Morningstar - Q1 2022 - 49

EXHIBIT 1
This exhibit shows (1) that the point for real
estate based on historical data is significantly
above the reverse-optimization line and (2)
how much lower the expected return on real
estate would be if we were reading it from
the reverse-optimization line. Both approaches
lead to implausible asset-class assumptions
for real estate.
Using Geometric Mean
The problem is that the index that I used to
represent the asset class is not based on market
values but rather on appraisal values. The
appraisal process results in valuations that are
much smoother than market values. In
other words, the historical standard deviation of
the appraisal-based index understates the
standard deviation of real estate.
While the appraisal process leads to
smoothed short-term returns, over the long run,
they should accurately reflect property
values. Hence, the historical long-term rate of
growth of the real estate index, its historical
geometric mean, should be similar to
what it would be had it been measured with
market data. So, to find a plausible set
of assumptions regarding real estate, I take
the historical geometric mean of the index
as the expected geometric mean of real
estate and treat both the expected return and
standard deviation of real estate as
variables to solve.
Solving Two Equations for Two Unknowns
To solve for both the expected return
and standard deviation, we need two equations
in these two variables. One equation gives
the relationship between expected
geometric mean, expected return, and standard
deviation when returns are lognormally
distributed.6
a + b i
i
i =
B
E =Ri
GM =
(1 + ER)2
(1 + ER)2 + SD2
6 Kaplan, Paul D. 2014. " Expected Return and Expected Geometric Mean. " Morningstar magazine, August/September.
7 Holding all else equal includes holding correlations fixed. However, because of appraisal smoothing, the correlations between real estate and stock and bonds are certainly understated.
Nevertheless, I take these correlations as given so as not to further complicate the exercise I present here.
RPt = ∑ m
RPt = + RˆPt
S =
P =
S + u
2
+ ut
P V P
Stocks
2
Bonds
P(
T)
P(
)(
D(y; c, T) =
∑ n
j=1
( ) (1+
j -f
q
c
q
morningstar.com/products/magazine
y
)
y
q
i =1 xPj Rjt
+Standard Deviation
et
o
P(y; c, T) = ∑(1+
j=1
c
q
y
q
)
+ 1+
(
)
f
y
q
f-j
P(y; c, T)
)
A
- 1
iB
I call this plot of the combinations of expected
return and standard deviations that have
the same geometric mean an isogeometric mean
curve. In EXHIBIT 3 , I plot the isogeometric
mean curve for real estate and label it as such.
5
Stocks, Bonds, and Real Estate: Historical Summary Statistics
Asset Class
Stocks
Bonds
Real Estate
Benchmark
Source: Morningstar.
Benchmark
Weight
(%)
33.33
33.33
33.33
100.00
Geometric
Mean
(%)
10.59
5.15
8.54
8.54
Arithmetic
Mean
(%)
12.36
5.30
8.67
8.78
Standard
Deviation
(%)
20.16
5.57
5.29
7.19
Beta
2.62
0.25
0.14
Correlation With:
Stocks
1
0.06
-0.07
1 -
Bonds Real Estate
-
1
0.02
-
-
-
1
-
EXHIBIT 2
Reverse-Optimization Lines Based on Data and on Inferred Values for Real Estate
Stocks
Bonds
Real Estate (Data)
Real Estate (Inferred)
Benchmark
7
Expected Return
14%
Source: Morningstar.
Beta
0.5
1.0
1.5
2.0
2.5
3.0
where:
GM = expected geometric mean
ER = expected return
SD = standard deviation
Data
ISOGEOMETRIC MEAN CURVE
In reverse optimization, the betas on each asset
class depend on the standard deviation of
all asset classes. Hence, holding all else equal,
varying the standard deviation of real estate
leads to a second relationship between expected
return and standard deviation for real estate.7
In EXHIBIT 3 , I plot this resulting relationship as
the curve labeled " Reverse Optimization. "
Inferred
f = n- qT
n = qT
n
P =
c
q
∑
-j
+ 1+
(
y
q
)
-n
(
1+
y
q
)
f
j=1
Arithmetic Mean
14%
7
The inferred values of the expected return and
standard deviation of real estate are at the point of
10
15
n
f-j
20
- qT
+ 1+T
(
y
q
Expected Return
14%
)
49
- qT
USING INFERRED VALUES FOR REAL ESTATE
USING DATA ON REAL ESTATE
N
R EVERSE O
PTIMIZATIO
ONTIER
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Morningstar - Q1 2022

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