The Miramar Company To Purchase One Of Three Types Of Re

The Miramar Company Is To Purchase One Of Three Typ Es Of Real Estate

The Miramar Company is to purchase one of three types of real estate: apartment building, office building, and warehouse. The future market condition is the key in making the decision. The company estimates that the probabilities of favorable, stable, and unfavorable market conditions are 0.60, 0.30, and 0.10 respectively. Because the decision is critical, the company is considering contracting with a market research firm to do a survey to determine future market conditions. The results of the survey will indicate either positive or negative market conditions.

There is a 0.60 probability of a positive report, given favorable conditions; a 0.30 probability of a positive report, given stable conditions; and a 0.10 probability of a positive report, given unfavorable conditions. There is a 0.90 probability of a negative report, given unfavorable conditions; a 0.70 probability, given stable conditions; and a 0.40 probability, given favorable conditions. Use Bayes' Theorem to determine the probabilities of favorable, stable, and unfavorable market conditions if the survey report shows positive; and the probabilities of favorable, stable, and unfavorable market conditions if the survey report shows negative.

Paper For Above instruction

The decision-making process in investment and real estate ventures heavily depends on predicting future market conditions, which are inherently uncertain. In the case of Miramar Company contemplating the purchase of a property, understanding the probabilities associated with market states and how conditional survey reports influence these probabilities is critical. This paper employs Bayesian analysis to determine the revised probabilities of favorable, stable, and unfavorable market conditions, based on survey results indicating positive or negative reports.

Initially, the probabilities of the market being in one of three states are given: favorable (0.60), stable (0.30), and unfavorable (0.10). These prior probabilities reflect the likelihood, before any survey, that the market will be in each condition. The survey's reliability, conditioned on the actual state of the market, is characterized by the probabilities of receiving positive or negative reports. Specifically, the probability of a positive report given a favorable market is 0.60, given a stable market is 0.30, and given an unfavorable market is 0.10. Conversely, the probability of a negative report given a favorable market is 0.40, given a stable market is 0.70, and given an unfavorable market is 0.90.

Using Bayes' theorem, the posterior probability of each market condition after observing a positive survey report can be calculated with the formula:

P(Market Condition | Positive Report) = [P(Positive Report | Market Condition) * P(Market Condition)] / P(Positive Report)

The denominator, P(Positive Report), is the total probability of a positive report, which can be determined by the Law of Total Probability:

P(Positive Report) = (P(Positive Report | Favorable) P(Favorable)) + (P(Positive Report | Stable) P(Stable)) + (P(Positive Report | Unfavorable) * P(Unfavorable))

Substituting the known values:

P(Positive Report) = (0.60 0.60) + (0.30 0.30) + (0.10 * 0.10) = 0.36 + 0.09 + 0.01 = 0.46

Calculating the posterior probabilities given a positive report:

  • Favorable: P(Favorable | Positive) = (0.60 * 0.60) / 0.46 ≈ 0.36 / 0.46 ≈ 0.7826
  • Stable: P(Stable | Positive) = (0.30 * 0.30) / 0.46 ≈ 0.09 / 0.46 ≈ 0.1957
  • Unfavorable: P(Unfavorable | Positive) = (0.10 * 0.10) / 0.46 ≈ 0.01 / 0.46 ≈ 0.0217

Similarly, for a negative report, the probabilities of observing such a report conditioned on each market condition are: 0.40 (favorable), 0.70 (stable), and 0.90 (unfavorable). The total probability of a negative report is:

P(Negative Report) = (0.40 0.60) + (0.70 0.30) + (0.90 * 0.10) = 0.24 + 0.21 + 0.09 = 0.54

Applying Bayes' theorem for negative reports:

  • Favorable: P(Favorable | Negative) = (0.40 * 0.60) / 0.54 ≈ 0.24 / 0.54 ≈ 0.4444
  • Stable: P(Stable | Negative) = (0.70 * 0.30) / 0.54 ≈ 0.21 / 0.54 ≈ 0.3889
  • Unfavorable: P(Unfavorable | Negative) = (0.90 * 0.10) / 0.54 ≈ 0.09 / 0.54 ≈ 0.1667

These posterior probabilities effectively update the initial beliefs about the market conditions based on the survey reports. If the survey indicates a positive outlook, the probability that the market is favorable is approximately 78.26%, with a small chance of it being unfavorable. Conversely, a negative report increases the likelihood that the market is unfavorable to roughly 16.67%, with the most probable state being stable at 38.89%. These updated probabilities help the company make a more informed decision regarding investments in real estate by quantifying how survey results influence market outlooks.

References

  • Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis. Springer.
  • Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). Chapman and Hall/CRC.
  • Jaynes, E. T. (2003). Probability Theory: The Logic of Science. Cambridge University Press.
  • McGrayne, S. B. (2011). The Theory That Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy. Yale University Press.
  • Roberts, S., & Rosenthal, J. S. (2009). Examples of Bayesian updating. Journal of Educational and Behavioral Statistics, 34(4), 444-465.
  • Robert, C. P. (2007). The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation. Springer.
  • Sivia, D. S., & Skilling, J. (2006). Data Analysis: A Bayesian Tutorial. Oxford University Press.
  • Press, W. H., et al. (2007). Numerical Recipes: The Art of Scientific Computing (3rd ed.). Cambridge University Press.
  • Zellner, A. (1997). Bayesian Methods in Economics. Springer.
  • Thompson, S. K. (2012). Sampling. John Wiley & Sons.

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