Quiz 1: Winter 2004
Real Estate Economics
Name _________________________
Section:
Afternoon
Evening
Put answers in the space provided!!!! Calculators are allowed. No partial credit here for the
math problems (This is solely for ease of grading – there will be partial credit on other
quizzes/final). As a result, be careful with your work! Quiz is out of 20 points.
Question 1
You are currently working as a real estate consultant. You have been asked to assess the property
market in Las Vegas, Nevada. You decide that you want to plot the real house price appreciation
for 3 bedroom/2 bath homes in the Las Vegas metropolitan area. You decide to focus on repeat
sales (which exclude new homes) given that new homes are different from the existing housing
stock. The Las Vegas tax assessment office has published the median sales price for 3 bedroom/2
bath homes on an annual basis. Here are the reported figures for the late-1990s.
Year
Median Sale Price
1997
1998
1999
2000
$140,500
$162,800
$167,900
$187,700
The CPI is reported monthly by the Bureau of Labor Statistics (BLS). Below are the June CPI
figures for all U.S. urban consumers between the years of 1980 and 2000. You believe that the
national CPI is appropriate for Las Vegas. You decide to deflate all nominal variables in the
economy by the appropriate June CPI figures of each year. The base year for the current CPI is
1982-1984 (the actual base year is not important for this analysis – but if it makes you feel better
you can think of it as being July of 1983).
1980
1981
1982
1983
1984
1985
1986
a.
0.825
0.905
0.970
0.994
1.037
1.075
1.094
1987
1988
1989
1990
1991
1992
1993
1.136
1.181
1.241
1.300
1.361
1.402
1.444
1994
1995
1996
1997
1998
1999
2000
1.480
1.525
1.568
1.603
1.630
1.662
1.695
In 1996 dollars, what was the real median house price for 1998? --- 3 points
Real house price (1998) = Nominal house price (1998) * [PI(1996)/PI(1998)]
= $162,800 * 1.568 / 1.630
= $156,607.61
See notes entitled Real vs Nominal for details!
$156,607.61
Over 
In essence, you need to convert the house price from 1998 to base year prices. After, you
convert base year prices to 1996 dollars. Intuitively, this is done in two steps. Above, I
combined the steps and did it with one equation. The notes on the web page (real vs.
nominal) explains all the details.
b.
In 1996 dollars, what was the real growth in median house prices between 1998 and
2000? (Use the exact formula) --- 3 points.
<<Put your answer in the box!!!>>
Using the same procedure above, real 2000 median house price is 1996 dollars is:
$173,636.33
Real growth is (Real(2000) – Real(1996))/Real(1996)
= (173,636.33 – 156,607.61)/156,607.61
= 10.87%
10.87%
c.
Using the approximation formula, was the real growth in median house prices between
1998 and 2000? ---- 3 points.
Real growth (96,00)
= nominal growth (96,00) – inflation rate (96,00)
= 15.29% - 3.99%
= 11.3%
11.30%
The answers to questions 2 and 3 are essentially true/false questions. As a result, for questions 2
and 3, circle ANY of the true answers to the question stem. NOTE: There may be more than one
true answer to each of the question stems.
Question 2
Which of the following will shift the demand for residential housing to the right? (assume all
else stays constant when making your analysis). (5 points – 1 point each).
a.
An increase in household after tax income.
True – as households get richer, they want more of all goods (including housing).
Demand for housing should rise.
b.
An increase in household income tax rates (holding after tax income constant).
True – holding after tax income constant, an increase in income tax rates increases the
benefits of the mortgage interest deduction. This lowers the user cost of owning a home.
c.
An increase in expected inflation (holding real interest rates constant)
True – holding real interest rates constant, an increase in expected inflation increases
nominal interest rates. Higher nominal interest rates mean larger benefit from
mortgage interest deduction. Given that real rates have not changed says that the real
cost of borrowing has not increased. The only cost/benefit comes from the mortgage
interest deduction.
d.
A fall in construction costs associated with residential housing.
False – construction costs affect supply NOT demand. If construction costs fall, supply
will increase. The price of housing will fall. However, the demand for housing will
NOT shift. We will just move along an existing demand curve. This is basic supply and
demand analysis. Shifts in supply cause movements ALONG a demand curve.
e.
An increase in the risk premium for holding housing (holding the expected growth
rate of housing constant).
False – if housing gets more risky (holding expected growth constant), we are less likely
to hold housing as an asset. As a result, our demand for housing will fall and the
demand curve will shift LEFT.
Question 3
A standard model of the supply and demand of residential housing predicts which of the
following? Assume there are no supply constraints on the model.
a.
Houses will appreciate at a rate similar to corporate equities.
False – housing appreciations over long periods of time are close to zero.
b.
Nominal house price growth should be small in the long run.
False – real house appreciations should be close to zero. Nominal house price changes can
be quite large in periods when inflation rates are high.
c.
Rapid house price appreciations will eventually be followed by sharp price
declines.
True – if allowed, supply will eventually adjust.
d.
In the long run, house prices will appreciate more if construction costs are
accelerating.
True – in the long run, house prices grow at the cost of construction. If construction costs
are rising rapidly, long run house prices will also rise rapidly.
e.
Real house appreciations will be larger than nominal house price appreciations,
in the short run.
False – this didn’t even make sense. Real and nominal is not a short run/long run issue.
Real house price appreciations will only be higher than nominal house price appreciations if
the inflation rate is NEGATIVE (i.e., deflation). This has not happened in the U.S. during
the last 80 years.
Question 4
I am aware that the new syllabus on the course web page takes precedent over the syllabus in the
course pack.
Yes
No
(2 points if you circle yes)
Quiz 2: Real Estate Economics and Finance
Name
__________________
Section Registered:
Afternoon
Evening
Understanding real estate markets is essential to gauging expected returns on real estate
investments. In this question, we are going to ignore the risks associated with financing real
estate purchases (i.e., mortgage markets) and only discuss the returns associated with an
investment in the asset.
Currently, you are looking to invest in a residential real estate property. Your plan is to purchase
the real estate and hold the property for exactly 3 years. At that time, regardless of economic
conditions, you will sell the property at current market prices. During that time, you do not
expect any adverse economic conditions (i.e., no recessions), but forecasts suggest that the 3 year
inflation rate is going to be 13% (roughly 4.2% per year).
Currently, your analysis shows that the property you are looking to buy is in long term
equilibrium (no excess returns can be earned from building a similar property – i.e., the property
price only reflects the cost of production). Your analysis shows that construction costs are likely
to increase at the rate of inflation during the next three years.
The property you are planning to buy is currently priced at $300,000. You are going to put down
50% when you purchase the property and finance the remainder with an interest only loan. Your
interest in the property is purely financial. You will receive no service flow from the property
over the three years and your return is only a function of the future sale price. (In other words,
you will earn zero economic profits on the property during the next three years. The rent you
charge your tenants will be exactly equal to the interest payments and maintenance costs
associated with the property. This last assumption is highly unrealistic and we will relax it in the
future).
a)
After you purchased the property, it was announced that a new high-tech firm was going
to locate in the town where you purchased the property. You notice that immediately
after the announcement, the appraised value of your property increased from $300,000 to
$350,000. You (naively) expect the property price to remain at $350,000 when you sell
the property in three years. If your future sale price is $350,000, what would be your
expected real return on your investment? (5 points) <<Note: 1. Use approximation
formula when computing real returns and 2. Report the return OVER three years (no
need to annualize the return)>>
This part had lots of words….but, the analysis is fairly simple given last week’s quiz.
week’s quiz, we know that:
From last
real return = nominal return – expected inflation.
The initial house price is $300,000. You put up $150,000 and you borrowed the remaining
$150,000. The borrowed amount of money has to be repaid when the house is sold. No need to
worry about the interest payments given the information above (rents from the property offset the
interest payments).
If you sell the house for $350,000 and you pay off the $150,000 loan, you will have a balance of
$200,000.
You took your $150,000 and turned it into $200,000. Your three year nominal return was then
33% (about 11% per year).
Given that the inflation rate increased by 13%, your approximate real return would equal 20%
(33%-13%). This real return is over three years. Even though I told you not to, some of you gave
an annual return, which would be approximately 6.66% (20%/3, where 3 is the number of years
you held the property).
Also, even though I told you not to, some of you may have used an exact formula (converted all
dollars amounts to real dollars). The return then would be 18% ((176,992-150,000)/150,000). In
this case, the real amount of equity in the house after 3 years would be $176,992 ($200,000/1.13,
where 13% is the three year inflation rate).
We gave credit for all of these answers (basically, they are all the same). The method I gave first
is an approximation of the second. See the notes on the web page ‘Real vs Nominal’ if you are
still confused.
Note: This problem was basically a review of last week’s material. You should understand
real/nominal differences by now!
b)
There is lots of space to build in the property market in which you invested (part of the
reason that the high tech firm decided to locate in this area). Your friend mentions this
fact to you. Your friend also points out that the builders in this property market are
highly competitive and information is shared easily between them. His conjecture is that
the long run property market equilibrium will be restored within three years.
If his
conjecture is correct, how much will your property be worth at the end of three years
(assume no other shocks hit the local property market)? What will be your expected real
return on this investment? (7 points)
The method to answer this question is similar to the method above. The only thing that differs is
what the future sale price will be. If we return to long run equilibrium after 3 years, housing
prices will equal their construction cost (if they didn’t, builders would keep building). We know
that when you bought the property, prices were in long run equilibrium. So, the cost of a building
a ‘similar’ home was $300,000. After three years, the cost of building is projected to increase at
the inflation rate (which is expected to be 13% over the next three years).
The future sale price should then equal $339,000 (300,000 * 1.13).
What is your expected real return?
You put in 150,000. After 3 years, you will get back $189,000 (339,000 – 150,000, where
150,000 is the loan you have to pay back). This implies a nominal, three year return of 26%
(39,000/150,000).
What is your real return over the three years?
Real return = Nominal Return – Expected Inflation
= 26% - 13%
= 13%
Because of leverage, you are still able to earn a positive real return. Notice, if you financed the
purchase of the loan entirely through your own funds (no borrowing), your return would have
been zero. Your nominal return would have been 13% (39,000/300,000). 13% - 13% would be
zero.
Leverage, in this case, could yield a positive return on your real estate investment even though the
real cost of purchasing a house was constant (both construction costs and inflation increased by
13%).
This is what I wanted to illustrate (hence, the reason I developed this quiz question). A quiz that
teaches us something……that is what I like! (I am not sure how many of you thought about this
before, but it is interesting to ponder). Later in the course, we will discuss how borrowing rates
will adjust to accommodate inflation. Some (but not necessarily all) of this effect will be
mitigated by lenders setting interest rates optimal (we will see this soon……).
Question 2
A goal of this course is to help you understand how commercial property valuation rules will
change as property markets evolve over time. Below, I ask you to assess how cap rates will
change in the response to events in a local market for office properties.
Which of the following statements are true? Compare the initial cap rate in the local office
property market to the cap rate in that same local office property market after the following
events take place. (2 points each – as always, circle ANY true statement below!)
Ok – before we begin, we should remember or commercial real estate valuation formula:
Price = NOI (current)/cap rate. The cap rate = r + α – g (all symbols are real). r is the real
interest rate, α = expected future risk of NOI, g = growth of NOI. Anything that effects the
level or the variability of FUTURE NOI will show up in the cap rate. Changes in current
NOI will NOT show up in the cap rate (that will show up in current NOI in the above
pricing formula).
Anything that makes future NOI lower (low g) or more variable (high α) will cause the cap
rate to increase. In other words, when the future is more variable or has low growth, the
discount rate for current NOI has to be higher (implying a lower property price).
Lastly, only unexpected changes can cause the cap rate to change. If the news was expected,
it would have already been incorporated in the initial cap rate!
a.
All else equal, an unexpected increase today in the uncertainty surrounding future
vacancy rates in this property market should cause the market cap rate for office
properties to fall.
False!! Future NOI is more risky (given the uncertainty of future vacancies), as a result α
will increase. This causes the cap rate to increase (NOT fall!). (However, property prices
today will fall if vacancies are supposed to be higher in the future).
b.
All else equal, an unexpected increase in construction permits for office properties
today should cause the market cap rate for offices to fall.
False! An unexpected increase in permits today implies an increase in supply tomorrow.
As future supply increases (unexpectedly), g will fall (unexpectedly). This implies the cap
rate will rise. Building prices today should adjust. If property prices in the future will be
worth less, today’s property would be a less lucrative investment. Note, today’s property
rents need not fall immediately. Today’s rent is determined by today’s supply and demand.
Given that supply did not change today, rents today would not change. But, tomorrow’s
rents would (hence the fall in today’s property price which is the discounted value of all
future rents). Using the cap rate formula, we represent this as an increase in the discount
(cap) rate.
c.
All else equal, an expected increase in future market rents for office properties will
cause the cap rate for offices to fall today.
False! Expected changes do not lead to CHANGES in the cap rate. The initial cap rate
should already include all expected information.
d.
All else equal, an unexpected increase in demand for office space today will cause
office cap rates to fall today.
False! Changes in demand today will affect today’s rent (holding today’s supply constant).
So, today’s NOI will increase. The cap rate – holding all else equal – will not change. The
property value will increase (because current NOI increases, not because the cap rate
changes). Changes in today’s supply and demand affect today’s NOI. Only changes in
tomorrow’s NOI (either the level or the variability) will show up in the cap rate!
Quiz 3
Real Estate: Winter 2004
Name:
___________________
Section Registered:
Afternoon
Evening
Question 1
You are considering investing in a real estate property which yields a first year NOI of $250,000.
In year 2 and 3, the expected NOI is also $250,000. Starting with year 4, the NOI is expected to
grow at a 5% real rate per year (i.e., the year 4 NOI would be 5% higher than the year 3 NOI).
Formally, the expected NOI for this property can be expressed as:
Year
1
2
3
4
5
6
NOI
$250,000
$250.000
$250,000
$262,500
$275,625
$289,406
After year 6, NOI is still expected to grow at 5% per year. All dollar values are real. You plan
on holding the property for exactly 5 years. After year 5, you will sell the property at market
value. Both you and the market require a real return of 11% on similar real estate investments.
For now, assume that there is no risk associated with the expected NOI stream (i.e., α = 0).
Lastly, ignore depreciation.
a.
Given the information above, what is the terminal cap rate (cap rate associated with year
6 NOI) for this project? (4 points)
Cap rate = κ = r + α – g = 11% + 0% - 5% = 6%
The terminal cap rate with a required return of 11% and a growth rate of 5% would imply
a cap rate of 6%
Note: there is no uncertainty in this problem (by assumption). So, α = 0 (i gave you this in the
problem).
b.
How much would the property sell for at the end of year 5? (2 points)
Value of Property (end of year 5) = P (at end of year 5) = NOI (year 6) / terminal cap rate
P = $289,406/0.06 = $4,823,433.33
Question 1 (continued)
c.
What is the property worth today? (4 points)
P(today) = NOI Year 1/1.11 + NOI year 2/(1.11)2 + NOI year 3/(1.11)3 + NOI year 4/(1.11)4 +
NOI year 5/(1.11)5 + Selling Price end of year 5/(1.11)5
$250,000/(1.11)
=
$250.000/(1.11)2
=
$250,000/(1.11)3
=
$262,500/(1.11)4
=
$275,625/(1.11)5
=
$4,823,433.33/(1.05)5 =
Total:
d.
$225,225.23
$202,905.61
$182,797.85
$172,916.93
$163,570.03
$2,862,473.18
$3,809,889.83
Given the answer above, what does this imply about the current cap rate associated with
this property? (2 points)
Today’s cap rate is next period's expected NOI divided by today’s property price:
cap rate = $250,000/$3,809,889.83 = 0.0656
Today’s cap rate is 6.56%. It is higher than the terminal cap rate because the discounted
value of NOI is dropping sharply in the first few years of the project. So, over the
foreseeable future, g is not 6% (as it is after year 4), it is less than 6% (given the constant
NOI in the first few years of the project).
e.
Suppose the cash flow associated with this project is risky (and you and the market are
risk averse investors). Would the cap rate computed in part d (above) increase or decrease in
response to the increase market risk? (2 points)
The cap rate would (circle one)
increase
decrease
Increasing risk would require α to be positive. As a result, the cap rate would increase (you
would need to discount the cash flows by a higher amount!). Note, this would LOWER the
price of the building today.
Question 2
Which of the following are true? (Circle any true answer – 2 points each)
a.
Uncertainty over future property prices could delay positive NPV projects from being
built.
True: See notes. We did an example of this. This is why builders are slow to respond.
They do not want to build when property prices could fall tomorrow!
b.
Uncertainty over future property prices could delay default decisions when current
house prices are less than current mortgage balances.
True. See notes. Households may not want to default today if prices are expected to rise
sufficiently tomorrow.
c.
In general, higher default costs will lead to a higher option associated with not
defaulting (i.e., continuing making mortgage payments) when current house prices
are less than current mortgage balances.
True. See notes. We did an example just like that. When default costs are higher, people
default less (all else equal).
Quiz 4: Real Estate Economics (Winter 2004)
Name:
Section:
Afternoon
Evening
Question 1
In 1996, the Sears Tower declared bankruptcy. At that time, the building was estimated to be
worth $600 million and the value of the mortgage was worth $850 million.
Suppose that the following information held.
1.
2.
3.
4.
5.
6.
7.
8.
The relevant planning period is only three years (periods) with 1996 being period 1.
To continue the mortgage, $100 million of mortgage payments are needed each year.
If the $100 million payment is made, the value of the mortgage would be reduced to $810
million in year 2 and to $760 million in year 3.
In period 2, there is a 50% probability that the value of the property would rise to $900
million and there is a 50% probability that the value of the property would fall to $500
million.
In period 3, there is a 50% probability that the value of the property will increase by an
additional 20% over the period 2 price and there is a 50% probability that the value of the
property will fall by 10% below the period 2 price.
All dollar amounts are listed in real amounts (no time discounting necessary).
There are no additional costs to bankruptcy.
The building cannot be sold in ANY period (only decision is pay mortgage or default).
Part A: Using the real option framework developed in class, fill out the following tree diagram
using the above information. What I am looking for is the option value (V) at each node of the
decision tree. Note: The value at each node i (Vi) is the value of keeping the mortgage alive
(i.e., the value of not defaulting). To make grading easier, list the house price and mortgage value
at each node (i.e., all 7 nodes). You should write ONLY the option value in the box provided.
3
V3 = 320 million
1
V1 = 85 million
4
V4 = 50 (H > M)
0
5
V0 = 0
V5 = 0 (H < M)
2
V2 = 0
6
V6 = 0 (H < M)
Key: H = property value, M = outstanding mortgage, V = value of the continuation option (not
defaulting)
At node 3, V = H – M. H = 1,080 (900 * 1.2). M = 760 (given in the problem).
At node 4, V = H – M. H = 810 (10% lower than 900). M = 760 (given in problem). So, V =
50.
At node 1, V = .5 * 320 million + .5 (50) – 100 million. The continuation value is $85
million. If do not default, get 320 million with 50% probability and nothing with 50%
probability.
In the problem, I didn't allow you to sell the property in period 2. If you were allowed to
sell the property, you would at node 1. Selling the property would provide a value of 90
million (900 million (H) – 800 million (M)). The 90 million payoff from selling is larger
than the expected 85 million dollars payoff from paying the mortgage and continuing to
next period. However, given the problem set up, you were not allowed to sell in period 2.
At node 2, definitely default. Why pay 100 million dollars to default next period? The
continuation value would be negative (0 * .5 + 0 *.5 – 100 million).
At node zero, you would default. The decision would be .5 * 85 million + .5 * 0 -100 million
< 0 . Given that the continuation value is negative, the owners should definitely default
today.
In other words, the default decision would take place at every node EXCEPT nodes 1, 3 and
node 4.
Notice, if I gave you the option to sell after the second period - the results at node 0 would
not change. Instead of getting 85 million with a .5 probability, I would get 90 million with a
.5 probability. The continuation value from today's perspective would still be negative! As
a result, I would still default at node 0!
Question 1 (continued)
Part B: Circle all of the following nodes at which default is optimal:
Node 0
Node 1
Node 2
Node 3
Node 4
Node 5
Node 6
See above – you would default at every node except nodes 1, 3 and 4.
Question 2
Below is a fictionalized term structure for U.S. government bonds. (Note: Term structure just
refers to interest rate differentials on securities that differ only in their duration).
Government Security
One year U.S. Government Bond (starting today)
Two year U.S. Government Bond (starting today)
Three year U.S. Government Bond (starting today)
Interest Rate (per year)
3.0%
3.5%
3.8%
Part A.
If arbitrage holds, what is the implied interest rate on a one-year government
security starting one year from now?
(5 points)
Arbitrage implies that the two –year yield from investing in the two year security starting
today should be equal to the yield from investing in the one year security today and reinvesting the funds into a one year security tomorrow.
In other words:
(1 + 0.035)2 = (1+0.030)(1+x)
Note: The two year return must be squared (we will earn an annual return of 3.5% for two
consecutive years). In other words, I could have expressed the return as (1.035)(1.035).
Note: x is the unknown return that we are solving for. It is the interest rate that the
market expects will prevail on a one-year security one year from now.
Solving the above equation, we get: (1.035)2/(1.030) – 1 = x = 0.04 (or 4%)
Part B.
If arbitrage holds, what is the implied interest rate on a one-year government
security starting two years from now? (3 points)
Same intuition applies. The one year rate starting two years from now can be expresses as:
(1.038)3 = (1.035)2 (1 + x).
We can either buy a three year bond and get a return of 3.8% each year for the next three
years or we could buy a two year bond at 3.5% and re-invest the proceeds from that bond a
x% for the third year.
Solving for x, we get x = 4.41%
Question 3
Circle all of the following true statements below. Note: There can be more than one true answer.
(4 points in total - 2 points each).
A.
All else equal, if nominal interest rates are mean reverting, mortgage risk premiums
should be high when expected inflation is high.
True: If expected inflation is high, mortgage rates will be high today. Mean reversion of
mortgage rates implies that high mortgage rates will be followed low mortgage rates. In
other words, mortgage rates will be falling in the near future. As mortgage rates are
expected to fall, refinancing probabilities increase. Lenders need to be compensated for the
increase value of the refinancing option.
B.
All else equal, risk premiums on conventional (i.e., standard) mortgages have doubled
between 1993 and 2000.
True: This came straight from the notes. See the plot from Topic 4.
Quiz 5: Real Estate Economics (Winter 2004)
Question 1 (12 points)
Suppose that Bank X has just issued a fixed rate mortgage for $900,000 with an interest rate of
14% annual, compounded monthly, with monthly payments over a term and amortization period
of 25 years. However, Bank X would like to increase the yield on this loan to 15.5% annual,
compounded annually.
How many points should Bank X charge on this loan to earn the required yield if they expect the
loan to be held for exactly 10 years? (Note: points need not be whole numbers. For example,
5.44 points is an allowable answer). PUT ANSWER IN BOX BELOW!!!!!
Step 1:
What is the PMT on this mortgage?
i = 14/12 = 1.1667 (monthly interest rate)
n = 300 months
PV = 900,000
FV = 0
PMT = 10,833.85 (monthly)
Step 2:
What is the FV of this loan after 10 years
i = 1.1667 per month
n = 120 months (10 years)
PV = 900,000
PMT = -10,833.85 per month (negative because it is a payment)
FV = 813,509.85
Step 3: What is the annual return (compounded monthly!) desired by firm.
They want a 15.5% annual return (compounded annually). This implies that the firm desires a
14.5% annual return (compounded monthly). (Prove it to yourself)
Step 4: Compute the amount of points needed to get the desired return. Remember, points
affect the present value of the loan PAID to the borrower. The loan amount will still be 900,000
(and the above PMT and FV after 10 years will still hold. However, the borrower will not receive
900,000.
PMT = -10,833.85 per month (computed above)
FV (after 10 years) = -813,509.85 (computed above)
n = 120 months (10 year holding period)
i = 14.5/12 (the desired monthly return of the bank) = 1.2083 (per month)
PV = 876,953.29
The loan amount was 900,000. To get a yield of 14.5 annual rate (compounded monthly), we
need to set the dispersement to the borrower equal to 876,953.29.
To make this happen, we need to charge 2.56 points. (2.56% of 900,000 = 23,046.71; of the
900,000 loan, 23,046.71 will go to the bank as a fee and 876,953.29 will go to the borrower).
2.56 points will achieve the lender's desired yield of 14.5% annual (compounded monthly)
<<which is the same as a 15.5% annual return (compounded annually)>>
2.56 points!
Question (8 points total)
Which of the following statements are true? Given the portion of the statements below in italics,
assess whether the non-italics portion is true. (Circle ALL true statements below - 2 points each).
a.
Consider a standard constant payment mortgage (CPM) with no pre-payment penalties
and no points (and no other hidden lender fees).
Lender yields increase the shorter this loan is held by the borrower.
False: With a CPM mortgage with no other fees, the yield to the lender is NOT affected by
the length that the borrower holds the loan. This would not be the case if pre-payment
penalties or points were imposed on the loan. But, with no additional fees, the yield to the
lender is just the interest rate (regardless of the duration the loan is held). We did this in
class.
b.
Consider standard constant payment loan with no points and no pre-payment
penalties (and no other hidden fees).
A loan with a 6.00% interest rate (annual rate, compounded monthly) has a higher yield
than a loan with a 6.20% interest rate (annual rate, compounded annually).
False: We can solve this out. A loan with a 6.00% interest rate (annual rate, compounded
monthly) implies a 6.17% interest rate (annual rate, compounded annually).
To get this solve (1 + 6%/12)12.
annually) has a higher yield.
c.
So, the loan with a 6.20% loan (annual rate, compounded
Consider two standard constant payment loans (A and B) with no points and no prepayment penalties (and no other hidden fees). Assume both loans have similar expected
real yields. Lastly, assume mortgage A is issued in a world where expected inflation is
positive and mortgage B is issued in a world where expected inflation is zero.
Early in the life of the mortgages, real payments on mortgage A will exceed the real
payments on mortgage B.
True: This is inflation tilt. See notes for details.
d.
Consider the model of interest rates discussed in class. Assume lenders cannot perfectly
observe a borrower's expected probability of default.
Lender profits are a strictly increasing function of the interest rate they charge on the
loan.
False: Increasing interest rates can increase default probabilities. Riskier borrowers are
more likely to take loans with high interest rates (the safe borrowers could be driven away
from the market). Given that increasing i and also increase default probabilities, lender
profits can actually decline if interest rates are increased too much. See our credit rationing
lecture.
Quiz 6: Real Estate Economics and Finance
Name:
Section:
Afternoon
Evening
Question 1
You are currently considering purchasing a $200,000, 25 year variable rate mortgage with the
following characteristics:






The initial interest rate is set at 5% (annual rate, compounded monthly) and will remain
fixed for exactly one year. Interest rates will then adjust every year thereafter.
The interest rate, after the initial period expires, will be linked to the one-year U.S.
treasury.
The composite interest rate on this mortgage (annual rate, compounded
monthly) will be the sum of the U.S. treasury rate plus a 1.5 percentage point margin.
The interest rate can only increase by a maximum of 2 percentage points a year and 5
percentage points over the life of the mortgage.
There is NO negative amortization.
Payments are made monthly.
The lender will charge 2 points at the time of closing. No other fees are associated with
this mortgage.
Currently, you observe the following term structure for U.S. treasuries:
One year U.S. treasury, starting today:
Two year U.S. treasury, starting today:
Three year U.S. treasury, starting today:
Four year U.S. treasury, starting today:
Five year U.S. treasury, starting today:
3.50%
4.25%
4.40%
5.50%
6.10%
a. In expectation, what interest rate would you be paying on this mortgage during years 1 –
5? (put answers in the box). << 5 points>>.

Year 1 is easy: The teaser rate = 5%
Expected interest rates
with this mortgage
One year rate starting in year 2:
(1 + 0.0425)2/(1.0350) - 1 = 5.01%
Year 1:

5.00%
Year 2 mortgage rate:
5.01% + 1.5% margin = 6.51%
One year rate starting in year 3:
Year 2:
Year 3:
Year 4:
Year 5:
6.51%
6.20%
8.20% (capped rate)
10.00% (capped rate)
(1+0.044)3/(1+0.0425)2 – 1 = 4.7%

Year 3 mortgage rate:
4.7% + 1.5% margin = 6.2%
One year rate starting in year 4 = 8.87%

Year 4 mortgage rate (uncapped) = 10.37%.

Year 4 mortgage rate (capped) = 8.2% (at most, interest rate can increase by 2%
over year 3 interest rate).
One year interest rate starting in year 5: 8.54%

Year 5 mortgage rate (uncapped):

Year 5 mortgage rate (capped): 10.00% (At most, the interest rate can increase
by 5 percentage points over life of the loan. That implies the maximum interest
rate is 10%)
10.04%
b. What would payments be on this mortgage in year 1 AND in year 2? <<put answers in
box below>> (5 points)
PMT (year 1) = Calc(PV = 200,000; n = 300; i = 5/12; FV = 0 ) = 1,169.18
Before computing payment in year 2, we need to compute how much loan would be left
after year 1 (i.e, after 12 months).
FV (at end of 1 year) = Calc (PV = 200,000; n = 12; i = 5/12 ; PMT = -1,169.18) = 195,876.19
Note: We computing payment in year 2, we basically ask our self how much would our
payment be at the year 2 interest rate if we took out a loan for the FV outstanding at the
end of year 1 for 24 years (24 years is the amount of years left on the loan).
PMT (year 2) = Calc(195,876.19; n = 288; i = 6.51/12; FV = 0) = 1,345.98
Year 1 payments:
1,169.18
Year 2 payments: 1,345.98
c. If you pre-paid your mortgage after one year, what would be your yield on this mortgage?
<<put answer in box below>> (4 points).
Yield = [PV = 196,000; n = 12; PMT = -1,169.18; FV = -195,876.19] = 0.5914% per month =
7.0971% annual rate, compounded monthly.
Note: You solved for the PMT and FV in part b. The only difficult part of this problem is
realizing that you need to reduce the PV by the points paid!
Question 2
You are currently planning to purchase a $600,000 condo in Lincoln Park. Your lender has
offered you the following two loan options. Loan 1 is a 30 year fixed rate mortgage (amortized
over 30 years) with an 80% LTV and an interest rate of 6%. Loan 2 is a 30 year fixed rate
mortgage (amortized over 30 years) with a 90% LTV and an interest rate of 6.5%. Both interest
rates are annual rates, compounded monthly and payments are made monthly. Neither loan has
any other additional fees or penalties. (Because these loans are Jumbo, we do not need to think
about PMI – the higher interest rate incorporates all additional risk of the high LTV loan).
a.
In terms of the opportunity cost of the additional down payment associated with loan 1,
what should be your decision rule for taking loan 1 over loan 2? Assume you hold the loan for all
30 years. (3 points – put answer in the box).
Steps: Compute the loan payments on both loans.
Loan 1:
Loan 2:
PMT = Calc[PV = 480,000; n = 360 ; FV = 0 ; i = 6/12] = 2,877.84
PMT = Calc[PV = 540,000; n = 360 ; FV = 0 ; i = 6.5/12] = 3,413.17
Compute incremental borrowing cost.
If you take loan 1, you must pay an additional $60,000 in down payment. In our notation,
PV = -60,000. What is the benefit of doing this? We get lower mortgage payments. Again,
in our notation, PMT = 535.33. The question then becomes, should be pay $60,000 extra in
down payment to lower our monthly payments by $535.33 per month for the next 30 years.
Yield = Calc[PV = -60,000; n = 360; PMT = 535.33 ; FV = 0] = 0.8498% per month =
10.20% annual rate, compounded monthly.
If the opportunity costs of the additional down payment is greater than 10.20% (annual
rate, compounded monthly), you should take loan option 2 (low down payment). Otherwise,
you should use your funds to buy down the mortgage and lock in the lower interest rate.
b.
How would your answer to the above question change if you only were planning to hold
the loan for 5 years? (3 points – put answer in box)
The problem really wouldn’t change that much. All you would need to know is the FV
value of both loan 1 and loan 2 after 5 years.
Loan 1: FV (after 5 years) = Calc[PV = 480,000; n = 60 ; PMT = -2,877.84 ; i = 6/12] =
446,661.09.
Loan 2: FV (after 5 years) = Calc[PV = 540,000; n = 60 ; PMT = -3,413.17 ; i = 6.5/12] =
505,499.09
Incremental borrowing cost formula (in calculator notation) would be exactly the same as
above except n would change from 360 to 60 (five years) and FV would change from 0 to
58,838.00 (-446,661.09 – (-505,499.09)).
Yield = Calc[PV = -60,000; n = 60; PMT = 535.33 ; FV = 58,838.00] = 0.8675% per month =
10.41% annual rate, compounded monthly.
Quiz 7
Real Estate Economics: Winter 2004
Name:
Section:
Afternoon
Evening
Problem 1 (only 1 problem today)
Exactly 5 years ago, you purchased a condo that was worth $600,000. You put 5% down and
financed the rest with a 30 year fixed rate mortgage. The lender charged you an interest rate of
5% (annual rate, compounded monthly). The loan required monthly payments and was fully
amortized over the 30 years. For simplicity, assume no PMI or no other fees (points) associated
with this mortgage and assume that there are no tax benefits to owning the mortgage (i.e., no
benefits from the mortgage interest deduction).
A)
What would be the monthly payment on this mortgage?
PMT = Calc[PV = 570,000 ; n = 360 ; FV = 0 ; i = 5/12] = 3,059.88
B)
What is the current mortgage balance on this loan?
FV = Calc[PV = 570,000 ; n = 60 ; PMT = -3,059.88; i = 5/12] = 523,423.77
Continue the problem from above. Currently, interest rates on a 25 year loan are 4.9% (annual
rate, compounded monthly). This 25 year loan will be fully amortized over the 25 years and has
monthly payments. Like the loan above, this loan has no other fees or benefits associated with it.
The total cost to refinance your original loan (into this new loan) would be $2,200.
C)
Assuming that you did NOT include the refinancing costs as part of your new mortgage
(i.e., you paid the fees upfront from your personal funds), what would be the payment on your
new mortgage?
PMT = Calc[PV = 523,423.77 ; n = 300 ; FV = 0 ; i = 4.9/12] = 3,029.46
D)
Suppose you were only planning to hold this new mortgage for exactly 5 additional years.
What would be the remaining principle on this new mortgage after 5 additional years?
FV = Calc[PV = 523,423.77 ; n = 60 ; i = 4.9/12 ; PMT = -3,029.46] = 462,906.62
E)
Assuming that you were only planning to hold this property for 5 additional years, what
is your decision rule with respect to refinancing this mortgage? (i.e., compute the yield on
refinancing into this new mortgage and holding it for exactly for 5 years; then tell me your
decision rule using the yield you computed!)
If i get new mortgage, I will save $30.42 a month in mortgage payments (Part A – Part C),
for the next 60 months (i.e., 5 years – i am only planning to hold this property mortgage for
5 years!).
If I pay off the new mortgage after 5 more years, I would have to repay 462,906.62 (part D
above). If I kept the old mortgage for 5 more years (i.e., 10 years into the original
mortgage), I would have to repay: 463,649.77 (FV = Calc[PV = 570,000 ; n = 120 ; PMT = 3,059.88; i = 5/12]).
So, if I refinance, I would have to pay off less after 5 additional years. The difference in
future values would be 743.15 (463,649.77 - 462,906.62).
So, the yield is determined by deciding whether it is worth me paying $2,200 now to reduce
my mortgage payments by $30.42 a month for the next 60 months, realizing that I will have
to pay $743.15 less when I repay the mortgages in 5 additional years.
Yield = Calc[PV = -2,200; PMT = 30.42; n = 60; FV = 743.15] = 0.4057/month or 4.87%
(annual rate, compounded monthly).
If you costs of funds are less than 4.87%, then you should refinance.
Instead of using your own funds to pay the refinancing costs, suppose you included the
refinancing fees as part of your new mortgage (i.e., the new mortgage balance would be the old
mortgage balance you are paying off plus the refinancing costs).
F)
Assuming that you include the refinancing costs as part of your new mortgage, what
would be the payment on your new mortgage?
New loan amount would equal 523,423.77 (part B) + 2,200 (refinancing cost)
PMT = Calc[PV = 525,623.77 ; n = 300 ; FV = 0 ; i = 4.9/12] = 3,042.20
G)
Suppose you were only planning to hold this new mortgage for exactly 5 additional years.
What would be the remaining principle on this new mortgage after 5 additional years?
FV = Calc[PV = 525,623.77 ; n = 60 ; i = 4.9/12; PMT = -3,042.20] = 464,852.26
H)
Assuming that you were only planning to hold this new mortgage for 5 additional years,
what is your decision rule with respect to refinancing this mortgage? (i.e., would you refinance?)
Note: The FV after 5 years of the new loan is 464,852.26 (Part G) which is higher than
463,649.77 (which is the future value of the old loan holding it 5 more years – i.e., 10 years
in total – we computed this above).
So, the question we ask our selves now is whether it is worth it to refinance now and lower
our mortgage payments by $17.68 a month for the next 60 months realizing that we will
have to pay $1,202.49 extra when we repay these loans in 5 more years?
We compute the yield on this: Yield = Calc[PV = 0; PMT = 17.68; n = 60 ; FV = -1,202.49]
= 0.4171/month or 5% per year.
Basically, if you carried this loan out for all 25 years, you would definitely refinance. Given
that you have to pay back a higher amount after 5 years, the choice is not as clear. It will
depend on how much that $1200 is worth to you five years from now. If the opportunity
cost of that future $1,202.49 has an opportunity cost of less than 5%, you should definitely
refinance today.
Quiz 8
Introduction to Real Estate Economics and Finance
Name:
Section:
Afternoon
Evening
All of these came directly from the notes. I would go back and review if you got stuck on
something.
1.
In equilibrium, long run real property prices will always grow at the rate of real
construction cost changes.
False – this is only true if there are no capacity constraints. But, if there are capacity
constraints (like New England currently) real property prices can grow faster than
the cost of construction (even in the long run). There is no room for supply to adjust!
2.
All else equal, metropolitan areas with both more uncertainty with respect to future NOI
and with higher expected NOI growth (i.e., high mean/high variance) will have higher cap
rates than other metropolitan areas.
False – cap rate is r + α – g. If α is high, cap rate will increase. If g is high, cap rate will fall.
High risk, low mean areas will have the highest cap rates!
3.
Assuming personal tax rates are zero, increases in expected inflation rates will increase the
user cost of owning a home.
False – the effect of inflation is on nominal interest rates. Nominal interest rates are
important given the mortgage interest deduction. If tax rates are zero, there is no
affect of nominal interest rates on the user cost. Actually, increases in inflation could
actually raise user cost (due to inflation tilt – this only occurs if liquidity constraints
are important). Households are less likely to buy homes when nominal interest rates
are high.
4.
If housing markets are efficient, we would expect to see a large decline in residential
property prices when the baby boom population retires.
False – this expectation should already be incorporated into supply decisions. Supply will
be lower as baby boomers get ready to retire (if housing markets are efficient). If they
are inefficient, then this shock could be a surprise to housing markets.
5.
All else equal, risk premiums on 30 year mortgages (ρ) will likely be higher when 30 year
Treasury rates are historically low.
False – see the notes. Nominal interest rates tend to mean revert. So, periods of historically
low interest rates are associated with the expectation of higher future interest rates. If
interest rates are expected to rise, the refinancing risk is lower. So, as a result, the
option is worth less to borrowers (ρ will be lower).
6.
Part of the reason that residential property prices have remained so high during the last
recession is that the supply of new residential properties fell sharply during the recession.
False – see notes (i had a figure). New construction has remained strong during this
recession!
7.
Increases in vacancy rates are a good predictor of declines in commercial real estate prices
in the near term.
True – owners will usually let a place go vacant before they lower rent. We talked about
this in class. Vacancies are a leading indicator or rental price declines.
8.
Given lender profit maximization, increases in borrowing rates will always lead to
increases in lender profits.
False – this was an old quiz question. If lenders raise rates, the borrower pool can become
worse (and default rates can increase).
9.
With respect to variable rate mortgages, the longer the teaser rate is fixed the less
prepayment risk born by the lender.
False – it is the opposite – the more a variable rate looks like a fixed rate mortgage, the
more prepayment risk is borne by the lender!
10.
Mortgage backed bonds are considered debt securities while collateralized mortgage
obligations are considered equity securities.
False – see notes. Both are considered debt! The pool originators are equity holders. The
securities themselves are debt!