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## Duration

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**Duration**Riccardo Colacito**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?**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**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**Duration is**where Duration: Calculation**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?)**Just use the formula**… or three years Duration is…**Question**• What happens if the YTM decreases? • Would you expect the duration to increase or decrease? • Why?**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**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.**Question**• What is the duration of a zero coupon bond? • The duration of a zero coupon bond is equal to its maturity**A perpetuity**• Definition: a security that pays a constant coupon forever • Its maturity is infinite! • Its duration is**Duration vs maturity**• Holding the coupon rate constant, a bond’s duration generally increases with time to maturity**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.**Uses of Duration**• Summary measure of length or effective maturity for a portfolio • Measure of price sensitivity for changes in interest rate • Immunization of interest rate risk (next class)**Duration/Price Relationship**• Take derivative of price wrt YTM • Hence the following holds as an approximation**More compact formula**where Modified Duration**Fact #1**• Prices of long-term bonds are more sensitive to interest rate changes than prices of short term bonds**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**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.**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.**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**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**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.**Intuition**• The higher the YTM, the lower the duration • The lower the duration, the lower the price sensitivity • High YTM implies low price sensitivity**Convexity**• The formula is an approximation of the more precise relation**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