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Homework answers / question archive / You've learned how to compute price elasticity with respect to changes in interest rates (modified duration) of a bond under the flat term structure of interest rates, and parallel shifts in the term structure
You've learned how to compute price elasticity with respect to changes in interest rates (modified duration) of a bond under the flat term structure of interest rates, and parallel shifts in the term structure. In this problem, you are asked to compute the price elasticity of a bond under a non-flat yield curve. The present value (price) of a bond is B =[# 1 (1+r)" CF Now consider the parallel shift of the yield curve by x, i.e. rt + rt+, Vt. Then the price elasticity of the bond with respect to parallel shifts in the yield curve can be written as PE = 1 db=t=1 (1+rojet txCF We can now approximate price changes in response to a small parallel shift Ar in the yield curve as ΔΒ -Βx (PE| x Δη Note that when the yield curve is flat, the absolute value of price elasticity |PE| equals modified duration. Consider the following term structure, which is upward sloping: 1-yr 2-yr 3-yr 4-yr 5-yr 2.1% 2.4% 2.67% 2.88% 3.06% A 5-year Treasury note (T-note) has a face value of $100 and a 2.62% coupon rate (assume annual payments). (a) Compute the price of the T-note. 98.09 (b) Suppose that the yield curve suddenly shifted down by 0.25% (this is a parallel shift, all points on the yield curve shift by the same amount). Compute the new price of the T-note and report the price change relative to the original price computed in (a). 99.14 X
Given:
Face Value = $ 100
Coupon rate = 2.62% (Annual coupon)
YTM structure:
| Year | Yield |
| 1 | 2.10% |
| 2 | 2.40% |
| 3 | 2.67% |
| 4 | 2.88% |
| 5 | 3.06% |
Coupon Payment = Coupon rate * Face Value
Hence, Coupon Payment = 2.62% * 100 = 2.62
Solution b:
Yield curve suddenly shifted down by 0.25%, (Parallel shift and all points on the uyield curve shift by the same amount)
New Yield = Old Yield - 0.25%
Hence, the New Ytm Structure is:
| Year | Old Yield | New Yield |
| 1 | 2.10% | 1.8500% |
| 2 | 2.40% | 2.1500% |
| 3 | 2.67% | 2.4200% |
| 4 | 2.88% | 2.6300% |
| 5 | 3.06% | 2.8100% |
The formula for the discounitng factor is :
DF = 1/ (1+r)n
Where, r is the yield, n is the period
| Year | Old Yield | New Yield | DF | DF |
| 1 | 2.10% | 1.8500% | 1/(1+1.85%)1 | 0.981836 |
| 2 | 2.40% | 2.1500% | 1/(1+2.15%)2 | 0.958348 |
| 3 | 2.67% | 2.4200% | 1/(1+2.42%)3 | 0.930777 |
| 4 | 2.88% | 2.6300% | 1/(1+2.63%)4 | 0.901369 |
| 5 | 3.06% | 2.8100% | 1/(1+2.81%)5 | 0.870609 |
Now, Let us multiply the Coupon Payment values with the discounting factors to get the PV of the Payments. Note that in Year 5, the Value of the Face bond is also included.
| Year | Old Yield | New Yield | DF | Payments | PV of Payments | PV of Payments |
| 1 | 2.10% | 1.8500% | 0.981836 | 2.62 | 0.981836 * 2.62 | 2.57241 |
| 2 | 2.40% | 2.1500% | 0.958348 | 2.62 | 0.958348 * 2.62 | 2.510872 |
| 3 | 2.67% | 2.4200% | 0.930777 | 2.62 | 0.930777 * 2.62 | 2.438636 |
| 4 | 2.88% | 2.6300% | 0.901369 | 2.62 | 0.901369 * 2.62 | 2.361587 |
| 5 | 3.06% | 2.8100% | 0.870609 | 2.62 | 0.870609 * 2.62 | 2.280996 |
| 5 | 3.06% | 2.8100% | 0.870609 | 100 | 0.870609 * 100 | 87.06091 |
| Total PV of Payments= | 99.22541 |
Hence, the Price of the Bond = 99.22541 = 99.23
The Price change (difference) between the payments in case a and b is:
$ 99.23 - $ 98.09 = $ 1.14
