Interesting math+social problem

[I’m capturing this from a chat session with a friend. I mostly edited out my friend’s words for privacy reasons, so it might sound a little disjointed.]

Here’s an interesting math+social problem I have been thinking about on and off. There’s no practical application for it (for me, right now anyway) but it’s something interesting to think about.

Let’s say N people decide to own a house together. Based on how much each person puts down, and how much they pay per month toward the “household” bills, how would you determine what fraction of property belongs to each person? (The situation could be roommates who want to quit renting and buy a house together, or someone who already owns a house and wants to share the house while actually giving the other person a chance to earn equity instead of just paying rent.)

The formula would have to cover how much each person “owns” at any given time, in case they leave or someone else joins. In the simplest case, let’s say N=4, each person puts in the same amount toward a down payment, and each person pays the same amount per month into the household account, they would each own 25%.

My current thinking is that the total amount contributed by all four of them (down pmt, mortgage, bills) is the base, and each person should “own” a percentage corresponding to the total amount they deposited. The house would hopefully go up in value over time, but because of the bills, the total value of the “household” is not equal to what each person put in. Like a corporation operating at a loss.

Then to that you add questions like, what if 3 people work and one stays home, do you count “sweat equity”, at the going rate of professional child care and house cleaning? Perhaps there’s an agreement that if the stay-at-home person does substantially more chores than the others, that would be counted as a contribution at an agreed-on value.

Then you have the case where someone outside the household helps out by paying some of the down payment. That person might own a fraction of the property but doesn’t get the benefit of living in it. Similarly, the two people in the master bedroom get more benefit from the house than the two in the smaller bedroom. Also similar, maybe two people share a room and the other two people each get one room of their own.

Perhaps there’s a “baseline” value that they’re receiving (and paying for) — something like the going rate for a share rental of that size and configuration, and the “investor” who is contributing but not living in the house has a 0 baseline. The “baseline” amount represents the value you’re getting out of living there, and if you’re just paying the “baseline” amount you’re pretty much just a renter, and not building equity. (In fact, I would probably want to keep new people as “you’re just renting here” until like 6 or even 12 months.)

As the final wrinkle, in the Heinlein model, a child who comes of age automaticaly “owns” a share, so everything would be padded a bit to allow for that. (A child who comes of age has presumably been providing value to the household, so would have earned a stake, which confers upon majority. Do your chores every week :)

Hopefully, if done correctly, if someone were out of a job they could just not pay anything for a couple months and their previous investment would just get eroded a bit. At the same time, if someone buys (or brings with them) a new appliance for the house, they would get credit for it as a contribution and their share would go up a little.

OK. time for bed. Will think about this more later.

0 thoughts on “Interesting math+social problem

  1. traveller_blues

    One of the ways to perhaps fairly handle this model is creating a limited liability corporation based on the household with the house as an asset. Issue private ‘shares’ of stock in the house, with everyone’s payment purchasing X shares of the house. In the interest of round numbership:

    mortgage payment + insurance payments / 200 (~50 increments) = share price
    So if your mortgage were say, 1800 and the insurance ran 450, your share price is 11.25. For round number’s sake, let’s call this a household of four people, all with jobs, and let’s say I pay 600 a month. It’s a little more than a quarter, but I don’t feel like writing 562.50 on a check every month. My ‘stock’ purchase is 53 1/3 shares a month.

    Dissolving the Partnership.
    Let’s say that after a year of paying this, I have 640 shares, and I get a new job in India. I can’t possibly expect my housemates to cough up a year’s worth of my paid mortgage to buy me out. It is still an investment, but I am no longer receiving the dividend (living there) that I would get by doing so.

    Meanwhile, back at the ranch, everyone’s new payment is 2250/3, or 750, which at the current share value nets each person remaining 66 2/3 shares of stock a month if things were being paid in an equality fashion.

    5 years in, one partner gets hitched and moves out. Let’s say for argument’s sake they were only paying 500 a month the entire time, so (500/11.25)*60=2,666 2/3 shares.

    20 years to the day, the folks remaining decide to sell the house.
    Person A, that left after year 1, owns 640 shares of stock for paying $7200.
    Person B, who left after 5 years, owns 2,666 2/3 shares of stock, for paying 30000.

    You can get the number of shares the house generates per year by dividing the total value of the house by the stock price (2250*12)/11.25 = 2400, which makes sense for person A to own slightly more than a quarter since they’re paying slightly more than a quarter.

    Assuming the other two paid the balance equally, Persons C and D therefore own (48,000 shares – 2,666 2/3 – 640) split two ways, or 22,347 shares apiece.

    The total amount of payments for mortgage + insurance was 540,000, but only 432,000 of that is towards the actual mortgage, and let’s just say that 300,000 of that went to equity. The house sells for 100,000 more than the original purchase price; well, add that to the equity and divide by the number of shares, and that’s a share value of 8 1/3 a share. (Insurance is not included in the value of a home). Of that 100,000 profit, person A gets 5,331; person B gets 22,213.

    Well, that’s a bit of a loss.

    Let’s just say the house price doubles: 300,000 more = 12.5 a share. Person A now gets 8,000 (profit of $800) and person B gets 33,333.25 (profit of 3,333). Persons C and D get 279,337 apiece out of the sale value of the house, on an investment of 251,403.75, a profit of 27,933 apiece.

    If this seems a bit unfair, consider that the value of a company is its assets (physical goods, etc) and does not count any costs paid to taxes, insurance, and interest either. Having to wait until the house is sold (20 years in this example) may seem a bit excessive, but it’s a net return of greater than zero than you’d get if you were just renting.

    Extending on this model, repair and improvement payments can either be given a share value for the person(s) who pay them, or just assumed as one-time expenses.

    Peculiarities: If Someone Wants to Work for the Rent

    Finally, to address non-paying members of a household, find the value of an equivalent cleaning service and add that to the monthly payments. Divide the number of tasks on that list by the value of only that payment, and that is the number of shares each task on that list is worth (with some tasks being worth more than others). This means that a person doing $150/month worth of cleaning work is earning far fewer shares than someone paying a share of the mortgage, yes, but the team is getting the value of that from what would be a cleaning service if nothing was being done.

    How’s that work?


    1. traveller_blues

      One addenda: If someone wants to be bought out early, there should be a contract clause in there somewhere for that. It’s likely going to put a hardship on the remaining members, but it’d work this way:

      Since at the beginning of a mortgage, interest paid tends to outweigh equity gained, payout is at the reduced value of equity/outstanding shares, with the remaining members -purchasing- shares of stock at that value from the person who wants to be bought out. Say after 3 years, I (as person A) want to get bought out, and the equity on the house is only 18,000. Well, there are currently 7200 (2400*3 years) shares of stock outstanding, so the price per share is $2.5. I get back $1600 on my $7200 investment, but hey, I got my money now.

      This of course means that if the party signed on for a first 5 years, interest only loan, I get zero.


      1. nekodojo

        You are right, it can be tough when someone decides to leave. There should probably be some money put aside from the start in the “contingency fund”, and any new residents should be encouraged to buy into the collective similar to the “down payment” the original folks put in. That way if someone leaves, they can be paid some of their value back, over an agreed-on period that doesn’t break the back of the remaining members, or they can elect to keep some value as investment. Even if there isn’t a down-payment, there should be a savings goal built in to the payments so the group becomes more resilient over time.

        Another site I found recommended having an ownership agreement in writing, not necessarily as a corporation but just as an agreement between individuals, and listing the “common” property as well (including any major appliances, tv, stereo, etc) and what should happen to them on dissolution. They recommended rewriting the ownership agreement every year but that seemed excessive to me.

        My thinking is that if people can manage to live together and not kill each other, this would be a way for them to get a slightly better house than they would each on their own, and to have some sort of equity similar to purchasing a house rather than just having one person/couple be the “owner” and the others be “renters”. There are probably other benefits too, like shared meals, socializing, etc… but sharing a home is certainly not for everyone.

    2. nekodojo

      Yes, this is almost exactly what I was thinking. The “basic” formula I was thinking of was, paying $1 toward the household earns you 1 share, and the expectation is that your share would be worth significantly less than one dollar, but still worth something, especially compared to renting.

      I found another site that had a similar arrangement, but the difference there was: 1. food, utilities, anything that a renter would normally pay would not count toward the ownership formula; 2. the non-ownership bills were split up according to ability to pay (as this was intended for a group family, not “just roommates”) and 3. they didn’t mention exactly who pays the mortgage other than to say “Part of each person’s overall monthly contribution is tagged for ownership expenses.” I understood this to mean that if you make less, you were expected to pay less, but you would also end up owning less of the assets at the end (but still something). The members could elect to put more in and would end up owning more shares.

      If applied to a shared living arrangement where the members don’t think of themselves as a family, it would probably make more sense to have the bills shared equally, but the important part is that the parties all agree ahead of time. My original thought was to have something that worked well regardless of whether the parties decide they want to share the burden as a family or just be co-owners and not “a family”.

      I think their sharp division between the “bills” and the “ownership” is probably similar to what I was trying to do with the “baseline”… if you occupy the largest bedroom, you’re getting a bit more benefit out of the house, so if you are paying the same as everyone else, and you end up owning the same percentage, you got the better deal. With a baseline, a roommate who makes less money might be content with the smaller bedroom, and could still have the option of paying $50 over the baseline and earning 50 shares.

  2. kethry

    Okay I am a bit cynical, but I just don’t think it works out if one person is doing “sweat equity”. One person paying more because they have the bigger bedroom is one thing.

    I feel its best that the 4th person pays the equal amount and they together everyone puts in to pay for a cleaning service.
    If you mean sweat equity by “building a new bathroom” or something else that add value to the house, I can see it. But not just maintenance (like cleaning and decorating). And if you go by the sweat equity route, then put it all in writing “Person is responsible for the following tasks on a weekly schedule”. Its just too easy for each partner to feel they are getting the fuzzy end of the lollipop if you don’t all agree in the beginning and stick to the agreement.

    1. nekodojo

      Understood. The “sweat equity” concept makes more sense in a family than in a “just roommates” situation. For example, a committed triad could probably afford to have one member stay home and watch the kids full-time, as they might do in a traditional marriage. But if you answer an ad for a share rental and then say “Hey I was thinking, instead of paying you rent, how about if I just do all your chores for you instead?” they would not waste much time rushing you out the door.

      I could definitely see the benefits of living in a three-income household with a stay-at-home mom or dad in residence. Economies of scale and all that.

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