Score One for Probability Theory

Published at 11:49 on 7 April 2014

So, I’ve been on the Island for a year it’s lease renewal time for me. I have two options: a normal lease at $1400/mo, or month-to-month at $1600/mo. The penalty for breaking a lease early is a fixed $1500/mo.

Wow, a fixed penalty of $1500/mo that’s less than the normal rent of $1600/mo without a lease. It seems like a no-brainer. Not so fast! Time to run the math.

The way to do it is with what is called expected value, essentially a weighted average taken by enumerating all possible scenarios then multiplying a scenario’s cost by its probability. After much thought, I chose a Poisson distribution with λ = 4 months as an educated guess.

I’ve been keeping an eye on the local real estate market and I actually think it will be more like three months, but I’m being a little pessimistic in case the entire spring and summer fly by and I end up buying nothing. In that case, things will get dead in the coming winter.  So, given those assumptions, here’s what happens when I find a home so many months into my continuing tenancy here:

MONTH       PROB        CUM      LEASE     M-to-M
    1     0.0183     0.0183    1427.47    1600.00
    2     0.0733     0.0916    1484.25    1575.54
    3     0.1465     0.2381    1491.58    1462.45
    4     0.1954     0.4335    1359.71    1230.32
    5     0.1954     0.6288    1086.19     917.64
    6     0.1563     0.7851     754.07     603.03
    7     0.1042     0.8893     457.11     350.16
    8     0.0595     0.9489     244.25     180.90
    9     0.0298     0.9786     116.24      83.83
   10     0.0132     0.9919      49.76      35.13
   11     0.0053     0.9972      11.39      13.41
   12     0.0019     0.9991       6.67       4.70
TOTAL                          8488.69    8057.11

PROB is the probability I will find something that month and CUM is the cumulative probability (i.e. the chance I find something that month or in an earlier month). Odd; what’s going on here?

The issue is the percentage cost. $1500 isn’t much a chunk of 11 month’s rent, but as time goes on, it becomes a bigger and bigger chunk of the remaining rent if one quits early:

    1     9.74
    2    10.71
    3    11.90
    4    13.39
    5    15.31
    6    17.86
    7    21.43
    8    26.79
    9    35.71
   10    53.57
   11   100.00
   12     0.00

Why 100% instead of 107.14% for quitting in month 11? Simple: only a fool would pay $1500 in penalty fees when it’s cheaper to pay $1400 to rent an unneeded, empty apartment for an extra month.

Still feel like I’m pulling my own leg here, using lots of math when common sense says it must be the other way? Consider the case where I find something in five months (that’s the point where the odds become in my favor of finding something).

With a lease, I pay 5 × $1400 + $1500 = $8500.

With no lease, I pay 5 × $1600 = $8000.

Sure, there’s a chance I’ll pay more, if I end up spending another year here and not finding anything. But the odds seem to be in favor of my paying less. In fact, the lease is only to my advantage if it takes eight or more months to do something I expect to do in four or five months.

Mine is a corporate landlord; doubtless they’ve run this sort of analysis themselves and deliberately crafted their lease-renewal offer to have a seductive yet economically disadvantageous option to it, knowing they can expect to pocket on average around $400 of pure profit each time they sucker a tenant into agreeing to it.

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