Part II: Quantitative Analysis

You are a homeowner that has accepted a new job. The new job requires a commute that is undesirably long, but not impossible. You would like to sell your home and move into a new one nearer your job. Your discount rate for the funds received from the sale is $8\%($annualized$).$ Potential buyers will come, view and inspect the home, and negotiate a price offer. Your decision is to set $($in your mind$)$ the lowest such offer you are willing to accept. For simplicity, we will assume your decision must be an increment of $10$K $($i$.$e$.,500$K$,510$K$,520$K etc.$)$

We will not go into the details of the negotiating process, and just assume that the eventual price offers come from a random process described below.

$4.$ For$-$Sale$-$by$-$Owner. If you sell on your own, you will generate price offers that are uniformly distributed between $[500$K$,550$K$]$ USD, and will attract one offer every $2$ months. However, as commission, the buyer$$s agent will get $3\%$ of the sale price $($you will get the other $97\%).$

a$.$ To maximize the expected present value of the sales proceeds, what is the lowest price you should be willing to accept $($again$,$ restricting attention to increments of $$10$K$)?$

b$.$ Using your solution to part $($a$),$ what is the expected present value of the sales proceeds?

c$.$ Using your solution to $($a$),$ what is the expected value of the sale price?

d$.$ Comment on how the expected sale price is not a sufficient metric for the value of setting any particular lowest$-$acceptable$-$price.

Because time value of money.

$5.$ Hiring an Agent. If you hire an agent, you will pay $6\%$ of the sale price in total commissions $(3\%$ to buyer agent, $3\%$ to your agent, and you will get the other $94\%).$

a$.$ If nothing else changes, how do your answers to Question $1$ change? So$,$ if your agent provides no benefit other than the convenience of handling the transaction, what do you pay $($in expected present value terms$)$ for this convenience?

b$.$ Suppose seller$-$agents have different skill levels: $1,2,3,$ or $4.$ More skilled agents help position the house more attractively, generating offers at higher prices and greater frequency. An agent of skill level $$L$$ will generate offers that are uniformly distributed between $[500$K$,550$K$+($L$-1)12$K$]$ USD, and will attract L offers every $2$ months. What is the minimum skill level required for an agent to be worth the commission in your case $($assuming you do not value the $$convenience$$ aspect at all$)?$

$6.$ The Agent$$s NPV$.$ Suppose you hire a seller agent of the type you identified in Question $3($b$).$ The agent must split the commission with the buyer$$s agent, and with the agent$$s parent firm. So$,$ your agent takes home only $2\%$ of the sales price. The agent discounts future payments at the same $8\%,$ but also incurs costs associated with the marketing of your home $($e$.$g$.,$ targeted online ads, open houses, the opportunity cost of the agent$$s time$/$effort$),$ that come out to $$1,000$ per month.

a$.$ What is the minimum acceptable price offer that maximizes the seller$-$agent$$s expected net present value $($taking into account the commission and the operating costs$)$ from your listing?

b$.$ It is often argued that agents know the local market better than sellers. Suppose that you are unsure if the price range for your house is as described above, or with every number shifted down by as much as $$20$K $($i$.$e$.,$ on your own, prices between $[480$K$,530$K$]$; with an L$-$agent prices between $[[480$K$,530$K$+($L$-1)12$K$]).$ If your agent attempts to maximize as described in part $($a$),$ what would the agent want you to believe about the price range for your home $($holding fixed that the true price range is as described in Question $5($b$))?$

c$.$ Why are the incentives of the seller and the seller$-$agent not perfectly aligned? Can you propose a different way to structure the agent$$s compensation so that their optimal decisions are closer to one another$$s$?$