NAIF - Numerical Analysis of Investment Functions

The arguments of the investment function are of the type xy_z where x is a string of three characters which denotes the type of argument and the following (integer) numbers y and z stand for possible specifiers. The meaning of the specifiers depend on the type of argument

ret#1

stands for the realized return #1 steps in the past. Specifier #1 can take positive integer values 1,2,3,etc. Notice that due to the timing convention, the contemporaneous value of return cannot be an argument of the investment function.

yld#1

stands for the realized dividend yield #1 steps in the past. Specifier #1 can take non-negative integer values 0,1,2,3,etc.

ewr#1_#2

stands for the Exponential Weighting Moving Average estimator of the #1-th central moment of realized returns computed with the weight parameter "lambda" equal to 0.#2. Specifier #1 can take positive integer values 1,2,3,etc. Specifier #2 can take any non-negative integer value 0,1,2,3,etc. Notice that parameter "lambda" belongs to the interval [0,1).

ewy#1_#2

stands for the Exponential Weighting Moving Average estimator of the #1-th central moment of realized yields computed with the weight parameter "lambda" equal to 0.#2. Specifier #1 can take positive integer values 1,2,3,etc. Specifier #2 can take any non-negative integer value. Notice that parameter "lambda" belongs to the interval [0,1).

cwr#1_#2

stands for the Constant Weighting Moving Average estimator of the #1-th central moment of realized returns computed on the last #2 time steps. Both specifiers #1 and #2 can take positive integer values 1,2,3,etc.

cwy#1_#2

stands for the Constant Weighting Moving Average estimator of the #1-th central moment of realized dividend yields computed on the last #2 time steps. Both specifiers #1 and #2 can take positive integer values 1,2,3,etc.

The following line

f=ret1+cwy3_12,x=.3,w=10

defines an agent who invests, at each time step, a share of wealth in the risky asset which is proportional to the sum of the last realized return (ret1) and the third central moment of the realized yields estimated through CWMA (cwy3) on the last 12 (₁₂) time steps. The initial agent's endowment is 10 (w=10) of which 30% is invested in the risky security (x=.3) and the remaining in the riskless one.

A small modification

f=ret1+cwy3_12,x=.3,w=10,eps=.01

leads to a definition of analogous agent, but now, at each time step, a random component is drawn from an uniform distribution on [-.005,.005] and added to the agents' investment function.

The line

f=ret1+cwy3_12,x=.3,w=10,eps=.01,u=0.99,l=0.01

describes the same agent, whose investment choice is truncated in order to be in interval [0.01,0.99]. (This guarantees the boundedness of the dynamics!)

A group of similar agent can be defined prepending to the agent definition an integer number, which denotes the number of agents in the group, followed by a column ':'. For instance

50:f=ret1+cwy3_12,x=.3,w=10,eps=.01,u=0.99,l=0.01

stands for a group of fifty agents similar to the agent discussed above. Notice that due to the i.i.d. nature of the random component of the investment function, the investment decisions of the different agents of the group usually will not be identical.

Each agent i is given an initial wealth level (w_i) and an initial share of wealth (x_i) invested in the risky assets. These quantities can be explicitly specified using the parameter w and x in the configuration file. If not specified, their default values are 1 and .5, respectively. From these quantities, the initial portfolio composition of each agent is determined as

\[ A_i = x_i w_i / \sum_j x_j w_j \]

\[ B_i = (1-x_i) w_i / \sum_j x_j w_j \]

where \(A_i\) and \(B_i\) stand for the initial amount of risky and riskless security, respectively. These values should be positive in order the initial market conditions could be defined. Otherwise, the message "Agents specification does not allow to define initial price." is printed.

The initial price (for time t=0) of the risky security, according to the initialization of the agents' portfolio defined above, is set to 1.

The generation of the first dividend (for time t=1) depends on the market process specified in the command line. If the process is described by means of the yield, then yield "yld1" is accordingly generated and first dividend coincides with "yld1" since p₀=1. If the process is described by means of the dividend (i.e. if the dividend is assumed to be growing), initial value of the dividend is set to .001.

The first price (for time t=1) is determined using the agents' investment functions, already.

Moreover, an initial "fake" histories of both price returns and dividend yields are generated. The lengths of these histories are set to a value large enough to provide an initial value for each estimator (see below). The returns and yields are identically and independently extracted from an uniform distribution. The mean and variance of the distributions for the return and yield can be specified from the command line. Both parameters have a default value of 0.

The initial value assigned to agent's forecast variables depends on their nature. Past returns variables (ret, yld) and equally weighted averages (cwr and cwy) are computed on the initial return or yield history, respectively, generated according to the procedure described above. Conversely, to the EWMA estimators (ewr and ewy), which are (in principle) functions of an infinite set of returns, a value is assigned which is equal to the last return raised to the order of the estimator. For example, if \(r_0\) is the last return of the initial history of returns, ewr1 is set to \(r_0\), ewr2 is set to \(r_0^2\), ewr3 to \(r_0^3\), etc. Notice that the memory parameter "lambda" doesn't play any role in the initial value of the estimator.