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Finance & Structuring for DS

IEOR 4574

Spring  2022

Homework 1

Duration & its Application  

Background

Duration (Macaulay) can be used to estimate the change in the price of a bond for small changes in its yield. Let D be the durations of a bond with price of P and yield of Y. If the yield changes by a small amount ΔY the change in the price of the bond will be approximately

       ΔP ≈ -D * P * ΔY

For illustration, suppose P= 101, Y = 1.75, and D= 3.36. If the yield drops by 2 bps to 1.73 the price of the bond will increase by approximately  

ΔP ≈ -3.36 * 101 * (-0.0002) = 0.0679 and the new bond price will be 101.0679.

Suppose the face value of this bond was $1,000 and you had 1,500 of them. The value of your portfolio when the price was 101 would be (101/100)*$1,000*1,500 = $1,515,000. If the yield drops to 1.73 the value of your portfolio will increase by approximately

    (0.0679/100)*$1000*1500 = $1,018.5. Similarly if the yield increases by 2 bps the bond price will drop by ΔP ≈ -3.36 * 101 * (0.0002) = -0.0679 and the value of your portfolio will fall by approximately $1,018.5.

      Modified Duration is Macaulay duration/ (1+ yield) and using this number provides a closer approximation. For the above example the

Modified Duration (MD) = D/(1+Y) = 3.36/(1+0.0175) = 3.3022. Using this number in the above calculations will provide a closer approximation.

              If the change in yield is large say 20 bps or 50 bps, then the error of approximation will be rather large and to improve the accuracy one has to include another term that is called Convexity.

Application

Following are the 2 year T₂ and 10 year T₁₀ Treasury prices on 9^th, 14^th, and 16^th of September 2021. Note that in the case of Treasury a price of say 99²⁴ means  99+24/32 =  99.75

              Price

Bond Coupon 9/9/21 9/14/21 9/16/21  

   T₂     3/8 99 ²⁶ 99 ¹⁷  99 ²⁴

   T₁₀    1 3/4 98 ¹⁸ 98 ¹²  99 ⁰⁸

1. Calculate the yield of T₂ and T₁₀ on each of the above dates. The Coupons are paid semi-annually and assume the first period is ½ of the year in all cases (i.e., ignore the passage of a two weeks that shorten the first period)

2. Calculate the Duration of T₂ and T₁₀  on 9/9/21

3. Using Duration and change in yield of T₂ and T₁₀  on 9/14/21 and 9/16/21 estimate the price of these bonds on those days and determine the difference between the actual prices and estimated prices.

4. Repeat part 3 using Modified Duration.