value $1,080.57(k=9%)

Solution
Try 7%
$1,250 = $100(PVIFA7%,15) + $1,000(PVIF7%, 15)
$1,250 = $910.79 + $362.4
= $1,273.19 [Rate is low!]
YTM = 7.22%
What is the meaning of V=$1250 at YTM=7.22%?

Decision making
With the given rate of return (7.22%), If the bond is sold at a price above the bond value ( $1250) it is known as overvalued and investor should not buy the bond, or can sell existing bonds if any.
With the given rate of return (7.22%), If the bond is sold at a price below the bond value($1250) it is known as undervalued and a wise decision is to invest in those bonds, or should not sell existing bonds if any.

Determining the YTM: Approximation Method
C = dollar amount of interest
MV = face value of bond
P = price of bond
n = number of years to maturity
In the previous example,

FIRST RELATIONSHIP
The value of the bond is inversely related to changes in the investor’s required rate of return (current interest rate) kb
If kd decrease , the value of the bond will increase
If kd increase , the value of the bond will decrease
FIVE IMPORTANT RELATIONSHIPS

FIVE IMPORTANT RELATIONSHIPS
SECOND RELATIONSHIP
The market value (Po) will be less than the par value (M) if the investor’s required rate of return (kb) is above the coupon rate (I), but it will be valued above the par value if the investor’s required rate of return (kb) is below the coupon rate (I),
If kd = C , then MV = Po (sold at par)
If kd > C , then MV > Po (discount bond)
If kd < C , then MV < Po (premium bond)

FIVE IMPORTANT RELATIONSHIPS
THIRD RELATIONSHIP
As the maturity approaches, the market value of the bond approaches its par value
FOURTH RELATIONSHIP
Change in price due to changes in interest rates
Long term bond have greater interest rate risk than do short –term bonds
Low coupon rate bonds have more price risk than high coupon rate bonds

Relationship between bond value & interest rate

FIVE IMPORTANT RELATIONSHIPS
FIFTH RELATIONSHIP
The sensitivity of a bond’s value to changing - depends on:
Length of time to maturity
The pattern of the cash flows provided by the bond

Exercise: YTM with semiannual coupon
Suppose a bond with a 10% coupon rate and semiannual coupons, has a face value of $1000, 20 years to maturity and is selling for $1197.93.
Is the YTM more or less than 10%?
What is the semiannual coupon payment?
How many periods are there?
Calculate the YTM.

Solution