Minimizing the probability of lifetime ruin under borrowing constraints

Bayraktar, Erhan; Young, Virginia R.

doi:10.1016/j.insmatheco.2006.10.015

Cited by 46 publications

(54 citation statements)

References 27 publications

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“…The convexity ofv implies its left and right derivatives D ±v exists (in R for interior points and in R ∪ {±∞} for boundary points) and are non-decreasing. 7 Since we have showed 0 ≤û ≤ p and we know p ′ 0 (w s ) = 0, an argument exactly the same as Remark 3.1 yields…”

Section: Regularitymentioning

confidence: 70%

“…For any C 2 function ϕ and π, θ ∈ R, define Proof. Same as [7], we let ∆ be the "coffin state" and [b, ∞)∪{∆} be the one point compactification of [b, ∞). Define the extension of u to [b, ∞) ∪ {∆} by assigning u(∆) = 0.…”

Section: Verificationmentioning

confidence: 99%

“…We mention that Jacka in an earlier work [22] considered a finite-fuel problem of very similar form. Subsequent variants of Young's work include but not limited to adding borrowing constraints [7], assuming consumption is ratcheted [8], allowing stochastic consumption [9] and stochastic volatility [4]. In all previous works, there is a fixed risky asset model; that is, the investor is certain about the evolution and distribution of the risky asset price.…”

Section: Introductionmentioning

confidence: 99%

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Minimizing the Probability of Lifetime Ruin Under Ambiguity Aversion

Bayraktar¹,

Zhang²

2015

SIAM J. Control Optim.

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Abstract. We determine the optimal robust investment strategy of an individual who targets at a given rate of consumption and seeks to minimize the probability of lifetime ruin when she does not have perfect confidence in the drift of the risky asset. Using stochastic control, we characterize the value function as the unique classical solution of an associated Hamilton-Jacobi-Bellman (HJB) equation, obtain feedback forms for the optimal investment and drift distortion, and discuss their dependence on various model parameters. In analyzing the HJB equation, we establish the existence and uniqueness of viscosity solution using Perron's method, and then upgrade regularity by working with an equivalent convex problem obtained via the Cole-Hopf transformation. We show the original value function may lose convexity for a class of parameters and the Isaacs condition may fail. Numerical examples are also included to illustrate our results.

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Section: Regularitymentioning

confidence: 70%

Section: Verificationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Minimizing the Probability of Lifetime Ruin Under Ambiguity Aversion

Bayraktar¹,

Zhang²

2015

SIAM J. Control Optim.

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“…In both papers, the individual seeks to minimize the probability that wealth reaches some point b > 0 before she dies. Young (2004) places no restriction on the optimal investment strategy, while Bayraktar and Young (2006) extend Young's work to two cases: (1) The individual may not borrow any money; and (2) the individual may borrow money but only at a rate higher than the rate earned by an investment in the riskless asset. Fleming and Zariphopoulou (1991) consider the latter setting in the problem of maximizing expected utility of consumption under power utility.…”

Section: Introductionmentioning

confidence: 99%

Correspondence between lifetime minimum wealth and utility of consumption

Bayraktar

Young

2007

Finance Stoch

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We establish when the two problems of minimizing a function of lifetime minimum wealth and of maximizing utility of lifetime consumption result in the same optimal investment strategy on a given open interval O in wealth space. To answer this question, we equate the two investment strategies and show that if the individual consumes at the same rate in both problems -the consumption rate is a control in the problem of maximizing utility -then the investment strategies are equal only when the consumption function is linear in wealth on O, a rather surprising result. It, then, follows that the corresponding investment strategy is also linear in wealth and the implied utility function exhibits hyperbolic absolute risk aversion.

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“…• In the next section, we remove the leveraging entirely by prohibiting borrowing of the riskless asset, as in Bayraktar and Young (2007a). Bayraktar and Young (2007a) consider the problem of minimizing the probability of lifetime ruin under the constraint that the individual cannot borrow; however, they consider only the case for which the ruin level x = 0.…”

Section: Probability Of Ruin At Deathmentioning

confidence: 99%

Minimizing the lifetime shortfall or shortfall at death

Bayraktar

Young

2009

Insurance: Mathematics and Economics

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We find the optimal investment strategy for an individual who seeks to minimize one of four objectives: (1) the probability that his wealth reaches a specified ruin level before death, (2) the probability that his wealth reaches that level at death, (3) the expectation of how low his wealth drops below a specified level before death, and (4) the expectation of how low his wealth drops below a specified level at death. Young (2004) showed that under criterion (1), the optimal investment strategy is a heavily leveraged position in the risky asset for low wealth.In this paper, we introduce the other three criteria in order to reduce the leveraging observed by Young (2004). We discovered that surprisingly the optimal investment strategy for criterion (3) is identical to the one for (1) and that the strategies for (2) and (4) are more leveraged than the one for (1) at low wealth. Because these criteria do not reduce leveraging, we completely remove it by considering problems (1) and (3) under the restriction that the individual cannot borrow to invest in the risky asset.

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Minimizing the probability of lifetime ruin under borrowing constraints

Cited by 46 publications

References 27 publications

Minimizing the Probability of Lifetime Ruin Under Ambiguity Aversion

Minimizing the Probability of Lifetime Ruin Under Ambiguity Aversion

Correspondence between lifetime minimum wealth and utility of consumption

Minimizing the lifetime shortfall or shortfall at death

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