In finance, the weighted-average life (WAL) of an amortizing loan or amortizing bond, also called average life, is the weighted average of the times of the principal repayments: it's the average time until a dollar of principal is repaid.
Contents
- WAL of classes of loans
- Related concepts
- Applications
- Examples
- Total Interest
- Proof
- Computing WAL from amortized payment
- References
In a formula,
where:
If desired,
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.
Related concepts
WAL should not be confused with the following distinct concepts:
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.
On a 30-year amortizing loan, paying equal amounts monthly, one has the following WALs, for the given annual interest rates (and corresponding monthly payments per $100,000 principal balance, calculated via an amortization calculator and the formulas below relating amortized payments, total interest, and WAL):
Note that as the interest rate increases, WAL increases, since the principal payments become increasingly back-loaded. WAL is independent of the principal balance, though payments and total interest are proportional to principal.
For a coupon of 0%, where the principal amortizes linearly, the WAL is exactly half the tenor plus half a payment period, because principal is repaid in arrears (at the end of the period). So for a 30-year 0% loan, paying monthly, the WAL is
Total Interest
WAL allows one to easily compute the total interest payments, given by:
where r is the annual interest rate and P is the initial principal.
This can be understood intuitively as: "The average dollar of principal is outstanding for the WAL, hence the interest on the average dollar is
Proof
More rigorously, one can derive the result as follows. To ease exposition, assume that payments are monthly, so periodic interest rate is annual interest rate divided by 12, and time
Then:
Total interest is
where
Working backwards,
For instance, if the principal amortized as $100, $80, $50 (with paydowns of $20, $30, $50), then the sum would on the one hand be
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
Similarly, the total interest as percentage of principal is given by