VIEWS: 6 PAGES: 40 POSTED ON: 11/26/2011
Duration Riccardo Colacito 1 Motivational example • Two bonds sell today with the same payment schedule – Time 1: $10 – Time 2: $20 • However one has a YTM of 1% and the other one has a YTM of 99% • What is the temporal distribution of the payments? 2 Example (cont’d) Time Payment PV (YTM=1%) PV (YTM=99%) 1 10 9.9 (33.6%) 5.025 (49.9%) 2 20 19.6 (66.4%) 5.050 (50.1%) 3 Effective maturity? • Both bonds have maturity 2 years • But they differ in the temporal distribution of the (present value of) payments • Their effective maturities should differ, i.e. the second one should have shorter maturity 4 Duration • A measure of the effective maturity of a bond • The weighted average of the times until each payment is received, with the weights proportional to the present value of the payment • Duration is always shorter or equal to maturity for all bonds 5 Duration: Calculation • Duration is T D t wt t 1 where CFt (1 y ) t wt Bond Price 6 Duration: an example • Consider the following bond – Maturity: 4 years – Par: $500 – Coupon: $80 (once per year) – YTM: 38.5% – Price is $287.205 (remember how to compute this?) 7 Compute the duration Present Value Year CF weight CF 1 80 57.76 0.20 2 80 41.71 0.15 3 80 30.11 0.10 4 580 157.63 0.55 8 Duration is… • Just use the formula D 1.20 2 .15 3.10 4 .55 3 • … or three years 9 Cash flows and their present values 700 600 500 400 CF Dicounted CF 300 200 100 0 1 2 3 4 10 Question • What happens if the YTM decreases? • Would you expect the duration to increase or decrease? • Why? 11 Duration: another example • Consider the following bond – Maturity: 4 years – Par: $500 – Coupon: $80 (once per year) – YTM: 15.6% 38.5% – Price is $505.641 $287.205 12 How are the weights affected? 700 600 500 400 300 200 100 0 1 2 3 4 13 What happens to the duration? • You now attach a relatively higher weight to cash flows that happen further in the future • That is: the duration goes up! • Can verify that the duration is now 3 years and a quarter. 14 Question • What is the duration of a zero coupon bond? • The duration of a zero coupon bond is equal to its maturity 15 A perpetuity • Definition: a security that pays a constant coupon forever • Its maturity is infinite! • Its duration is 1 y D y 16 Duration vs maturity • Holding the coupon rate constant, a bond’s duration generally increases with time to maturity 17 Duration vs maturity 35 30 Zero-coupon bond 25 20 15 10 Par=$1,000, Coupon=$30 (yearly), YTM=5% 5 0 0 5 10 15 20 25 30 35 18 Duration vs Maturity (cont’d) 35 30 Zero-coupon bond 25 20 Par=$1,000, Coupon=$30 (yearly), YTM=15% 15 10 5 Perpetuity 0 0 5 10 15 20 25 30 35 19 Why does this happen? • As the YTM increases, the present value of the face value at high maturities gets smaller and smaller. • That is the bond starts looking as a perpetuity with a very high yield to maturity. 20 Figure 10.3 Duration as a Function of Maturity 21 Uses of Duration 1. Summary measure of length or effective maturity for a portfolio 2. Measure of price sensitivity for changes in interest rate 3. Immunization of interest rate risk (next class) 22 Duration/Price Relationship • Take derivative of price wrt YTM P T C Par t t 1 T (1 y ) t 1 (1 y ) (1 y )T 1 1 T C Par t (1 y )t T (1 y )T 1 y t 1 1 D P 1 y • Hence the following holds as an approximation P (1 y ) D P 1 y 23 Modified Duration • More compact formula P y D* P where D D * 1 y 24 Fact #1 • Prices of long-term bonds are more sensitive to interest rate changes than prices of short term bonds 25 0 500 1000 1500 2000 2500 3000 3500 4000 0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05 0.05 0.06 0.06 0.07 0.07 0.08 0.08 0.09 0.09 0.1 0.1 0.11 0.11 0.12 0.12 Fact #1 0.13 0.13 0.14 0.14 0.15 0.15 0.16 0.16 0.17 0.17 0.18 0.18 0.19 0.19 30 10 26 Intutition • The higher the maturity, the higher the duration • The higher the duration, the higher the price sensitivity • Higher maturity implies higher price sensitivity to interest rate risk 27 Fact #2 • The sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases. • Equivalently, interest rate risk is less than proportional to bond maturity. 28 0 1000 2000 3000 4000 5000 6000 0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05 0.05 0.06 0.06 0.07 0.07 0.08 0.08 0.09 0.09 0.1 0.1 0.11 0.11 0.12 0.12 Fact #2 0.13 0.13 0.14 0.14 0.15 0.15 0.16 0.16 0.17 0.17 0.18 0.18 0.19 0.19 50 30 10 29 Intuition • Duration increases less than proportionally with maturity (e.g. see slide 18) • Hence price sensitivity to interest rate risk increases less than proportionally with maturity. 30 Fact #3 • Prices of high coupon bonds are less sensitive to changes in interest rates than prices of low coupon bonds. • Or equivalently, interest rate risk is inversely related to the bond’s coupon rate 31 -100 100 300 500 700 900 1100 1300 1500 0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05 0.05 0.06 0.06 0.07 0.07 0.08 0.08 0.09 0.09 0.1 0.1 0.11 0.11 0.12 0.12 Fact #3 0.13 0.13 0.14 0.14 0.15 0.15 0.16 0.16 0.17 0.17 0.18 0.18 0.19 0.19 10 100 32 Intuition • The higher the coupon, the lower the duration • Lower duration implies lower price sensitivity. • The higher the coupon, the lower the price sensitivity to interest rate risk 33 Fact #4 • The sensitivity of a bond’s price to a change in its yield is inversely related to the yield to maturity at which the bond currently is selling. 34 Fact #4 2000 1800 Percentage Price change =(1341-1000)/1000=34.1% 1600 1400 1200 Percentage Price change =(769-610)/610=26% 1000 800 600 400 200 0 0.001 0.011 0.021 0.031 0.041 0.051 0.061 0.071 0.081 0.091 0.101 0.111 0.121 0.131 0.141 0.151 0.161 0.171 0.181 0.191 Busi 580 - Investments 35 Intuition • The higher the YTM, the lower the duration • The lower the duration, the lower the price sensitivity • High YTM implies low price sensitivity 36 Convexity • The formula P y D* P is an approximation of the more precise relation P y D convexity y * 1 2 P 2 37 Figure 10.6 Bond Price Convexity 38 The higher the convexity, the better it is for the investor 39 Yet another fact • An increase in a bond’s yield to maturity results in a smaller price change than a decrease in yield of equal magnitude 2000 1800 1600 1400 1200 1000 800 600 400 200 0 0.001 0.011 0.021 0.031 0.041 0.051 0.061 0.071 0.081 0.091 0.101 0.111 0.121 0.131 0.141 0.151 0.161 0.171 0.181 0.191 40