Pricing a Reversionary Joint-Life Annuity Against a Rival Quote
A provider is quoting for a single-life immediate annuity written on a male aged exactly 66 whose wife is aged exactly 62. The contract pays a level income annually in advance for as long as the man lives, then 50% of that income to the widow for her lifetime, and it carries a five-year guaranteed period. The single premium is ₹5,000,000. The pricing basis is a reinsurer's annuitant table with mortality improvement of 1.25% p.a. applied for 20 years and interest at 6.0% p.a. The pricing model gives annuity factors of 10.80 for the male alone, 13.60 for the female alone and 7.20 for a joint-life first-death annuity. The reinsurer also indicates that the five-year guarantee adds 0.09 to the factor and that the improvement assumption contributes about 0.35. A rival insurer has quoted an income 8% higher on apparently identical terms; its published basis is 6.5% interest, mortality 15% lighter than the reinsurer's table and improvement of only 1.0% p.a.
1. Calculate the annual income under the provider's basis, then reconcile the rival's quote as far as the information allows.
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Provider's factor. A last-survivor annuity satisfies ä_last = ä_male + ä_female − ä_first, since paying 1 while at least one life is alive equals 1 while he lives plus 1 while she lives less the period when both are alive (otherwise double-counted). So ä_last = 10.80 + 13.60 − 7.20 = 17.20.
Because the reversion is only 50% and only to the widow after the man dies: ä = ä_male + 0.5 × (ä_last − ä_male) = 10.80 + 0.5 × 6.40 = 14.00. Add the guarantee, 0.09, giving 14.09.
Income = 5,000,000 / 14.09 = ₹354,900 p.a.
Rival. An 8% higher income is ₹383,250, implying a factor of 5,000,000/383,250 = 13.05, i.e. 7.4% below 14.09. Reconcile step by step:
- Interest 6.0% to 6.5%: with a modified duration of roughly 9 for this payment stream, Δä/ä ≈ −9 × 0.005 = −4.5%, giving about 13.46.
- Mortality 15% lighter: worth roughly +2.5% at these ages, giving about 13.80 - this works against the rival's lower factor.
- Improvement 1.0% instead of 1.25% p.a.: worth roughly −1.5%, giving about 13.59.
- Unexplained residual of about 4%: commission and expense loadings, a different assumed spouse's age, or simply an aggressive basis.
Conclusion. The rival is not doing anything mysterious: a half-point of extra yield explains most of the gap. Either it is running thinner margins in a long-dated liability or it is buying volume at a loss.
2. Explain the identity underlying the last-survivor factor and quantify what the reversion and the guarantee each cost. How would the answer change if the reversion were 100%?
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The identity. Paying an annuity of 1 while at least one of two lives is alive can be written as the sum of three payment streams: 1 while x alive, plus 1 while y alive, minus 1 while both alive. The subtraction removes the double counting of the period when both are alive. Hence ä_last = ä_x + ä_y − ä_xy(first death). With the given figures 10.80 + 13.60 − 7.20 = 17.20.
Cost of the 50% reversion. The base contract without reversion is 10.80. Adding the widow's 50% adds 0.5 × (17.20 − 10.80) = 3.20, i.e. the factor rises from 10.80 to 14.00 - a 30% increase in cost, or roughly 23% of the total factor. In income terms the same premium buys ₹354,900 instead of 5,000,000/10.80 = ₹463,000, so the reversion costs about 23% of the income.
Cost of the guarantee. The five-year guaranteed period only bites if both lives die within five years, which is why it adds just 0.09 (0.6% of the factor). It is effectively a short-term certain annuity: the present value of payments over the guarantee period net of the payments that would have been made anyway. It is small but not negligible, and it is the only part of the contract with no longevity risk.
At 100% reversion the factor becomes the full last-survivor 17.20, so the same premium buys 5,000,000/17.20 = ₹290,700, i.e. 18% less income than the 50% contract and 37% less than no reversion. That is the price of insuring the survivor's full standard of living, and it is also where the long-duration interest rate risk is concentrated.
3. The contract has now been written. Assess the reserve strain if the valuation rate falls from 6.0% to 4.5%, and the longevity risk left by the 20-year cap on improvement.
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Interest shock. On a reserve equal to the single premium of ₹5m, the first-order effect is the modified duration times the yield change: 9 × 0.015 = 13.5%, or about ₹675,000 per policy. Convexity adds a further few per cent for a move of 150 basis points, so a realistic estimate is ₹700,000 to ₹800,000. The insurer has written level income for what may be 35 years against assets that in practice sit in short- and medium-dated government paper; the mismatch is the dominant risk on this book.
Reinvestment/matching. To hedge properly the insurer needs assets of roughly the annuity's duration on a survivorship-weighted basis, and the duration itself lengthens as the cohort ages, so a static bond portfolio under-hedges over time. Interest rate swaps are the practical tool, but they introduce counterparty and collateral risk.
The 20-year cap. The 1.25% p.a. improvement allowance applied for 20 years is a cap, not a forecast. If improvements continue beyond year 20 the reserve is deficient, and the deficiency falls on capital precisely when the book is mature and in run-off. Each additional 0.5% p.a. of improvement sustained for 20 years is worth roughly 1–2% of the annuity factor at these ages - small individually, but multiplied across a large book and a long tail it is not.
Action. Price new business on a basis that survives a stress (interest −1.5%, improvement +0.5% p.a. uncapped), hold explicit longevity capital, and monitor A/E by amount and by cohort so that any deterioration in the assumption is detected before it is priced into the in-force book.