Average Life is a financial metric used in the analysis of fixed-income securities, such as bonds or mortgage-backed securities. It represents the average time it takes for the present value of future cash flows from the security to be repaid to the investor.

There are two commonly used measures of average life:

1. **Macaulay Duration:**

– The Macaulay Duration represents the weighted average time to receive the bond’s cash flows, considering both coupon payments and the return of principal at maturity.

– It is calculated as the sum of the present value of each cash flow divided by the current bond price.

– Mathematically, it is expressed as:

\[ \text{Macaulay Duration} = \frac{\sum_{t=1}^{T} \frac{t \cdot C_t}{(1 + r)^t} + \frac{T \cdot FV}{(1 + r)^T}}{\text{Current Bond Price}} \]

where:

– \(C_t\) is the cash flow at time \(t\),

– \(r\) is the periodic yield or discount rate,

– \(T\) is the number of periods (or time to maturity),

– \(FV\) is the face value of the bond.

2. **Modified Duration:**

– Modified Duration is a modified version of Macaulay Duration that measures the sensitivity of a bond’s price to changes in interest rates.

– It is calculated as the Macaulay Duration divided by \(1 + \frac{r}{n}\), where \(n\) is the number of compounding periods per year.

– Mathematically, it is expressed as:

\[ \text{Modified Duration} = \frac{\text{Macaulay Duration}}{1 + \frac{r}{n}} \]

These measures provide a way for investors to assess interest rate risk and make informed decisions based on the potential impact of interest rate changes on the value of their fixed-income investments.

In the context of mortgage-backed securities, the average life is often used to estimate the period over which the principal will be repaid, taking into account prepayments and the impact of interest rate changes on the timing of cash flows.

It’s important to note that average life measures are approximations, and actual cash flows may vary due to factors such as prepayments, call provisions, or changes in interest rates. Investors use these metrics as tools for risk assessment and portfolio management in the fixed-income market.