1. Bond Valuation
Chapter 6
Miss Faith Moono Simwami

2. Bond Valuation Important Terms Types of Bonds Valuation of Bonds
Handling Semiannual
Compounding
Bond Price - Yield Relationship
Determining the interest rate on
a bond Yield to Maturity (YTM)
1 - 2

3. Important Bond Terms
A bond is a long-term debt instrument issued by a corporation or government.
The Maturity Value (MV) [ Face value] Par Value or Nominal Value: of a bond is the stated value. A bond has par value, also known as face value of the bond which represents the original amount borrowed by the issuer of the bond. (‘Original price’).
In the case of a US bond, the face value is usually $1,000.
Yield to Maturity: The rate of return that an investor would earn if he bought the bond at its current market price and held it until maturity. Alternatively, it represents the discount rate which equates the discounted value of a bond's future cash flows to its current market price.
Yield to Call: The rate of return that an investor would earn if he bought a callable bond at its current market price and held it until the call date, given that the bond was called on the call date.

4. Important Bond Terms
The bond’s coupon rate is the stated rate of interest; the (annual interest payment divided by the bond’s face value). This is the rate of interest that the bond issuer promises to pay on the bond. It is a percentage of the par value. E.g. K1000 par value Bond with interest rate of 10%.
Coupon Payment: This is the monetary value of interest that is paid on the bond at specified times or periods yearly or semi annually or quarterly etc. The coupon payment for a K1000 par value bond at 10% is K100. During the term of the bond only interest is paid, while the principle is repaid at maturity.
The discount rate is the rate at which future cash flow streams from a given asset are discounted to the present, hence deriving the price or value of the asset.

5. Important Bond Terms the face value plus one year's interest.
Maturity Period: this is time from when the bond is issued to the time it
matures. The maturity period of bonds varies from 5 year-10year and
more.
Maturity Date: A bond has a maturity date which is the date on which the par value is supposed to be repaid.
Call Provision: This allows the issuers of the bond to call back the bond
before its maturity date. Usually there is a premium for calling a bond
called a Call premium.
Call Price: The amount of money the issuer has to pay to call a callable
bond (there is a premium for calling the bond early). When a bond first
becomes callable, i.e., on the call date, the call price is often set to equal
the face value plus one year's interest.

6. Perpetual Bond Example
Different Types of Bonds
A perpetual bond is a bond that never matures. It has an infinite life.
Where:
I = the interest payment (Coupon)
Kd= market interest rate applicable on bonds
V = I / kd
Perpetual Bond Example
Bond P has a K1,000 face value and provides an 8% annual coupon. The appropriate discount rate is 10%. What is the value of the perpetual bond?
I = K1,000 ( 8%) = $80.
kd = 10%.
V = I / kd = $80 / 10% = $800.

7. Different Types of Bonds
A coupon-paying bond is a coupon paying bond with a finite life.
Vb = I (PVIFA kd, t ) + MV (PVIF kd, t )
Where :I = the interest payment (Coupon)
MV= maturity value of the bond at maturity
kd= market interest rate applicable on bonds
t = number of years until the bond matures.
This declines each year after the bond is issued.
Coupon Bond Example 1.
Bond C has a K1,000 face value and provides an 8% annual coupon for 30 years. The appropriate discount rate is 10%. What is the value of the coupon bond?
Write out the calculations for PVIFA and PVIF
V = K80 (PVIFA10%, 30) + K1,000 (PVIF10%, 30)
V = K80 (9.427) + K1,000 (.057)
V = K K = K

8. Coupon Bond Example
Bond Z has a Par value K1000, coupon rate 10%, maturity
10years, market rate for bonds 15%.
Find the value of the bond five years after it had been issued.
NOTE: The time take is the time remaining before the bond matures and not the original maturity of 10 years.
I=K1000 x 0.1= K100, t=5years , r=15%
Vb =100(PVIFA 15%, 5years) (PVIF 15%, 5years)
Vb = 100(3.3522) +1000(0.4972) =
=K832.42
coupon-paying bond -Valuation for interest compounded more than once per annum:
When compounding occurs more than once per year the formulas are modified as follows:
Where m = is the number of compounding periods.
Divide the interest rate kd by the number of periods of compounding (m) and multiply the time by the same number of compounding periods.

9. Example: Semiannual Compounding
Most bonds pay interest twice a year (1/2 of the annual coupon).
Adjustments needed:
A coupon bond adjusted for semi-annual compounding.
Divide kd by 2
Multiply t by 2
Divide I by 2
Semiannual Coupon Bond Example
Bond C has a K1,000 face value and provides an 8% semi-annual coupon for 15 years. The appropriate discount rate is 10% (annual rate). What is the value of the coupon bond?
Explain why I is $40. Semi-Annual (FV/8%)
V = K40 (PVIFA5%, 30) + K1,000 (PVIF5%, 30)
= K40 (15.373) + K1,000 (.231)
= K K = K845.92

10. Different Types of 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. MV
= MV (PVIF kd, t)
V =
(1 + kd)t or
Zero-Coupon Bond Example
Bond Z has a K1,000 face value and a 30 year life. The appropriate discount rate is 10%. What is the value of the zero-coupon bond?
V = K1,000 (PVIF10%, 30)= K1,000 (0.057) = K57.00
Lulumbi. Talk about issuers of zero coupon bonds.

11. Yield to Maturity (YTM) Finding the YTM by formula:
Yield to Maturity (YTM) is the rate of return earned on a bond if it is held to maturity. This return is important to investors who want to buy bonds in the market; they would like to know how much they will earn from a bond if they bought it and held it to maturity. This rate of return is what is compared to the rates on other bonds in the market to determine whether it’s worthwhile to buy or not.
The YTM is derived from the Bond Value Formula:
The YTM is identical to the total annual rate of return (Kd).
Finding the YTM by formula:
Where;
MV is the maturity value or par value,
Vb is the market price for the bond,
N is the time until the bond matures and
PMT is the coupon payment.
When a bond is called it is possible to determine a Yield to Call (YTC). In this case MV will be replaced with Call Price (CP)

12. Example1 Suppose one offered a 14 year,15% coupon and K1000 par value bond at a price of What rate of interest would one earn on his investment if he held it to maturity (YTM)?
To find the Yield to Maturity one could solve using either of the two equation 1 2
Solution to Example 1.
(From the Financial Calculator or using the trail and error methods, Kd = 10%) The YTM is identical to the total annual rate of return , the Kd discussed in the preceding section.
This rate is called Yield to maturity(YTM), and it is the interest rate discussed by bond traders when they talk about rates of return in the Market.

13. Determining the interest rate on a bond Yield to Maturity (YTM)
Example 2 A company has a bond issued at par K1000 and 2 years after being issued
the bond is being sold for K The coupon rate on the bonds is 10%
and had a maturity of 10 years. Find the YTM? =
Example 3. Suppose these bonds had a call provision and were called after 2 years. Find the YTC if the call premium is 10% of par value.
Note:
Call price =
CP =MV(1+R)
=1000(1.1)= K1100 =

14. Current Yield (CY)
This is the return earned by the bond based on its current market price. It is found by dividing the coupon with the bond current market price.
Example1. If a bond issued at par was selling for K900 and paid a coupon of K100. Find the Current Yield (CY)?

15. Bond Price - Yield Relationship
Bond prices in the market are affected by changes in the interest rates. The following are the observation for prices when rates change up and down in the market.
When the going rate of interest is below the coupon rate a bond will sell above par value. Such a bond is called a Premium Bond.
When the Market rate of interest is above the coupon rate a bond will sell below par value. Such a bond is called a discount Bond.   Lu

16. Summary on: Bond Price - Yield Relationship
When the Market rate of interest (kd) is equal to the coupon rate a bond will sell at par value.  
Summary on: Bond Price - Yield Relationship
Discount Bond – The market required rate of return exceeds the coupon rate (Par > P0 ).
Premium Bond – The coupon rate exceeds the market required rate of return (P0 > Par).
Par Bond – The coupon rate equals the market required rate of return (P0 = Par).

17. Bond Price-Yield Relationship (Rising and Falling interest Rates)
Example: If the required rate of return on a 15 year, 10% annual coupon paying bond (K 1,000 Par Value) has risen from 10% to 12%.
Therefore, the bond price has fallen from K1,000 to K864. (K on calculator)
When interest rates fall, then the market required rates of return fall and bond prices will rise.
Example: Assume that the required rate of return on a 15 year, 10% annual coupon paying bond falls from 10% to 8%. What happens to the bond price?
The required rate of return on a 15 year, 10% coupon paying bond has fallen from 10% to 8%.
Therefore, the bond price has risen from K 1,000 to K 1, 171.
(K1, on calculator)

18. The Role of Bond Maturity
The longer the bond maturity, the greater the change in bond price for a given change in the market required rate of return.
Assume that the required rate of return on both the 5 and 15 year, 10% annual coupon paying bonds fall from 10% to 8%. What happens to the changes in bond prices?
The required rate of return on both the 5 and 15 year, 10% annual coupon paying bonds has fallen from 10% to 8%.
The 5 year bond price has risen from $1,000 to $1,080 for the 5 year bond (+8.0%).
The 15 year bond price has risen from K1,000 to K1,171 (+17.1%). Twice as fast!
The Role of the Coupon Rate
Coupon rate. This is the annual rate of interest payable on the bond.
For the owner of a bond, the higher the coupon rate, the higher the interest payments the owner receives. The rate is set at the time the bond is issued and generally does not change. Most bonds make interest payments semiannually, although some bonds are offered with monthly and quarterly payments.