Bonds ValuationPERTEMUAN 17-18

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

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

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1

91.613)05.1(

000,110

PV

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

Valuing a level coupon bond

• Example: If r = 5%

• Note: If P0 >: 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%

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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

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

Sensitivity of zero-coupons to interest rate

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50,00

100,00

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200,00

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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

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):

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100

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rt

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1)1(

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Duration = Maturity

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

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 = .0581×1 + .0553×2 + .0527 ×3 + .0502 ×4 + .7835 ×5

• = 4.44

• For coupon bonds, duration < maturity

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%

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Duration

P

P

1

Difference due to convexity

Duration -mathematics

• If the interest rate changes:

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

• As:

• we get:

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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

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20,00

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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

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Spot rates

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

Maturity Price1-year 95.242-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%

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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:

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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

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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

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

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

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