# Abstracts

**Optimal investment with stochastic mortality and stochastic interest rates**

*Jan De Kort (University of Amsterdam, The Netherlands)*

Tuesday June 3, 11:00-11:30 | session 1.2 | Portfolio Optimization | room CD

This paper concerns the the optimal asset allocation of an investor in the presence of stochastic interest rates and a nonnegative stochastic mortality rate. A complete market setting is assumed where wealth can be invested in a zero-coupon bond, a survival bond, a stock and the money-market account. Survival bonds, which pay the difference between realized and expected mortality for a given population, can be thought of as an insurance against changes in survival probabilities. The investor, whose lifetime is uncertain, derives utility from accumulated wealth at his/her retirement date and utility from intermediate consumption. The mortality of the agent is modelled by a doubly stochastic Poisson process; the mortality rate and the short rate are assumed to follow independent Cox-Ingersoll-Ross processes. Using dual methods it will be shown that this problem has a closed-form solution. Conditions for existence of an optimal solution will be provided in terms of the model parameters. To derive these, we extend results by [1], [2] and [3]. We use a result by [4] to establish finiteness of the value function.

[1] G. Deelstra, M. Grasselli and P.F. Koehl (2000). Optimal investment strategies in a CIR framework. J. Appl. Prob. 37, 936-946.

[2] H. Kraft (2003). Optimal portfolios with stochastic interest rates and defaultable assets. Springer.

[3] H. Kraft (2009). Optimal portfolios with stochastic short rate: Pitfalls when the short rate is non-Gaussian or the market price of risk is unbounded. J. of Theoretical and Applied Finance 12, 767-796.

[4] T.R. Hurd and A. Kuznetsov (2008). Explicit formulas for Laplace transforms of stochastic integrals. Markov Processes and Related Fields 14, 277--290.

**Optimal Portfolio Choice with Multiple Benchmarks**

*Jan Vecer (Frankfurt School of Finance and Management, Germany)*

Tuesday June 3, 11:30-12:00 | session 1.2 | Portfolio Optimization | room CD

The objective of many portfolio managers is to beat a specific benchmark. This benchmark is typically chosen to be the stock market. The performance of the fund is then compared with a performance of the benchmark, say a stock index SP500 for a fund that invest in US stocks. From the no arbitrage arguments, it is impossible to beat the stock index for sure, thus such an investment strategy can guarantee success only with a certain probability. The problem how to maximize the probability of beating a specific index by a certain percentage has already been widely studied in the previous literature. Thus we focus our attention to another drawback of such a strategy. The fund manager can still beat the stock index, but in the situation of a market downturn, his strategy can significantly under perform the money market, making the investor worse off in comparison to a conservative strategy of holding the currency.

We can formulate the investment problem in the following way, we want the investment fund $X$ to have at least $\alpha$ units of the stock market $S$ and at least $\beta$ units of the money market $M$ at the end of the monitoring period $T$. Thus the objective is $$ X(T) \geq \max(\alpha S(T), \beta M(T)). $$ There is a region of values $a$ and $b$ such that this objective can be satisfied for sure and we find this set of feasible values in the geometric Brownian motion model. Certainly any values above 1 for both $a$ and $b$ are not feasible, this would create an arbitrage opportunity. However, for an arbitrary choice of $a$ and $b$, one can identify a portfolio that beats both benchmarks with the largest possible probability using the techniques of quantile hedging.

We also find the trading strategy that would deliver the objective portfolio for any $a$ and $b$ in the feasible set. Since both values of $a$ and $b$ must be below 1, there is a possibility that the resulting portfolio would under perform both the stock and the money market. We explicitly compute this probability. As it turns out, this probability of underperformance of both benchmarks converges to zero as time goes to infinity.

**A performance evaluation of weight-constrained conditioned portfolio optimisation using a new numerical scheme for multisignal problems**

*Jang Schiltz (University of Luxembourg, Luxembourg)*

Tuesday June 3, 12:00-12:30 | session 1.2 | Portfolio Optimization | room CD

Within a traditional context of myopic discrete-time mean-variance portfolio optimisation, the problem of conditioned optimisation, in which predictive information about returns contained in a signal is used to inform the choice of portfolio weights, was first expressed and solved in concrete terms by Ferson and Siegel. An optimal control formulation of conditioned portfolio problems was proposed and justified by Boissaux and Schiltz. This opens up the possibility of solving variants of the basic problem that do not allow for closed-form solutions through the use of standard numerical algorithms used for the discretisation of optimal control problems.

The present paper contributes to the empirical literature on this topic. We compare the performance of strategies resulting from conditioned optimisation and using several possible indicators for signalling purposes, to that obtained using standard approaches to portfolio investment. In particular, we report on both ex ante improvements to the accessible efficient frontier as measured through the typical associated metrics such as the Sharpe ratio, and ex post results affected, most notably, by specification errors regarding the relationship between signal and returns. We then discuss different problem parameters, examine their impact on performance and check whether significant ex post improvements may be achieved through optimal parameter selection.

This analysis implies the solving of optimal control problems involving a multidimensional objective function integral. To that effect, we propose a very simple direct collocation discretisation scheme suitable for the numerical solution of problems of this type. A convergence result is established to show that the scheme is consistent with multidimensional Pontryagin Principle relations in several important respects. Whilst the discussion focuses on the two-dimensional case, the simplicity of the scheme allows for easy generalisation. We carry out a backtest using real-world data and confirm that its results validate our proposed numerical scheme.

**Discrete and continuous-time expert opinions for portfolio optimization with partial information**

*Jörn Sass (University of Kaiserslautern, Germany)*

Tuesday June 3, 12:30-13:00 | session 1.2 | Portfolio Optimization | room CD

In a financial market with partial information we solve and compare utility maximization problems which include expert opinions on the underlying unobservable factors. We consider an investor who wants to maximize expected utility of terminal wealth obtained by trading in a financial market consisting of one riskless asset and several stocks. Stock returns are driven by a Brownian motion and the drift process is either an Ornstein-Uhlenbeck process (OUP) or a continuous time Markov chain (CTMC), independent of that Brownian motion. Thus the drift is hidden and has to be estimated from the observed stock prices. The best estimates given the observations are the Kalman and Wonham filters, respectively.

However, to improve the estimate, an investor may rely on expert opinions providing a noisy estimate of the current state of the drift. This reduces the variance of the filter and thus improves expected utility. It can be seen as a continuous time version of the classical Black-Litterman approach. Frey/Gabih/Wunderlich (2012) solve the case of an underlying CTMC. As an approximation, also expert opinions arriving continuously in time can be introduced. This allows for more explicit solutions for the portfolio optimization problem. Davis/LLeo (2013) consider this approach for an underlying OUP.

In this talk we present and solve the utility maximization problems for the two missing cases, an underlying CTMC with continuous expert opinions and an OUP with time-discrete opinions. We analyze relations and differences of the continuous-time and discrete-time approaches and discuss the value of the additional information.

References:

Davis, M., Lleo, S. (2013): Black-Litterman in continuous time: The case for filtering. Quantitative Finance Letters, to appear.

Frey, R. Gabih, A., Wunderlich, R. (2012) Portfolio optimization under partial information with expert opinions. International Journal of Theoretical and Applied Finance 15/1.