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Bond Price Volatility

• Zvi Wiener
• Based on Chapter 4 in Fabozzi
• Bond Markets, Analysis and Strategies

2
You Open a Bank!

• You have 1,000 customers.
• Typical CD is for 1-3 months with 1,000.
• You pay 5 on these CDs.
• A local business needs a 1M loan for 1 yr.
• The business is ready to pay 7 annually.
• What are your major sources of risk?
• How you can measure and manage it?

3
Money Manager
value
0 t D
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8 Coupon Bond
Zero Coupon Bond
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Price-Yield for option-free bonds
price
yield
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Taylor Expansion

• To measure the price response to a small change
  in risk factor we use the Taylor expansion.
• Initial value y0, new value y1, change ?y

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Derivatives
F(x)
x
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Properties of derivatives
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Zero-coupon example
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Example
y10, ?y0.5
T P0 P1 ?P 1 90.90 90.09 -0.45 2 82.64 81.16 -1
.79 10 38.55 35.22 -8.65
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Property 1

• Prices of option-free bonds move in OPPOSITE
  direction from the change in yield.
• The price change (in ) is NOT the same for
  different bonds.

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Property 2

• For a given bond a small increase or decrease in
  yield leads very similar (but opposite) changes
  in prices.
• What does this means mathematically?

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Property 3

• For a given bond a large increase or decrease in
  yield leads to different (and opposite) changes
  in prices.
• What does this means mathematically?

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Property 4

• For a given bond a large change in yield the
  percentage price increase is greater than the
  percentage decrease.
• What does this means mathematically?

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What affects price volatility?

• Linkage
• Credit considerations
• Time to maturity
• Coupon rate

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Bond Price Volatility

• Consider only IR as a risk factor
• Longer TTM means higher volatility
• Lower coupons means higher volatility
• Floaters have a very low price volatility
• Price is also affected by coupon payments
• Price value of a Basis Point (PVBP) price change
  resulting from a change of 0.01 in the yield.

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Duration and IR sensitivity
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Duration

• F. Macaulay (1938)
• Better measurement than time to maturity.
• Weighted average of all coupons with the
  corresponding time to payment.
• Bond Price Sum CFt/(1y)t
• suggested weight of each coupon
• wt CFt/(1y)t /Bond Price
• What is the sum of all wt?

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Duration

• The bond price volatility is proportional to the
  bonds duration.
• Thus duration is a natural measure of interest
  rate risk exposure.

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Modified Duration

• The percentage change in bond price is the
  product of modified duration and the change in
  the bonds yield to maturity.

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Duration
22
Duration
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Duration
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Measuring Price Change
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The Yield to Maturity

• The yield to maturity of a fixed coupon bond y is
  given by

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Macaulay Duration

• Definition of duration, assuming t0.

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Macaulay Duration
A weighted sum of times to maturities of each
coupon.

• What is the duration of a zero coupon bond?

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Meaning of Duration
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Parallel shift
r
T
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Comparison of two bonds

• Coupon bond with duration 1.8853
• Price (at 5 for 6m.) is 964.5405
• If IR increase by 1bp
• (to 5.01), its price will fall to 964.1942, or
• 0.359 decline.

• Zero-coupon bond with equal duration must have
  1.8853 years to maturity.
• At 5 semiannual its price is
• (1,000/1.053.7706)831.9623
• If IR increase to 5.01, the price becomes
• (1,000/1.05013.7706)831.66
• 0.359 decline.

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Duration
D
Zero coupon bond
15 coupon, YTM 15
Maturity
0 3m 6m 1yr 3yr 5yr 10yr 30yr
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Example

• A bond with 30-yr to maturity
• Coupon 8 paid semiannually
• YTM 9
• P0 897.26
• D 11.37 Yrs
• if YTM 9.1, what will be the price?
• ?P/P - ?y D
• ?P -(?y D)P -9.36
• P 897.26 - 9.36 887.90

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What Determines Duration?

• Duration of a zero-coupon bond equals maturity.
• Holding ttm constant, duration is higher when
  coupons are lower.
• Holding other factors constant, duration is
  higher when ytm is lower.
• Duration of a perpetuity is (1y)/y.

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What Determines Duration?

• Holding the coupon rate constant, duration not
  always increases with ttm.

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Convexity
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Example

• 10 year zero coupon bond with a semiannual yield
  of 6

The duration is 10 years, the modified duration
is
The convexity is
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Example
If the yield changes to 7 the price change is
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FRM-98, Question 17

• A bond is trading at a price of 100 with a yield
  of 8. If the yield increases by 1 bp, the price
  of the bond will decrease to 99.95. If the yield
  decreases by 1 bp, the price will increase to
  100.04. What is the modified duration of this
  bond?
• A. 5.0
• B. -5.0
• C. 4.5
• D. -4.5

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FRM-98, Question 17
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FRM-98, Question 22

• What is the price of a 10 bp increase in yield on
  a 10-year par bond with a modified duration of 7
  and convexity of 50?
• A. -0.705
• B. -0.700
• C. -0.698
• D. -0.690

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FRM-98, Question 22
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Portfolio Duration

• Similar to a single bond but the cashflow is
  determined by all Fixed Income securities held in
  the portfolio.

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Bond Price Derivatives

• D - modified duration, dollar duration is the
  negative of the first derivative

Dollar convexity the second derivative, C -
convexity.
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Duration of a portfolio
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ALM Duration

• Does NOT work!
• Wrong units of measurement
• Division by a small number

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Duration Gap

• A - L C, assets - liabilities capital

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ALM Duration

• A similar problem with measuring yield

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• Do not think of duration as a measure of time!

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• Key rate duration
• Principal component duration
• Partial duration

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Very good question!

• Cashflow
• Libor in one year from now
• Libor in two years form now
• Libor in three years from now (no principal)
• What is the duration?

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Home Assignment

• What is the duration of a floater?
• What is the duration of an inverse floater?
• How coupon payments affect duration?
• Why modified duration is better than Macaulay
  duration?
• How duration can be used for hedging?

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Home AssignmentChapter 4

• Ch. 4 Questions 1, 2, 3, 4, 15.
• Calculate duration of a consul (perpetual bond).

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End Ch. 4
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Understanding of Duration/Convexity

• What happens with duration when a coupon is paid?
• How does convexity of a callable bond depend on
  interest rate?
• How does convexity of a puttable bond depend on
  interest rate?

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Callable bond

• The buyer of a callable bond has written an
  option to the issuer to call the bond back.
• Rationally this should be done when
• Interest rate fall and the debt issuer can
  refinance at a lower rate.

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Puttable bond

• The buyer of a such a bond can request the loan
  to be returned.
• The rational strategy is to exercise this option
  when interest rates are high enough to provide an
  interesting alternative.

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Embedded Call Option
r
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Embedded Put Option
regular bond
r
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Convertible Bond
Payoff
Stock
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Timing of exercise

• European
• American
• Bermudian
• Lock up time

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Macaulay Duration
Modified duration
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Bond Price Change
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Duration of a coupon bond
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Exercise

• Find the duration and convexity of a consol
  (perpetual bond).
• Answer (1y)/y.

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FRM-98, Question 29

• A and B are perpetual bonds. A has 4 coupon, and
  B has 8 coupon. Assume that both bonds are
  trading at the same yield, what can be said about
  duration of these bonds?
• A. The duration of A is greater than of B
• B. The duration of A is less than of B
• C. They have the same duration
• D. None of the above

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FRM-97, Question 24

• Which of the following is NOT a property of bond
  duration?
• A. For zero-coupon bonds Macaulay duration of the
  bond equals to time to maturity.
• B. Duration is usually inversely related to the
  coupon of a bond.
• C. Duration is usually higher for higher yields
  to maturity.
• D. Duration is higher as the number of years to
  maturity for a bond selling at par or above
  increases.

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FRM-99, Question 75

• You have a large short position in two bonds with
  similar credit risk. Bond A is priced at par
  yielding 6 with 20 years to maturity. Bond B has
  20 years to maturity, coupon 6.5 and yield of
  6. Which bond contributes more to the risk of
  the portfolio?
• A. Bond A
• B. Bond B
• C. A and B have similar risk
• D. None of the above

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Portfolio Duration and Convexity

• Portfolio weights

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Example

• Construct a portfolio of two bonds A and B to
  match the value and duration of a 10-years, 6
  coupon bond with value 100 and modified duration
  of 7.44 years.
• A. 1 year zero bond - price 94.26
• B. 30 year zero - price 16.97

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• Barbel portfolio consists of very short and very
  long bonds.
• Bullet portfolio consists of bonds with similar
  maturities.
• Which of them has higher convexity?

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FRM-98, Question 18

• A portfolio consists of two positions. One is
  long 100 of a two year bond priced at 101 with a
  duration of 1.7 the other position is short 50
  of a five year bond priced at 99 with a duration
  of 4.1. What is the duration of the portfolio?
• A. 0.68
• B. 0.61
• C. -0.68
• D. -0.61

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FRM-98, Question 18
Note that 100 means notional amount and can be
misunderstood.
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Useful formulas
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Volatilities of IR/bond prices

• Price volatility in End 99 End 96
• Euro 30d 0.22 0.05
• Euro 180d 0.30 0.19
• Euro 360d 0.52 0.58
• Swap 2Y 1.57 1.57
• Swap 5Y 4.23 4.70
• Swap 10Y 8.47 9.82
• Zero 2Y 1.55 1.64
• Zero 5Y 4.07 4.67
• Zero 10Y 7.76 9.31
• Zero 30Y 20.75 23.53

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Duration approximation

• What duration makes bond as volatile as FX?
• What duration makes bond as volatile as stocks?
• A 10 year bond has yearly price volatility of 8
  which is similar to major FX.
• 30-year bonds have volatility similar to equities
  (20).

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Volatilities of yields

• Yield volatility in , 99 ?(?y/y) ?(?y)
• Euro 30d 45 2.5
• Euro 180d 10 0.62
• Euro 360d 9 0.57
• Swap 2Y 12.5 0.86
• Swap 5Y 13 0.92
• Swap 10Y 12.5 0.91
• Zero 2Y 13.4 0.84
• Zero 5Y 13.9 0.89
• Zero 10Y 13.1 0.85
• Zero 30Y 11.3 0.74