Introduction to Real Estate
Finance and Investments
11.431/15.426
Walter N. Torous
Center for Real Estate
MIT
Fall 2023
1 / 26
Calculating Present Value
I
Consider an investment project that with certainty will pay
you $1 in one period of time.
I
What is the present value of this investment?
I
PV < $1. Why?
2 / 26
I
Put PV into an “equivalent” security that earns a return of r
so that you have $1 in one period of time:
PV × (1 + r ) = $1.
We can now calculate PV :
PV =
$1
1+r
where
r ≡ opportunity cost of capital.
⇒ Given a well-functioning capital market, by making this
investment you forgo the opportunity of investing in
equivalent securities.
3 / 26
Net Present Value
I
I
But there are costs to acquiring an investment project.
Let’s define:
C0 (< 0) ≡ cost today
C1 ≡ cash inflow one period from today
C1
PV =
.
1+r
I
The KEY point to note is that there are two ways to receive
C1 in one period of time:
I
I
I
through the project – at a cost of C0 today
or through the capital market – at a cost of PV today.
Comparing these costs today gives the project’s Net Present
Value (NPV):
NPV = C0 +
C1
.
1+r
4 / 26
Opportunity Cost of Capital - Example #1
I
Project:
I
I
Invest $1000 now
Receive certain $1100 after one year
I
Assume investors can obtain 5% safe return.
I
Decision:
I
I
Invest because 10% project return exceeds 5% opportunity
cost;
Invest because present value of $1100 next year exceeds $1000
now:
PRESENT VALUE =
$1100
= $1048
1.05
NET PRESENT VALUE = −$1000 + $1048 = $48
Firm value increased by $48. Why?
5 / 26
Opportunity Cost of Capital - Example #2
I
Project:
I
I
Invest $1000 now
Receive uncertain $1100 after one year
I
Assume investors can buy “equally” risky securities with
forecasted return of 15%.
I
Decision:
I
I
Don’t invest because 10% project return is less 15%
opportunity cost;
Don’t invest because present value of $1100 next year is less
than $1000 now:
PRESENT VALUE =
$1100
= $957
1.15
NET PRESENT VALUE = −$1000 + $957 = −$43
Firm value decreased by $43. Why?
6 / 26
Calculating Returns
I
We can calculate the return to a one period project by the
following formula:
Return =
Profit
.
Investment
Example #1
Return =
$1100 − $1000
= 10% > 5%
$1000
Example #2
Return =
$1100 − $1000
= 10% < 15%
$1000
7 / 26
Two Rules
I
These simple examples motivate two fundamental rules for
investment decision making:
I
I
NPV rule: Invest in positive NPV projects.
IRR (Internal Rate of Return) Rule: Invest in projects offering
returns in excess of the opportunity cost of capital.
8 / 26
Multiple Cash Flows
X $1 after 1 period at r1 =5% ; PV1 =
$1
=$.952
(1.05)1
$1
=$.873
(1.07)2
X $1 after 2 periods at r2 =7% ; PV2 =
∴ $1 after 1 period and $1 after 2 periods
PV1+2 =
$1
(1.05)1
+
$1
(1.07)2
=$1.826
Why?
You can add dollars today with dollars today!
9 / 26
Calculating Present Value: Multiple Cash Flows
I
General Discounted Cash Flow Formula
PV =
I
C1
C2
C3
+
+
+ ...
2
(1 + r1 ) (1 + r2 )
(1 + r3 )3
How to Simplify:
I
Assume r1 = r2 = r3 = . . . ≡ r
⇒ PV =
C2
C3
C1
+
+
+ ...
(1 + r ) (1 + r )2
(1 + r )3
I
Make simplifying assumptions about the cash flows C1 , C2 , . . ..
I
So simple that EXCEL built-in functions are not needed!
10 / 26
Perpetuity
I
A perpetuity is an investment which makes constant payments
(C1 = C2 = C3 = . . . ≡ C ) forever.
PV
=
1
PV
(1 + r )
=
C
C
C
+
+
+ ...
2
(1 + r ) (1 + r )
(1 + r )3
C
C
C
+
+
+ ...
2
3
(1 + r )
(1 + r )
(1 + r )4
⇒
PV −
1
PV
(1 + r )
=
PV
=
C
(1 + r )
C
r
11 / 26
Example: Valuing an Apartment Building
An apartment building has 100 identical units that rent at $1000
per month with building operating expenses paid by the landlord
(gross lease) equal to $500 per month. On average, there is a 5%
vacancy rate. You don’t expect rents and operating costs to
increase over time. The opportunity cost of capital is 12% per
year. How much is the property worth?
PV
=
$570, 000 $570, 000 $570, 000
+
+
+ ...
1.12
1.122
1.123
=
$570, 000
.12
= $4, 750, 000.
12 / 26
Growing Perpetuity
I
A growing perpetuity is an investment which makes payments
that grow at a constant rate g (< r ) forever.
PV
=
(1 + g)
PV
(1 + r )
=
C
C (1 + g) C (1 + g)2
+
+
+ ...
(1 + r )
(1 + r )2
(1 + r )3
C (1 + g) C (1 + g)2
+
+ ...
(1 + r )2
(1 + r )3
⇒
PV −
I
I
(1 + g)
PV
(1 + r )
=
PV
=
C
(1 + r )
C
r −g
What is the value of a growing perpetuity if g > r ?
Referred to as Gordon growth model in finance and real estate
applications.
13 / 26
Example: Valuing an Apartment Building (continued)
By how much does the value of the apartment building change if
you now assume that it’s Net Operating Income (NOI) will grow at
3% per year?
PV
=
$570, 000 $570, 000 × 1.03 $570, 000 × 1.032
+
+
+ ...
1.12
1.122
1.123
=
$570, 000
.12 − .03
= $6, 333, 333.
14 / 26
Multiples
In general
Value of Perpetuity =
Cash Flow
r −g
which can be re-written as
1
× Cash Flow
r −g
= Multiple × Cash Flow
1
where Multiple =
.
r −g
Value of Perpetuity =
I
I
Multiples are based on comparable transactions.
Multiples depend on the anticipated growth rate and riskiness
of cash flows:
I
I
As g ↑ → Multiple ↑
As r ↑ → Multiple ↓
15 / 26
Example: Ferrari’s 2015 IPO
I
Is Ferrari an auto manufacturer or a luxury brand?
Auto Manufacturer
BMW
Daimler
Tata Motors
Tesla
Toyota
Median
Luxury Brand
Burberry
LVMH
Richemont
Median
Enterprise Value/EBITDA
8.2
8.9
4.9
3298.3
11.0
8.9
Enterprise Value/EBITDA
10.4
12.8
14.5
12.8
16 / 26
Example (continued)
I
If Luxury Brand:
Ferrari EBITDA
Multiple
Enterprise Value
Ferrari Debt
Implied Equity Value
# of shares outstanding
Implied Share Price
I
e687 MM
12.8x
e8793.6 MM
e2300 MM
e6493.6 MM
189 MM
e34.36=$39.08
Epilogue
I
Ferrari’s IPO priced on October 20, 2015 at $52/share
I
‘RACE’ opened the next day on the NYSE at $60/share and
closed at $55/share
17 / 26
Example: Porsche’s 2022 IPO
Porsche Prospectus
Geographically, Porsche's revenue exposure is well diversified: the NAM
region accounts for about 27% of revenues, Europe accounts for 29% and
Asia is about 32%.
Valuation
Personally, I believe it is reasonable to value Porsche based on an annuity
that anchors on the carmaker's 2021 EBITDA annuity, with a 10% discount
rate and a 3% terminal growth rate. This would equal a $105.7 billion
enterprise value ($7.4 billion / (10% - 3%), and an EV/EBITDA multiple of
x14.4 respectively. For reference, Tesla is valued at a one-year forward
EV/EBITDA of x43. And Ferrari's EV/EBITDA valuation multiple is x24.
Accordingly, Porsche's IPO valuation of about x12.9 EV/EBITDA provides an
attractive risk/reward for investors. And as a consequence, I personally
believe that Porsche shares will surge on the IPO.
Interestingly, according to a recent Bloomberg report, Porsche shares in
the unregulated grey market were trading at as much as 17% above the
9/28/22
IPO top-range price estimate of EURSeeking
82.5. ThisAlpha
could indicate
that Porsche
shares could jump significantly in the first day of trading, as a larger retail
investor community could rush to buy shares in the highly popular
sportscar maker.
Risks
The valuation for Porsche's IPO, I argue, provides a reasonable risk/reward
18 / 26
Valuing an Apartment Building (continued)
I
When you assumed that rents and expenses would not grow
over time, a multiple of 8.33× was applied to the building’s
NOI.
PV = 8.33 × $570, 000 = $4, 750, 000
I
When you assumed that NOI would grow at 3% per year, a
multiple of 11.11× was applied to the building’s NOI.
PV = 11.11 × $570, 000 = $6, 333, 333
Why is a higher multiple being applied?
I
These multiples are related to the building’s cap rate.
19 / 26
I
Cap Rates
Capitalization or “cap” rates play an important role in
commercial real estate.
I
Cap rates are a way of quoting observed property prices in
relation to expected first year asset-level income:
cap rate =
I
I
NOI1
; building’s cash flow yield.
V
Analogous to dividend yield for common stock.
Rearranging:
V=
NOI1
cap rate
which is the “direct capitalization” method of valuation.
I
Corresponds to Gordon growth model where
cap rate ≡ r − g.
20 / 26
Valuing an Apartment Building (continued)
I
A cap rate = 12% was used when you assumed that rents and
expenses would not grow over time:
$4, 750, 000 =
I
$570, 000
.12
A cap rate = 9% was used when you assumed that NOI would
grow at 3% per year.
$6, 333, 333 =
$570, 000
.09
Notice that a lower cap rate is associated with a more
valuable property.
21 / 26
Valuing an Apartment Building (continued)
I
Be careful - cap rates apply only to fully stabilized properties!
I
Assume the apartment building will be 50% occupied by the
end of year 1, 75% occupied by the end of year 2, and will
reach its “stabilized” occupancy rate of 95% by the end of
year 3 when NOI will grow at 3% per year thereafter.
I
PV
if you capped NOI at 50% occupancy, you are implicitly
assuming that the unrented space has no value and will never
be rented.
1
$570, 000
.50 × $600, 000 .75 × $600, 000
+
+
{
}
2
2
(1.12)
(1.12)
(1.12)
.09
= $5, 675, 489.
=
22 / 26
I
Here are recent cap rates for various property types:
Office
Retail
Industrial
Multifamily
I
Cap Rate
5.33%
5.84%
4.54%
4.55%
Multiple of NOI
18.76x
17.12x
22.03x
21.98x
Why do cap rates vary across these property types?
23 / 26
U.S. commercial real estate cap rates 2024 | Statista
https://www.statista.com/statistics/245008/us-commercial-property-cap-...
Cap rates vary over time:
Source: https://www.statista.com/statistics/245008/us-commercial-property-cap-rates/
Commercial real estate capitalization rates in the United States from 2012 to 2022 with a forecast until 2024, by property type
7%
6.5%
Cap rates
6%
5.5%
5%
4.5%
4%
3.5%
2012
2013
2014
2015
Retail
2016
Office
2017
2018
Industrial
2019
2020
2021
2022
2023* 2024*
Multifamily
Additional Information
© Statista 2023
24 / 26
Do changes in cap rates explain all of the changes in property
Empirical cap rates and market values:
value?
RCA Cap Rates & Price Index: New York City Apartment Properties
Cap Rates haven’t been the whole story of the 5X price run-up: 1/.081=12.3, 1/.047=21.3,
10
21.3/12.3=1.7. Rents (NOI) have been an even bigger part (in NYC): 5.0/1.7=2.9.
25 / 26
Let’s look at some numbers:
500−100
100
I
NYC apartments appreciated by
I
cap rates fell from 8.1% to 4.7%
I
I
an increase in value from
21.3−12.3
= 73%
12.3
1
.081
= 400%
= 12.3 to
1
.047
= 21.3 or
∴ an increase in rents (NOI) was an even bigger part of the
reason for the increase in NYC apartment values
26 / 26