arXiv subjects

Thomas S. Salisbury

Publications and source records attributed to Thomas S. Salisbury.

At least 19 recordsLinked to original sources

arXiv2025

Equitable Longevity Risk Sharing or, the raison d'être for a First Nations Pension Plan

We investigate the extent to which groups with elevated mortality rates ex ante might opt out of guaranteed national pensions in favour of demographically aligned plans, which we label equitable longevity risk sharing (ELRiS) pools, even if this involves accepting some idiosyncratic risk. Technically, this paper develops a stochastic model of retirement income within an ELRiS structure that is calibrated to equate the discounted expected utility of a guaranteed national pension. The practical motivation for developing this alternative is that working members of First Nations peoples of Canada: (1) experience much higher mortality rates than average over their entire life cycle, and (2) some are actually allowed by current legislation to opt out of the Canada Pension Plan (CPP). We then demonstrate that under reasonable economic preferences and parameters, a sub-group with a 10-year life expectancy gap relative to the population could attain equivalent lifetime utility by contributing a mere two-thirds to the plan, even if they were pooled with only 30 members. For a longevity gap of 20 years, such as between an Indigenous male versus a non-Indigenous female, the contribution rate falls to less than a third. The difference between the statutory and mandatory contribution rates to a guaranteed national pension and those needed within these self-sustaining pools is an implicit subsidy from Indigenous to non-Indigenous. From a policy perspective, this paper aspires to jump-start a conversation that sparks a change in a status quo, which is obviously unfair and inequitable.

econ.GN↗

arXiv2025

Bifurcation in optimal retirement

We study optimal consumption and retirement using a Cobb-Douglas utility and a simple model in which an interesting bifurcation arises. With high wealth, individuals plan to retire. With low wealth they plan to never retire. At a critical level of initial wealth they may choose to defer this decision, leading to a continuum of wealth trajectories with identical utilities.

q-fin.PM↗

arXiv2024

The Riccati Tontine: How to Satisfy Regulators on Average

This paper presents a new type of modern accumulation-based tontine, called the Riccati tontine, named after two Italians: mathematician Jacobo Riccati (b. 1676, d. 1754) and financier Lorenzo di Tonti (b. 1602, d. 1684). The Riccati tontine is yet another way of pooling and sharing longevity risk, but is different from competing designs in two key ways. The first is that in the Riccati tontine, the representative investor is expected -- although not guaranteed -- to receive their money back if they die, or when the tontine lapses. The second is that the underlying funds within the tontine are deliberately {\em not} indexed to the stock market. Instead, the risky assets or underlying investments are selected so that return shocks are negatively correlated with stochastic mortality, which will maximize the expected payout to survivors. This means that during a pandemic, for example, the Riccati tontine fund's performance will be impaired relative to the market index, but will not be expected to lose money for participants. In addition to describing and explaining the rationale for this non-traditional asset allocation, the paper provides a mathematical proof that the recovery schedule that generates this financial outcome satisfies a first-order ODE that is quadratic in the unknown function, which (yes) is known as a Riccati equation.

q-fin.MF↗

arXiv2023

A greedy algorithm for habit formation under multiplicative utility

We consider the problem of optimizing lifetime consumption under a habit formation model, both with and without an exogenous pension. Unlike much of the existing literature, we apply a power utility to the ratio of consumption to habit, rather than to their difference. The martingale/duality method becomes intractable in this setting, so we develop a greedy version of this method that is solvable using Monte Carlo simulation. We investigate the behaviour of the greedy solution, and explore what parameter values make the greedy solution a good approximation to the optimal one.

q-fin.PM↗

arXiv2022

Percolation of terraces, and enhancements for the orthant model

We study a model of an i.i.d.~random environment in general dimensions $d\ge 2$, where each site is equipped with one of two environments. The model comes with a parameter $p$ which governs the frequency of the first environment, and for each dimension $d$ there is a critical parameter $p_c(d)$ at which there is a phase transition for the geometry of a particular connected cluster (the cluster is infinite for all $p$). We use the celebrated methodology of enhancements in this novel setting to prove that $p_c(d)$ is strictly monotone in $d$ for this model. To do so we study the discrete geometry and percolation theory of higher-dimensional structures called terraces.

math.PR↗

arXiv2021

Refundable income annuities: Feasibility of money-back guarantees

Refundable income annuities (IA), such as cash-refund and instalment-refund, differ in material ways from the life-only version beloved by economists. In addition to lifetime income they guarantee the annuitant or beneficiary will receive their money back albeit slowly over time. We document that refundable IAs now represent the majority of sales in the U.S., yet they are mostly ignored by insurance and pension economists. And, although their pricing, duration, and money's-worth-ratio is complicated by recursivity which will be explained, we offer a path forward to make refundable IAs tractable. A key result concerns the market price of cash-refund IAs, when the actuarial present value is grossed-up by an insurance loading. We prove that price is counterintuitively no longer a declining function of age and older buyers might pay more than younger ones. Moreover, there exists a threshold valuation rate below which no price is viable. This may also explain why inflation-adjusted IAs have all but disappeared.

q-fin.PR↗

arXiv2019

Phase transitions for degenerate random environments

We study a class of models of i.i.d.~random environments in general dimensions $d\ge 2$, where each site is equipped randomly with an environment, and a parameter $p$ governs the frequency of certain environments that can act as a barrier. We show that many of these models (including some which are non-monotone in $p$) exhibit a sharp phase transition for the geometry of connected clusters as $p$ varies.

math.PR↗

arXiv2019

A shape theorem for the orthant model

We study a particular model of a random medium, called the orthant model, in general dimensions $d\ge 2$. Each site $x\in \Z^d$ independently has arrows pointing to its positive neighbours $x+e_i$, $i=1,\dots, d$ with probability $p$ and otherwise to its negative neighbours $x-e_i$, $i=1,\dots, d$ (with probability $1-p$). We prove a shape theorem for the set of sites reachable by following arrows, starting from the origin, when $p$ is large. The argument uses subadditivity, as would be expected from the shape theorems arising in the study of first passage percolation. The main difficulty to overcome is that the primary objects of study are not stationary, which is a key requirement of the subadditive ergodic theorem.

math.PR↗

arXiv2018

Retirement spending and biological age

We solve a lifecycle model in which the consumer's chronological age does not move in lockstep with calendar time. Instead, biological age increases at a stochastic non-linear rate in time like a broken clock that might occasionally move backwards. In other words, biological age could actually decline. Our paper is inspired by the growing body of medical literature that has identified biomarkers which indicate how people age at different rates. This offers better estimates of expected remaining lifetime and future mortality rates. It isn't farfetched to argue that in the not-too-distant future personal age will be more closely associated with biological vs. calendar age. Thus, after introducing our stochastic mortality model we derive optimal consumption rates in a classic Yaari (1965) framework adjusted to our proper clock time. In addition to the normative implications of having access to biological age, our positive objective is to partially explain the cross-sectional heterogeneity in retirement spending rates at any given chronological age. In sum, we argue that neither biological nor chronological age alone is a sufficient statistic for making economic decisions. Rather, both ages are required to behave rationally.

q-fin.MF↗

arXiv2018

The implied longevity curve: How long does the market think you are going to live?

We use life annuity prices to extract information about human longevity using a framework that links the term structure of mortality and interest rates. We invert the model and perform nonlinear least squares to obtain implied longevity forecasts. Methodologically, we assume a Cox-Ingersoll-Ross (CIR) model for the underlying yield curve, and for mortality, a Gompertz-Makeham (GM) law that varies with the year of annuity purchase. Our main result is that over the last decade markets implied an improvement in longevity of of 6-7 weeks per year for males and 1-3 weeks for females. In the year 2004 market prices implied a $40.1\%$ probability of survival to the age 90 for a 75-year old male ($51.2\%$ for a female) annuitant. By the year 2013 the implied survival probability had increased to $46.1\%$ (and $53.1\%$). The corresponding implied life expectancy has increased (at the age of 75) from 13.09 years for males (15.08 years for females) to 14.28 years (and 15.61 years.) Although these values are implied directly from markets, they are consistent with demographic projections. Similar to implied volatility in option pricing, we believe that our implied survival probabilities (ISP) and implied life expectancy (ILE) are relevant for the financial management of assets post-retirement and very important for the optimal timing and allocation to annuities; procrastinators are swimming against an uncertain but rather strong longevity trend.

q-fin.MF↗

arXiv2016

Conditions for ballisticity and invariance principle for random walk in non-elliptic random environment

We study the asymptotic behaviour of random walks in i.i.d. non-elliptic random environments on $\mathbb{Z}^d$. Standard conditions (and proofs) for ballisticity and the central limit theorem require ellipticity. We use oriented percolation and martingale arguments to give non-trivial local conditions for ballisticity and an annealed invariance principle in the non-elliptic setting.

math.PR↗

arXiv2016

Optimal retirement income tontines

Tontines were once a popular type of mortality-linked investment pool. They promised enormous rewards to the last survivors at the expense of those died early. And, while this design appealed to the gambling instinc}, it is a suboptimal way to generate retirement income. Indeed, actuarially-fair life annuities making constant payments -- where the insurance company is exposed to longevity risk -- induce greater lifetime utility. However, tontines do not have to be structured the historical way, i.e. with a constant cash flow shared amongst a shrinking group of survivors. Moreover, insurance companies do not sell actuarially-fair life annuities, in part due to aggregate longevity risk. We derive the tontine structure that maximizes lifetime utility. Technically speaking we solve the Euler-Lagrange equation and examine its sensitivity to (i.) the size of the tontine pool $n$, and (ii.) individual longevity risk aversion $γ$. We examine how the optimal tontine varies with $γ$ and $n$, and prove some qualitative theorems about the optimal payout. Interestingly, Lorenzo de Tonti's original structure is optimal in the limit as longevity risk aversion $γ\to \infty$. We define the natural tontine as the function for which the payout declines in exact proportion to the survival probabilities, which we show is near-optimal for all $γ$ and $n$. We conclude by comparing the utility of optimal tontines to the utility of loaded life annuities under reasonable demographic and economic conditions and find that the life annuity's advantage over the optimal tontine is minimal. In sum, this paper's contribution is to (i.) rekindle a discussion about a retirement income product that has been long neglected, and (ii.) leverage economic theory as well as tools from mathematical finance to design the next generation of tontine annuities.

q-fin.MF↗

arXiv2016

How round are the complementary components of planar Brownian motion?

Consider a Brownian motion $W$ in ${\bf C}$ started from $0$ and run for time 1. Let $A(1),A(2),\dots$ denote the bounded connected components of ${\bf C}-W([0,1])$. Let $R(i)$ (resp. $r(i)$) denote the out-radius (resp. in-radius) of $A(i)$ for $i\in\bf N$. Our main result is that ${\bf E}[\sum_i R(i)^2|\log R(i)|^θ]<\infty$ for any $θ<1$. We also prove that $\sum_i r(i)^2|\log r(i)|=\infty$ almost surely. These results have the interpretation that most of the components $A(i)$ have a rather regular or round shape.

math.PR↗

arXiv2016

Notes on oriented percolation

These notes fill in results about oriented percolation that are required for the paper [3] ("Forward clusters for degenerate random environments"). Since these are essentially modifications of results found in other sources (but adapted to the model we particularly need), there is no intention to publish these.

math.PR↗

arXiv2015

Uniqueness for Volterra-type stochastic integral equations

We study uniqueness for a class of Volterra-type stochastic integral equations. We focus on the case of non-Lipschitz noise coefficients. The connection of these equations to certain degenerate stochastic partial differential equations plays a key role.

math.PR↗

arXiv2013

Conditioning super-Brownian motion on its boundary statistics, and fragmentation

We condition super-Brownian motion on "boundary statistics" of the exit measure $X_D$ from a bounded domain $D$. These are random variables defined on an auxiliary probability space generated by sampling from the exit measure $X_D$. Two particular examples are: conditioning on a Poisson random measure with intensity $βX_D$ and conditioning on $X_D$ itself. We find the conditional laws as $h$-transforms of the original SBM law using Dynkin's formulation of $X$-harmonic functions. We give explicit expression for the (extended) $X$-harmonic functions considered. We also obtain explicit constructions of these conditional laws in terms of branching particle systems. For example, we give a fragmentation system description of the law of SBM conditioned on $X_D=ν$, in terms of a particle system, called the backbone. Each particle in the backbone is labeled by a measure $\tildeν$, representing its descendants' total contribution to the exit measure. The particle's spatial motion is an $h$-transform of Brownian motion, where $h$ depends on $\tildeν$. At the particle's death two new particles are born, and $\tildeν$ is passed to the newborns by fragmentation.

math.PR↗

arXiv2013

Optimal Retirement Tontines for the 21st Century: With Reference to Mortality Derivatives in 1693

Historical tontines promised enormous rewards to the last survivors at the expense of those who died early. While this design appealed to the gambling instinct, it is a suboptimal way to manage longevity risk during retirement. This is why fair life annuities making constant payments -- where the insurance company is exposed to the longevity risk -- induces greater lifetime utility. However, tontines do not have to be designed using a winner-take-all approach and insurance companies do not actually sell fair life annuities, partially due to aggregate longevity risk. In this paper we derive the tontine structure that maximizes lifetime utility, but doesn't expose the sponsor to any longevity risk. We examine its sensitivity to the size of the tontine pool; individual longevity risk aversion; and subjective health status. The optimal tontine varies with the individual's longevity risk aversion $γ$ and the number of participants $n$, which is problematic for product design. That said, we introduce a structure called a natural tontine whose payout declines in exact proportion to the (expected) survival probabilities, which is near-optimal for all $γ$ and $n$. We compare the utility of optimal tontines to the utility of loaded life annuities under reasonable demographic and economic conditions and find that the life annuity's advantage over tontines, is minimal. We also review and analyze the first-ever mortality-derivative issued by the British government, known as King Williams's tontine of 1693. We shed light on the preferences and beliefs of those who invested in the tontines vs. the annuities and argue that tontines should be re-introduced and allowed to co-exist with life annuities. Individuals would likely select a portfolio of tontines and annuities that suit their personal preferences for consumption and longevity risk, as they did over 320 years ago.

q-fin.PM↗

arXiv2013

Forward clusters for degenerate random environments

We consider connectivity properties and asymptotic slopes for certain random directed graphs on $Z^2$ in which the set of points $C_o$ that the origin connects to is always infinite. We obtain conditions under which the complement of $C_o$ has no infinite connected component. Applying these results to one of the most interesting such models leads to an improved lower bound for the critical occupation probability for oriented site percolation on the triangular lattice in 2 dimensions.

math.PR↗