RE I: Class Note #9
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Real Estate Investments I
(Business 33450)
Winter Quarter, 2023
Instructor: . Pagliari, Jr.
Key Take-Aways:
• Law of one price.
• Risk measures:
Volatility (σ ),
Correlation (ρx,y ),
Prob(target return), and
Total v. systematic risk.
• Risk diversification, via # assets and/or time periods.
• Mean reversion; distance from trend?
• Types of risk forecasting:
Single-point estimates,
Sensitivity analysis,
Scenario forecasting, and
Monte Carlo simulation.
• Price convexity and capitalization-rate volatility.
• The variance “drain.”
• Pricing example: high- v. low-barrier markets.
Real Estate Investments I
Instructor: . Pagliari, Jr.
Class Notes – Week #9:
Risk/Return Analysis: A Probabilistic Approach
Table of Contents
I. INVESTMENTS INVOLVE ASSUMPTIONS ……………………………………………………………….. 1
II. THE OBJECTIVE: IDENTIFY FAVORABLE RISK-ADJUSTED RETURNS ………………………… 4
III. MEASURES OF RISK & RETURN …………………………………………………………………………. 11
IV. SOME STYLIZED EXAMPLES – ASSESSING RISK …………………………………………………….. 24
V. SINGLE-POINT ESTIMATES ← THE BANE OF REAL ESTATE PRACTICE …………………… 51
VI. SENSITIVITY ANALYSIS …………………………………………………………………………………….. 51
VII. SCENARIO FORECASTING …………………………………………………………………………………. 53
VIII. MONTE CARLO SIMULATION ……………………………………………………………………………. 56
IX. A NOTE ON PRICE CONVEXITY & CAPITALIZATION-RATE VOLATILITY …………………. 75
X. BEWARE THE “VARIANCE DRAIN” …………………………………………………………………….. 81
XI. REAL ESTATE PRICING EXAMPLE: HIGH- V. LOW-BARRIER MARKETS …………………… 84
XII. RISK-ADJUSTED RETURNS: SOME BASIC CONCEPTS & THOUGHTS ………………………… 97
XIII. SOME ADDITIONAL THOUGHTS ON MONTE CARLO SIMULATION…………………………. 114
“Doubt is not a pleasant condition, but certainty is absurd.” Voltaire
“It is difficult to make predictions, especially about the future.”
I. Investments Involve Assumptions
{about the future and the confidence we place in those assumptions}
A. We tend to be overconfident in our estimate of likely ranges of results. (See quiz results.)
B. Some processes are better handled with groups (see quiz results) and other better handled
by individuals.
C. “Experts” can be as wrong as non-experts (see below).
D. The utility of certain sums can be seen by the quiz examples where an individual has the
opportunity to either select: a) a certain sum with no risk, or b) the outcome of a coin toss,
where “heads” results in an amount equal to twice the certain sum and “tails” results zero.
Note the asymmetry of risk.
E. The risk/return trade-off of investing is a personal matter and depends on the investor’s risk
tolerance (often as a f(aging)).
OVERCONFIDENCE AMONG EXPERTS*
“Heavier-than-air flying machines are impossible.”
– British mathematician, physicist and
president of the British Royal Society, c. 1895.
“Reagan doesn’t have the presidential look.”
United Artists Executive – dismissing the idea
that be offered the starring role in the movie
The Best Man, 1964.
“A severe depression like that of 1920-01 is outside the range of probability.”
Harvard Economic Letter – weekly letter, November 16, 1929.
“We know on the authority of Moses, that longer ago
than six thousand years, the world did not exist.”
– 1843-1546, German leader of the Protestant Reformation.
“Impossible!”
Jimmy “the Greek” Snyder – odds maker, when asked
whether he thought could last six rounds in his
upcoming bout with World Heavyweight Champion Sonny Liston, 1964.
“They couldn’t hit an elephant at this dist___.”
General John B. Sedgwick – Union Army Civil War officer’s last words,
uttered during the Battle of Spotsylvania, 1864.
* J. and .H. Schoemaker, Decision Traps: Ten Barriers to Brilliant
Decision Making and How to Overcome Them ( , : Simon &
Schuster, Inc., 1989) p.74.
{Arts/Cinema}
{Economics}
{Religion}
{Military}
More recently, consider:
• Bernanke forecast a
continuation of the
“great moderation) in
2004 (four years before
• the so-called experts
inability to forecast the
2016 presidential
election, or
• Yellen’s assertion (2021)
that inflation was
“transitory.”
Wow! Have you
seen some past
presidents?
overconfidence
F. Consider another element: the “Availability Heuristic” – the process of judging size or
frequency by the ease with which instances come to mind (e.g., your impression of the
frequency of divorce among Hollywood celebrities and/or sex scandals among politicians is
likely to be biased (upwards) by how easily you retrieve such incidents from memory,
because of their notoriety in the popular press).1
G. Interesting paradox2 of the Availability Heuristic: If you are required to list a significant
number of such instances, the heuristic may affect you in reverse (i.e., increasing the required
number of recalled incidents may become difficult; this difficulty leads to a downward bias
in the judgment of size or frequency).
1. The initial academic examples included:
• people believe that they use their bicycles less often when they are asked to recall
many rather than few instances, and
• people are less confident in a choice when they are asked to produce more
arguments to support it rather than fewer arguments.
2. A quasi-academic application: a UCLA professor asked different groups of students
to list ways to improve the course (varying the required number of improvements
among groups); the students/groups who listed fewer ways to improve the class rated
the class higher!
H. Q : What might be real-world applications of this paradox in a real estate-investment setting?
What might be some caveats? 3 A : Consider a fund’s pre-marketing period. Ask a potential
(lead) investor for no more than 1 or 2 suggestions as to how to improve a particular fund
1 This section adopted from: , Thinking, Fast and Slow, : Farrar, Strauss and
Giroux, 2011, pp. 129-136.
2 See: , et al., “Ease of Retrieval as Information: Another Look at the Availability Heuristic,”
Journal of Personality and Social Psychology, vol. 61, no. 2, 1991, pp. 195-202.
3 A robust critique of behavioral economics can be found in chapters 8 and 9 of . Epstein, Skepticism
and Freedom: A Modern Case for Classical Liberalism, Chicago: The University of Chicago Press, 2003.
https://psycnet.apa.org/record/1991-33131-001
II. The Objective: Identify Favorable Risk-Adjusted Returns
A. As sophisticated real estate investors, our objective is to seek out superior risk-adjusted
returns. As a starting point, we need to consider the equilibrium risk/return tradeoff:
B. However, identifying such opportunities in practice is considerably less precise.
1. We have spent much of the first half of the quarter attempting to understand the
return-generating process for commercial real estate (and its many variations):
= + + ∆ = ∇
Illustration of Return & Risk
The Basis for the
Volatility
Recall: = i f
Equilibrium
2. While past returns are self-evident, future returns4 can only be estimated:
( ) ( ) ( ) ( )( )
E E f E N E
= + + ∆ = ∇
3. In other words, future returns are conditional on the realizations of such parameters.
[More said later about conditional expectations and variations.]
C. We need to quantify risk as well as return ← the focus of this week’s class!
D. In the parlance of institutional investors (and their consultants and investment managers),
we are seeking investment opportunities that offer “positive alpha”:
E. In practice, this equilibrium is constantly changing (e.g., before the GFC, immediately after the
4 Even though it mostly addresses predictions made outside the investment arena, ’s The Signal
and Noise: Why So Many Predictions Fail – But Some Don’t (Penguin Press: , 2012) is a highly
accessible overview of improving your predictive abilities.
Illustration of Alpha vis-a-vis Equilibrium Return & Risk
Volatility
These notes have so far
done little to distinguish
total risk (σ) from
systematic risk (β) – partly
this reflects the difficulties
that private market-assets
(real estate or otherwise)
have in satisfying the
underlying assumptions of,
say, the CAPM.
GFC, recovery from the GFC; now (COVID)?); therefore, practitioners need to monitor the
ever-changing risk/return profiles of properties in the market.
F. In practice, it is also common to prepare the following sort of historical performance analysis
of various markets (in this case, for NCREIF apartment property-level returns, grouped by
metropolitan areas, for the 15-year period ended 4th quarter 2020):
Source: NCREIF Analytics
https://www.ncreif.org/member-home/analytics/npi-3-Interactive-Performance/
Some surprises?
• Nashville (α = 3.7%)
• (α = –3.0%)
Past is not prologue!
https://www.ncreif.org/member-home/analytics/npi-3-Interactive-Performance/
G. As you may recall from weeks #6-7, I dissent from the received wisdom regarding a linear
risk/return tradeoff for risky private-market assets. My objection is to the underlying
assumption that the cost of debt (kd) is constant across all leverage ratios; instead, I argue that
both theory and practice indicate a (convex) curvi-linear relationship between the cost of debt
and the leverage ratio:
0% 15% 30% 45% 60% 75% 90%
Loan-to-Value Ratio
Recall: Illustration of Cost of Indebtedness as f(LTV)
for a Given Maturity Date
Risk-free Rate
Mortgage Interest Rate
Default Risk (δ) Premium
Structural Differences (γ) in Payment Schedules, Servicing Fees, Etc .
H. If so, then there is a (concave) curvi-linear risk/return tradeoff for risky private-market assets.
And, the “law of one price” argues that – in equilibrium – all risky assets lie on this risk/return
continuum, as illustrated below:
0% 10% 20% 30% 40% 50% 60% 70%
Volatility of Expected Returns
Illustration of the Law of One Price
Unlevered Core Real Estate
Core with 25% Leverage
Core with 50% Leverage
Core with 75% Leverage
Through financial leverage, we can
transform basic, core RE investment
into “higher-octane” opportunities.
A note on the volatility of levered equity:
I. In order to tie this concept into earlier cases, consider how the State & Main cases (both Parts
I and II) ought to lie on this risk/return continuum. Moreover, yet more leverage (than that
suggested in Part II of the case) could be applied, thereby transforming the transaction from
a value-added project to an opportunistic one:
0% 10% 20% 30% 40% 50% 60% 70%
Volatility of Expected Returns
Illustration of the Law of One Price | Risky Investments in Equilibrium
Unlevered Core Real Estate
Unlevered State & Main
State & Main with 33% Leverage
State & Main with 67% Leverage
Value-Added
Opportunistic
An aside: The case assumed
leverage of ≈ 50%.
J. Finally, recall that neither Part I nor Part II of the State and Main case asked you to explicitly
consider – let alone quantify – the riskiness of that investment.5 So, as an illustration consider
three different risk profiles for the (unlevered) investment:
K. Trying to assess and quantify risk is a difficult task. However, you are paid as an investment
manager to assess/quantify both return and risk.6
L. But before we can assess and quantify the risk of our real estate investments, we will first need
to explore a few concepts – see next section.
5 This could have been Part III of the case: asking you to model the riskiness of σa and, therefore, σe. See
graphic immediately above.
6 For a review of corporate risk-taking, see: Campello and Kankanhalli, “Corporate Decision Making under
Uncertainty: Review and Future Research Directions,” NBER working paper, December 2022.
0% 10% 20% 30% 40% 50% 60% 70%
Volatility of Expected Returns
Illustration of the Law of One Price | Risky Investments in Equilibrium
Unlevered Core Real Estate
Unlevered State & Main with Different Risk Profiles
State & Main with 33% Leverage
State & Main with 67% Leverage
Value-Added
Opportunistic
REJECTACCEPT This is the essence of the
investor’s problem: estimating
not only returns, but risk as well!
~~~~~~~~~~~~~~~~~~
Part III of the State & Main case
could have asked you to model
the riskiness of ka and/or ke.
https://www.nber.org/system/files/working_papers/w30733/w30733.pdf?utm_campaign=PANTHEON_STRIPPED&%3Butm_medium=PANTHEON_STRIPPED&%3Butm_source=PANTHEON_STRIPPED
https://www.nber.org/system/files/working_papers/w30733/w30733.pdf?utm_campaign=PANTHEON_STRIPPED&%3Butm_medium=PANTHEON_STRIPPED&%3Butm_source=PANTHEON_STRIPPED
III. Measures of Risk & Return
A. This section should be a review from other classes. If you’re uncomfortable with this
section, review the optional material at the end of this class note and/or materials from
earlier classes.
B. Single-period returns can be written (in the traditional (stock and bond) manner) as:
Income change in market value
Beginning market value
Income ending market value – beginning market value
Beginning market value
where: periodic, nominal return.
C. Multiple-period returns can be computed in one of two ways:
1. Arithmetic Average:
where: average nominal return per period, and
nominal return in period .
+ + + k k k kk =
2. Geometric Average:
( ) ( ) ( ) ( )
1 1 1 … 1 1
k = k k k k
With repeated “draws” (e.g., long-run returns), the geometric average is often
preferred because it provides a more accurate measurement when periodic
returns are volatile. See Exhibit 1.
Exhibit 1: Measures of Risk and Returns
Year 1 Year 2
Initial Investment Value $10,000 $5,000
Income & Appreciation (5,000) 5,000
Ending Investment Value $5,000 $10,000
Return per Period -50.0% 100.0%
Arithmetic Return: 25.0%
Geometric Return: 0.0%
( )( )2 1 50% 1 100% 1= − + −
t 0 t 1 t 2
This is the classic example.
Another is: you’re down 40%
one period and up 60% the
next – except, you’re worse off
than when you started.
D. Risk can be defined in a variety of ways, including:
1. the probability of loss,
2. the probability of receiving less than what was expected, and/or
3. the variance of actual or expected returns.
E. Traditional financial theory7 uses the third definition (see §II. C. 3.) for estimating the risk
of a single project or a portfolio of projects.
F. Two related measures of risk are:
1. Variance:
( ) ( ) ( ) ( ) ( )
where : variance of returns.
k k + k k + k k + k k
2. Standard Deviation:
returns. ofdeviation standard : where =
7 The use of the standard deviation is partly predicated on the assumption that the distribution of returns can
be described as multi-variate normal. For example, see: . Fama, Foundations of Finance, Basic
Books: , 1976.
Can Sam Zell calculate the σ ? I doubt it.
Nevertheless, he has an exceptional sense for assessing (and pricing) risk. In fact, I’ve heard
him refer to himself as a “risk arbiter.”
The point is: In your career, it is likely that you’ll need to develop both a statistical sense of
risk as well as a more visceral approach. In my view, each approach reinforces the other!
G. While other distributions are certainly available to us, the normal distribution8 – see Exhibit
2 – has convenient properties:
1. the mean (or average) = the median,
2. familiar bell-shaped curve,
3. two parameters (µ, σ) describe the entire distribution, and
4. means of distributions are additive: E(k1 + k2) = E(k1) + E(k2).
H. Assuming a normal distribution, the standard deviation has certain attractive statistical
properties:
1. ± one standard deviation ≈ 68% of the outcomes,9
2. ± two standard deviations ≈ 95% of the outcomes, and
3. ± three standard deviations ≈ 99% of the outcomes.
8 For those intrigued by the history of mathematics, please see: , “The Evolution of the Normal
Distribution,” Mathematics Magazine, April 2006, pp. 96-113, which notes the contributions of Bernoulli,
Fermat, Galileo, Galton, Gauss, Laplace and Pascal, among other famous mathematicians. [If the random
variable, X, follows a normal distribution, it is typically written as X ~N(µ,σ2).]
9 We can also determine frequency associated with either the left or the right tail from three facts about the
normal distribution: a) 100% of the outcomes lie beneath the curve, b) 50% of the outcomes lie on either side
of the mean, and c) since ≈ 68% of the outcomes lie within ± one standard deviation of the mean, ≈ 32% of
the outcomes lie outside of one or the other standard deviation. Therefore, the frequency associated with either
tail (i.e., ± one standard deviation from the mean) ≈ (100% – 68%)/2 ≈ 50% – 34% ≈ 16%.
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