In
finance, the weighted-average life (WAL) of an
amortizing loan
In banking and finance, an amortizing loan is a loan where the principal of the loan is paid down over the life of the loan (that is, amortized) according to an amortization schedule, typically through equal payments.
Similarly, an amortizing b ...
or amortizing bond, also called average life,
[PIMCO glossary](_blank)
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The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the times of the ''principal repayments'': it's the average time until a dollar of principal is repaid.
In a formula,
:
where:
* is the (total) principal,
* is the principal repayment that is included in payment , hence
* is the fraction of the total principal that is included in payment , and
* is the time (in years) from the calculation date to payment .
If desired, can be expanded as for a monthly bond, where is the fraction of a month between settlement date Settlement date is a securities industry term describing the date on which a trade (bonds, equities, foreign exchange, commodities, etc.) settles. That is, the actual day on which transfer of cash or assets is completed and is usually a few days a ...
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
Option-adjusted spread (OAS) is the yield spread which has to be added to a benchmark yield curve to discount a security's payments to match its market price, using a dynamic pricing model that accounts for embedded options. OAS is hence model-de ...
model.
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 value
In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has inte ...
s'' 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
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
, 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
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimodal d ...
: 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
The Interpolated Spread or I-spread or ISPRD of a bond is the difference between its yield to maturity and the linearly interpolated yield for the same maturity on an appropriate reference yield curve. The reference curve may refer to government ...
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
An amortization calculator is used to determine the periodic payment amount due on a loan (typically a mortgage), based on the amortization process.
The amortization repayment model factors varying amounts of both interest and principal into eve ...
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 years.
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 , and now one multiplies by the principal to get total interest payments."
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 (time in years is period number in months, over 12).
Then:
:
Total interest is
:
where is the principal outstanding at the ''beginning'' of period ''i'' (it's the principal on which the ''i'' interest payment is based). The statement reduces to showing that . Both of these quantities are the time-weighted total principal of the bond (in periods), and they are simply different ways of slicing it: the sum counts how ''long'' each dollar of principal is outstanding (it slices ''horizontally''), while the counts how much principal is outstanding ''at each point in time'' (it slices ''vertically'').
Working backwards, , and so forth: the principal outstanding when ''k'' periods remain is exactly the sum of the next ''k'' principal payments. The principal paid off by the last (''n''th) principal payment is outstanding for all ''n'' periods, while the principal paid off by the second to last ((''n'' − 1)th) principal payment is outstanding for ''n'' − 1 periods, and so forth. Using this, the sums can be re-arranged to be equal.
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 , and on the other hand would be . This is demonstrated in the following table, which shows the amortization schedule, broken up into principal repayments, where each column is a , and each row is :
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 and the total interest payments are , so the WAL is:
:
Similarly, the total interest as percentage of principal is given by :
:
Notes and references
* {{citation
, title=The handbook of fixed income securities
, first=Frank J.
, last=Fabozzi
, year=2000
, isbn=0-87094-985-3
See also
* Amortization calculator
An amortization calculator is used to determine the periodic payment amount due on a loan (typically a mortgage), based on the amortization process.
The amortization repayment model factors varying amounts of both interest and principal into eve ...
* Amortization schedule
An amortization schedule is a table detailing each periodic payment on an amortizing loan (typically a mortgage), as generated by an amortization calculator. Amortization refers to the process of paying off a debt (often from a loan or mortgage) ov ...
* Amortizing loan
In banking and finance, an amortizing loan is a loan where the principal of the loan is paid down over the life of the loan (that is, amortized) according to an amortization schedule, typically through equal payments.
Similarly, an amortizing b ...
Fixed income analysis
Bond valuation