本文的主要目標是找出房地產投資組合的最適時點，並建立濾嘴法則來鎖定收益。近年來各國房價暴漲，房地產投資客未必自住，但他們通常抓緊時機收取租金並鎖定投資收益。文獻中最相關的研究關注在房地產的最適持有期間，而我們的目標是強化投資客按照濾嘴法則賣房的意願。首先，我們確立最適停利準則的解析解，充分引用幾何布朗運動作為房價過程，並引入首次通過時間機率密度函數給最適停利準則。此外，我們考慮相應於不同投資者風險偏好的恆定型相對風險趨避(CRRA)效用特徵。再者，我們允許一階自回歸過程保有高持續性的影響。我們的研究發現，投資客重新意識未來房價可能帶有自回歸的傾向。他們通常沒有達到滿足點，並期待不斷上漲的房價。只有在維納過程中，投資客才會提前達到滿足點。第三，我們發現投資環境的限制迫使投資客不僅承擔房屋的持有成本，還得承受隨時間不斷推移的房地產折舊效應。最後，利用比較靜態得知風險趨避程度較低的投資客可能傾向於長期持有房地產，因為他們實際的最適停利準則很有可能會是意料之外的暴漲。然而，高風險趨避投資客偏好賣低房價，同時他們仍可以賺取穩定的租金收入。即使投資客面臨投資期間不斷增加房屋的持有成本，投資環境恰好抵銷高房價帶來的效益。

The main goal of this paper is to find out the optimal time in real estate portfolio and establish the filter rule as locking in gain. In the recent years, prices have recently shot up in many countries. Real estate investors do not necessarily provide for residential use, but they usually collect rent revenues and lock in their gains at an appropriate time. The most relevant research in the literature focuses optimal holding period on real estate, but our aim reinforces that investors sell the house in accordance with the filter rules. Firstly, we determine a closed-form solution of optimal stop-gain rule and fully cite a geometric Brownian motion as housing price process and then introduce the first passage time into the optimal stop-gain rule. Also, we consider the characteristics of the constant relative risk aversion (CRRA) utility which is corresponding to investors’ risk preferences. Secondly, we allow for highly persistent process in the first order process. Our study found that investors restart to tag along the probability of estate prices tend to autoregression in the future. They usually get barely up to bliss point and look forward the continuously increasing housing price. Only for Wiener process, investors have attained in advance to a satiation point. Thirdly, we found that the limit of investment climate forced investors to bear not only their own holding cost, but also estate depreciation effect in the course of time. Finally, comparative statics analysis indicates that investors with lower risk aversion probably tend to hold their house for a long time since their actual optimal stop-gain rule may boost over the one which is imagined. However, the higher risk-averse investors prefer to sell a lower estate price while they can still maintain a stable rent revenues. Even if estate investors face the pressure of carrying cost have increased continuously throughout all investment period, investment climate just offset benefit of high estate price.

Abstract..........................................................................................................................ii
Abstract (Chinese).........................................................................................................iii
List of Tables..................................................................................................................vi
List of Figures................................................................................................................vii
Chapter 1 Introduction....................................................................................................1
Chapter 2 Literature Review...........................................................................................3
Chapter 3 Methodology.................................................................................................5
3.1 Dynamics with a geometric Brownian motion...........................................................5
3.1.1 An application of geometric Brownian motion with existing of
Wiener process...............................................................................................................5
3.1.2 The sum of continuous and discrete free cash flows on real estate
investment.......................................................................................................................6
3.1.3 Definition of a stop-gain rule and the expected net present value
of risk-free cash flows.....................................................................................................8
3.1.4 The technique of Laplace transform combines the stop-gain rule
with the first hitting time density.....................................................................................8
3.1.5 The characteristics of the power CRRA utility family............................................10
3.1.6 Definition of a stop-gain rule and the expected net present value
of cash flows for risk-averse investors.......................................................................... 12
3.1.7 Simulations of dynamics in a geometric Brownian motion....................................13
3.2 Dynamics with the autoregressive model................................................................14
3.2.1 An application of the first-order autoregression...................................................14
3.2.2 The first order process in the actual real-world probability..................................15
3.2.3 Financial applications of change of measures......................................................16
3.2.4 The forecasted terminal value of the future real estate prices.............................17
3.2.5 Definition of a stop-gain rule and the expected net present value
of cash flows under the two types of all probability measure........................................17
3.2.6 The volatility of dynamics in the stochastic process............................................19
3.2.7 Simulations of dynamics in the first order process..............................................20
3.3 Optimal time of selling the house with filter and executing optimal stop-
gain rule..........................................................................................................................20
Chapter 4 Monte Carlo Methods...................................................................................22
4.1 Applying Monte Carlo methods and simulating the dynamics of
stochastic process under the risk-averse measure.......................................................24
4.1.1 The simulations of the stochastic process............................................................24
4.1.2 Effect on mean of CRRA utility in timing of sales..................................................25
4.1.3 Effect on mean of CRRA utility and carrying cost in sale timing...........................32
4.2 Applying Monte Carlo methods and simulating the dynamics of
stochastic process under the risk-neutral measure.......................................................36
4.2.1 The simulations of the stochastic process...........................................................37
4.2.2 Effect on expected net present value of risk-free cash flows in
timing of sales................................................................................................................38
Chapter 5 Comparative Statics Analysis for CRRA Utility.............................................44
5.1 Simulations of comparative statics in a geometric Brownian motion......................44
5.1.1 Excluding holding cost of circumstance from investing in real
estate.............................................................................................................................44
5.1.2 Introducing carrying cost into the decision filter..................................................45
5.2 Simulations of comparative statics in the first order process.................................47
5.2.1 Investing in real estate without considering the state of housing
market............................................................................................................................47
5.2.2 The pressure of holding cost inside the decision filter........................................48
Chapter 6 Conclusion............................................................................................ .......51
Reference.......................................................................................................................53
Appendix A.....................................................................................................................54
Appendix B.....................................................................................................................57
Appendix C.....................................................................................................................57
Appendix D.....................................................................................................................58
Appendix E.....................................................................................................................59

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