# Revisiting Investability of Heritage Properties through Indexation and Portfolio Frontier Analysis

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Research Methodology

^{T}vW, where W is a column vector containing the weights of different assets in the portfolio. V is the covariance matrix, and W

^{T}is the transpose of the matrix W.

_{1}to W

_{n}are the weights of assets 1 to n in the portfolio, and ${\sigma}_{xy}$ is the covariance between assets x and y. Note that ${\sigma}_{1.n}$

^{2}means the covariance itself. The covariance matrix is required to calculate portfolio variance. By knowing that ρ is the correlation between two assets, the correlation matrix is multiplied by the diagonal matrix of standard deviations, it can be transformed into a covariance matrix. Subsequently, the portfolio variance is obtained by multiplying the covariance matrix by the weights of selected assets.

## 4. Results and Discussions

## 5. Conclusions and Recommendation

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**): Asset Allocation of Portfolio_CV (2009–2019). (

**b**): Asset Allocation of Portfolio_HR (2009–2019).

**Figure 3.**Performance Analysis Between Conventional Portfolio and Diversified Portfolio Based on Efficient Frontiers (2009–2019).

Description | Unit of Measurement | Data Types | Sources | |
---|---|---|---|---|

Dependent variable | Transacted price of heritage properties | Ringgit Malaysia (logarithm form) | Numerical | Valuation and Property Services Department (JPPH) |

Independent variables | Land area | Square meter (logarithm form) | Numerical | Valuation and Property Services Department (JPPH) |

Share (ownership) | Value | Numerical | ||

Transaction period (yearly basis) | Dummy Variable, where 1 = if the units of heritage properties were purchased at respective year, t 0 = if the units of heritage properties were purchased at base year, 0 | Categorical | ||

Location | Latitude and Longitude (polynomial form) | Numerical | Valuation and Property Services Department (JPPH) and Google Map | |

Types of shop | Dummy Variable, where 1= typical shop house, 0 = residential unit which permitted for commercial use | Numerical | Valuation and Property Services Department (JPPH) and Google Streetview | |

Building Condition | Dummy Variable, where 1 = Good, 0 = otherwise 1 = Average, 0 = otherwise 0 = Bad condition | Categorical | Valuation and Property Services Department (JPPH) and Google Streetview |

Types | Explanatory Notes |
---|---|

Linear Regression | The linear regression model is the typical model that adjusting the weights (coefficients of variables) w by optimizing the problem with the cost function below: |

$\sum _{i=1}^{M}}{\left({y}_{i}-\overline{{y}_{i}}\right)}^{2$ | |

$\mathrm{where}{y}_{i}$$\mathrm{is}\mathrm{the}\mathrm{expected}\mathrm{output}\mathrm{from}\mathrm{the}\mathrm{sample}i\mathrm{and}\overline{{y}_{i}}$ is the predicted value from the model. The linear regression is usually simply derived as | |

$\overline{{y}_{i}}={\displaystyle \sum _{j=0}^{p}}{w}_{j}\times {x}_{ij}$ | |

where j is the number of variables. | |

Ridge | Ridge regression is the shrinkage model of linear regression that supports multicollinearity analysis for overfitting reduction by introducing a slight bias in the estimates. The procedure trades away much of the variance in exchange for a little bias for a better coefficient when multicollinearity is present. This can be achieved by introducing the lambda to the cost function to the linear regression as follow. |

$\sum _{i=1}^{M}}{\left({y}_{i}-\overline{{y}_{i}}\right)}^{2}={\displaystyle \sum _{i=1}^{M}}{\left({y}_{i}-{\displaystyle \sum _{j=0}^{p}}{w}_{j}\times {x}_{ij}\right)}^{2}+\lambda {\displaystyle \sum _{j=0}^{p}}{w}_{j}{}^{2$ | |

Lasso | Lasso is another shrinkage model that is similar to the Ridge regression that aims to increase the performance with a simpler model with the effect on the selection of variables. This is achieved by adding penalization to the weights of the variables as defined as follows: |

$\sum _{i=1}^{M}}{\left({y}_{i}-\overline{{y}_{i}}\right)}^{2}={\displaystyle \sum _{i=1}^{M}}{\left({y}_{i}-{\displaystyle \sum _{j=0}^{p}}{w}_{j}\times {x}_{ij}\right)}^{2}+\lambda {\displaystyle \sum _{j=0}^{p}}\left|{w}_{j}\right|$ | |

Multilasso | The multi-task Lasso model is a Lasso model improved by jointly enforcing the selected features to be the same across tasks to stabilize the feature selection task. This can be achieved by optimizing the summing of the cost function across tasks K. |

$\sum _{k=1}^{K}}{\displaystyle \sum _{i=1}^{M}}{\left({y}_{i}{}^{k}-{\overline{{y}_{i}}}^{k}\right)}^{2}={\displaystyle \sum _{k=1}^{K}}{\displaystyle \sum _{i=1}^{M}}{\left({y}_{i}{}^{k}-{\displaystyle \sum _{j=0}^{p}}{w}_{j}{}^{k}\times {x}_{ij}{}^{k}\right)}^{2}+\lambda {\displaystyle \sum _{j=0}^{p}}{\mathrm{max}}_{k}\left|{w}_{j}{}^{k}\right|$ | |

ElasticNet | The feature selection task by the Lasso model is based on the data causing it to be unstable with the bias in the samples. Elastic Net is aimed to optimize the solution by combining both the penalization methods of Lasso and Ridge for balancing the feature selection and the outlier elimination tasks in the data. The Elastic Net incorporates penalties from both L1 and L2 regularization by introducing the mixing ratio α into the cost function of the problem as follows: |

$\sum _{i=1}^{M}}{\left({y}_{i}-\overline{{y}_{i}}\right)}^{2}=\frac{{{\displaystyle \sum}}_{i=1}^{M}{\left({y}_{i}-{{\displaystyle \sum}}_{j=0}^{p}{w}_{j}{x}_{ij}\right)}^{2}}{2n}+\lambda \left(\frac{1-\alpha}{2}{\displaystyle \sum _{j=0}^{p}}\left|{w}_{j}\right|+\alpha {\displaystyle \sum _{j=0}^{p}}{w}_{j}{}^{2}\right)$ | |

MultiTaskElasticNet | Similar to the Multi-Task Lasso, the Multi-Task Elastic Net solves the regression problem by training with a mixed L1, L2-norm, and L2 for regularization to estimate the sparse coefficients for multiple regression problems jointly to achieve optimization across tasks. |

Least Angle | The Least Angle Regression is similar to the traditional forward selection method but it is less greedy throughout the algorithm process. This is achieved by increasing the estimated parameters in a direction equiangular to each one’s correlations with the residual rather than including variables at each step. The algorithm steps are as follow: Step 1: Start with r = y and W = 0, $\mathrm{Step}2:\mathrm{Find}\mathrm{the}\mathrm{predictor}{x}_{j}$ that is most correlated with r $\mathrm{Step}3:\mathrm{Increase}\mathrm{the}{w}_{j}$$\mathrm{in}\mathrm{the}\mathrm{direction}\mathrm{of}\mathrm{sign}{r}^{T}{x}_{j}$$\mathrm{until}\mathrm{some}\mathrm{other}\mathrm{competitor}{x}_{k}$ has a greater or equal correlation with current residue. $\mathrm{Step}4:\mathrm{Move}({w}_{j},{w}_{k}$$)\mathrm{in}\mathrm{the}\mathrm{joint}\mathrm{least}\text{-}\mathrm{squares}\mathrm{direction}\mathrm{for}({x}_{j},{x}_{k}$$)\mathrm{until}\mathrm{some}{w}_{l}$ has a greater or equal correlation with the current residual. Step 5: Continue until the desired number of predictors has entered in the model and obtain the ordinary least square solution after the p-th iteration. |

p-Value (Linear Regression/OLS) | Coefficients | Expected Sign of Coefficients | |||||
---|---|---|---|---|---|---|---|

1% Significance | 5% Significance | 10% Significance | Total Number | Positive (+) | Negative (-) | ||

$\mathrm{lnLAND}$ | 10 | 0 | 0 | 10 | 10 | 0 | + |

$\mathrm{SHARE}$ | 10 | 0 | 0 | 10 | 10 | 0 | + |

$\mathrm{lat}$ | 9 | 0 | 0 | 9 | 10 | 0 | +/− |

$\mathrm{lon}$ | 8 | 1 | 1 | 10 | 10 | 0 | +/− |

${\mathrm{lat}}^{2}$ | 10 | 0 | 0 | 10 | 0 | 10 | +/− |

${\mathrm{lon}}^{2}$ | 6 | 2 | 2 | 10 | 10 | 0 | +/− |

$\mathrm{latlon}$ | 8 | 0 | 2 | 10 | 0 | 10 | +/− |

$\mathrm{YEAR}$ | 10 | 0 | 0 | 10 | 9 | 1 | +/− |

$\mathrm{GOOD}$ | 10 | 0 | 0 | 10 | 10 | 0 | + |

$\mathrm{AVG}$ | 2 | 0 | 0 | 2 | 10 | 0 | + |

$\mathrm{SHOP}$ | 5 | 2 | 2 | 9 | 10 | 0 | + |

Types of Regression Technique | Mean Square Error (MSE) | Standard Deviation | |
---|---|---|---|

Cross Validation ^{1} | True Sample ^{2} | ||

Linear Regression | 0.55 | 0.40 | 0.32 |

Ridge | 0.42 | 0.41 | 0.12 |

Lasso | 0.59 | 0.41 | 0.17 |

Multilasso | 0.59 | 0.60 | 0.17 |

ElasticNet | 0.74 | 0.75 | 0.16 |

MultiTaskElasticNet | 0.55 | 0.56 | 0.18 |

LeastAngle | 0.70 | 0.70 | 0.16 |

^{1}The cross-validation approach has segregated 20% of the data of each sample randomly for model testing. This action is repeated 10 times with different datasets from the same sample in order to get a consistent MSE for each model.

^{2}The true sample means that the dataset is fully used for developing a price index model.

REIT_INDEX | STOCK_INDEX | MHPI | PIHPI_TH | PPHPI | PIHPI_HR | Bond | |
---|---|---|---|---|---|---|---|

REIT_INDEX | 1.000 | ||||||

STOCK_INDEX | 0.695 * | 1.000 | |||||

MHPI | 0.455 | 0.275 | 1.000 | ||||

PIHPI_TH | 0.498 | 0.500 | 0.847 * | 1.000 | |||

PPHPI | 0.287 | 0.170 | 0.918 * | 0.799 * | 1.000 | ||

PIHPI_HR | 0.730 * | 0.704 * | 0.584 | 0.788 * | 0.405 | 1.000 | |

Bond | −0.862 * | −0.570 | −0.531 | −0.440 | −0.407 | −0.513 | 1.000 |

Types of Investment Asset | Return (Geometric Mean) | Risk (Std. Deviation) | Sharpe Ratio | PerformanceRank |
---|---|---|---|---|

MHPI | 7.62% | 3.63% | 1.19 | 1 |

PPHPI | 7.27% | 4.95% | 0.80 | 2 |

REIT_INDEX ^{1} | 7.71% | 8.06% | 0.55 | 3 |

PIHPI_HR | 14.66% | 24.23% | 0.47 | 4 |

PIHPI_TH | 5.43% | 5.36% | 0.40 | 5 |

STOCK_INDEX ^{2} | 2.63% | 10.09% | −0.07 | 6 |

Bond ^{3} | 3.29% | 0.15% | - | - |

^{1}The REIT_INDEX (base year = 2010) is computed based on the stock prices of REIT companies that had been listed in Bursa Malaysia between 2009 and 2019. This study selected 10 REITs (AXREIT, AMFIRST, YTLREIT, HEKTAR, MQREIT, ALQAREIT, ATRIUM, ARREIT, TWREIT, and UOAREIT) to represent its market performance for the study period.

^{2}The base year of KLCI was adjusted to the year 2010, namely STOCK_INDEX. The purpose of such adjustments is to maintain consistency among the investment assets for a comparative study.

^{3}Bond yield of three-year Malaysian Government Securities (MGS).

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## Share and Cite

**MDPI and ACS Style**

Cheng, C.T.; Ling, G.H.T.; Gan, Y.-S.; Wong, W.F.; Lai, K.S.
Revisiting Investability of Heritage Properties through Indexation and Portfolio Frontier Analysis. *Risks* **2021**, *9*, 91.
https://doi.org/10.3390/risks9050091

**AMA Style**

Cheng CT, Ling GHT, Gan Y-S, Wong WF, Lai KS.
Revisiting Investability of Heritage Properties through Indexation and Portfolio Frontier Analysis. *Risks*. 2021; 9(5):91.
https://doi.org/10.3390/risks9050091

**Chicago/Turabian Style**

Cheng, Chin Tiong, Gabriel Hoh Teck Ling, Yee-Siang Gan, Wai Fang Wong, and Kong Seng Lai.
2021. "Revisiting Investability of Heritage Properties through Indexation and Portfolio Frontier Analysis" *Risks* 9, no. 5: 91.
https://doi.org/10.3390/risks9050091