Duration is a measure used in finance and investment to estimate the sensitivity of the price of a fixed-income security, such as bonds, to changes in interest rates. It provides investors with a way to assess the potential impact of interest rate changes on the value of their fixed-income investments. Duration is a key concept in bond portfolio management and risk analysis.

Key points about duration include:

1. **Definition:** Duration is a weighted average time it takes for the bond’s cash flows (interest and principal payments) to repay its price. It is expressed in terms of years.

2. **Interest Rate Sensitivity:** Duration helps investors gauge how sensitive the price of a bond or a bond portfolio is to changes in interest rates. The higher the duration, the more sensitive the bond’s price is to interest rate movements.

3. **Macaulay Duration:** The Macaulay duration is the most common form of duration and represents the weighted average time to receive the bond’s cash flows, with weights based on the present value of each cash flow. It is named after economist Frederick Macaulay.

\[ \text{Macaulay Duration} = \frac{\sum_{t=1}^{n} t \times \frac{C_t + F}{(1 + YTM)^t}}{\text{Current Bond Price}} \]

Where:

– \( t \) = time period,

– \( C_t \) = cash flow at time \( t \),

– \( F \) = face value of the bond,

– \( YTM \) = yield to maturity.

4. **Modified Duration:** Modified duration is a modified version of Macaulay duration that provides a percentage change in the bond’s price for a 1% change in yield. It is a useful measure for estimating interest rate risk.

\[ \text{Modified Duration} = \frac{\text{Macaulay Duration}}{(1 + \frac{YTM}{n})} \]

Where:

– \( n \) = number of compounding periods per year.

5. **Key Relationship:** The relationship between duration and interest rate sensitivity is inverse. As interest rates rise, bond prices generally fall, and the duration helps estimate the extent of that price decline.

6. **Duration of a Bond Portfolio:** For a bond portfolio, the duration is the weighted average of the durations of individual bonds, with weights based on the market value of each bond in the portfolio.

7. **Immunization:** Duration plays a crucial role in immunization strategies, where investors seek to match the duration of their assets with their liabilities to minimize interest rate risk.

8. **Limitations:** Duration assumes a linear relationship between bond prices and interest rates, which may not hold under extreme market conditions. Additionally, it does not account for changes in cash flow patterns due to prepayments or calls.

Investors and portfolio managers use duration as a tool for risk management and to make informed decisions regarding interest rate risk in their fixed-income investments.