Money Networks Manage your money with synchronous-reactive money networks. By Adam Cataldo.

Money Networks

Manage your money with synchronous-reactive money networks.

By Adam Cataldo

Outline

I. Discrete-Event Money Models

II. Synchronous Reactive Money Networks

III. Two-Way Functions

IV. Money Network Properties

V. N-Way Functions

Discrete-Event Money Models

Transactions happen at discrete points in time, but time, as we know it, is not discretized.

This discrete event model describes this behavior.

This model is accurate but awkward to work with.

DE Savings Account

Present/Future Value problems

An account opened today with $m has present value (PV) $m.

If it is compounded annually at interest rate i, after t years, the account is worth PV(1+i)^t.

I call this amount the future value (FV) after t years.

PV/FV function Given an interest rate i and a time t, I

can calculate PV from FV or FV from PV as follows:

(1 )tFV PV i

(1 )tFV

PVi

PV/FV function in SR domain

Given the present value, the PV/FV function returns the future value.

Given the future value, the function returns the present value.

For an initial investment of PV left to mature, we use:

f(PV) = PV * (1 + i) ^ t g(FV) = FV / (1 + i)^t

Specifically

( , ) ( , )

( , ) ( , ( ))

( , ) ( ( ), )

( , ) ( , )

F

F PV PV f PV

F FV g FV FV

F PV FV PV FV

In Ptolemy II

In Ptolemy II

In Ptolemy II

PV

In Ptolemy II

PV FV

Calculating PV from FV

Two-Way Function The two-way function is a

generalization of the PV/FV function in the SR domain.

The function has two inputs x and y. Either y = f(x) or x = g(y), depending

on which input is known first.

Form of the Two-Way Function

( , ) ( , )

( , ) ( , ( ))

( , ) ( ( ), )

( , ) ( , )

F

F x x f x

F y g y y

F x y x y

`

Two-Way Function Properties

The two-way function is monotonic. That is, if (a,b) (c,d) then F(a,b) F(c,d).

The two-way function is continuous. That is, F(V(a,b)) = V F(a,b) for any chain.

This means F has a least fixed point for any two signals.

Another Two-Way Function

Account with monthly investments m:

0

0

( ) / (1 )

( ) (1 )

tk

k

tk

k

m FV FV i

FV m m i

Another Two-Way Function

In terms of x, y, f, and g:

( ) ( )

( ) ( )

f x m x

g y FV y

In Ptolemy II

This two-way function will not currently run, because the Ptolemy expression language cannot not support the “sum” function.

If it did however...

In Ptolemy II

Money Networks A money network is any synchronous-

reactive network used to calculate monetary values.

A money network describes an investment situation.

This network allows fast redefining of inputs and outputs (present and future values).

N-Way Function Recall the situation where monthly

investments of m are made.

1

0

1( )

(1 )

( ) (1 )

t

nn

tk

k

PV m mi

FV m m i

N-Way Function If we know either m, PV, or FV, we

know all three values, because

1

0

1( ) 1/

(1 )

( ) 1/ (1 )

t

nn

tk

k

m PVi

m FV i

N-Way Function

and

1

0

1( ) ( )

(1 )

( ) ( ) (1 )

t

nn

tk

k

PV FV m FVi

FV PV m PV i

N-Way Function This suggests a generalization of the

Two-Way Function to n signals. If one signal is known, all other signals

equal a function of that signal. Otherwise, the signals do not change.

Three-Way Function Example

2 3

1 3

1 2

1 1 1 1

2 2 2 2

3 3 3 3

( , , ) ( , ( ), ( ))

( , , ) ( ( ), , ( ))

( , , ) ( ( ), ( ), )

x x

x x

x x

F x x f x f x

F x g x x g x

F x h x h x x

( , , ) ( , , )F a b c a b c,otherwise

Ptolemy Model(not implemented)

Number of Functions For the simple three-way function, we

require 6 functions. In general, we require n(n-1) functions

for an n-way function. We can reduce this number to n when

all the functions are invertible. This function is one such function.

Key Result: Networks on N-Way Functions

If a set of N-way functions is connected in a graph, knowing the value along exactly one edge determines the values at all other edges of the graph.

This value can be set by another function, such as the constant function in Ptolemy.

Conclusions

Money networks make it possible to determine several present and future values based on a single value.

The same money network can be used to determine different values.

In a connected network, knowing a single value determines all others.

Future Work: Improving Money Networks

Build a library of money network functions.

Improve the GUI representation of money networks in Ptolemy II.

Extend the Ptolemy expression language to handle more general expressions, such as “sum”