The 0.90 Multiplier: Graduating an Annuitant Portfolio
An Indian life insurer is reviewing the mortality basis for its immediate annuity book. During calendar year 2023 about 8,500 pensioner annuities were in force at some point, contributing 7,150 life-years of central exposed to risk, and 396 deaths were reported. Applying the insurer's current standard annuitant table gives 439 expected deaths. Lives range from age 60 to 99, though exposures at ages 96 and above are below 20 life-years at each age, and no deaths were recorded at ages 97 and 98. When the same comparison is weighted by annuity amount rather than by life, the ratio of actual to expected deaths falls to 0.85. The valuation actuary proposes to value the whole book on the standard table with an across-the-board multiplier of 0.90 and an interest rate of 6.5%, arguing that the aggregate ratio 'proves' the portfolio is 10% lighter than standard. The appointed actuary is uneasy, especially because a reinsurer has just quoted new business on a materially lighter basis.
1. Critique the proposed 0.90 multiplier. What does the ratio 396/439 actually measure, and what would you check before adopting it for valuation?
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What the ratio measures. The standardised mortality ratio is 396/439 = 0.902, but it is an aggregate, unstandardised comparison. It mixes two things: the portfolio's age distribution and its mortality level. Because the standard table's own age weights differ from this book's, an insurer with exactly standard mortality but a younger mix would still show an SMR below 1. Direct standardisation (apply the crude age-specific rates to the standard table's exposure) or an age-by-age analysis is required before any multiplier is defensible.
Sampling error. Treating deaths as Poisson, sd(A) ≈ √439 ≈ 21, so se(SMR) ≈ 21/439 ≈ 4.8%. A 95% interval is roughly 0.81 to 0.99. The data are consistent with 10% lighter mortality, but equally with 2% lighter or with standard mortality; 0.902 is not a precise estimate and the proposed 0.90 embeds no measurable margin.
Lives versus amounts. The amount-weighted ratio of 0.85 shows the book is heterogeneous: larger annuities experience lighter mortality (wealthier annuitants, later purchase, different socio-economic mix). A single lives-weighted multiplier over-reserves small policies and under-reserves the large ones that dominate the liability; since the reserve is an amount-weighted quantity, this is a solvency-relevant distortion.
Data checks. Confirm the denominator counts lives in force correctly, that death notifications are not lagged into the next year, and that annuities ceasing for other reasons (commutation, fraud, unreported deaths) are not left in the exposure.
2. Recommend a graduation approach and set out how you would test it. How much smoothing is appropriate for a book of this size?
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Why graduate. Crude rates at individual ages are unusable for valuation. At an age with 500 life-years and 12 deaths the crude rate is 0.024 with a relative standard error of about 1/√12 ≈ 29%; at ages 96+ the exposure is so thin that one death moves the rate by several percentage points. Graduation removes random fluctuation, produces a smooth series for interpolation and extrapolation, and exposes genuine anomalies.
Method choice. With only 396 deaths, parsimony matters. A Gompertz–Makeham law, μ(x) = A + Bc^x, uses three parameters over roughly 40 ages and extrapolates sensibly, but cannot reproduce an accident hump, cohort effects or the flattening of mortality at the oldest ages. A logistic/Perks form, or a Makeham with a separate check on ages 90+, addresses old-age flattening. Non-parametric smoothing (Whittaker–Henderson, or a spline with a penalty chosen by cross-validation) is more flexible but can chase noise at the thin ages. My preference is a parametric fit below age 90, with ages 95+ grouped or fitted to a separate logistic tail, cross-checked against the reinsurer's table.
Tests. (i) Chi-square of fit, Σ(A−E)²/E, compared with χ² on n − k degrees of freedom, remembering the statistic must be adjusted for the degrees of freedom consumed in graduation. (ii) Signs test: the number of positive deviations is Binomial(n, 0.5). (iii) Cumulative deviations test, which detects a systematic bias disguised by alternating errors. (iv) Runs test for serial pattern. (v) Graphical comparison against crude rates with confidence bands.
Trade-off. There is no unique 'correct' graduation: smoothness buys stability at the cost of bias, adherence buys fidelity at the cost of noise. Tests inform judgement; they do not replace it.
3. The 0.90 multiplier is adopted at 6.5% interest. How sensitive is the annuity reserve to a 10% understatement of mortality, and is the interest assumption a safe offset?
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Direction and size. Lighter mortality raises the annuity value. The effect is not proportional: if survival probabilities are raised from tpx to (tpx)^0.9, the annuity factor at age 65 (ä ≈ 10.5 at 6.5%) rises by only about 2–2.5%, roughly a quarter of a year's annuity. The curvature is the point - a 10% cut in the force of mortality does not produce a 10% (or even 5%) increase in the liability, and the effect grows with age and duration. The exact figure requires recalculation, not scaling.
Is interest a safe offset? No, for three reasons. First, the two margins act in opposite directions: a high interest assumption reduces the reserve, so any overstatement of yield is a hidden release of margin, and 6.5% is aggressive for a long-dated, level, non-escalating liability funded at current yields. Second, a level annuity has a long duration; if yields fall, the reserve rises sharply and the mortality and interest risks compound rather than offset. Third, the mortality margin is not merely imprecise, it is mis-allocated: the lives-based multiplier gives too little reserve on the large policies where the money is.
Recommendation. Base the reserve on graduated rates (or the reinsurer's table for new business) with an explicit, quantified longevity improvement allowance, for example 1.5% p.a. for 20 years, hold explicit capital for improvement beyond that, and report A/E ratios by amount annually so the basis is tested against experience rather than asserted.