This is the expected number of months of rent that an illegal subletter expects to get:

- (1-

*p*

^{12N})/(1-

*p*)

*p*is the probability of not getting caught in a given month and

*N*is the rental life of the unit.

This expression may not quite look right, but I assure you it can be obtained by summing

- Sum

_{[k=0 to 12N-1] }(

*k*+1)

*p*

^{k}(1-

*p*)

- Sum

_{[k=12N to ∞] }12

*N p*

^{k}(1-

*p*)

*As for why they are equal, I don't have any deep intuition beyond just working it out. I was so skeptical, I did the sum explicitly in Excel and did a comparison. It does check out.*]

This, multiplied by the monthly rental, is the minimum fine that should be levied to make illegal subletters indifferent between obeying the law and trying their luck. Raising it a bit more would tip the scales in favour of the law.

Since the rental life of flats tend to be long, for any small probability of getting caught in any given month, the expected number of months of rental accrued is very high. For instance, if the probability of not getting caught in a given month is 0.995, then for a flat with a rental life of 20 years, the probability of not getting caught at all is about 0.3 (and the expected number of months of rent accrued is 139.94).

At a glance:

Expected Months of Rental Collected | Rental Life N (years) | ||||
---|---|---|---|---|---|

5 | 10 | 15 | 20 | ||

p | 0.9975 | 55.7815 | 103.784 | 145.0924 | 180.6402 |

0.995 | 51.9478 | 90.4027 | 118.8694 | 139.9422 | |

0.9925 | 48.4603 | 79.3076 | 98.9434 | 111.4425 | |

0.99 | 45.2843 | 70.062 | 83.6192 | 91.0371 | |

0.9875 | 42.3889 | 62.3175 | 71.6868 | 76.0916 | |

0.985 | 39.7462 | 55.796 | 62.277 | 64.8941 | |

0.9825 | 37.3314 | 50.2742 | 54.7615 | 56.3172 | |

0.98 | 35.1223 | 45.5731 | 48.6828 | 49.6081 | |

0.9775 | 33.0989 | 41.5482 | 43.7051 | 44.2557 | |

0.975 | 31.2434 | 38.083 | 39.5803 | 39.9081 | |

0.9725 | 29.5396 | 35.083 | 36.1233 | 36.3185 | |

0.97 | 27.9731 | 32.4714 | 33.1947 | 33.311 | |

0.9675 | 26.5311 | 30.1855 | 30.6888 | 30.7582 | |

0.965 | 25.2019 | 28.174 | 28.5246 | 28.5659 | |

0.9625 | 23.9751 | 26.395 | 26.6392 | 26.6639 | |

0.96 | 22.8412 | 24.8136 | 24.9839 | 24.9986 | |

0.9575 | 21.7919 | 23.4011 | 23.5199 | 23.5287 | |

0.955 | 20.8194 | 22.1337 | 22.2166 | 22.2219 | |

0.9525 | 19.9171 | 20.9914 | 21.0493 | 21.0525 | |

0.95 | 19.0786 | 19.9576 | 19.998 | 19.9999 |

Probability of Non-detection | Rental Life N (years) | ||||
---|---|---|---|---|---|

5 | 10 | 15 | 20 | ||

p | 0.9975 | 0.8605 | 0.7405 | 0.6373 | 0.5484 |

0.995 | 0.7403 | 0.548 | 0.4057 | 0.3003 | |

0.9925 | 0.6365 | 0.4052 | 0.2579 | 0.1642 | |

0.99 | 0.5472 | 0.2994 | 0.1638 | 0.0896 | |

0.9875 | 0.4701 | 0.221 | 0.1039 | 0.0489 | |

0.985 | 0.4038 | 0.1631 | 0.0658 | 0.0266 | |

0.9825 | 0.3467 | 0.1202 | 0.0417 | 0.0144 | |

0.98 | 0.2976 | 0.0885 | 0.0263 | 7.84E-3 | |

0.9775 | 0.2553 | 0.0652 | 0.0166 | 4.25E-3 | |

0.975 | 0.2189 | 0.0479 | 0.0105 | 2.30E-3 | |

0.9725 | 0.1877 | 0.0352 | 6.61E-3 | 1.24E-3 | |

0.97 | 0.1608 | 0.0259 | 4.16E-3 | 6.69E-4 | |

0.9675 | 0.1377 | 0.019 | 2.61E-3 | 3.60E-4 | |

0.965 | 0.1179 | 0.0139 | 1.64E-3 | 1.93E-4 | |

0.9625 | 0.1009 | 0.0102 | 1.03E-3 | 1.04E-4 | |

0.96 | 0.0864 | 7.46E-3 | 6.44E-4 | 5.56E-5 | |

0.9575 | 0.0738 | 5.45E-3 | 4.03E-4 | 2.97E-5 | |

0.955 | 0.0631 | 3.98E-3 | 2.52E-4 | 1.59E-5 | |

0.9525 | 0.0539 | 2.91E-3 | 1.57E-4 | 8.46E-6 | |

0.95 | 0.0461 | 2.12E-3 | 9.78E-5 | 4.50E-6 |

To use these tables, one might first make a ballpark estimate of the probability of an illegal subletter never getting caught given he/she illegally rents out part of his/her flat for some number of years. Using the second table, a value for

*p*can be estimated. Then the first table can be used to identify the right ballpark of a deterrent fine.

In any event, all this tells us that since it is very unlikely to ever get caught, the rental that an illegal subletter expects to accrue is extremely high. Therefore, unless there are stiff penalties such as large enough fines or repossession of the flat by HDB, illegal subletters will continue to ply their trade.

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