The goal of this project is to solve the optimal portfolio choice problem of an investor. The investor allocates her wealth to various risky assets (stocks) and a riskless bond. The allocations are not all permitted, since the agent has to respect a budget constraint. In fact,  she cannot go beyond a certain credit line. This credit line is specified technically as  a borrowing constraint against future labor income.  This means that when trading, the agent can borrow only up to a certain limit, provided by the expectation of her future labor income.

Labor income is not constant, it varies with time. Its time evolution is modeled by an equation, which is stochastic because noise is taken into account.  The evolution of labor income here incorporates the stickiness feature of the wages, also called nominal rigidity.

Sticky wages is an important macroeconomic concept. As systematically outlined by  Keynes  in The General Theory of Employment, Interest and Money, 1936, stickiness refers to the fact that wages and prices do not adjust immediately to shocks (i.e. sudden changes) in the economy. As one can easily imagine,  since then a vast economic literature on the topic has appeared.

Labor income evolution specifications  (read: the equation which governs it) are key to build up a consistent and reliable model.  If we only know that a solution to the labor income equation exists, this is acceptable to mathematicians, but of little use. So, it is crucial to choose in the model an equation which has  a friendly, explicit solution and meets empirical features.  A category of processes well known in Econometrics, the ARMA processes, are solution to specific evolution equations – they offer a class of general models.  As pointed out in several papers, there is empirical evidence that  ARMA processes offer satisfactory models precisely for stochastic labor income,  as they also incorporate stickiness.

The equation for labor income that we choose in this paper can be seen as a limit of ARMA processes. Why passing to the limit is necessary? Because ours is a continuous time setting! In our framework, the agent  can trade continuously in time. It is as if the exchange were always open. Our equation, describing how labor income changes, is technically called stochastic delay differential equation (SDDE) . Since it is a limit of ARMA processes, our SDDE seems a sensible choice to describe labor income evolution in continuous time models.

Continuous time SDDEs are however infinite dimensional: the involved parameters are infinite. Therefore, it is apparent that the ability to solve a SDDE  is an issue! In fact, one always aims to find out explicit solutions,  but here is rarely the case due to the complexity of the setup.

Let’s focus on who are the simplest possible delay equations: linear ones. To explain a little, linear delay equation is an (evolution) equation in which the contribution of the past to the present is linear.  A simple example of linearity which gives ‘stickiness’ is: my labor income today is  3 + half of my yesterday labor income +half of my labor income of two days ago.  In this toy example, the function 0.5 of yesterday + 0.5 of two days ago is called weight function. The extent of memory of this example is just {-2,-1} meaning -2 days from now, -1 days from now. Note that here time is discrete, while we work in continuous time and, as we will see, the extent of memory will become an interval.

In the context of portfolio choice with sticky labor income, linear SDDE already appeared in the literature.  In the present project,  technically speaking, the stocks are  a multidimensional geometric Brownian motion and labor income follows a linear SDDE in which delay is present in one part of the equation only, called the drift.

The past is weighted by a function  φ on the bounded time window [-d, 0). The time window models the extent of memory of the system, up to d instants backward from now.

Since d is bounded and not infinity, we are in a  case of bounded memory.

This choice turns out to be appropriate. In fact, there is evidence of  bounded memory in labor income adjustment delays.

How do we measure φ? Is it easy? Absolutely not! An estimation of how the past influences the future, namely the estimation of the function φ is by no means easy.  And, if you think about it, how do you know that your salary today is precisely 3 +half of yesterday and half of two days ago? Could it be 3 +half of yesterday and a quarter of two days ago and a quarter of three days ago?

Therefore, we allow the delay weight φ to vary in a given set K where the function will plausibly belong.   We thus account for  possible misspecification of the weight function due to the complexity of the global economic dynamics. This set K represents the confidence set.

As for investment possibilities, as we already said, the  agent is allowed to borrow against future wages. A budget constraint is present. When the confidence set K is just one point, namely when the function φ is known with 100% confidence,  a risk averse investor  simply maximizes  expected (power) utility from consumption and bequest. The optimal portfolio resulting from this maximization will be her choice.

When K is general, we need a little more. We assume that the investor is  an uncertainty averse investor. She is not only risk averse, but also assumes that Nature will always choose the worst weight delay φ against her. That is, she thinks the weight which gives her the worst utility will come out of the hat. Then, the agent will take a so called maxmin approach (worst case approach). That is, she will first minimize over the delay weights φ in K, taking the worst,  and then maximize over all her feasible allocations. This is a prudent attitude indeed!

To conclude: we can solve our equations and get an explicit solution.

For economists and mathematicians we add a couple of lines here below.

We  show that the investor becomes observationally equivalent to one in the infinite dimensional Merton problem with worst  weight delay φ*.  The couple: optimal strategy  and φ*  are the unique solution to the corresponding  HJB-Isaacs equation.  This couple is also easily shown to be the solution of the minmax problem, and therefore is a saddle point of  the  zero-sum  game between the investor and the adverse player (Nature) .

This paper is written jointly with Fausto Gozzi and Margherita Zanella.