Weighted-average life

Wikipedia open wikipedia design.

In finance, the weighted-average life (WAL) of an amortizing loan or amortizing bond, also called average life,[1][2][3] is the weighted average of the times of the principal repayments: it's the average time until a dollar of principal is repaid.

In a formula,[4]

${\displaystyle {\text{WAL}}=\sum _{i=1}^{n}{\frac {P_{i}}{P}}t_{i},}$

where:

• ${\displaystyle P}$ is the (total) principal,
• ${\displaystyle P_{i}}$ is the principal repayment that is included in payment ${\displaystyle i}$, hence
• ${\displaystyle {\frac {P_{i}}{P}}}$ is the fraction of the total principal that is included in payment ${\displaystyle i}$, and
• ${\displaystyle t_{i}}$ is the time (in years) from the calculation date to payment ${\displaystyle i}$.

If desired, ${\displaystyle t_{i}}$ can be expanded as ${\displaystyle {\frac {1}{12}}(i+\alpha -1)}$ for a monthly bond, where ${\displaystyle \alpha }$ is the fraction of a month between settlement date and first cash flow date.

WAL of classes of loans

In loans that allow prepayment, the WAL cannot be computed from the amortization schedule alone; one must also make assumptions about the prepayment and default behavior, and the quoted WAL will be an estimate. The WAL is usually computed from a single cash-flow sequence. Occasionally, a simulated average life may be computed from multiple cash-flow scenarios, such as those from an option-adjusted spread model.[5]

Related concepts

WAL should not be confused with the following distinct concepts:

Bond duration
Bond duration is the weighted-average time to receive the discounted present values of all the cash flows (including both principal and interest), while WAL is the weighted-average time to receive simply the principal payments (not including interest, and not discounting). For an amortizing loan with equal payments, the WAL will be higher than the duration, as the early payments are weighted towards interest, while the later payments are weighted towards principal, and further, taking present value (in duration) discounts the later payments.
Time until 50% of the principal has been repaid
WAL is a mean, while "50% of the principal repaid" is a median; see difference between mean and median. Since principal outstanding is a concave function (of time) for a flat payment amortizing loan, less than half the principal will have been paid off at the WAL. Intuitively, this is because most of the principal repayment happens at the end. Formally, the distribution of repayments has negative skew: the small principal repayments at the beginning drag down the WAL (mean) more than they reduce the median.
Weighted-average maturity (WAM)
WAM is an average of the maturity dates of multiple loans, not an average of principal repayments.

Applications

WAL is a measure that can be useful in credit risk analysis on fixed income securities, bearing in mind that the main credit risk of a loan is the risk of loss of principal. All else equal, a bond with principal outstanding longer (i.e., longer WAL) has greater credit risk than a bond with shorter WAL. In particular, WAL is often used as the basis for yield comparisons in I-spread calculations.

WAL should not be used to estimate a bond's price-sensitivity to interest-rate fluctuations, as WAL includes only the principal cash flows, omitting the interest payments. Instead, one should use bond duration, which incorporates all the cash flows.

Examples

The WAL of a bullet loan (non-amortizing) is exactly the tenor, as the principal is repaid precisely at maturity.

230 100 80 50
1 × 20 20
2 × 30 30 30
3 × 50 50 50 50

Computing WAL from amortized payment

The above can be reversed: given the terms (principal, tenor, rate) and amortized payment A, one can compute the WAL without knowing the amortization schedule. The total payments are ${\displaystyle An}$ and the total interest payments are ${\displaystyle An-P}$, so the WAL is:

${\displaystyle {\text{WAL}}={\frac {An-P}{Pr}}}$

Similarly, the total interest as percentage of principal is given by ${\displaystyle {\text{WAL}}\times r}$:

${\displaystyle {\text{WAL}}\times r={\frac {An-P}{P}}}$

Notes and references

• Fabozzi, Frank J. (2000), The handbook of fixed income securities, ISBN 0-87094-985-3