1. Property Cycle Dynamics Estimating Property Cycle Turning Points John MacFarlane School of Computing and Mathematics University of Western Sydney Email: j.macfarlane@uws.edu.auj.macfarlane@uws.edu.au

2. EXHIBIT 1:From Pyhrr et al 1999

3. EXHIBIT 2:From Mueller and Laposa (1994)

4. EXHIBIT 3: Sydney CBD Vacancy Rate (%), 1970 – 2008

5. EXHIBIT 4: Equal Phases and symmetric

6. EXHIBIT 5: Equal Phases but non-symmetric within each phase

7. EXHIBIT 6:Asymmetric Phases

8. EXHIBIT 10Property cycle as a circle or clock

9. Cycle Representation A CIRCLE (Clock) is the ideal representation for cyclic behaviour. Physics:Particle moving in a circle at constant angular velocity. The vertical displacement at time T gives standard sine/cosine curve. Easy for respondents to indicate where they believe we are on the cycle by indicating the appropriate time (3, 6 10, etc.)

10. Australian Property Directions Survey

12. Problems in Representing Cyclic Behaviour

30. An Alternative Model Sine/Cosine Model: Particle moving in a circle at constant angular velocity. Alternative (generalised) Model: Angular velocity varies with the angle Easiest such model:

31. Gives rise to periodic solutions a = b = 0 reduces to sine/cosine model No closed form solution for the general case

33. With no closed form expression for the alternative model, fitting needs to be done using a 2-equation iterative process