Portfolio optimization under dynamic risk constraints: Continuous vs. discrete time trading
-
Imke Redeker
Abstract
We consider an investor facing a classical portfolio problem of optimal investment in a log-Brownian stock and a fixed-interest bond, but constrained to choose portfolio and consumption strategies that reduce a dynamic shortfall risk measure. For continuous- and discrete-time financial markets we investigate the loss in expected utility of intermediate consumption and terminal wealth caused by imposing a dynamic risk constraint. We derive the dynamic programming equations for the resulting stochastic optimal control problems and solve them numerically. Our numerical results indicate that the loss of portfolio performance is not too large while the risk is notably reduced. We then investigate time discretization effects and find that the loss of portfolio performance resulting from imposing a risk constraint is typically bigger than the loss resulting from infrequent trading.
A Proof of Lemma 2.3
Under the assumption that the portfolio-proportion strategy
From the equation above we obtain that
Recall, the Expected Loss at time t is defined by
This expectation can be calculated as follows. Let
denote the probability density function of Z; then
where
For the integral
where
and
Using the change of variables technique yields
where
Note that
B Proof of Lemma 3.1
Given an investment-consumption strategy
Note that given a probability level
We have
and
where
We have used that the random variable
Since the Dynamic Value at Risk is the smallest l satisfying the above inequality, we obtain
where
The Tail Conditional Expectation at time
where
and
yields that the above inequality
We obtain
where we have used that the random variable
Finally, the Dynamic Tail Conditional Expectation can be written as
where
In order to prove the statement for the Dynamic Expected Loss, we use that the wealth at time
This expectation can be calculated as follows. Let
denote the probability density function of Z; then
where
and
The integral
where
The integral
Using the change of variables method yields
where
Note that
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© 2017 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Portfolio optimization under dynamic risk constraints: Continuous vs. discrete time trading
- On risk measuring in the variance-gamma model
- Distortion risk measures, ROC curves, and distortion divergence
- EM algorithm for Markov chains observed via Gaussian noise and point process information: Theory and case studies
- Optimal expected utility risk measures
Artikel in diesem Heft
- Frontmatter
- Portfolio optimization under dynamic risk constraints: Continuous vs. discrete time trading
- On risk measuring in the variance-gamma model
- Distortion risk measures, ROC curves, and distortion divergence
- EM algorithm for Markov chains observed via Gaussian noise and point process information: Theory and case studies
- Optimal expected utility risk measures
