One of the most popular risk management strategies for wheat producers is varietal diversification. Previous studies proposed a mean-variance model as a tool to optimally select wheat varieties. However, this study suggests that the mean–expected shortfall (ES) model (which is based on a downside risk measure) may be a better tool because variance is not a correct risk measure when the distribution of wheat variety yields is multivariate nonnormal. Results based on data from Texas Blacklands confirm our conjecture that the mean-ES framework performs better in term of selecting wheat varieties than the mean-variance method.

This paper addresses the financing of public pensions in a stochastic environment. Traditionally, funded and unfunded pension schemes have been viewed as opposite solutions for the first pillar of public pensions. However, more recently countries as Sweden and Poland have explored mixed solutions that combine pay-as-you-go (PAYG) with funding mechanisms. The aims of this paper are to examine the rationality of such a combination using portfolio theory arguments and to find the optimal split of the contributions between the two systems. We first introduce the classical deterministic model leading to the well-known Samuelson–Aaron rule according to which diversification is never optimal. We then introduce different stochastic models in which the main processes (wage growth, population growth, financial rate of return) are random. In particular, we obtain conditions on parameters to justify diversification and explicit optimal sharing between PAYG and funding. We also introduce the possibility of investing in several financial assets and explore the impact of introducing systematic longevity risk.

This research shows that a portfolio of wheat varieties could enhance profitability and reduce risk over the selection of a single variety for Kansas wheat producers. Many Kansas wheat farmers select varieties solely based on published average yields. This study uses portfolio theory from business investment analysis to find the optimal, yield-maximizing and risk-minimizing combination of wheat varieties in Kansas.

We study stochastic dynamic investment games in continuous time between two investors (players) who have available two different, but possibly correlated, investment opportunities. There is a single payoff function which depends on both investors’ wealth processes. One player chooses a dynamic portfolio strategy in order to maximize this expected payoff, while his opponent is simultaneously choosing a dynamic portfolio strategy so as to minimize the same quantity. This leads to a stochastic differential game with controlled drift and variance. For the most part, we consider games with payoffs that depend on the achievement of relative performance goals and/or shortfalls. We provide conditions under which a game with a general payoff function has an achievable value, and give an explicit representation for the value and resulting equilibrium portfolio strategies in that case. It is shown that non-perfect correlation is required to rule out trivial solutions. We then use this general result explicitly to solve a variety of specific games. For example, we solve a probability maximizing game, where each investor is trying to maximize the probability of beating the other's return by a given predetermined percentage. We also consider objectives related to the minimization or maximization of the expected time until one investor's return beats the other investor's return by a given percentage. Our results allow a new interpretation of the market price of risk in a Black-Scholes world. Games with discounting are also discussed, as are games of fixed duration related to utility maximization.

We study a variety of optimal investment problems for objectives related to attaining goals by a fixed terminal time. We start by finding the policy that maximizes the probability of reaching a given wealth level by a given fixed terminal time, for the case where an investor can allocate his wealth at any time between n + 1 investment opportunities: n risky stocks, as well as a risk-free asset that has a positive return. This generalizes results recently obtained by Kulldorff and Heath for the case of a single investment opportunity. We then use this to solve related problems for cases where the investor has an external source of income, and where the investor is interested solely in beating the return of a given stochastic benchmark, as is sometimes the case in institutional money management. One of the benchmarks we consider for this last problem is that of the return of the optimal growth policy, for which the resulting controlled process is a supermartingale. Nevertheless, we still find an optimal strategy. For the general case, we provide a thorough analysis of the optimal strategy, and obtain new insights into the behavior of the optimal policy. For one special case, namely that of a single stock with constant coefficients, the optimal policy is independent of the underlying drift. We explain this by exhibiting a correspondence between the probability maximizing results and the pricing and hedging of a particular derivative security, known as a digital or binary option. In fact, we show that for this case, the optimal policy to maximize the probability of reaching a given value of wealth by a predetermined time is equivalent to simply buying a European digital option with a particular strike price and payoff. A similar result holds for the general case, but with the stock replaced by a particular (index) portfolio, namely the optimal growth or log-optimal portfolio.

Dynamic asset allocation strategies that are continuously rebalanced so as to always keep a fixed constant proportion of wealth invested in the various assets at each point in time play a fundamental role in the theory of optimal portfolio strategies. In this paper we study the rate of return on investment, defined here as the net gain in wealth divided by the cumulative investment, for such investment strategies in continuous time. Among other results, we prove that the limiting distribution of this measure of return is a gamma distribution. This limit theorem allows for comparisons of different strategies. For example, the mean return on investment is maximized by the same strategy that maximizes logarithmic utility, which is also known to maximize the exponential rate at which wealth grows. The return from this policy turns out to have other stochastic dominance properties as well. We also study the return on the risky investment alone, defined here as the present value of the gain from investment divided by the present value of the cumulative investment in the risky asset needed to achieve the gain. We show that for the log-optimal, or optimal growth policy, this return tends to an exponential distribution. We compare the return from the optimal growth policy with the return from a policy that invests a constant amount in the risky stock. We show that for the case of a single risky investment, the constant investor's expected return is twice that of the optimal growth policy. This difference can be considered the cost for insuring that the proportional investor does not go bankrupt.

We derive optimal gambling and investment policies for cases in which the underlying stochastic process has parameter values that are unobserved random variables. For the objective of maximizing logarithmic utility when the underlying stochastic process is a simple random walk in a random environment, we show that a state-dependent control is optimal, which is a generalization of the celebrated Kelly strategy: the optimal strategy is to bet a fraction of current wealth equal to a linear function of the posterior mean increment. To approximate more general stochastic processes, we consider a continuous-time analog involving Brownian motion. To analyze the continuous-time problem, we study the diffusion limit of random walks in a random environment. We prove that they converge weakly to a Kiefer process, or tied-down Brownian sheet. We then find conditions under which the discrete-time process converges to a diffusion, and analyze the resulting process. We analyze in detail the case of the natural conjugate prior, where the success probability has a beta distribution, and show that the resulting limit diffusion can be viewed as a rescaled Brownian motion. These results allow explicit computation of the optimal control policies for the continuous-time gambling and investment problems without resorting to continuous-time stochastic-control procedures. Moreover they also allow an explicit quantitative evaluation of the financial value of randomness, the financial gain of perfect information and the financial cost of learning in the Bayesian problem.

The set of efficient portfolios in an asset liability model is discussed in detail. The occurence of liabilities leads to a parallel shift of the efficient set. Under an appropriate assumption, the shift vector can be decomposed in different components. For the special case, where the investor is a pension fund, it is shown how shortfall constraints can be reconciled with efficiency. Finally, optimality conditions for the market portfolio are derived.

This paper provides an introduction to asset pricing theory and its applications in non-life insurance. The first part of the paper presents a basic review of asset pricing models, including discrete and continuous time capital asset pricing models (the CAPM and ICAPM), arbitrage pricing theory (APT), and option pricing theory (OPT). The second part discusses applications in non-life insurance. Among the insurance models reviewed are the insurance CAPM, discrete time discounted cash flow models, option pricing models, and more general continuous time models. The paper concludes that the integration of actuarial and financial theory can provide major advances in insurance pricing and financial management.