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##### Subject 3. Empirical Duration

The purpose of calculating duration is to estimate a bond's interest-rate risk. However, a more general way to think about the risk associated with a bond is that it is comprised of two elements: an equity risk and an interest-rate risk. For corporate bonds that have a high credit rating (i.e., investment-grade bonds), the interest-rate risk is the dominant risk. In contrast, for corporate bonds with a low credit rating (i.e., non-investment-grade or high-yield corporate bonds), the equity risk is more likely to be dominant. Hence, the duration measure as calculated by formula (analytical duration) may not be a good measure of interest-rate risk for such bonds.

Empirical duration and analytical duration are two different approaches to measuring the sensitivity of a bond's price to changes in interest rates. While both measures are used to assess interest rate risk, they differ in their calculation methods and underlying assumptions.

Empirical Duration

Empirical duration, also known as historical duration or effective duration, is calculated using historical data on a bond's price and yield movements. It is based on observed changes in market prices and yields over a specific time period. The formula for empirical duration is:

Empirical Duration = (δP / P) / δY

where δP is the change in the bond's price, P is the original price, and δY is the change in yield.

Empirical duration reflects the bond's historical sensitivity to changes in interest rates based on observed market behavior. It considers both price changes and yield changes, providing a measure of the bond's average price volatility over a specific historical period.

The advantages of empirical duration include that the estimate does not rely on theoretical formulas and analytic assumptions; the investor only needs a reliable series of bond prices and a reliable series of Treasury yields. It is a better measure especially under stressed market conditions, for a portfolio consisting of a variety of different bonds from different issuers.

Disadvantages include that a reliable series of a bond's price may not be available, and the series of prices that is available might not be market based, but rather modelled or matrix priced (the price is based on similar security).

Analytical Duration

Analytical duration, also known as modified duration, is a theoretical measure calculated based on the bond's cash flows and yield. It is derived from the bond's present value formula and assumes a linear relationship between price and yield changes.

Analytical duration provides an estimate of the bond's price sensitivity to changes in yield based on its cash flows and yield-to-maturity. It assumes that the relationship between price and yield is linear, which may not hold for bonds with complex cash flow structures or significant embedded options.

While both empirical duration and analytical duration are useful for assessing interest rate risk, it's important to note that empirical duration is based on historical data and reflects past market behavior, while analytical duration is a theoretical measure based on the bond's characteristics and assumptions about price-yield relationships.