# Valuation of Real Estate Investments through Fuzzy Logic

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

- ${\mu}_{A}(X)=1\text{}\Rightarrow x\in A$,
- ${\mu}_{A}(X)=0\text{}\Rightarrow x\notin A$.

- $\exists x\in R\text{}such\text{}that\text{}\mu (x|A)=1$ (normality),
- $\mu (x|A)\ge min\left\{\mu ({x}_{1}|A),\mu ({x}_{2}|A)\text{}\right\}\forall x\in [{x}_{1},{x}_{2}]$ (convexity).

- $\mu (x|A)=({a}_{1},{f}_{1}(y|A)/\text{}{a}_{2},\text{}{a}_{3}\text{}/\text{}{f}_{2}(y|A),{a}_{4}),$,

- ${a}_{1}<{a}_{2}<{a}_{3}<{a}_{4}$,
- ${f}_{1}(y|A)$ is a monotone increasing function for $0\le Y\le 1$ with ${f}_{1}(0|A)={a}_{1}$,
- ${f}_{1}(1|A)={a}_{2}$,
- ${f}_{2}(y|A)$ is a monotone decreasing function for $0\le Y\le 1$ with ${f}_{2}(0|A)={a}_{4}$,
- ${f}_{2}(1|A)={a}_{3}$.

_{1}and f

_{2}such that:

_{2}= a

_{3}, the fuzzy number is triangular and it could be identified by means of the triple $({a}_{1},{a}_{2}={a}_{3},{a}_{4})$.

^{n}with values in R, and if A

_{1}, A

_{2}… A

_{n}, are fuzzy numbers, then it is possible to define the fuzzy number B = f(A

_{1}, A

_{2,}…, A

_{n}) with following membership function:

**Sum**. A ⊕ B is a fuzzy number such that:μ (x|A ⊕ B) = (a_{1}+ b_{1}, f_{1}(y|A) + f_{1}(y|B)/a_{2}+ b_{2},a_{3}+ b_{3}/f_{2}(y|A) + f_{2}(y|B),a_{4}+ b_{4}).**Difference**. A Θ B is a fuzzy number such that:μ (x|A Θ B) = (a_{1}− b_{4}, f_{1}(y|A) − f_{2}(y|B)/a_{2}− b_{3},a_{3}− b_{2}/f_{2}(y|A) − f_{1}(y|B),a_{4}+ b_{1}).**Product by a scalar k**. k ⊗ A is a fuzzy number such that:μ (x| k ⊗ A) = (ka_{1}, kf_{1}(y|A)/ka_{2},ka_{3}/kf_{2}(y|A),ka_{4}).**Product**. A ⊗ B is a fuzzy number such that:μ (x|A ⊗ B) = (a_{1}b_{1}, f_{1}(y|A)f_{1}(y|B)/a_{2}b_{2},a_{3}b_{3}/f_{2}(y|A)f_{2}(y|B),a_{4}b_{4}).**Exponentiation scalar k**. A^{k}is a fuzzy number such that:μ (x|A^{k}) = (a^{k}_{1}, f_{1}(y|A)^{k}/a^{k}_{2},a^{k}_{3}/f_{2}(y|A)^{k},a^{k}_{4}).

_{1}− b

_{4}), f

_{1}(y|A) − f

_{2}(y|B)/g(a

_{2}− b

_{3}),g(a

_{3}− b

_{2})/g(f

_{2}(y|A) − f

_{1}(y|B)),g(a

_{4}+ b

_{1})),

## 3. From Fuzzy Numbers to Fuzzy Financial Mathematics

^{−n}.

_{1}(y|P) = (f

_{1}(y|S) (1+ f

_{2}(y|r))

^{−n}, f

_{2}(y|P) = f

_{2}(y|S) (1+ f

_{1}(y|r))

^{−n}),

_{1}(y|P) = (f

_{1}(y|S) (1+ f

_{2}(y|r))

^{−n}, f

_{2}(y|P) = f

_{2}(y|S) (1+ f

_{2}(y|r))

^{−n}).

_{1}= f

_{1}(0|P); p

_{2}= f

_{1}(1|P); p

_{3}= f

_{2}(1|P); p

_{4}= f

_{2}(0|P).

_{0}, S

_{1}, …, S

_{n}, with a rate r, is calculated, using crisp mathematic, as follows:

_{0}, S

_{1}, …, S

_{n}, N = NPV(S,n) with a membership function equal to:

- $\mathsf{\Gamma}({u}_{0},\dots ,{u}_{w},u,w)={{\displaystyle \sum}}_{i=0}^{w}{u}_{i}{(1+v)}^{-i}.$

_{i}are positive, for i ≥ 1, the calculation of Net Present Value can be simplified using the following membership function:

_{i}) indicates the membership function of NPV(S,n

_{i}).

_{0}, S

_{1}, S

_{2},…, S

_{n}) be the cash flow for an investment project; then, IRR can be defined as the interest rate such that:

_{0}+ S

_{1}(1 + r)

^{−1}+ S

_{2}(1 + r)

^{−2}+ …+ S

_{n}(1 + r)

^{−n}= 0.

_{0}is a negative fuzzy number, while all S

_{i}for i > 0 are positive fuzzy numbers, the fuzzy IRR can be defined, for extension of relation (1), as the fuzzy interest rate r such that:

_{0}⊕ (S

_{1}⊗ (1 ⊕ ρ ) − 1) ⊕ (S

_{2}⊗ (1 ⊕ ρ )

^{−2}) ⊕ ... ⊕ (S

_{n}⊗ (1 ⊕ ρ) − ν) = 0,

_{1}(y|r) and f

_{2}(y|r) can be defined with following relations:

_{1}(y|r) is growing, f

_{2}(y|r) is decreasing, and f

_{1}(y|r) ≤ f

_{2}(y|r); these conditions not are always verifiable, and, for this reason, generally a valid IRR does not exist for a fuzzy cash flow. However, there are some conditions that guarantee the existence and uniqueness of this rate. In fact, if S is a set of fuzzy cash flow composed by trapezoidal numbers such that:

_{k}(y) = 1 + f

_{k}(y|r), we obtain the following polynomial equations:

_{2}(y|S

_{0})x

_{2}(y)

^{n}+ f

_{1}(y|S

_{1})x

_{2}(y)

^{n −}

^{1}+ f

_{1}(y|S

_{2})x

_{2}(y)

^{n −}

^{2}+ … + f

_{1}(y|S

_{n}) = 0,

_{1}(y|S

_{0})x

_{1}(y)

^{n}+ f

_{2}(y|S

_{1})x

_{2}(y)

^{n −}

^{1}+ f

_{2}(y|S

_{2})x

_{1}(y)

^{n −}

^{2}+ … + f

_{2}(y|S

_{n}) = 0.

_{k}(y) for each equation, from which f

_{k}(y|r) = x

_{k}(y) – 1.

_{1}(y|r) and f

_{2}(y|r) are obtained, which represent the cutting functions of searched solutions.

## 4. Case Study

- GSM: Gross Square Meters;
- PP: Purchase Price;
- $\overline{CMR}$: Current Market Rent;
- $\overline{PI}$: Projected increase in market rent per year;
- $\overline{MC}$: Management Costs (calculated on effective gross income);
- $\overline{CPI}$: Consumer Price Index.

- ${\overline{TOE}}^{t}={{\displaystyle \sum}}_{i=1}^{N}{\overline{TOE}}_{i}^{(t)}$,
- ${\overline{SOE}}^{t}={{\displaystyle \sum}}_{i=1}^{N}{\overline{SOE}}_{i}^{(t)}$.

^{(T)}the mortgage balance at year t, the cash flow becomes:

_{o}as a fuzzy negative number, each S

_{i}for i > 0 being a fuzzy positive number, the IRR can be obtained as a fuzzy interest rate r such that:

_{0}⊕ ( S

_{1}⊗ (1 ⊕ ρ) − 1) ⊕ ( S

_{2}⊗ (1 ⊕ ρ ) − 2) ⊕ ... ⊕ (S

_{n}⊗ (1 ⊕ ρ ) − ν) = 0.

- Increase property taxes (5%; 10%; 15%),
- Increase insurance costs (3%; 3.5%; 4.5%),
- Increase utilities costs (4.5%; 5%; 5%),
- Increase doorman costs (2%; 3%; 3.5%),
- Increase maintenance costs (2%; 3%; 4.5%),
- Increase management costs (4.5%; 5%; 5.5%).

**Before Tax IRR: (27,78%, 19,44%, 8,09%) (compared with 19.64% of crisp amount)**,

**After Tax IRR: (26,54%, 14,31%, −7,43%) (compared with 14.54% of crisp amount)**.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Key parameters | Deterministic | Fuzzy |
---|---|---|

Purchase Price ($) | 8,500,000 | 8,500,000; 8,500,000; 8,500,000 |

Unit Market Rent ($/sqm) | 15.0 | 13.5; 15.0; 16.5 |

Gross Area (sqm) | 100,000 | 100,000; 100,000; 100,000 |

Annual Rate of Growth of Unit Market Rent (%)—PI | 4.0 | 3.0; 4.0; 5.0 |

Management Cost (% of EGI)—MC | 5.0 | 4.5; 5.0; 5.5 |

Annual rate of Growth of Prices (%)—CPI | 4.0 | 3.0; 4.0; 5.0 |

Vacancy Rate (%)—VI | 5.0 | 4.5; 5.0; 5.5 |

Depreciation Rate (%) | 2.2 | 2.2; 2.2; 2.2 |

Tax Rate on Income (%) | 36 | 30; 36; 40 |

Tax Rate on Capital Gain (%) | 28 | 25; 28; 30 |

Sale Price $ | 9,500,000 | 8,000,000; 9,500,000; 11,000,000 |

Before Tax Discount Rate % | 18 | 16.0; 18.0; 18.5 |

After Tax Discount Rate % | 13 | 12.0; 13.0; 13.5 |

Tenant | Sqm | Unit Market Rent | Total Market Rent | End of Contract | % Increase Percentage |
---|---|---|---|---|---|

1 | 30,000 | 14.00 | 420,000 | 3 | 50.00 |

2 | 25,000 | 14.00 | 350,000 | 3 | 50.00 |

3 | 15,000 | 14.00 | 210,000 | 3 | 50.00 |

4 | 10,000 | 14.50 | 145,000 | 4 | 50.00 |

5 | 10,000 | 14.50 | 150,000 | 5 | 50.00 |

6 | 6,000 | 15.00 | 90,000 | 5 | 50.00 |

Total | 96,000 | 15.00 | 1,365,000 |

Tenant | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 |
---|---|---|---|---|---|

1 | 420,000 | 428,000 | 436,968 | 506,189 | 516,313 |

420,000 | 426,300 | 432,694 | 442,554 | 449,193 | |

420,000 | 428.400 | 436.968 | 506.189 | 516.313 | |

420,000 | 430.500 | 441.262 | 573.024 | 587.350 | |

2 | 350,000 | 357,000 | 364,140 | 421,824 | 430,260 |

350,000 | 355.250 | 360.579 | 368.795 | 374.327 | |

350,000 | 357.000 | 364.140 | 421.824 | 430.260 | |

350,000 | 358.750 | 367.719 | 477.520 | 489.458 | |

3 | 210,000 | 214,200 | 218,484 | 253,094 | 258,156 |

210,000 | 213.150 | 216.347 | 221.277 | 224.596 | |

210,000 | 214.200 | 218.484 | 253.094 | 258.156 | |

210,000 | 215.250 | 220.631 | 286.512 | 293.675 | |

4 | 145,000 | 147,900 | 150,858 | 153,875 | 175,479 |

145,000 | 147.175 | 149.383 | 151.623 | 151.944 | |

145,000 | 147.900 | 150.858 | 153.875 | 175.479 | |

145,000 | 148.625 | 152.341 | 156.149 | 200.559 | |

5 | 150,000 | 153,000 | 156,060 | 159,181 | 162,365 |

150,000 | 152.250 | 154.534 | 156.852 | 159.205 | |

150,000 | 153.000 | 156.060 | 159.181 | 162.365 | |

150,000 | 153.750 | 157.594 | 161.534 | 165.572 | |

6 | 90,000 | 91,800 | 93,636 | 95,509 | 97,419 |

90,000 | 91.350 | 92.720 | 94.111 | 95.523 | |

90,000 | 91,800 | 93,636 | 95,509 | 97,419 | |

90,000 | 92,250 | 94,556 | 96,920 | 99,343 | |

Total | 1,365,000 | 1,392,000 | 1,420,146 | 1,589,672 | 1,639,992 |

1,365,000 | 1,385,475 | 1,406,257 | 1,435,213 | 1,454,787 | |

1,365,000 | 1,392,300 | 1,420,146 | 1,589,672 | 1,639,992 | |

1,365,000 | 1,399,125 | 1,434,103 | 1,751,660 | 1,835,957 |

Operative Costs | $/sqm | % increase Planned |
---|---|---|

Tax Property | 1.55 | 2.0 |

1.5 | ||

2.0 | ||

2.5 | ||

Insurance | 0.15 | 3.5 |

3.0 | ||

3.5 | ||

4.5 | ||

Utilities | 1.25 | 5.0 |

4.5 | ||

5.0 | ||

5.5 | ||

Doorman | 0.80 | 3.0 |

2.0 | ||

3.0 | ||

3.5 | ||

Maintenance | 0.70 | 3.0 |

2.0 | ||

3.0 | ||

4.5 |

Operative Costs | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 |
---|---|---|---|---|---|

Tax Property | 148,800 | 151,776 | 154,812 | 156,908 | 161,066 |

148,800 | 151,032 | 153,297 | 155,597 | 157,931 | |

148,800 | 151,776 | 154,812 | 156,908 | 161,066 | |

148,800 | 152,520 | 156,333 | 160,241 | 164,247 | |

Insurance | 14,400 | 14,904 | 15,426 | 15,966 | 16,524 |

14,400 | 14,832 | 15,277 | 15,735 | 16,207 | |

14,400 | 14,904 | 15,426 | 15,966 | 16,524 | |

14,400 | 15,048 | 15,725 | 16,433 | 17,172 | |

Utilities | 120,000 | 126,000 | 132,300 | 138,915 | 145,861 |

120,000 | 125,400 | 131,043 | 136,940 | 143,102 | |

120,000 | 126,000 | 132,300 | 138,915 | 145,861 | |

120,000 | 126,600 | 133,563 | 140,909 | 148,659 | |

Doorman | 76,800 | 79,104 | 81,477 | 83,921 | 86,439 |

76,800 | 78,336 | 79,903 | 81,501 | 83,131 | |

76,800 | 79,104 | 81,477 | 83,921 | 86,439 | |

76,800 | 79,488 | 82,270 | 85,150 | 88,130 | |

Maintenance | 67,200 | 69,216 | 71,292 | 73,431 | 75,634 |

67,200 | 68,544 | 69,915 | 71,313 | 72,739 | |

67,200 | 69,216 | 71,292 | 73,431 | 75,634 | |

67,200 | 70,224 | 73,384 | 76,686 | 80,137 | |

Total | 427,200 | 441,000 | 455,307 | 470,141 | 485,524 |

427,200 | 438,144 | 449,435 | 461,086 | 473,111 | |

427,200 | 441,000 | 455,307 | 470,141 | 485,524 | |

427,200 | 443,880 | 461,275 | 479,419 | 498,346 |

**Table 6.**Costs to be paid to the property when the maximum limit established in the contract is exceeded.

Tenant | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 |
---|---|---|---|---|---|

1 | 13,500 | 17,813 | 22,283 | 26,919 | 31,726 |

13,500 | 16,920 | 20,448 | 24,089 | 27,847 | |

13,500 | 17,813 | 22,283 | 26,919 | 31,726 | |

13,500 | 18,712 | 24,149 | 29,818 | 35,733 | |

2 | 11,250 | 14,844 | 18,569 | 22,433 | 26,439 |

11,250 | 14,100 | 17,040 | 20,075 | 23,206 | |

11,250 | 14,844 | 18,569 | 22,433 | 26,439 | |

11,250 | 15,594 | 20,124 | 24,849 | 29,778 | |

3 | 6750 | 8906 | 11,142 | 13,460 | 15,863 |

6750 | 8460 | 10,224 | 12,045 | 13,924 | |

6750 | 8906 | 11,142 | 13,460 | 15,863 | |

6750 | 9356 | 12,074 | 14,909 | 17,867 | |

4 | 2000 | 3438 | 4928 | 6473 | 8075 |

2000 | 3140 | 4316 | 5530 | 6782 | |

2000 | 3438 | 4928 | 6473 | 8075 | |

2000 | 3737 | 5550 | 7439 | 9411 | |

5 | 0 | 1438 | 2928 | 4473 | 6075 |

0 | 1140 | 2316 | 3530 | 4782 | |

0 | 1438 | 2928 | 4473 | 6075 | |

0 | 1737 | 3550 | 5439 | 7411 | |

6 | 0 | 0.863 | 1757 | 2684 | 3645 |

0 | 0.684 | 1390 | 2118 | 2869 | |

0 | 0.863 | 1757 | 2684 | 3645 | |

0 | 1042 | 2130 | 3264 | 4447 | |

Total | 33,500 | 47,400 | 61,607 | 76,441 | 91,824 |

33,500 | 38,708 | 43,895 | 49,053 | 54,176 | |

33,500 | 47,400 | 61,607 | 76,441 | 91,824 | |

33,500 | 55,916 | 79,415 | 107,105 | 129,881 |

Cost or Income | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 |
---|---|---|---|---|---|

Total Initial Rent | 1,365,000 | 1,329,300 | 1,420,146 | 1,589,672 | 1,639,992 |

1,365,000 | 1,385,475 | 1,406,457 | 1,435,213 | 1,454,787 | |

1,365,000 | 1,329,300 | 1,420,146 | 1,589,672 | 1,639,992 | |

1,365,000 | 1,399,125 | 1,434,103 | 1,751,660 | 1,835,957 | |

Vacancy | 0 | 0 | 0 | 79,484 | 82,000 |

0 | 0 | 0 | 64,585 | 65,465 | |

0 | 0 | 0 | 79,484 | 82,000 | |

0 | 0 | 0 | 96,341 | 100,978 | |

Operative Costs | 427,200 | 441,000 | 455,307 | 470,141 | 485,524 |

427,200 | 438,144 | 449,435 | 461,086 | 473,111 | |

427,200 | 441,000 | 455,307 | 470,141 | 485,524 | |

427,200 | 443,880 | 461,275 | 479,419 | 498,346 | |

Refunded Costs | 33,500 | 47,400 | 61,607 | 76,441 | 91,824 |

33,500 | 38,708 | 43,895 | 49,053 | 54,176 | |

33,500 | 47,400 | 61,607 | 76,441 | 91,824 | |

33,500 | 55,916 | 79,415 | 107,052 | 129,881 | |

Management Costs | 68,250 | 69,615 | 71,007 | 75,509 | 77,900 |

61,425 | 62,346 | 63,282 | 60,249 | 60,921 | |

68,250 | 69,615 | 71,007 | 75,509 | 77,900 | |

75,075 | 76,952 | 78,876 | 92,789 | 97,377 | |

Net Operating Income | 903,050 | 929,985 | 955,439 | 1,040,979 | 1,086,393 |

896,225 | 909,087 | 921,841 | 834,050 | 837,498 | |

903,050 | 929,985 | 955,439 | 1,040,979 | 1,086,393 | |

909,875 | 948,815 | 988,926 | 1,251,459 | 1,341,105 |

Income, Amount of Mortgage, BTCF | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 |
---|---|---|---|---|---|

Net Operating Income | 903,050 | 928,439 | 955,439 | 1,040,979 | 1,086,393 |

896,225 | 909,087 | 921,841 | 834,050 | 837,498 | |

903,050 | 928,439 | 955,439 | 1,040,979 | 1,086,393 | |

948,815 | 948,815 | 948,815 | 1,251,459 | 1,341,105 | |

Fixed Amount of Mortgage | 698,885 | 698,885 | 698,885 | 698,885 | 698,885 |

698,885 | 698,885 | 698,885 | 698,885 | 698,885 | |

698,885 | 698,885 | 698,885 | 698,885 | 698,885 | |

698,885 | 698,885 | 698,885 | 698,885 | 698,885 | |

Before Tax Cash Flow | 204,165 | 230,100 | 256,554 | 342,094 | 387,508 |

197,340 | 210,202 | 222,956 | 135,165 | 138,613 | |

204,165 | 230,100 | 256,554 | 342,094 | 387,508 | |

210,990 | 249,930 | 290,077 | 552,574 | 642,220 |

Before Tax Cash Flow (year 5) | 387,508 |

138,613 | |

387,508 | |

642,220 | |

Sale Price | 9,500,000 |

8,000,000 | |

9,500,000 | |

11,000,000 | |

Fractionated Capital to Refund | 5,315,735 |

5,315,735 | |

5,315,735 | |

5,315,735 | |

Total Before Tax Cash Flow (year 5) | 4,571,735 |

2,822,840 | |

4,571,745 | |

6,326,447 |

Year | Cash Flow | Present Value |
---|---|---|

1 | 204,165 | 173,021 |

197,340 | 166,532 | |

204,165 | 173,021 | |

210,990 | 181,888 | |

2 | 230,100 | 165,254 |

210,202 | 149,693 | |

230,100 | 165,254 | |

249,930 | 185,739 | |

3 | 256,554 | 156,147 |

222,956 | 133,988 | |

256,554 | 156,147 | |

290,077 | 185,840 | |

4 | 342,094 | 176,449 |

135,165 | 68,547 | |

342,094 | 176,449 | |

552,574 | 305,182 | |

5 | 4,571,735 | 1,998,347 |

2,822,840 | 1,208,077 | |

4,571,735 | 1,998,347 | |

6,326,447 | 3,012,104 | |

Initial Investment | 2,550,000 | |

2,550,000 | ||

2,550,000 | ||

2,550,000 | ||

Before Tax Net Present Value | 119,218 | |

−823,164 | ||

119,218 | ||

1,320,752 |

Item | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 |
---|---|---|---|---|---|

Net Operating Income | 903,050 | 928,439 | 955,439 | 1,040,979 | 1,086,393 |

896,225 | 909,087 | 921,841 | 834,050 | 837,498 | |

903,050 | 928,439 | 955,439 | 1,040,979 | 1,086,393 | |

948,815 | 948,815 | 948,815 | 1,251,459 | 1,341,105 | |

Interest Share | 595,000 | 584,612 | 573,184 | 560,614 | 546,787 |

595,000 | 584,612 | 573,184 | 560,614 | 546,787 | |

595,000 | 584,612 | 573,184 | 560.614 | 546,787 | |

595,000 | 584,612 | 573,184 | 560,614 | 546,787 | |

Depreciation | 187,000 | 187,000 | 187,.000 | 187,000 | 187,000 |

170,000 | 170,000 | 170,000 | 170,000 | 170,000 | |

187,000 | 187,000 | 187,000 | 187,000 | 187,000 | |

204,000 | 204,000 | 204,000 | 204,000 | 204,000 | |

Taxable Income | 121,050 | 157,373 | 195,255 | 293,365 | 352,606 |

97,225 | 120,476 | 144,657 | 64,436 | 86,711 | |

121,050 | 157,373 | 195,255 | 293,365 | 352,606 | |

144,875 | 194,203 | 245,778 | 520,845 | 624,318 | |

Tax Income | 43,578 | 56,654 | 70,292 | 105,611 | 126,938 |

29,167 | 36,143 | 43,397 | 20,831 | 26,013 | |

43,578 | 56,654 | 70,292 | 105,611 | 126,938 | |

57,950 | 77,681 | 98,311 | 208,338 | 249,727 | |

BTCF | 204,165 | 230,100 | 256,554 | 342,094 | 387,508 |

197,340 | 210,202 | 222,956 | 135,165 | 138,613 | |

204,165 | 230,100 | 256,554 | 342,094 | 387,508 | |

210,990 | 249,930 | 290,077 | 552,574 | 642,220 | |

Tax | 43,578 | 56,654 | 70,292 | 105,611 | 126,938 |

29,167 | 36,143 | 43,397 | 20,831 | 26,013 | |

43,578 | 56,654 | 70,292 | 105,611 | 126,938 | |

57,950 | 77,681 | 98,311 | 208,338 | 249,727 | |

ATCF | 160,587 | 173,446 | 186,446 | 236,483 | 260,570 |

139,390 | 132,521 | 124,645 | −73,173 | −111,114 | |

160,587 | 173,446 | 186,445 | 236,483 | 260,570 | |

181,823 | 213,787 | 246,680 | 531,743 | 616,207 |

Sale Price | 9,500,000 |

8,000,000 | |

9,500,000 | |

11,000,000 | |

Purchase Price | 8,500,000 |

8,500,000 | |

8,500,000 | |

8,000,000 | |

Cumulated Depreciation | 935,000 |

850,000 | |

935,000 | |

1,020,000 | |

Increase of Capital | 1,935,000 |

350,000 | |

1,935,000 | |

3,520,000 | |

ATCF (year 5) | 260,570 |

−111,114 | |

260,570 | |

616,207 | |

Financed Capital to Be Refunded | 5,315,773 |

5,315,773 | |

5,315,773 | |

5,315,773 | |

Tax on Increase of Capital | 541,800 |

87,500 | |

541,800 | |

1,056,000 | |

Total ATCF (Year 5) | 3,902,997 |

1,517,113 | |

3,902,997 | |

6,212,934 |

Year | Cash Flow | Present Value |
---|---|---|

1 | 160,587 | 142,113 |

139,390 | 122,811 | |

160,587 | 142,113 | |

181,823 | 162,342 | |

2 | 173,446 | 135,833 |

132,521 | 102,871 | |

173,446 | 135,833 | |

213,787 | 170,430 | |

3 | 186,262 | 129,089 |

124,645 | 85,249 | |

186,262 | 129,089 | |

246,680 | 175,582 | |

4 | 236,483 | 145,039 |

−73,173 | −44,093 | |

236,483 | 145,039 | |

531,743 | 337,932 | |

5 | 3,902,997 | 2,118,390 |

1,517,113 | 805,450 | |

3,902,997 | 2,118,390 | |

6,212,934 | 3,525,386 | |

Initial investment | 2,550,000 | |

2,550,000 | ||

2,550,000 | ||

2,550,000 | ||

After Tax Net Present Value | 120,465 | |

−1,477,712 | ||

120,465 | ||

1,821,672 |

Year | Before Tax Cash Flow | ||
---|---|---|---|

0 | −2,550,000 | −2,550,000 | −2,550,000 |

1 | 207,199 | 214,024 | 220,849 |

2 | 221,549 | 239,959 | 258,300 |

3 | 220,971 | 266,413 | 311,781 |

4 | 134,788 | 351,953 | 572,668 |

5 | 2,796,664 | 4,554,122 | 6,317,398 |

Year | After Tax Cash Flow | ||
---|---|---|---|

0 | −2,550,000 | −2,550,000 | −2,550,000 |

1 | 147,476 | 168,850 | 190,352 |

2 | 142,722 | 181,737 | 220,405 |

3 | 116,196 | 194,568 | 270,643 |

4 | −79,347 | 244,734 | 553,610 |

5 | 1,485,752 | 3,883,800 | 6,205,134 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Del Giudice, V.; De Paola, P.; Cantisani, G.B.
Valuation of Real Estate Investments through Fuzzy Logic. *Buildings* **2017**, *7*, 26.
https://doi.org/10.3390/buildings7010026

**AMA Style**

Del Giudice V, De Paola P, Cantisani GB.
Valuation of Real Estate Investments through Fuzzy Logic. *Buildings*. 2017; 7(1):26.
https://doi.org/10.3390/buildings7010026

**Chicago/Turabian Style**

Del Giudice, Vincenzo, Pierfrancesco De Paola, and Giovanni Battista Cantisani.
2017. "Valuation of Real Estate Investments through Fuzzy Logic" *Buildings* 7, no. 1: 26.
https://doi.org/10.3390/buildings7010026