Simple Explicit Formula for Near-Optimal Stochastic Lifestyling

In lifecycle economics the Samuelson paradigm (Samuelson, 1969) states that optimal investment is in constant proportions out of lifetime wealth (composed of current savings and future income). It is well known that in the presence of credit constraints this paradigm no longer applies. Instead, optimal lifecycle investment gives rise to so-called stochastic lifestyling (Cairns et al., 2006), whereby for low levels of accumulated capital it is optimal to invest fully in stocks and then gradually switch to safer assets as the level of savings increases. In stochastic lifestyling not only does the ratio between risky and safe assets change but also the mix of risky assets varies over time. While the existing literature relies on complex numerical algorithms to quantify optimal lifestyling the present paper provides a simple formula that captures the main essence of the lifestyling effect with remarkable accuracy

Simple Explicit Formula for Near-Optimal Stochastic Lifestyling

In life-cycle economics, the Samuelson paradigm (Samuelson, 1969) states that the optimal investment is in constant proportions out of lifetime wealth composed of current savings and the present value of future income. It is well known that in the presence of credit constraints this paradigm no longer applies. Instead, optimal life-cycle investment gives rise to so-called stochastic lifestyling (Cairns et al., 2006), whereby for low levels of accumulated capital it is optimal to invest fully in stocks and then gradually switch to safer assets as the level of savings increases. In stochastic lifestyling not only does the ratio between risky and safe assets change but also the mix of risky assets varies over time. While the existing literature relies on complex numerical algorithms to quantify optimal lifestyling, the present paper provides a simple formula that captures the main essence of the lifestyling effect with remarkable accuracy

Target-driven investing: Optimal investment strategies in defined contribution pension plans under loss aversion

Assuming loss aversion, stochastic investment and labour income processes, and a path-dependent target fund, we show that the optimal investment strategy for defined contribution pension plan members is a target-driven 'threshold' strategy. With this strategy, the equity allocation is increased if the accumulating fund is below target and decreased if it is above. However, if the fund is sufficiently above target, the optimal investment strategy switches discretely to 'portfolio insurance'. We show that under loss aversion, the risk of failing to attain the target replacement ratio is significantly reduced compared with target-driven strategies derived from maximising expected utility.Defined Contribution Pension Plan; Investment Strategy; Loss Aversion; Target Replacement Ratio; Threshold Strategy, Portfolio Insurance, Dynamic Programming

A model of pension portfolios with salary and surplus process

Magister Scientiae - MScEssentially this project report is a discussion of mathematical modelling in pension funds, presenting sections from Cairns, A.J.D., Blake, D., Dowd, K., Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans, Journal of Economic Dynamics and Control, Volume 30, Issue 2006, Pages 843-877, with added details and background material in order to demonstrate the mathematical methods. In the investigation of the management of the investment portfolio, we only use one risky asset together with a bond and cash as other assets in a continuous time framework. The particular model is very much designed according to the members’ preference and then the funds are invested by the fund manager in the financial market. At the end, we are going to show various simulations of these models. Our methods include stochastic control for utility maximisation among others. The optimisation problem entails the optimal investment portfolio to maximise a certain power utility function. We use MATLAB and MAPLE programming languages to generate results in the form of graphs and tables.South Afric

Target-driven investing: Optimal investment strategies in defined contribution pension plans under loss aversion

Assuming the loss aversion framework of Tversky and Kahneman (1992), stochastic investment and labour income processes, and a path-dependent fund target, we show that the optimal investment strategy for defined contribution pension plan members is a target-driven 'threshold' strategy, whereby the equity allocation is increased if the accumulating fund is below target and is decreased if it is above. However, if the fund is sufficiently above target, the optimal investment strategy switches to 'portfolio insurance'. We show that the risk of failing to attain the target replacement ratio is significantly lower with target-driven strategies than with those associated with the maximisation of expected utility

Target-driven investing: Optimal investment strategies in defined contribution pension plans under loss aversion

Assuming loss aversion, stochastic investment and labour income processes, and a path-dependent target fund, we show that the optimal investment strategy for defined contribution pension plan members is a target-driven 'threshold' strategy. With this strategy, the equity allocation is increased if the accumulating fund is below target and decreased if it is above. However, if the fund is sufficiently above target, the optimal investment strategy switches discretely to 'portfolio insurance'. We show that under loss aversion, the risk of failing to attain the target replacement ratio is significantly reduced compared with target-driven strategies derived from maximising expected utility

Stochastic modelling in bank management and optimization of bank asset allocation

>Magister Scientiae - MScThe Basel Committee published its proposals for a revised capital adequacy framework(the Basel II Capital Accord) in June 2006. One of the main objectives of this framework is to improve the incentives for state-of-the-art risk management in banking, especially in the area of credit risk in view of Basel II. The new regulation seeks to provide incentives for greater awareness of differences in risk through more risk-sensitive minimum capital requirements based on numerical formulas. This attempt to control bank behaviour has a heavy reliance on regulatory ratios like the risk-based capital adequacy ratio (CAR). In essence, such ratios compare the capital that a bank holds to the level of credit, market and operational risk that it bears. Due to this fact the objectives in this dissertation are as follows. Firstly, in an attempt to address these problems and under assumptions about retained earnings, loan-loss reserves, the market and shareholder-bank owner relationships, we construct continuous-time models of the risk-based CAR which is computed from credit and market risk-weighted assets (RWAs) and bank regulatory capital (BRC) in a stochastic setting. Secondly, we demonstrate how the CAR can be optimized in terms of equity allocation. Here, we employ dynamic programming for stochastic optimization, to obtain and verify the results. Thirdly, an important feature of this study is that we apply the mean-variance approach to obtain an optimal strategy that diversifies a portfolio consisting of three assets. In particular, chapter 5 is an original piece of work by the author of this dissertation where we demonstrate how to employ a mean-variance optimization approach to equity allocation under certain conditions

Stochastic control, numerical methods, and machine learning in finance and insurance

We consider three problems motivated by mathematical and computational finance which utilize forward-backward stochastic differential equations (FBSDEs) and other techniques from stochastic control. Firstly, we review the case of post-retirement annuitization with labor income in framework of optimal stochastic control and optimal stopping. We apply the martingale approach to a Cobb–Douglas type utility maximization problem. We have proved the theoretical existence and uniqueness of an optimal solution. Several analyses are made based on the simulations for the optimal stopping choice and strategies. Secondly, We review the convolution method in backward stochastic differential equations (BSDEs) framework and study the application of convolution method to Heston model. We provide an easy representation of the Heston characteristic function that avoids the discontinuities caused by branch rotations in the logarithm of complex functions and is able to be applied in calibration. We proposed two convolution schemes to the Heston model and provide the error analysis that shows the error orders of discretization and truncation. We review two error control methods and improve the accuracy on the boundaries. Numerical results comparing to a Fourier method and an integration method is provided. Thirdly, we review the forecasting problem in bond markets. Our data include both U.S. Treasuries and coupon bonds from twelve corporate issuers. We apply the arbitrage-free model in predicting the yields and the prices of coupon bonds in a sequential model with the Kalman filter, the extended Kalman filter and the particle filter. We implement the arbitrage penalty and obtain the optimal dynamic parameterization using deep neural networks. The purpose of the prediction is to examine the effect of arbitrage penalty and the forecasting performance on different time horizons. Our result shows that the arbitrage-free penalty has improving performance on short time period but downgrading performance on long time period. We provide analysis on the prediction errors, the distribution of errors, and the average excess return. The predicted bond prices shows the prediction errors have non-Gaussian distribution, excess kurtosis, and fat tails. Future works will be from two aspects, refine the importance sampling by non-parametric distribution and refine the term structure model with jump process and credit risk