Finance 300Financial Markets

Lecture 14

Fall, 2001©

Professor J. Petry

http://www.cba.uiuc.edu/broker/fin300/fin300pp.htm

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Exam PreparationMid-term #2 next class

– Will be a touch more challenging• Chapter IV (Debt Market), Chapter V (Money Market)

– True & false, matching, and problems, other MC– All Multiple Choice format; length comparable, but heavier emphasis on problems

—I will provide a formula sheet, which I will post by Thursday. – Only covers new material

• Everything in Ch IV except:– realized yield calculation on page 162.– Immunization calculation done in answers to TTD (IV-18, D)

• All definitions, concepts in Ch V, including yield definitions—i.e. what is included in Bankers Discount Yield, what is not—but no problems to actually calculate. No bid/ask or auction problems (example on page 192 is out for instance). However, all institutional details of primary, secondary markets and instruments will be on the exam.

• Series 7 like questions are legit in both chapters.

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Bond PricingThe Term Structure of Interest Rates (cont’d)

Deriving the Term Structure from Coupon Bonds

Months to Zero Coupon Bonds Coupon Bondsmaturity Yld Price Yld Price

6 8.000% B

12 8.300% C

18 E D = A - B - C 8.900% A

24 H G = F - B - C - D 8.920% F*Solve A using price formula for coupon bonds

**Solve B, C using price formula for zero coupon bond formula

***Solve E, H using same as B, C except solving for yield

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Duration & VolatilityVolatility

– The tendency of a security price or market index to change due to the changes in market conditions.

• Volatility = Δ price / price

Duration– A measure of the effective maturity of a bond, defined as the weighted average of

the times until each payment, with weights proportional to the present value of the payment.

– Bond price volatility and duration are directly related.– Modified duration measures the volatility in response to a 1% change in interest

rates.• Volatility = Δ price/price = -[duration / (1 + yield)] * Δi

– Duration trading strategies would include increasing duration exposure ahead of expected decreases in interest rates, (or decreasing ahead of interest rate increases) to maximize price impact on your holdings.

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Duration Calculation

Vol=Δprice/price=-[duration/(1+ yield)]*Δi=4.27/1.07*1%=-3.9907%Modified Duration = D* = duration/(1+yield)=3.9907 yearsEx: 1% change (decline) in interest rates would result in what change in bond price? From 7% to 6% ; -D* x -.01 =-3.9907 x -.01 = 3.9907% ; 1082 x 1.039907 = $1125.18

Duration Calculation5 year 9% bond (annual coupon payment)

durationyear coupon NPV @ 7% NPV/V NPV/V*t

1 90 84.112 0.078 0.0782 90 78.609 0.073 0.1453 90 73.467 0.068 0.2044 90 68.661 0.063 0.2545 1090 777.155 0.718 3.5916 1082.00 1.00 4.27

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Pricing Error from ConvexityPrice

Yield

Duration

Pricing Error from

Convexity

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Correction for Convexity

)(21* 2yConvexityyD

P

P

Modify the pricing equation:

Convexity is Equal to:

N

tt

t tty

CFP 1

22 )1(y)(1

1

Where: CFt is the cashflow (interest and/or principal) at time t.

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Duration & VolatilityConvexity

– The curvature of the price-yield relationship of a bond.• Duration and volatility are a linear estimate of a convex relationship.

Convexity is a correction to the duration formula, which adjusts for the convexity of the relationship. The convexity correction is particularly important for measuring large changes in the price-yield relationship.

• Convexity increases with lower coupon rates, longer maturity and lower yield.

• To more properly estimate the price change due to a change in interest rates we add the convexity correction to the duration estimate:

ΔP/P = -modified duration Δy + ½ Convexity Δy2

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Duration & Convexity Calcs

Duration & Convexity Calculation5 year 9% bond (annual coupon payment)

duration convexityyear coupon NPV @ 7% NPV/V NPV/V*t NPV/V*t(t+1)

1 90 84.112 0.078 0.078 0.1552 90 78.609 0.073 0.145 0.4363 90 73.467 0.068 0.204 0.8154 90 68.661 0.063 0.254 1.2695 1090 777.155 0.718 3.591 21.5486 1082.00 1.00 4.27 24.22

Volatility=Δ price/price =-[duration / (1 + yield)] * Δi = 4.27/1.07*1% =-3.9907%Corrected ΔP/P = -modified duration Δy + ½ Convexity Δy2

= -3.9907 x –0.01 + .5 (24.22) (-.01)2 = .039907+.001211=.041118%Check this against actual price change

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Duration & Convexity Calcs

Volatility = Δ price/price = -[duration / (1 + yield)] * Δi = Corrected ΔP/P = -modified duration Δy + ½ Convexity Δy2

Duration & Convexity Calculation5 year 5% bond (annual coupon payment)

duration convexityyear coupon NPV @ 7% NPV/V NPV/V*t NPV/V*t(t+1)

123456

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Duration & VolatilityExample

– Assume a 30 year bond, 8% coupon and initial yield to maturity of 8%. The bonds duration is 11.37 years.

• (What does this mean?) – Convexity for this bond is 212.4.

• (What does this mean?)– If yields move from 8% to 10%, how much would you expect

the price of this bond to move?– What would the price be at the new interest rate?– Check your answer by re-valuing the bond at the new yield.