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

Derivatives F(x) x Zvi Wiener Fabozzi Ch 4

Properties of derivatives Zvi Wiener Fabozzi Ch 4

Zero-coupon example Zvi Wiener Fabozzi Ch 4

Example y=10%, y=0.5% T P0 P1 P 1 90.90 90.09 -0.45% 1 90.90 90.09 -0.45% 2 82.64 81.16 -1.79% 10 38.55 35.22 -8.65% Zvi Wiener Fabozzi Ch 4

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

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

Duration Zvi Wiener Fabozzi Ch 4

Duration Zvi Wiener Fabozzi Ch 4

Duration Zvi Wiener Fabozzi Ch 4

Measuring Price Change Zvi Wiener Fabozzi Ch 4

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

Duration D Zero coupon bond 15% coupon, YTM = 15% Maturity 0 3m 6m 1yr 3yr 5yr 10yr 30yr Zvi Wiener Fabozzi Ch 4

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

Convexity r $ Zvi Wiener Fabozzi Ch 4

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

Example If the yield changes to 7% the price change is Zvi Wiener Fabozzi Ch 4

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

Duration of a portfolio Zvi Wiener Fabozzi Ch 4

ALM Duration Does NOT work! Wrong units of measurement Division by a small number Zvi Wiener Fabozzi Ch 4

Duration Gap A - L = C, assets - liabilities = capital Zvi Wiener Fabozzi Ch 4

ALM Duration A similar problem with measuring yield Zvi Wiener Fabozzi Ch 4

Do not think of duration as a measure of time! Zvi Wiener Fabozzi Ch 4

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

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

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

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

Embedded Put Option r puttable bond regular bond Zvi Wiener Fabozzi Ch 4

Convertible Bond Stock Payoff Convertible Bond Straight Bond Stock Zvi Wiener Fabozzi Ch 4

Macaulay Duration Modified duration Zvi Wiener Fabozzi Ch 4

Bond Price Change Zvi Wiener Fabozzi Ch 4

Duration of a coupon bond Zvi Wiener Fabozzi Ch 4

Portfolio Duration and Convexity Portfolio weights Zvi Wiener Fabozzi Ch 4

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

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

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 Zvi Wiener Fabozzi Ch 4

Useful formulas Zvi Wiener Fabozzi Ch 4

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 Zvi Wiener Fabozzi Ch 4

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

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 Zvi Wiener Fabozzi Ch 4