Copyright 2015 by Diane S. Docking 1 Bond Valuation

Copyright 2015 by Diane S. Docking 1

Bond Valuation

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.

2


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


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?


6

Solution to Example: Bond Price

30

1,000

= PV


= 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:


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.)

8

Solution to Example: Bond Price

30

1,000

= PV


= 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:


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

12


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


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


Impact of Coupon Rates onPrice Volatility

Bond Value

Bond Value

Interest Rate

Low-Coupon Bond

High-Coupon Bond


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


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?



Solution to Example 1: Bond Valuation If current market rates are 7%, what should be the price of these

bonds?



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?


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?


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



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%


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%


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.


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?


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%


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?


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%


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?


Solution to Example 3: Determining Realized Rate of Return

1.

2.

FV = 10,500PV = 10,000Pmt = 1,000n = 2 i/y = 12.3546%

FV = 9,400PV = 10,000Pmt = 1,000n = 2 i/y = 7.1029%

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Copyright 2015 by Diane S. Docking 1 Bond Valuation