Kenneth R. Vetzal’s research while affiliated with University of Waterloo and other places

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Publications (81)


Figure 1. Frontiers generated from the synthetic market. Parameters based on real CRSP index, real 30-day T-bills (see Table 1). Tontine case is as in Table 3. The No Tontine case uses the same scenario, but with no tontine gains, and no fees. The Const q, Const p case has q = 40, p = 0.10, with no tontine gains, which is the best result from Table 4, assuming no tontine gains, and no fees. Units: thousands of dollars.
Figure 6. Optimal EW-ES. Heat map of controls: fraction in stocks and withdrawals, computed from Problem EW-ES (7.1). Real capitalization weighted CRSP index, and real 30-day T-bills. Scenario given in Table 3. Control computed and stored from the Problem 7.2 in the synthetic market. q min = 40, q max = 80 (per year). κ = 0.18. W * = 385. = −10 −4 . Normalized withdrawal (q − q min )/(q max − q min ). Units: thousands of dollars.
Estimated annualized parameters for double exponential jump diffusion model. Value- weighted CRSP index, 30-day T-bill index deflated by the CPI. Sample period 1926:1-2020:12.
Optimal performance of a tontine overlay subject to withdrawal constraints
  • Article
  • Full-text available

November 2023

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37 Reads

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3 Citations

Astin Bulletin

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Kenneth R. Vetzal

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Graham Westmacott

We consider the holder of an individual tontine retirement account, with maximum and minimum withdrawal amounts (per year) specified. The tontine account holder initiates the account at age 65 and earns mortality credits while alive, but forfeits all wealth in the account upon death. The holder wants to maximize total withdrawals and minimize expected shortfall at the end of the retirement horizon of 30 years (i.e., it is assumed that the holder survives to age 95). The holder controls the amount withdrawn each year and the fraction of the retirement portfolio invested in stocks and bonds. The optimal controls are determined based on a parametric model fitted to almost a century of market data. The optimal control algorithm is based on dynamic programming and the solution of a partial integro differential equation (PIDE) using Fourier methods. The optimal strategy (based on the parametric model) is tested out of sample using stationary block bootstrap resampling of the historical data. In terms of an expected total withdrawal, expected shortfall (EW-ES) efficient frontier, the tontine overlay dramatically outperforms an optimal strategy (without the tontine overlay), which in turn outperforms a constant weight strategy with withdrawals based on the ubiquitous four per cent rule.

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Figure 12.1: Frontiers generated from the synthetic market. Parameters based on real CRSP index, real 30-day T-bills (see Table 8.1). Tontine case is as in Table 10.1. The No Tontine case uses the same scenario, but with no tontine gains, and no fees. The Const q, Const p case has q = 40, p = 0.10, with no tontine gains, which is the best result from Table 11.1, assuming no tontine gains, and no fees. Units: thousands of dollars.
Figure 12.2: Effect of varying fees charged for the Tontine, basis points (bps) per year. Frontiers generated from the synthetic market. Parameters based on real CRSP index, real 30-day T-bills (see Table 8.1). Base case Tontine is as in Table 10.1 (fees 50 bps per year). The No Tontine case uses the same scenario, but with no tontine gains, and no fees. Units: thousands of dollars.
Figure 12.3: Effect of randomly varying group gain G (Section 2.2). Frontiers generated from the synthetic market. Parameters based on real CRSP index, real 30-day T-bills (see Table 8.1). Base case Tontine (G = 1.0) is as in Table 10.1. Random G case uses the control computed for the base case, but in the Monte Carlo simulation, G is normally distributed with mean one and standard deviation 0.1. Units: thousands of dollars.
Constant weight, constant withdrawals, synthetic market results. No tontine gains. Stock index: real capitalization weighted CRSP stocks; bond index: real 30-day T-bills. Parameters from Table 8.1. Scenario in Table 10.1. Units: thousands of dollars. Statistics based on 2.56 × 10 6 Monte Carlo simulation runs. T = 30 years.
Optimal performance of a tontine overlay subject to withdrawal constraints

November 2022

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25 Reads

We consider the holder of an individual tontine retirement account, with maximum and minimum withdrawal amounts (per year) specified. The tontine account holder initiates the account at age 65, and earns mortality credits while alive, but forfeits all wealth in the account upon death. The holder desires to maximize total withdrawals, and minimize the expected shortfall, assuming the holder survives to age 95. The investor controls the amount withdrawn each year and the fraction of the investments in stocks and bonds. The optimal controls are determined based on a parametric model fitted to almost a century of market data. The optimal control algorithm is based on dynamic programming and solution of a partial integro differential equation (PIDE) using Fourier methods. The optimal strategy (based on the parametric model) is tested out of sample using stationary block bootstrap resampling of the historical data. In terms of an expected total withdrawal, expected shortfall (EW-ES) efficient frontier, the tontine overlay greatly outperforms an optimal strategy (without the tontine overlay), which in turn outperforms a constant weight strategy with withdrawals based on the ubiquitous four per cent rule.


OPTIMAL CONTROL OF THE DECUMULATION OF A RETIREMENT PORTFOLIO WITH VARIABLE SPENDING AND DYNAMIC ASSET ALLOCATION

July 2021

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13 Reads

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3 Citations

Astin Bulletin

We extend the Annually Recalculated Virtual Annuity (ARVA) spending rule for retirement savings decumulation (Waring and Siegel (2015) Financial Analysts Journal , 71 (1), 91–107) to include a cap and a floor on withdrawals. With a minimum withdrawal constraint, the ARVA strategy runs the risk of depleting the investment portfolio. We determine the dynamic asset allocation strategy which maximizes a weighted combination of expected total withdrawals (EW) and expected shortfall (ES), defined as the average of the worst 5% of the outcomes of real terminal wealth. We compare the performance of our dynamic strategy to simpler alternatives which maintain constant asset allocation weights over time accompanied by either our same modified ARVA spending rule or withdrawals that are constant over time in real terms. Tests are carried out using both a parametric model of historical asset returns as well as bootstrap resampling of historical data. Consistent with previous literature that has used different measures of reward and risk than EW and ES, we find that allowing some variability in withdrawals leads to large improvements in efficiency. However, unlike the prior literature, we also demonstrate that further significant enhancements are possible through incorporating a dynamic asset allocation strategy rather than simply keeping asset allocation weights constant throughout retirement.


Figure 10.3: Heat map of controls computed from solving the pre-commitment EW-ES problem (6.2) for κ = 2.5 with ARVA withdrawals based on the scenario from Table 9.1. The stabilization parameter in equation (7.9) is = −10 −4 .
Optimal control of the decumulation of a retirement portfolio with variable spending and dynamic asset allocation

January 2021

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228 Reads

We extend the Annually Recalculated Virtual Annuity (ARVA) spending rule for retirement savings decumulation to include a cap and a floor on withdrawals. With a minimum withdrawal constraint, the ARVA strategy runs the risk of depleting the investment portfolio. We determine the dynamic asset allocation strategy which maximizes a weighted combination of expected total withdrawals (EW) and expected shortfall (ES), defined as the average of the worst five per cent of the outcomes of real terminal wealth. We compare the performance of our dynamic strategy to simpler alternatives which maintain constant asset allocation weights over time accompanied by either our same modified ARVA spending rule or withdrawals that are constant over time in real terms. Tests are carried out using both a parametric model of historical asset returns as well as bootstrap resampling of historical data. Consistent with previous literature that has used different measures of reward and risk than EW and ES, we find that allowing some variability in withdrawals leads to large improvements in efficiency. However, unlike the prior literature, we also demonstrate that further significant enhancements are possible through incorporating a dynamic asset allocation strategy rather than simply keeping asset allocation weights constant throughout retirement.


OPTIMAL ASSET ALLOCATION FOR DC PENSION DECUMULATION WITH A VARIABLE SPENDING RULE

April 2020

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38 Reads

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10 Citations

Astin Bulletin

We determine the optimal asset allocation to bonds and stocks using an annually recalculated virtual annuity (ARVA) spending rule for DC pension plan decumulation. Our objective function minimizes downside withdrawal variability for a given fixed value of total expected withdrawals. The optimal asset allocation is found using optimal stochastic control methods. We formulate the strategy as a solution to a Hamilton–Jacobi–Bellman (HJB) Partial Integro Differential Equation (PIDE). We impose realistic constraints on the controls (no-shorting, no-leverage, discrete rebalancing) and solve the HJB PIDEs numerically. Compared to a fixed-weight strategy which has the same expected total withdrawals, the optimal strategy has a much smaller average allocation to stocks and tends to de-risk rapidly over time. This conclusion holds in the case of a parametric model based on historical data and also in a bootstrapped market based on the historical data.


Management of Portfolio Depletion Risk through Optimal Life Cycle Asset Allocation

June 2019

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83 Reads

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21 Citations

North American Actuarial Journal

Members of defined contribution (DC) pension plans must take on additional responsibilities for their investments, compared to participants in defined benefit (DB) pension plans. The transition from DB to DC plans means that more employees are faced with these responsibilities. We explore the extent to which DC plan members can follow financial strategies that have a high chance of resulting in a retirement scenario that is fairly close to that provided by DB plans. Retirees in DC plans typically must fund spending from accumulated savings. This leads to the risk of depleting these savings, that is, portfolio depletion risk. We analyze the management of this risk through life cycle optimal dynamic asset allocation, including the accumulation and decumulation phases. We pose the asset allocation strategy as an optimal stochastic control problem. Several objective functions are tested and compared. We focus on the risk of portfolio depletion at the terminal date, using such measures as conditional value at risk (CVAR) and probability of ruin. A secondary consideration is the median terminal portfolio value. The control problem is solved using a Hamilton-Jacobi-Bellman formulation, based on a parametric model of the financial market. Monte Carlo simulations that use the optimal controls are presented to evaluate the performance metrics. These simulations are based on both the parametric model and bootstrap resampling of 91 years of historical data. The resampling tests suggest that target-based approaches that seek to establish a safety margin of wealth at the end of the decumulation period appear to be superior to strategies that directly attempt to minimize risk measures such as the probability of portfolio depletion or CVAR. The target-based approaches result in a reasonably close approximation to the retirement spending available in a DB plan. There is a small risk of depleting the retiree’s funds, but there is also a good chance of accumulating a buffer that can be used to manage unplanned longevity risk or left as a bequest.


Defined Contribution Pension Plans: Who Has Seen the Risk?

April 2019

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149 Reads

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7 Citations

Journal of Risk and Financial Management

The trend towards eliminating defined benefit (DB) pension plans in favour of defined contribution (DC) plans implies that increasing numbers of pension plan participants will bear the risk that final realized portfolio values may be insufficient to fund desired retirement cash flows. We compare the outcomes of various asset allocation strategies for a typical DC plan investor. The strategies considered include constant proportion, linear glide path, and optimal dynamic (multi-period) time consistent quadratic shortfall approaches. The last of these is based on a double exponential jump diffusion model. We determine the parameters of the model using monthly US data over a 90-year sample period. We carry out tests in a synthetic market which is based on the same jump diffusion model and also using bootstrap resampling of historical data. The probability that portfolio values at retirement will be insufficient to provide adequate retirement incomes is relatively high, unless DC investors adopt optimal allocation strategies and raise typical contribution rates. This suggests there is a looming crisis in DC plans, which requires educating DC plan holders in terms of realistic expectations, required contributions, and optimal asset allocation strategies.


Optimal Asset Allocation for Retirement Saving: Deterministic Vs. Time Consistent Adaptive Strategies

January 2019

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33 Reads

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34 Citations

Applied Mathematical Finance

We consider optimal asset allocation for an investor saving for retirement. The portfolio contains a bond index and a stock index. We use multi-period criteria and explore two types of strategies: deterministic strategies are based only on the time remaining until the anticipated retirement date, while adaptive strategies also consider the investor’s accumulated wealth. The vast majority of financial products designed for retirement saving use deterministic strategies (e.g., target date funds). In the deterministic case, we determine an optimal open loop control using mean-variance criteria. In the adaptive case, we use time consistent mean-variance and quadratic shortfall objectives. Tests based on both a synthetic market where the stock index is modelled by a jump-diffusion process and also on bootstrap resampling of long-term historical data show that the optimal adaptive strategies significantly outperform the optimal deterministic strategy. This suggests that investors are not being well served by the strategies currently dominating the marketplace.


Citations (59)


... In that spirit, one natural choice would be to build a generative adversarial network (GAN) employing adapted distances as loss functions. Xu et al. [Xu+20] introduce the COT-GAN, using causal Wasserstein and optimal values of multistage optimization problems, like mean-variance portfolio optimization [FV22], log-utility maximization [Mer75], and optimal stopping [BCJ19]. On real market datasets, such as S&P 500 and VIX, conditional TC-VAE enables us to generate paths, as many as possible and as long as possible. ...

Reference:

Time-Causal VAE: Robust Financial Time Series Generator
Multi-Period Mean Expected-Shortfall Strategies: ‘Cut Your Losses and Ride Your Gains’
  • Citing Article
  • June 2023

Applied Mathematical Finance

... This is not a novel assumption and is in line with the mental bucketing idea proposed by Shefrin and Thaler (1988). The use of this assumption within literature targeting similar problems is also common (see Forsyth et al. (2022)). Of course, the objective of the optimal control is to make running out of savings an unlikely event. ...

Optimal Performance of a Tontine Overlay Subject to Withdrawal Constraints
  • Citing Article
  • January 2023

SSRN Electronic Journal

... At this stage, the retiree can choose to maximize the withdrawal amount while maintaining a high median wealth to meet the conditions for buying a life annuity after 15 years. Meanwhile, [12] focused on how to optimize retirement withdrawal through dynamic asset allocation and variable spending rules. This paper presented an extended annual recalculation virtual annuity (ARVA) rule that includes upper and lower limits on the amount of withdrawal. ...

OPTIMAL CONTROL OF THE DECUMULATION OF A RETIREMENT PORTFOLIO WITH VARIABLE SPENDING AND DYNAMIC ASSET ALLOCATION
  • Citing Article
  • July 2021

Astin Bulletin

... There are two strands of research. A small body of research proposes rules for how the weight in risky assets changes conditional on both age and the value of financial assets, including Basu, Byrne, and Drew (2011), Leung (2011), Forsythe andVetzal (2019) and Kobor and Muralidhar (2020). A larger body of research examines deterministic 'glide paths' for the adjustment of risky asset weight based on age alone. ...

Optimal Asset Allocation for Retirement Saving: Deterministic Vs. Time Consistent Adaptive Strategies
  • Citing Article
  • January 2019

Applied Mathematical Finance

... They emphasize that the addition of inflation rates makes the previously riskless asset risky and the lack of tools to hedge against inflation risk results in heightened return risk. Other pertinent studies in this field include Han and Hung (2012), Yao et al. (2013), Guan and Liang (2014), Chen and Delong (2015), Menoncin and Vigna (2017), Tang et al. (2018), Dong and Zheng (2020), Xu et al. (2020), Forsyth et al. (2020), Wang et al. (2021), Chen et al. (2023), and Wei and Yang (2023), among others. ...

OPTIMAL ASSET ALLOCATION FOR DC PENSION DECUMULATION WITH A VARIABLE SPENDING RULE
  • Citing Article
  • April 2020

Astin Bulletin

... where W t − 0 =Ŵ t − 0 := w 0 > 0 and n = 0, ..., N rb − 1. Since active funds often have restrictions on leverage and short-selling (see for example Forsyth et al. (2019); Ni et al. (2024)), these constraints are included in the formulation. ...

Management of Portfolio Depletion Risk through Optimal Life Cycle Asset Allocation
  • Citing Article
  • June 2019

North American Actuarial Journal

... There is also the risk of inadequate retirement income under the DC schemes given that the burden is entirely borne by the plan participants whose savings decisions are full of flaws (Forsyth & Vetzal, 2019). They recommended a behavioural change regarding the required contributions, expectations and the optimal asset allocation mechanisms. ...

Defined Contribution Pension Plans: Who Has Seen the Risk?

Journal of Risk and Financial Management

... Boado-Penas et al. (2020) assert that countries are better off under a mixed pension system from a DBS to a DCS. Moreover, Wanger (2021) argues that a DBS guarantees retirement income adequately but is risky and expensive to manage, whereas a DCS comes with the risk of inadequate retirement income since the portfolio value in the retirement stage may not be sufficient to provide an adequate retirement income, except if the DCS investors applied optimal allocation strategies and raised the contribution rate (Forsyth & Vetzal, 2019). Bulow (1982) observes that a corporate pension liability, which is an employment benefit, can be analysed based on the valuation of an ordinary corporate bond, which depends on the terms of the contract, dates and amounts of interest and principal payments, call prices, seniority of the debt and property alienation to security holders. ...

Defined Contribution Pension Plans: Who Has Seen the Risk?
  • Citing Article
  • January 2019

SSRN Electronic Journal

... The historical asset returns time series are inflation-adjusted using inflation data from the US Bureau of Labor Statistics 10 . For the purposes of obtaining the parameters for (5.1) in Subsections (5.1) and (5.2), we use the same calibration methodology as outlined in Dang and Forsyth (2016); Forsyth and Vetzal (2017), and assume the jump dynamics of the Kou (2002) model. ...

Dynamic mean variance asset allocation: Tests for robustness
  • Citing Article
  • September 2017

International Journal of Financial Engineering

... Roberts (2015) and Demerjian (2017) find that covenants serve as a trigger for renegotiation. Debt covenants can alleviate adverse information problems, mitigate agency conflicts, and reduce the cost of renegotiation and the risk to borrowers' future cash flows (Smith and Warner 1979;Leland 1994;Douglas et al. 2016). Borrowers may therefore include protective covenants in bond contracts to shield themselves from high default costs and renegotiation during the high policy uncertainty period. ...

Cash flow volatility and corporate bond yield spreads
  • Citing Article
  • February 2016

Review of Quantitative Finance and Accounting