Morningstar - Q1 2022 - 48

Strategies
Reading Between the Lines
Sometimes historical returns are not
enough to go on.
QUANT U
Paul D. Kaplan
I don't think it's an exaggeration to say that
Harry Markowitz's mean-variance optimization
model is both the foundation of finance
and its most important development. Yet, it can
be difficult to use in practice because of
the demands it places on the practitioner to
come up with the expected return of each asset,
the standard deviation of return of each
asset, and the correlation of returns between
every pair of assets. While it is common practice
to estimate these parameters from summary
statistics on historical returns, doing so can lead
to problems because it assumes that the
future will be just like the past. Thus, to form
more forward-looking parameters, practitioners
sometimes need to go beyond pure statistics
to infer some factors from the data and
additional relationships that they expect to hold
true. In other words, they need to read
between the lines.
Reverse Optimization
One method of reading between the lines is to
use reverse optimization.1
and solving for optimal portfolios, reverse
optimization assumes that a benchmark portfolio
is optimal and solves for the expected returns.
To do this, from the standard deviations and
correlations of returns and from the benchmark
portfolio weights, first calculate the beta of
each asset class i:
i
i =
B
E =Ri
GM =
B
a + b i
where:
i
(1 + ER)2
(1 + ER)2 + SD2
- 1
= the standard deviation of return on
asset class i
= the standard deviation of return on the
benchmark portfolio
RPt = ∑ m
RPt
iB = + RˆPt
The relationship between beta and expected return
is linear. Letting ERi
S =
on asset class i, and a and b be parameters to be
determined, we have:
R =SP
P =
i
i =
B
B
E =Ri
A =SP
Reverse optimization
takes the standard deviations of correlations
of returns on a set of asset classes as given.2
But rather than taking expected returns as given
GM =
a + b i
2
2
A RSP
P
(1 + ER)2
(1 + ER)2 + SD2
- 1
Thus, a plot of beta versus expected return will be
a straight line, which I refer to as a reverseoptimization
line.
RPt = ∑ m
1 I have previously used reverse optimization in Quant U. See Kaplan, P.D. 2020. " Solving the Asset-Location Problem, Part I. " Morningstar, Q3.
2 These parameters are often estimated from historical returns.
3 Sharpe, William F. 1974. " Imputing Expected Security Returns From Portfolio Composition. " Journal of Financial Quantitative Analysis, Vol. 9, No 3. (June), PP 463-472.
4 Kaplan, Paul D. 1995. " Reverse Mean-Variance Optimization for Real Estate Asset-Allocation Parameters. " Real Estate Investing in the 1990s. Charlottesville, VA: Association for Investment
Management and Research (now CFA Institute). Chapter 12 in Kaplan, Paul D., ed., Frontiers of Modern Asset Allocation. 2012. Hoboken, N.J.: John Wiley & Sons.
RPt = + RˆPt
˜ = F RF
RW
S = W
5 For consistency with the article cited in footnote 4, I use the summary statistics reported there, except for geometric mean, which I calculate from arithmetic mean and standard deviation
using the formula I present later in this article. (I do this for internal consistency.) While the period that I used there, 1926-92, is from three decades ago, the summary statistics on
stocks and intermediate-term government bonds are similar to those for the period 1926-2020. See Ibbotson, Roger G. 2021. 2021 Stock, Bonds, Bills, and Inflation (SBBI) Yearbook. Chicago:
Duff & Phelps, A Kroll Business.
P =
R =SP
B
,
48
Morningstar Q1 2022
A =SP
S + u
2
max U EW T + B ) VW
A
(
P R(SP
B
s.t. 2
P
A
A A' = ,
2 RSP
,
2
P
(T +A
A
B ))
B' = 1 - , ≥ 0,
B ≥ 0
D
c
q
D
+ ut
˜ + VP W P
H RH
˜
-
W
L RL
˜
i =1 xPj Rjt
A
+ et
o
Pc
D
D
P iB
RSP
S + u
2
P V P
2
denote the expected return
+ ut
i =1 xPj Rjt
= the correlation of returns between asset
class i and the benchmark portfolio
A
iB
One way to determine a and b is to assume an
expected return for the benchmark and for
one of the asset classes. Another is to assume an
expected return for two of the asset classes.
Either way, once values are set for a and b,
the above equation can be used to calculate the
expected return of all asset classes.
When Reverse Optimization Is Not Enough
The method of reverse optimization was introduced
by Nobel laureate William Sharpe in 1974.3
However, it was not sufficient in modeling direct
investment in residential real estate.4
To see why, consider the summary statistics on
large-cap U.S. stocks, intermediate-term
government bonds, and direct residential real
estate in EXHIBIT 1 .5
Note that in the absence of a specific benchmark,
I have simply set the weights of each asset
class to 33.33%.
+ et
According to these statistics, direct residential real
estate has less risk than intermediate-term U.S.
government bonds, with an annual average return
that is more than 3% higher. If we were to use
these statistics as optimization inputs, real estate
would dominate bonds across a large part of
the efficient frontier.
o
D
On the other hand, if we were to use reverse
optimization, because the beta on real estate is
less than that of bonds, we would conclude
that the expected return on real estate is less than
that of bonds. I illustrate this in EXHIBIT 2 ,
where I plot the reverse-optimization line using
the statistics presented in EXHIBIT 1 as the
line labeled " Using Data on Real Estate. " (I'll
discuss the other line later.) I formed this
line by treating the historical arithmetic means
on stocks and bonds as expected returns.
f
n
P
P
P
f
n
c
q
P
=
P
https://www.morningstar.com/authors/528/paul-kaplan https://www.nxtbook.com/nxtbooks/morningstar/magazine_2020q3/index.php#/p/48
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