Bond Valuation
Content
Bond Valuation
Learning Module
1
Definitions
Par or Face Value
The amount of money that is paid to the bondholders at maturity. For most bonds this amount is $1,000. It also generally represents the amount of money borrowed by the bond issuer.
Coupon Rate
The coupon rate, which is generally fixed, determines the periodic coupon or interest payments. It is expressed as a percentage of the bond's face value. It also represents the interest cost of the bond to the issuer.
2
Definitions
Coupon Payments
The coupon payments represent the periodic interest payments from the bond issuer to the bondholder. The annual coupon payment is calculated by multiplying the coupon rate by the bond's face value. Since most bonds pay interest semiannually, generally one half of the annual coupon is paid to the bondholders every six months.
Maturity Date
The maturity date represents the date on which the bond matures, i.e., the date on which the face value is repaid. The last coupon payment is also paid on the maturity date.
3
Definitions
Original Maturity
The time from when the bond was issued until its maturity date.
Remaining Maturity
The time currently remaining until the maturity date.
Call Date
For bonds which are callable, i.e., bonds which can be redeemed by the issuer prior to maturity, the call date represents the earliest date at which the bond can be called.
4
Definitions
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.
Required Return
The rate of return that investors currently require on a bond.
5
Definitions
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.
6
Bond Valuation
Bonds are valued using time value of money concepts. Their coupon, or interest, payments are treated like an equal cash flow stream (annuity). Their face value is treated like a lump sum.
7
Example
Assume Hunter buys a 10-year bond from the KLM corporation on January 1, 2003. The bond has a face value of $1000 and pays an annual 10% coupon. The current market rate of return is 12%. Calculate the price of this bond today. Draw a timeline $1000 + $100 $100 $100 $100 $100 $100 $100 $100 $100 $100
1.
? ?
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Example
2.
First, find the value of the coupon stream Remember to follow the same approach you use in time value of money calculations. You can find the PV of a cash flow stream
PV = $100/(1+.12)1 + $100/(1+.12)2 + $100/ (1+.12)3 + $100/(1+.12)4 + $100/(1+.12)5 + $100/(1+.12)6 + $100/(1+.12)7 + $100/(1+.12)8 + $100/(1+.12)9+ $100/(1+.12)10 PVA = $100 * {[1-(1+.12)-10 ]/.12}
Or, you can find the PV of an annuity
PV = $565.02
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Example
3.
Find the PV of the face value
PV = CFt / (1+r)t PV = $1000/ (1+.12)10 PV = $321.97
3.
Add the two values together to get the total PV
$565.02 + $321.97 = $886.99
Alternatively, on your calculator
PMT = 100 FV = 1000 n = 10 i = 12 PV = ? Note that if the payments had been semiannual, PMT=50, FV=1000, n=20, i=6, PV=?=$885.30.
10
Realized Return
Sometimes you will be asked to find the realized rate of return for a bond. This is the return that the investor actually realized from holding a bond. Using time value of money concepts, you are solving for the required rate of return instead of the value of the bond.
11
Example
Doug purchased a bond for $800 5-years ago and he sold the bond today for $1200. The bond paid an annual 10% coupon. What is his realized rate of return?
n
PV = Σ [CFt / (1+r)t]
t= 0
$800 = [$100/(1+r) + $100/(1+r)2 + $100/(1+r)3 + $100/(1+r)4 + $100/(1+r)5] + [$1200/(1+r)5] To solve, you need use a “trail and error” approach. You plug in numbers until you find the rate of return that solves the equation. The realized rate of return on this bond is 19.31%.
12
Example
This is much easier to find using a financial calculator: n=5 PV = -800 FV = 1200 PMT = 100 i = ?, this is the realized rate of return on this bond Note that if the payments had been semiannual, n=10, PV=-800, FV=1200, PMT=50, i=?=9.47%. Thus, the realized return would have been 2 * 9.47% = 18.94%.
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Learning Module
1
Definitions
Par or Face Value
The amount of money that is paid to the bondholders at maturity. For most bonds this amount is $1,000. It also generally represents the amount of money borrowed by the bond issuer.
Coupon Rate
The coupon rate, which is generally fixed, determines the periodic coupon or interest payments. It is expressed as a percentage of the bond's face value. It also represents the interest cost of the bond to the issuer.
2
Definitions
Coupon Payments
The coupon payments represent the periodic interest payments from the bond issuer to the bondholder. The annual coupon payment is calculated by multiplying the coupon rate by the bond's face value. Since most bonds pay interest semiannually, generally one half of the annual coupon is paid to the bondholders every six months.
Maturity Date
The maturity date represents the date on which the bond matures, i.e., the date on which the face value is repaid. The last coupon payment is also paid on the maturity date.
3
Definitions
Original Maturity
The time from when the bond was issued until its maturity date.
Remaining Maturity
The time currently remaining until the maturity date.
Call Date
For bonds which are callable, i.e., bonds which can be redeemed by the issuer prior to maturity, the call date represents the earliest date at which the bond can be called.
4
Definitions
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.
Required Return
The rate of return that investors currently require on a bond.
5
Definitions
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.
6
Bond Valuation
Bonds are valued using time value of money concepts. Their coupon, or interest, payments are treated like an equal cash flow stream (annuity). Their face value is treated like a lump sum.
7
Example
Assume Hunter buys a 10-year bond from the KLM corporation on January 1, 2003. The bond has a face value of $1000 and pays an annual 10% coupon. The current market rate of return is 12%. Calculate the price of this bond today. Draw a timeline $1000 + $100 $100 $100 $100 $100 $100 $100 $100 $100 $100
1.
? ?
8
Example
2.
First, find the value of the coupon stream Remember to follow the same approach you use in time value of money calculations. You can find the PV of a cash flow stream
PV = $100/(1+.12)1 + $100/(1+.12)2 + $100/ (1+.12)3 + $100/(1+.12)4 + $100/(1+.12)5 + $100/(1+.12)6 + $100/(1+.12)7 + $100/(1+.12)8 + $100/(1+.12)9+ $100/(1+.12)10 PVA = $100 * {[1-(1+.12)-10 ]/.12}
Or, you can find the PV of an annuity
PV = $565.02
9
Example
3.
Find the PV of the face value
PV = CFt / (1+r)t PV = $1000/ (1+.12)10 PV = $321.97
3.
Add the two values together to get the total PV
$565.02 + $321.97 = $886.99
Alternatively, on your calculator
PMT = 100 FV = 1000 n = 10 i = 12 PV = ? Note that if the payments had been semiannual, PMT=50, FV=1000, n=20, i=6, PV=?=$885.30.
10
Realized Return
Sometimes you will be asked to find the realized rate of return for a bond. This is the return that the investor actually realized from holding a bond. Using time value of money concepts, you are solving for the required rate of return instead of the value of the bond.
11
Example
Doug purchased a bond for $800 5-years ago and he sold the bond today for $1200. The bond paid an annual 10% coupon. What is his realized rate of return?
n
PV = Σ [CFt / (1+r)t]
t= 0
$800 = [$100/(1+r) + $100/(1+r)2 + $100/(1+r)3 + $100/(1+r)4 + $100/(1+r)5] + [$1200/(1+r)5] To solve, you need use a “trail and error” approach. You plug in numbers until you find the rate of return that solves the equation. The realized rate of return on this bond is 19.31%.
12
Example
This is much easier to find using a financial calculator: n=5 PV = -800 FV = 1200 PMT = 100 i = ?, this is the realized rate of return on this bond Note that if the payments had been semiannual, n=10, PV=-800, FV=1200, PMT=50, i=?=9.47%. Thus, the realized return would have been 2 * 9.47% = 18.94%.
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