## Exam (2015)

This is the exam for the 2015 ensae course.

## Complete Markets, Static Case

In this post, I introduce the notion of market completeness in the context of a two period model. I also review the notion of a stochastic discount factor (SDF). The absence of arbitrage coupled with market completeness imply the existence of a unique strictly positive SDF. In this context, the investor chooses wealth/consumption in each state of the world so as to equalize marginal rates of substitution across states and the ratio of the values the SDF takes in the corresponding states. In other words, the marginal value of wealth/consumption in each state is proportional to the SDF, the constant of proportionality being the same for all states . For a given level of initial wealth and a given utility function, the SDF determines the end wealth/consumption distribution.

## Dynamic Models (Continuous Time)

In this post, I look at continuous time models. Price dynamics are described and wealth dynamics are derived. I then look at dynamic programming, describing the Hamilton Jacobi Bellman partial differential equation. I apply this to the case of constant investment opportunities and CRRA utility (including log utility). The results described in discrete time are shown to hold here as well. I then look at cases where investment opportunities fluctuate, reflecting the dynamics of a scalar Markovian process. In the first example, the driving variable is the (stochastic) cash rate. This model allows to discuss the role of bonds as hedging instruments within portfolios. In the second example, the cash rate is constant but the equity premium fluctuates and the equity index is more volatile in the short term than in the long term. This makes the optimal equity weight dependent on the investment horizon.

## Stochastic Calculus at the Speed of Light

In this post, I briskly review stochastic calculus. I concentrate on continuous martingales. Continuous martingales have trajectories with infinite first order variation and finite quadratic variation. This forces the development of a specific integration theory. I introduce continuous semimartingales, for which I spell out the Ito formula. I then consider the geometric Brownian motion and the Ornstein Uhlenbeck process.

## Dynamic Programming

In this short post, I sketch the derivation of the dynamic programming principle in discrete time.

## Dynamics, Discrete Time

Having looked at static investment problems, I now turn to dynamics starting with the discrete time case. Where static modeling is restricted to buy and hold policies, the dynamic setup allows rebalancing. Dynamic programming is introduced in a Markovian context with intermediate consumption. Some concrete problems with a terminal CRRA utility function are then solved. In the case of i.i.d. returns, the optimal policy is seen to be independent of the investment horizon, an initially counterintuitive result. It entails rebalancing to constant weights.The growth optimal portfolio which corresponds to log-utility is then considered, both when returns are i.i.d. and when they depend on a Markovian state variable. The optimal investment policy is often called the Kelly rule.

## Static Portfolio Choice

In this section of the course, I review the static portfolio choice problem. The investor chooses a portfolio structure which is then left alone. The investment criterion is the expected utility of wealth at a terminal date. I briefly review specifications for the utility function together with risk aversion concepts. I look at the case of constant absolute risk aversion and normal returns, with or without labour income. I then introduce mean variance preferences, linking them to expected utility. Mean variance with and without a risk free asset is studied. The link between mean variance preferences and the expected returns/beta relationship is explained (the key ingredient of the CAPM). I then touch on the implementation problem.

## Foreword to the ENSAE Course in Portf. Choice

The category ‘ENSAE’ groups posts that correspond to a course tought at ENSAE on portfolio choice. This post gives contains references and warnings.

## The Volatility-Drag

I review the impact on the future price level of random noise in the return process. This effect operates through the non-linearity of compounding. I look at how an increased level of volatility changes the price trajectories. Normally distributed (say) return variability lowers the median of the price distribution and skews it towards the upside. Over time, the skew increases as compounding operates. The majority of price trajectories are depressed by volatility. The mean price level is however unaffected as the lower trajectories are made up by a few ‘explosive’ ones. This will be important to bear in mind when discussing portfolio choice and the effect of portfolio rebalancing.

## Discrete Time Lognormal Dynamics

This post introduces discrete time price dynamics in a log-normal setup. This framework allows an easy translation of some of the continuous time results discussed on this site.