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

2. Outline I. Discrete-Event Money Models II. Synchronous Reactive Money Networks III. Two-Way Functions IV. Money Network Properties V. N-Way Functions

3. 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.

4. DE Savings Account

5. 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.

6. PV/FV function  Given an interest rate i and a time t, I can calculate PV from FV or FV from PV as follows:

7. 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

8. Specifically

9. In Ptolemy II

13. Calculating PV from FV

14. 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.

15. Form of the Two-Way Function

16. `

17. 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.

18. Another Two-Way Function  Account with monthly investments m:

19. Another Two-Way Function  In terms of x, y, f, and g:

20. 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...

21. In Ptolemy II

22. 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).

23. N-Way Function  Recall the situation where monthly investments of m are made.

24. N-Way Function  If we know either m, PV, or FV, we know all three values, because

25. N-Way Function and

26. 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.

27. Three-Way Function Example

28. Ptolemy Model (not implemented)

29. 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.

30. 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.

31. 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.

32. 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”