What is the difference between Macaulay, Modified and Effective duration?

Macaulay duration is a measure of a bond's price sensitivity to changes in interest rates, expressed in years. It takes into account the timing and size of all of the bond's cash flows, including interest payments and the principal repayment at maturity, and calculates the present value-weighted average time until each cash flow is received. 

Macaulay duration is useful for determining the average maturity of a bond's cash flows and for comparing the interest rate risk of bonds with different maturities and coupon rates.

It is calculated as the Macaulay duration divided by 1 plus the bond's yield to maturity, or as the percentage change in the bond's price for a given percentage change in the market interest rate.

Modified duration measures the sensitivity of a bond's price to changes in its yield to maturity, assuming a parallel shift in the yield curve. In other words, it measures how much the bond's price would change for a given change in its yield to maturity, while holding all other factors constant. 

This measure is useful for estimating the potential impact of changes in interest rates on a bond's price.

Modified duration is always less than Macaulay duration.

Effective duration, on the other hand, takes into account the possibility of changes in the shape of the yield curve, rather than assuming a parallel shift. 

It calculates the weighted average time until the bond's cash flows are received, with the present value of each cash flow calculated using market interest rates plus or minus a specified change in interest rates. 

This measure reflects the impact of both changes in interest rates and changes in the yield curve on a bond's price, making it a more accurate measure of a bond's interest rate risk.

As a result, effective duration is generally considered to be a more accurate measure of a bond's interest rate risk, especially for bonds with embedded options or other complex structures like callable and putable bonds.

To summarize, modified duration measures yield duration by assuming a parallel shift in the yield curve, while effective duration measures curve durationbytakingintoaccountchangesintheshapeoftheyield curve.

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