Journal of Real Estate Finance and Economics, 11: 119-135 (1995)
0 1995 Kluwer Academic Publishers
The Present Value Model with Time-Varying Discount
Rates: Implications for Commercial Property Valuation
and Investment Decisions
DAVID GELTNER
University of Cincinnati, Department of Finance, College of Business Administmtion, Lindner Hall,
Cincinnati, OH 45221-0195
JJANPING ME1
New York University, Department of Finance, Leonard N. Stem School of Business, Management Education
Center, 44 West 4th Street, Suite $190, New York, New York 10012-1126
Abstract
A vector autoregressive model is developed for predicting cash flow and returns in the private (unsecuritized)
commercial property markets. The model predicts both of these variables quite well during the sample period.
The forecasting model is then used to develop a simple “buy/sell” rule for identifying property market value
peaks aud troughs. An improved present value model, taking account of the predictability of property returns,
is described and found to track historical market values much more closely than does either the appraisal-based
index or the traditional present value model with constant expected returns. Analysis in this paper suggests that
most of the change in commercial property market values has been due to changes in expected returns, rather
than to changes in expected future operating cash flows.
Key Wordsr valuation, investment, present value model, timing, cycles, discount rates
The present value model underlies all of modern financial economics, and lies at the heart
of commercial property valuation and real estate investment decision making. Traditionally
this model is applied by forecasting property net cash flows and discounting those cash
flows at a constant discount rate. In this model the discount rate is meant to represent the
expected return (that is, the internal rate of return or total return) to an investment in the
property, thereby reflecting the opportunity cost of capital. In practice, the discount rate
used in the present value model as applied to property valuation has not changed much
over time, largely because analysts have not known how to quantify changes in the market’s
expected return on pr0perty.i
Recently, evidence has mounted that asset value changes in the securities markets are
not consistent with the constant discount rate model or the constant expected return assumption? Campbell and Mei (1993), among others, have found that changes in stock prices
over time are due more to changes in the market’s required total returns (including price
changes), than to changes in the market’s cash flow expectations.3 Furthermore, changes
in the market’s expected return can be forecasted to some extent, as has been demonstrated
by Liu and Mei (1992, 1994) in the case of REITs, small stocks, and large stocks. This
implies that asset price changes are more predictable than was previously thought, and
that turning points in asset price cycles may be somewhat identifiable in advance!
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GELTNER AND ME1
The fact that expected returns change, and that these changes are forecastable, has important implications for property valuation in a discounted cash flow framework. Forecasts
of market return requirements (i.e., of discount rates in the present value model) can be
combined with forecasts of cash flow from operations to provide an improved present value
model. Such a model would, in principle, take account of the predictability of future asset
price changes, as well as the predictability of the operating cash flows the property will
generate.
In the present study we apply some of the methodology developed in the above-cited
papers to analyze the returns to commercial real estate in the private (i.e., unsecuritized)
property markets. In particular, we develop a vector autoregressive (VAR) model that simultaneously forecasts both the future operating cash flow and the discount rate (expected total
return) for commercial property based on the currently observable values of these and other
variables. This forecasting model is then used to develop a simple investment timing rule
to help provide decision guidance in making strategic buy, sell, or hold decisions for real
estate as an investment asset class. We also apply this forecasting model to demonstrate
the improved present value model of property valuation, which allows for time-varying
discount rates (i.e., forecasted expected total returns). This is done by applying the VAR
forecasting model retrospectively to historical data to simulate the present values, which
would have been obtained at each point in time from the improved present value model.
This is done by embedding the historical cash flow and return forecasts from the VAR
model within the log-linear present value model developed by Shiller and Campbell (1988).
Finally, we perform a type of sensitivity analysis, to see how important the use of return
forecasting and the improved present value model is in practice. We simulate the historical
present value calculations for commercial property both with and without allowing for variable return expectations in the present value model. From this exercise we make two major
observations: First, most of the volatility across time in the simulated historical present
value is due to changes in property total return expectations, rather than to changes in property operating cash flow expectations. Second, the improved present value model approximates the unsmoothed historical market values much better than do either the appraisalbased index or the traditional present value model without time-varying discount rates.
Some of the issues explored in the present paper were examined in a different context
by Mei and Liu’s (1994) article on market timing. However, the present paper differs from
that article in several important respects. First, Mei and Liu study securitized real estate
stocks, while the present paper analyzes commercial real estate in the private property market. The very different microstructure of the securitized and private market arenas impart
different asset valuation and risk/return characteristics to the private property market. Second, commensurate with the different nature of the private property market, the frequency
of return interval and return prediction horizon is quite different between the two studies.
While Mei and Liu examine high-frequency (monthly) time variation in expected returns,
the present paper focuses on low-frequency (annual) time variation in expected returns.
A third difference between the present paper and that of Mei and Liu (1994) is that the
present study examines forecasting not only of the expected returns but also of the operating cash flows of the asset class over time, and places both the return forecast and cash
flow forecast in the valuation framework of the improved present value model. In this way,
we seek to provide in the present study a comprehensive valuation model for commercial
THE PRESENT VALUE MODEL WITH TIME-VARYING DISCOUNT RATES
121
property assets in the private property markets. It is our hope that such a model can help
real estate appraisers, in particular, to incorporate the influence of changing market conditions (real estate yields, net operating cash flows, REIT returns) into the appraisal process.
1. Methodology and Data
In this paper we study the aggregate market for institutional-grade commercial property
in the United States during the period 1975-1992. Annual frequency return and cash flow
data is obtained for the private commercial property markets by splicing the PRISA Index
together with the Russell-NCREIF Index. PRISA returns are used from 1975 through 1977,
with Russell-NCREIF returns from 1978 through 1992.5 These return series include both
appreciation and current income return components. We unsmooth the appreciation returns
to correct for both disaggregate level appraisal smoothing and aggregate level index construction effects such as temporal aggregation. The unsmoothing procedure is that of Geltner
(1993), which does not assume that the underlying real estate market values are informationally efficient. In particular, appreciation returns (first differences of log values as of
the end of the calendar year) are unsmoothed using the following reverse filter:
gt = <g; -
(0.6k-,YO.4,
where gr is the observed appraisal-based index appreciation return in year c, and g, is the
unsmoothed return. Figure 1 shows the historical nominal value levels of US commercial
property, as measured by both the appraisal-based (PRISA/Russell-NCREIF) index and
the unsmoothed market value index, during the 1975-1992 period (end-of-year values).
An index of the level of the net operating income (NOI) generated by PRISA/NCREIF
properties is obtained by applying the current income yield series to the property value
level series? This results in the NO1 Index portrayed in Figure 2, which we use as a representation of the historical cash flow level for commercial property.7
Commercial Properly Market Values (PRISA/Russell-NCREF)
Nommd Appreciation Vaiue Levels
Figugure I. Historical private property market smoothed and unsmoothed nominal value levels used in the current study.
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GELTNER AND MEI
figure 2. Historical commercial property cash flow levels used in the current study.
In addition to the above-described private property market data, we also employ data
on REm returns in the public stock exchanges, as represented by the NAREIT Index.8 Several
recent studies have indicated that REIT returns may provide a leading indicator of returns
in the private property markets (see, e.g., Giliberto, 1990; Gyourko-Keim, 1992; BarkhamGeltner, 1995). It therefore makes sense for us to incorporate REIT return data in our forecasting model.
The data described above is used in a five-variable vector autoregression (VAR) model.
The two variables we need to forecast to implement the improved present value model are
the (unsmoothed) commercial property market total return and the cash flow? In addition
to these two variables, we include three others in the model: the REIT return, the appraisalbased index return, and the current income yield of the appraisal-based index. These three
variables are selected in part because they are easily observed by market participants. Also,
as noted, there is some a ptiori reason to believe these variables will be useful in forecasting
property returns or cash flow levels. lo The historical time series statistics for these five
variables over the 19751992 period are presented in Table 1.
Table 1. Historical statistics of VAR model variables (annual nominal
values: 1975-1992).
Variable
Mean
Std. Dev.
Real Estate Market Total Return*
.0800
.1009
Real Estate Cash Flow**
.1085
.0124
Appraisal Yield
.0760
.0081
Appraisal Total Return
.0870
.0695
REIT Total Return
.1358
.1554
*Unsmoothed from the appraisal-based n%rns, as described in Section 1.
**Aggregate cash flow level expressed as a fraction of the aggregate property value level as of the end of 1974.
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THE PRESENT VALUE MODEL WITH TIME-VARYING DISCOUNT RATES
Table 2. VAR model estimation results. Estimated coefficients (t-ratio).
Independent Variables
Const
Mkt. Ret,
CF,
Appr. Yld,
REIT Ret,
Appr. Rett
Adj R*
-1.48
(2.42)
.721
(1.51)
3.752
(1.54)
16.376
(2.89)
,100
(1.33)
- 1.824
(1.70)
.556
.05
(1.80)
-.Oll
(0.48)
.749
(6.67)
- ,423
(1.61)
.009
.lOl
(2.03)
.937
(2.44)
Aw. W+I
.04
(2.15)
-.004
(0.24)
- .207
(2.54)
,702
(3.69)
,008
(3.06)
-.o@l
(0.11)
.922
REIT %+I
.83
(0.62)
-1.500
(1.45)
-6.653
(1.26)
-1.216
(0.10)
-.128
(0.78)
2.683
(1.15)
,110
Aw. I%+I
-.56
(2.25)
,288
(1.48)
1.348
(1.36)
6.292
(2.72)
.044
(1.42)
- .076
(0.17)
,843
Dep. Var.
Mkt. Reti+,
Cash Flow,+i
_
Mkt. Ret. =Property market nominal total return obtained by “unsmoothing” the appraisal-based return.
Cash Flow =Property nominal net operating income level obtained from income and appreciation return components of appraisal-based index.
Appr. Yld. =Current income return component of appraisal-based index.
REIT Ret. =NAREIT All-REIT index total nominal return.
Appr. Ret. =Appraisal-based index total nominal return (PRISAIRussell-NCREIF).
All data annual calendar year (4th quarter to 4th quarter). Observations 1975-1992.
The VAR model relates the value of each of the five endogenous variables to the preceding
year’s value of all the five variables (including itself). The VAR process is thus a straightforward multivariable extension of univariate autoregressive modeling. The VAR model
system of regression equations estimated from the 18 annual observations of each of the
five variables is presented in Table 2 ?I The model generally tits the historical data quite
well, with high adjusted R* statistics on all the regressions except that for the REIT
returns.‘* The unsmoothed property market returns are predicted with an adjusted R* of
56%, and the property cash flow levels with an adjusted R* of 94%?3 This compares to
an adjusted R* of 17.5% obtained by Liu and Mei (1992) in a latent variables model predicting monthly REIT returns. It thus appears that, at least within the historical data, private
property market returns are more predictable than securitized real estate returns as represented by REITs.
2. Results of the Return and Cash Flow Forecasting Analysis
Figure 3 displays the history and VAR model forecast of the three nominal total returns:
the property market returns, the appraisal-based returns, and the REIT returns. Figure 4
shows the history and forecast of the property cash flow level, or NOI. It is apparent in
the figures that the forecasting model is predicting a near-term bottoming out of the real
estate market, first in market values (with substantial positive total returns forecast in 1993
GELTNER AND ME1
124
Figure 3. History and VAR model forecast of real estate returns.
VAR Mode( Forecast cd Cash Fbw
,* of N0rnd Roprcl NO1 Level
II
‘75-‘92=tistwy
‘93-201%Forecast
16 -
Figure 4.
History and VAR model forecast of property cash flow level (NOI).
and 1994), and then in the cash flow (which bottoms in 1994). While the near-term return
forecast displays considerable volatility, the longer-term forecast reveals the underlying trend,
which appears to be an approximately 20-year cycle in both the returns and the cash flows,
with the return cycle leading the cash flow cycle by several years. This underlying longterm trend is seen more clearly in the 57-year forecast of return and cash flow shown in
Figure 5.
THE PRESENT VALUE MODEL WITH TIME-VARYING DISCOUNT RATES
VAR W 57-Year Fcwxst From 1992
125
Data
Figure 5. Forecast of property market returns and cash flow levels, through year 2050. (Cash flow as fraction
of 1974 property market value.)
3. Using the Forecasting Model to Assist in Investment Timing Decisions
Under the classical efficient markets paradigm, which underlay most academic thinking
about asset markets into the 198Os, little attention was paid to the importance of timing
of investment decisions. As asset prices already supposedly reflected all available relevant
information, there was little to be gained by trying to time investment buy and sell decisions, and little hope of being able to consistently beat the market.
More recently, academic research has called into question the informational efficiency
of the stock markets (see, e.g., Shleifer and Summers, 1990) and the traditional view that
expected returns on assets are constant over time (see, e.g., Fama and French, 1988;
Campbell and Mei, 1993). There is reason to believe, and some evidence to support, the
idea that private (unsecuritized) property markets are sluggish, or even less informationally
efficient, than the securities markets (see, e.g., Barkham and Geltner, 1995). While the
nature and degree of informational inefficiency in asset markets is still controversial, there
is wider agreement that expected returns change over time in ways that are somewhat predictable, even in the stock marketi More specifically, as noted above, there is evidence
that the expected return on securitized real estate (as represented by REIT stocks) varies
substantially over time (see Liu and Mei, 1992, 1994; and Mei and Lee, 1994), and our
findings described in section 1 indicate even greater predictability in the unsecuritized property markets.
This suggests that, in addition to the traditional strategic concerns about diversification
and mean/variance portfolio optimization, investment managers need to seriously consider
questions of timing. If the market is somewhat predictable, then there may be times that
are particularly good for buying real estate (in general, as an asset class, with still the
need to perform careful due-diligence at the level of the individual deal), and conversely
there may be times that are particularly bad for buying real estate (or particularly good
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GELTNER AND MEI
Figure 6. Historical investment timing signals based on the VAR model and simple timing decision rule.
for selling it). Real estate investors with constant risk aversion ought to try to use predictability of prices and returns to try to identify when these buy and sell times occur, so as
to exploit opportunities in the expected returns. Forecasting models, such as the VAR described here, can help in this regard.
As an example, consider the following simple timing rule, based on the VAR model prediction of real estate market returns. If, using information available at time r, the model predicts
the next two consecutive years (t f 1 and t + 2) will have above-average returns, then
we call it a buy signal. If it predicts two consecutive years of below-average returns, we
call it a sell signal. Otherwise, the model is ambiguous, perhaps suggesting a hold or waitand-see strategy. Figure 6 shows the signals this decision rule would have given in the past
using the VAR model of Table 2 applied to historical data .i5 Figure 6 shows the actual historical property market values (unsmoothed, as of the end of each year), and the buy/sell
recommendations at the corresponding points in time. The timing rule and model work
pretty well (at least, in the historical sample), as it signals buys in most of the down years
and sells in most of the up years. The long string of buy signals in the late 1970s and
early 1980s clearly presaged the real estate boom, while the string of sell signals in the
mid-to-late 1980s was warning about the crash to come in the early 1990s. Interestingly,
at the end of 1992 the first buy signal in 12 years (since 1980) was issued.
4. Improving the Present Value Model of Property Valuation
In addition to assisting with investment timing decisions at the strategic level, the cash flow
and return forecasting model developed here can be used to improve property valuation.
THE PRESENT VALUE MODEL WITH TIME-VARYING DISCOUNT RATES
127
Traditionally the present value model has been used in commercial property valuation with
the cash flows in the numerators being forecasted as expectations, but the discount rate
in the denominators being treated, in effect, as a deterministic constant. Recognizing that
the expected returns, or discount rates, applied by the market are not constant, and can
be forecasted with some degree of accuracy (particularly in the case of the private property
market), suggests that the traditional present value model can be improved by allowing for
time-varying discount rates reflecting a forecast (such as our VAR model) of those rates.
There is, however, a technical problem in incorporating forecasted returns in the present
value model, which does not arise when forecasting only cash flows. The expected returns
enter the traditional present value model in a nonlinear fashion. For example, in the traditional present value model the present value of the cash flow which is expected to occur
two years from now, is given by
PV = Eo[CW{(l + rd(l + rd)l
where CF2 is the cash flow two years from now; rl and rz are the one-year returns for
the upcoming year 1 and the next year (year 2); and E,,[ .] is the expectation as of today
(i.e., the forecast) of these random variables (the future values of CF2, rl, and r2 are uncertain as of time 0). However, when we forecast cash flows and returns, we obtain l&[CF2],
&[rl], and &[r2], not Eo[CF2/{(1 + rl)(l + r2)}]. The expectation of a product is not
equal to the product of the expectations, because of Jensen’s Inequality. Thus, we cannot
use our forecast of expected returns in the traditional present value model formula.“j
Fortunately, Campbell and Shiller (1988) have developed an alternate formulation that
allows the return forecast to enter the valuation linearly. While their log-linear present value
model is only an approximation of the actual present value, it is a very close approximation
in most practical circumstances. The present value model which we must therefore use
to incorporate forecasted time-varying discount rates, is thus the following:
Pt = k/U - P) + (1 - P)
2 P'JS[d~+l+jl -5 dJVr,+l+jl
j=O
j=O
(2)
where
Pt = the log of the present value of the asset as of time t
P= a constant slightly less than unity (equal to one divided by the quantity one
plus the dividend payout rate)
k = a constant that is a nonlinear function of p17
&[4+l+jl = the expectation (forecast), as of time t, of the log of the of the cash flow in
time t + 1 + j
&h+l+j 1 = the expectation (forecast), as of time t, of the one-period (continuouslycompounded) return in period t + 1 + j
While this formula looks complicated, it is not at all difficult to work with in practice.
Note that the expected cash flows enter the valuation formula positively (larger expected
cash flows increase the present value), and the expected returns play the role of the discount
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GELTNER AND ME1
- Simulated Hstonxl Present Values
t Histonxl Apprmsal-Based Index
Figure Z Comparison of the improved present value model versus actual (unsmoothed) property market values
and appraisal-based index values.
rate in the traditional formula, entering here linearly in an additive (negative) term instead
of nonlinearly in the denominators of the traditional formula. Thus, the intuition is quite
clear: the larger the expected future cash flow, the higher the present value; and the larger
the expected (i.e., required) returns, the lower will be the present value.
We have applied our VAR model forecasts of real estate operating cash flows and return
expectations to the historical data, and then used the resulting forecasts (as of the past historical points of time) in formula 2. This gives us a simulated historical series of improved
present values, that is, present value taking account of the time-variability and predictability
of return expectations in the real estate market. After taking antilogs to convert the log
values back to straight levels (and indexing to a value of unity in 1975), the resulting historical
value series is plotted in Figure 7. Figure 7 also shows actual (unsmoothed) historical property market values, as well as the appraisal-based index values, during the 1975-1992 period.
Of course, the present value model was, in a sense, calibrated on the historical market
values, as the VAR model was estimated using historical data and developed to predict the
market value returns. So a good fit is not surprising. Nevertheless, the results portrayed
in Figure 7 suggest that the VAR forecasting model and the improved present value formulation hold good prospects for improving real estate valuation. We note in particular how
closely the present value model tracks the market value, and how it reflected the fall in
market values before this was portrayed in the appraisal-based index.
5. Using the Model to Explore the Nature of Real Estate Risk
If we believe that the VAR model developed here captures well the structure of real estate
market expectations, then we can use this model, in the present value formulation described
in Section 4, to explore the nature of real estate risk. In particular, the model allows us
to decompose the volatility of property market values to see what portion of the change
THE PRESENT VALUE MODEL WITH TIME-VARYING DISCOUNT RATES
129
over time in property values comes from changes in investor expectations about future cash
flows, versus what proportion comes from changes in investor expectations (or requirements)
about real estate returns.
Our findings in this regard are displayed in Figure 8. Figure 8 shows the simulated historical value index traced out by the improved present value model. First the model is applied as in Section 4 with both the cash flows and returns variable and forecasted by the
VAR model. Then the model is applied with the cash flows held artificially constant, allowing
only the return expectations to vary through time according to the forecasts of the VAR
model applied to the historical data. Finally, the returns are held artificially constant with
only the forecasted cash flows allowed to vary through time according to the VAR model
forecast. Notice that the model with constant cash flows but variable returns tracks very
closely the unrestricted model (which, in turn, closely tracks the actual historical market
values, as we saw in Figure 7). In contrast, the present value model with the returns held
constant (similar to the traditional present value approach) does not track closely at all
to the other series. Indeed, when we remove the variability in the return expectations, we
remove almost all of the volatility from the present value modeLis
This finding is interesting in several respects. For one thing, it suggests rather strongly
that accounting for time-variability in returns is quite important in real estate market valuation. The traditional approach may be seriously misleading as a representation of market
value. On the other hand, recall that the VAR model suggested that market values in real
estate tend to be mean-reverting (as seen in the cyclicality in the return forecasts). This
suggests that much of the volatility in the market value may be temporary or transient in
nature, in that property market values will eventually tend to revert toward the long-run
trend value. This long-run trend value, which might be termed a fundamental value, may
be better modeled by the constant-return present value formulation, which depends purely
on the forecasted operating cash flow fundamentals of the property. Figure 8 suggests that
the great boom and bust that commercial property markets experienced in the 1980s was
Usng
Simulated Historical Real Estate Present Values
VAR-Forecasted Cash Flow (CF) & Retmm (R) m Present V&K Model
Figure 8. The effect of varying cash flow expectations and varying return expectations in the present value model
of commercial property.
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GELTNER AND ME1
caused primarily by changes in return expectations rather than by changes in expectations
about operating cash flows. While such swings in return expectations do not necessarily
reflect irrationality on the part of investors, or informational inefficiency phenomena such
as speculative bubbles and fads, such phenomena might at least partially lie behind the
wide and somewhat persistent changes in return expectations found in this study.
In any case, careful real estate investors should attempt to keep track of what is driving
changes in property market values: Is it cash flow fundamentals or investor preferences
(and “sentiments”?) regarding return expectations? The type of analysis and tools described
in this report should help investors to interpret the market.
6. Summary and Conclusion
It might be said that at the macro or strategic level of large-scale institutional investment
decision making, there are two broad issues that must be addressed. The first issue is how
to efficiently and effectively diversify the investment portfolio. This is the issue addressed
by Modem Portfolio Theory and the various quantitative analytical tools that are based
generally on the efficient market hypothesis. That is, the tools used in addressing the diversification issue tend to assume a lack of return predictability or a stability in market return
expectations, so that cross-sectional asset allocation considerations take precedence over
market timing considerations. The second issue is investment timing, the notion that there
may be some times when it makes particular sense to buy or sell certain asset classes,
precisely because return expectations are not constant and price changes may be somewhat
predictable. It is this second issue with which this paper has been concerned.
We have sought to describe some tools and procedures that can be useful in addressing
the timing issue regarding direct investment in real estate, that is, commercial property
traded in the private (unsecuritized) property market. We have developed a VAR forecasting
model for predicting expected returns and cash flows for commercial property, and shown
how this model can be applied to develop a simple buy/sell decision rule to aid in the market
timing decison. We have also shown how the forecasts of expected total returns and operating
cash flows can be applied in the present value framework for valuing property. Finally,
we have used this approach to develop a simulated historical series of real estate present
values, to explore the nature of real estate risk.
It appears from this analysis that most of the change in property value over time has
resulted from changes in return expectations (or requirements), perhaps reflecting changes
in investor risk perceptions or risk preferences regarding the asset class rather than from
changes in operating cash flow expectations. Our findings also suggest that returns in private
property markets are more persistent and more predictable than those of securitized real
estate as represented by REITs.
Acknowledgments
The authors appreciate the financial support of the Real Estate Research Institute (RERI).
The authors appreciate the assistance of Lloyd Lynford and Charlie Wurtzebach. Any opinions and conclusions expressed in this paper are those of the authors, not necessarily shared
by the RERI.
THE PRESENT VALUE MODEL WITH TIME-VARYING DISCOUNT RATES
131
Appendix: The VAR Forecasting Model and the Present Value Model of Asset Value
A.I. l7ze Present Value Model with i’?me-Varying Discount Rates
In general, asset prices and returns are affected by changing expectations about both cash
flows and required returns. A technical difficulty is that the standard present value relation
is nonlinear when expected returns vary through time. This makes it intractable except
in a few special cases.
Campbell and Shiller (1988) propose a log-linear approximation to the standard model.
They argue that the approximation is both tractable and surprisingly accurate. Campbell
and Shiller originahy derived their approximation for a beginning-of-period (cum cash flow)
asset price, but we will work with an end-of-period (ex cash flow) price, which is more
standard in the finance literature.*’ We define the one-period log holding return on the
asset as: rt+] = log(P,+i + D,+i) - log(PJ, where P, is the asset price measured at the
end of period t (ex cash flow), and D, is the cash flow paid during period t. The right
side of this identity is a nonlinear function of the log asset price and the log cash flow;
it can be approximated, using a first-order Taylor expansion:
rt+l =
k + Ppt+l + (1 - P)dt+l - Pt
(A.1)
where lowercase letters are used for logs. The parameter p is the average ratio of the asset
price to the sum of the asset price and the cash flow, a number slightly smaller than one,
and the constant k is a nonlinear function of p. Equation A.1 replaces the log of the sum
of price and cash flow with a weighted average of log price and log cash flow.
Equation A.1 can be thought of as a difference equation relating pt to pt+i, d,+i, and rt+l.
It holds ex post, but it also holds ex ante as an expectational difference equation. Campbell
and Shiller impose the terminal condition that limi+, JI$Lp’p,+J = 0. This condition rules
out rational bubbles that would cause explosive behavior of the log asset price. With this
terminal condition, the ex ante version of (A.l) can be solved forward to obtain the equation 2 in the main body of the paper.
Pt = k/(1 - P> + (1 - P)
5j=Od&[dt+l+jl -2j=O pjWrt+l+jl
(-4.2)
This equation is useful because it enables one to calculate the effect on the asset price of
a change in expected asset returns. It says that the log asset price pt can be written as an
expected discounted value of all future cash flows dt+i+j less future returns rt+i+j, discounted at the constant rate p plus a constant k/(1 - p). If the asset price is high today, this
must mean that future expected cash flows are high unless returns are expected to be low
in the future. Note that (A.2) is not an economic model, but has been derived by approximating an identity and imposing a terminal condition. It is best thought of as a consistency
Al. This section is modified from Campbell and Mei (1993). We replace stock and dividends in their derivation
by asset and cash flows.
GELTNER AND ME1
132
condition that must be satisfied by any reasonable set of expectations. For simplicity, we
will label the first sum in equation (A.2): r], and the second sum as: vr.
A.2. Empirical Proxies for Future Expectations
To compute the price of a property, we need to construct empirical proxies for expectations
about future cash flows and returns. To do this, we assume the economy is determined by a
vector of state variables xt. We assume that this vector has L elements xlt, 1 = 1 . . . L,
the first of which is the cash flow (rent) to the property and the second of which is the
rate of return. The other elements are variables known to the market by the end of period t,
such as REIT returns, appraisal-based returns, and appraisal-based yields.
Next we assume that the state vector follows a first-order VAR.
xt+1 = Ax, + e,+l.
(A.3)
The assumption that the VAR is first order is not restrictive since a higher-order VAR can
always be rewritten in first-order form as discussed by Campbell and Shiller (1988) among
others. The matrix A is known as the companion matrix of the VAR.
Given the VAR model, revisions in long-horizon expectations of x~+~ are
(A.4)
Next we define X1 to be an L-element vector whose first element is one and whose other
elements are all zero. This vector picks the cash flow out of the state vector. And we define
X, to be an L-element vector whose second element is one and whose other elements are
all zero. This vector picks the return out of the state vector. Then equation A.2 implies
that the components of asset prices can be written as follows:
?c = X,PA(I - PA)-lx,,
vr = &pA(I - pA)-‘x,
(A.%
Expected future cash flows and returns are determined by the movements in the economic
state variables, and the matrix A governing the dynamic relation among the state variables.
The term pA(1 - pA)-l xt, which appears in the above expressions, represents the expectation at time t in the discounted multiperiod forecast of the state vector into the infinite
future. Appropriate elements are then taken from this state vector forecast revision to form
the components of asset values.
A.3. Data Constraints on the V2R Model Used in the Present Paper
While the first-order VAR structure has broad appeal in theory, it is important to note
some practical limitations imposed by the reality of the empirical data available to study
private commercial property markets in the United States today. Ideally, one would prefer
THE PRESENT VALUE MODEL WITH TIME-VARYING DISCOUNT RATES
133
to experiment with higher-order lag structures within the variables of the model, and one
would test the forecasting ability of the model using observation of out-of-sample fit (such
as rolling regressions). With only 18 datapoints, neither of these approaches have been
possible in the present paper. Given the data constraints, it should be recognized that the
model specification and empirical parameter estimates obtained in the present paper may
differ considerably from what would be obtained with more data.
Notes
1. For example, the Korpacz Yield Indicator, a benchmark widely cited in the industry, has varied barely &50
basis points (between roughly 11.5% and 12.5%) over the past five years, a period when short-term interest
rates have dropped nearly 600 basis points, and long-term interest rates have dropped over 300 basis points.
(The Korpacz Yield Indicator is published by Peter F. Korpacz & Associates, Inc., in Frederick, Maryland.
It is an average of the IRRs, i.e., expected total return on a multiyear commercial real estate investment,
reported by approximately a dozen national institutional investors and advisors surveyed by Korpacz Associates.)
Similar findings have been reported in the periodic investor surveys published by Salomon Brothers (see,
e.g., “Real Estate Risk & Return: 1991 Survey Results,” Salomon Brothers, Inc., March 1992).
2. See, for example, Keim and Stambaugh (1986), Fama and French (1988), Ferson and Harvey (1991), and
Campbell and Ammer (1993).
3. The 1987 stock market rise and crash is a famous recent episode of price changes that seem to be unrelated
(or out of all proportion) to reasonable contemporaneous changes in future earnings or dividends expectations.
Our findings in this paper suggest that the rise and fall of commercial property during the 1980s and early
1990s may be another example of this phenomenon.
4. When the market’s near-term expected returns are high, this implies that current prices are relatively low
compared to what they are expected to be in the near future, thus predicting an increase in asset prices. When
expected returns are low, this suggests current prices are high, predicting a future decline or slower growth
in prices.
5. The PRISA Index is published by the Prudential Realty Group, Newark, New Jersey. The Russell-NCREIF
Index is published by the National Council of Real Estate Investment Fiduciaries, in cooperation with the
Frank Russell Company, Tacoma, Washington. PRISA returns are used to extend the analysis back further
in time, beyond the 1978 commencement date of the Russell-NCREIF data. While PRISA returns begin a
few years prior to 1975, both PRISA and NAREIT return indices appear to be unreliable before 1975. Annual
frequency returns ate used to avoid some noise and smoothing problems in higher-frequency data (see Geltner,
1993).
6. NOI, = y,V,_t, where yt is the current income return component reported in the appraisal-based indices, and
Vi_, is the previous year’s value level (accumulated compound appreciation returns) from the same index.
7. Recall that the NO1 includes cash flow from vintage leases as well as new leases, and that the indices used
here represent existing properties that are aging over time. Average market rents might be expected to show
both greater volatility and a larger positive nominal growth trend than we observe in Figure 2. However,
the NO1 depicted in Figure 2 should represent well the historical operating cash flow pattern obtained by an
investor in a typical commercial property, and is therefore appropriate for us to use in the present analysis.
8. The NAREIT Index is published by the National Association of Real Estate Investment Trusts, Washington, DC.
9. For technical reasons, the log of the cash flow level is actually used in the present value model (to generate
a log present value that is then exponentiated to convert back to present value levels). However, in the present
analysis the VAR model is structured to forecast the cash flow (NOI) in straight levels. Also, we have kept
to working with nominal values in the present version of this paper, for the sake of simplicity and clarity
of presentation. It must be pointed out here. that because the inflation factor in the cash flow forecast and
required return forecast exactly offset each other in the present value mode, our analysis (equation 2) is valid
even though there is inflation in the economy.
10. For example, several studies, including those of REITs by Liu and Mei (1992, 1994), have found that yields
tend to be good predictors of returns.
134
GELTNER AND ME1
11. Due to lack of data, our model specification is somewhat constrained. In particular, it has not been possible
to work with higher-order lags in the VAR structure or the variables. As a result, some serial correlation
in the residuals remains with the model specification used here (e.g., - 37 X first-order autocorrelation in
the residuals of the property market return). It should be recognized that better data would allow superior
empirical models to be developed that might have different dynamic characteristics than the model used in
the present paper. It must be recognized that the objective of the present paper is demonstrative, rather than
definitive. We seek to demonstrate how time variation of future cash flows and required return may be taken
into account in property valuation.
12. It is not surprising that REIT returns are not very predictable, due to the informational efficiency of the stock
market.
13. See the appendix for a detailed description of the VAR approach.
14. Predictability of returns does not necessarily imply informational inefficiency. See, for example, Campbell
and Shiller (1987, 1988). Investors’ preferences (e.g., risk tolerance) may change over time in ways that are
somewhat predictable. Furthermore, it is empirically difficult to distinguish market inefficiency from timevarying expected returns. From a practical perspective, however, the distinction between these two cases may
not matter much for many investors. Whether the predictability is caused by informational inefftciency or
time-varying preferences, both of these two cases create market-timing opportunities for a portfolio manager
with constant risk aversion, i.e., making it possible to beat certain investment performance benchmarks by
taking risk at times when the risk/return trade-off is most favorable.
15. The model is applied in each year only to the data available as of that year. However, the model itself was
estimated using the full 1975-1992 sample of data. Thus, the performance of the forecasting model and timing
decision rule depicted in Figure 6 assumes that the parameters of the VAR system do not change over time.
16. All of this would not matter if we assumed that r, = rt , a deterministic constant, which is the implicit assump
tion in the traditional present value formula.
17. k = -log(p) - (1 - p)log(llp - 1).
18. This finding for annual frequency private property market returns contrasts to some extent with Liu and Mei’s
(1994) findings regarding monthly frequency REIT returns. However, the methodology and focus of the present
paper is somewhat different from that of Liu and Mei, which makes direct comparisons difftcult.
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