copyright 2015 by diane s. docking 1 bond valuation
TRANSCRIPT
Copyright 2015 by Diane S. Docking
Learning ObjectivesTo understand the cash flow characteristics of a bond. how the price of a bond is determined. why the price of a bond changes. that the price/yield curve of an option-free bond is
convex. that the two characteristics of a bond that affect its price
volatility are its coupon and its maturity. why the yield to maturity is used as a measure of a
bond’s return. the importance of the reinvestment rate in realizing the
yield to maturity.
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Copyright 2015 by Diane S. Docking 3
Bond Valuation
Bonds are simply valued as the PV of future cash flows
As market interest rates ↑ (↓),
bond prices ↓ (↑)
Bond Valuation The present value of a bond (Vb) can be written as:
Par = the par or face value of the bond, usually $1,000
INT = the annual interest (or coupon) payment
T = the number of years until the bond matures
r = the annual interest rate (often called yield to maturity (ytm))
Assumes semi-annual interest payments.
The present value of a bond (Vb) can be written as:
Par = the par or face value of the bond, usually $1,000
INT = the annual interest (or coupon) payment
T = the number of years until the bond matures
r = the annual interest rate (often called yield to maturity (ytm))
Assumes semi-annual interest payments.
2T
2T
2
12
(r/2))(1
Par
)2(r
))2(r(111
2
INT
))2/(1())2/(1(
1
2
T
tT
t
br
Par
r
INTV
Copyright 2015 by Diane S. Docking 4
Example : Bond Price
A $1,000 face value bond, 6% coupon, 3-year maturity is available for purchase today. Interest is paid semi-annually.
If current market rates are 6%, what should be the price of this bond?
If current market rates are 8%, what should be the price of this bond?
Copyright 2015 by Diane S. Docking 5
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Solution to Example: Bond Price
30
1,000
= PV
Copyright 2015 by Diane S. Docking
= 30/(1.03)129.13= 30/(1.03)228.28
837.49
$1,000.00
= 30/(1.03)3
10 2 3 4 5 6
30 30 30 30 30
27.45
26.65
25.88
25.12
= 30/(1.03)4
= 30/(1.03)6
= 1,000/(1.03)6
= 30/(1.03)5
This problem can more easily be solved using the financial calculator:
Copyright 2015 by Diane S. Docking 7
FV = $1,000N = 3 years x 2 = 6PMT = 1,000 x 6%/2 = $30I/Y = 6%/2 = 3%CPT PV = $1,000
Bond is selling at aPAR value.
Solution to Example: Bond Price (cont.)
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Solution to Example: Bond Price
30
1,000
= PV
Copyright 2015 by Diane S. Docking
= 30/(1.04)128.85= 30/(1.04)227.74
790.31
$947.58
= 30/(1.04)3
10 2 3 4 5 6
30 30 30 30 30
26.67
25.64
24.66
23.71
= 30/(1.04)4
= 30/(1.04)6
= 1,000/(1.04)6
= 30/(1.04)5
This problem can more easily be solved using the financial calculator:
Copyright 2015 by Diane S. Docking 9
FV = $1,000N = 3 years x 2 = 6PMT = 1,000 x 6%/2 = $30I/Y = 8%/2 = 4%CPT PV = $947.58
Bond is selling at aDISCOUNT to face value.
Solution to Example: Bond Price (cont.)
10Copyright 2015 by Diane S. Docking
Relationship Between Interest Rates and Bond PricesCoupon:Maturity:
rm Bond A Price Δ Bond B Price Δ Bond C Price Δ Bond D Price Δ
0% $1,300.00 $34.66 $3,500.00 $513.57 $1,150.00 $32.07 $2,250.00 $367.141% $1,265.34 $33.52 $2,986.43 $418.58 $1,117.93 $31.00 $1,882.86 $294.912% $1,231.82 $32.42 $2,567.84 $342.86 $1,086.93 $29.96 $1,587.94 $237.943% $1,199.40 $31.36 $2,224.99 $282.28 $1,056.97 $28.96 $1,350.00 $192.884% $1,168.04 $30.34 $1,942.71 $233.65 $1,028.01 $28.01 $1,157.12 $157.125% $1,137.70 $29.36 $1,709.06 $194.46 $1,000.00 $1,000.006% $1,108.34 $28.42 $1,514.60 $162.76 $972.91 ($27.09) $871.35 ($128.65)7% $1,079.93 $27.51 $1,351.83 $137.01 $946.71 ($26.20) $765.44 ($105.91)8% $1,052.42 $26.63 $1,214.82 $116.01 $921.37 ($25.35) $677.77 ($87.68)9% $1,025.79 $25.79 $1,098.81 $98.81 $896.84 ($24.53) $604.76 ($73.01)10% $1,000.00 $1,000.00 $873.11 ($23.73) $543.60 ($61.16)11% $975.02 ($24.98) $915.34 ($84.66) $850.13 ($22.97) $492.05 ($51.55)12% $950.83 ($24.20) $842.38 ($72.96) $827.89 ($22.24) $448.33 ($43.72)13% $927.38 ($23.44) $779.13 ($63.25) $806.36 ($21.53) $411.02 ($37.32)14% $904.67 ($22.72) $723.99 ($55.15) $785.51 ($20.85) $378.97 ($32.05)15% $882.65 ($22.02) $675.63 ($48.36) $765.31 ($20.20) $351.26 ($27.71)16% $861.31 ($21.34) $633.00 ($42.63) $745.74 ($19.57) $327.16 ($24.10)17% $840.62 ($20.69) $595.20 ($37.79) $726.78 ($18.96) $306.06 ($21.09)18% $820.56 ($20.06) $561.53 ($33.67) $708.42 ($18.37) $287.49 ($18.57)19% $801.11 ($19.46) $531.38 ($30.15) $690.61 ($17.80) $271.04 ($16.45)20% $782.24 ($18.87) $504.26 ($27.12) $673.36 ($17.26) $256.39 ($14.65)
Assume semi-annual interest payments.
5%25 years
10%3 years
10%25 years
5%3 years
11Copyright 2015 by Diane S. Docking
Relationship Between Interest Rates and Bond PricesCoupon:Maturity:
rm Bond A Price Δ Bond B Price Δ Bond C Price Δ Bond D Price Δ
0% $1,300.00 $34.66 $3,500.00 $513.57 $1,150.00 $32.07 $2,250.00 $367.141% $1,265.34 $33.52 $2,986.43 $418.58 $1,117.93 $31.00 $1,882.86 $294.912% $1,231.82 $32.42 $2,567.84 $342.86 $1,086.93 $29.96 $1,587.94 $237.943% $1,199.40 $31.36 $2,224.99 $282.28 $1,056.97 $28.96 $1,350.00 $192.884% $1,168.04 $30.34 $1,942.71 $233.65 $1,028.01 $28.01 $1,157.12 $157.125% $1,137.70 $29.36 $1,709.06 $194.46 $1,000.00 $1,000.006% $1,108.34 $28.42 $1,514.60 $162.76 $972.91 ($27.09) $871.35 ($128.65)7% $1,079.93 $27.51 $1,351.83 $137.01 $946.71 ($26.20) $765.44 ($105.91)8% $1,052.42 $26.63 $1,214.82 $116.01 $921.37 ($25.35) $677.77 ($87.68)9% $1,025.79 $25.79 $1,098.81 $98.81 $896.84 ($24.53) $604.76 ($73.01)10% $1,000.00 $1,000.00 $873.11 ($23.73) $543.60 ($61.16)11% $975.02 ($24.98) $915.34 ($84.66) $850.13 ($22.97) $492.05 ($51.55)12% $950.83 ($24.20) $842.38 ($72.96) $827.89 ($22.24) $448.33 ($43.72)13% $927.38 ($23.44) $779.13 ($63.25) $806.36 ($21.53) $411.02 ($37.32)14% $904.67 ($22.72) $723.99 ($55.15) $785.51 ($20.85) $378.97 ($32.05)15% $882.65 ($22.02) $675.63 ($48.36) $765.31 ($20.20) $351.26 ($27.71)16% $861.31 ($21.34) $633.00 ($42.63) $745.74 ($19.57) $327.16 ($24.10)17% $840.62 ($20.69) $595.20 ($37.79) $726.78 ($18.96) $306.06 ($21.09)18% $820.56 ($20.06) $561.53 ($33.67) $708.42 ($18.37) $287.49 ($18.57)19% $801.11 ($19.46) $531.38 ($30.15) $690.61 ($17.80) $271.04 ($16.45)20% $782.24 ($18.87) $504.26 ($27.12) $673.36 ($17.26) $256.39 ($14.65)
Assume semi-annual interest payments.
10% 10% 5% 5%3 years 25 years 3 years 25 years
FV = 1,000N = 25 x 2 = 50PMT = 1,000 x 10%/2 = $50I/Y = 8%/2 = 4%PV = $1,214.82
Price Change when interest rates go from 10% to 8 % = +$214.82
FV = 1,000N = 25 x 2 = 50PMT = 1,000 x 10%/2 = $50I/Y = 12%/2 = 6%PV = $842.38
Price Change when interest rates go from 10% to 12 % = −$157.62
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Copyright 2015 by Diane S. Docking
0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%$0
$500
$1,000
$1,500
$2,000
$2,500
$3,000
$3,500
$4,000
Bond A, 10%, 3-yr.
Bond B, 10%, 25-yr.
Bond C, 5%, 3-yr.
Bond D, 5%, 25-yr.
Bonds: Interest & Price Relationship
Price
Market Interest Rates
Copyright 2015 by Diane S. Docking 13
Sensitivity of Bond Prices to Interest Rate Movements Price-Sensitive Bonds
1. _______ relationship between interest rates and prices of bonds
2. _______ maturity—more price variation for a change in interest rates
3. _______ coupon rate bonds are more price sensitive (the principal is a greater % of current value)
4. _______________ bonds most sensitive
5. Price sensitivity is _______ for declining rates than for increasing rates
Impact of Maturity on Price Volatility
Absolute Value of Percent Change in aBond’s Price for aGiven Change inInterest Rates
Absolute Value of Percent Change in aBond’s Price for aGiven Change inInterest Rates
Time to Maturity
Short time tomaturity – low volatility
Vo
lati
lity Long time to
maturity – high volatility
Copyright 2015 by Diane S. Docking 14
Impact of Coupon Rates onPrice Volatility
Bond Value
Bond Value
Interest Rate
Low-Coupon Bond
High-Coupon Bond
Copyright 2015 by Diane S. Docking 15
Impact of r on Price VolatilityBond Price
Interest Rate
How does volatility change with interest rates?
Price volatility is inversely related to the level of the initial interest rate
r
Copyright 2015 by Diane S. Docking 16
Example 1: Bond Valuation
BBB Manufacturers has outstanding bonds with a $1,000,000 face value. The coupon rate on the bonds is 5%, interest is paid semi-annually, and the bonds mature in 10 years.
If current market rates are 7%, what should be the price of these bonds?
If current market rates are 3%, what should be the price of these bonds?
Copyright 2015 by Diane S. Docking 17
Solution to Example 1: Bond Valuation If current market rates are 7%, what should be the price of these
bonds?
If current market rates are 3%, what should be the price of these bonds?
Copyright 2015 by Diane S. Docking 18
FV = $1,000,000N = 10 years x 2 = 20PMT = 1,000,000 x 5%/2 = $25,000I/Y = 7%/2 = 3.5%CPT PV = $857,875.97
FV = $1,000,000N = 10 years x 2 = 20PMT = 1,000,000 x 5%/2 = $25,000I/Y = 3%/2 = 1.5%CPT PV = $1,171,686.39
Bonds are selling at aDISCOUNT to face value.
Bonds are selling at aPREMIUM to face value.
Example 2: Bond Valuation
Mary bought a bond when it was issued by Mattress Co. 14 years ago. The bond, which has a $1,000 face value and a coupon rate of 10%, matures in 6 years. Interest is paid semi-annually.
If the yield on similar risk investments is 14%, what is the current market value (price) of the bond?
Suppose the yield on similar risk investments is only 8%. What is the current market value (price) of the bond?
Copyright 2015 by Diane S. Docking 19
Solution to Example 2: Bond Valuation If the yield on similar risk investments is 14%, what is the current
market value (price) of the bond?
Suppose the yield on similar risk investments is only 8%. What is the current market value (price) of the bond?
Copyright 2015 by Diane S. Docking 20
FV = $1,000N = 6 years x 2 = 12PMT = 1,000 x 10%/2 = $50I/Y = 14%/2 = 7%CPT PV = $841.15
FV = $1,000N = 6 years x 2 = 12PMT = 1,000 x 10%/2 = $50I/Y = 8%/2 = 4%CPT PV = $1,093.85
Bond is selling at aDISCOUNT to face value.
Bond is selling at aPREMIUM to face value.
Bond Prices Summary
Premium bond: if Coupon > market rate; then Price > Par
Discount bond: if Coupon < market rate; then Price < Par
Par bond: if Coupon = market rate; then Price = Par
Premium bond: if Coupon > market rate; then Price > Par
Discount bond: if Coupon < market rate; then Price < Par
Par bond: if Coupon = market rate; then Price = Par
Copyright 2015 by Diane S. Docking 21
Copyright 2015 by Diane S. Docking 22
Finding Bond Yields (market rates): Yield to Maturity The Yield to Maturity (YTM) – is the average rate
of return you earn per year if you buy a bond and hold it until it matures.
Example: A $1,000 face value bond, 6% coupon, 3-year maturity is available for purchase today at a price of $852.48. Interest is paid semi-annually. What is the bond’s YTM?
PV = -$852.48 FV = $1,000N = 3 years x 2 = 6PMT = 1,000 x 6%/2 = $30Cpt I/Y = 6% x 2 = 12%
Copyright 2015 by Diane S. Docking 23
Finding Bond Yields (market rates): Holding Period Yield The Holding Period Yield (HPY) – is the average
rate of return you earn per year if you buy a bond and then sell it before it matures.
Example: A $1,000 face value bond, 6% coupon, 3-year maturity is available for purchase today at a price of $852.48. Interest is paid semi-annually. You purchase the bond and sell it 1 year later for $900? During the year you received 2 interest payments. What is your holding period yield?
PV = -$852.48 FV = $900N = 1 years x 2 = 2PMT = 1,000 x 6%/2 = $30Cpt I/Y = 6.2222% x 2 = 12.44%
Copyright 2015 by Diane S. Docking 24
Realized Rates of Returns
Rate of Return: we can decompose returns into two pieces:
gi
P
PPc
t
tt
1CReturn
tc P
Ci where = current yield, and
t
tt
P
PP 1g = capital gains.
Copyright 2015 by Diane S. Docking 25
Example: Determining Realized Rate of Return
Union Corporation’s 30-year bonds currently pay an annual interest payment of $100.00 per every $1,000 face value. Bonds are currently selling at par. Assume you purchase $10,000 of Union bonds at today’s market price. Time passes and at the end of 1 year, the bond’s are selling for 105% of par. If you sell the bonds in one year, what is your annual rate of return on this investment?
Copyright 2015 by Diane S. Docking 26
Solution to Example: Determining Realized Rate of Return
return totalgains capitalY)interest(C
11
15%5%%01
000,10
500,1
000,10
500
000,10
000,1
000,10
000,10500,10
000,10
000,1
t
tt
t P
PP
P
CR
gi
P
PPc
t
tt
1CReturn
FV = 10,500PV = 10,000Pmt = 1,000n = 1 i = 15%
Copyright 2015 by Diane S. Docking 27
Example 2: Determining Realized Rate of Return
Union Corporation’s 30-year bonds currently pay an annual interest payment of $100.00 per every $1,000 face value. Bonds are currently selling at par. Assume you purchase $10,000 of Union bonds at today’s market price. Time passes and at the end of 1 year, the bond’s are selling for 94% of par. If you sell the bonds in one year, what is your annual rate of return on this investment?
Copyright 2015 by Diane S. Docking 28
Solution to Example 2: Determining Realized Rate of Return
return totalloss capitalinterest
11
4%%6-%01
000,10
400
000,10
600
000,10
000,1
000,10
000,10400,9
000,10
000,1
t
tt
t P
PP
P
CR
FV = 9,400PV = 10,000Pmt = 1,000n = 1 i = 4%
Copyright 2015 by Diane S. Docking 29
Example 3: Determining Realized Rate of Return
1. Union Corporation’s 30-year bonds currently pay an annual interest payment of $100.00 per every $1,000 face value. Bonds are currently selling at par. Assume you purchase $10,000 of Union bonds at today’s market price. Time passes and at the end of 2 years, the bond’s are selling for 105% of par. If you sell the bonds in two years, what is your annual rate of return on this investment?
2. Union Corporation’s 30-year bonds currently pay an annual interest payment of $100.00 per every $1,000 face value. Bonds are currently selling at par. Assume you purchase $10,000 of Union bonds at today’s market price. Time passes and at the end of 2 years, the bond’s are selling for 94% of par. If you sell the bonds in two years, what is your annual rate of return on this investment?