Macaulay Duration (traditionally just called Duration)
The formula usually used to calculate a bond's basic duration is the
Macaulay duration, which was created by Frederick Macaulay in 1938,
although it was not commonly used until the 1970s. Macaulay duration is
calculated by adding the results of multiplying the present value of
each cash flow by the time it is received and dividing by the total
price of the security.
Modified Duration
Modified duration is a modified version of the Macaulay model that accounts for changing interest rates.
Because they affect yield, fluctuating interest rates will affect
duration, so this modified formula shows how much the duration changes
for each percentage change in yield. For bonds without any embedded
features, bond price and interest rate move in opposite directions, so
there is an inverse relationship between modified duration and an
approximate 1% change in yield. Because the modified duration formula
shows how a bond's duration changes in relation to interest rate
movements, the formula is appropriate for investors wishing to measure
the volatility of a particular bond.
Effective Duration
The modified duration formula discussed above assumes that the expected
cash flows will remain constant, even if prevailing interest rates
change; this is also the case for option-free fixed-income securities.
On the other hand, cash flows from securities with embedded options or
redemption features will change when interest rates change. For
calculating the duration of these types of bonds, effective duration is
the most appropriate.