The formula for duration (assuming a parallel shift in interest rates) and for spread duration (assuming a parallel shift in credit spreads) produce the same result.

(a) How is empirical duration estimated?

(b) Is empirical duration typically greater than or less than the analytical duration computed through the formula for spread duration? Why?

(c) Is the difference between empirical duration and the formula for analytical duration generally greater for investment grade or non-investment grade bonds? Why?