# Optimal Asset Allocation in a High Inflation Regime:
a Leverage-feasible Neural Network Approach

Chendi Ni  Cheriton School of Computer Science, University of Waterloo,
Waterloo, N2L 3G1, Canada<EMAIL_ADDRESS>Yuying Li  Cheriton School
of Computer Science, University of Waterloo, Waterloo, N2L 3G1, Canada,
<EMAIL_ADDRESS>Peter Forsyth  Cheriton School of Computer Science,
University of Waterloo, Waterloo, N2L 3G1, Canada<EMAIL_ADDRESS>
###### Abstract

We study the optimal multi-period asset allocation problem with leverage
constraints in a persistent, high-inflation environment. Based on filtered
high-inflation regimes, we discover that a portfolio containing an equal-
weighted stock index partially stochastically dominates a portfolio containing
a capitalization-weighted stock index. Assuming the asset prices follow the
jump diffusion model during high inflation periods, we establish a closed-form
solution for the optimal strategy that outperforms a passive strategy under
the cumulative quadratic tracking difference (CD) objective. The closed-form
solution provides insights but requires unrealistic constraints. To obtain
strategies under more practical considerations, we consider a constrained
optimal control problem with bounded leverage. To solve this optimal control
problem, we propose a novel leverage-feasible neural network (LFNN) model that
approximates the optimal control directly. The LFNN model avoids high-
dimensional evaluation of the conditional expectation (common in dynamic
programming (DP) approaches). We establish mathematically that the LFNN
approximation can yield a solution that is arbitrarily close to the solution
of the original optimal control problem with bounded leverage. Numerical
experiments show that the LFNN model achieves comparable performance to the
closed-form solution on simulated data. We apply the LFNN approach to a four-
asset investment scenario with bootstrap resampled asset returns. The LFNN
strategy consistently outperforms the passive benchmark strategy by about 200
bps (median annualized return), with a greater than 90% probability of
outperforming the benchmark at the terminal date. These results suggest that
during persistent inflation regimes, investors should favor short-term bonds
over long-term bonds, and the equal-weighted stock index over the cap-weighted
stock index.

##  1 Introduction

Since the global outbreak of COVID-19 in March 2020, we have observed
significant world-wide inflation. Particularly, from May 2021 to February 2023
(the latest information as of the time of writing), the 12-month change in the
CPI index in the U.S. has not been lower than 5% (Bureau of Labor Statistics,
2023). Before the pandemic, the U.S. economy has seen almost four decades of
low inflation. The sudden shift from a long-term low-inflation environment to
a high-inflation environment causes large uncertainty and volatility in the
financial market. In 2022, the technology-heavy NASDAQ stock index recorded a
yearly return of $-33.10\%$ (NASDAQ, 2023).

Equally concerning is the uncertainty around the duration of this round of
high inflation. Some believe that the geopolitical tensions and the COVID-19
pandemic will overturn the trend of globalization and lead to global supply
chain restructuring (Javorcik, 2020) which may cause a higher cost of
production in the foreseeable future. Moreover, Ball et al. (2022) suggests
that the future inflation rate may remain high if the unemployment rate
remains low.

In this article, we do not attempt to make predictions about the future
inflation situation. Instead, we look into historical high-inflation regimes,
and try to answer the following question: how should investors invest in a
persistent high-inflation environment? More specifically, we formulate the
investment problem as a multi-period asset allocation problem, in which an
investment manager rebalances an active portfolio periodically, with the goal
of finding asset allocation strategies that perform well under specific
investment criteria during high-inflation regimes.

To gain some initial insights into investing during high-inflation regimes, in
Section 2, we study the performance of fixed-mix (constant weight) strategies
on bootstrap resampled data from historical high-inflation periods. Fixed-mix
strategies allocate a fixed fraction of the portfolio wealth to the underlying
assets and are common choices of benchmark strategies for large sovereign
wealth funds (CPP Investments, 2022; Norges Bank, 2022). Our preliminary
findings show that the fixed-mix strategies with the equal-weighted stock
index achieves partial first-order stochastic dominance over fixed-mix
strategies which use the cap-weighted stock index during high-inflation
regimes.

In Section 3, we seek an active investment strategy (i.e. a dynamic asset
allocation), so that the active portfolio would outperform a fixed-mix
benchmark portfolio consistently throughout the investment horizon. We assume
that the real (inflation adjusted) asset returns during a high-inflation
regime follow stochastic processes and treat the asset allocation decisions as
the control of a dynamic system. The multi-period asset allocation problem
then becomes a stochastic optimal control problem.

To solve the stochastic optimal control problem, in Section 3.4, we study a
two-asset case under a synthetic market assumption. The two assets are a stock
index, and a constant maturity bond index. There is a large amount of extant
literature on closed-form solution methods for beating a stochastic benchmark
under synthetic market assumptions (Browne, 1999, 2000; Tepla, 2001; Basak et
al., 2006; Davis and Lleo, 2008; Lim and Wong, 2010a; Oderda, 2015; Alekseev
and Sokolov, 2016; Al-Aradi and Jaimungal, 2018a). In many of these articles,
the objective function involves a log-utility function, e.g. the log wealth
ratio. Under the log wealth ratio formulation, it is often hard to accommodate
a fixed stream of cash injections, which is a common characteristic of open-
ended funds. Forsyth et al. (2022) consider a scenario where a fixed amount of
cash injections is allowed and provides a closed-form solution under a
cumulative quadratic tracking difference (CD) objective as well as the
assumption that the stock price follows a double exponential jump-diffusion
model and the bond price is deterministic. We extend the results in Forsyth et
al. (2022) by considering the case that both the stock index and the bond
index follow jump-diffusion models. While this introduces additional
complexity to the problem, we believe the assumption that the bond index price
is stochastic and has jumps is appropriate under a high-inflation scenario. In
Section 3.4, we present the closed-form solution under the CD objective
derived under the synthetic model. The closed-form solution provides insights
for understanding the behaviour of the optimal control (allocation strategy).

However, to derive the closed-form solution, some unrealistic assumptions such
as continuous rebalancing, infinite leverage, and continued trading when
insolvent, are required. Ideally, we would like to solve the stochastic
optimal control problem under realistic constraints, i.e., discrete
rebalancing, limited leverage, and no trading when insolvent. To solve a
discrete-time multi-period asset allocation problem, it is common to use the
dynamic programming (DP) approach to convert a multi-step optimization problem
into multiple single-step optimization problems. However, van Staden et al.
(2023) point out that dynamic programming-based approaches require the
evaluation of a high-dimensional performance criterion to obtain the optimal
control which is comparatively low-dimensional. This means that solving the
discrete-time problem numerically using dynamic programming-based techniques
(for example numerical solutions to the corresponding PIDE (Wang and Forsyth,
2010), or reinforcement learning (RL) techniques (Dixon et al., 2020; Park et
al., 2020; Lucarelli and Borrotti, 2020; Gao et al., 2020)) are inefficient
and are computationally prone to known issues such as error amplification over
recursions (Wang et al., 2020).

Acknowledging these limitations, in Section 3.7, we propose to use a single
neural network model to approximate the optimal control, which converts the
multi-period optimization problem into a single finite-dimensional
optimization problem, which is solved by learning the parameters of the neural
network model on data sets of asset returns using standard optimization
methods. This direct approximation of the control exploits the lower
dimensionality of the optimal control and bypasses the problem of solving
high-dimensional conditional expectations associated with DP methods. We note
that the idea of using a neural network to directly approximate the control
process is also used in Han et al. (2016); Buehler et al. (2019); Tsang and
Wong (2020); Reppen et al. (2022), in which they propose a stacked neural
network approach which includes individual sub-networks for each rebalancing
step. In contrast, we propose a single shallow neural network that includes
time as an input feature, and thus avoids the need to have multiple sub-
networks for each rebalancing step and greatly reduces the computational and
modeling complexity. Furthermore, using time as a feature in the neural
network approximation function is consistent with the observation that (under
assumptions) the optimal control is known to be a continuous function of time,
which we discuss in detail in Section 3.4.

In terms of the modeling framework, the closest work is Li and Forsyth (2019);
Ni et al. (2022), which also use a single neural network to approximate the
controls under an optimal defined-contribution plan problem. However, the
neural network models proposed in Li and Forsyth (2019); Ni et al. (2022) are
limited to solving long-only investment problems, in which leverage is not
permitted. In contrast, we consider the more practical investment scenario
that allows a certain amount of leveraging and shorting in the investment
portfolio to mimic the investment behavior of large sovereign wealth funds.
The leverage assumption introduces additional constraints to the optimization
problem, and thus requires a new model architecture. Particularly, in Section
3.7, we propose the novel leverage-feasible neural network (LFNN) model that
converts the leverage-constrained optimization problem into an unconstrained
optimization problem. In Section (3.8), we prove that, under benign
assumptions, the solution of the unconstrained optimization problem
(parameterized by the LFNN model) can approximate the optimal control
arbitrarily well, which mathematically justifies the validity of the LFNN
approach.

In addition, compared to Li and Forsyth (2019); Ni et al. (2022), we consider
the more practical cumulative tracking difference-based investment objective
that evaluates the relative performance of the active portfolio versus the
benchmark portfolio throughout the investment horizon, instead of only
considering the performance at the terminal time of the investment horizon.

In Section 4, we discuss various numerical experiments and results. In Section
4.1, we first compare the leverage-constrained investment strategy computed
using the LFNN model with the clipped form of the closed-form solution on
synthetic market data. We show that the LFNN model produces a strategy that
achieves comparable performance against the closed-form solution, thus
substantiating the LFNN approach. In Section 4.2, we further apply the neural
network model to a more realistic case with four underlying assets, which
include the equal-weighted stock index, the cap-weighted stock index, the
30-day U.S. treasury bill index, and the 10-year U.S. treasury bond index. We
conduct numerical experiments on bootstrap resampled data under the cumulative
quadratic tracking shortfall (CS) objective and obtain a leverage-constrained
strategy that outperforms the benchmark with more 2% higher median
(annualized) internal rate of return (IRR), and more than 90% probability of
achieving a higher terminal wealth.

We summarize our contributions below:

  1. (i)

We assess the performance of the passive investment strategies on resampled
data from filtered historical inflation periods. We find that the fixed-mix
strategy with the equal-weighted stock index partially stochastically
dominates the fixed-mix strategy with the cap-weighted stock index in
resampled high-inflation regimes. This suggests that the equal-weighted stock
index is preferred over the cap-weighted stock index during high inflation
periods.

  2. (ii)

To discover dynamic allocation strategies that outperform the passive fixed-
mix strategies in high-inflation regimes, we formulate the problem as a
stochastic optimal control problem with suitable objective functions under
leverage constraints. To gain intuition about the behavior of the optimal
controls, we derive the closed-form solution under a jump-diffusion asset
price model and continuous rebalancing for a two-asset case. The closed-form
solution provides important insights into the properties of the optimal
control as well as meaningful interpretations of the neural network models
that approximate the controls.

  3. (iii)

We propose to represent the control directly by a neural network
representation, so that the stochastic optimal control problem can be solved
numerically under realistic constraints such as discrete rebalancing and
limited leverage. Particularly, we propose the novel leverage-feasible neural
network (LFNN) model that allows us to convert the original leverage-
constrained optimization problem into an unconstrained optimization problem.

  4. (iv)

We prove that with a suitable choice of the hyperparameter of the LFNN model,
the solution of the parameterized unconstrained optimization problem can
approximate the optimal control arbitrarily well. This provides a mathematical
justification for the validity of the LFNN approach.

  5. (v)

We benchmark the performance of the LFNN model with the clipped form of the
closed-form solution and show that the LFNN model achieves comparable
objective function values, and the allocation strategy from the LFNN model
achieves highly comparable performance as the clipped form and the closed-form
solution. The results provide empirical validation for the neural network
approach.

  6. (vi)

In a more realistic stochastic market scenario, we apply the neural network
method to bootstrap resampled asset returns with four underlying assets,
including the equal-weighted and cap-weighted stock index, and the
30-day/10-year treasury bond indexes. The dynamic strategy that follows the
learned LFNN model outperforms the fixed-mix benchmark strategy consistently
throughout the investment horizon, with a 2% higher median (annualized)
internal rate of return (IRR), and more than 90% probability of achieving a
higher terminal wealth. Furthermore, the learned allocation strategy suggests
that the equal-weighted stock index and short-term bonds are preferable
investment assets during high-inflation regimes.

##  2 Passive investing in historical high-inflation periods

Inflation has been a topic of much recent concern. Before the recent COVID-19
pandemic, we have just observed a long period of benign inflation, and low
(real) short-term interest rates. Some would argue that this has led to a
bubble in asset prices. Now, after the coronavirus crisis, we are seeing high
inflation statistics in most of the world. Going forward, we have to be
cognizant of the risk of inflation. If we enter into a long period of even
moderate inflation, will the traditional passive approach using a mix of stock
indexes and moderate-term bonds still work?

In this section, we first filter the historical time series (1926-2022) to
uncover periods of high, sustained inflation. We concatenate these high-
inflation regimes to produce asset return series which have been observed in
inflationary times. We then examine the performance of various passive
investment strategies on bootstrapped returns from the high-inflation series,
aiming to gain a crude understanding of high-inflation investing. Our
preliminary finding is that an investor should use a mix of short-term bonds
and an equal-weighted stock index, to mitigate inflationary effects.

###  2.1 Filtering historical inflation regimes

We use monthly data from the Center for Research in Security Prices (CRSP)
over the 1926:1-2022:1 period.111The date convention is that, for example,
1926:1 refers to January 1, 1926.222More specifically, results presented here
were calculated based on data from Historical Indexes, ©2022 Center for
Research in Security Prices (CRSP), The University of Chicago Booth School of
Business. Wharton Research Data Services (WRDS) was used in preparing this
article. This service and the data available thereon constitute valuable
intellectual property and trade secrets of WRDS and/or its third-party
suppliers. We also use the U.S. CPI index, also supplied by CRSP.

Our objective is to select high-inflation periods as determined by the CPI
index. Monthly data is quite volatile, so we use the following filtering
procedure. We use a moving window of $k$ months, and we determine the
cumulative CPI index log return (annualized) in this window. If the cumulative
annualized CPI index log return is greater than a cutoff, then all the months
in the window are flagged as part of a high-inflation regime. Note that some
months may appear in more than one moving window. Any months which do not meet
this criterion are considered to be in low-inflation regimes. See Algorithm
A.1 in Appendix A.1 for the pseudo-code.

This approach requires the specification of the cutoff and the window size.
The average annual inflation over the period 1926:1-2022:1 was 2.9%. In other
words, inflation of about 3% was normal historically. Federal Reserve
policymakers have been targeting the inflation rate of 2% over the long run to
achieve maximum employment and price stability (The Federal Reserve, 2011).
After some experimentation, we use a cutoff of 5% as the threshold for high
inflation, which is more than double the Fed’s target inflation rate. In
addition, we use the moving window size of 5 years (see Appendix A.2 for more
discussion). This uncovers two inflation regimes: 1940:8-1951:7 and
1968:9-1985:10, which correspond to well-known market shocks (i.e. the second
world war, and price controls; the oil price shocks and stagflation of the
seventies).

Table 2.1 shows the average annual inflation over the two regimes identified
from the moving-window filter.

Time Period | Average Annualized Inflation
---|---
1940:8-1951:7 | .0564
1968:9-1985:10 | .0661
Table 2.1: Inflation regimes determined using a five-year moving window with a
cutoff inflation rate of $0.05$.

For possible investment assets, we consider the 30-day U.S. T-bill index (CRSP
designation “t30ind”). In addition, we construct a constant maturity 10-year
U.S. treasury index.333The 10-year treasury index was generated from monthly
returns from CRSP back to 1941 (CRSP designation “b10ind”). The data for
1926-1941 are interpolated from annual returns in Homer and Sylla (1996). The
10-year treasury index is constructed by (a) buying a 10-year treasury at the
start of each month, (b) collecting interest during the month, and then (c)
selling the treasury at the end of the month. We repeat the process at the
start of the next month. The gains in the index then reflect both interest and
capital gains and losses. We also study the cap-weighted stock index (CapWt)
and the equal-weighted stock index (EqWt), also from CRSP.444The
capitalization-weighted total returns have the CRSP designation “vwretd”, and
the equal-weighted total returns have the CRSP designation “ewretd”. The CRSP
indexes are total return indexes, which include all distributions for all
domestic stocks trading on major U.S. exchanges. All of these various indexes
are in nominal terms, so we adjust them for inflation by using the U.S. CPI
index, also supplied by CRSP.

We calculate the average historical returns of the assets during the two
inflation periods (see details in Appendix A.3), and find that the equal-
weighted stock index has a higher average return and higher volatility than
the cap-weighted stock index. In addition, we find that the 30-day T-bill
index has similar average return as the 10-year T-bond index, but much lower
volatility. This seems to indicate that the T-bill index is the better choice
of defensive asset during high inflation. Therefore, we consider the equal-
weighted stock index, the cap-weighted stock index and the 30-day T-bill index
for the following experiment.

###  2.2 Bootstrap resampling

We choose to test allocation strategies using stationary block bootstrap
resampling (Politis and Romano, 1994). See Appendix B.1 for detailed pseudo-
code for bootstrap resampling. Compared to the traditional bootstrap method,
the block bootstrap technique preserves the local dependency of data within
blocks. The stationary block bootstrap uses random blocksizes which preserves
the stationarity of the original time series data. Briefly, each bootstrap
resample consists of (i) selecting a random starting date in the historical
return series, (ii) then selecting a block (of random size) of consecutive
returns from this start date, and (iii) repeating this process until a sample
of the total desired length is obtained.

An important parameter is the expected blocksize, which, informally, is a
measure of serial correlation in the return data. A challenge in using block
bootstrap resampling is the need to choose a single blocksize for multiple
underlying time series data so that the bootstrapped data entries for
different assets are synchronized in time. In the following experiments, we
use the expected blocksize of 6 months for all time series data. However, we
have compared different numerical experiments using a range of blocksizes,
including i.i.d. assumptions (i.e. expected blocksize equal to one month), and
find that the results are relatively insensitive to blocksize, as discussed in
more detail in Appendix B.2.

Typically, the bootstrap technique resamples from data sourced from one
contiguous segment of historical periods. However, the moving-window filtering
algorithm has identified two non-contiguous historical inflation regimes. To
apply the bootstrap method, there are two intuitive possibilities: 1)
concatenate the two historical inflation regimes first, then bootstrap from
the concatenated combined series, or 2) bootstrap within each regime (i.e.,
using circular block bootstrap resampling within each regime), then combine
the bootstrapped resampled data points. We have experimented with both
methods, and find that the difference is minimal (see Appendix B.3). In this
article, we adopt the first method, i.e., we concatenate the historical
regimes first, then bootstrap from the combined series. This method is also
adopted by Anarkulova et al. (2022), where stock returns from different
countries are concatenated and the bootstrap is applied to the combined data.

###  2.3 Comparison of passive strategies

In this section, we compare the performances of two fixed-mix strategies. The
first strategy, the “EqWt” strategy, maintains a 70% allocation to the equal-
weighted index, and 30% allocation to the 30-day T-bill index. The second
strategy, the “CapWt” strategy, maintains a 70% allocation to the cap-weighted
index, and 30% allocation to the 30-day T-bill index.

Investment horizon $T$ (years) | 10
---|---
Equity market indexes | CRSP cap-weighted/equal-weighted index (real)
Bond index | 30-day T-bill (U.S.) (real)
Index Samples | Concatenated 1940:8-1951:7, 1968:9-1985:10
Initial portfolio wealth | 100
Rebalancing frequency | Monthly
Table 2.2: Investment scenario.

In the numerical experiment, we consider the investment scenario described in
Table 2.2. Briefly, we begin with an initial wealth of 100 for both
strategies, with no further cash injections and withdrawals. The investment
horizon is 10 years, with monthly rebalancing to maintain the constant weights
in the portfolio. We evaluate the investment results by examining the
distribution of the final wealth $W(T)$ at $T=10$ years. We evaluate the two
strategies on $10,000$ block bootstrap resampled data samples (Politis and
Romano, 1991; Dichtl et al., 2016; Anarkulova et al., 2022) from the
concatenated CRSP combined time series from 1940:8-1951:7 and 1968:9-1985:10,
with an expected blocksize of six months. We then compare the CDF (cumulative
distribution functions) of the terminal wealth of the EqWt strategy and the
CapWt strategy.

Before we discuss the experiment results, we quickly review the concept of
partial stochastic dominance. Suppose two investment strategies $A$ and $B$
are evaluated on a set of data samples under the same investment scenario. We
consider the CDFs of terminal wealth $W$ associated with both strategies.
Specifically, we denote the CDF of strategy A by $\text{CDF}_{A}(W)$ and that
of strategy B by $\text{CDF}_{B}(W)$. Let $W_{T}$ be the random wealth at time
$T$ and $W$ be a possible wealth realization, then we can interpret
$\text{CDF}_{A}(W)$ as

$\displaystyle\text{CDF}_{A}(W)$ $\displaystyle=$ $\displaystyle
Prob(W_{T}\leq W)~{}.$ (2.1)

Following Atkinson (1987); van Staden et al. (2021), we define partial first-
order stochastic dominance.

######  Definition 2.1 (Partial first order stochastic dominance).

Given an investment strategy A which generates a CDF of terminal wealth $W$
given by $F_{A}(W)$, and a strategy B with CDF $F_{B}(W)$, then strategy $A$
partially stochastically dominates strategy B (to first order) in the interval
$(W_{lo},W_{hi})$ if

$\displaystyle F_{A}(W)$ $\displaystyle\leq$ $\displaystyle F_{B}(W),\;\forall
W\in(W_{lo},W_{hi})$ (2.2)

with strict inequality for at least one point in $(W_{lo},W_{hi})$.

The arguments for relaxing the usual definition of stochastic dominance are
given in (Atkinson, 1987; van Staden et al., 2021). Consider first the case of
the upper bound $W_{hi}$. Given some initial wealth $W_{0}$, if $W_{hi}\gg
W_{0}$, then an investor may not be concerned that strategy $A$ underperforms
strategy $B$ at these very high wealth values. In this case, the investor is
fabulously wealthy.

Turning attention to the lower bound $W_{lo}$, we consider two cases. Suppose
that $W_{lo}\ll W_{0}$, Assume $F_{A}(W_{lo})=F_{B}(W_{lo})$. As an extreme
example, suppose $W_{lo}=$ two cents. The fact that strategy $B$ has a higher
probability of ending up with one cent, compared with strategy $A$ is cold
comfort, and not particularly interesting. On the other hand, suppose
$F(W_{lo})\ll 1$. Again, an investor may not be interested in events with
exceptionally low probabilities.

Figure 2.1: Cumulative distribution function of final real wealth $W$ at
$T=10$ years, bootstrap resampling expected blocksize six months, $10,000$
resamples (Appendix A.1). $T=10$ years. Data: concatenated returns,
1940:8-1951:7, 1968:9-1985:10. Scenario described in Table 2.2.

This seems to be the case for our experiment here. In Figure 2.1, we compare
the CDFs of the terminal wealth of the EqWt strategy and the CapWt strategy.
Remarkably, the EqWt strategy appears to partially stochastically dominate the
CapWt strategy, since the CDF curve of the EqWt strategy almost appears to be
entirely on the right side of the CDF curve of the CapWt strategy, except at
very low probability values. In fact, close examination shows that the curves
cross at the point $F_{EqWt}(W_{lo})=F_{CapWt}(W_{lo})\simeq.02$, with a
slight underperformance of the EqWt strategy compared to the CapWt strategy in
this extreme left tail.

The fact that the EqWt strategy partially stochastically dominates the CapWt
strategy seems to suggest that the equal-weighted stock index is the better
choice for the stock index than the cap-weight stock index during high
inflation times. In a later numerical experiment in Section 4.2, where the
admissible stock indexes include both cap-weighted and equal-weighted indexes,
we show that the computed optimal strategy almost always ignores the cap-
weighted stock index and picks the equal-weighted stock index, which further
confirms the argument that the equal-weighted stock index is the preferred
asset for investors during high inflation scenarios.

However, if we examine more recent data555Since about 2010. Of course, this is
outside a period of sustained high inflation. the situation is not so clear
(Taljaard and Mare, 2021), since the equal-weighted index appears to
underperform. However, Taljaard and Mare (2021) suggests that this is due to
the recent market concentration in tech stocks.666As of February, 2023, Apple,
Microsoft, Amazon and Alphabet (A and C) in total comprised 17% of the market
capitalization of the S&P 500. In fact, a plausible explanation for the
outperformance (historically) of an equal-weighted index is that this is
simply due to the small-cap effect, which was not widely known until about
1981 (Banz, 1981). Plyakha et al. (2021) acknowledge that the equal-weighted
index has significant exposure to the size factor. However, Plyakha et al.
(2021) argue that the equal-weighted index also has a larger exposure to the
value factor. In addition, there is a significant alpha effect due to the
contrarian strategy of frequent rebalancing to equal weights. It would appear
to be simplistic to dismiss an equal weight strategy on the grounds that this
is simply a small cap effect that has become less effective.

##  3 Dynamic outperforming benchmark under bounded leverage

###  3.1 Sovereign wealth funds and benchmark targets

So far, we have examined the performance of passive portfolios in bootstrap
resampled sustained inflation regimes. In practice, instead of taking a
passive approach, some of the largest sovereign wealth funds often adopt an
active management philosophy and use passive portfolios as the benchmark to
evaluate the efficiency of active management. For example, the Canadian
Pension Plan (CPP) uses a base reference portfolio of 85% global equity and
15% Canadian government bonds (CPP Investments, 2022). Another example is the
Government Pension Fund Global of Norway (also known as the oil fund) managed
by Norges Bank Investment Management (NBIM), which uses a benchmark index
consisting of 70% equity index and 30% bond index.777The Ministry of Finance
of Norway sets the allocation fraction between the equity index and the bond
index. It gradually raised the weight for equities from 60% to 70% from
2015-2018. The benchmark equity index is constructed based on the market
capitalization for equities in the countries included in the benchmark. The
benchmark index for bonds specifies a defined allocation between government
bonds and corporate bonds, with a weight of 70 percent to government bonds and
30 percent to corporate bonds (Norges Bank, 2022).

However, the excess return that these well-known sovereign wealth funds have
achieved over their respective passive benchmark portfolios cannot be
described as impressive. In the 2022 fiscal year report, CPP claims to have
beaten the base reference portfolio by an annualized 80 bps after fees over
the past 5 years (CPP Investments, 2022). On the other hand, NBIM reports a
mere average of 27 bps of annual excess return over the benchmark over the
last decade (see Table 3.1). It is worth noting that these behemoth funds
achieve seemingly meager results by hiring thousands of highly paid investment
professionals and spending billions of dollars on day-to-day operations. For
example, the CPP 2021 annual report (CPP Investments, 2021) lists personnel
costs as CAD 938 million, for 1,936 employees, which translates to average
costs of about CAD 500,000 per employee-year.

Year | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | Average
---|---|---|---|---|---|---|---|---|---|---|---
Excess return (%) | 0.21 | 0.99 | -0.77 | 0.45 | 0.15 | 0.70 | -0.30 | 0.23 | 0.27 | 0.74 | 0.27
Table 3.1: Norges Bank Investment Management, relative return to benchmark
portfolio

The stark contrast between the enormous spending of sovereign wealth funds and
the meager outperformance of the funds relative to the passive benchmark
portfolios is probably provocative to taxpayers and pensioners who invest
their hard-earned money in the funds. Equally concerning is the potential of a
long, persistent inflation regime and the funds’ ability to consistently beat
the benchmark portfolio in such times. After all, both the CPP Investments and
NBIM were established in the late 1990s, a decade after the last long
inflation period ended in the mid-1980s.

These concerns prompt us to ask the following question: in a presumed
persistent high-inflation environment, can a fund manager find a simple asset
allocation strategy that consistently beats the benchmark passive portfolios
by a reasonable margin (preferably without spending billions of dollars in
personnel costs)?

###  3.2 Mathematical formulation

In this section, we mathematically formulate the problem of outperforming a
benchmark. Let $[t_{0}(=0),T]$ denote the investment horizon, and let $W(t)$
denote the wealth (value) of the portfolio actively managed by the manager at
time $t\in[t_{0},T]$. We refer to the actively managed portfolio as the
“active portfolio”. Furthermore, let $\hat{W}(t)$ denote the wealth of the
benchmark portfolio at time $t\in[t_{0},T]$. To ensure a fair assessment of
the relative performance of the two portfolios, we assume both portfolios
start with an equal initial wealth amount $w_{0}>0$, , i.e.,
$W(t_{0})=\hat{W}(t_{0})=w_{0}>0$.

Technically, the admissible sets of underlying assets for the active and
passive portfolio need not be identical. However, for simplicity, we assume
that both the active portfolio and the benchmark portfolio can allocate wealth
to the same set of $N_{a}$ assets.

Let vector $\bm{S}(t)=(S_{i}(t):i=1,\cdots,N_{a})^{\top}\in\mathbb{R}^{N_{a}}$
denote the asset prices of the $N_{a}$ underlying assets at time
$t\in[t_{0},T]$. In addition, let vectors
$\bm{p}(t)=(p_{i}(t):i=1,\cdots,N_{a})^{\top}\in\mathbb{R}^{N_{a}}$ and
$\bm{\hat{p}}(t)=(\hat{p}_{i}(t):i=1,\cdots,N_{a})^{\top}\in\mathbb{R}^{N_{a}}$
denote the fraction of wealth allocated to the $N_{a}$ underlying assets at
time $t\in[t_{0},T]$, respectively, for the active portfolio and the benchmark
portfolio. From a control theory perspective, the allocation vectors $\bm{p}$
and $\hat{\bm{p}}$ can be regarded as the controls of the system, as they
determine how the wealth of the portfolios evolves over time. We will seek to
find the optimal feedback control. In other words, the closed loop controls
(allocation decisions) are assumed to also depend on the value of the state
variables (e.g. portfolio wealth). Therefore, we consider $\bm{p}$ and
$\hat{\bm{p}}$ as functions of time as well as the state variables.
Mathematically,
$\bm{p}(\bm{X}(t))=(p_{i}(\bm{X}(t)):i=1,\cdots,N_{a})^{\top}\in\mathbb{R}^{N_{a}}$
and
$\bm{\hat{p}}(\hat{\bm{X}}(t))=(\hat{p}_{i}(\hat{\bm{X}}(t)):i=1,\cdots,N_{a})^{\top}\in\mathbb{R}^{N_{a}}$,
where ${\bm{X}(t)}\in\mathcal{X}\subseteq\mathbb{R}^{N_{x}}$ and
$\hat{\bm{X}}(t)\in\hat{\mathcal{X}}\subseteq\mathbb{R}^{N_{\hat{x}}}$ are the
state variables taken into account by the active portfolio and the benchmark
portfolio respectively. Since the controls depend on time $t$, here we include
$t$ in $\bm{X}(t)$ and $\hat{\bm{X}}(t)$ for notational simplicity. In this
article, we consider the particular problem of outperforming a passive
portfolio, in which $\bm{X}(t)=\big{(}t,W(t),\hat{W}(t)\big{)}^{\top}$.
Without further specification, we use $\bm{X}(t)$ to represent
$\big{(}t,W(t),\hat{W}(t)\big{)}^{\top}$ in the rest of the article.

We assume that the active portfolio and the benchmark portfolio follow the
same rebalancing schedule denoted by $\mathcal{T}\subseteq[t_{0},T]$. In the
case of discrete rebalancing, $\mathcal{T}\subset[t_{0},T]$ is a discrete set.
In the case of continuous rebalancing, $\mathcal{T}=[t_{0},T]$, i.e.,
rebalancing happens continuously throughout the entire investment horizon.

Additionally, we assume both portfolios follow the same deterministic sequence
of cash injections, defined by the set
$\mathcal{C}=\\{c(t),\;t\in\mathcal{T}_{c}\\}$, where
$\mathcal{T}_{c}\subseteq[t_{0},T]$ is the schedule of the cash injections.
When $\mathcal{T}_{c}$ is a discrete injection schedule, $c(t)$ is the amount
of cash injection at $t$. In the case of continuous cash injection, i.e.,
$\mathcal{T}=[t_{0},T]$, $c(t)$ is the rate of cash injection at $t$, i.e.,
the total cash injection amount during $[t,t+dt]$ is $c(t)dt$, where $dt$ is
an infinitesimal time interval. For simplicity, we assume that
$\mathcal{T}_{c}=\mathcal{T}$, so that the cash injections schedule is the
same as the rebalancing schedule. At $t\in\mathcal{T}$, $W(t)$ and
$\hat{W}(t)$ always denote the wealth after the cash injection (assuming there
is a cash injection event happening at $t$).

The active and benchmark strategies, respectively, are defined as the sequence
of the allocation fractions following the rebalancing schedule.
Mathematically, the active and benchmark strategy are defined by sets

$\mathcal{P}=\\{\bm{p}(\bm{X}(t)),\;t\in\mathcal{T}\\},\quad\text{and}\quad\hat{\mathcal{P}}=\\{\bm{\hat{p}}(\hat{\bm{X}}(t)),\;t\in\mathcal{T}\\}.$
(3.1)

Denote $\mathcal{A}$ as the set of admissible strategies, which reflects the
investment constraints on the controls. Mathematically, let
{$\mathcal{X}_{i}$: $i=1,\cdots,k$} be a partition of $\mathcal{X}$ (the state
variable space), i.e.

$\begin{cases}\bigcup_{i=1}^{k}\mathcal{X}_{i}=\mathcal{X},\\\
\mathcal{X}_{i}\bigcap\mathcal{X}_{j}=\varnothing,\forall 1\leq i<j\leq
k,\end{cases}$ (3.2)

and {$\mathcal{Z}_{i}\subseteq\mathbb{R}^{N_{a}}$: $i=1,\cdots,k$} be the
corresponding range of feasible controls such that any feasible control
$\bm{p}$ satisfies

$\bm{p}(\bm{x})\in\mathcal{Z}_{i},\forall\bm{x}\in\mathcal{X}_{i},\;\forall
i\in\\{1,\cdots,k\\}.$ (3.3)

We say that strategy $\mathcal{P}$ is an admissible strategy, i.e.,
$\mathcal{P}\in\mathcal{A}$, if and only if

$\mathcal{P}=\Big{\\{}\bm{p}(\bm{X}(t)),\;t\in\mathcal{T}\Big{|}\bm{p}(\bm{X}(t))\in\mathcal{Z}_{i},\;\text{if
}\bm{X}(t)\in\mathcal{X}_{i}\Big{\\}}$ (3.4)

Consider a discrete rebalancing schedule
$\mathcal{T}=\\{t_{j},\;j=0,\cdots,N\\}$ with $N$ rebalancing events, where
$t_{0}<t_{1}<\cdots<t_{N}=T$.888Technically, at $t=t_{0}$, the manager makes
the initial asset allocation, rather than a “rebalancing” of the portfolio.
However, despite the different purposes, a rebalancing of the portfolio is
simply a new allocation of the portfolio wealth. Therefore, for notational
simplicity, we include $t_{0}$ in the rebalancing schedule. Then, the wealth
evolution of the active portfolio and the benchmark portfolio can be described
by the equations

$\begin{cases}W(t_{j+1})=\Big{(}\sum\limits_{i=1}\limits^{N_{a}}p_{i}(\bm{X}(t_{j}))\cdot\frac{S_{i}(t_{j+1})-S_{i}(t_{j})}{S_{i}(t_{j})}\Big{)}W(t_{j})+c(t_{j+1}),\;j=0,\cdots,N-1,\\\
\hat{W}(t_{j+1})=\Big{(}\sum\limits_{i=1}\limits^{N_{a}}\hat{p}_{i}(\hat{\bm{X}}(t_{j}))\cdot\frac{S_{i}(t_{j+1})-S_{i}(t_{j})}{S_{i}(t_{j})}\Big{)}\hat{W}(t_{j})+c(t_{j+1}),\;j=0,\cdots,N-1.\end{cases}$
(3.5)

In the continuous rebalancing case, $\mathcal{T}=[t_{0},T]$. Let $dS_{i}(t)$
denote the instantaneous change in price for asset $i$,
$i\in[1,\cdots,N_{a}]$.999For illustration purposes, here we assume
$S_{i}(t),i\in[1,\cdots,N_{a}]$ follow standard diffusion processes, i.e., no
jumps. We will discuss the case with jumps in detail in Section 3.4. Then, at
$t\in\mathcal{T}=[t_{0},T]$, the wealth dynamics of the active portfolio and
the benchmark portfolio, following their respective strategies $\mathcal{P}$
and $\hat{\mathcal{P}}$, can be described by the equations

$\begin{cases}dW(t)=\Big{(}\sum\limits_{i=1}\limits^{N_{a}}p_{i}(\bm{X}(t))\cdot\frac{dS_{i}(t)}{S_{i}(t)}\Big{)}W(t_{j})+c(t)dt,\\\
d\hat{W}(t)=\Big{(}\sum\limits_{i=1}\limits^{N_{a}}\hat{p}_{i}(\hat{\bm{X}}(t))\cdot\frac{dS_{i}(t)}{S_{i}(t)}\Big{)}\hat{W}(t_{j})+c(t)dt.\end{cases}$
(3.6)

Let sets $\mathcal{W}_{\mathcal{P}}=\\{W(t),t\in\mathcal{T}\\}$ and
$\hat{\mathcal{W}}_{\hat{\mathcal{P}}}=\\{\hat{W}(t),t\in\mathcal{T}\\}$
denote the wealth trajectories of the active portfolio and the benchmark
portfolio following their respective investment strategies $\mathcal{P}$ and
$\hat{\mathcal{P}}$. Let
$F(\mathcal{W}_{\mathcal{P}},\hat{\mathcal{W}}_{\hat{\mathcal{P}}})\in\mathbb{R}$
denote an investment metric that measures the performances of the active and
benchmark strategies, based on their respective wealth trajectories.

In this article, we assume that the asset prices
$\bm{S}(t)\in\mathbb{R}^{N_{a}}$ are stochastic. Then, the wealth trajectories
$\mathcal{W}_{\mathcal{P}}$ and $\hat{\mathcal{W}}_{\hat{\mathcal{P}}}$ are
also stochastic, as well as the performance metric
$F(\mathcal{W}_{\mathcal{P}},\hat{\mathcal{W}}_{\hat{\mathcal{P}}})$, which
measures the relative performance of the active strategy with respect to the
benchmark strategy. Therefore, when investment managers target to optimize an
investment metric, the evaluation is often via taking the expectation of the
random metric.

Let
$\mathbb{E}_{\mathcal{P}}^{(t_{0},w_{0})}[F(\mathcal{W}_{\mathcal{P}},\hat{\mathcal{W}}_{\hat{\mathcal{P}}})]$
denote the expectation of the value of the performance metric $F$, with
respect to a given initial wealth $w_{0}=W(0)=\hat{W}(0)$ at time $t_{0}=0$,
following an admissible investment strategies $\mathcal{P}\in\mathcal{A}$, and
the benchmark investment strategy $\hat{\mathcal{P}}$. The benchmark strategy
is often pre-determined and known, hence we keep the benchmark strategy
$\hat{\mathcal{P}}$ implicit in this notation for simplicity. Going forward,
we refer to
$\mathbb{E}_{\mathcal{P}}^{(t_{0},w_{0})}[F(\mathcal{W}_{\mathcal{P}},\hat{\mathcal{W}}_{\hat{\mathcal{P}}})]$,
the expectation of a desired performance metric, as the (investment) objective
function.

Finally, we can mathematically define the investment objective (the
optimization problem) associated with a general performance metric $F$ as

$\text{(Optimization
problem)}\quad\inf_{\mathcal{P}\in\mathcal{A}}\mathbb{E}_{\mathcal{P}}^{(t_{0},w_{0})}\big{[}F(\mathcal{W}_{\mathcal{P}},\hat{\mathcal{W}}_{\hat{\mathcal{P}}})\big{]}.$
(3.7)

###  3.3 Choice of investment objective

The first step to designing a proper outperforming investment objective is to
clarify the definition of beating the benchmark. In the context of measuring
the performance of the portfolio against the benchmark, a common metric is the
tracking error, which measures the volatility of the difference in returns,
i.e.,

$\text{Tracking error}=stdev(R-\hat{R}),$ (3.8)

where $R$ denotes the return of the active portfolio, and $\hat{R}$ denotes
the return of the benchmark portfolio. Note that the returns of the active
portfolio and the benchmark portfolio are determined from their respective
wealth trajectories ($\mathcal{W}_{\mathcal{P}}$ and
$\hat{\mathcal{W}}_{\hat{\mathcal{P}}}$) that are evaluated under the same
investment horizon and same market conditions. The tracking error measures the
volatility of the difference in returns over the investment horizon. A
criticism of the tracking error is that it only measures the variability in
the difference in returns, but does not reflect the magnitude of the return
difference itself. For example, an active strategy with a constant negative
return difference over the investment horizon would yield a better tracking
error than an active strategy with a positive but volatile return difference.
For this reason, many prefer the tracking difference (Johnson et al., 2013;
Hougan, 2015; Charteris and McCullough, 2020; Boyde, 2021), which is defined
as the annualized difference between the active portfolio’s cumulative return
and the benchmark portfolio’s cumulative return over a specific period. Note
that both tracking error and tracking difference metrics measure the return
difference of the active portfolio over the benchmark portfolio. In other
words, these metrics measure how closely the return of the active portfolio
tracks the return of the benchmark portfolio. In practice, if an investment
manager aims to achieve a certain annualized relative return target, e.g.,
$\beta$, then the tracking difference metric may not be appropriate. To
address this, van Staden et al. (2022) suggests the investment objective

$\inf_{\mathcal{P}\in\mathcal{A}}\mathbb{E}_{\mathcal{P}}^{(t_{0},w_{0})}\Big{[}\Big{(}W(T)-e^{\beta
T}\hat{W}(T)\Big{)}^{2}\Big{]},$ (3.9)

where $W(T)$ and $\hat{W}(T)$ are the respective terminal wealth of the active
portfolio and the benchmark portfolio at terminal time $T$, and $\beta$ is the
annualized relative return target.

The optimal control problem (3.9) aims to produce an active strategy that
minimizes the quadratic difference between $W(T)$ and the terminal portfolio
value target of $e^{\beta T}\hat{W}(T)$. In other words, the optimal control
policy tries to outperform the benchmark portfolio by a total factor of
$e^{\beta T}$ over the time horizon $[0,T]$, which is equivalent to an
annualized relative return of $\beta$. The quadratic term of the difference
incentivizes the terminal wealth of the active portfolio $W(T)$ to closely
track the elevated target $e^{\beta T}$.

It is worth noting that the relative return target $\beta$ can be intuitively
interpreted as the manager’s willingness to take more risks. As
$\beta\downarrow 0$, the optimal solution to problem (3.9) is simply to mimic
the benchmark strategy. However, as $\beta$ grows larger, the manager needs to
take on more risk (for more return) in order to beat the benchmark portfolio
by the relative return target rate.

A criticism of the investment objective (3.9) is that it is symmetrical in
terms of the outperformance and underperformance of $W(T)$ relative to the
elevated target $e^{\beta T}\hat{W}(T)$. This is a common issue for
volatility-type measures, such as the Sharpe ratio (Ziemba, 2005). In
practice, investors may favor outperformance more than underperformance, while
still aiming to track the elevated target closely. Acknowledging this, instead
of (3.9), Ni et al. (2022) propose the following asymmetrical objective
function,

$\inf_{\mathcal{P}\in\mathcal{A}}\mathbb{E}_{\mathcal{P}}^{(t_{0},w_{0})}\Bigg{[}\Big{(}\min(W(T)-e^{\beta
T}\hat{W}(T),0)\Big{)}^{2}+\max\big{(}W(T)-e^{\beta
T}\hat{W}(T),0\big{)}\Bigg{]}.$ (3.10)

The investment objective (3.10) penalizes the outperformance (of $W(T)$
relative to the elevated target $e^{\beta T}\hat{W}(T)$) linearly but the
underperformance quadratically, thus encouraging the optimal policy to favor
outperformance more than underperformance when necessary. Note that the use of
objective function (3.10) does not permit closed-form solutions and requires
machine learning techniques (Ni et al., 2022) to derive the desired optimal
strategy numerically.

Another criticism of the investment objective (3.9) and (3.10) is that both
are only concerned with the relative performance at terminal time $T$.
However, in reality, investment managers are often required to report
intermediate portfolio performance internally or externally at regular time
intervals. Instead of achieving the annualized relative return target when
reviewing the portfolio performance at the end of the investment horizon, it
is more desirable for managers to consistently achieve the relative return
target throughout the entire investment horizon. Therefore, it is often
important for investment managers to set an investment objective function to
control the deviation of the wealth of the portfolio from the target along a
market scenario within the investment horizon. For this reason, van Staden et
al. (2022) propose the following cumulative quadratic tracking difference (CD)
objectives

$\displaystyle(CD(\beta)):\quad$
$\displaystyle\inf_{\mathcal{P}\in\mathcal{A}}\mathbb{E}_{\mathcal{P}}^{(t_{0},w_{0})}\Bigg{[}\int_{t_{0}}^{T}\Big{(}W(t)-e^{\beta
t}\hat{W}(t)\Big{)}^{2}dt\Bigg{]},\;\text{if }\mathcal{T}=[t_{0},T],$ (3.11)
$\displaystyle(CD(\beta)):\quad$
$\displaystyle\inf_{\mathcal{P}\in\mathcal{A}}\mathbb{E}_{\mathcal{P}}^{(t_{0},w_{0})}\Bigg{[}\sum_{t\in\mathcal{T}\cup\\{T\\}}\Big{(}W(t)-e^{\beta
t}\hat{W}(t)\Big{)}^{2}\Bigg{]},\;\text{if
}\mathcal{T}\subseteq[t_{0},T],\mathcal{T}\text{ discrete}.$ (3.12)

Here, note that objective (3.11) is for the continuous rebalancing case, and
(3.12) for discrete rebalancing. Both (3.11) and (3.12) measure the cumulative
deviation of the wealth of the active portfolio relative to the target, along
a market scenario within the entire investment horizon. Therefore, they
measure the intermediate performance deviations effectively. However, similar
to (3.9), (3.11) and (3.12) penalize outperformance and underperformance
symmetrically. Therefore, we also consider following cumulative quadratic
shortfall (CS) objectives that only penalize the shortfall (underperformance
with respect to the target)

$\displaystyle(CS(\beta)):\quad$
$\displaystyle\inf_{\mathcal{P}\in\mathcal{A}}\mathbb{E}_{\mathcal{P}}^{(t_{0},w_{0})}\Bigg{[}\int_{t_{0}}^{T}\Big{(}\min\big{(}W(t)-e^{\beta
t}\hat{W}(t),0\big{)}\Big{)}^{2}dt+\epsilon W(T)\Bigg{]},\;\text{if
}\mathcal{T}=[t_{0},T],$ (3.13) $\displaystyle(CS(\beta)):\quad$
$\displaystyle\inf_{\mathcal{P}\in\mathcal{A}}\mathbb{E}_{\mathcal{P}}^{(t_{0},w_{0})}\Bigg{[}\sum_{t\in\mathcal{T}\cup\\{T\\}}\Big{(}\min\big{(}W(t)-e^{\beta
t}\hat{W}(t),0\big{)}\Big{)}^{2}+\epsilon W(T)\Bigg{]},\;\text{if
}\mathcal{T}\subseteq[t_{0},T],\mathcal{T}\text{ discrete}.$
$\displaystyle(CS(\beta)):\quad$ (3.14)

Here (3.13) and (3.14) are the investment objectives for the continuous
rebalancing and discrete rebalancing cases respectively. $\epsilon$ is a small
regularization parameter to ensure that problem (3.13) and (3.14) are well-
posed. A more detailed comparison of the CD and CS objective functions can be
found in Appendix F.

###  3.4 Closed-form solution for CD problem

In this section, we present the closed-form solution to the CD problem (3.11)
under several assumptions. The closed-form solution not only provides us with
insights for understanding the CD-optimal controls for problem (3.11), but
also serves as the baseline for understanding the numerical results derived
from the neural network method (discussed in later sections). Specifically, in
this section, we consider the case that all asset prices follow jump-diffusion
processes and portfolios with cash injections, which are aspects not
frequently considered in benchmark outperformance literature Browne (1999,
2000); Tepla (2001); Basak et al. (2006); Yao et al. (2006); Zhao (2007);
Davis and Lleo (2008); Lim and Wong (2010b); Oderda (2015); Zhang and Gao
(2017); Al-Aradi and Jaimungal (2018b); Nicolosi et al. (2018); Bo et al.
(2021).

We first summarize the assumptions for obtaining the closed-form solution to
CD problem (3.11).

######  Assumption 3.1.

(Two assets, no friction, unlimited leverage, trading in insolvency, constant
rate of cash injection) The active portfolio and the benchmark portfolio have
access to two underlying assets, a stock index, and a constant-maturity bond
index. Both portfolios are rebalanced continuously, i.e.,
$\mathcal{T}=[t_{0},T]$. There is no transaction cost and no leverage limit.
Furthermore, we assume that trading continues in the event of insolvency,
i.e., when $W(t)<0$ for some $t\in[t_{0},T]$. Finally, we assume both
portfolios receive constant cash injections with an injection rate of $c$,
which means during any time interval $[t,t+\Delta
t]\subseteq[t_{0},T],\forall\Delta t>0$, both portfolios receive cash
injection amount of $c\Delta t$.

######  Remark 3.1.

(Remark on Assumption 3.1) For illustration purposes, we assume only two
underlying assets. However, the technique for deriving the closed-form
solution can be extended to multiple assets. We remark that unlimited leverage
is unrealistic, and is only assumed for deriving the closed-form solution. In
Section 4.1.1, we will discuss the technique for handling the leverage
constraint in more detail. We also acknowledge that it is not realistic to
assume that the manager can continue to trade and borrow when insolvent.
However, this assumption is typically required for obtaining closed-form
solutions, see Zhou and Li (2000); Li and Ng (2000) for the case of a multi-
period mean-variance asset allocation problem. In Section 4.1.1, we will
discuss the impact of insolvency and its handling in experiments.

######  Assumption 3.2.

(Fixed-mix benchmark strategy) We assume that the benchmark strategy is a
fixed-mix strategy (also known as the constant weight strategy). We assume the
benchmark always allocates a constant fraction of
$\hat{\varrho}\;(\in\mathbb{R})$ of the portfolio wealth to the stock index,
and a constant fraction of $1-\hat{\varrho}$ to the bond index. Let
$\hat{\bm{\varrho}}=(\hat{\varrho},1-\hat{\varrho})^{\top}\in\mathbb{R}^{2}$
denote the vector of allocation fractions to the stock index and the bond
index, the benchmark strategy is the fixed-mix strategy defined by
$\hat{\mathcal{P}}=\\{\hat{\bm{p}}(\hat{\bm{X}}(t))=\big{(}\hat{p}_{1}(\hat{\bm{X}}(t)),\hat{p}_{2}(\hat{\bm{X}}(t))\big{)}^{\top}\equiv\hat{\bm{\varrho}},\;\forall
t\in\mathcal{T}\\}$.

Finally, we assume the stock index price and bond index price follow the jump-
diffusion processes described below.

######  Assumption 3.3.

(Jump-diffusion processes) Let $S_{1}(t)$ and $S_{2}(t)$ denote the deflated
(adjusted by inflation) price of the stock index and the bond index at time
$t\in[t_{0},T]$. We assume $S_{i}(t),\;i\in\\{1,2\\}$ follow the jump-
diffusion processes

$\frac{dS_{i}(t)}{S_{i}(t^{-})}=(\mu_{i}-\lambda_{i}\kappa_{i}+{r_{i}\cdot\textbf{1}_{S_{i}(t^{-})<0}})dt+\sigma_{i}dZ_{i}(t)+d\Big{(}\sum_{k=1}^{\pi_{i}(t)}(\xi_{i}^{(k)}-1)\Big{)},\;i=1,2.$
(3.15)

Here $\mu_{i}$ are the (uncompensated) drift rate, $\sigma_{i}$ is the
diffusive volatility, $Z_{1}(t),Z_{2}(t)$ are correlated Brownian motions,
where $\mathbb{E}[dZ_{1}(t)\cdot dZ_{2}(t)]=\rho dt$. $r_{i}$ are the
borrowing premiums when $S_{i}(t^{-})$ is negative.101010Intuitively, there is
a premium for shorting an asset. In the closed-form solution derivation, we
assume $r_{i}=0$. $\pi_{i}(t)$ is a Poisson process with positive intensity
parameter $\lambda_{i}$. $\\{\xi_{i}^{(k)},\;k=1,\cdots,\pi_{i}(t)\\}$ are
i.i.d. positive random variables that describe jump multipliers associated
with the assets. If a jump occurs for asset $i$ at time $t\in(t_{0},T]$, its
underlying price jumps from $S_{i}(t^{-})$ to $S_{i}(t)=\xi_{i}\cdot
S_{i}(t^{-})$.111111For any functional $\psi(t)$, we use the notation
$\psi(t^{-})$ as shorthand for the left-sided limit $\psi(t^{-})=\lim_{\Delta
t\downarrow 0}\psi(t-\Delta t)$. $\kappa_{i}=\mathbb{E}[\xi_{i}-1]$. $\xi_{i}$
and $\pi_{i}(t)$ are independent of each other. Moreover, $\pi_{1}(t)$ and
$\pi_{2}(t)$ are assumed to be independent.121212See Forsyth (2020) for the
discussion on the empirical evidence for stock-bond jump independence. Also
note that the assumption of independent jumps can be relaxed without technical
difficulty if needed (Kou, 2002), but will significantly increase the
complexity of notations.

######  Remark 3.2.

(Motivation for jump-diffusion model) The assumption of stock index price
following a jump-diffusion model is common in the financial mathematics
literature (Merton, 1976; Kou, 2002). In addition, we follow the practitioner
approach and directly model the returns of the constant maturity bond index as
a stochastic process, see for example Lin et al. (2015); MacMinn et al.
(2014). As in MacMinn et al. (2014), we also assume that the constant maturity
bond index follows a jump-diffusion process. During high-inflation regimes,
central banks often make rate hikes to curb inflation, which causes sudden
jumps in bond prices (Lahaye et al., 2011). We believe this is an appropriate
assumption for bonds in high-inflation regimes.

Under the jump-diffusion model (3.15), the wealth processes for the active
portfolio and benchmark portfolio are

$\begin{cases}dW(t)=\Big{(}\sum\limits_{i=1}\limits^{N_{a}}p_{i}(\bm{X}(t^{-}))\cdot\frac{dS_{i}(t)}{S_{i}(t^{-})}\Big{)}W(t_{j}^{-})+cdt,\\\
d\hat{W}(t)=\Big{(}\sum\limits_{i=1}\limits^{N_{a}}\hat{p}_{i}(\hat{\bm{X}}(t^{-}))\cdot\frac{dS_{i}(t)}{S_{i}(t^{-})}\Big{)}\hat{W}(t_{j}^{-})+cdt,\end{cases}$
(3.16)

where $t\in(t_{0},T]$, $W(t_{0})=\hat{W}(t_{0})=w_{0}$ and
$X(t^{-})=(t,W(t^{-}),\hat{W}(t^{-}))^{\top}\in\mathbb{R}^{3}$ is the state
variable vector.

We now derive the closed-form solution of the CD problem (3.11) under
Assumption 3.1, 3.2 and 12. We first present the verification theorem for the
HJB integro-differential equation (PIDE) satisfied by the value function and
the optimal control of the CD problem (3.11).

######  Theorem 3.1.

(Verification theorem for CD problem (3.11) For a fixed $\beta>0$, assume that
for all $(t,w,\hat{w},\hat{\varrho})\in[t_{0},T]\times\mathbb{R}^{3}$, there
exists a function
$V(t,w,\hat{w},\hat{\varrho}):[t_{0},T]\times\mathbb{R}^{3}\mapsto\mathbb{R}$
and
$p^{*}(t,w,\hat{w},\hat{\varrho}):[t_{0},T]\times\mathbb{R}^{3}\mapsto\mathbb{R}^{2}$
that satisfy the following two properties. (i) $V$ and $\bm{p}^{*}$ are
sufficiently smooth and solve the HJB PIDE (3.17), and (ii) the function
$\bm{p}^{*}(t,w,\hat{w},\hat{\varrho})$ attains the pointwise infimum in
(3.17) below

$\begin{cases}\frac{\partial V}{\partial t}+(w-e^{\beta
t}\hat{w})^{2}+\inf\limits_{\bm{p}\in\mathbb{R}^{2}}H(\bm{p};t,w,\hat{w},\hat{\bm{\varrho}})=0,\\\
V(T,w,\hat{w},\hat{\varrho})=0,\end{cases}$ (3.17)

where

$\displaystyle H(\bm{p};t,w,\hat{w},\hat{\bm{\varrho}})=$
$\displaystyle\big{(}w\cdot\bm{\alpha}^{\top}\bm{p}+c\big{)}\cdot\frac{\partial
V}{\partial
w}+\big{(}\hat{w}\cdot\bm{\alpha}^{\top}\hat{\bm{\varrho}}+c\big{)}\cdot\frac{\partial
V}{\partial\hat{w}}-\Big{(}\sum\limits_{i}\lambda_{i}\Big{)}\cdot
V(t,w,\hat{w},\hat{\varrho})$
$\displaystyle+\frac{w^{2}}{2}\cdot\big{(}\bm{p}^{\top}\bm{\Sigma}\bm{p}\big{)}\cdot\frac{\partial^{2}V}{\partial
w^{2}}+\frac{\hat{w}^{2}}{2}\cdot\big{(}\hat{\bm{\varrho}}^{\top}\bm{\Sigma}\hat{\bm{\varrho}}\big{)}\cdot\frac{\partial^{2}V}{\partial\hat{w}^{2}}+w\hat{w}\cdot\big{(}{\bm{p}}^{\top}\bm{\Sigma}\hat{\bm{\varrho}}\big{)}\cdot\frac{\partial^{2}V}{\partial
w\partial\hat{w}}$
$\displaystyle+\sum\limits_{i}\lambda_{i}\int_{0}^{\infty}V(w+p_{i}w(\xi-1),\hat{w}+\hat{p}_{i}\hat{w}(\xi-1),t,\hat{\varrho})f_{\xi_{i}}(\xi)d\xi.$
(3.18)

Here
$\bm{\alpha}=(\mu_{1}-\lambda_{1}\kappa_{1},\mu_{2}-\lambda_{2}\kappa_{2})^{\top}$
is the vector of (compensated) drift rates,
$\bm{\Sigma}=\begin{bmatrix}\sigma_{1}^{2}&\rho\sigma_{1}\sigma_{2}\\\
\rho\sigma_{1}\sigma_{2}&\sigma_{2}^{2}\end{bmatrix}$ is the covariance
matrix, and $f_{\xi_{i}}$ is the density function for $\xi_{i}$.

Then, under Assumption 3.1, 3.2 and 12, $V$ is the value function and
$\bm{p}^{*}$ is the optimal control for the CD problem (3.11).

###### Proof.

See Appendix C.1 ∎

Define several auxiliary variables

$\begin{cases}\kappa_{i}^{(2)}=\mathbb{E}\big{[}(\xi_{i}-1)^{2}\big{]},\quad(\sigma_{i}^{(2)})^{2}=(\sigma_{i})^{2}+\lambda_{i}\kappa_{i}^{(2)},\;i\in\\{1,2\\},\\\
\vartheta=\sigma_{1}\sigma_{2}\rho-(\sigma_{2}^{(2)})^{2},\quad\gamma=(\sigma_{1}^{(2)})^{2}+(\sigma_{2}^{(2)})^{2}-2{\sigma_{1}\sigma_{2}}\rho,\\\
\phi=\frac{(\mu_{1}-\mu_{2})(\mu_{1}-\mu_{2}+\vartheta)}{\gamma},\quad\eta=\frac{(\mu_{1}-\mu_{2}+\theta)^{2}}{\gamma}-(\sigma_{2}^{(2)})^{2},\end{cases}$
(3.19)

then we have the following proposition regarding the optimal control to
problem (3.11).

######  Proposition 3.1.

(CD-optimal control) Suppose Assumption 3.1, 3.2 and 12 are applicable, then
the optimal control fraction of the wealth of the active portfolio to be
invested in the stock index for the $CD(\beta)$ problem (3.11) is given by
${p}^{*}(t,w,\hat{w},\hat{\varrho})\in\mathbb{R}$, where

$p^{*}(t,w,\hat{w},\hat{\varrho})=\frac{1}{W^{*}(t)}\Bigg{[}\frac{(\mu_{1}-\mu_{2})}{\gamma}h(t;\beta,c)+\frac{(\mu_{1}-\mu_{2}+\vartheta)}{\gamma}\Big{(}g(t;\beta)\hat{W}(t)-W^{*}(t)\Big{)}+g(t;\beta)\hat{W}(t)\cdot\hat{\varrho}\Bigg{]}.$
(3.20)

Here $W^{*}(t)$ denotes the wealth process of the active portfolio from (3.6)
following control
$\bm{p}^{*}(t,W^{*}(t),\hat{W}(t),\hat{\varrho})=\Big{(}p^{*}(t,W^{*}(t),\hat{W}(t),\hat{\varrho}),1-p^{*}(t,W^{*}(t),\hat{W}(t),\hat{\varrho})\Big{)}^{\top}$,
where $p^{*}$ is the optimal stock allocation described in (3.20), and
$\hat{W}(t)$ is the wealth process of the benchmark portfolio following the
fixed-mixed strategy described in Assumption 3.2. Here, $h$ and $g$ are
deterministic functions of time,

$g(t;\beta)=-\frac{D(t;\beta)}{2A(t)},\quad\quad
h(t;\beta,c)=-\frac{B(t;\beta,c)}{2A(t)},$ (3.21)

where $A,D$ and $B$ are deterministic functions defined as

$A(t)=\frac{e^{(2\mu_{2}-\eta)(T-t)}-1}{(2\mu_{2}-\eta)},\quad\quad
D(t;\beta)=2e^{\beta
T}\Big{(}\frac{e^{-\beta(T-t)}-e^{(2\mu_{2}-\eta)(T-t)}}{2\mu_{2}-\eta+\beta}\Big{)},$
(3.22)

and

$\displaystyle B(t;\beta,c)$
$\displaystyle=\frac{2c}{2\mu_{2}-\eta}\Big{(}\frac{e^{(2\mu_{2}-\eta)(T-t)}-e^{(\mu_{2}-\phi)(T-t)}}{\mu_{2}+\phi-\eta}-\frac{e^{(\mu_{2}-\phi)(T-t)}-1}{\mu_{2}-\phi}\Big{)}$
$\displaystyle+\frac{2ce^{\beta
T}}{2\mu_{2}-\eta+\beta}\Big{(}\frac{e^{(\mu_{2}-\phi)(T-t)}-e^{-\beta(T-t)}}{\mu_{2}-\phi+\beta}-\frac{e^{(2\mu_{2}-\eta)(T-t)}-e^{(\mu_{2}-\phi)(T-t)}}{\mu_{2}+\phi-\eta}\Big{)}.$
(3.23)

###### Proof.

See Appendix C.2. ∎

####  3.4.1 Insights from CD-optimal control

The CD-optimal control (3.20) provides insights into the behaviour of the
optimal allocation policy. For ease of exposition, we first establish the
following properties of $g(t;\beta)$ and $h(t;\beta,c)$.

######  Corollary 3.1.

(Properties of $g(t;\beta)$) The function $g(t;\beta)$ defined in (3.21) has
the following properties for $t\in[t_{0},T]$ and $\beta>0$:

  1. (i)

For fixed $t\in[t_{0},T]$, $g(t;\beta)$ is strictly increasing on
$\beta\in(0,\infty)$.

  2. (ii)

For fixed $\beta>0$, $g(t;\beta)$ is strictly increasing on $t\in[t_{0},T]$.

  3. (iii)

$g(t;\beta)$ admits the following bounds:

$e^{\beta t}\leq g(t;\beta)\leq e^{\beta T}.$ (3.24)

###### Proof.

See Appendix C.3. ∎

######  Corollary 3.2.

(Properties of $h(t;\beta,c)$) The function $h(t;\beta,c)$ defined in (3.21)
has the following properties for $t\in[t_{0},T]$, $\beta>0$ and $c\geq 0$:

  1. (i)

For fixed $t\in[t_{0},T]$ and $c>0$, $h(t;\beta,c)$ is strictly increasing on
$\beta\in(0,\infty)$.

  2. (ii)

$h(t;\beta,c)\geq 0$,
$\forall(t,\beta,c)\in[t_{0},T]\times(0,\infty)\times[0,\infty)$.

  3. (iii)

For fixed $t\in[t_{0},T]$ and $\beta>0$, $h(t;\beta,c)$ is strictly increasing
on $c\in[0,\infty)$. $h(t;\beta,0)\equiv 0$. Moreover, $h(t;\beta,c)\propto
c$, i.e. $h(t;\beta,c)$ is proportional to $c$.

###### Proof.

See Appendix C.3. ∎

In order to analyze the closed-form solution, we make the following
assumptions.

######  Assumption 3.4.

(Drift rates of the two assets) We assume that the drift rates of the stock
and the bond index $\mu_{1}$ and $\mu_{2}$ satisfy the following properties,

$\mu_{1}-\mu_{2}>0,\quad\mu_{1}-\mu_{2}+\vartheta>0,$ (3.25)

where $\vartheta$ is defined in (3.19).

######  Remark 3.3.

(Remark on drift rate assumptions) The first inequality $\mu_{1}-\mu_{2}>0$
indicates that the stock index has a higher drift rate than the bond index,
which is a standard assumption.131313In fact, in this two-asset case, this
assumption does not cause loss of generality. The second inequality
$\mu_{1}-\mu_{2}+\vartheta>0$ is also practically reasonable. $\vartheta$ is a
variance term that is usually on a smaller scale compared to the drift rates.
In reality, it is unlikely that $\mu_{1}-\mu_{2}>0$ but
$\mu_{1}-\mu_{2}+\vartheta\leq 0$.141414For reference, based on the calibrated
jump-diffusion model (3.15) on historical high-inflation regimes,
$\mu_{1}=0.051,\mu_{2}=-0.014,\vartheta=-0.00024$, and thus both inequalities
are satisfied.

Now we proceed to summarize the insights from the CD-optimal control (3.20).
The first obvious observation is that the CD-optimal control is a contrarian
strategy. This can be seen from the fact that fixing time and the wealth of
the benchmark portfolio $\hat{W}(t)$, the allocation to the more risky stock
index decreases when the wealth of the active portfolio $W^{*}(t)$ increases.

If we take a deeper look at (3.20), we can see that the CD-optimal control
consists of two components: a cash injection component $p_{cash}^{*}$ and a
tracking component $p_{track}^{*}$. Mathematically,

$p^{*}(t,w,\hat{w},\hat{\varrho})=p_{cash}^{*}(t,w,\hat{w})+p_{track}^{*}(t,w,\hat{w},\hat{\varrho}),$
(3.26)

where

$\begin{cases}p_{cash}^{*}(t,w,\hat{w})=\frac{1}{W^{*}(t)}\Bigg{[}\frac{(\mu_{1}-\mu_{2})}{\gamma}h(t;\beta,c)\Bigg{]},\\\
p_{track}^{*}(t,w,\hat{w},\hat{\varrho})=\frac{1}{W^{*}(t)}\Bigg{[}\frac{(\mu_{1}-\mu_{2}+\vartheta)}{\gamma}\Big{(}g(t;\beta)\hat{W}(t)-W^{*}(t)\Big{)}+g(t;\beta)\hat{W}(t)\cdot\hat{\varrho}\Bigg{]}.\end{cases}$
(3.27)

Based on Assumption 3.4 and Corollary 3.2, the cash injection component
$p_{cash}$ is always non-negative. Furthermore, from Corollary 3.2, we know
that the stock allocation from the cash injection component is proportional to
the cash injection rate $c$. In addition, as $t\uparrow T$, $h(t;\beta,c)$
increases, and thus the stock allocation from the cash injection component
also increases with time.

On the other hand, the tracking component $p_{track}$ does not depend on the
cash injection rate $c$, but only concerns the tracking performance of the
active portfolio. One key finding is that

$\begin{cases}p_{track}^{*}(t,w,\hat{w},\hat{\varrho})\geq\hat{\varrho},\qquad\text{if
}W^{*}(t)\leq g(t;\beta)\hat{W}(t),\\\
p_{track}^{*}(t,w,\hat{w},\hat{\varrho})<\hat{\varrho},\qquad\text{if
}W^{*}(t)>g(t;\beta)\hat{W}(t).\end{cases}$ (3.28)

This means that the CD-optimal control uses $g(t;\beta)\hat{W}(t)$ as the true
target for the active portfolio to decide if the active portfolio should take
more or less risk than the benchmark portfolio. This is a key observation,
since the CD objective function (3.11) measures the difference between $W(t)$
and $e^{\beta t}\hat{W}(t)$. One would naively think that the optimal strategy
would be based on the deviation from $e^{\beta t}\hat{W}(t)$. In contrast,
from Corollary 3.1, we know that the true target $g(t;\beta)\hat{W}(t)$ used
for decision making is greater than $e^{\beta t}\hat{W}(t)$. The insight from
this observation is that, if the manager wants to track an elevated target
$e^{\beta t}\hat{W}(t)$, she should aim higher than the target itself.

###  3.5 Leverage constraints

In practice, large pension funds such as the Canadian Pension Plan often have
exposures to alternative assets, such as private equity (CPP Investments,
2022). Unfortunately, due to practical limitations, we only have access to
long-term historical returns of publicly traded stock indexes and treasury
bond indexes. Although controversial, some literature suggests that returns on
private equity can be replicated using a leveraged small-cap stock index
(Phalippou, 2014; L’Her et al., 2016). Following this line of argument, we
allow managers to take leverage to invest in public stock index funds to
roughly mimic the pension fund portfolios with some exposure to private
equities.

Essentially, taking leverage to invest in stocks requires borrowing additional
capital, which incurs borrowing costs. For simplicity, we assume the borrowing
activity is represented by shorting some bond assets within the portfolio, and
thus the manager is required to pay the cost of shorting these shortable
assets. We assume that the cost consists of two parts: the returns of the
shorted assets, and an additional borrowing premium (rate depends on specific
investment scenarios) so that the total borrowing cost reflects both the
interest rate environment (the return of shorted bond assets) and is
reasonably estimated (with the added borrowing premium).

Following the notation from Section 3.2, we assume that the total $N_{a}$
underlying assets are divided into two groups. The first group of $N_{l}$
assets are long-only assets, which we index by the set $\\{1,\cdots,N_{l}\\}$.
The second group of $N_{a}-N_{l}$ assets are shortable assets that can be
shorted to create leverage and are indexed by the set
$\\{N_{l}+1,\cdots,N_{a}\\}$. Recall the notation of $p_{i}(\bm{X}(t)$ for the
allocation fraction for asset $i$ at time $t$.

For long-only assets, the wealth fraction needs to be non-negative, hence we
have

$\text{(Long-only constraint):}\quad p_{i}(\bm{X}(t))\geq
0,\;i\in\\{1,\cdots,N_{l}\\},\;t\in\mathcal{T}..$ (3.29)

Furthermore, the total allocation fraction for all assets should be one.
Therefore, the following summation constraint needs to be satisfied

$\text{(Summation
constraint):}\quad\sum_{i=1}^{N_{a}}p_{i}(\bm{X}(t))=1,\;t\in\mathcal{T}.$
(3.30)

In practice, due to borrowing costs (from taking leverage) and risk management
mandates, the use of leverage is often constrained. For this reason, we cap
the maximum leverage by introducing a constant $p_{max}$, which represents the
total allocation fraction for long-only assets. Therefore,

$\text{(Maximum leverage
constraint):}\quad\sum_{i=1}^{N_{l}}p_{i}(\bm{X}(t))\leq
p_{max},\;t\in\mathcal{T}.$ (3.31)

Finally, we make the following assumption on the scenario of shorting multiple
shortable assets.

######  Assumption 3.5.

(Simultaneous shorting) If one shortable asset has a negative weight, other
shortable assets must have nonpositive weights. Mathematically, this
assumption can be expressed as

$\text{(Simultaneous shorting constraint):}\begin{cases}\;p_{i}(\bm{X}(t))\leq
0,\;\forall i\in\\{N_{l}+1,\cdots,N_{a}\\},\;\text{if
}\sum_{i=1}^{N_{l}}p_{i}(\bm{X}(t))>1,\;t\in\mathcal{T}\\\
\;p_{i}(\bm{X}(t))\geq 0,\;\forall i\in\\{N_{l}+1,\cdots,N_{a}\\},\;\text{if
}\sum_{i=1}^{N_{l}}p_{i}(\bm{X}(t))\leq 1,\;t\in\mathcal{T}\end{cases}.$
(3.32)

######  Remark 3.4.

(Remark on Assumption 3.32) This assumption avoids the ambiguity between the
long-only assets and shortable assets in scenarios that involve leverage. When
leveraging occurs, all shortable assets are treated as one group to provide
the needed liquidity to achieve the desired leverage level.

The above constraints consider scenarios with non-negative portfolio wealth.
Before we proceed to the handling of the negative portfolio wealth scenarios,
we first define the following partition of the state space $\mathcal{X}$,

######  Definition 3.1.

(Partition of state space) We define
$\big{\\{}\mathcal{X}_{1},\mathcal{X}_{2}\big{\\}}$ to be a partition of the
state space $\mathcal{X}$, such that

$\begin{cases}\mathcal{X}_{1}=\Big{\\{}x=(t,W(t),\hat{W}(t))^{\top}\in\mathcal{X}\Big{|}W(t)\geq
0\Big{\\}},\\\
\mathcal{X}_{2}=\Big{\\{}x=(t,W(t),\hat{W}(t))^{\top}\in\mathcal{X}\Big{|}W(t)<0\Big{\\}}.\end{cases}$
(3.33)

Intuitively, we separate the state space $\mathcal{X}$ into two regions by the
wealth of the active portfolio, one with non-negative wealth and the other
with negative wealth. Then, we present the following assumption concerning the
negative wealth (insolvency) scenarios.

######  Assumption 3.6.

(No trading in insolvency) If the wealth of the active portfolio is negative,
then all long-only asset positions should be liquidated, and all the debt
(i.e. the negative wealth) is allocated to the least-risky shortable asset (in
terms of volatility). Particularly, without loss of generality, we assume all
debt is allocated to asset $N_{l}+1$. Let
$\bm{e}_{i}\in\mathbb{R}^{N_{a}}=(0,\cdots,0,1,0,\cdots,0)^{\top}$ denote the
standard basis vector of which the $i$-th entry is 1 and all other entries are
0. Then, we can formulate this assumption as follows.

$\text{(No trading in insolvency):}\quad
p(\bm{X}(t))=\bm{e}_{N_{l}+1},\quad\text{if }\bm{X}(t)\in\mathcal{X}_{2}.$
(3.34)

######  Remark 3.5.

(Remark on Assumption 3.34) Essentially, when the portfolio wealth is
negative, we assume the debt is allocated to a short-term bond asset and
accumulates over time.

Summarizing the constraints, the corresponding space of feasible control
vector values $\mathcal{Z}$ and the admissible strategy set $\mathcal{A}$ are

(3.36)

Note that the conditional constraints in (LABEL:control_space_lev) and (3.36)
cannot be formulated into a standard constrained optimization problem.

###  3.6 Neural network method

####  3.6.1 Motivation for neural network method

In Section 3.4, we derive the closed-form solution under the jump-diffusion
model, by obtaining the first-order optimality of the corresponding PIDE of
the problem. The closed-form solution requires several unrealistic assumptions
such as continuous rebalancing, unlimited leverage, and trading in insolvency.
Furthermore, the closed-form solution is specific to the investment objective
defined in the CD problem (3.11). Ideally, we would like to solve the general
investment problem (3.7) under realistic constraints, such as discrete
rebalancing and limited leverage (i.e., leverage constraints discussed in
Section 3.5). Unfortunately, for such cases, closed-form solutions rarely
exist. Therefore, we often need to solve the problem numerically.

One method of solving a discrete-time multi-period optimal asset allocation
problem utilizes dynamic programming (DP) based reinforcement learning (RL)
algorithms. For example, Dixon et al. (2020); Park et al. (2020); Lucarelli
and Borrotti (2020); Gao et al. (2020) use Q-learning algorithms to solve the
discrete-time multi-period optimal allocation problem. In general, if there
are $N_{a}$ assets to invest in, then the use of Q-learning involves
approximation of an action-value function (“Q” function) which is a
${(2N_{a}+1)}$-dimensional function (van Staden et al., 2023) which represents
the conditional expectation of the cumulative rewards at an intermediate
state.151515Intuitively, the dimensionality comes from tracking the allocation
in the $N_{a}$ assets for both the active portfolio and benchmark portfolio
when evaluating the changes in wealth of both portfolios over one period in
the action-value function. Meanwhile, the optimal control is a mapping from
the state space to the allocation fractions to the assets. If the state space
is relatively low-dimensional, 161616For example, the state space of problem
(3.11) with assumptions of a fixed-mix strategy is a vector in
$\mathbb{R}^{3}$. then the DP-based approaches are potentially unnecessarily
high-dimensional.

Instead of using dynamic programming methods, Han et al. (2016); Buehler et
al. (2019); Tsang and Wong (2020); Reppen et al. (2022) propose to approximate
the optimal control function by neural network functions directly. In
particular, they propose a stacked neural network approach that essentially
uses a sub-network to approximate the control at every rebalancing step.
Therefore, the number of neural networks required grows linearly with the
number of rebalancing periods.

Note that in the taxonomy of Powell (2023), this method has been termed Policy
Function Approximation (PFA).

In this article, we follow the lines of Li and Forsyth (2019); Ni et al.
(2022) and propose a single neural network to approximate the optimal control
function. The direct representation of the control function avoids the high-
dimensional approximation required in DP-based methods. In addition, we
consider time $t$ as an input feature (along with the wealth of the active
portfolio and benchmark portfolio), therefore avoiding the need for multiple
sub-networks in the stacked neural network approach.

Mathematically, the numerical solution to the general problem (3.7) requires
solving for the feedback control $\bm{p}$. We approximate the control function
$\bm{\theta}$ by a neural network function
$f(\bm{X}(t);\bm{\theta}):\mathcal{X}\mapsto\mathbb{R}^{N_{a}}$, where
$\bm{\theta}\in\mathbb{R}^{N_{\bm{\theta}}}$ represents the parameters of the
neural network (i.e., weights and biases). In other words,

$\bm{p}(\bm{X}(t))\simeq f(\bm{X}(t);\bm{\theta})\equiv f(\cdot;\bm{\theta}).$
(3.37)

Then, the optimization problem (3.7) can be converted to solving the following
optimization problem.

$\text{(Parameterized optimization
problem):}\quad\inf_{\bm{\theta}\in\mathcal{Z}_{\bm{\theta}}}\mathbb{E}_{f(\cdot;\bm{\theta})}^{(t_{0},w_{0})}\big{[}F(\mathcal{W}_{\bm{\theta}},\hat{\mathcal{W}}_{\hat{\mathcal{P}}})\big{]}.$
(3.38)

Here $\mathcal{W}_{\bm{\theta}}$ is the wealth trajectory of the active
portfolio following the neural network approximation function parameterized by
$\theta$. $\mathcal{Z}_{\bm{\theta}}\subseteq\mathbb{R}^{N_{\bm{\theta}}}$ is
the feasibility domain of the parameter $\bm{\theta}$, which is translated
from the constraints of the original problem, e.g., (LABEL:control_space_lev)
and (3.36). Mathematically,

$\mathcal{Z}_{\bm{\theta}}=\Bigg{\\{}\bm{\theta}:\begin{cases}f(\bm{X}(t);\bm{\theta})\in\mathcal{Z},\qquad\text{if
}\bm{X}(t)\in\mathcal{X}_{1},\\\
f(\bm{X}(t);\bm{\theta})\in\big{\\{}\bm{e}_{N_{l}+1}\big{\\}},\;\text{if
}\bm{X}(t)\in\mathcal{X}_{2}.\end{cases}\Bigg{\\}}$ (3.39)

Here $\mathcal{Z}$ is defined in (LABEL:control_space_lev) and
$\big{\\{}\mathcal{X}_{1},\mathcal{X}_{2}\big{\\}}$ is the partition of the
state space $\mathcal{X}$ defined in Definition 3.1.

Note here that $\mathcal{Z}_{\bm{\theta}}$ is dependent on the structure of
the neural network function $f(\cdot;\bm{\theta})$. Intuitively,
$\mathcal{Z}_{\bm{\theta}}$ is the preimage of $\mathcal{Z}$ and
$\big{\\{}\bm{e}_{N_{l}+1}\big{\\}}$ under $f(\cdot;\bm{\theta})$. Specific
neural network model design may result in
$\mathcal{Z}_{\bm{\theta}}=\mathbb{R}^{N_{\bm{\theta}}}$, which means (3.38)
becomes an unconstrained optimization problem. For long-only investment
problems, the only constraints are the long-only constraint (3.29) and the
summation constraint (3.30). Previous work has proposed a neural network
architecture with a softmax activation function at the last layer so that the
output (vector of allocation fractions) automatically satisfies the two
constraints, and thus $\mathcal{Z}_{\bm{\theta}}=\mathbb{R}^{N_{\bm{\theta}}}$
and problem (3.38) becomes an unconstrained optimization problem (see, e.g.,
Li and Forsyth (2019); Ni et al. (2022)). However, as discussed in Section
3.5, we consider the more complicated case where leverage and shorting is
allowed. The problem thus involves more constraints than the long-only case
does and therefore requires a new model architecture to convert the
constrained optimization problem to an unconstrained problem. We will discuss
the design of the leverage-feasible neural network (LFNN) model in the next
section, and how the LFNN model achieves this goal.

It is worth noting that for the particular CD problem (3.12) and CS problem
(3.14), our technique may be formulated to appear similar to policy gradient
methods in RL literature (Silver et al., 2014) on a high level. Examples of
policy gradient methods in financial problems include Coache and Jaimungal
(2021), in which the authors develop an actor-critic algorithm for portfolio
optimization problems with convex risk measures. However, there are two main
differences between our proposed methodology and policy gradient algorithms.
Firstly, we assume that the randomness of the environment (i.e., asset
returns) over the entire investment horizon is readily available upfront
(e.g., through calibration of parametric models or resampling of historical
data), which is a common assumption adopted by practitioners when backtesting
investment strategies. On the other hand, RL literature often considers an
unknown environment, and the algorithms focus on the exploration of the agent
to learn from the unknown environment and thus may be unnecessarily
complicated for our use case. Secondly, our proposed methodology is not
limited to the cumulative reward framework in RL and thus is more universal
and suitable for problems in which the investment objective cannot be easily
expressed in the form of a cumulative reward.

###  3.7 Leverage-feasible neural network (LFNN)

In this section, we demonstrate the leverage-feasible neural network (LFNN)
model design, which yields
$\mathcal{Z}_{\bm{\theta}}=\mathbb{R}^{N_{\bm{\theta}}}$ for leverage
constraints defined in equation (LABEL:control_space_lev), and converts a
constrained optimization problem (3.38) to an unconstrained problem.

Let vector $\bm{x}=(t,W(t),\hat{W}(t))^{\top}\in\mathcal{X}$ be the feature
(input) vector. We first define a standard fully-connected feedforward neural
network (FNN) function $\tilde{f}:\mathcal{X}\mapsto\mathbb{R}^{N_{a}+1}$ as
follows:

$(\text{FNN}):\quad\begin{cases}h^{(1)}_{j}=\text{Sigmoid}\Big{(}\sum_{i=1}^{N_{x}}x_{i}\theta^{(1)}_{ij}+b^{(1)}_{j}\Big{)},\;j=1,\cdots,N_{h}^{(1)},\\\
h^{(k)}_{j}=\text{Sigmoid}\Big{(}\sum_{i=1}^{N_{h}^{(k-1)}}h^{(k-1)}_{i}\theta^{(k)}_{ij}+b^{(k)}_{j}\Big{)},\;j=1,\cdots,N_{h}^{(k)},\;\forall
k\in\\{2,\cdots,K\\},\\\
o_{j}=\sum_{i=1}^{N_{h}^{(K)}}h_{i}\theta^{(K+1)}_{ij},\;j=1,\cdots,N_{a}+1,\\\
{\tilde{f}(\bm{x};\bm{\theta}):=(o_{1},\cdots,o_{N_{a}+1})^{\top}.}\end{cases}$
(3.40)

Here Sigmoid($\cdot$) denotes the sigmoid activation function. $K$ denotes the
number of hidden layers, $h^{k}_{j}$ denotes the value of the $j$-th node in
the $k$-th hidden layer, and $N_{h}^{(k)}$ is the number of nodes in the
$k$-th hidden layer.
$\bm{\theta}^{(k)}=(\theta^{(k)}_{ij})\in\mathbb{R}^{N_{h}^{(k)}\times
N_{h}^{(k-1)}}$ and $\bm{b}^{(k)}=(b^{(k)}_{j})\in\mathbb{R}^{N_{h}^{(k)}}$
are the (vectorized) weight matrix and bias vector for the $k$-th
layer,171717For $k=K+1$, define $N_{h}^{(K+1)}=N_{a}+1$ so
$\bm{\theta}^{(K+1)}$ is well-defined. and the final parameter vector of the
neural network is
$\bm{\theta}=(\bm{\theta}^{(1)},\bm{b}^{(1)},\cdots,\bm{\theta}^{(K)},\bm{b}^{(K)},\bm{\theta}^{(K+1)})^{\top}\in\mathbb{R}^{N_{\bm{\theta}}}$,
where $N_{\bm{\theta}}=\sum_{k=1}^{K+1}N_{h}^{(k)}\cdot
N_{h}^{(k-1)}+\sum_{k=1}^{K}N_{h}^{(k)}$.

Building upon $\tilde{f}$, we propose the following leverage-feasible neural
network (LFNN) model $f:\mathcal{X}\mapsto\mathbb{R}^{N_{a}}$:

$(\text{LFNN}):\quad
f(\bm{x};\bm{\theta}):=\psi\Big{(}\tilde{f}(\bm{x};\bm{\theta}),\bm{x}\Big{)}\in\mathbb{R}^{N_{a}}.$
(3.41)

Here, $\psi(\cdot)$ is the leverage-feasible activation function. For
$\bm{o}=(o_{1},\cdots,o_{N_{a}+1})^{\top}\in\mathbb{R}^{N_{a}+1}$ and
$\bm{p}=(p_{1},\cdots,p_{N_{a}})^{\top}\in\mathbb{R}^{N_{a}}$,
$\psi:{(\bm{o},\bm{x})\in\mathbb{R}^{N_{a}+1}\times{\mathcal{X}}}\mapsto\bm{p}\in\mathbb{R}^{N_{a}}$
is defined by

$\psi(\bm{o},\bm{x})=\bm{p}=\begin{cases}\begin{cases}l=p_{max}\cdot\text{Sigmoid}(o_{N_{a}+1}),\\\
p_{i}=l\cdot\frac{e^{o_{i}}}{\sum_{k=1}^{N_{l}}e^{o_{k}}},\;i\in\\{1,\cdots,N_{l}\\},\qquad\qquad\qquad\text{if
}\bm{x}\in\mathcal{X}_{1},\\\
p_{i}=(1-l)\cdot\frac{e^{o_{i}}}{\sum_{k=N_{l}+1}^{N_{a}}e^{o_{k}}},\;i\in\\{N_{l}+1,\cdots,N_{a}\\},\end{cases}\\\
\bm{e}_{N_{l}+1},\;\;\;\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{if
}\bm{x}\in\mathcal{X}_{2}.\\\ \end{cases}$ (3.42)

Note that $N_{l}$ is the number of long-only assets and $p_{max}$ is the
maximum leverage allowed. We show that the leverage-feasible activation
function $\psi$ has the following property.

######  Lemma 3.1.

(Decomposition of $\psi$) The leverage-feasible function $\psi$ defined in
(3.42) has the function decomposition that

$\psi(\bm{o},\bm{x})=\varphi(\zeta(\bm{o}),\bm{x}),$ (3.43)

where

$\begin{cases}\zeta:\mathbb{R}^{N_{a}+1}\mapsto\tilde{\mathcal{Z}},\zeta(o)=\Bigg{(}\text{Softmax}\Big{(}(o_{1},\cdots,o_{N_{l}})\Big{)},\text{Softmax}\Big{(}(o_{N_{l}+1},\cdots,o_{N_{a}})\Big{)},p_{max}\cdot\text{Sigmoid}(o_{N_{a}+1})\Bigg{)}^{\top},\\\
\varphi:\tilde{\mathcal{Z}}\times{\mathcal{X}}\mapsto\mathbb{R}^{N_{a}},\varphi(z)=\Big{(}z_{N_{a}+1}\cdot(z_{1},\cdots,z_{N_{l}}),(1-z_{N_{a}+1})\cdot(z_{N_{l}+1},\cdots,z_{N_{a}})\Big{)}^{\top}\cdot\textbf{1}_{\bm{x}\in\mathcal{X}_{1}}+\bm{e}_{N_{l}+1}\cdot\textbf{1}_{\bm{x}\in\mathcal{X}_{2}},\end{cases}$
(3.44)

and

$\tilde{\mathcal{Z}}=\Bigg{\\{}z\in\mathbb{R}^{N_{a}+1},\sum_{i=1}^{N_{l}}z_{i}=1,\sum_{i=N_{l}+1}^{N_{a}}z_{i}=1,z_{N_{a}+1}\leq
p_{max},z_{i}\geq 0,\forall i\Bigg{\\}}.$ (3.45)

###### Proof.

This is easily verifiable by definition of $\psi$ in (3.42). ∎

######  Remark 3.6.

(Remark on Lemma 3.1) The leverage-feasible activation function $\psi$
corresponds to a two-step decision process described by $\zeta$ and $\varphi$.
Intuitively, $\zeta$ first determines the internal allocations within long-
only assets and shortable assets, as well as the total leverage. Then,
$\varphi$ converts the internal allocations and total leverage into final
allocation fractions, which depend on the wealth of the active portfolio.

With the LFNN model outlined above, the parameterized optimization problem
(3.38) becomes an unconstrained optimization problem. Specifically, we present
the following theorem regarding the feasibility domain
$\mathcal{Z}_{\bm{\theta}}$ associated with the LFNN model (3.41).

######  Theorem 3.2.

(Unconstrained feasibility domain) The feasibility domain
$\mathcal{Z}_{\bm{\theta}}$ defined in (3.39) associated with the LFNN model
(3.41) is $\mathbb{R}^{N_{\bm{\theta}}}$.

###### Proof.

See Appendix D.1. ∎

Following Theorem 3.2, the constrained optimization problem (3.7) can be
transformed into the following unconstrained optimization problem

$\text{(Unconstrained parameterized
problem)}\quad\inf_{\bm{\theta}\in\mathbb{R}^{N_{\bm{\theta}}}}\mathbb{E}_{f(\cdot;\bm{\theta})}^{(t_{0},w_{0})}\big{[}F(\mathcal{W}_{\bm{\theta}},\hat{\mathcal{W}}_{\hat{\mathcal{P}}})\big{]}.$
(3.46)

###  3.8 Mathematical justification for LFNN approach

By approximating the feasible control with a parameterized LFNN model, we have
shown that the initially constrained optimization problem is transformed into
an unconstrained optimization problem, which is computationally more
implementable.

However, an important question remains: does solving the parameterized
unconstrained optimization problem (3.46) yield the optimal control of the
original problem (3.7)? In other words, suppose $\bm{\theta}^{*}$ is the
solution to (3.46), is $f(\cdot;\bm{\theta}^{*})$ the solution to (3.7)?

In this section, we prove that under benign assumptions and appropriate
choices of the hyperparameter of the LFNN model (3.41), solving the
unconstrained problem (3.46) provides an arbitrarily close approximation the
original problem (3.7). We start by establishing the following lemma.

######  Lemma 3.2.

(Structure of feasible control) Any feasible control function
$p:\mathcal{X}\mapsto\mathbb{R}^{N_{a}}$ has the function decomposition

$p(x)=\varphi(\omega(x),x),$ (3.47)

where $\varphi:\tilde{\mathcal{Z}}\times\mathcal{X}\mapsto\mathbb{R}^{N_{a}}$
is defined in (3.44) and $\omega:\mathcal{X}\mapsto\tilde{\mathcal{Z}}$.

###### Proof.

See Appendix D.2. ∎

Next, we propose the following benign assumptions on the state space and the
optimal control.

######  Assumption 3.7.

(Assumption on state space and optimal control)

(Admissible set): &

  1. (i)

The space of state variables $\mathcal{X}$ is a compact set.

  2. (ii)

Following Lemma 3.2, the optimal control
$p^{*}:\mathcal{X}\mapsto\mathbb{R}^{N_{a}}$ has the decomposition
$p^{*}(x)=\varphi(\omega^{*}(x),x)$ for some
$\omega^{*}:\mathcal{X}\mapsto\tilde{\mathcal{Z}}$. We assume $\omega^{*}\in
C(\mathcal{X},\tilde{\mathcal{Z}})$.

######  Remark 3.7.

(Remark on Assumption 3.8) In our particular problem of outperforming a
benchmark portfolio, the state variable vector is
$X(t)=(t,W(t),\hat{W}(t))^{\top}\in\mathcal{X}$ where $t\in[0,T]$. In this
case, assumption (i) is equivalent to the assumption that the wealth of the
active portfolio and benchmark portfolio are bounded, i.e.
$\mathcal{X}=[0,T]\times[w_{min},w_{max}]\times[w_{min},\hat{w}_{max}]$, where
$w_{min},w_{max}$ and $\hat{w}_{min},\hat{w}_{max}$ are the respective wealth
bounds for the portfolios. Intuitively, assumption (ii) states that the
decision process for the optimal control to obtain the internal allocation
fractions within the long-only assets, shortable assets, and the total
leverage is a continuous function. This is a natural extension of the long-
only case, in which it is commonly assumed that the allocation within long-
only assets are a continuous function of state variables.

Finally, we present the following theorem.

######  Theorem 3.3.

(Approximation of optimal control) Following Assumption 3.8,
$\forall\epsilon>0$, there exists $N_{h}\in\mathbb{N}$, and
$\bm{\theta}\in\mathbb{R}^{N_{\bm{\theta}}}$ such that the corresponding LFNN
model $f(\cdot;\bm{\theta})$ described in (3.41) satisfies the following:

$\sup_{x\in\mathcal{X}}\|f(x;\bm{\theta})-p^{*}(x)\|<\epsilon.$ (3.48)

###### Proof.

See Appendix D.2. ∎

Theorem 3.3 shows that given any arbitrarily small distance, there exists a
suitable choice of the hyperparameter of the LFNN model (e.g. number of hidden
layers and nodes), and a parameter vector $\bm{\theta}$, such that the
corresponding parameterized LFNN function is within this distance of the
optimal control function.181818The distance is defined in (3.48), i.e. the
supremum of the pointwise distance over the extended state space
$\mathcal{X}$. In other words, with a large enough LFNN model (in terms of
number of hidden nodes), solving the unconstrained parameterized problem
(3.46) approximately solves the original optimization problem (3.7) with any
required precision.

##  4 Numerical experiments

In this section, we discuss several numerical experiments and their results.
Since the numerical experiments involve the solution and evaluation of the
optimal parameters $\bm{\theta}^{*}$ of the LFNN model (3.41) in problem
(3.46), we briefly review how the parameters are computed in experiments.

In numerical experiments, the expectation in (3.46) is approximated by using a
finite set of samples of the set $\bm{Y}=\\{Y^{(j)}:j=1,\cdots,N_{d}\\}$,
where $N_{d}$ is the number of samples, and $Y^{(j)}$ represents a time series
sample of joint asset return observations
$R_{i}(t),\;i\in\\{1,\cdots,N_{a}\\}$, observed at
$t\in\mathcal{T}$.191919Note that the corresponding set of asset prices can be
easily inferred from the set of asset returns, or vice versa. Mathematically,
problem (3.46) is approximated by

$\quad\inf_{\bm{\theta}\in\mathbb{R}^{N_{\bm{\theta}}}}\Bigg{\\{}\frac{1}{N_{d}}\sum_{j=1}^{N_{d}}F(\mathcal{W}^{(j)}_{\bm{\theta}},\hat{\mathcal{W}}^{(j)}_{\hat{\mathcal{P}}})\Bigg{\\}}.$
(4.1)

Here $\mathcal{W}^{(j)}_{\bm{\theta}}$ is the wealth trajectory of the active
portfolio following the LFNN parameterized by $\bm{\theta}$, and
$\hat{\mathcal{W}}^{(j)}$ is the wealth trajectory of the benchmark portfolio
following the benchmark strategy $\hat{\mathcal{P}}$, both evaluated on
$Y^{(j)}$, the $j$-th time series sample.

Based on our experiments, we find that a shallow neural network model is
capable of achieving good results. Specifically, we use an LFNN model with one
single hidden layer with 10 hidden nodes, i.e., $K=1$ and $N_{h}^{(1)}=10$. We
use the 3-tuple vector $(t,W_{\bm{\theta}}(t),\hat{W}(t))^{\top}$ as the input
(feature) to the LFNN network. At $t\in[t_{0},T]$, $W_{\bm{\theta}}(t)$ is the
wealth of the active portfolio of the strategy that follows the LFNN model
parameterized by $\bm{\theta}$, and $\hat{W}(t)$ is the wealth of the
benchmark portfolio.

Then, the optimal parameter $\bm{\theta}^{*}$ can be numerically obtained by
solving problem (4.1) using standard optimization algorithms such as ADAM
(Kingma and Ba, 2014). This process is commonly referred to as “training” of
the neural network model, and $\bm{Y}$ is often referred to as the training
data set (Goodfellow et al., 2016).

Once $\bm{\theta}^{*}$ is numerically obtained, the resulting optimal strategy
$f(\cdot;\bm{\theta}^{*})$ is evaluated on a separate “testing” data set
$\bm{Y}^{test}$, which contains a different set of samples generated from
either the same distribution of the training process or a different process
(depending on experiment purposes) so that the “out-of-sample” performance of
$f(\cdot;\bm{\theta}^{*})$ is assessed.

###  4.1 Comparing LFNN with closed-form solution

In this section, we compare the performance of the strategy following the
learned shallow LFNN model (which we refer to as the “neural network strategy”
from now on) with the closed-form solution (3.20), and provide empirical
validation of the LFNN approach.

####  4.1.1 Approximate form under realistic assumptions

We first note that the closed-form solution $p^{*}$ defined in (3.20) is
obtained under several unrealistic assumptions, namely continuous rebalancing,
unlimited leverage, and continuing trading in insolvency.202020Note that we
consider a two-asset scenario here, thus the scalar $p^{*}\in\mathbb{R}$
(allocation fraction for the stock index) fully describes the allocation
strategy $\bm{p}^{*}$, since $\bm{p}^{*}=(p^{*},1-p^{*})^{\top}$. In practice,
investors have constraints such as discrete rebalancing, limited leverage, and
no trading when insolvent. For a meaningful comparison, instead of comparing
the neural network strategy with the closed-form solution $p^{*}$ directly, we
compare the neural network strategy with an easily obtainable approximation to
the closed-form solution which satisfies realistic constraints.

In particular, we consider an equally-spaced discrete rebalancing schedule
$\mathcal{T}_{\Delta t}$ defined as

$\mathcal{T}_{\Delta t}=\Big{\\{}t_{i}:\;i=0,\cdots,N\Big{\\}},$ (4.2)

where $t_{i}=i\Delta t$, and $\Delta t=T/N$. Then, the clipped form
$\bar{p}_{\Delta t}:\mathcal{T}_{\Delta
t}\times\mathbb{R}^{3}\mapsto\mathbb{R}$ is defined as

$(\text{Clipped form}):\quad\bar{p}_{\Delta t}(t_{i},\bar{W}_{\Delta
t}(t_{i}),\hat{W}_{\Delta
t}(t_{i}),\hat{\varrho})=\min\Bigg{(}\max\Big{(}p^{*}(t_{i},\bar{W}_{\Delta
t}(t_{i}),\hat{W}_{\Delta
t}(t_{i}),\hat{\varrho}),p_{min}\Big{)},p_{max}\Bigg{)}.$ (4.3)

Here $[p_{min}=0,p_{max}>1]$ is the allowed leverage range, $\bar{W}_{\Delta
t}(t_{i})$ is the wealth of the active portfolio at $t_{i}$ following
$\bar{p}_{\Delta t}$ from $t_{0}$ to $t_{i}$, $\hat{W}_{\Delta t}(t_{i})$ is
the wealth of the benchmark portfolio at $t_{i}$ following the fixed-mix
strategy described by constant allocation fraction $\hat{\varrho}$, but only
rebalanced discretely according to $\mathcal{T}_{\Delta t}$. Clearly, the
allocation strategy from $\bar{p}_{\Delta t}$ follows the discrete schedule of
$\mathcal{T}_{\Delta t}$, and satisfies the leverage constraint that
$\bar{p}_{\Delta t}\in[p_{min}=0,p_{max}]$. $p_{\Delta t}$ approaches the
closed-form solution $p^{*}$ as $\Delta t\downarrow
0,p_{min}\downarrow-\infty$ and $p_{max}\uparrow\infty$. We note that a
similar clipping idea is explored in Vigna (2014) in the context of closed-
form solutions for multi-period mean-variance asset allocation. However, it
should be emphasized that the clipped form $\bar{p}_{\Delta t}$ with finite
$(p_{min},p_{max})$ is a feasible, but in general sub-optimal, control of the
leverage-constrained CD problem (3.12).

We then address the assumption that trading continues when insolvent, i.e.,
when the wealth of the portfolio reaches zero. While necessary for the
mathematical derivation of the closed-form solution, we acknowledge that this
is by no means reasonable for practitioners. Under the continuous rebalancing
case (no jumps), if the control (allocation) is bounded, it is shown that the
wealth of the portfolio can never be negative (Wang and Forsyth, 2012).
However, with discrete rebalancing, even with a bounded control, as long as
the upper bound $p_{max}>1$, it is theoretically possible that the portfolio
value becomes negative. We address this assumption by applying an overlay on
strategies so that in the case of insolvency, we assume the manager liquidates
the long-only positions and allocate the debt (negative wealth) to a shortable
(bond) asset (consistent with Assumption 3.34) to allow outstanding debt to
accumulate until the end of the investment horizon. Going forward, when we
refer to any strategy (e.g. neural network strategy, clipped form), we mean
the strategy with this overlay applied. We remark that in practice, this
overlay has little effect. In numerical experiments with 10,000 samples of
observed wealth trajectories (based on calibrated jump-diffusion model or
bootstrap resampled data paths), we do not observe any single wealth
trajectory that ever hits negative wealth for any strategy (e.g., neural
network strategy, clipped form, etc).

In summary, the clipped form satisfies the realistic constraints, and is a
comparable benchmark for the neural network strategy. In the following
section, we will numerically compare the performance of the clipped form, the
neural network strategy, and the closed-form solution.

####  4.1.2 Performance of neural network strategy

We assess and compare the performance of the neural network strategy and the
clipped form, we assume the following investment scenario described in Table
4.1.

Investment horizon $T$ (years) | 10
---|---
Assets | CRSP cap-weighted index (real)/30-day T-bill (U.S.) (real)
Index Samples | Concatenated 1940:8-1951:7, 1968:9-1985:10
Initial portfolio wealth/annual cash injection | 100/10
Rebalancing frequency | Monthly, quarterly, semi-annually, annually
Maximum leverage | 1.3
Benchmark equity percentage | 0.7
Outperformance target rate $\beta$ | 1% (100 bps)
Table 4.1: Investment scenario.

We assume the stock index and the bond index prices follow the jump-diffusion
model (3.15).212121We remind the reader that the closed-form solution is
derived under the jump-diffusion model. Furthermore, we assume that the prices
follow a double exponential jump model (Kou, 2002; Kou and Wang, 2004), i.e.,
for the jump variable $\xi_{i}$, $y_{i}=\log(\xi_{i})$ follows the double
exponential distribution with density functions $g_{i}(y_{i})$ defined as
follows

$g_{i}(y_{i})=\nu_{i}\iota_{i}e^{-\iota_{i}y_{i}}\textbf{1}_{y_{i}\geq
0}+(1-\nu_{i})\varsigma_{i}e^{-\varsigma_{i}y_{i}}\textbf{1}_{y_{i}<0},\;i=1,2.$
(4.12)

where $\nu_{i}$ is the probability for an upward jump, and $\iota_{i}$ and
$\varsigma_{i}$ are parameters that describe the upward jump and downward jump
respectively. The double exponential jump-diffusion model allows the
flexibility of modeling asymmetric upward and downward jumps in asset prices,
which is an appropriate assumption for inflation regimes.

Using the threshold technique (Mancini, 2009; Cont et al., 2011; Dang and
Forsyth, 2016), we calibrate the double exponential jump-diffusion models to
the historical high-inflation periods described in Section 2. The calibrated
parameters can be found in Appendix E. Then, we sample a training data set
$\bm{Y}$ and a testing data set $\bm{Y}^{test}$ from the calibrated model,
each with 10,000 samples.

The neural network strategy follows the LFNN model learned on $\bm{Y}$. We
then evaluate the performance of the neural network strategy and the
approximate form (4.3) on the testing data set $\bm{Y}^{test}$. Specifically,
we compare the value of the CD objective function (3.12) for the neural
network strategy and the clipped form on $\bm{Y}^{test}$. In particular, this
training/testing process is repeated for various rebalancing frequencies from
monthly to annually, as described in Table 4.1.

Strategy | $\Delta t=1$ | $\Delta t=1/2$ | $\Delta t=1/4$ | $\Delta t=1/12$ | $\Delta t=0$
---|---|---|---|---|---
Clipped form | 545 | 504 | 479 | 467 | 461 (extrapolated)
Neural network | 537 | 498 | 476 | 464 | 458 (extrapolated)
Closed-form solution |  |  |  |  | 418
Table 4.2: CD objective function values. Results shown are evaluated on
$\bm{Y}^{test}$, the lower the better.

In Table 4.2, we can see that the neural network strategy consistently
outperforms the clipped form in terms of the objective function value for all
rebalancing frequencies. From Table 4.2 we can see that the objective function
values of both the neural network strategy and the clipped form converge at
roughly a first order rate as $\Delta t\downarrow 0$. Assuming this to be
true, we extrapolate the solution to $\Delta t=0$ using Richardson
extrapolation. These extrapolated values are estimates of the exact value of
the continuous-time CD objective function (3.11) for the clipped form and the
neural network strategy. We can see that the neural network strategy still
outperforms the clipped form in terms of the extrapolated objective function
value. We can also see that the extrapolated neural network objective function
value is lower than the (suboptimal) clipped form extrapolated value, but, of
course, larger than the the unconstrained closed-form solution.

(a) Clipped form

(b) Neural network strategy

Figure 4.1:  Stock allocation fraction w.r.t. tracking ratio
$W(t)/\Big{(}e^{\beta t}\hat{W}(t)\Big{)}$ and time $t$. Results are based on
the evaluation on the testing data set $\bm{Y}^{test}$, and monthly
rebalancing (i.e. $\Delta t=1/12$).

Finally, we compare the allocation strategies from the neural network strategy
with the clipped form. Specifically, in Figure 4.1, we consider the case of
monthly rebalancing and present the scatter plots of the allocation fraction
in the stock index with respect to time $t$ and the ratio between the wealth
of the active portfolio $W(t)$ and the elevated target $e^{\beta
t}\hat{W}(t)$. For simplicity, we call this ratio the “tracking ratio”. We
plot the 3-tuple $\Big{(}\frac{W(t)}{e^{\beta
t}\hat{W}(t)},t,p_{1}(W(t),\hat{W}(t),t)\Big{)}$ (obtained from the evaluation
of the strategies on samples from $\bm{Y}^{test}$) by using time $t$ as the
x-axis, the tracking ratio $\frac{W(t)}{e^{\beta t}\hat{W}(t)}$ as the y-axis,
and the values of the corresponding allocation fraction to the cap-weighted
index $p_{1}(W(t),\hat{W}(t),t)$ to color the scattered dots on the plot. A
darker shade of the color indicates a higher allocation fraction.

As we can see from Figure 4.1, the stock allocation fraction of the neural
network strategy behaves similarly to the stock allocation fraction from the
clipped form. Both strategies invest more wealth in the stock when the
tracking ratio is lower, which is consistent with the insights we obtained in
Section 3.4.1. In addition, the transition patterns of the allocation fraction
of the two strategies are also highly similar. One can almost draw an
imaginary horizontal dividing line around $\frac{W(t)}{e^{\beta
t}\hat{W}(t)}=0.9$ that separates high stock allocation and low stock
allocation for both strategies.

We remark that a common criticism towards the use of neural networks is about
the lack of interpretability compared to more interpretable counterparts such
as the regression models (Rudin, 2019). In this section, we see that the
neural network strategy closely resembles the closed-form solution for the CD
objective. The closed-form solution, in turn, complements the neural network
model and offers an alternative way of interpreting results obtained from the
neural network.

###  4.2 Case study under realistic market scenario

####  4.2.1 Learning from bootstrap resampled data

So far, our discussion has focused on scenarios in which asset prices follow a
parametric model (specifically, the jump-diffusion model (3.15)). While the
assumption of parametric models brings many benefits, most notably allowing us
to derive the closed-form solution, we also acknowledge that the assumption of
the parametric model has several limitations.

However, the parameter estimation for a parametric model is very difficult
(Black, 1993). Even for a simple geometric Brownian motion (GBM) model,
estimating the drift rate accurately is error prone and requires the data to
cover a long historical period (Brigo et al., 2008). More complicated models,
such as the double exponential jump-diffusion model we have discussed in
detail, introduce more components to the stochastic model and thus require
estimation of the extra parameters. Furthermore, parametric models implicitly
introduce many assumptions concerning the true stochastic model for asset
prices, which are sometimes debatable.

Acknowledging the above limitations of parametric market data models, we turn
to the non-parametric method of bootstrap resampling as a data-generating
process for numerical experiments. Unlike parametric models, non-parametric
models such as bootstrap resampling aim to not make assumptions about the
parametric form of the asset price dynamics. Intuitively speaking, the
bootstrap resampling method randomly chooses data points from the historical
time series data and reassembles them into new paths of time series data. The
bootstrap was initially proposed as a statistical method for estimating the
sampling distribution of statistics (Efron, 1992). We use it as a data-
generating procedure, as the philosophy behind bootstrap resampling is
consistent with the idea that “history does not repeat, but it rhymes.”. The
bootstrap resampling provides an empirical distribution, which seems to be the
least prejudiced estimate possible of the underlying distribution of the data-
generating process. We also note that bootstrap resampling is widely adopted
by practitioners (Alizadeh and Nomikos, 2007; Cogneau and Zakamouline, 2013;
Dichtl et al., 2016; Scott and Cavaglia, 2017; Shahzad et al., 2019; Cavaglia
et al., 2022; Simonian and Martirosyan, 2022) as well as academics (Anarkulova
et al., 2022).

As we have discussed in Section 2.2, we bootstrap from inflation-adjusted
returns from the concatenated series consisting of two identified high-
inflation regimes: 1940:8-1951:7 and 1968:9-1985:10. In the subsequent
numerical experiments, we use the stationary block bootstrap resampling (see
Appendix B.1) to generate the training and testing data sets. The stationary
block bootstrap resampling method is a special form of block bootstrap
resampling that preserves the weak stationarity of the original time series
data (Politis and Romano, 1994), while taking into account serial correlation
effects in the time series.

####  4.2.2 Experiment results

The details of the investment specification are given in Table 4.3. Briefly,
the active portfolio and benchmark portfolio begin with the same initial
wealth of 100 at $t_{0}=0$. Both portfolios are rebalanced monthly. The
benchmark portfolio is rebalanced to maintain the weight of 70% in the equal-
weighted stock index and 30% in the 30-day U.S. T-bill index. We select this
fixed-mix portfolio based on our observation that the equal-weighted stock
index shows superior performance to the cap-weighted stock index during high-
inflation environments, as discussed in Section 2.222222After all, it is not
interesting to try to beat a weak benchmark, such as a 70% cap-weighted stock
index/30% bond benchmark. The investment horizon is 10 years, and there is an
annual cash injection of 10 for both portfolios, evenly divided over 12
months.

We consider an empirical case in which we allow the manager to allocate
between four investment assets: the equal-weighted stock index, the cap-
weighted stock index, the 30-day U.S. T-bill index, and the 10-year U.S.
T-bond index. We assume that the stock indexes and the 10-year T-bond index
are long-only assets. The manager can short the T-bill index to take leverage
and invest in the long-only assets (with maximum total leverage of 1.3). In
this experiment, we assume the borrowing premium rate is zero. Essentially, we
assume that the manager can borrow short-term funding to take leverage at the
same cost as the treasury bill. This may be a reasonable assumption for
sovereign wealth funds, as they are state-owned and enjoy a high credit
rating. We remark that the borrowing premium does not really affect the
results significantly (see Appendix G for a more detailed discussion). The
annual outperformance target $\beta$ is set to be 2% (i.e. 200 bps per year).

We use the bootstrap resampling algorithm (see Appendix B.1) to generate a
training data set $\bm{Y}$ and a testing data set $\bm{Y}^{test}$ (both with
10,000 resampled paths) from the concatenated index samples from two
historical inflation regimes: 1940:8-1951:7 and 1968:9-1985:10, using an
expected blocksize of 6 months. The testing data set $\bm{Y}^{test}$ is
generated using a different random seed as the training data set $\bm{Y}$, and
thus the probability of seeing the same sample in $\bm{Y}$ and $\bm{Y}^{test}$
is near zero (see Ni et al. (2022) for proof).

It is worth noting that in this experiment, we train the LFNN model (3.41) on
$\bm{Y}$ under the discrete-time CS objective (F.1), instead of the CD
objective (3.12). As discussed in Section 3.3, the CS objective function only
penalizes underperformance relative to the elevated target. Numerical
comparisons of the two objective functions suggest that the CS objective
function indeed yields more favorable investment results than the CD objective
(see Appendix F).

Investment horizon $T$ (years) | 10
---|---
Equity market indexes | CRSP cap-weighted/equal-weighted index (real)
Bond index | CRSP 30-day/10-year U.S. treasury index (real)
Index samples for bootstrap | Concatenated 1940:8-1951:7, 1968:9-1985:10
Initial portfolio wealth/annual cash injection | 100/10
Rebalancing frequency | Monthly
Maximum leverage | 1.3
Outperformance target rate $\beta$ | 2%
Table 4.3: Investment scenario.

(a) Percentiles of wealth ratio $\frac{W(t)}{\hat{W}(t)}$

(b) Terminal wealth ratio

(c) CDF of IRR

Figure 4.2:  Percentiles of wealth ratio $\frac{W(t)}{\hat{W}(t)}$ and CDF of terminal wealth ratio $\frac{W(T)}{\hat{W}(T)}$ and internal rate of return (IRR). Results are based on the evaluation of the learned neural network model on $\bm{Y}^{test}$.  Strategy | Median[$W_{T}$] | E[$W_{T}$] | std[$W_{T}$] | 5th Percentile | Median IRR (annual)
---|---|---|---|---|---
Neural network | 364.2 | 403.4 | 211.8 | 136.3 | 0.078
Benchmark | 308.5 | 342.9 | 165.0 | 149.0 | 0.056
Table 4.4: Statistics of strategies. Results are based on the evaluation
results on the testing data set.

From Figure 2(a), we observe that, on average, the neural network strategy
consistently outperforms the benchmark in terms of the wealth ratio
$W(t)/\hat{W}(t)$. The mean and median wealth ratio increases smoothly and
consistently over time. In terms of tail performance (20th percentile), the
neural network strategy initially trails behind the benchmark, but eventually
recovers over time, and ends up with 10% greater wealth at the terminal time.
This shows that the neural network strategy manages tail risk well.

Another metric that might be of great interest to managers is the distribution
of the terminal wealth ratio $\frac{W(T)}{\hat{W}(T)}$. This metric is
concerned with the relative performance of the strategy at the end of the
investment period. We can see from Figure 2(b) that there is a more than 90%
chance that the neural network strategy outperforms the benchmark strategy in
terms of terminal wealth. This is a surprisingly good result, considering that
the terminal wealth ratio is not directly targeted in the objective function
(F.1).

Since there are constant cash injections in the portfolios, it may be suitable
to use the internal rate of return (IRR) to measure the annualized performance
of the portfolio. As we can see from Figure 2(c), the neural network strategy
has a more than 90% chance of producing a higher IRR. The median IRR of the
neural network strategy is slightly more than 2% higher than the median IRR of
the benchmark strategy, which is in line with the choice of the target
outperformance rate $\beta=0.02$. This indicates that the neural network model
achieves the desired target performance in most of the outcomes.

We remark that the neural network has a worse extreme left tail compared to
the benchmark strategy. As listed in Table 4.4, the 5th percentile of the
terminal wealth of the neural network strategy is lower than that of the
benchmark strategy. Indeed, as we will see shortly, the neural network
strategy takes more risk than the benchmark strategy, as it allocates a higher
fraction of wealth to the equal-weighted stock index, the riskier asset, than
the benchmark portfolio. Therefore, in a persistent bear market in which the
stocks perform badly, the neural network strategy suffers more than the
benchmark strategy. However, such scenarios happen with a low probability. As
shown in Figure 2(b), the neural network strategy outperforms the benchmark in
terms of terminal value with an extremely high probability of over 90%.

At this point, readers may wonder how the neural network strategy performs
asset allocation to achieve such good results. Below we present the allocation
strategies for the neural network strategy.

Figure 4.3: Mean allocation fraction over time, evaluated on $\bm{Y}^{test}$

We first plot the mean allocation fraction for the four assets over time in
Figure 4.3. From Figure 4.3, the first interesting observation is that the
neural network strategy does not allocate wealth to the cap-weighted stock
index on average. While seemingly surprising at first, this is actually in
line with the fact that the equal-weighted stock index has significantly
higher real returns than the cap-weighted stock index during historical high-
inflation periods (see Appendix A.3). Since the objective is to outperform a
benchmark strategy heavily invested in the equal-weighted index (70%), it does
not make sense to allocate wealth to a weaker index in the active portfolio.

The second observation from Figure 4.3 concerns the evolution of the mean bond
allocation fractions. As we can see from Figure 4.3, on average, the neural
network strategy shorts the 30-day T-bill index in the first two years. During
the same period, the strategy takes on some leverage and invests heavily in
the equal-weighted stock index. Intuitively speaking, such behavior indicates
that the neural network strategy takes risks early on in order to establish an
advantage over the benchmark strategy. After the first two years, the
allocation to the 10-year T-bond decreases, and the allocation to the 30-day
T-bill ramps up. The decrease in the 10-year bond allocation coincides with
the decrease in the allocation to the equal-weighted index, indicating that
the initial allocation to the T-bond was simply due to the need for leveraging
- we assume that the manager can only short the T-bill for leveraging, thus
making the 10-year bond the only defensive asset available during the first
two years. As we can see from Figure 4.3, once leverage is not needed in later
years, the neural network strategy picks the T-bill over the 10-year bond.

Overall, while the stock allocation eventually decreases over time, the neural
network strategy maintains more than 80% allocation to the equal-weighted
stock index on average. This is not surprising, since beating an aggressive
benchmark with 70% allocation to the equal-weighted stock index requires
taking risk. Despite having a higher allocation to riskier assets, the neural
network strategy achieves good results compared to the benchmark strategy, as
shown in Figure 4.2.

Finally, we note that the neural network strategy learned under high-inflation
regimes performs very well, somewhat surprisingly, on low-inflation testing
data sets, demonstrating the robustness of the strategy. Interested readers
may find more discussion in Appendix H.

##  5 Conclusion

In this paper, we studied a leverage-constrained multi-period asset allocation
problem during high inflation periods.

We first identified the high inflation periods from historical data using a
moving-window filtering algorithm. To gain intuition about high-inflation
investing, we tested the passive fixed-mix strategies on the bootstrap
resampled asset returns from the identified inflation regimes. We discovered
empirically that passive strategies with the equal-weighted stock index
partially stochastically dominate passive strategies with the cap-weighted
stock index.

To mimic the behavior of large sovereign wealth funds, we then studied
leverage-constrained multi-period asset allocation problem with the objective
of outperforming a passive benchmark strategy.

Under the assumption that both asset prices follow jump-diffusion models, we
derived a closed-form solution for a two-asset case under the cumulative
tracking difference (CD) objective function. In order to obtain a closed-form
solution, additional assumptions such as continuous rebalancing, unlimited
leverage, and continued trading in insolvency are required. The closed-form
solution provides useful insight for understanding the behavior of the optimal
control. Notably, in order to track the elevated target, the optimal control
needs to aim higher than the target when making allocation decisions.

To address the unrealistic assumptions (e.g., continuous rebalancing,
unlimited leverage, trading if insolvent) required by the closed-form
solution, we then proposed a novel leverage-feasible neural network (LFNN)
model that approximates the optimal control directly. The direct approximation
of a low-dimensional control bypasses the need for high-dimensional
approximation of conditional expectations (which is required in dynamic
programming approaches). In addition, the LFNN model converts the leverage-
constrained optimization problem into an unconstrained optimization problem.
Furthermore, we justify the validity of the LFNN approach by proving that
solution to the parameterized unconstrained optimization problem can
approximate the solution to the original constrained optimization problem with
arbitrary precision.

In numerical experiments, we showed that the LFNN model achieves comparable
results to an approximate form of the closed-form solution on synthetic data
under the CD objective. This validates the capability of the LFNN (even with a
shallow structure) to represent the optimal control accurately. In turn, the
closed-form solution improves the interpretability of the neural network
model.

Then, we studied an empirical investment case with four assets. We evaluated
the LFNN model on bootstrap resampled data and showed that the neural network
strategy achieves consistent outperformance over the benchmark strategy
throughout the investment period. Specifically, the neural network strategy
achieves a 2% higher median IRR over the benchmark strategy, and a higher
terminal wealth with more than 90% probability. The allocation strategy
suggests that managers should favor equal-weighted stock index over cap-
weighted stock index and short-term bonds over long-term bonds during a high-
inflation period.

##  Appendix A Moving-window inflation filter

###  A.1 Filtering algorithm

Algorithm A.1 presents the pseudocode for the moving-window filtering
algorithm.

Data:

CPI[i]; $i=1,\ldots,N$ /* CPI Index */

Cutoff /* High inflation cutoff: annualized */

$\Delta t$ /* CPI index time interval */

$K$ /* smoothing window size */

Result: Flag[i]; $i=1,\ldots,N$ /* = 1 high-inflation month; = 0 otherwise */

/* initialization */

Flag[i] =0; $i=1,\ldots,N$;

for (  _$i=1,\ldots,N-K$_ ) $\\{$

if  _$\log(CPI[i+K]/CPI[i])/(K*\Delta t) >$Cutoff_ then

for (  _$j=0,\ldots,K$_ ) $\\{$

Flag[i+j] = 1 ;

$\\}$

end if

$\\}$

Algorithm A.1 Pseudocode window inflation filter

###  A.2 Effect of moving window size

Figure A.1 shows the filtering results for windows of size 12, 60, and 120
months. We can see that the five-year window produces two obvious inflation
regimes: 1940:8-1951:7 and 1968:9-1985:10, which correspond to well-known
market shocks (i.e. the second world war, and price controls; the oil price
shocks and stagflation of the seventies). Increasing the window size to 10
years results in similar-looking plots as the five-year window size, but the
number of months in each window increases, and the average inflation rate is
lower. Since our objective is to determine the effect of high-inflation
periods on allocation strategies, we choose the five-year window size.

Figure A.1: high-inflation regimes, using the moving-window method, with the
window size shown. The cutoff for high-inflation regimes was $0.05$. High-
inflation months have a label value of one, and low-inflation months have a
label value of zero. CPI data identified from the historical period
1926:1-2022:1.

###  A.3 Asset performance during high inflation

To gain some intuition on the behavior of asset returns during the inflation
periods, we assume that each real (adjusted by CPI index) index follows
geometric Brownian motion (GBM). For example, given an index with value $S$,
then

$\displaystyle dS$ $\displaystyle=$ $\displaystyle\mu S~{}dt+\sigma S~{}dZ~{}$
(A.1)

where $dZ$ is the increment of a Wiener process. We use maximum likelihood
estimation to fit the drift rate $\mu$ (expected arithmetic return) and
volatility $\sigma$ in each regime, for each index, as shown in Table A.1. We
also show a series constructed by: converting the indexes in each regime to
returns, concatenating the two return series, and converting the concatenated
return series back to an index. This concatenated index does not, of course,
correspond to an actual historical index, but is a pseudo-index constructed
from high-inflation regimes. This amounts to a worst-case sequence of returns
in terms of the duration of historical inflation periods, that could plausibly
be expected during a long period of high inflation.

Index | $\mu$ | $\sigma$ | $\mu-\sigma^{2}/2$
---|---|---|---
| 1940:8-1951:7
CapWt | 0.079 | 0.140 | .069
EqWt | 0.145 | 0.190 | .127
10 Year Treasury | -0.035 | 0.036 | -.036
30-day T-bill | -0.050 | 0.029 | -.050
| 1968:9-1985:10
CapWt | 0.026 | 0.164 | .013
EqWt | 0.065 | 0.220 | .041
10 Year Treasury | 0.011 | 0.093 | .007
30-day T-bill | 0.009 | 0.012 | .009
| Concatenated: 1940:8-1951:7 and 1968:9 - 1985:10
CapWt | 0.049 | 0.156 | .038
EqWt | 0.098 | 0.209 | .076
10 Year Treasury | -0.008 | 0.076 | -.011
30-day T-bill | -0.014 | 0.022 | -.014
Table A.1: GBM parameters for the indexes shown. All indexes are real
(deflated). $\mu$ is the expected annualized arithmetic return. $\sigma$ is
the annualized volatility. ($\mu-\sigma^{2}/2$) is the annualized mean
geometric return.

It is striking that in each historical inflation regime (i.e., 1940:8-1951:7
and 1968:9-1985:10) in Table A.1, the drift rate $\mu$ for the equal-weighted
index is much larger than the drift rate for the cap-weighted index. We can
observe that the mean geometric return for the cap-weighted index, in the
period 1968:9-1985:10, was only about one percent per year.

It is also noticeable that bonds performed very poorly in the period
1940:8-1951:7. As well, during the period 1968:9-1985:10, there was
essentially no term premium for 10-year treasuries, compared with 30-day
T-bills. In addition, the 10-year treasury index had much higher volatility
compared to the 30-day T-bill index. Looking at the concatenated series, it
appears that 30-day T-bills are arguably the better defensive asset here since
the volatility of this index is quite low (but with a negative (real) drift
rate).

Consequently, in the following, we will focus attention on 30-day T-bills, the
cap-weighted index, and the equal-weighted index.

##  Appendix B Bootstrap resampling

###  B.1 Stationary block bootstrap algorithm

Algorithm B.1 presents the pseudocode for the stationary block bootstrap. See
Ni et al. (2022) for more discussion.

/* initialization */

bootstrap_samples = [ ];

/* loop until the total number of required samples are reached */

while  _True_ do

/* choose random starting index in [1,…,N], N is the index of the last
historical sample */

index = UniformRandom( 1, N );

/* actual blocksize follows a shifted geometric distribution with the expected
value of exp_block_size */

blocksize = GeometricRandom( $\frac{1}{exp\\_block\\_size}$ );

for (  _$i=0;\ i <blocksize;\ i=i+1$_ ) $\\{$

/* if the chosen block exceeds the range of the historical data array, do a
circular bootstrap */

if  _index + i $>$ N_ then

bootstrap_samples.append( historical_data[ index + i - N ] );

else

bootstrap_samples.append( historical_data[ index + i ] );

end if

if  _bootstrap_samples.len() == number_required_ then

return bootstrap_samples;

end if

$\\}$

end while

Algorithm B.1 Pseudocode for stationary block bootstrap

###  B.2 Effect of blocksize

As discussed, we will use bootstrap resampling (Politis and Romano, 1994;
Politis and White, 2004; Patton et al., 2009; Dichtl et al., 2016; Anarkulova
et al., 2022), to analyze the performance of using the equal-weighted index
compared to the cap-weighted index, during periods of high inflation (our
concatenated series: 1940:8-1951:7, 1968:9-1985:10).

First, we examine the effect of the expected blocksize parameter in the
bootstrap resampling algorithm. We will use a paired sampling approach, where
we simultaneously draw returns from the bond and stock indexes.232323This
preserves correlation effects. The algorithm in Politis and White (2004) was
developed for single asset time series. It is therefore important to assess
the effect of the blocksize on numerical results. In Table B.1, we examine the
effect of different blocksizes on the statistics of stationary block bootstrap
resampling.

Expected blocksize (months) | Median[$W_{T}$] | E[$W_{T}$] | std[$W_{T}$] | 5th Percentile
---|---|---|---|---
1 | 170.9 | 191.6 | 97.6 | 78.6
3 | 174.6 | 202.9 | 120.4 | 69.3
6 | 174.2 | 204.2 | 125.9 | 66.8
12 | 175.5 | 204.4 | 124.2 | 67.9
24 | 179.2 | 205.1 | 118.4 | 68.7
Table B.1: Effect of expected blocksize, on the statistics of the final wealth
$W(T)$ at $T=10$ years. Constant weight, scenario in Table 2.2. Equity weight:
0.7, rebalanced monthly. Bond index: 30-day T-bill. Equity index: equal-
weighted. Concatenated series: 1940:8-1951:7, 1968:9-1985:10 (high-inflation
regimes). All quantities are real (inflation-adjusted). Initial wealth 100.
Bootstrap resampling, $10,000$ resamples).

Perhaps a more visual way of analyzing the effect of the expected blocksize is
shown in Figure B.1, where we show the cumulative distribution function (CDF)
of the final wealth after 10 years, for different blocksizes. We show the CDF
since this gives us a visualization of the entire final wealth distribution,
not just a few summary statistics.

Since the data frequency is at one-month intervals, specifying a geometric
mean expected blocksize of one month means that the blocksize is always a
constant one month. This effectively means that we are assuming that the data
is i.i.d. However, the one-month results are an outlier, compared to the other
choices of expected blocksize. There is hardly any difference between the CDFs
for any choice of expected blocksize in the range of 3-24 months. In this
article, we use an expected blocksize of 6 months.

Figure B.1:  Cumulative distribution function (CDF), final wealth $W(T)$ at
$T=10$ years. The effect of expected blocksize. Constant weight, scenario in