FINANCE4. Bond Valuation

Professeur André Farber

Solvay Business School

Université Libre de Bruxelles

Fall 2007

MBA 2007 Bonds |2

Review: present value calculations

• Cash flows: C1, C2, C3, … ,Ct, … CT

• Discount factors: DF1, DF2, … ,DFt, … , DFT

• Present value: PV = C1 × DF1 + C2 × DF2 + … + CT × DFT

TT

Tt

t

t

r

C

r

C

r

C

r

CPV

)1(...

)1(...

)1()1( 22

2

1

1

TT

tt

r

C

r

C

r

C

r

CPV

)1(...

)1(...

)1()1( 221

If r1 = r2 = ...=r

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Review: Shortcut formulas

• Constant perpetuity: Ct = C for all t

• Growing perpetuity: Ct = Ct-1(1+g)

r>g t = 1 to ∞

• Constant annuity: Ct=C t=1 to T

• Growing annuity: Ct = Ct-1(1+g)

t = 1 to T

r

CPV

gr

CPV

1

))1(

11(

Trr

CPV

))1(

)1(1(1

T

T

r

g

gr

CPV

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Bond Valuation

• Objectives for this session :

– 1.Introduce the main categories of bonds

– 2.Understand bond valuation

– 3.Analyse the link between interest rates and bond prices

– 4.Introduce the term structure of interest rates

– 5.Examine why interest rates might vary according to maturity

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Zero-coupon bond

• Pure discount bond - Bullet bond

• The bondholder has a right to receive:

• one future payment (the face value) F

• at a future date (the maturity) T

• Example : a 10-year zero-coupon bond with face value $1,000

•

• Value of a zero-coupon bond:

• Example :

• If the 1-year interest rate is 5% and is assumed to remain constant

• the zero of the previous example would sell for

TrPV

)1(

1

91.613)05.1(

000,110

PV

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Level-coupon bond

• Periodic interest payments (coupons)

• Europe : most often once a year

• US : every 6 months

• Coupon usually expressed as % of principal

• At maturity, repayment of principal

• Example : Government bond issued on March 31,2000

• Coupon 6.50%

• Face value 100

• Final maturity 2005

• 2000 2001 2002 2003 2004 2005

• 6.50 6.50 6.50 6.50 106.50

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Valuing a level coupon bond

• Example: If r = 5%

• Note: If P0 > F: the bond is sold at a premium

• If P0 <F: the bond is sold at a discount

• Expected price one year later P1 = 105.32

• Expected return: [6.50 + (105.32 – 106.49)]/106.49 = 5%

49.1067835.01003295.45.61005.6 5505.0 dAP

TTrTT

dACrr

C

r

C

r

CP

100

)1(

100

)1(...

)1(1 20

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When does a bond sell at a premium?

• Notations: C = coupon, F = face value, P = price

• Suppose C / F > r

• 1-year to maturity:

• 2-years to maturity:

• As: P1 > F

FPrF

C

Fr

FCP

00 1

1

1

r

PCP

1

10 with

r

FCP

11

FrF

C

Fr

FCP

1

1

10

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A level coupon bond as a portfolio of zero-coupons

• « Cut » level coupon bond into 5 zero-coupon

• Face value Maturity Value

• Zero 1 6.50 1 6.19

• Zero 2 6.50 2 5.89

• Zero 3 6.50 3 5.61

• Zero 4 6.50 4 5.35

• Zero 5 106.50 5 83.44

• Total 106.49

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Bond prices and interest rates

Bond prices fall with arise in interest rates and rise with a fall ininterest rates

0,00

20,00

40,00

60,00

80,00

100,00

120,00

140,00

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%

Interest rate

Bo

nd

pri

ce

MBA 2007 Bonds |11

Sensitivity of zero-coupons to interest rate

0,00

50,00

100,00

150,00

200,00

250,00

300,00

350,00

400,00

450,00

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%

Interest rate

Bo

nd

pri

ce

5-Year

10-Year

15-Year

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Duration for Zero-coupons

• Consider a zero-coupon with t years to maturity:

• What happens if r changes?

• For given P, the change is proportional to the maturity.

• As a first approximation (for small change of r):

trP

)1(

100

Pr

t

rr

t

rt

dr

dPtt

1)1(

100

1)1(

1001

rr

t

P

P

1

Duration = Maturity

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Duration for coupon bonds

• Consider now a bond with cash flows: C1, ...,CT

• View as a portfolio of T zero-coupons.

• The value of the bond is: P = PV(C1) + PV(C2) + ...+ PV(CT)

• Fraction invested in zero-coupon t: wt = PV(Ct) / P

• •

• Duration : weighted average maturity of zero-coupons

D= w1 × 1 + w2 × 2 + w3 × 3+…+wt × t +…+ wT ×T

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Duration - example

• Back to our 5-year 6.50% coupon bond.

Face value Value wt

Zero 1 6.50 6.19 5.81%

Zero 2 6.50 5.89 5.53%

Zero 3 6.50 5.61 5.27%

Zero 4 6.50 5.35 5.02%

Zero 5 106.50 83.44 78.35%

Total 106.49

• Duration D = .0581×1 + 0.0553×2 + .0527 ×3 + .0502 ×4 + .7835 ×5

• = 4.44

• For coupon bonds, duration < maturity

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Price change calculation based on duration

• General formula:

• In example: Duration = 4.44 (when r=5%)

• If Δr =+1% : Δ ×4.44 × 1% = - 4.23%

• Check: If r = 6%, P = 102.11

• ΔP/P = (102.11 – 106.49)/106.49 = - 4.11%

rr

Duration

P

P

1

Difference due to convexity

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Duration -mathematics

• If the interest rate changes:

• Divide both terms by P to calculate a percentage change:

• As:

• we get:

)(1

...)(1

2)(

1

1

)(...

)()(

21

21

T

T

CPVr

TCPV

rCPV

r

dr

CdPV

dr

CdPV

dr

CdPV

dr

dP

))(

...)(

2)(

1(1

11 21

P

CPVT

P

CPV

P

CPV

rPdr

dP T

P

CPVT

P

CPV

P

CPVDuration T )(

...)(

2)(

1 21

r

Duration

Pdr

dP

1

1

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Yield to maturity

• Suppose that the bond price is known.

• Yield to maturity = implicit discount rate

• Solution of following equation:Ty

FC

y

C

y

CP

)1(...

)1(1 20

0,00

20,00

40,00

60,00

80,00

100,00

120,00

140,00

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%

Interest rate

Bo

nd

pri

ce

MBA 2007 Bonds |18

Yield to maturity vs IRR

The yield to maturity is the internal rate of return (IRR) for an investment in a bond.

123456789

1011

A B C D E F G HYield to maturity - illustration

Coupon 7%Face value 100Maturity 6 yearsPrice 105

0 1 2 3 4 5 6Cash flows -105 7 7 7 7 7 107

Yield to maturity 5.98% B11. =IRR(B9:H9)

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Asset Liability Management

• Balance sheet of financial institution (mkt values):

• Assets = Equity + Liabilities → ∆A = ∆E + ∆L

• As: ∆P = -D * P * ∆r (D = modified duration)

-DAsset * A * ∆r = -DEquity * E * ∆r - DLiabilities * L * ∆r

DAsset * A = DEquity * E + DLiabilities * L

( )Equity Asset Asset Liabilities

LD D D D

E

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Examples

Value MDuration Value MDurationAssets 100 3 Equity 10 21

Liabilities 90 1

Value MDuration Value MDurationAssets 100 15 Equity 10 -30

Liabilities 90 20

SAVING BANK

LIFE INSURANCE COMPANY

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• Immunization: DEquity = 0

• As: DAsset * A = DEquity * E + DLiabilities * L

• DEquity = 0 → DAsset * A = DLiabilities * L

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Spot rates

• Spot rate = yield to maturity of zero coupon

• Consider the following prices for zero-coupons (Face value = 100):

Maturity Price

1-year 95.24

2-year 89.85

• The one-year spot rate is obtained by solving:

• The two-year spot rate is calculated as follow:

• Buying a 2-year zero coupon means that you invest for two years at an average rate of 5.5%

%51

10024.95 1

1

rr

%5.5)1(

10085.89 22

2

rr

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Measuring spot rate

Bond Coupon Maturity Price YTMB1 5.00 1 99.06 6.00%B2 9.00 2 103.70 6.96%B3 6.50 3 97.54 7.45%B4 8.00 4 100.36 7.89%

Data:

99.06 = 105 * d1

103.70 = 9 * d1 + 109 * d2

97.54 = 6.5 * d1 + 6.5 * d2 + 106.5 * d3

100.36 = 8 * d1 + 8 * d2 + 8 * d3 + 108 * d4

To recover spot prices, solve:

Maturity Disc Fac. Spot rate1 0.9434 6.00%2 0.8734 7.00%3 0.8050 7.50%4 0.7350 8.00%

Solution:

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Forward rates

• You know that the 1-year rate is 5%.

• What rate do you lock in for the second year ?

• This rate is called the forward rate

• It is calculated as follow:

• 89.85 × (1.05) × (1+f2) = 100 → f2 = 6%

• In general:

(1+r1)(1+f2) = (1+r2)²

• Solving for f2:

• The general formula is:

111

)1(

2

1

1

22

2

d

d

r

rf

11)1(

)1( 11

1

t

tt

t

tt

t d

d

r

rf

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Forward rates :example

• Maturity Discount factor Spot rates Forward rates

• 1 0.9500 5.26

• 2 0.8968 5.60 5.93

• 3 0.8444 5.80 6.21

• 4 0.7951 5.90 6.20

• 5 0.7473 6.00 6.40

• Details of calculation:

• 3-year spot rate :

• 1-year forward rate from 3 to 4

%80.51)8444.0

1(

)1(

18444.0 3

1

333

rr

%21.618444.0

8968.011

)1(

)1(

3

22

2

33

3

d

d

r

rf

MBA 2007 Bonds |26

Term structure of interest rates

• Why do spot rates for different maturities differ ?

• As

• r1 < r2 if f2 > r1

• r1 = r2 if f2 = r1

• r1 > r2 if f2 < r1

• The relationship of spot rates with different maturities is known as the term structure of interest rates

Time to maturity

Spotrate

Upward sloping

Flat

Downward sloping

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Forward rates and expected future spot rates

• Assume risk neutrality

• 1-year spot rate r1 = 5%, 2-year spot rate r2 = 5.5%

• Suppose that the expected 1-year spot rate in 1 year E(r1) = 6%

• STRATEGY 1 : ROLLOVER

• Expected future value of rollover strategy:

• ($100) invested for 2 years :

• 111.3 = 100 × 1.05 × 1.06 = 100 × (1+r1) × (1+E(r1))

• STRATEGY 2 : Buy 1.113 2-year zero coupon, face value = 100

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Equilibrium forward rate

• Both strategies lead to the same future expected cash flow

• → their costs should be identical

• In this simple setting, the foward rate is equal to the expected future spot rate

f2 =E(r1)

• Forward rates contain information about the evolution of future spot rates

)1)(1(

100))(1)(1(

)1(

1113.1100

21112

2 frrEr

r