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Example 4. Compare price sensitivities to yield changes for a 1-year and a 10-year zero coupon bonds with the face value of $1,000, both initially yielding 5%:

V_{b_zero,1yr,5%} = $1,000/1.05 = $952.38 V_{b_zero,10yrs,5%} = $1,000/1.05¹⁰ =$1,000/1.6289 = $613.91

Of course, the 10-year zero is worth much less than the one-year zero because $1,000 ten years from now is less valuable than $1,000 one year from now (the discount rate is the same). We learned it in Ch. 4 (TVM), so this is not the main point of the exercise.

Lets now assume the yield falls from 5% to 4% for both bonds. The new prices are:

V_{b_zero,1yr,4%} = $1,000/1.04 = $961.54 V_{b_zero,10yrs,4%} = $1,000/1.04¹⁰ =$1,000/1.4802 = $675.56

Both bonds increased in price because we decreased the discount rate, causing the present value to go up. This is also nothing new we have established the inverse relation between bond prices and interest rates earlier in this chapter. The point of this exercise is to see how much the two bonds prices changed compared to one another. Let's find percentage changes (%'s) in the prices:

%Price = (NewPrice - OldPrice)/OldPrice:

%V_{b_zero,1yr,4%to5%} = (961.54 952.38)/952.38 = 0.96%

%V_{b_zero,10yrs,4%to5%} = (675.56 613.91)/613.91 = 10.04%

If the yield of the bonds from the above example went up from 5% to 6%, the 1-year bond price would ( type "fall" or "rise") by (blank)%, and the 10-year bond price would (fall or rise) by (blank)%. Round to 0.01%, drop the % symbol in the answer.

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