We are considering buying a parcel of land that has been zoned for multifamily housing. We

have determined that the best use of this land is to build condominiums.

Our first decision involves the scale of our investment. The size of the lot dictates that we can build

three units per floor and our issue is whether to build a two$-$ or three$-$story building.

A two$-$story building would have six units and would cost $$80,000$ per unit to build. A three$-$story

building would have nine units and would cost $$90,000$ per unit to build. Cost per unit increases

as the building gets taller, largely because of additional foundation and elevator costs. We will

assume that condo investors earn their entire retum through capital gains $($rather than keeping the

condos and renting them out$).$

Our second decision involves timing. The current value of a one$-$unit condominium is $$100,000,$

but the housing market is volatile. We assume that, with equal likelihood, the price of a

condominium in one year will be either $$150,000$ or $$90,000.$

The risk$-$free rate of interest is $10\%$ and $Tc=0.$

Questions:

$(1)$ What is the static NPV of immediate development of a two$-$story building? Assume that the

building will be built imenediatelely and the condos sold immediately

$(2)$ What is the static NPV of immediate development of a three$-$story building? Assume that the

$(3)$ If the choice between a two$-$story building and a three$-$story building were made today, which

should we choose?

Erom this point forward $($i$.$e$.$ when answering parts $(4)$ to $(1)),$ assume that we will wait one vear

and that we will make our decision after observing condominium prices at $t=1.$

$(4)|$What is the net value of the two$-$story building at year $1,$ if condominium prices rise? Assume

that the building will be and the condos will be sold exactly at $t=1($one year from today$)$

and that construction costs will be the same in one year as they are today.

$(5)$ What is the net value of the three$-$story building at year $1,$ if condominium prices rise? Assume

that the building will be and the condos will be sold exactly at $t=1($one year from today$)$

and that construction costs will be the same in one year as they are today.

$(6)$ What is the optimal choice at year $1$ if condominium prices rise? $($Hint: make your choice

based on your answers to parts $(4)$ and $(5))$

$(7)$ What is the net value of the two$-$story building at year $1,$ if condominium prices decline?

today$)$ and that construction costs will be the same in one year as they are today.

$(8)$ What is the net value of the three$-$story building at year $1,$ if condominium prices decline?

Assume that the building will be builland the condos will be sold exactly at $t=1($one year from

today$)$ and that construction costs will be the same in one year as they are today.

$(9)$ What is the optimal choice at year $1$ if condominium prices decline? $($Hint: make your choice

based on your answers to parts $(7)$ and $(8))$

$(10)$ What is the value of the development today if the decision between the two$-$story and the

three$-$story building will be made in one year $($i$.$e$.$ at $t=1)?$ Use a one$-$step binomial model with

one condominium as the underlying asset $,$ and $VdL-$$$90,000.$

Hint: You can check your work by solving the binomial option problem using both available

methods: the replicating portfolio approach and the risk$-$neutral probability approach. If your

solutions are correct, the two results should be equal.

$(11)$ It is optimal to delay the construction of the condominiums by one year and make our decision

at $t=1$ instead of deciding today. True or false?