What is average life (weighted average life)?
– Definition: Average life (also called weighted average life or weighted average maturity) measures the mean time until principal on a debt issue is repaid. It counts only principal repayments — not interest — and is expressed in years (or months). It is useful for instruments that return principal before final maturity, such as amortizing bonds, mortgage-backed securities (MBS), and asset-backed securities (ABS).
Why it matters (short list)
– Tells you how quickly your principal is returned.
– Helps compare amortizing securities with different repayment patterns.
– Affects reinvestment risk: earlier principal return may force reinvestment at different rates.
– Helps assess sensitivity to prepayment and default behavior in MBS/ABS.
Key jargon (first-use definitions)
– Principal: the outstanding loan or face amount that must be repaid.
– Amortization: schedule of regular principal (and interest) payments that reduce outstanding principal over time.
– Maturity: the contractual final date when remaining principal is due.
– Prepayment risk: the chance borrowers repay principal earlier than scheduled (shortens average life).
– Default risk: the chance borrowers fail to make payments, reducing expected recoveries.
How to calculate average life (step-by-step)
1. Obtain the amortization or payment schedule showing each principal payment and the date (or time in years from now).
2. For each scheduled principal payment i, record:
– principal_i = amount of principal repaid at time_i
– time_i = timing of that payment expressed in years (use fractions if needed)
3. Compute weighted sum = sum over i of (principal_i × time_i).
4. Compute average life = weighted sum ÷ total original principal (or outstanding principal at issue).
Formula (plain text): Average life = (Σ principal_i × time_i) / Total principal
Worked numeric example
Assume a 4-year amortizing note with face amount $400 and principal repayments as follows:
– Year 1: $100
– Year 2: $120
– Year 3: $80
– Year 4: $100
Step-by-step:
– Weighted sum = (100 × 1) + (120 × 2) + (80 × 3) + (100 × 4)
– Weighted sum = 100 + 240 + 240 + 400 = 980
– Average life = 980 ÷ 400 = 2.45 years
Interpretation: On average, a dollar of principal is outstanding for about 2.45 years even though the final maturity is 4 years.
Special considerations and limits
– Prepayment behavior: For MBS/ABS, borrower prepayments shorten average life and reduce future interest received; prepayment speed assumptions materially change the calculation.
– Default and recovery: Defaults reduce expected principal returns; average life computed from scheduled payments ignores losses from defaults unless adjusted for expected recoveries.
– Callable or sinking-fund features: Call options, sinking funds, and other contractual features can cause accelerated principal repayment; include expected exercise behavior when estimating average life.
– Units: Express time consistently (years or months). If using months, divide final result by 12 to get years.
– Use scenario analysis: Because prepayment/default behavior is uncertain, compute average life under multiple scenarios (
compute average life under multiple scenarios (e.g., slow / base / fast prepayment speeds) and report a range of average lives rather than a single point estimate.
Practical steps, checklist, and worked examples
Formula (principal-weighted average time)
– Average life (AL) = sum_{t=1..T} [ t * PR_t ] / P0
– t = time of principal repayment measured consistently (years or months).
– PR_t = principal repaid at time t (scheduled principal + any unscheduled principal such as prepayment or recovered principal).
– P0 = original principal outstanding at time 0.
– If you measure t in months, divide the result by 12 to express AL in years.
Checklist — data you need
1. Full principal cash-flow schedule (PR_t) by period, including scheduled amortization and any expected unscheduled principal (prepayments, calls, recoveries).
2. Consistent timing convention (e.g., payments at year-end or month-end).
3. Scenario definitions for prepayment/default (slow, base, fast, stressed).
4. Calculator or spreadsheet to compute weighted average.
Step-by-step calculation (use a spreadsheet)
1. Lay out periods t (1, 2, …, T).
2. Enter PR_t in each period (absolute currency amounts or percent of original principal).
3. Multiply each PR_t by its period number t.
4. Sum the products across all periods.
5. Divide by original principal P0.
6. If t was in months, divide final result by 12 for years.
Worked numeric example 1 — simple amortizing loan (no prepayment)
– Original principal P0 = 100 (units).
– 4-year loan with equal principal repayments: PR_1 = PR_2 = PR_3 = PR_4 = 25.
– AL = (1*25 + 2*25 + 3*25 + 4*25) / 100 = (25*(1+2+3+4))/100 = (25*10)/100 = 250/100 = 2.5 years.
Interpretation: On average, principal is returned at 2.5 years.
Worked numeric example 2 — single-year large prepayment
– Same original loan P0 = 100 with scheduled PR_t = 25 each year.
– Assume at the end of year 1 the borrower prepays 50% of remaining principal.
– After scheduled PR_1 = 25, remaining = 75. Prepay 50% of 75 = 37.5 additional principal at