9.12 Effect of Increased Yield on Duration
Problem:
Using the same bond from before, let’s calculate the duration if the yield increased from 10% to 12%.
| Period (p) | Coupon (10%; semi) | PVF @ .06 | PVCF | p × PVCF |
|---|---|---|---|---|
| 1 | 50 | .9434 | $47.17 | $47.17 |
| 2 | 50 | .8900 | 44.50 | 89.00 |
| 3 | 50 | .8396 | 41.98 | 125.94 |
| 4 | 50 | .7921 | 39.61 | 158.44 |
| 5 | 50 | .7473 | 37.37 | 186.85 |
| 6 | 50 | .7050 | 35.25 | 211.50 |
| 7 | 50 | .6651 | 33.26 | 232.82 |
| 8 | 50 | .6274 | 31.37 | 250.96 |
| 9 | 50 | .5919 | 29.60 | 266.40 |
| 10 | 50 | .5584 | 27.92 | 279.20 |
| 1,000 | .5584 | 558.40 | 5,584.00 | |
| Total | $1,500 | 926.43 | 7,432.28 |
| Macaulay Duration | = (p × PVCF) ÷ (Price) ÷ 2 |
| = (7,432.28 ÷ 926.43) ÷ 2 = 4.01 years |
Notes:
- This compares with a Macaulay-Duration of 4.05 years at the lower discount rate of 10%. That is, when YTM increases, Duration goes down.
- YTM affects the PV of both the numerator and dominator (in the Macaulay duration formula above), but as the numerator contains multipliers (i.e., the periods), it goes down relatively more than the denominator.
- In this problem, the numerator went from $8,107 (see prior page) to $7,432, a drop of 8.33%. The denominator declined from $1,000 to $926, a smaller relative decline of 7.4%. If the numerator goes down relatively more than the denominator, the Duration figure will also go down – as it has from 4.05 to 4.01.
- Dollar Price and Duration are positively related – relative to YTM.
