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**Bond Price Volatility Zvi Wiener Based on Chapter 4 in Fabozzi**

Bond Markets, Analysis and Strategies Fall-02

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**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? Zvi Wiener Fabozzi Ch 4

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**Money Manager value Original plan New plan Market shock 0 t D**

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8% Coupon Bond Zero Coupon Bond Zvi Wiener Fabozzi Ch 4

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**Price-Yield for option-free bonds**

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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: Zvi Wiener Fabozzi Ch 4

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Derivatives F(x) x Zvi Wiener Fabozzi Ch 4

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**Properties of derivatives**

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Zero-coupon example Zvi Wiener Fabozzi Ch 4

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**Example y=10%, y=0.5% T P0 P1 P 1 90.90 90.09 -0.45%**

% % % Zvi Wiener Fabozzi Ch 4

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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. Zvi Wiener Fabozzi Ch 4

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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? Zvi Wiener Fabozzi Ch 4

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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? Zvi Wiener Fabozzi Ch 4

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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? Zvi Wiener Fabozzi Ch 4

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**What affects price volatility?**

Linkage Credit considerations Time to maturity Coupon rate Zvi Wiener Fabozzi Ch 4

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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. Zvi Wiener Fabozzi Ch 4

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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/(1+y)t ] suggested weight of each coupon: wt = CFt/(1+y)t /Bond Price What is the sum of all wt? Zvi Wiener Fabozzi Ch 4

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Duration The bond price volatility is proportional to the bond’s duration. Thus duration is a natural measure of interest rate risk exposure. Zvi Wiener Fabozzi Ch 4

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Modified Duration The percentage change in bond price is the product of modified duration and the change in the bond’s yield to maturity. Zvi Wiener Fabozzi Ch 4

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Duration Zvi Wiener Fabozzi Ch 4

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Duration Zvi Wiener Fabozzi Ch 4

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Duration Zvi Wiener Fabozzi Ch 4

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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 Zvi Wiener Fabozzi Ch 4

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**Macaulay Duration Definition of duration, assuming t=0. Zvi Wiener**

Fabozzi Ch 4

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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? Zvi Wiener Fabozzi Ch 4

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Meaning of Duration r $ Zvi Wiener Fabozzi Ch 4

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**Parallel shift r upward move Current TS Downward move T Zvi Wiener**

Fabozzi Ch 4

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**Comparison of two bonds**

Coupon bond with duration Price (at 5% for 6m.) is $ If IR increase by 1bp (to 5.01%), its price will fall to $ , or 0.359% decline. Zero-coupon bond with equal duration must have years to maturity. At 5% semiannual its price is ($1,000/ )=$ If IR increase to 5.01%, the price becomes: ($1,000/ )=$831.66 0.359% decline. Zvi Wiener Fabozzi Ch 4

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**Duration D Zero coupon bond 15% coupon, YTM = 15% Maturity**

0 3m 6m 1yr 3yr 5yr 10yr 30yr Zvi Wiener Fabozzi Ch 4

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**Example A bond with 30-yr to maturity Coupon 8%; paid semiannually**

YTM = 9% P0 = $897.26 D = Yrs if YTM = 9.1%, what will be the price? P/P = - y D* P = -(y D*)P = -$9.36 P = $ $9.36 = $887.90 Zvi Wiener Fabozzi Ch 4

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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 (1+y)/y. Zvi Wiener Fabozzi Ch 4

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**What Determines Duration?**

Holding the coupon rate constant, duration not always increases with ttm. Zvi Wiener Fabozzi Ch 4

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Convexity r $ Zvi Wiener Fabozzi Ch 4

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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 Zvi Wiener Fabozzi Ch 4

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**Example If the yield changes to 7% the price change is Zvi Wiener**

Fabozzi Ch 4

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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 If the yield decreases by 1 bp, the price will increase to What is the modified duration of this bond? A. 5.0 B. -5.0 C. 4.5 D. -4.5 Zvi Wiener Fabozzi Ch 4

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FRM-98, Question 17 Zvi Wiener Fabozzi Ch 4

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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 B C D Zvi Wiener Fabozzi Ch 4

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FRM-98, Question 22 Zvi Wiener Fabozzi Ch 4

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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. Zvi Wiener Fabozzi Ch 4

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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. Zvi Wiener Fabozzi Ch 4

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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 Zvi Wiener Fabozzi Ch 4

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**Duration Gap A - L = C, assets - liabilities = capital Zvi Wiener**

Fabozzi Ch 4

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**ALM Duration A similar problem with measuring yield Zvi Wiener**

Fabozzi Ch 4

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

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**Principal component duration Partial duration**

Key rate duration Principal component duration Partial duration Zvi Wiener Fabozzi Ch 4

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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? Zvi Wiener Fabozzi Ch 4

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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? Zvi Wiener Fabozzi Ch 4

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**Home Assignment Chapter 4**

Ch. 4: Questions 1, 2, 3, 4, 15. Calculate duration of a consul (perpetual bond). Zvi Wiener Fabozzi Ch 4

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End Ch. 4 Zvi Wiener Fabozzi 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? Zvi Wiener Fabozzi Ch 4

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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. Zvi Wiener Fabozzi Ch 4

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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. Zvi Wiener Fabozzi Ch 4

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**Embedded Call Option r regular bond strike callable bond Zvi Wiener**

Fabozzi Ch 4

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**Embedded Put Option r puttable bond regular bond Zvi Wiener**

Fabozzi Ch 4

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**Convertible Bond Stock Payoff Convertible Bond Straight Bond Stock**

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**Timing of exercise European American Bermudian Lock up time Zvi Wiener**

Fabozzi Ch 4

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Macaulay Duration Modified duration Zvi Wiener Fabozzi Ch 4

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Bond Price Change Zvi Wiener Fabozzi Ch 4

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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: (1+y)/y. Zvi Wiener Fabozzi Ch 4

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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 Zvi Wiener Fabozzi Ch 4

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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. Zvi Wiener Fabozzi Ch 4

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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 Zvi Wiener Fabozzi Ch 4

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**Portfolio Duration and Convexity**

Portfolio weights Zvi Wiener Fabozzi Ch 4

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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 Zvi Wiener Fabozzi Ch 4

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

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? Zvi Wiener Fabozzi Ch 4

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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 What is the duration of the portfolio? A. 0.68 B. 0.61 C D Zvi Wiener Fabozzi Ch 4

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**FRM-98, Question 18 Note that $100 means notional amount**

and can be misunderstood. Zvi Wiener Fabozzi Ch 4

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Useful formulas Zvi Wiener Fabozzi Ch 4

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**Volatilities of IR/bond prices**

Price volatility in % End 99 End 96 Euro 30d Euro 180d Euro 360d Swap 2Y Swap 5Y Swap 10Y Zero 2Y Zero 5Y Zero 10Y Zero 30Y Zvi Wiener Fabozzi Ch 4

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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%). Zvi Wiener Fabozzi Ch 4

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**Volatilities of yields**

Yield volatility in %, 99 (y/y) (y) Euro 30d Euro 180d Euro 360d Swap 2Y Swap 5Y Swap 10Y Zero 2Y Zero 5Y Zero 10Y Zero 30Y Zvi Wiener Fabozzi Ch 4

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