FINANCE 4. Bond Valuation Professeur André Farber Solvay Business School Université Libre de...

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

MBA 2007 Bonds |3

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

MBA 2007 Bonds |4

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

MBA 2007 Bonds |5

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

MBA 2007 Bonds |6

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

MBA 2007 Bonds |7

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

MBA 2007 Bonds |8

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

MBA 2007 Bonds |9

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

MBA 2007 Bonds |10

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

MBA 2007 Bonds |12

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

MBA 2007 Bonds |13

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

MBA 2007 Bonds |14

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

MBA 2007 Bonds |15

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

MBA 2007 Bonds |16

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

MBA 2007 Bonds |17

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)

MBA 2007 Bonds |19

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

MBA 2007 Bonds |20

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

MBA 2007 Bonds |21

• Immunization: DEquity = 0

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

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

MBA 2007 Bonds |22

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

MBA 2007 Bonds |23

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:

MBA 2007 Bonds |24

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

MBA 2007 Bonds |25

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

MBA 2007 Bonds |27

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

MBA 2007 Bonds |28

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