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Bond Price Volatility. Zvi Wiener Based on Chapter 4 in Fabozzi Bond Markets, Analysis and Strategies. 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.

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Bond price volatility l.jpg

Bond Price Volatility

Zvi Wiener

Based on Chapter 4 in Fabozzi

Bond Markets, Analysis and Strategies

http://pluto.mscc.huji.ac.il/~mswiener/zvi.html


You open a bank l.jpg
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?

Fabozzi Ch 4


Money manager l.jpg

Original plan

New plan

Market shock

Money Manager

value

0 t D

Fabozzi Ch 4


8 coupon bond l.jpg
8% Coupon Bond

Zero Coupon Bond

Fabozzi Ch 4


Price yield for option free bonds l.jpg
Price-Yield for option-free bonds

price

yield

Fabozzi Ch 4


Taylor expansion l.jpg
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:

Fabozzi Ch 4


Derivatives l.jpg
Derivatives

F(x)

x

Fabozzi Ch 4



Zero coupon example l.jpg
Zero-coupon example

Fabozzi Ch 4


Example l.jpg
Example

y=10%, y=0.5%

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%

Fabozzi Ch 4


Property 1 l.jpg
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.

Fabozzi Ch 4


Property 2 l.jpg
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?

Fabozzi Ch 4


Property 3 l.jpg
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?

Fabozzi Ch 4


Property 4 l.jpg
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?

Fabozzi Ch 4


What affects price volatility l.jpg
What affects price volatility?

  • Linkage

  • Credit considerations

  • Time to maturity

  • Coupon rate

Fabozzi Ch 4


Bond price volatility16 l.jpg
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.

Fabozzi Ch 4



Duration l.jpg
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/(1+y)t ]

  • suggested weight of each coupon:

  • wt = CFt/(1+y)t /Bond Price

  • What is the sum of all wt?

Fabozzi Ch 4


Duration19 l.jpg
Duration

  • The bond price volatility is proportional to the bond’s duration.

  • Thus duration is a natural measure of interest rate risk exposure.

Fabozzi Ch 4


Modified duration l.jpg
Modified Duration

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

Fabozzi Ch 4


Duration21 l.jpg
Duration

Fabozzi Ch 4


Duration22 l.jpg
Duration

Fabozzi Ch 4


Duration23 l.jpg
Duration

Fabozzi Ch 4



The yield to maturity l.jpg
The Yield to Maturity

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

Fabozzi Ch 4


Macaulay duration l.jpg
Macaulay Duration

  • Definition of duration, assuming t=0.

Fabozzi Ch 4


Macaulay duration27 l.jpg
Macaulay Duration

  • What is the duration of a zero coupon bond?

A weighted sum of times to maturities of each coupon.

Fabozzi Ch 4


Meaning of duration l.jpg

$

r

Meaning of Duration

Fabozzi Ch 4


Parallel shift l.jpg

upward move

Current TS

Downward move

Parallel shift

r

T

Fabozzi Ch 4


Comparison of two bonds l.jpg

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.

Comparison of two bonds

Fabozzi Ch 4


Duration31 l.jpg
Duration

D

Zero coupon bond

15% coupon, YTM = 15%

Maturity

0 3m 6m 1yr 3yr 5yr 10yr 30yr

Fabozzi Ch 4


Example32 l.jpg
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

Fabozzi Ch 4


What determines duration l.jpg
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 (1+y)/y.

Fabozzi Ch 4


What determines duration34 l.jpg
What Determines Duration?

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

Fabozzi Ch 4


Convexity l.jpg

$

r

Convexity

Fabozzi Ch 4


Example36 l.jpg
Example

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

The duration is 10 years, the modified duration is:

The convexity is

Fabozzi Ch 4


Example37 l.jpg
Example

If the yield changes to 7% the price change is

Fabozzi Ch 4


Frm 98 question 17 l.jpg
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

Fabozzi Ch 4


Frm 98 question 1739 l.jpg
FRM-98, Question 17

Fabozzi Ch 4


Frm 98 question 22 l.jpg
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

Fabozzi Ch 4


Frm 98 question 2241 l.jpg
FRM-98, Question 22

Fabozzi Ch 4


Portfolio duration l.jpg
Portfolio Duration

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

Fabozzi Ch 4


Bond price derivatives l.jpg
Bond Price Derivatives

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

Dollar convexity = the second derivative,

C - convexity.

Fabozzi Ch 4



Alm duration l.jpg
ALM Duration

  • Does NOT work!

  • Wrong units of measurement

  • Division by a small number

Fabozzi Ch 4


Duration gap l.jpg
Duration Gap

  • A - L = C, assets - liabilities = capital

Fabozzi Ch 4


Alm duration47 l.jpg
ALM Duration

  • A similar problem with measuring yield

Fabozzi Ch 4



Slide49 l.jpg

Fabozzi Ch 4


Very good question l.jpg
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?

Fabozzi Ch 4


Home assignment l.jpg
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?

Fabozzi Ch 4


Home assignment chapter 4 l.jpg
Home AssignmentChapter 4

  • Ch. 4: Questions 1, 2, 3, 4, 15.

  • Calculate duration of a consul (perpetual bond).

Fabozzi Ch 4


End ch 4 l.jpg
End Ch. 4

Fabozzi Ch 4


Understanding of duration convexity l.jpg
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?

Fabozzi Ch 4


Callable bond l.jpg
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.

Fabozzi Ch 4


Puttable bond l.jpg
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.

Fabozzi Ch 4


Embedded call option l.jpg

regular bond

strike

callable bond

Embedded Call Option

r

Fabozzi Ch 4


Embedded put option l.jpg

puttable bond

Embedded Put Option

regular bond

r

Fabozzi Ch 4


Convertible bond l.jpg

Stock

Convertible Bond

Straight Bond

Convertible Bond

Payoff

Stock

Fabozzi Ch 4


Timing of exercise l.jpg
Timing of exercise

  • European

  • American

  • Bermudian

  • Lock up time

Fabozzi Ch 4


Macaulay duration61 l.jpg
Macaulay Duration

Modified duration

Fabozzi Ch 4


Bond price change l.jpg
Bond Price Change

Fabozzi Ch 4



Exercise l.jpg
Exercise

  • Find the duration and convexity of a consol (perpetual bond).

  • Answer: (1+y)/y.

Fabozzi Ch 4


Frm 98 question 29 l.jpg
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

Fabozzi Ch 4


Frm 97 question 24 l.jpg
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.

Fabozzi Ch 4


Frm 99 question 75 l.jpg
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

Fabozzi Ch 4


Portfolio duration and convexity l.jpg
Portfolio Duration and Convexity

  • Portfolio weights

Fabozzi Ch 4


Example69 l.jpg
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

Fabozzi Ch 4


Slide70 l.jpg

Modified duration

  • Barbel portfolio consists of very short and very long bonds.

  • Bullet portfolio consists of bonds with similar maturities.

  • Which of them has higher convexity?

Fabozzi Ch 4


Frm 98 question 18 l.jpg
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

Fabozzi Ch 4


Frm 98 question 1872 l.jpg
FRM-98, Question 18

Note that $100 means notional amount

and can be misunderstood.

Fabozzi Ch 4


Useful formulas l.jpg
Useful formulas

Fabozzi Ch 4


Volatilities of ir bond prices l.jpg
Volatilities of IR/bond prices

  • Pricevolatility 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

Fabozzi Ch 4


Duration approximation l.jpg
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%).

Fabozzi Ch 4


Volatilities of yields l.jpg
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

Fabozzi Ch 4