Erik’s Final
Real Estate Economics and Finance
Winter 2004
Name (print):
_______________________________________
Name (signature):
_______________________________________
Which mail folder would you like your test to be put in? (Circle one - if you do not circle
any - I will hold the exam in my office for exactly one month. If you do not pick them up by
then, I will dispose of them at that time.)
Campus
Evening
Weekend

As always, that honor code rules are in effect. You know the routine. All the usual
disclaimers apply. By signing above, you are pledging to uphold the GSB’s honor code.

You have 2 hour and 45 minutes for the exam.

Use calculator notation on all problems for the exam (it will help with partial credit).

For discussion problems, explain - but do not be wordy - the more you say, the greater
the chance that you can say something wrong - answer the question and move on.

You are allowed:
One Piece of Paper - Handwritten - Not Photo Copied - Both Sides
Use of a Financial Calculator

Marina and I have lots of grading to do (with this class and the macro class, I am not
sure how quickly we will have them back to you – it could be up to 12 days). I will send
out a message when they are in your mail folders.

Good Luck!
Part 1: True/False/Uncertain (5 questions @ 6 points each; 30 points total).
For each of the following parts, discuss whether the non-italic part of the problem is True, False
or Uncertain. In this part of the exam, we will give NO credit if you just write true (even if the
answer is actually true). 100% of your grade will come from your explanation. Your explanation
NEEDS to be brief. We will take points off for excessive verbiage. Most of problems can be
answered in 3-5 sentences!
A.
During 1998 and 2002, the interest rate on commercial property mortgages has
fallen sharply. Yet, commercial property cap rates across the U.S. have increased
slightly during that same time period. I received a call from a reporter from a major
business publication. He argued that these two facts are inconsistent. He said that a
decline in commercial property mortgage rates should cause cap rates to fall. Assess
whether the following statement is true/false/uncertain.
A decline in commercial property mortgage rates will be associated with a fall in the
cap rate.
False/Uncertain. The cap rate is defined as r + α – g. The way we defined the cap rate is as
a function of the real interest rate r. However, cap rates are often expressed as nominal
variables. But, even if nominal interest rates fell, the cap rate would increase if α increased
or if g fell. Recently, α increased (as risk in the economy increased due to the recession) and
g (the growth rate in expected future NOI) fell. So, given that α and g were changing, it is
not necessary that a fall in discount rates (r) would call the cap rate to fall.
B.
In order to avoid paying corporate taxes, REITs are required by law to payout 9095% of their taxable income to their investors in the form of dividends. Given this
information, assess whether the following statement is true/false/uncertain.
Given they have to payout almost all of their taxable income, REITs are forced to
grow either by issuing new debt or issuing new equity.
So false. While REITs do have to pay out their taxable income, their cash flow is always
much larger given their large depreciation allowances. As a result, REITs have the option
to grow using their own cash flow.
1
C.
There has been lots of talk during the last year suggesting that residential properties
are experiencing a bubble. Some academics and policy makers point to the fact that
housing prices have risen at unprecedented rates during the last 4 years. With this
in mind, discuss whether the following statement is true/false/uncertain.
A rapid rise in house prices is a likely signal that residential property markets are
experiencing a bubble.
False/Uncertain. While a rapid rise in house prices is consistent with a bubble, it is not
conclusive. Prices could increase in the short run if there is a large demand shock given
that supply is fixed (think San Francisco in the late 1990s).
D.
The secondary market has been innovating on ways for mortgage investors to better
manage different types of risks. IO and PO strips (chapter 18) have been a recent
innovation to manage interest rate risk. With this in mind, discuss whether the
following statement is true/false/uncertain.
PO strips increase in value when interest rates fall.
True. See text for rationale. (Low interest rates imply more refinancing so the investor gets
more cash flow in earlier periods—that is valuable to the investor).
E.
You wish to borrow $500,000 to finance the purchase of a residential property. You
are offered two mortgage options. Option 1 is a 30 year fixed rate mortgage
(amortized fully over 30 years, with monthly payments) with 2 points and an interest
rate of 5.5% (annual rate, compounded monthly). Option 2 is a 30 year fixed rate
mortgage (amortized fully over 30 years, with monthly payments) with 0 points an
interest rate of 5.65% (annual rate, compounded monthly). Neither loan has any
additional fees associated with it. Given the above information, assess whether the
following statement is true/false/uncertain.
You would prefer option 1 over option 2 if you planned to hold the mortgage for the
full 30 years.
False. In class, I told you that if you hold a loan longer, you may prefer a loan with no
points. However, given the set up of this problem, you would ALWAYS prefer the loan
with no points. Do this out! (i.e., solve for the yield for both loans if you held the loan to
term).
2
Part II: Short Answer Question
irrelevant for 2005
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Problem 1: Yields with Prepayment Penalties (12 points)
Suppose that Bank X has just issued a fixed rate mortgage for $500,000 with an interest rate of
10% 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 10.75% annual,
compounded annually.
What pre-payment penalty should Bank X place on this loan to earn the required yield if they
expect the loan to be held for exactly 10 years? (Note: pre-payment penalties need not be whole
numbers. For example, 1.99% pre-payment penalty is an allowable answer). PUT ANSWER IN
BOX BELOW!!!!!
Step 1: Compute payment on loan with no penalties:
PMT = Calc[PV = 500,000; i = 10/12 ; n = 300 ; FV = 0] = -4,543.50
Step 2: Compute FV after 10 years:
FV = Calc[PV = 500,000 ; i = 10/12 ; n = 120 ; PMT = -4,543.50] = -422,807.58
Step 3: Compute desired yield as an annual yield, compounded monthly
(1+i/12)12 = 1+0.1075 ---- solve for i; i = 10.25
Step 4: Compute the desired future value that would yield the desired yield of 10.25%
(annual rate, compounded monthly)
FV = Calc[PV = 500,000 ; i = 10.25/12 ; n = 120 ; PMT = -4,543.50] = -443,338.85
Step 5: Compute the pre-payment penalty as a rate:
(443,338.85 – 422,807.58)/422,807.58 = 4.86%
Note: Prepayment penalties are formulated as a fraction of the future value (SEE NOTES
FOR A DISCUSSION). Not all of you got that part – it is ok. We still gave credit.
For instance, some of you computed the pre-payment penalty as a dollar amount (that is ok
as well): The dollar amount would just be 20,531.27 (443,338.85 – 422,807.58).
Also, some of you took the rate as being a fraction of the INITIAL LOAN BALANCE
(instead of the amount that you will prepay). We also gave credit for that. If you had a
penalty of 20,531.27, we didn't care what you divided it by.
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Problem 2: Optimal Refinancing (14 points)
5 years ago your partnership purchased a commercial property. To partially finance that property,
the partnership took out a 20-year fixed rate mortgage for $14 million. This loan had an interest
rate of 6.75% (annual rate, compounded semi-annually), semi-annual payments and was to be
fully amortized over the 20 years. The mortgage had no points or other fees associated with it.
However, there is a $1 million prepayment penalty embedded in the mortgage (i.e., if you prepay
the mortgage at any time, your partnership must bear a $1 million penalty).
Interest rates have been declining over the past 5 years. Your partnership has put you in charge
of the refinancing decision. Today (at the end of year 5), what is the maximum interest rate that
you would have to be offered in order to make refinancing a profitable decision for your
partnership? In other words, at what interest rate would refinancing become a positive NPV
decision?
Additional assumptions:
 All the loans you are considering are 15 year mortgages with semi-annual payments
(fully amortized over the remaining 15 years).
 All the loans you are considering have interest rates which are annual rates, compounded
semi-annually.
 The refinancing penalty will be rolled over into the new loan (i.e., it will not come out of
your pocket – you will add the penalty to your new loan balance).
 There are no other costs associated with refinancing. This implies that there are no out of
pocket costs associated with refinancing.
 You are expecting to hold the new mortgage for the remaining 15 years.
Put your answer in the box below (the answer should be an interest rate -- annual rate,
compounded semi-annually!). (HINT: Think this problem through. It may seem hard, but it is
rather easy. Like most problems, it can be done in 3 or 4 steps). <<I would think about it on the
scrap paper at the end and then transpose below>>.
Step 1: Compute Payment on original loan:
PMT = Calc[PV = 14 million, i = 6.75/2; n = 40 ; FV = 0] = -642,928.26
Step 2: Compute FV after 5 years
FV = Calc[PV = 14 million ; i = 6.75/2 ; n = 10 ; PMT = -642,928.26] = 12,102,150.84
Step 3: Compute yield at which you would be indifferent between paying the refinancing
cost and keeping the same payment stream. In reality, you would like to pay the
refinancing cost and get a LOWER refinancing stream. That is why it is the MAXIMUM
interest rate you would pay.
Yield = Calc[PV = 13,012,150.84; FV = 0 ; n = 30; PMT = -642,928.26] =
2.755%/semi-annually = 5.51% annual (compounded semi-annually)
Note: When computing yield, the PV of new loan will be FV of old loan ($12,102,150.84)
plus $1 million prepayment penalty.
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Note: Some of you tried to do an incremental borrowing cost analysis on this problem. It
was not necessary given that there was no PV and no FV. No money came out of your
pocket. No additional money had to be pre-paid at the end. All you are trying to do is
minimize payments.
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Problem 3: CMO Payouts in Year 1 (10 points)
Your firm has decided to issue $95 million of CMO securities based on a pool of $100 million of
mortgages. Each mortgage in the pool has a term and amortization period of 10 years. The
interest rate on all mortgages in the pool is 8% annual rate, compounded annually. Mortgage
payments are made annually.
Your firm has decided to issue the following CMO tranches on this pool of mortgages:
Tranche A
Tranche B
Tranche Z
35 million of bonds
15 million of bonds
45 million of bonds
@
@
@
5.75% interest rate
7.00% interest rate
8.00% interest rate
All interest rates are annual rates, compounded annually. The bonds make 1 payment per year (at
the end of the year).
This CMO has a payout structure exactly the same as we went over in class (and exactly the same
as discussed with the homework). For simplicity, we will assume that there will be no
prepayments into this pool.
In year 1, what is the total payout to Tranche A investors, Tranche B investors, Tranche Z
investors, and the payout to the pool originator (i.e., your firm)? What is the balance of the debt
owed to Z at the end of period 1? What is the balance of debt owed to the A investor at the end of
period 1? Make sure you show your work (and/or discuss your intuition).
This problem was very easy (especially if you did the homework). This was the easiest
problem on the test.
A gets there own interest and (Z's interest and the entire pool's principal) – the latter two
reduce A's initial principal.
Step 1:
Year 1 Analysis for the ENTIRE mortgage pool
Outstanding Debt
100 million


PMT
14,902,948.87
Interest
8 million
Principal
6,902,948.97
PMT = Calc[100 million, i = 8 ; n = 10; FV = 0]
Interest = 8% * 100 million
Interest owed to each tranche:
A:
B:
Z:
$2,012,500
$1,050,000
$3,600,000
Payments to each investor:
A:
B:
Z
Equity:
12,515,449 = (2,012,500 + 3,600,000 + 6,902,948.97)
1,050,000
0
8 million – 2,012,500 – 1,050,000 – 3,600,000 =
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Notes: A gets all their interest, all Z’s interest, and all the pool’s principal (in
period 1).
Equity holder gets: All residual interest (interest to the pool – less interest it pays
out).
Z gets nothing (until A and B are fully paid off)
B just gets their interest
A’s loan balance at the end of year 1 will be: 35 million – 3,600,000 – 6,902,948.97 =
24,497,052 (you subtract off the interest to Z and the principal of the pool).
Z’s loan balance at the end of year 1 will be 48,600,00 (45 million + deferred interest of
3,600,000).
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Problem 4: Lease vs. Buy Analysis (12 points)
Your firm is considering expanding its production facilities by opening a new plant in Western
Michigan. After some preliminary analysis, you find that there is a vacant facility that is exactly
what your firm was looking for. You have two options available with respect to your expansion.
Option 1:
Option 2:
Buy this existing production facility. The current sale price is $40 million.
Get a 10 year lease on this property for $1 million a year (assume constant rent
over the life of the lease).
Regardless of which option is chosen, the following information holds.




$3 million is needed to retool the old production facility to meet your firm’s current
needs. This is a one time upfront cost needed before production can begin.
Annual sales from the production facility are estimated to be constant at $8 million per
year.
Other costs associated with the sale of production are estimated to be constant at $3.6
million per year (these costs include maintenance, insurance and property tax on the
property).
The tax rate on income from your business will be 34%. Assume a zero capital gains tax
on the sale of property.
If you purchase the property, you can get a 10 year interest rate only loan for up to 75% of the
purchase price for 6.9% (annual rate, compounded annually). Interest payments would be made
annually and the entire initial loan amount would have to be repaid at the end of year 10. Also,
you are allowed to take an annual depreciation allowance of $1 million if you own the property.
You and your firm have decided that you would borrow the full 75% if you purchased the
property. You also decided that you are going to expand your production facility. You and your
firm have a planning horizon of exactly 10 years. After 10 years, if you own the production
facility, you plan on selling the property for $45 million. When you sell the property, you have to
repay your interest only mortgage. For simplicity, assume that you plan on divesting yourself
totally of this project after 10 years.
a.
Given the property price and the first year NOI, what is the implied cap rate associated
with this property?
I gave credit for basically all answers here (i.e., everyone got a free 4 points if you said
something sensible).
My answer was:
NOI = Revenues – Cost (excluding interest and taxes)
Some of you had "after tax NOI" instead of above (basically, the cash flow to the equity
holder as opposed to the total project's cash flow). I gave credit to all sensibly defined NOI.
My answer was: NOI = 4.4 million. Cap rate = 11% = 4.4 million / 40 million.
Some of you had cap rate = 4.69% (1.8776 million / 40 million).
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b.
The only decision you and your firm have is whether you should buy the production
facility or rent it. Using our incremental borrowing cost analysis, what is the decision
rule for whether you should buy or rent the property (assuming you are divesting your
interest of the property in exactly 10 years)?
What is cash flow if buy property:
Revenues:
Costs:
8 million
- 3.6 million
NOI (my definition)
Less:
Less:
4.4 million
Interest on Loan 2.07 million
Depreciation
2.33 million
(before tax cash flow to equity holder)
1.00 million
(depreciation is deducted for tax purposes)
Taxable Income
Less:
Taxes
1.33 million
0.4522 million
After Tax Income
0.8776 million
Plus:
1.00 million
Depreciation
After Tax Cash Flow
So:
(30 million loan * 6.9%)
(taxable income * tax rate of 34%)
(add depreciation back in)
1.8776 million
If you buy the property:
PV
PMT
n
FV
= -13 million (3 million of upfront costs + 10 million out of pocket to buy
building)
= 1.8776 million (see above)
= 10
(have project for 10 years)
= 15 million
(sell building for 45 million and pay off 30 million loan)
10
What is cash flow from leasing:
Revenues:
Costs:
8 million
- 3.6 million
NOI (my definition)
Less:
Less:
Lease Payment
Depreciation
Taxable Income
Less:
Taxes
1.0 million
3.4 million
(before tax cash flow to equity holder)
0.00 million
(no depreciation expense for renters)
3.4 million
1.56 million
After Tax Income
So:
4.4 million
(taxable income * tax rate of 34%)
2.244 million
If you buy the property:
PV
PMT
n
FV
= -3 million (3 million of upfront costs)
= 2.244 million (see above)
= 10
(have project for 10 years)
= 0 million
(nothing has residual value of if lease)
Incremental analysis (subtract leasing from owning)
PV
PMT
n
FV
= -10 million (more out of pocket expenses upfront with owning)
= -0.3664 million (less cash flow each period if renting)
= 10
= 15 million (more cash flow from residual value from owning)
Solve for the yield and get approximately 1.1% (annual rate, compounded annually).
This says that if the opportunity cost of your funds is LESS than 1.1% , you should buy.
Otherwise, you should lease. Given that in the real world, the opportunity cost of funds is usually
greater than 1.1%, you would likely always lease.
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Problem 5: Interest Rates, Leverage and Default (14 points)
Suppose that you are considering buying a piece of property worth $265,000 today. You plan on
holding this property for exactly one period. Below is the probability distribution for expected
property prices next period – these expectations are held by both you and the lender:
Probability
5%
40%
40%
10%
5%
Price
$340,000
$300,000
$285,000
$220,000
$190,000
You are considering financing this property with a one-period interest only loan. In other words,
in the next period, you will have to pay back all the loan principal plus interest on that principal.
For simplicity, let’s assume expected inflation is zero. This assumption implies that no time
discounting is necessary (a dollar today is worth a dollar tomorrow).
The lender is currently offering you only two loan options (both interest rates are annual rates,
compounded annually).
Loan 1 has a 75% LTV with an interest rate of 8.01%.
Loan 2 has an 85% LTV with an interest rate of 15.31%
Next period, you have the option to default if there is there is negative equity in your house (this
is your only reason for default). In other words, if House Value > Balance + Interest Owed.
For simplicity, let’s assume that there are no costs of default to the borrower. Additionally, let’s
assume that the lender will get 50% of the house value if the borrower defaults. (In other words, if
the house is worth $265,000 and the borrower defaults, the lender is expected to recoup $132,500
after they liquidate the house). Lastly, let's assume that you have already made the decision to
buy the house – your only decision is what loan to get.
A.
If you choose Loan 1, what is the expected probability that you will default?
You will default when what you owe next period is LARGER than your house price next
period. What will you owe next period?


Today, you borrow 75% of 265,000 = 198,750
Tomorrow, you will have to payback: 214,669.90 (198,750 * 1.0801). In
other words, if house price < than $214,669.90 you will default!
This implies that you will default when house prices are $190,000. In all other instances,
you are better off not defaulting. Your probability of default is then: 5%
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Problem 5 (continued)
B.
If you choose Loan 1, what is your expected return on this project (from today's
perspective, realizing that you may default tomorrow)?
I did this by computing my “equity” at each future house price:
Probability
5%
40%
40%
10%
5%
Price
340,000
300,000
285,000
220,000
190,000
Interest + Principal
214,669.90
214,669.90
214,669.90
214,669.90
214,669.90
Total
Equity
125,330.12
85,330.12
70,330.12
5,330.12
0
Value
6,266.51
34,132.05
28,132.05
533.01
0
69,063.62
Your expected value of this project next period is 69,063.62
You invest 66,250 today.
Your additional profit is 2,813.62.
Your return is 2,813.62/69,063.62 = 4.07%
Note: Value = Equity * probability
Note: When prices are 190,000, equity equals zero (because you will default)
C.
If you choose Loan 1, what is the expected return to the lender (the lender’s actual
yield)? Again, answer this from today's perspective realizing that the lender knows
that you may default tomorrow.
Return to the lender:
This formula comes from class:
(1+r)B = (1+i) B(1 – π) + π * ½ * H(default)
Where π is the probability of default and H(default) is the residual value of the property in
the default state. Notice, given the problem set up, you get ½ the house value in default. B
is the amount of the loan. r is their actual return and i is the interest rate they charge.
Your job is to solve for r. You have i, π, B and H(default). Plug them into the equations
and you get:
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r = 5% (the lenders effective return is 5%)
D.
If you choose Loan 2, what is your expected probability of default?
You would borrow 225,250 if you choose loan option 2. Next period you will owe 259,736
(1.1531 * 225,250)
In this case, you will default anytime your house value is less than 259,736. Given the
information above, that implies you will default 15% of the time.
E.
If you choose Loan 2, what is your expected return on this project (from today's
perspective)?
I did this by computing my “equity” at each future house price:
Probability
5%
40%
40%
10%
5%
Price
340,000
300,000
285,000
220,000
190,000
Interest + Principal
259,735.78
259,735.78
259,735.78
259,735.78
259,735.78
Total
Equity
80,264.23
40,264.23
25,264.23
0
0
Value
4,013.21
16,105.69
10,105.69
0
0
30,224.59
Note: Your equity value is 0 in two states (because you will default when house prices are
220,000 and 190,000).
Note: Value = Equity * probability
Your expected value of this project next period is 30,224.59
You invest 39,750 today.
Your additional profit is -9,525.41. (You don't even get your original investment back!).
Your return is -9,525.41/39,750 = -23.96% (your return is negative under this scenario!)
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F.
If you choose Loan 2, what is the expected return to the lender (the lender’s actual
yield)? Again, from today's perspective given that there is a chance of default
tomorrow.
This is a little more complex than the formula in part C above (only because there are two
house prices that the lender can get if the borrower defaults.
Let's define π1 as the probability that the house price will be 190,000. Let's call 190,000, H1.
Let's define π2 as the probability that the house price will be 220,000. Let's call 220,000, H2.
The lender's return is now defined as:
(1+r)B = (1+i) B(1 – π1 – π2 ) + π1* ½ * H1 + π2* ½ * H2
where i is the interest rate the lender chargers us if we borrow B (15.29%), B is the amount
borrowed (225,250) and the π's and H's are defined as above.
Like before, you will solve this equation for r (their expected return on lending B dollars
knowing that default is possible).
Solving this out, you will get 5% (some of you got 5.01% or 4.99%, they are all 5% in my
eyes. The difference is just due to rounding errors).
G.
Which loan would you prefer? Which loan would the lender prefer you choose?
Borrower's will prefer Loan 1 (gets a higher yield). Lenders are indifferent between the two
loans, they are both yielding 5%. The lender has undone the incentive to leverage by
raising the interest rate to compensate themselves for the additional risk. Given this
stream, borrowers are better going with less leverage!
This was a cool problem (it was my favorite on the exam). You had to think your way
through it. Basically, it showed us how lenders can set interest rates given expectations of
future house prices. The can choose an interest rate to compensate themselves for the
default premium! Even though default probabilities differed between the two loans, the
lender still earned the same return! They were able to protect themselves from the default
probabilities by choosing the appropriate interest rate.
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TEST GRADE BREAKDOWN
Part I:
(True/False/Uncertain: 30 points total)
____________
Part II:
(Discussion Question: 8 points total)
____________
Part III:
(Problem 1: 12 points total)
____________
Part IV:
(Problem 2: 14 points total)
____________
Part V:
(Problem 3: 10 points total)
____________
Part VI:
(Problem 4: 12 points total)
____________
Part VII:
(Problem 5: 14 points total)
____________
Total (out of 100)
___________
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