What does Macaulay duration of say 3.7235 years mean for 4 year 5% annual pay bond trading at par?
Modified duration - for a 20 year 4% annual pay bond trading at par. for a 25 basis point change in yield, the change comes to 13.6. This implied that for a 1% change in yield, the bond value changes by 13.6%.
Current price (Vo) = 1000, Price at 4.25% (V+) = 966.76, Price at 3.75% (V-) = 1034.74
After applying the formula, the value comes to 13.6. So, for 1% change in YTM, price of bond changes by 13.6%. However, price at 5.25% is 847, and price at 4.25% is 966.76, which shows a change of 15%. How can this be not 13.6?
It says that the present-value-weighted-average time to receipt of cash flows is 3.7235 years.
And, given the relationship between Macaulay duration and modified duration, it says that for a small change in the YTM, the percentage price change will be approximately -3.5462 (= 3.7235 / 1.05) times the percentage yield change.
A modified duration of 13.6 (years) means that for a 1% change in yield, the price change will be approximately -13.6%. It’s approximate because you’re using a straight line to try to compute the price change along the price-yield curve. So,
The actual price change will differ from the duration approximation, and the greater the yield change, the greater the difference is likely to be, and
The calculated duration will be different for different starting YTMs (or, equivalently, different starting prices. The percentage price change when the YTM changes from 3.75% to 4.25% will not be the same as (half of) the percentage price change when the YTM changes from 4.25% to 5.25%.