# Strategic mean-variance investing under mean-reverting stock returns

Søren Fiig Jarner
Danish Labour Market Supplementary Pension Fund (ATP)

(March, 2021)

###### Abstract

In this report we derive the strategic (deterministic) allocation to bonds and
stocks resulting in the optimal mean-variance trade-off on a given investment
horizon. The underlying capital market features a mean-reverting process for
equity returns, and the primary question of interest is how mean-reversion
effects the optimal strategy and the resulting portfolio value at the horizon.
In particular, we are interested in knowing under which assumptions and on
which horizons, the risk-reward trade-off is so favourable that the value of
the portfolio is effectively bounded from below on the horizon. In this case,
we might think of the portfolio as providing a stochastic excess return on top
of a “guarantee” (the lower bound).

Deriving optimal strategies is a well-known discipline in mathematical
finance. The modern approach is to derive and solve the Hamilton-Jacobi-
Bellman (HJB) differential equation characterizing the strategy leading to
highest expected utility, for given utility function. However, for two reasons
we approach the problem differently in this work. First, we wish to find the
optimal strategy depending on time only, i.e., we do not allow for
dependencies on capital market state variables, nor the value of the portfolio
itself. This constraint characterizes the strategic allocation of long-term
investors. Second, to gain insights on the role of mean-reversion, we wish to
identify the entire family of extremal strategies, not only the optimal
strategies. To derive the strategies we employ methods from calculus of
variations, rather than the usual HJB approach.

_Keywords:_ Deterministic strategies, mean-variance optimization, mean-
reverting stock returns, calculus of variations.

###### Contents

  1. 1 Introduction
    1. 1.1 Outline
  2. 2 Preliminaries
    1. 2.1 Capital market model
    2. 2.2 Portfolio dynamics
    3. 2.3 Portfolio distribution
  3. 3 Optimal mean-variance strategies
    1. 3.1 Optimal rate strategies
      1. 3.1.1 Interpretation of the optimal bond strategy
      2. 3.1.2 Closed form expressions for the extremal mean and variance
    2. 3.2 Optimal equity strategies
      1. 3.2.1 Characterization of extremal equity strategies
      2. 3.2.2 Optimal equity strategies with positive mean-variance trade-off
      3. 3.2.3 Extremal equity strategies with negative mean-variance trade-off
    3. 3.3 Optimal joint strategies
      1. 3.3.1 Dependent risk factors
  4. 4 Numerical illustrations
    1. 4.1 Optimal rate strategies
      1. 4.1.1 Yield curves
      2. 4.1.2 Risk-reward profiles
      3. 4.1.3 Portfolio distributions
      4. 4.1.4 Illustration of optimal strategies
    2. 4.2 Optimal equity strategies
      1. 4.2.1 Volatility profiles and calibration of mean-reversion
      2. 4.2.2 Risk-reward profiles
      3. 4.2.3 Illustration of optimal strategies
      4. 4.2.4 Interior wedges of extremal strategies
  5. 5 Concluding remarks
  6. A Optimal equity strategies with positive mean-variance trade-off
  7. B Overview of extremal equity strategies
    1. B.1 Extremal equity strategies of type II and III
  8. C Proofs of results in the main text
  9. D Risk and reward statistics
    1. D.1 Optimal rate strategies
    2. D.2 Optimal equity strategies

##  1 Introduction

This is a technical document providing the theoretical foundation for a study
of the risk-reward trade-off on long horizons in a capital market under which
equity returns are mean-reverting. The problem is stated and solved in
continuous-time based on the model analyzed in Jarner and Preisel (2017), see
also Munk et al. (2004).

We are particularly interested in studying how the risk-reward trade-off
depends on the degree of mean-reversion and the horizon, and under which
assumptions the trade-off is so favourable that the (optimal) portfolio is
effectively bounded from below. In this case, one might argue that the
portfolio can be used as a “hedge” for a guaranteed payment, equal to the
lower bound of the portfolio.

The optimal strategy itself is of course also of interest. In a model without
mean-reversion it is optimal to hold a constant share of equities over time.
This is intuitively clear, since under this assumption a given equity return
has the same probability and the same effect on the (final) portfolio value
regardless of when it occurs. In other words, it is equally likely and equally
bad to lose, say, 10% of your investment on the first day as it is to lose it
on the last day.

Under mean-reverting equity returns the situation is more complicated.
Intuitively we would expect early losses to have a smaller effect than late
losses, since an early loss has a higher chance of being “reverted” than a
late loss. Qualitatively, this ought to imply a higher (optimal) equity
exposure in the beginning of the period and a lower equity exposure at the end
of the period. Quantitatively, however, we need to take account of the fact
that we only benefit from mean-reversion if we maintain an equity exposure
throughout. Loosely speaking, to benefit from the assumed mean-reversion we
need to hold equities when they (hopefully) rebound later on. Essentially,
“early” risk is better rewarded than “late” risk, but only if we take “late”
risk. Mathematically, the optimal strategy needs to balance early and late
risk taking account of the benefit late risk has on early risk.

We are looking for optimal strategies depending on time only. More precisely,
the strategy is allowed to depend on the initial value of the state variables
only. In particular, the strategy is not allowed to depend on the
unobservable, stochastic excess return on equities as it changes over the
period, nor on the value of the portfolio. This assumption makes the strategy
less tailored to the specific capital market model used to derive it, and it
is therefore reasonable to expect that the strategy is also close to optimal
under other, less stylized, types of mean-reversion. The assumption also
ensures that the strategies can be implemented and tested in practice, without
the need to identify latent variables. Christiansen and Steffensen (2015,
2018) find optimal, deterministic strategies motivated by the fact that
marketed life-cycle products are deterministic, but apart from their work only
few results exist in the literature.

Focusing on deterministic strategies implies that the portfolio is log-
normally distributed on all horizons, in the model employed. The problem can
thereby be stated as optimizing the mean for given variance of the (log)
portfolio value on the horizon of interest. We focus primarily on optimal
equity strategies, but we also consider optimal rate strategies and jointly
optimal strategies. The optimal equity strategies are derived by methods from
calculus of variations in three steps. First, we derive an integral equation
characterizing the optimal strategy. Second, we use the integral equation to
derive a second order differential equation, which gives us the general
solution. Finally, we derive and solve a system of equations for determining
the constants of the general solution. The solution is explicit up to the
presence of a Lagrange multiplier, which when varied gives the entire family
of extremal, including optimal, strategies for varying levels of variance. We
also provide illustrations of the results, discussing in particular the role
of mean-reversion.

###  1.1 Outline

The rest of the paper is organized as follows. In Section 2 we set up the
model and provide the distributional results on the portfolio value on future
horizons needed for the optimization, in Section 3 we formulate and solve
three optimization problems, finding optimal rate and equities strategies,
both separately and jointly. The results are illustrated and discussed in
Section 4, and Section 5 summarizes and concludes on the findings. All proofs
and technical details are in the appendices.

##  2 Preliminaries

In this section we state the underlying capital market model. We refer to the
model as a factor model due to its intended use as a sparse representation of
generic “rates” and ”equities”. Accordingly, we interpret the resulting
strategies as profiles for generic interest and equity risk, respectively. How
to obtain this risk in practise, i.e., the selection of specific bonds and
stocks, is outside the scope of the model.

After introducing the capital market model, we state a distributional result
for portfolios resulting from time-dependent strategies of the kind we will be
considering. The formulation and solution of the optimization problem rely on
this result and the accompanying integral representation.

###  2.1 Capital market model

The capital market model is described in Jarner and Preisel (2017), see also
Munk et al. (2004), but for ease of reference we restate it here. The model of
Jarner and Preisel (2017) also contains realized inflation and break-even
inflation (BEI) curves for pricing inflation-indexed bonds, and inflation
swaps. Here, however, we disregard inflation and include only nominal interest
rates and equities in the model.

Hence, the model features

  * •

A stochastic short rate ($r_{t}$)

  * •

A bond market of all maturities with stochastic risk premium
($\lambda_{t}^{r}$)

  * •

A stochastic equity index ($S_{t}$)

  * •

A mean-reverting equity risk premium ($x_{t}$)

Since we are interested in portfolio optimization we need the so-called ’real
world’ dynamics of the state variables of the model (sometimes referred to as
the $P$-dynamics as opposed to the $Q$-dynamics used for pricing). In addition
to these, we also need the evolution of the term structure of interest rates
(the yield curve) for pricing bonds.

We assume that the short (nominal) interest rate follows an Ornstein-Uhlenbeck
process,

$\displaystyle dr_{t}=\kappa(\bar{r}-r_{t})dt+\sigma_{r}dW^{r}_{t},$ (1)

where $\bar{r}$ is the long-run mean of the short interest rate, $\kappa$
describes the degree of mean reversion, $\sigma_{r}$ is the interest rate
volatility, and $W^{r}$ is a standard Brownian motion.

The stock index (total return index) is assumed to evolve according to the
dynamics

$\displaystyle\frac{dS_{t}}{S_{t}}=(r_{t}+x_{t})dt+\sigma_{S}dW^{S}_{t},$ (2)

where $r_{t}$ is the short rate from (1), $x_{t}$ is the time-varying risk
premium (expected excess return) from investing in stocks, $\sigma_{S}$ is the
stock index volatility, and $W^{S}$ is a standard Brownian motion. We further
assume that the risk premium follows an Ornstein-Uhlenbeck process,

$\displaystyle dx_{t}=\alpha(\bar{x}-x_{t})dt-\sigma_{x}dW^{S}_{t},$ (3)

where $\bar{x}$ denotes the long-run equity risk premium, $\alpha$ describes
the degree of mean reversion towards this level, and $\sigma_{x}$ is the risk
premium volatility. We assume joint normality of the two Brownian motions
$W^{r}$ and $W^{S}$ with correlation coefficient $\rho$.

Note that the stock index and the risk premium processes are locally perfectly
negatively correlated, i.e., a stock return above or below its expected value
will “cause” a change in the (future) risk premium in the opposite direction.
This interaction induces a mean-reversion in the stock returns over time.

Finally, we assume that the term structure of interest rates is of the form
considered by Vasicek (1977). Specifically, we assume that the (arbitrage
free) price at time $t$ of a zero-coupon bond maturing at time $T\geq t$ is
given by

$\displaystyle p_{t}(T)=\exp\left\\{G(\Delta)-H(\Delta)r_{t}\right\\},$ (4)

with $\Delta=T-t$,

$\displaystyle H(\Delta)$
$\displaystyle=\frac{1}{a}\left(1-\exp\\{-a\Delta\\}\right),$ (5)
$\displaystyle G(\Delta)$
$\displaystyle=\left(b-\frac{\sigma_{r}^{2}}{2a^{2}}\right)\left(H(\Delta)-\Delta\right)-\frac{\sigma_{r}^{2}}{4a}H^{2}(\Delta),$
(6)

and where $a$ and $b$ are parameters controlling the slope and level of the
yield curves.111The price of a zero-coupon bond can be obtained by the usual
risk-neutral valuation formula,
$p_{t}(T)=\operatorname{\mathsf{E}}_{t}^{Q}\left[-\int_{t}^{T}r_{s}ds\right]$,
where $dr_{t}=a(b-r_{t})dt+\sigma_{r}d\bar{W}^{r}_{t}$ with $\bar{W}^{r}$
being a $Q$-Brownian motion. The specification corresponds to the market price
of interest rate risk being equal to

$\lambda^{r}_{t}=\\{(a-\kappa)r_{t}+\kappa\bar{r}-ab\\}/\sigma_{r}.$ (7)

As shown in Section 5.1 of Jarner and Preisel (2017), the price dynamics of
$p_{t}(T)$ for fixed $T$ has the form

$\frac{dp_{t}(T)}{p_{t}(T)}=(r_{t}-\lambda^{r}_{t}\Psi(a,T-t)\sigma_{r})dt-\Psi(a,T-t)\sigma_{r}dW^{r}_{t},$
(8)

where

$\displaystyle\Psi(a,t)\equiv\int_{0}^{t}e^{-au}du=\begin{cases}t&\mbox{for
}a=0,\\\ \frac{1}{a}\left(1-e^{-at}\right)&\mbox{for }a\neq 0.\\\ \end{cases}$
(9)

From (8) we see that the (negative) market price of interest rate risk,
$-\lambda^{r}_{t}$, can be interpreted as the risk-reward trade-off at a given
point in time, i.e., as the excess return per unit of volatility. It is the
continuous-time analogue to the Sharpe ratio used in portfolio construction
and benchmarking, cf. Sharpe (1966, 1994). In general, the market price of
interest rate risk depends on the current short rate, in particular, the
market price of interest rate risk is stochastic. For subsequent use, we
define the market price of equity risk as the excess return per unit of
volatility when investing in equities,

$\lambda^{S}_{t}=x_{t}/\sigma_{S}.$ (10)

###  2.2 Portfolio dynamics

We now consider the dynamics of a portfolio exposed to (interest) rate and
equity risk. In general, the dynamics of the portfolio value, $V_{t}$, is
given by

$\frac{dV_{t}}{V_{t}}=\left(r_{t}+f_{t}^{\prime}\lambda_{t}\right)dt+f_{t}^{\prime}dW_{t},$
(11)

where $f_{t}=(f^{r}_{t},f^{S}_{t})^{\prime}$ is the vector of (factor)
exposures to rate and equity risk, respectively,
$\lambda_{t}=(\lambda_{t}^{r},\lambda_{t}^{S})^{\prime}$ is the vector of
market prices of risk given by (7) and (10), and
$W_{t}=(W_{t}^{r},W_{t}^{S})^{\prime}$ is the vector of driving Brownian
motions.

The exposures are measured in volatility and formula (11) succinctly states
that the excess return, i.e., the return in excess of the short rate, is the
market prices of risk weighted by the exposure. As an example, if
$f^{r}_{t}\equiv 0$ and $f^{S}_{t}\equiv\sigma_{S}$ we get

$\frac{dV_{t}}{V_{t}}=\left(r_{t}+\sigma_{S}\lambda^{S}_{t}\right)dt+\sigma_{S}dW^{S}_{t}=\left(r_{t}+x_{t}\right)dt+\sigma_{S}dW^{S}_{t},$
(12)

which is equal to the dynamics of the equity index, as expected.

In general, assuming only that the strategies are adapted and sufficiently
regular, we have the following integral representation of the portfolio
process

$V_{t}=V_{0}\exp\left\\{\int_{0}^{t}r_{s}ds+\int_{0}^{t}f_{s}^{\prime}\lambda_{s}ds-\frac{1}{2}\int_{0}^{t}f_{s}^{\prime}Cf_{s}ds+\int_{0}^{t}f_{s}^{\prime}dW_{s}\right\\},$
(13)

where $C$ is the (instantaneous) correlation matrix given by

$C=\begin{pmatrix}1&\rho\\\ \rho&1\end{pmatrix}.$ (14)

This follows by an application of the multi-dimensional version of Itô’s
lemma, see e.g. Proposition 4.18 of Björk (2009). Let
$X_{t}=\int_{0}^{t}r_{s}ds+\int_{0}^{t}f_{s}^{\prime}\lambda_{s}ds-\frac{1}{2}\int_{0}^{t}f_{s}^{\prime}Cf_{s}ds+\int_{0}^{t}f_{s}^{\prime}dW_{s}$,
and use Itô’s lemma to calculate the differential of $V_{t}=G(X_{t})$, where
$G(x)=V_{0}\exp(x)$,

$\displaystyle dV_{t}$ $\displaystyle=$ $\displaystyle\frac{\partial
G}{\partial x}dX_{t}+\frac{1}{2}\frac{\partial^{2}G}{\partial
x^{2}}dX_{t}dX_{t}$ $\displaystyle=$ $\displaystyle
V_{t}\left[\left(r_{t}+f_{t}^{\prime}\lambda_{t}-\frac{1}{2}f_{t}^{\prime}Cf_{t}\right)dt+f_{t}^{\prime}dW_{t}\right]+\frac{1}{2}V_{t}\left[\left(f^{r}_{t}\right)^{2}+\left(f^{S}_{t}\right)^{2}+2\rho
f^{r}_{t}f^{S}_{t}\right]dt$ $\displaystyle=$ $\displaystyle
V_{t}\left[\left(r_{t}+f_{t}^{\prime}\lambda_{t}\right)dt+f_{t}^{\prime}dW_{t}\right],$

which is the same as (11). Since the right-hand side of (13) equals $V_{0}$
for $t=0$ we conclude (under standard regularity conditions) that (13) is the
solution to (11) as claimed. In the second equality of the above calculations
we have used the formal multiplication rules:
$dW^{r}_{t}dW^{r}_{t}=dW^{S}_{t}dW^{S}_{t}=dt$ and $dW^{r}_{t}dW^{S}_{t}=\rho
dt$.

###  2.3 Portfolio distribution

As mentioned in the introduction, we are aiming at finding the optimal risk-
reward trade-off for a deterministic strategy, i.e., a strategy that depends
on time only. We are aided in this task by the fact that under this
restriction, the portfolio is log-normally distributed at every horizon. In
this section we derive the necessary distributional results.

We need the following integral representations for the risk premium, the short
rate and the integrated short rate from Section 3 of Jarner and Preisel
(2017):

$\displaystyle x_{t}$ $\displaystyle=\bar{x}+e^{-\alpha
t}(x_{0}-\bar{x})-\sigma_{x}\int_{0}^{t}e^{-\alpha(t-s)}dW^{S}_{s},$ (15)
$\displaystyle r_{t}$ $\displaystyle=\bar{r}+e^{-\kappa
t}(r_{0}-\bar{r})+\sigma_{r}\int_{0}^{t}e^{-\kappa(t-s)}dW^{r}_{s},$ (16)
$\displaystyle\int_{0}^{t}r_{s}ds$
$\displaystyle=t\bar{r}+(r_{0}-\bar{r})\Psi(\kappa,t)+\sigma_{r}\int_{0}^{t}\Psi(\kappa,t-s)dW^{r}_{s},$
(17)

where $\Psi$ is given by (9). Since $\lambda^{r}_{t}$ and $\lambda^{S}_{t}$
are linear in $r_{t}$ and $x_{t}$, respectively, it follows from the
expressions above that the integrals at the right-hand side of (13) are either
deterministic, or stochastic integrals with deterministic integrands and
Brownian integrators. From this observation, the claimed log-normality
follows; the mean and variance are given by the following theorem.

######  Theorem 2.1.

Assume that $f_{t}=(f^{r}_{t},f^{S}_{t})^{\prime}$ is deterministic (depends
on time only). For given $T\geq 0$, the portfolio value at time $T$ can be
expressed as

$V_{T}=V_{0}\exp\left\\{m^{0}_{T}+m^{r}_{T}+m^{S}_{T}-\rho\int_{0}^{T}f^{r}_{u}f^{S}_{u}du+\int_{0}^{T}h_{u}^{r}dW^{r}_{u}+\int_{0}^{T}h^{S}_{u}dW^{S}_{u}\right\\},$
(18)

where

$\displaystyle m^{0}_{T}$
$\displaystyle=T\bar{r}+(r_{0}-\bar{r})\Psi(\kappa,T),$ (19) $\displaystyle
m^{r}_{T}$
$\displaystyle=\frac{a(\bar{r}-b)}{\sigma_{r}}\int_{0}^{T}f^{r}_{s}ds+\frac{a-\kappa}{\sigma_{r}}\int_{0}^{T}e^{-\kappa
s}(r_{0}-\bar{r})f^{r}_{s}ds-\frac{1}{2}\int_{0}^{T}\left(f^{r}_{s}\right)^{2}ds,$
(20) $\displaystyle m^{S}_{T}$
$\displaystyle=\frac{1}{\sigma_{S}}\int_{0}^{T}f^{S}_{s}\left(\bar{x}+e^{-\alpha
s}(x_{0}-\bar{x})\right)ds-\frac{1}{2}\int_{0}^{T}\left(f^{S}_{s}\right)^{2}ds,$
(21) $\displaystyle h^{r}_{u}$
$\displaystyle=\sigma_{r}\Psi(\kappa,T-u)+f^{r}_{u}+(a-\kappa)\int_{u}^{T}f^{r}_{s}e^{-\kappa(s-u)}ds,$
(22) $\displaystyle h^{S}_{u}$
$\displaystyle=f^{S}_{u}-\frac{\sigma_{x}}{\sigma_{S}}\int_{u}^{T}f^{S}_{s}e^{-\alpha(s-u)}ds.$
(23)

In particular, $V_{T}/V_{0}$ is log-normally distributed with mean and
variance given by

$\displaystyle\mu_{T}$
$\displaystyle=m^{0}_{T}+m^{r}_{T}+m^{S}_{T}-\rho\int_{0}^{T}f^{r}_{u}f^{S}_{u}du,$
(24) $\displaystyle\sigma_{T}^{2}$
$\displaystyle=\int_{0}^{T}\left(h_{u}^{r}\right)^{2}+\left(h^{S}_{u}\right)^{2}+2\rho
h^{r}_{u}h^{S}_{u}du.$ (25)

Note that the weight functions $h^{r}$ and $h^{S}$, which measure the impact
of the risk sources $W^{r}$ and $W^{S}$ on the horizon value of the portfolio,
contain both a direct and an indirect effect. Due to mean-reversion, the risk
(innovation) at time $u$ measured by $dW^{r}_{u}$ and $dW^{S}_{u}$,
respectively, has both a direct effect which depends on the exposure at time
$u$ (the terms $f^{r}_{u}$ and $f^{S}_{u}$ in $h^{r}_{u}$ and $h^{S}_{u}$,
respectively) and an indirect effect which depends on the subsequent exposure
to the risk source (the integrals from $u$ to $T$). The direct effect depends
only on the current exposure, while the indirect effect depends over the
(future) exposure on the remaining horizon.

We also note that the indirect effect for rates disappears if $a=\kappa$,
which corresponds to the case where the market price of interest rate risk is
constant, $\lambda^{r}_{t}\equiv\kappa(\bar{r}-b)/\sigma_{r}$. Thus, it is not
the mean-reverting nature of the short rate process itself, but rather the
state dependent market price of risk, that causes the indirect effect for
rates. For equities, the indirect effect is caused by the feedback mechanism
from equity risk to (future) equity risk premia. The size of this effect
depends on the “loading” ratio $\sigma_{x}/\sigma_{S}$ and the degree of
persistence, $\alpha$, in the equity risk premium process.

In principle, Theorem 2.1 can be used for joint optimization of the mean-
variance trade-off at a given horizon. However, to make the computations
easier to follow and to highlight the structure of the solution in special
cases, we solve the problem in stages under various simplifying assumptions.
We are primarily interested in the effect of mean-reverting equity returns and
consequently we keep this feature, while we make the simplifying assumptions
that the market price of interest rate risk is constant ($a=\kappa$) and that
rate and equity factors are independent ($\rho=0$).

We state without proof a simplification of Theorem 2.1 to be used in the
following.

######  Corollary 2.2.

Assume that $f^{r}_{t}$ is deterministic, $f^{S}_{t}\equiv 0$ and $a=\kappa$.
For given $T>0$, $V_{T}/V_{0}$ is log-normally distributed with mean and
variance given by

$\displaystyle\mu_{T}$
$\displaystyle=m^{0}_{T}+\lambda^{r}\int_{0}^{T}f^{r}_{s}ds-\frac{1}{2}\int_{0}^{T}\left(f^{r}_{s}\right)^{2}ds,$
(26) $\displaystyle\sigma_{T}^{2}$
$\displaystyle=\int_{0}^{T}\left[\sigma_{r}\Psi(\kappa,T-s)+f^{r}_{s}\right]^{2}ds,$
(27)

where $m^{0}_{T}$ is given by (19) of Theorem 2.1, and
$\lambda^{r}=\kappa(\bar{r}-b)/\sigma_{r}$.

##  3 Optimal mean-variance strategies

We are now ready to state the optimization problem(s) and derive the solution.
As previously announced we proceed in stages: rates only, equities only, and
finally combined rates and equities assuming independence of the two factors.

###  3.1 Optimal rate strategies

We first want to find the deterministic, rates only investment strategies that
lead to the optimal (log) mean-variance trade-off on a given horizon. We
consider only the case of a constant market price of interest rate risk,
corresponding to $a=\kappa$. Clearly, a constant market price of interest rate
risk is a mathematical simplification. However, at this stage we are primarily
interested in getting an intuition for the structure of the optimal strategies
in a simple setup.222The solution technique developed in the next section for
handling mean-reverting stock returns can be used to solve the rates only
problem for general $a$, but the resulting strategies are less intuitive and
harder to interpret.

Corollary 2.2 gives the mean and variance of the log portfolio value and the
problem can therefore be phrased as

Problem 1:  Find $f^{r}$ such that (26) is maximized for given value of (27).

For given horizon, $T$, the solution to Problem 3.1 takes the form of a family
of strategies, one for each value of the variance $\sigma_{T}^{2}$. Members of
this family are referred to as optimal bond strategies. More generally, we
refer to bond strategies maximizing, or minimizing, (26) for given value of
(27) as extremal bond strategies.

######  Theorem 3.1.

Assume $a=\kappa$ such that the market price of interest rate risk is
constant, $\lambda^{r}=\kappa(\bar{r}-b)/\sigma_{r}$. For given horizon $T>0$,
let $g^{r}_{s}=\sigma_{r}\Psi(\kappa,T-s)$ for $0\leq s\leq T$. The extremal
bond strategies on horizon $T$ are of the form

$f^{r}_{s}=\frac{\lambda^{r}+2\nu g^{r}_{s}}{1-2\nu}\quad(0\leq s\leq T),$
(28)

where $\nu$ depends on the prescribed value of the variance $\sigma_{T}^{2}$.

The optimal bond strategies have $\nu<1/2$, and the bond strategies minimizing
the mean for given value of the variance have $\nu>1/2$.

For $\nu=0$, Theorem 3.1 yields $f_{s}^{r}=\lambda^{r}$, which is the strategy
(globally) maximising (26). Recall from (8) that long positions in bonds
correspond to negative exposures to the underlying Brownian motion, $W^{r}$,
and that positive excess returns for bonds correspond to $\lambda^{r}<0$.
Thus, the maximizing strategy, $f_{s}^{r}=\lambda^{r}$, is long in bonds when
bonds have positive excess returns, and short in bonds when bonds have
negative excess returns. We typically expect positive excess returns for
bonds, but we cannot rule out the possibility of negative excess returns,
$\lambda^{r}>0$, neither mathematically, nor in practise. Also note, that the
maximising strategy amounts to a constant exposure to interest rate risk over
time; this is the only constant, optimal strategy.

The proof of Theorem 3.1 is a variational argument where $\nu$ plays the role
of a Lagrange multiplier introduced to handle the variance constraint. At an
extremal point (strategy) the gradients of the mean functional (the objective)
and the variance functional (the constraint) are parallel with proportionality
coefficient $-\nu$. If $\nu<0$ we are at a point where relaxing the
constraint, i.e., allowing a larger variance, leads to a larger mean.
Conversely, if $\nu>0$ we are at a point where a larger variance leads to a
smaller mean. With this in mind, we can categorize the optimal strategies of
Theorem 3.1 as follows.

In the limit $\nu$ tending to minus (or plus) infinity, we have the strategy
$f_{s}^{r}=-g_{s}^{r}$, which corresponds to the buy-and-hold strategy of a
bond that matures at time $T$, cf. (8). This strategy has $\sigma_{T}^{2}=0$,
and it is the optimal (and only) strategy with no variance on the horizon. For
$\nu<0$ we obtain optimal strategies where added risk is rewarded, i.e., as
$\nu$ is increased from minus infinity to zero we get optimal strategies with
higher variance and higher mean. For $\nu=0$ we obtain the maximal mean
possible. For $0<\nu<1/2$, we have optimal strategies where added risk is
penalized, i.e., strategies where the prescribed value of the variance can
only be achieved by exposures so high that the quadratic term in (26) impairs
the mean. Mathematically, these strategies are optimal in the sense that they
achieve the highest possible mean given the variance, but in practice they are
sub-optimal: for each of these strategies there exist strategies with lower
variance and identical or higher mean. Hence, for practical purposes we are
interested only in strategies of form (28) with $\nu\leq 0$.

####  3.1.1 Interpretation of the optimal bond strategy

The optimal strategy of Theorem 3.1 can be written
$f^{r}_{s}=a(\nu)(-g_{s}^{r})+(1-a(\nu))\lambda^{r}$, where
$a(\nu)=-2\nu/(1-2\nu)$. For $\nu\leq 0$, we have $0\leq a(\nu)<1$, and hence
we can interpret the strategy as a convex combination of two optimal
strategies: $T$-bonds and constant risk exposure of size $\lambda^{r}$. It is
natural to interpret the $T$-bonds as a risk-free hedging component, and the
constant risk exposure as a return-seeking component. This interpretation is
supported by the fact that we always hold a long, unlevered position in
$T$-bonds, while the constant risk exposure is obtained by either a long or
short bond position depending on the sign of the market price of interest rate
risk.

Strictly speaking, this is only one possible way to interpret the optimal
strategy. Since we are using a single factor rate model, all bond returns are
fully correlated and only the net interest rate exposure matters. Furthermore,
the exposure can be taken using bonds of any maturity (with varying degrees of
leverage). Still, interpreting (28) as a combination of variance reducing
$T$-bonds ("hedging") and additional long, or short, return-seeking bond
positions ("investments") provides a useful intuition. The structure and the
interpretation closely resemble the classic two-fund separation theorem of
modern portfolio theory, see Markowitz (1952); Tobin (1958); Merton (1972).

For $0<\nu<1/2$ we still have a linear, but no longer a convex, combination of
"hedging" and "investment" strategies. More precisely, for $0<\nu<1/2$ we have
$a(\nu)<0$ and $1-a(\nu)>1$. Hence, in this situation we have a short position
in $T$-bonds and a long return-seeking position, larger than the optimal level
of $\lambda^{r}$. Thus we are borrowing risk-free funds (as seen from time
$T$) and investing these in the optimal return-seeking strategy. Finally, the
extremal strategies minimizing the mean have $\nu>1/2$ which correspond to
$a(\nu)>1$ and $1-a(\nu)<0$. Essentially, we are spending the variance budget
on borrowing funds as expensively as possible and investing them in risk-free
$T$-bonds.

####  3.1.2 Closed form expressions for the extremal mean and variance

For given $\nu$, we can find the portfolio mean and variance by computing the
integrals of (26) and (27), respectively, after substituting $f^{r}_{s}$ with
the extremal strategy of (28). Due to the simple form of the extremal strategy
it is possible to evaluate these integrals analytically, and we will do so
here. However, apart from special cases, e.g., $\nu=\pm\infty$, or $\nu=0$,
the resulting expressions are hard to interpret. Instead, in Section 4.1 we
will investigate numerically the risk-reward profile obtained by varying
$\nu$.

To express the results we introduce the function $\Upsilon$ given by

$\displaystyle\Upsilon(a,t)\equiv\int_{0}^{t}\Psi^{2}(a,s)ds=\begin{cases}\frac{t^{3}}{3}&\mbox{for
}a=0,\\\ \frac{1}{2a^{3}}\left(-3+2at+4e^{-at}-e^{-2at}\right)&\mbox{for
}a\neq 0.\\\ \end{cases}$ (29)

Using this notation we can express the zero-coupon bond price as

$p_{t}(T)=\exp\left\\{-\Delta
b-\Psi(a,\Delta)(r_{t}-b)+\frac{\sigma_{r}^{2}}{2}\Upsilon(a,\Delta)\right\\},$
(30)

where $\Delta=T-t$, cf. Section 5.1 of Jarner and Preisel (2017). This
expression is valid for all values of $a$. However, recall that the extremal
strategies of Theorem 3.1 are derived under the simplifying assumption
$a=\kappa$, such that $\lambda^{r}$ is constant.

We can now evaluate (26) with $f_{s}^{r}$ given by (28)

$\displaystyle\mu_{T}$
$\displaystyle=\underbrace{T\bar{r}+\Psi(\kappa,T)(r_{0}-\bar{r})}_{m_{T}^{0}}+\underbrace{\frac{T(\lambda^{r})^{2}+2\nu(T-\Psi(\kappa,T))(\bar{r}-b)}{1-2\nu}}_{\lambda^{r}\int_{0}^{T}f_{s}^{r}ds}-\frac{1}{2}\int_{0}^{T}(f_{s}^{r})^{2}ds$
$\displaystyle=T\left[\frac{\bar{r}}{1-2\nu}-\frac{2\nu
b}{1-2\nu}+\frac{(\lambda^{r})^{2}}{1-2\nu}-\frac{(\lambda^{r})^{2}}{2(1-2\nu)^{2}}-\frac{2\nu(\bar{r}-b)}{(1-2\nu)^{2}}\right]+$
(31)
$\displaystyle\quad\Psi(\kappa,T)\left[r_{0}-\frac{\bar{r}}{1-2\nu}+\frac{2\nu
b}{1-2\nu}+\frac{2\nu(\bar{r}-b)}{(1-2\nu)^{2}}\right]-\frac{\sigma_{r}^{2}}{2}\left(\frac{2\nu}{1-2\nu}\right)^{2}\Upsilon(\kappa,T),$
(32)

where we have used that $\lambda^{r}\sigma_{r}/\kappa=\bar{r}-b$. Evaluation
of the integral in (27) with $f_{s}^{r}$ given by (28) gives us the extremal
variance

$\displaystyle\sigma_{T}^{2}$
$\displaystyle=\int_{0}^{T}[\sigma_{r}\Psi(\kappa,T-s)+f_{s}^{r}]^{2}ds=\frac{\int_{0}^{T}[\lambda^{r}+\sigma_{r}\Psi(\kappa,T-s)]^{2}ds}{(1-2\nu)^{2}}$
$\displaystyle=\frac{T[(\lambda^{r})^{2}+2(\bar{r}-b)]-\Psi(\kappa,T)2(\bar{r}-b)+\sigma_{r}^{2}\Upsilon(\kappa,T)}{(1-2\nu)^{2}}.$
(33)

It follows from (33) that there are two extremal strategies for any positive
specification of the variance, with the pair of Lagrange multipliers
characterising the strategies being of the form $\nu=1/2\pm\eta$ for $\eta>0$.
The strategy with $\nu<1/2$ is mean optimizing, and the strategy with
$\nu>1/2$ is mean minimizing. In the limit for $\nu$ tending to plus or minus
infinity, we have $\sigma_{T}^{2}=0$, i.e., $V_{T}$ is constant, and

$\frac{V_{T}}{V_{0}}=\exp(\mu_{T})=\exp\left\\{Tb+\Psi(\kappa,T)(r_{0}-b)-\sigma_{r}^{2}\Upsilon(\kappa,T)/2\right\\}=\frac{1}{p_{0}(T)},$
(34)

where the first equality follows from Corollary 2.2, the second equality
follows from (31)—(32), and the last equality follows from (30) with
$a=\kappa$. Thus, as expected, the return on the risk free strategy equals the
return on a zero-coupon bond maturing at the horizon.

###  3.2 Optimal equity strategies

In this section we set out to find the equity exposure strategy that leads to
the optimal (log) mean-variance trade-off for excess returns on a given
horizon. One might think of this as an overlay strategy to an underlying bond
strategy, e.g., one of the strategies derived in Section 3.1. In particular,
the equity strategy can act as an overlay to the hedging $T$-bond strategy.

For reasons stated earlier, we are interested in deterministic, or strategic,
equity exposure profiles only, i.e., we allow the strategy to depend on time
and the initial equity risk premium only. In contrast to Problem 3.1, we solve
the equity problem in full generality taking account of the mean-reverting
risk premia. Mathematically, this makes the present problem much harder. The
solution is presented in three steps: an integral equation characterizing the
strategy, a general solution, and explicit formulas for the coefficients in
the general solution.

Assuming independence between the two risk sources ($\rho=0$), or
alternatively no interest rate exposure ($f_{s}^{r}=0$), the excess return due
to equity exposure is well-defined. The (excess) mean and variance due to
equity exposure is given by the terms $m^{S}_{T}$ and
$\int_{0}^{T}h^{S}_{u}dW^{S}_{u}$, respectively, of Theorem 2.1.

For ease of notation and in this section only, we drop superscript $S$ on all
quantities relating to equities. We also introduce a short-hand notation for
the expected market price of equity risk at future times

$\xi_{s}=\operatorname{\mathsf{E}}\left[\lambda_{s}\right]=\frac{1}{\sigma_{S}}\operatorname{\mathsf{E}}\left[x_{s}\right]=\frac{1}{\sigma_{S}}\left(\bar{x}+e^{-\alpha
s}(x_{0}-\bar{x})\right).$ (35)

Using this notation we obtain from Theorem 2.1 the following corollary, which
serves to formulate the problem.

######  Corollary 3.2.

Assume $\rho=0$. For given $T>0$, the excess return of $V_{T}/V_{0}$ due to
equity exposure, $f$, is log-normally distributed with mean and variance given
by

$\displaystyle\mu_{T}$
$\displaystyle=\int_{0}^{T}\xi_{s}f_{s}-\frac{1}{2}f_{s}^{2}ds,$ (36)
$\displaystyle\sigma_{T}^{2}$
$\displaystyle=\int_{0}^{T}\left[f_{u}-\frac{\sigma_{x}}{\sigma_{S}}\int_{u}^{T}f_{s}e^{-\alpha(s-u)}ds\right]^{2}du.$
(37)

The problem of this section can now be phrased as

Problem 2:  Find $f$ ($=f^{S}$) such that (36) is maximized for given value of
(37).

Similarly to Problem 3.1, for given horizon, $T$, the solution to Problem 3.2
takes the form of a family of strategies, one for each value of the variance
$\sigma_{T}^{2}$. Members of this family are referred to as optimal equity
strategies. More generally, we refer to equity strategies (locally)
maximising, or minimizing, (36) for given value of (37) as extremal equity
strategies.

Before discussing the solution to Problem 3.2 in detail we start by noting
that the global (unconstrained) maximum of $\mu_{T}$ is achieved for
$f_{s}=\xi_{s}$ yielding $\hat{\mu}_{T}=\frac{1}{2}\int_{0}^{T}\xi_{s}^{2}ds$;
the maximum is obtained by optimizing the integrand in (36) at each $s$
individually. In particular, the maximal excess return is finite and generally
also strictly positive (except in the degenerate cases $x_{0}=\bar{x}=0$, or
$x_{0}=\alpha=0$). Let us denote by $\hat{\sigma}_{T}^{2}$ the variance
associated with the optimal mean, i.e., (37) evaluated with $f_{s}=\xi_{s}$.
Since $\hat{\mu}_{T}$ is the global maximum it is also the constrained maximum
under the variance constraint $\sigma_{T}^{2}=\hat{\sigma}_{T}^{2}$. In
particular, $f=\xi$ is an optimal equity strategy.

We typically expect a positive risk-reward trade-off, meaning that the larger
the risk (variance) the larger the reward (mean). Hence, we typically expect
that as we increase the prescribed value of the variance the associated
optimal mean also increases. This however is true only up to a certain point,
namely $\hat{\sigma}_{T}^{2}$. As we vary the variance from $0$ to
$\hat{\sigma}_{T}^{2}$ the optimal mean increases from $0$ to $\hat{\mu}_{T}$,
but if we increase the prescribed value of the variance further the optimal
mean starts to decrease. In other words, demanding a variance beyond
$\hat{\sigma}_{T}^{2}$ is counter-productive in the sense that it can only be
achieved by investing so much in equities that it impairs the mean log-return.
Mathematically we can characterize the optimal strategies for any prescribed
non-negative variance, but in practise we are interested only in optimal
strategies with variance at most $\hat{\sigma}_{T}^{2}$.

####  3.2.1 Characterization of extremal equity strategies

The double integral in (37) is a manifestation of the global nature of the
problem by which later equity exposure reduces the variance arising from
earlier exposure. The double integral also implies that Problem 3.2 does not
immediately conform with problems solvable by the Euler-Lagrange equation of
calculus of variations. However, we can still use a variational argument to
arrive at the following integral characterization of the extremal equity
strategies.

Extremal strategies are defined as stationary points (strategies) for the
constrained optimization problem. These strategies are candidates for the
optimal strategies, since optimal strategies are also extremal. However,
extremal strategies can also be minimizing strategies, or local extremal
strategies.

######  Lemma 3.3.

For given $T>0$, the extremal equity strategies satisfy

$\xi_{s}-f_{s}+2\nu
h_{s}-2\nu\frac{\sigma_{x}}{\sigma_{S}}\int_{0}^{s}h_{u}e^{-\alpha(s-u)}du=0\quad(0\leq
s\leq T),$ (38)

where

$h_{u}=f_{u}-\frac{\sigma_{x}}{\sigma_{S}}\int_{u}^{T}f_{s}e^{-\alpha(s-u)}ds\quad(0\leq
u\leq T),$ (39)

and $\nu$ is a Lagrange multiplier.

Note that $h$ defined in Lemma 3.3 is the same as $h^{S}$ of Theorem 2.1.

The extremal strategies are characterized by having the same mean-variance
trade-off for all (infinitesimal) perturbations of the strategy, i.e., if we
alter an extremal strategy slightly such that the variance changes by
$\epsilon$, say, then the mean changes by $\delta$, say, regardless of how we
alter the strategy. The Lagrange multiplier, $\nu$, is equal to minus the
(common) value of the mean-variance trade-off for the corresponding extremal
strategy; loosely speaking, $\nu=-\delta/\epsilon$. In principle, there could
be more than one solution to (38) for given $\nu$, but we will show later that
the solution (if it exists) is unique. Hence, the extremal strategies are
uniquely identified by their mean-variance trade-off.

Assuming sufficient regularity, differentiating (38) twice leads to a second-
order differential equation in $f$ from which the general form of the solution
can be inferred.

######  Lemma 3.4.

For given $T>0$, the extremal equity strategies satisfy

$Af^{\prime\prime}_{s}+Cf_{s}+D=0\quad(0\leq s\leq T),$ (40)

where

$A=1-2\nu,\quad
C=2\nu\left(\alpha-\frac{\sigma_{x}}{\sigma_{S}}\right)^{2}-\alpha^{2},\quad
D=\alpha^{2}\frac{\bar{x}}{\sigma_{S}}.$ (41)

Lemma 3.4 gives a surprisingly simple characterization of the extremal
strategies as solutions to a differential equation with constant coefficients
which do not depend on $T$, nor $x_{0}$. In general, the solution space to
(40) is two-dimensional, and it remains to identify the specific solution also
satisfying (38).

####  3.2.2 Optimal equity strategies with positive mean-variance trade-off

As noted in the introductory remarks to this section, the unconstrained
maximum to (36) is achieved by $f=\xi$ and this strategy is an optimal
strategy. We note that $f=\xi$ is also the (unique) solution to (38) when
$\nu=0$; this is to be expected, since a Lagrange multiplier vanishes at a
global maximum.

In the limit $\nu$ tending to minus infinity, we get the strategy $f=0$ with
$\mu_{T}=\sigma_{T}^{2}=0$. It is not obvious that this is the limiting
solution to (38), but it follows from the explicit solution given in Theorem
3.5 below. It also conforms with the intuition that the size of $\nu$ reflects
the investor’s risk aversion. Although, as we shall see in Section 4.2.4, this
intuition is only partly true.

Strategies satisfying (38) with $-\infty<\nu<0$ correspond to extremal
strategies with a positive mean-variance trade-off, i.e., strategies where
added risk is rewarded in terms of higher mean. Varying $\nu$ in this range
gives us all the strategies of practical relevance, ranging from the risk free
strategy to the strategy achieving the global maximum. In this section we give
an explicit expression for these strategies.333Strictly speaking, we only know
that the strategies are extremal and with rewarded risk. However, we cannot
rule out the possibility that for certain values of the variance constraint,
there could exist both an extremal strategy with rewarded risk and a better
(optimal) extremal strategy with unrewarded risk. This is conceivable because
whether or not risk is rewarded is a local property applying only to
infinitesimal changes in risk. A formal proof for the non-existence of this
possibility would require an analysis of the global structure of the solution
space, which is outside the scope of the present paper. In practise, however,
it is easily verified that extremal strategies with positive mean-variance
trade-off are indeed optimal; this can be observed from a mean-variance plot
of the family of all extremal strategies for given model parameters, cf.
Section 4

Hence, assume $\nu<0$. Under this assumption, $A>1$ and $C<0$. Note, that the
strict negativity of $C$ holds even if $\alpha=0$ (assuming $\sigma_{x}>0$ and
$\sigma_{S}>0$). From standard theory we know that the general solution to
(40) takes the form of a specific solution plus the general solution to the
associated, homogeneous differential equation,
$Af^{\prime\prime}_{s}+Cf_{s}=0$. The characteristic polynomial of the latter
is $Ar^{2}+C$, which, under the current assumption, has two distinct, real
roots $c_{1}=\sqrt{-C/A}$ and $c_{2}=-\sqrt{-C/A}$, say, and it follows that
the general solution to the homogeneous differential equation is of the form
$f_{s}=b_{1}e^{c_{1}s}+b_{2}e^{c_{2}s}$. Regarding the specific solution, we
note that $f_{s}=-D/C$ solves (40), since $C$ is non-zero. Thus, the general
solution to (40) is given by

$f_{s}=b_{0}+b_{1}e^{c_{1}s}+b_{2}e^{c_{2}s}\quad(0\leq s\leq T),$ (42)

where $b_{0}=-D/C$. This leaves the determination of the two coefficients
$b_{1}$ and $b_{2}$. To determine those we insert the general form of $f$
given by (42) into the left-hand side of (38). In general, this results in a
linear combination of four exponential functions with different exponents, and
a constant term. The requirement that all terms vanish gives us five equations
for determining the five constants in (42), in particular $b_{1}$ and $b_{2}$.
This programme is carried out in Appendix A. For ease of reference, the
following theorem summarizes the results (with superscript $S$ reintroduced).

######  Theorem 3.5.

Assume independent risk factors ($\rho=0$). For given horizon $T>0$, the
optimal equity strategies with positive mean-variance trade-off are of the
form

$f^{S}_{s}=b_{0}+b_{1}e^{c_{1}s}+b_{2}e^{c_{2}s}\quad(0\leq s\leq T),$ (43)

where $c_{1}=\sqrt{-C/A}>0$, $c_{2}=-\sqrt{-C/A}$, $b_{0}=-D/C$ with $A$, $C$,
and $D$ given by (41), and $\nu<0$ depends on the prescribed value of the
variance $\sigma_{T}^{2}$.

For $\alpha\neq\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$, the exponents
satisfy $c_{1}=|c_{2}|\neq|\alpha|$, and

$\begin{pmatrix}b_{1}\\\
b_{2}\end{pmatrix}=\begin{pmatrix}\frac{e^{c_{1}T}}{c_{1}-\alpha}&\frac{e^{c_{2}T}}{c_{2}-\alpha}\\\\[5.69054pt]
\frac{\sigma_{x}}{\sigma_{S}(c_{1}+\alpha)-\sigma_{x}}&\frac{\sigma_{x}}{\sigma_{S}(c_{2}+\alpha)-\sigma_{x}}\end{pmatrix}^{-1}\begin{pmatrix}\frac{\alpha\bar{x}}{\sigma_{S}\left(\alpha^{2}-2\nu(\alpha-\sigma_{x}/\sigma_{S})^{2}\right)}\\\\[5.69054pt]
-\frac{x_{0}}{\sigma_{S}}+\frac{\alpha\bar{x}}{\sigma_{S}}\frac{\alpha-2\nu(\alpha-\sigma_{x}/\sigma_{S})}{\alpha^{2}-2\nu(\alpha-\sigma_{x}/\sigma_{S})^{2}}\end{pmatrix}.$
(44)

For $\alpha=\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$, the exponents are
$c_{1}=\alpha$ and $c_{2}=-\alpha$, with

$b_{1}=0,\quad
b_{2}=\frac{-\frac{x_{0}}{\sigma_{S}}+\frac{\bar{x}}{\sigma_{S}}\left(1-\frac{4\nu}{1-2\nu}\left(e^{-\alpha
T}-1\right)\right)}{2\nu e^{-2\alpha T}-1}.$ (45)

Note that the equity risk premium process is transient for $\alpha=0$, and
exploding for $\alpha<0$, so for practical purposes we would consider using
the model only for $\alpha>0$. Nevertheless, it is interesting to learn that
the solution has the same structure regardless of the value of $\alpha$.
Recall that the global maximum is achieved by $f=\xi$ with $\xi$ given by
(35). This strategy is also of form (43) with either $c_{1}$ or $c_{2}$ equal
to $\alpha$, and $b_{0}=\bar{x}/\sigma_{S}$.

For $\nu$ tending to minus infinity, the exponents approach
$\pm(\alpha-\sigma_{x}/\sigma_{S})$. In particular, the exponents have a
finite limit. Further, for $\alpha\neq\sigma_{x}/\sigma_{S}$ the
$b$-coefficients all tend to $0$ for $\nu$ tending to minus infinity, while
for $\alpha=\sigma_{x}/\sigma_{S}$ the sum of the $b$-coefficients tends to
$0$ in the same limit.444We here give an outline of the argument. Write (44)
as $F(b_{1}\ b_{2})^{\prime}=B$. Using the established limits of $c_{1}$ and
$c_{2}$, we have that for $\alpha\neq\sigma_{x}/\sigma_{S}$, one of the bottom
entries of $F$ diverges while the other three entries have finite, non-zero
limits. Under the same condition on $\alpha$, the top entry of $B$ tends to
$0$, while the bottom entry has a finite limit. From this we can conclude that
$b_{1}$ and $b_{2}$ both tend to $0$. For the special case,
$\alpha=\sigma_{x}/\sigma_{S}$, both $c_{1}$ and $c_{2}$ tend to $0$ for $\nu$
tending to minus infinity. It follows that the top row of $F$ converges to
$(-1/\alpha\ -1/\alpha)=(-\sigma_{S}/\sigma_{x}\ -\sigma_{S}/\sigma_{x})$,
while the top entry of $B$ equals
$\bar{x}/\sigma_{x}(=b_{0}\sigma_{S}/\sigma_{x})$. From this we can conclude
that $b_{0}+b_{1}+b_{2}$ tends to $0$. We leave it to the reader to handle the
last special case, $\alpha=\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$. Thus, in
either case the limiting strategy is the risk-free strategy, $f=0$, as
previously claimed.

The optimal strategy has a simple structure, but apart from the observations
made already, the expressions in Theorem 3.5 appear too complex for intuitive
interpretation. Instead we will explore the optimal mean-variance trade-off
and the optimal strategies numerically in Section 4.2.

####  3.2.3 Extremal equity strategies with negative mean-variance trade-off

From a practical point of view, we are only interested in pursuing optimal
strategies with a positive mean-variance trade-off, and these are therefore
the main focus of the paper. However, it is very instructive to also study
extremal strategies with a negative mean-variance trade-off, corresponding to
$\nu>0$. These strategies include both optimal strategies with "excessive"
variance, minimizing strategies with lowest possible mean for given variance,
but also "interior" strategies, where the mean is strictly between the minimal
and maximal value for given variance. The latter class of strategies displays
an intriguing variety of equity profiles, exemplified in Section 4.2.4. In
this section we briefly discuss the different types of solution that can occur
for $\nu>0$. The actual strategies can be found in Appendix B, with Theorem
B.1 giving an overview of the full set of extremal strategies.

Depending on the type of roots to the characteristic polynomial, $Ar^{2}+C$,
the extremal strategy takes one of three forms. For distinct, real roots we
get the exponential solution already covered, for complex roots we get a
trigonometric solution of form

$f_{s}=b_{0}+b_{1}\sin(cs)+b_{2}\cos(cs),$ (46)

where $b_{0}=-D/C$ and $c=\sqrt{C/A}$, and in the special case where zero is a
double root we get a quadratic solution

$f_{s}=b_{0}+b_{1}s+b_{2}s^{2},$ (47)

where $b_{2}=-D/(2A)$.

The solution is exponential if $A$ and $C$ are of opposite signs, it is
trigonometric if $A$ and $C$ are of the same sign, and it is quadratic in the
special case $C=0$ and $A\neq 0$. In general, there is no extremal equity
strategy for $\nu=1/2$ ($A=0$). The domains for the respective solution types
are illustrated in Figure 11 of Appendix B. We know that the exponential
solution applies for $\nu<0$, but we see from the figure that it is in fact
also the "typical" solution for $\nu\geq 0$.

The quadratic solution defines the border between the exponential and
trigonometric solutions. For given model parameters, there is (at most) one
value of $\nu$ for which the extremal strategy is quadratic. Of course, one is
unlikely to encounter this solution in applications, but it is interesting to
note that quadratic solutions bridge the two main domains. This suggests that
extremal strategies in general might be well approximated by quadratic
strategies.

Finally, we note that the solution type depends only on $\nu$ and the mean
reversion ratio $\tilde{\alpha}=\alpha/[\sigma_{x}/\sigma_{S}]$, where
$\sigma_{x}/\sigma_{S}$ can be interpreted as the volatility of the market
price of equity risk, $\lambda^{S}_{t}=x_{t}/\sigma_{S}$. Thus, apart from
$\nu$, the determining factor for the type of solution is the ratio of mean
reversion strength to noise in the market price of equity risk process,

$d\lambda^{S}_{t}=\alpha\left(\bar{\lambda}^{S}-\lambda^{S}_{t}\right)dt-\frac{\sigma_{x}}{\sigma_{S}}dW^{S}_{t},$
(48)

where $\bar{\lambda}^{S}=\bar{x}/\sigma_{S}$ denotes the long-run market price
of equity risk.

###  3.3 Optimal joint strategies

Assuming independent risk factors ($\rho=0$), the effects of rate and equity
exposures can be separated and joint maximization can be performed based on
the results of the preceding two sections.

To phrase the problem, let us first state a simplified version of Theorem 2.1.

######  Corollary 3.6.

Assume $\rho=0$ and $a=\kappa$. For given $T>0$, $V_{T}/V_{0}$ is log–normally
distributed with mean and variance given by

$\displaystyle\mu_{T}$
$\displaystyle=m^{0}_{T}+\int_{0}^{T}\lambda^{r}f^{r}_{s}-\frac{1}{2}\left(f^{r}_{s}\right)^{2}ds+\int_{0}^{T}\xi_{s}f^{S}_{s}-\frac{1}{2}\left(f^{S}_{s}\right)^{2}ds,$
(49) $\displaystyle\sigma_{T}^{2}$
$\displaystyle=\int_{0}^{T}\left[\sigma_{r}\Psi(\kappa,T-u)+f^{r}_{u}\right]^{2}du+\int_{0}^{T}\left[f^{S}_{u}-\frac{\sigma_{x}}{\sigma_{S}}\int_{u}^{T}f^{S}_{s}e^{-\alpha(s-u)}ds\right]^{2}du.$
(50)

where $m^{0}_{T}$ is given by (19) of Theorem 2.1, $\xi_{s}$ is given by (35),
and $\lambda^{r}=\kappa(\bar{r}-b)/\sigma_{r}$.

The problem of this section can now be phrased as

Problem 3:  Find $(f^{r},f^{S})$ such that (49) is maximized for given value
of (50).

For given horizon, $T$, the solution to Problem 3.3 takes the form of a family
of pairs of rate and equity strategies, one for each value of the variance
$\sigma_{T}^{2}$. Members of this family are referred to as optimal pairs of
strategies. More generally, we refer to pairs of strategies (locally)
maximising, or minimizing, (49) for given value of (50) as extremal pairs of
strategies.

We note that the mean and variance of Corollary 3.6 are the sum of the means
and variances, respectively, of Corollaries 2.2 and 3.2. Further, the interest
rate strategy, $f^{r}$, and the equity strategy, $f^{S}$, affect separate
terms. This implies that the solutions to Problem 3.3 consist of the
previously derived, optimal strategies to Problems 3.1 and 3.2. A fortiori,
the extremal pairs consist of extremal bond strategies and extremal equity
strategies with the same Lagrange multiplier, $\nu$. Heuristically, the
argument is as follows.

Assume $(f^{r},f^{S})$ is an extremal pair of strategies. This implies that
the mean-variance trade-off is the same for all (infinitesimal) perturbations
of the strategy. Since if this were not the case, it would be possible to
combine two changes with different trade-offs to change the mean, while
preserving the variance, contradicting the assumed extremity. In particular,
the mean-variance trade-off is the same for all (infinitesimal) perturbations
of only the rate strategy, or only the equity strategy. But since the mean-
variance trade-off associated with $f^{r}$ is the same in Problem 3.3 as it is
in Problem 3.1, we conclude that $f^{r}$ is in fact an extremal bond strategy.
Similarly, $f^{S}$ is in fact an extremal equity strategy. Moreover, the mean-
variance trade-off of $f^{r}$ and $f^{S}$ must match, and since the trade-off
equals (minus) the Lagrange multiplier, $\nu$, we conclude that the extremal
strategies $f^{r}$ and $f^{S}$ must have the same value of $\nu$. The argument
can be made formal, but it is useful to keep the heuristic argument in mind.

######  Theorem 3.7.

Assume independent risk factors ($\rho=0$), and constant market price of
interest rate risk ($a=\kappa$). For given horizon $T>0$, the extremal pairs
of strategies consist of pairs with $f^{r}$ given by Theorem 3.1, and $f^{S}$
given by Theorem 3.5 (Theorem B.1) for the same value of $\nu\neq 1/2$.

For $\nu<0$, the pairs are optimal strategies with positive mean-variance
trade-off.

We note from Theorem 3.7 that all extremal rate and equity strategies belong
to one, and only one, extremal pair. For the rate strategies, we can compute a
priori the variance contribution by formula (33), but we have no similar
(simple) formula for the variance contribution of the equity strategies. Thus,
we cannot a priori say anything about the relative size of the variance
contributions from rates and equities in the extremal pairs. In particular, we
do not know in general whether the optimal pairs are balanced regarding rate
and equity risk, or whether one of the risk sources dominates. Of course, we
can answer this question numerically on a case by case basis by computing the
family of extremal pairs and the variance contribution from each risk source.

In addition to the optimal pairs of strategies, it is also of interest to
consider strategies with the same amount of rate and equity risk. Such risk
parity, or balanced, strategies might be considered more robust as they do not
rely on assumed differences in risk premia. In the present context, one could
consider pairs of strategies with the same variance contribution on the
horizon, i.e., pair up strategies solving Problems 3.1 and 3.2 for the same
value of $\sigma_{T}^{2}$ given by, respectively, (27) and (37). Again, this
pairing has to be done numerically on a case by case basis. We do not pursue
these ideas further in this paper.

####  3.3.1 Dependent risk factors

The case of dependent risk factors ($\rho\neq 0$) is mathematically rather
more challenging. In full generality, the problem amounts to maximising (24)
for given value of (25). We see from Theorem 2.1, that for $\rho\neq 0$ the
mean and variance of $\log(V_{T}/V_{0})$ are affected by terms depending on
both $f^{r}$ and $f^{S}$. Thus we cannot hope to solve the general problem by
"stitching" together partial solutions. In fact, it is not even clear how to
separate the mean and variance contributions arising from, respectively, the
rate and equity exposures.

Apart from the mathematical difficulties, we also argue that the resulting
strategies—if we were able to obtain them—are unlikely to be of much value in
practice. An assumed correlation between rate and equity risk factors is very
hard to verify in practice, and we therefore might be reluctant to pursue
strategies which exploit this correlation. From that perspective, the
independence assumption is the "neutral" assumption most often used in
practice.

Rather than deriving the general solution, we can alternatively test the
robustness of strategies derived under no correlation in an environment with
correlation. We can, e.g., establish the correlation range for which the
derived strategies are better than constant strategies. This in turn can be
used as a guide for when the derived strategies are near-optimal without the
need for an exact value of the correlation coefficient. We do not, however,
pursue this idea further.

As an indication of the complexity of the optimal strategies under dependent
risk factors we derive the optimal rate strategy for given equity exposure
$f^{S}$. Assuming $a=\kappa$, this amounts to maximizing the mean

$\mu_{T}=m^{0}_{T}+\int_{0}^{T}f^{r}_{s}\left[\lambda^{r}-\rho
f_{s}^{S}\right]ds-\frac{1}{2}\int_{0}^{T}\left(f^{r}_{s}\right)^{2}ds,$ (51)

for given value of the variance

$\sigma_{T}^{2}=\int_{0}^{T}\left[\sigma_{r}\Psi(\kappa,T-s)+f^{r}_{s}\right]\left[\sigma_{r}\Psi(\kappa,T-s)+f^{r}_{s}+2\rho
h_{s}^{S}\right]ds.$ (52)

These expressions follow from Theorem 2.1, see also Corollary 2.2 and Problem
3.1 for the original optimization problem. By a straightforward extension of
the proof of Theorem 3.1 we find that the optimal rate strategies on horizon
$T$ are of the form

$f_{s}^{r}=\frac{\lambda^{r}+2\nu g_{s}^{r}+\rho(2\nu
h_{s}^{S}-f_{s}^{S})}{1-2\nu}\quad(0\leq s\leq T),$ (53)

with $g_{s}^{r}=\sigma_{r}\Psi(\kappa,T-s)$ and $\nu<1/2$. We see that,
although computable, the strategy is complicated and with explicit reference
to the given equity exposure via both $f^{S}$ and $h^{S}$. This indicates that
the optimal pais of strategies are presumably very complicated indeed, and we
will not try to find them.

##  4 Numerical illustrations

In this section we provide numerical illustrations of the optimal risk-reward
profiles and the underlying optimal strategies. We cover optimal rate and
equity strategies separately, with an emphasis on the latter. Assuming
independent risk factors, the jointly optimal pais of strategies consist of
optimal rate and equity strategies with the same value of $\nu$, cf. Theorem
3.7. Thus, joint optimization consists of pairing the illustrated optimal rate
and equity strategies. We do not, however, explicitly consider joint
optimization in this section.

###  4.1 Optimal rate strategies

In the following we consider the optimal strategies of Section 3.1, i.e.,
optimal strategies when we are allowed to invest only in the interest rate
market. For illustrative purposes we consider two parameter sets,
corresponding to moderate and low market prices of interest rate risk, cf.
Table 1. For these parameter sets we show the risk-reward profile for the
entire family of extremal strategies and we give examples of optimal rate
strategies and the resulting portfolio distributions. We also consider
alternative risk and reward statistics connected to the portfolio
distributions.

####  4.1.1 Yield curves

First, we visualize the interest rate assumptions. The continuously compounded
zero-coupon yield for the period $[t,T]$, $r_{t}(T)$, is defined by the
relation $p_{t}(T)=\exp\\{-(T-t)r_{t}(T)\\}$. From (30) it follows

$r_{t}(T)=b+\frac{\Psi(a,\Delta)}{\Delta}(r_{t}-b)-\frac{\sigma_{r}^{2}}{2}\frac{\Upsilon(a,\Delta)}{\Delta},$
(54)

where $\Delta=T-t$, and $\Psi(a,0)/0=1$ and $\Upsilon(a,0)/0=0$ are defined by
continuity. The yield as a function of $T$ is referred to as the yield curve.
The yield curve and the corresponding curve of zero-coupon bond prices are
equivalent ways of representing the bond market, but we typically prefer the
former due to its more intuitive interpretation.

Parameter set |  $\kappa$ |  $\bar{r}$ |  $\sigma_{r}$ |  $a$ |  $b$ |  $-\lambda^{r}$
---|---|---|---|---|---|---
Moderate |  0.08 |  0.02 |  0.007 |  0.08 |  0.04 |  0.2286
Low |  0.08 |  0.02 |  0.007 |  0.08 |  0.03 |  0.1143
Table 1: Parameters governing the short-rate $P$-dynamics (1), $\kappa$,
$\bar{r}$, and $\sigma_{r}$, and pricing parameters, $a$ and $b$. Since
$a=\kappa$, the market price of interest rate risk is constant and equal to
$\lambda^{r}=\kappa(\bar{r}-b)/\sigma_{r}$. The last column shows
$-\lambda^{r}$ which is the continuous-time analogue to the Sharpe ratio,
i.e., excess return per unit of volatility for holding bonds.

Figure 1 shows yield curves corresponding to different values of the short
rate, $r_{0}$, for the two parameter sets in Table 1. The yield curves
represent the yield that can be locked in today (time $0$) by purchasing a
zero-coupon bond. In the left plot interest rate risk is rewarded higher than
in the right plot, and consequently the yields are higher. For example, if
$r_{0}=0\%$ (green curves) we can lock in a return of $r_{0}(20)=1.89\%$ per
year on a 20-year horizon when the market price of risk is moderate (left
plot), and a return of $r_{0}(20)=1.39\%$ when the market price of risk is low
(right plot). Note that since the short-rate $P$-dynamics are the same in the
two cases, a money-market account will give rise to the exact same return
distribution on any horizon in the two cases, while bond strategies will yield
higher returns in the "moderate" parameter set than in the "low" parameter
set.

Figure 1: Illustration of yield curves with moderate (left) and low (right)
market price of interest rate risk, cf. Table 1. For each parameter set, the
yield curve for five different values of the short-rate is shown,
$r_{0}=-2\%,\ 0\%,\ 2\%,\ 4\%,\ 6\%$.

####  4.1.2 Risk-reward profiles

Section 3.1.2 provides formulas for the extremal mean and variance, i.e., the
log-portfolio mean, $\mu_{T}$, and the log-portfolio variance,
$\sigma_{T}^{2}$, arising from following the extremal rate strategies of
Theorem 3.1. Figure 2 plots $\mu_{T}$ against $\sigma_{T}$ for $T=20$ for four
different values of the initial short-rate, $r_{0}$, using the same colour
scheme as in Figure 1. We refer to the curves as risk-reward profiles. The
upper part of the profiles represent maximizing strategies ($\nu<1/2$), and
the lower part of the profiles represent minimizing strategies ($\nu>1/2$),
where $\nu$ is the Lagrange multiplier of Theorem 3.1. The upper part of the
profiles consists of an initial upward-sloping part ($\nu<0$) where adding
risk increases the mean until the global maximum is achieved ($\nu=0$), and a
downward-sloping part ($0<\nu<1/2$) where risk is so high that adding further
risk decreases the mean. Of course, for practical purposes only strategies
with $\nu\leq 0$ are of interest.

It follows from Corollary 2.2 and expressions (31)—(33) that,

$V_{T}\stackrel{{\scriptstyle\cal{D}}}{{=}}V_{0}\exp\left(\mu_{T}+\sigma_{T}U\right)=\frac{V_{0}}{p_{0}(T)}\underbrace{\exp\left(\mu_{T}+\log(p_{0}(T))+\sigma_{T}U\right)}_{\mbox{independent
of $r_{0}$}}=\frac{V_{0}}{p_{0}(T)}Y_{T},$ (55)

where $U$ is a standard normal variate. Thus, the portfolio value on the
horizon can be interpreted as the amount that can be locked in with certainty
at time $0$ times a stochastic factor, $Y_{T}$. The stochastic factor depends
on the $P$-dynamics of the short-rate process, the market price of interest
rate risk, and $\nu$, but not on the initial value of the short-rate, $r_{0}$.
From this observation and (55) it follows that the profiles in Figure 2 are
(vertical) translations of each other, with the (vertical) distance between
two profiles being equal to the difference in log bond prices.

Figure 2: Risk-reward profiles, $(\sigma_{T},\mu_{T})$, for extremal rate
strategies on a horizon of $T=20$ years for moderate (left) and low (right)
market price of interest rate risk, cf. Table 1. The different profiles
correspond to $r_{0}$ of $-2\%$ (yellow), $0\%$ (green), $2\%$ (orange), and
$4\%$ (blue). For $r_{0}=0\%$, four risk-reward pairs are marked,
corresponding to $\nu=-\infty$ ($\bullet$), $\nu=-1$ ($\blacklozenge$),
$\nu=-1/4$ ($\blacktriangle$), and $\nu=0$ ($\blacksquare$); the underlying
portfolio distributions and strategies are shown in Figures 3 and 4,
respectively.

####  4.1.3 Portfolio distributions

We have marked four points on each of the profiles in Figure 2 for
$r_{0}=0\%$. Figure 3 illustrates the underlying distribution of $V_{T}$ (for
$V_{0}=1$); alternatively, we can interpret the plots as showing
$Y_{T}/p_{0}(T)$, i.e., the stochastic multiplier scaled by the 20-year bond
return that can be obtained when $r_{0}=0\%$. The thin black lines show the
median, $\exp(\mu_{T})$, of each distribution. As $\nu$ is increased from
$-\infty$ to $0$ the median increases, but the volatility also increases
markedly. In particular, when the market price of risk is low (right plot) the
increase in median is very modest compared to the additional risk. It follows
from (55) that, apart from scaling, these plots will look the same for other
values of $r_{0}$.

In Appendix D, Tables 5 and 6 quantify various risk and reward statistics for
$Y_{T}$ for a range of optimal rate strategies. The tables show the median as
a measure of the reward, along with three different risk measures:

  * •

The probability that we earn less than the risk-free investment,
$\operatorname{\mathsf{P}}(Y_{T}<1)$;

  * •

The expected size of the loss given that we experience a loss,
$\operatorname{\mathsf{E}}[1-Y_{T}|Y_{T}<1]$;

  * •

The expected size of the loss,
$\operatorname{\mathsf{E}}[1-Y_{T}]^{+}=\operatorname{\mathsf{E}}[1-Y_{T}|Y_{T}<1]\operatorname{\mathsf{P}}(Y_{T}<1)$.

Since $Y_{T}=1$ can be obtained with no risk by holding only $T$-bonds, we
interpret $Y_{T}<1$ as a loss relative to this risk-free strategy. The risk
measures therefore focus on the event $Y_{Y}<1$. We see that for given $\nu$,
the probability of earning less than the risk-free investment is substantially
higher when the market price of interest rate risk is low (Table 6) than when
it is moderate (Table 5), but both the conditional and unconditional losses
are smaller. This is due to the fact that the exposure depends on the market
price of risk, see Figure 4. Thus, when risk is less rewarded the exposure is
decreased and therefore as the distribution of $Y_{T}$ moves "to the right" it
also becomes more narrow.

Figure 3: The distribution of $V_{T}$ for $T=20$ years for four optimal rate
strategies when $V_{0}=1$, $r_{0}=0\%$ and the market price of interest rate
risk is moderate (left) and low (right), cf. Table 1. For $\nu=-\infty$,
$V_{T}=1/p_{0}(T)$ is constant which is marked by a vertical green line. The
thin black lines mark the median of each distribution. The underlying
strategies are shown in Figure 4.

####  4.1.4 Illustration of optimal strategies

Figure 4 shows the underlying strategies for the portfolio distributions of
Figure 3. The strategies are given by (28) of Theorem 3.1. The strategies give
rise to the stochastic multiplier, $Y_{T}$, which acts on the risk-free bond
return that can be locked in initially. The bond return depends on the initial
yield curve which in turn depends on the initial short-rate, $r_{0}$. The
strategies themselves, however, do not depend on $r_{0}$. In particular,
Figure 4 is valid for all interest rate levels. In other words, an investor
with a relative risk tolerance will choose the same strategy regardless of the
interest rate level and thereby obtain the same (stochastic) multiplier on the
risk-free investment. Conversely, an investor with an absolute risk tolerance
will choose a strategy that depends on the interest rate level, but still
within the family of optimal strategies.

In Figure 4 we plot the negated interest rate exposure ($-f_{s}^{r}$) as a
function time, rather than the interest rate exposure itself. The optimal
exposure is negative (corresponding to holding bonds) and the negated exposure
therefore gives a more intuitive plot with higher exposures implying larger
bond holdings. Also, for the optimal strategies the negated exposure is equal
to the (instantaneous) volatility of the portfolio.555It follows from (11)
that, in general, the (instantaneous) volatility of the portfolio at time $s$
is given by $\sqrt{[f_{s}^{r}]^{2}+[f_{s}^{S}]^{2}+2\rho f_{s}^{r}f_{s}^{S}}$.
With no equity risk, $f_{s}^{S}=0$, this reduces to $|f_{s}^{r}|$, which
equals $-f_{s}^{r}$ when $f_{s}^{r}$ is negative.

Recall from Section 3.1 that $\nu=-\infty$ corresponds to the risk-free, buy-
and-hold strategy where $V$ is fully invested in $T$-bonds, while $\nu=0$
corresponds to a constant risk exposure of $\lambda^{r}$ which is the strategy
achieving the (unconstrained) maximal mean. For $-\infty<\nu<0$ we get a
convex combination of these two strategies. Holding only $T$-bonds corresponds
to a (negated) exposure of $-f_{s}^{r}=\sigma_{r}\Psi(\kappa,T-s)$ with an
initial volatility of $0.007\Psi(0.08,20)=7\%$, decreasing to zero over time.
This (hedging) strategy does not depend on the market price of interest rate
risk, and it is therefore the same for the two parameter sets. All other
optimal strategies, however, depend on the market price of interest rate risk:
The higher the market price of risk, the higher the optimal exposure (for
given $\nu$).

Finally, Figure 4 also shows an example of an optimal strategy on the "wrong"
side of the risk-reward profile ($\nu=1/8$) where risk is so excessive that
the quadratic term in (26) impairs the mean. The exposure is still negative,
however, i.e., we are long in bonds. In contrast, for mean minimizing
strategies the exposure is fully or partly positive, corresponding to shorting
bonds; if drawn these strategies would lie below the risk-free strategy.

Figure 4: Optimal rate strategies for $T=20$ years when the market price of
interest rate risk is moderate (left) and low (right), cf. Table 1. The
symbols distinguishing four of the strategies are also used in Figures 2 and 3
where they mark the risk-reward pair, ($\sigma_{T},\mu_{T}$), and the
portfolio value at the horizon, $V_{T}$, respectively, when the strategies are
applied in a market with $r_{0}=0\%$.

###  4.2 Optimal equity strategies

We now turn attention to the optimal strategies of Section 3.2, i.e., optimal
strategies where we are allowed to invest only in the equity market. Recall
that we are optimizing excess returns on a given horizon and that we interpret
the resulting strategies as overlay strategies, i.e, strategies where the
exposure is financed by a corresponding short money-market position. In our
setup, an equity strategy acts as a stochastic multiplier to (the portfolio
value of) the underlying rate strategy, which in turn acts as a stochastic
multiplier to the risk-free, $T$-bond hedging strategy.

####  4.2.1 Volatility profiles and calibration of mean-reversion

We begin with an investigation of the model for equity returns (2)—(3). The
rationale for the model is that equity returns above/below the expected return
leads to a decrease/increase in the future risk premium, $x_{t}$. Thus, there
is a tendency for gains to follow losses, and vice versa, and this mean-
reversion feature effectively compresses the equity process, and prevents both
extreme loses and extreme gains over longer periods. The model is simple, yet
it can produce a variety of different volatility profiles for the equity
process. In fact, the simplicity of the model is deceptive and counter-
intuitive volatility profiles can easily arise even for "innocent looking"
parameter values.

There are two parameters of interest in the risk premium process: $\alpha$
controlling the degree of mean reversion, and $\sigma_{x}$ controlling the
size of feedbacks. At first sight, we are free to choose these parameters as
we please. A larger/smaller $\sigma_{x}$ causes a larger/smaller correction of
the equity premium, while a larger/smaller value of $\alpha$ causes a
shorter/longer "corrective" period. While mean-reversion decreases the
volatility of the equity process in the short term, excessive mean-reversion
($\sigma_{x}$ too large and/or $\alpha$ too small) actually increases
volatility in the long term. Intuitively, if mean-reversion parameters are set
too aggressively, the uncertainty that builds up in the risk premium process
can overshadow the short-term volatility reductions and lead to an overall
increase in volatility.

In the following we study this phenomenon more formally and give
recommendations on the degree of mean-reversion. We are interested in the
process of excess equity returns, which we denote by $\tilde{S}_{t}$. It
follows from Section 3 of Jarner and Preisel (2017) that

$\tilde{S}_{t}\equiv
S_{t}\exp\left\\{-\int_{0}^{t}r_{s}ds\right\\}=S_{0}\left\\{\int_{0}^{t}x_{s}ds-\frac{\sigma_{S}^{2}}{2}t+\sigma_{S}W^{S}_{t}\right\\},$
(56)

and, further, from Section 6 ibid. it follows that

$\operatorname{\mathsf{Var}}[\log\tilde{S}_{t}]=\sigma_{x}^{2}\Upsilon(\alpha,t)+\sigma_{S}^{2}t-2\sigma_{x}\sigma_{S}\Theta(\alpha,t),$
(57)

where $\Upsilon$ is given by (29) on page 29 of the present report, and
$\Theta$ is given by

$\displaystyle\Theta(a,t)\equiv\int_{0}^{t}\Psi(a,s)ds=\begin{cases}\frac{t^{2}}{2}&\mbox{for
}a=0,\\\ \frac{1}{a^{2}}\left(-1+at+e^{-at}\right)&\mbox{for }a\neq 0;\\\
\end{cases}$ (58)

the interested reader is referred to Appendix A of Jarner and Preisel (2017)
for background information on the expression for the variance. We note that
the variance expression in (57) contains both a positive (first) term arising
from the stochasticity of the risk premium itself, and a negative (last) term
due to the link between innovations and risk premia. The asymptotic rate of
increase of these two terms determine whether mean-reversion leads to an
increase or a decrease of the long-term variance—compared to a (Black-Scholes)
model with constant risk premium, $x_{t}=\bar{x}$.

For $\alpha>0$, we have from Section 6 of Jarner and Preisel (2017) that the
asymptotic (rate of) variance is given by

$\lim_{t\to\infty}\frac{\operatorname{\mathsf{Var}}[\log\tilde{S}_{t}]}{t}=\frac{\sigma_{x}^{2}}{\alpha^{2}}+\sigma_{S}^{2}-2\frac{\sigma_{x}\sigma_{S}}{\alpha}=\left(\sigma_{S}-\frac{\sigma_{x}}{\alpha}\right)^{2}=\sigma_{S}^{2}\left(1-\frac{1}{\tilde{\alpha}}\right)^{2},$
(59)

where $\tilde{\alpha}=\alpha/[\sigma_{x}/\sigma_{S}]$ is the mean-reversion
ratio encountered in Section 3.2.3. It follows from (59) that the asymptotic
variance is the same, or below, the variance of the Black-Scholes model if and
only if $\tilde{\alpha}\geq 1/2$.

Par. set |  $\sigma_{S}$ |  $\sigma_{x}$ |  $\alpha$ |  $\tilde{\alpha}$ |  Asym. vol. |  SD($x_{\infty}$)
---|---|---|---|---|---|---
1 |  0.15 |  0 |  0.06 |  $\infty$ |  0.15 |  0
2 |  0.15 |  0.003 |  0.06 |  3.00 |  0.10 |  0.0087
3 |  0.15 |  0.007 |  0.06 |  1.30 |  0.033 |  0.020
4 |  0.15 |  0.009 |  0.06 |  1 |  0 |  0.026
5 |  0.15 |  0.015 |  0.06 |  0.60 |  0.10 |  0.043
6 |  0.15 |  0.020 |  0.06 |  0.45 |  0.18 |  0.058
7 |  0.15 |  0.030 |  0.06 |  0.30 |  0.35 |  0.087

Table 2: Set of parameters (with varying values of $\sigma_{x}$) governing the
mean-reversion of equity returns. Also shown is the mean-reversion ratio
$\tilde{\alpha}=\alpha/[\sigma_{x}/\sigma_{S}]$, the asymptotic rate of
volatility, $|\sigma_{S}-\sigma_{x}/\alpha|$, and the standard deviation in
the stationary distribution of the risk premium, $\sigma_{x}/\sqrt{2\alpha}$.

Par. set |  $\sigma_{S}$ |  $\sigma_{x}$ |  $\alpha$ |  $\tilde{\alpha}$ |  Asym. vol. |  SD($x_{\infty}$)
---|---|---|---|---|---|---
A |  0.15 |  0.007 |  0.90 |  19.3 |  0.14 |  0.0052
B |  0.15 |  0.007 |  0.14 |  3.00 |  0.10 |  0.013
C |  0.15 |  0.007 |  0.06 |  1.30 |  0.033 |  0.020
D |  0.15 |  0.007 |  0.047 |  1 |  0 |  0.023
E |  0.15 |  0.007 |  0.020 |  0.43 |  0.20 |  0.035
F |  0.15 |  0.007 |  0.010 |  0.21 |  0.55 |  0.049
G |  0.15 |  0.007 |  0 |  0 |  $\infty$ |  $\infty$

Table 3: Sets of parameters (with varying values of $\alpha$) governing the
mean-reversion of equity returns. Also shown is the mean-reversion ratio
$\tilde{\alpha}=\alpha/[\sigma_{x}/\sigma_{S}]$, the asymptotic rate of
volatility, $|\sigma_{S}-\sigma_{x}/\alpha|$, and the standard deviation in
the stationary distribution of the risk premium, $\sigma_{x}/\sqrt{2\alpha}$.

The asymptotic volatility, i.e., the square-root of (59), is given by
$|\sigma_{S}-\sigma_{x}/\alpha|$. For $\tilde{\alpha}>1$,
$\sigma_{x}/\alpha<\sigma_{S}$ and the effect of mean-reversion can be
interpreted as reducing the (local) volatility. Conversely, for
$\tilde{\alpha}<1$, $\sigma_{x}/\alpha>\sigma_{S}$ and the volatility
reduction due to mean-reversion in a sense "overshoots" the (local)
volatility; if the overshoot is large enough mean-reversion in fact increases,
rather than reduces, volatility. Also note, that for $\tilde{\alpha}=1$ the
two terms are equal and stocks are asymptotically risk free!

Figure 5 shows the annualized volatility, i.e,
$(\operatorname{\mathsf{Var}}[\log\tilde{S}_{t}]/t)^{1/2}$, as a function of
time for the parameter sets in Tables 3 and 3. The highlighted, third
parameter set is inspired by the empirical estimates reported in Table 1 of
Munk et al. (2004) and it is the same in the two tables. With this parameter
set as a starting point, we vary the values of $\sigma_{x}$ and $\alpha$ in
Tables 3 and 3, respectively. In each case, we see that increased mean-
reversion lowers the asymptotic volatility, but only up to a certain point
(given by $\tilde{\alpha}=1$). Beyond that point the asymptotic volatility
starts to increase again and it eventually diverges as $\tilde{\alpha}$ tends
to zero. As mentioned above, this is due to the uncertainty that accumulates
in the risk premium process. In the last column of each table, the standard
deviation of the stationary distribution of the risk premium process is shown
as a measure of this uncertainty.

Clearly, not every volatility profile in Figure 5 is suitable for modelling
equity returns. It is noteworthy, that all parameter values appear reasonable
and in fact they all (except set A) are within one standard error of their
empirical estimates, cf. Munk et al. (2004). Thus, some kind of expert
judgment is needed in the calibration process. The theoretical foundation for
the model seems strongest for $\tilde{\alpha}>1$, and as a rule of thumb we
recommend using parameter sets satisfying this constraint. Parameter sets with
$\tilde{\alpha}\leq 1$ might be useful on shorter horizons, but on longer
horizons they can lead to counter-intuitive results, as illustrated later.

Figure 5: Illustration of the annualized volatility of log excess stock
returns, i.e., $(\operatorname{\mathsf{Var}}[\log\tilde{S}_{t}]/t)^{1/2}$ as a
function of $t$. The left plot illustrates the parameter sets in Table 3,
where $\sigma_{x}$ is varied; the right plot illustrates the parameter sets in
Table 3, where $\alpha$ is varied. The emphasized profile (3 and C) is the
same in the two plots. It is based on the empirical estimates reported by Munk
et al. (2004).

####  4.2.2 Risk-reward profiles

In this section we illustrate the optimal risk-reward trade-off that can be
obtained by equity strategies, and we give examples of the optimal strategies.
We also compare the optimal strategies with constant strategies; constant
strategies provide a natural benchmark due to their optimality in the Black-
Scholes model. In particular, it is instructive to see how the horizon and the
degree of mean-reversion affect the performance of optimal and constant
strategies differently.

We consider the two parameter sets given in Table 4, corresponding to a
moderate ($\tilde{\alpha}=1.30$) and high ($\tilde{\alpha}=0.60$) degree of
mean-reversion. The two sets are the same as sets 3 and 5 of Table 3 with an
added assumption on the risk premium in stationarity, $\bar{x}=0.045$. For
both parameter sets, the (expected) market price of equity risk in
stationarity is $\bar{\lambda}^{S}=0.30$. This is higher than the market price
of interest rate risk used in Section 4.1, cf. Table 1, but lower than values
typically used for equities.666Munk et al. (2004) reports on empirical
estimates of $\bar{x}$ and $\sigma_{S}$ of $0.0648$ and $0.1468$,
respectively. This corresponds to a market price of equity risk in stationary
of $\bar{\lambda}^{S}=0.44$, which is more in line with levels typically used.
However, we do not wish to overstate the return potential on stocks going
forward, and therefore we use more conservative return assumptions in this
paper. Note, that the market price of equity risk is stochastic, with sizeable
variations both between paths and within paths. This stochasticity is part of
the optimization problem and the main driver of the different (optimal)
strategies that arise.

Parameter set |  $\bar{x}$ |  $\sigma_{S}$ |  $\sigma_{x}$ |  $\alpha$ |  $\tilde{\alpha}$ |  $\bar{\lambda}^{S}$
---|---|---|---|---|---|---
Moderate |  0.045 |  0.015 |  0.007 |  0.06 |  1.30 |  0.30
High |  0.045 |  0.015 |  0.015 |  0.06 |  0.60 |  0.30
Table 4: Parameters governing the dynamics for equity returns (2)—(3). Also
shown is $\tilde{\alpha}=\alpha/[\sigma_{x}/\sigma_{S}]$ as a measure of mean
reversion, and the market price of equity risk in stationary,
$\bar{\lambda}^{S}=\bar{x}/\sigma_{S}$. The latter is the continuous-time
analogue to the Sharpe ratio, i.e., excess return per unit of volatility for
holding equities.

It follows from Corollaries 2.2 and 3.2, that we can represent the portfolio
value as

$V_{T}\stackrel{{\scriptstyle\cal{D}}}{{=}}\frac{V_{0}}{p_{0}(T)}Y_{T}Z_{T},$
(60)

where $Y_{T}$ and $Z_{T}$ are independent, and determined by the rate and
equity strategies, respectively. Further, $Z_{T}$ is log-normally distributed,
$\log Z_{T}\sim N(\mu_{T},\sigma_{T}^{2})$, where $\mu_{T}$ and
$\sigma_{T}^{2}$ are given by (36) and (37), respectively. Due to (60), we can
study the effect of the rate and equity strategies separately. In Appendix D,
Tables 7 and 8 show risk and reward statistics for $Z_{T}$ for a range of
optimal equity strategies.

Figures 6 and 7 plot $\mu_{T}$ against $\sigma_{T}$ on horizons 20 and 40
years, respectively, for the optimal equity strategies. The profiles are
constructed by varying $\nu$ from $-\infty$ to $1/2$, both values excluded.
For each of the selected values of $\nu$, the corresponding optimal strategy
is found by Theorem B.1, and $\mu_{T}$ and $\sigma_{T}^{2}$ are then computed
by numeric integration of (36) and (37), respectively, with the optimal
strategy inserted.777In fact, since $\tilde{\alpha}>1/2$ for the two parameter
sets under consideration and since $\nu<1/2$ the optimal strategies are all of
exponential form (type I), cf. Figure 11 of Appendix B, and Theorem 3.5 also
applies.

Each plot also shows the risk-reward trade-off for constant strategies,
$f_{s}^{S}=c$. For constant strategies the log-mean and log-variance of
$Z_{T}$ can be calculated explicitly as

$\displaystyle\mu_{T}$
$\displaystyle=\frac{c}{\sigma_{S}}\left[T\bar{x}+\frac{x_{0}-\bar{x}}{\alpha}\left(1-e^{-\alpha
T}\right)\right]-\frac{T}{2}c^{2},$ (61) $\displaystyle\sigma_{T}^{2}$
$\displaystyle=c^{2}\left[T\left(\frac{\tilde{\alpha}-1}{\tilde{\alpha}}\right)^{2}+2\frac{\tilde{\alpha}-1}{\alpha\tilde{\alpha}}\left(1-e^{-\alpha
T}\right)+\frac{1}{2\alpha\tilde{\alpha}^{2}}\left(1-e^{-2\alpha
T}\right)\right],$ (62)

assuming both $\alpha$ and $\tilde{\alpha}$ are non-zero. The dedicated reader
is encouraged to implement the variance formula and compare its behaviour as a
function of time to the variance of a constant strategy in a Black-Scholes
model, $\sigma_{T}^{2}=c^{2}T$.

Figure 6: Risk-reward profiles, $(\sigma_{T},\mu_{T})$, for optimal (thick)
and constant (thin) equity strategies on a horizon of $T=20$ years for
moderate (left) and high (right) degrees of mean-reversion, cf. Table 4. The
different profiles correspond to $x_{0}$ of $0.005$ (green), $0.045$ (orange),
and $0.085$ (blue). The global optimum is obtained for $\nu=0$
($\blacksquare$). The vertical dashed line marks $\sigma_{T}=0.2$, the
corresponding optimal and constant strategies are illustrated in Figure 8.

Figure 7: Risk-reward profiles, $(\sigma_{T},\mu_{T})$, for optimal (thick)
and constant (thin) equity strategies on a horizon of $T=40$ years for
moderate (left) and high (right) degrees of mean-reversion, cf. Table 4. The
different profiles correspond to $x_{0}$ of $0.005$ (green), $0.045$ (orange),
and $0.085$ (blue). The global optimum is obtained for $\nu=0$
($\blacksquare$). The vertical dashed line marks $\sigma_{T}=0.3$, the
corresponding optimal and constant strategies are illustrated in Figure 9.

We see from Figures 6 and 7 that for moderate levels of mean-reversion the
optimal strategies improve the mean only slightly compared to constant
strategies, at least for the strategies of interest, i.e., strategies with
$\nu\leq 0$. However, for high levels of mean-reversion substantial
improvements are obtained on the 40-year horizon. Indeed, the optimal risk-
reward profile in the right plot of Figure 7 is surprising. The profile is
very steep initially meaning that substantial equity gains can be achieved on
long horizons with very little risk. Mathematically, the high degree of mean-
reversion allows a very effective netting of equity fluctuations—but only on
long horizons, and only if the exposure is taken in the right way.

####  4.2.3 Illustration of optimal strategies

In Figures 8 and 9 we illustrate the optimal equity strategies with
$\sigma_{T}=0.2$ on 20 years horizon and $\sigma_{T}=0.3$ on 40 years horizon,
respectively. The optimal strategies exploit the dynamics of the risk premium
process, and different strategies will therefore be employed depending on the
initial value of the risk premium. Therefore, there are three optimal
strategies in each plot, corresponding to the three different values of
$x_{0}$. In contrast, we see from (62) that the volatility of a constant
strategy does not depend on $x_{0}$. Consequently, for given volatility target
the same constant strategy will be used regardless of $x_{0}$; the horizontal,
black line in each plot shows this strategy.

The natural scale (left axis) for the strategies is the exposure, $f^{S}_{s}$,
to the Brownian motion driving equity returns. When positive, the exposure can
be interpreted as the (local) volatility due to equity investments. To aid
interpretation we also show (right axis) the exposure in terms of the
corresponding equity share, i.e., $f^{S}_{s}/\sigma_{S}$. An equity share of,
say, 50 pct. means that the equity exposure should equal the exposure obtained
from investing half the portfolio in equities. Note, since all equity
exposures are financed by equivalent short positions in cash, there is no
upper limit on the equity exposure/share.

The optimal strategies are generally decreasing over time, except when the
initial risk premium is sufficiently low. We also know that for moderate mean-
reversion only little is gained from following the optimal strategy compared
to a constant strategy, even though the strategies are quite different.
However, with high mean-reversion and a long horizon optimal strategies
substantially outperform constant strategies. We see from the right plot of
Figure 9 that the optimal exposure is initially very high and declines rapidly
thereafter. When mean-reversion is high, this profile is very effective at
netting equity fluctuations over time, and constant strategies can only obtain
the same volatility target by having a much smaller exposure.

Figure 8: Optimal equity strategies with $\sigma_{T}=0.2$ for $T=20$ years
when the degree of mean-reversion is moderate (left) and high (right), cf.
Table 4. The different profiles correspond to $x_{0}$ of $0.005$ (green),
$0.045$ (orange), and $0.085$ (blue). The horizontal black line shows the
constant equity strategy that achieves $\sigma_{T}=0.2$.

Figure 9: Optimal equity strategies with $\sigma_{T}=0.3$ for $T=40$ years
when the degree of mean-reversion is moderate (left) and high (right), cf.
Table 4. The different profiles correspond to $x_{0}$ of $0.005$ (green),
$0.045$ (orange), and $0.085$ (blue). The horizontal black line shows the
constant equity strategy that achieves $\sigma_{T}=0.3$.

####  4.2.4 Interior wedges of extremal strategies

We conclude the numerical section with an illustration of a surprising
phenomenon due to excessive mean-reversion. The risk-reward profiles shown in
Section 4.1.2 have two branches. An upper branch consisting of optimal rate
strategies and a lower branch consisting of minimizing rate strategies. From
the calculations in Section 3.1.2 it follows that there can be no other
extremal rate strategies. For extremal equity strategies the situation is
considerably more complex.

The left plot of Figure 10 shows the risk-reward profile for the entire family
of extremal equity strategies ($\nu\neq 1/2$) for the parameter set of Table 4
with high mean-reversion. We show the profile for a horizon of $T=40$ years
and $x_{0}=\bar{x}$, but other horizons and initial risk premiums give similar
plots. The upper branch of the profile is formed by $\nu<1/2$; this part is
also shown in the right plot of Figure 7 (orange thick line). The remaining
extremal strategies, however, do not all line up in a lower branch, as might
be expected. Instead, an interior wedge of strategies is formed, in addition
to the lower branch of minimizing strategies. The wedge consists of locally
minimizing and maximizing strategies, i.e., strategies where, for given
variance target, the mean cannot be improved, either upwards or downwards, in
a vicinity of the strategies (in the underlying function space of strategies).
A similar wedge does not occur for the parameter set of Table 4 with moderate
mean-reversion.

Figure 10: Left plot shows the risk-reward profile, $(\sigma_{T},\mu_{T})$,
for the family of extremal strategies ($\nu\neq 1/2$) on a horizon of $T=40$
years for the parameter set of Table 4 with high mean-reversion and
$x_{0}=\bar{x}=0.045$. The upper branch corresponds to $\nu<1/2$, with the
global maximum attained for $\nu=0$ ($\blacksquare$); the interior wedge
corresponds to $1/2<\nu<15$ (rounded), and the lower branch corresponds to
$\nu>15$. The vertical dashed line marks $\sigma_{T}=0.5$, the four extremal
strategies attaining this level of variability are illustrated in the right
plot. The colors distinguish between globally (orange) and locally (blue)
minimizing/maximizing strategies.

To investigate further this phenomenon, the right plot of Figure 10 shows the
four different extremal strategies with $\sigma_{T}=0.5$. The optimal strategy
(1) is similar to the optimal strategy shown in Figure 9 for $\sigma_{T}=0.3$
(orange thick line). The "drastic" change in exposure over time of the optimal
strategy is, however, dwarfed by the radical change in exposure of the
minimizing strategy (4). Without a high degree of mean-reversion such a
strategy would lead to a very high variance, but high mean-reversion in
combination with declining (absolute) exposure all but cancel aggregate return
fluctuations.

The locally minimizing strategy (3) is essentially the mirror strategy of the
globally minimizing strategy: The same glidepath is used to control the
variance, but with positive rather than negative exposure. The reason this
strategy has a relatively low mean, is that the extreme leverage results in a
large quadratic term that impairs the mean, cf. (36). The locally maximizing
strategy (2) uses a dampened version of the same glidepath, but with an
overall lower exposure resulting in a lower quadratic term and a higher mean.

In summary, high degrees of mean-reversion implies the existence of very
aggressive glidepaths with low aggregate variance. Depending on the sign of
the exposure, these glidepaths can be used as building blocks to create either
very low, or high, but not optimal, portfolio means. It is the presence of
these glidepaths that generates a richer set of extremal strategies than
typically seen. Apparently, it requires a high degree of mean-reversion before
the interior wedge appears. We conjecture that $\tilde{\alpha}<1$ is a
necessary condition, but we leave this as an open research question.

When mean reversion is even higher, e.g., parameter set 7 in Table 3, a
multitude of interior wedges can appear with locally minimizing/maximizing
exposure profiles of trigonometric form (not illustrated). These periodic
profiles constitute yet another type of building block for obtaining
(relatively) low variance even when the absolute exposure is very high. The
structural richness of the extremal strategies is intriguing and further
insights about the role of mean-reversion can undoubtedly be uncovered, but we
leave that for others to explore.

##  5 Concluding remarks

In this report we have derived the mean-variance optimal investment strategies
in a two-factor model with stochastic interest rates and mean-reverting equity
returns, with the constraint that the strategies are allowed to depend on time
only. Such deterministic strategies are closer to practical use than their
state-dependent counterparts typically considered in the literature. We have
used techniques from calculus of variations to obtain explicit solutions for
optimal rate and equity strategies separately, and discussed how to combine
these strategies. For equity strategies a complete solution has been provided,
while optimal rate strategies have been derived under the simplifying
assumption of constant market price of interest rate risk. We have also
provided illustrations of the results, and discussed the calibration of the
mean-reversion component in detail.

Assuming independent risk factors, the distribution of the portfolio on the
horizon can be succinctly represented as

$V_{T}\stackrel{{\scriptstyle\cal{D}}}{{=}}\frac{V_{0}}{p_{0}(T)}Y_{T}Z_{T},$
(63)

where $Y_{T}$ and $Z_{T}$ are independent, log-normally distributed random
variates depending on the rate and equity strategies, respectively. Holding
only $T$-bonds and no equity risk corresponds to $Y_{T}=Z_{T}=1$ and thereby
no risk on the horizon. All rate and equity strategies, or combination of
these, deviating from this hedging strategy, entail a non-trivial risk-reward
trade-off captured by the distribution of the stochastic multipliers, $Y_{T}$
and $Z_{T}$. Appendix D contains tables with risk and reward statistics that
can be used to assess this trade-off. The strategies are parameterized by
$\nu$, which can be interpreted as a risk-aversion parameter, with $\nu=0$
corresponding to the strategy with maximal log-mean.

We conclude from Appendix D that on long horizons, say, 30 years and above
there exist equity strategies with substantial upside and very little risk of
accumulated losses ($Z_{T}<1$). When mean-reversion is high the strategies are
essentially risk free (Table 8), but also for moderate levels of mean-
reversion the risk is low for all but the most aggressive strategies (Table
7). In other words, on long horizons, (optimal) equity strategies can function
as return generating overlays with very low risk of performing below the risk-
free, hedging strategy.

The optimal equity strategies are typically decreasing over time. This
conforms with our intuitive understanding of the effect of mean-reversion.
Essentially, "early" exposure is better rewarded than "late" exposure, since
early losses are more likely to be (partly) compensated by subsequent excess
returns than losses towards the end of the period. However, we have
demonstrated numerically, that for moderate levels of mean-reversion constant
strategies perform close to optimally. Indeed, under no mean-reversion, i.e.,
the Black-Scholes model, constant strategies are optimal. Conversely, with
high levels of mean-reversion the optimal strategies outperform constant
strategies significantly, but they rely on rather extreme glidepaths which may
be hard to implement in practice.

##  Appendix A Optimal equity strategies with positive mean-variance trade-off

In this appendix we derive the (unique) solution to (38) of Lemma 3.3 for
$\nu<0$. These strategies correspond to the optimal equity strategies with
positive mean-variance trade-off, as discussed in the main text. We assume
throughout that $\sigma_{x}>0$ and $\sigma_{S}>0$, while $x_{0}$ and $\bar{x}$
can have any real value.

It follows from Lemma 3.4 that for $\nu<0$ the solution to (38), if it exists,
must be of form

$f_{s}=b_{0}+b_{1}e^{c_{1}s}+b_{2}e^{c_{2}s}\quad(0\leq s\leq T).$ (64)

The plan therefore is to rewrite (38) using this form of $f$, and from this
derive the values of the five constants. We first carry out this programme,
assuming $\alpha\neq 0$ and
$\alpha\neq\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$. The two special cases
for $\alpha$ are handled separately afterwards. Proof of uniqueness of the
stated solutions is given at the end of the appendix. We begin with the
following lemma, the conclusion of which we will need throughout.

######  Lemma A.1.

Assume $\nu<0$ and $\alpha\neq\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$. Let
$c_{1}=\sqrt{-C/A}$, where $A=1-2\nu$ and
$C=2\nu(\alpha-\frac{\sigma_{x}}{\sigma_{S}})^{2}-\alpha^{2}$. Then, $c_{1}>0$
and $c_{1}\neq|\alpha|$. Further, the equation

$\left(1+\frac{\sigma_{x}}{\sigma_{S}(c-\alpha)}\right)\left(1-\frac{\sigma_{x}}{\sigma_{S}(c+\alpha)}\right)=\frac{1}{2\nu}$
(65)

has two distinct solutions given by $c=\pm c_{1}$.

###### Proof.

In the expression for $C$, at least one of the terms
$(\alpha-\frac{\sigma_{x}}{\sigma_{S}})^{2}$ and $\alpha^{2}$ will be strictly
positive, with coefficients $2\nu$ and $-1$, respectively, which are both
strictly negative. This implies $C<0$, and since $A>1$ we have $-C/A>0$, and
thereby also $c_{1}>0$.

For the second statement, we first note that the assumption
$\alpha\neq\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$ implies that
$\alpha^{2}\neq(\alpha-\frac{\sigma_{x}}{\sigma_{S}})^{2}$. Now, consider

$c_{1}^{2}=-C/A=p(\nu)\alpha^{2}+(1-p(\nu))\left(\alpha-\frac{\sigma_{x}}{\sigma_{S}}\right)^{2},$
(66)

where $p(\nu)=1/(1-2\nu)$ with $0<p(\nu)<1$. Equation (66) expresses
$c_{1}^{2}$ as a strict convex combination of two distinct values. In
particular, $c_{1}^{2}\neq\alpha^{2}$, which is equivalent to
$c_{1}\neq|\alpha|$.

Considering equation (65) we note that the right-hand side is finite, while
the left-hand side diverges for $c$ approaching $\pm\alpha$. In the search for
a solution we can therefore assume $c\neq\pm\alpha$. Under this assumption and
introducing the short-hand notation $R=\sigma_{x}/\sigma_{S}>0$, we can
rewrite the left-hand side of (65) as follows

$\left(1+\frac{R}{c-\alpha}\right)\left(1-\frac{R}{c+\alpha}\right)=\frac{c-(\alpha-R)}{c-\alpha}\frac{c+(\alpha-R)}{c+\alpha}=\frac{c^{2}-(\alpha-R)^{2}}{c^{2}-\alpha^{2}}.$
(67)

Equating this to the right-hand side of (65) and rearranging terms, we see
that this is equivalent to $Ac^{2}+C=0$ with solution $c=\pm\sqrt{-C/A}=\pm
c_{1}$. Finally, since $c_{1}>0$ the solutions are distinct. ∎

The next two lemmas establish the (general) form of $h$ and $\int h$ as
defined in Lemma 3.3.

######  Lemma A.2.

Assume $f$ is of form (64) with $c_{1}\neq\alpha$, $c_{2}\neq\alpha$, and
$\alpha\neq 0$. For $0\leq u\leq T$,

$h_{u}\equiv
f_{u}-\frac{\sigma_{x}}{\sigma_{S}}\int_{u}^{T}f_{s}e^{-\alpha(s-u)}ds=d_{0}+d_{1}e^{c_{1}u}+d_{2}e^{c_{2}u}+d_{3}e^{\alpha
u},$ (68)

where

$\displaystyle d_{0}$
$\displaystyle=b_{0}\left[1-\frac{\sigma_{x}}{\sigma_{S}\alpha}\right],\quad
d_{1}=b_{1}\left[1+\frac{\sigma_{x}}{\sigma_{S}(c_{1}-\alpha)}\right],\quad
d_{2}=b_{2}\left[1+\frac{\sigma_{x}}{\sigma_{S}(c_{2}-\alpha)}\right],$ (69)
$\displaystyle d_{3}$ $\displaystyle=e^{-\alpha
T}\frac{\sigma_{x}}{\sigma_{S}}\left[\frac{b_{0}}{\alpha}-\frac{b_{1}}{c_{1}-\alpha}e^{c_{1}T}-\frac{b_{2}}{c_{2}-\alpha}e^{c_{2}T}\right].$
(70)

###### Proof.

This follows from inserting $f$ and evaluating integrals of the form
$\int_{u}^{T}e^{ks}ds$ for $k=\alpha$, $c_{1}-\alpha$, and $c_{2}-\alpha$. By
assumption, all three exponents are different from $0$, and the integral
evaluates to
$\frac{1}{k}\left[e^{ks}\right]_{u}^{T}=\frac{1}{k}(e^{kT}-e^{ku})$. We get,

$\displaystyle h_{u}=\ $ $\displaystyle
b_{0}+b_{1}e^{c_{1}u}+b_{2}e^{c_{2}u}-$
$\displaystyle\frac{\sigma_{x}}{\sigma_{S}}e^{\alpha
u}\left(-\frac{b_{0}}{\alpha}\left[e^{-\alpha
s}\right]_{u}^{T}+\frac{b_{1}}{c_{1}-\alpha}\left[e^{(c_{1}-\alpha)s}\right]_{u}^{T}+\frac{b_{2}}{c_{2}-\alpha}\left[e^{(c_{2}-\alpha)s}\right]_{u}^{T}\right),$

from which the result follows after expanding and collecting terms. ∎

######  Lemma A.3.

Assume $f$ is of form (64) with $c_{1}\neq\pm\alpha$, $c_{2}\neq\pm\alpha$,
and $\alpha\neq 0$. Let $h_{u}$ and $d_{0}$, $d_{1}$, $d_{2}$, and $d_{3}$ be
defined as in Lemma A.2. For $0\leq s\leq T$,

$\frac{\sigma_{x}}{\sigma_{S}}\int_{0}^{s}h_{u}e^{-\alpha(s-u)}du=a_{0}+a_{1}e^{c_{1}s}+a_{2}e^{c_{2}s}+a_{3}e^{\alpha
s}+a_{4}e^{-\alpha s},$ (71)

where

$\displaystyle a_{0}$
$\displaystyle=\frac{\sigma_{x}}{\sigma_{S}}\frac{d_{0}}{\alpha},\quad
a_{1}=\frac{\sigma_{x}}{\sigma_{S}}\frac{d_{1}}{c_{1}+\alpha},\quad
a_{2}=\frac{\sigma_{x}}{\sigma_{S}}\frac{d_{2}}{c_{2}+\alpha},\quad
a_{3}=\frac{\sigma_{x}}{\sigma_{S}}\frac{d_{3}}{2\alpha},$ (72)
$\displaystyle\quad a_{4}$
$\displaystyle=-\frac{\sigma_{x}}{\sigma_{S}}\left[\frac{d_{0}}{\alpha}+\frac{d_{1}}{c_{1}+\alpha}+\frac{d_{2}}{c_{2}+\alpha}+\frac{d_{3}}{2\alpha}\right].$
(73)

###### Proof.

This follows by the same argument as in the proof of Lemma A.2 using that, by
assumption, all exponents are different from $0$. We leave the details to the
reader. ∎

Note, that the assumption of Lemma A.2 is slightly weaker than in Lemma A.3.
In the former, we need to ensure that the exponents of $f$ are different from
$\alpha$, while in the latter we need to ensure that the exponents of $f$ are
also different from $-\alpha$. Both assumptions are typically satisfied,
except in certain special cases.

######  Proposition A.4.

Assume $\nu<0$, $\alpha\neq 0$, and
$\alpha\neq\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$. Condition (38) of Lemma
3.3 is satisfied for

$f_{s}=b_{0}+b_{1}e^{c_{1}s}+b_{2}e^{c_{2}s}\quad(0\leq s\leq T),$ (74)

where $c_{1}=\sqrt{-C/A}>0$, $c_{2}=-\sqrt{-C/A}$, $b_{0}=-D/C$ with
$A=1-2\nu$, $C=2\nu(\alpha-\frac{\sigma_{x}}{\sigma_{S}})^{2}-\alpha^{2}$, and
$D=\alpha^{2}\bar{x}/\sigma_{S}$. Further, the exponents satisfy
$c_{1}=|c_{2}|\neq|\alpha|$, and

$\begin{pmatrix}b_{1}\\\
b_{2}\end{pmatrix}=\begin{pmatrix}\frac{e^{c_{1}T}}{c_{1}-\alpha}&\frac{e^{c_{2}T}}{c_{2}-\alpha}\\\\[5.69054pt]
\frac{\sigma_{x}}{\sigma_{S}(c_{1}+\alpha)-\sigma_{x}}&\frac{\sigma_{x}}{\sigma_{S}(c_{2}+\alpha)-\sigma_{x}}\end{pmatrix}^{-1}\begin{pmatrix}\frac{\alpha\bar{x}}{\sigma_{S}\left(\alpha^{2}-2\nu(\alpha-\sigma_{x}/\sigma_{S})^{2}\right)}\\\\[5.69054pt]
-\frac{x_{0}}{\sigma_{S}}+\frac{\alpha\bar{x}}{\sigma_{S}}\frac{\alpha-2\nu(\alpha-\sigma_{x}/\sigma_{S})}{\alpha^{2}-2\nu(\alpha-\sigma_{x}/\sigma_{S})^{2}}\end{pmatrix}.$
(75)

###### Proof.

By Lemma A.1 we have that $c_{1}\neq\pm\alpha$, and thereby also
$c_{2}\neq\pm\alpha$. The assumptions of Lemmas A.2 and A.3 are thereby
satisfied, and we can use the conclusions of these lemmas to recast the left-
hand side of condition (38) as

$\displaystyle\xi_{s}-f_{s}$ $\displaystyle+2\nu
h_{s}-2\nu\frac{\sigma_{x}}{\sigma_{S}}\int_{0}^{s}h_{u}e^{-\alpha(s-u)}du$
$\displaystyle=\ $ $\displaystyle\left[\bar{x}+e^{-\alpha
s}(x_{0}-\bar{x})\right]/\sigma_{S}+2\nu\left(d_{0}+d_{1}e^{c_{1}s}+d_{2}e^{c_{2}s}+d_{3}e^{\alpha
s}\right)-$ $\displaystyle
b_{0}-b_{1}e^{c_{1}s}-b_{2}e^{c_{2}s}-2\nu\left(a_{0}+a_{1}e^{c_{1}s}+a_{2}e^{c_{2}s}+a_{3}e^{\alpha
s}+a_{4}e^{-\alpha s}\right)$ $\displaystyle=\ $
$\displaystyle\left[\bar{x}/\sigma_{S}-b_{0}+2\nu(d_{0}-a_{0})\right]+$ (76)
$\displaystyle\left[-b_{1}+2\nu(d_{1}-a_{1})\right]e^{c_{1}s}+\left[-b_{2}+2\nu(d_{2}-a_{2})\right]e^{c_{2}s}+$
(77) $\displaystyle\left[2\nu(d_{3}-a_{3})\right]e^{\alpha
s}+\left[(x_{0}-\bar{x})/\sigma_{S}-2\nu a_{4}\right]e^{-\alpha s}.$ (78)

Condition (38) states that the expression above should equal $0$ for all
$0\leq s\leq T$. Since the exponents in (76)–(78) are all non-zero and
distinct, this is satisfied if (and only if) the five expressions in square
brackets in (76)–(78) are all $0$. It remains to be shown that this is
fulfilled if (and only if) the parameters take the stated values. For the
current result we only need the "if"-part, but for the later uniqueness result
we also need the "only if"-part; to facilitate this argument we show slightly
more than needed below.

By Lemmas A.2 and A.3, we have that the expression in (76) is $0$ if and only
if

$b_{0}\left[2\nu\left(1-\frac{\sigma_{x}}{\sigma_{S}\alpha}\right)^{2}-1\right]=-\frac{\bar{x}}{\sigma_{S}},$
(79)

and that the two expressions in square brackets in (77) are zero if and only
if

$\displaystyle b_{1}$
$\displaystyle=b_{1}2\nu\left(1+\frac{\sigma_{x}}{\sigma_{S}(c_{1}-\alpha)}\right)\left(1-\frac{\sigma_{x}}{\sigma_{S}(c_{1}+\alpha)}\right),$
(80) $\displaystyle b_{2}$
$\displaystyle=b_{2}2\nu\left(1+\frac{\sigma_{x}}{\sigma_{S}(c_{2}-\alpha)}\right)\left(1-\frac{\sigma_{x}}{\sigma_{S}(c_{2}+\alpha)}\right).$
(81)

Further, assuming (80)–(81) hold and using
$\alpha\neq\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$, the two expressions in
square brackets in (78) are zero if and only if

$\begin{pmatrix}\frac{e^{c_{1}T}}{c_{1}-\alpha}&\frac{e^{c_{2}T}}{c_{2}-\alpha}\\\\[5.69054pt]
\frac{\sigma_{x}}{\sigma_{S}(c_{1}+\alpha)-\sigma_{x}}&\frac{\sigma_{x}}{\sigma_{S}(c_{2}+\alpha)-\sigma_{x}}\end{pmatrix}\begin{pmatrix}b_{1}\\\
b_{2}\end{pmatrix}=\begin{pmatrix}\frac{b_{0}}{\alpha}\\\
-\frac{x_{0}}{\sigma_{S}}+\frac{\bar{x}}{\sigma_{S}}-2\nu
b_{0}\frac{\sigma_{x}}{\sigma_{S}\alpha}\left(1-\frac{\sigma_{x}}{\sigma_{S}\alpha}\right)\end{pmatrix}.$
(82)

To reach the desired conclusion, we need to verify that (79)–(82) all hold.
Clearly, $b_{0}=-D/C$ satisfies (79), while (80) and (81) are satisfied by
Lemma A.1. Finally, with $b_{0}=-D/C$, the right-hand side of (82) is
equivalent to the rightmost column of (75), and it follows that (82) is also
satisfied. ∎

Proposition A.4 leaves two cases to be handled separately. These are covered
by the following propositions. The line of argument is the same, but the
specific calculations differ slightly from those in the proof of Proposition
A.4.

######  Proposition A.5.

Assume $\nu<0$ and $\alpha=0$. Condition (38) of Lemma 3.3 is satisfied for

$f_{s}=b_{1}e^{c_{1}s}+b_{2}e^{c_{2}s}\quad(0\leq s\leq T),$ (83)

where $c_{1}=\sqrt{-C/A}>0$, $c_{2}=-\sqrt{-C/A}$ with $A=1-2\nu$, and
$C=2\nu(\frac{\sigma_{x}}{\sigma_{S}})^{2}$. Further, the exponents satisfy
$c_{1}=|c_{2}|\neq 0$, and

$\begin{pmatrix}b_{1}\\\
b_{2}\end{pmatrix}=\begin{pmatrix}\frac{e^{c_{1}T}}{c_{1}}&\frac{e^{c_{2}T}}{c_{2}}\\\\[5.69054pt]
\frac{\sigma_{x}}{\sigma_{S}c_{1}-\sigma_{x}}&\frac{\sigma_{x}}{\sigma_{S}c_{2}-\sigma_{x}}\end{pmatrix}^{-1}\begin{pmatrix}0\\\\[5.69054pt]
-\frac{x_{0}}{\sigma_{S}}\end{pmatrix}.$ (84)

###### Proof.

We will only give a sketch of the proof. By computations similar to those of
Lemmas A.2 and A.3 we first obtain expressions for $h$ and $\int h$; the
computations are left to the reader. Using these expressions we recast the
left-hand side of condition (38) as

$\displaystyle\xi_{s}-f_{s}$ $\displaystyle+2\nu
h_{s}-2\nu\frac{\sigma_{x}}{\sigma_{S}}\int_{0}^{s}h_{u}du$ $\displaystyle=\ $
$\displaystyle
x_{0}/\sigma_{S}-b_{1}e^{c_{1}s}-b_{2}e^{c_{2}s}+2\nu\left(d_{1}e^{c_{1}s}+d_{2}e^{c_{2}s}+d_{3}\right)-$
$\displaystyle
2\nu\left(a_{1}e^{c_{1}s}+a_{2}e^{c_{2}s}+a_{3}s-a_{1}-a_{2}\right)$
$\displaystyle=\ $
$\displaystyle\left[-b_{1}+2\nu(d_{1}-a_{1})\right]e^{c_{1}s}+\left[-b_{2}+2\nu(d_{2}-a_{2})\right]e^{c_{2}s}+$
(85) $\displaystyle\left[-2\nu
a_{3}\right]s+\left[x_{0}/\sigma_{S}+2\nu(d_{3}+a_{1}+a_{2})\right],$ (86)

where

$\displaystyle d_{1}$
$\displaystyle=b_{1}\left[1+\frac{\sigma_{x}}{\sigma_{S}c_{1}}\right],d_{2}=b_{2}\left[1+\frac{\sigma_{x}}{\sigma_{S}c_{2}}\right],d_{3}=-\frac{\sigma_{x}}{\sigma_{S}}\left[\frac{b_{1}}{c_{1}}e^{c_{1}T}+\frac{b_{2}}{c_{2}}e^{c_{2}T}\right],$
$\displaystyle a_{1}$
$\displaystyle=\frac{\sigma_{x}}{\sigma_{S}}\frac{d_{1}}{c_{1}},\quad
a_{2}=\frac{\sigma_{x}}{\sigma_{S}}\frac{d_{2}}{c_{2}},\quad
a_{3}=\frac{\sigma_{x}}{\sigma_{S}}d_{3}.$

As in the proof of Proposition A.4, it follows that condition (38) is
satisfied if and only if the four terms in square brackets in (85) and (86)
are zero. This in turn is equivalent to the following set of conditions,

$\displaystyle b_{1}$
$\displaystyle=b_{1}2\nu\left(1+\frac{\sigma_{x}}{\sigma_{S}c_{1}}\right)\left(1-\frac{\sigma_{x}}{\sigma_{S}c_{1}}\right),\
b_{2}=b_{2}2\nu\left(1+\frac{\sigma_{x}}{\sigma_{S}c_{2}}\right)\left(1-\frac{\sigma_{x}}{\sigma_{S}c_{2}}\right),$
(87) $\displaystyle 0$
$\displaystyle=\frac{b_{1}}{c_{1}}e^{c_{1}T}+\frac{b_{2}}{c_{2}}e^{c_{2}T},\quad-\frac{x_{0}}{\sigma_{S}}=\frac{b_{1}\sigma_{x}}{\sigma_{S}c_{1}-\sigma_{x}}+\frac{b_{2}\sigma_{x}}{\sigma_{S}c_{2}-\sigma_{x}}.$
(88)

Since $\alpha=0\neq\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$, we can use Lemma
A.1 to conclude that (87) is satisfied, while it follows from (84) that (88)
is satisfied. ∎

Note that the solution stated in Proposition A.5 is in fact the same as in
Proposition A.4 with $\alpha=0$; it is only the argument for verifying the
solution that differs. This is due to the fact that in both cases the
exponents, $c_{1}$ and $c_{2}$, differ from $\pm\alpha$. For the other special
case, $\alpha=\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$, the exponents equal
$\pm\alpha$ which change the conditions somewhat.

######  Proposition A.6.

Assume $\nu<0$ and $\alpha=\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$.
Condition (38) of Lemma 3.3 is satisfied for

$f_{s}=b_{0}+b_{2}e^{-\alpha s}\quad(0\leq s\leq T),$ (89)

where

$b_{0}=\frac{\bar{x}}{\sigma_{S}(1-2\nu)},\quad
b_{2}=\frac{-\frac{x_{0}}{\sigma_{S}}+\frac{\bar{x}}{\sigma_{S}}\left(1-\frac{4\nu}{1-2\nu}\left(e^{-\alpha
T}-1\right)\right)}{2\nu e^{-2\alpha T}-1}.$ (90)

###### Proof.

For use in the later uniqueness result, we will prove a bit more than needed
to establish the current result. Consider $f$ of the form

$f_{s}=b_{0}+b_{1}e^{\alpha s}+b_{2}e^{-\alpha s}\quad(0\leq s\leq T).$ (91)

By computations similar to those of Lemmas A.2 and A.3 we recast the left-hand
side of condition (38) as

$\displaystyle\xi_{s}-f_{s}$ $\displaystyle+2\nu
h_{s}-2\nu\frac{\sigma_{x}}{\sigma_{S}}\int_{0}^{s}h_{u}e^{-\alpha(s-u)}du$
$\displaystyle=\ $ $\displaystyle\left[\bar{x}+e^{-\alpha
s}(x_{0}-\bar{x})\right]/\sigma_{S}+2\nu\left(-b_{0}+d_{1}e^{\alpha
s}+d_{2}e^{\alpha s}s\right)-$ $\displaystyle b_{0}-b_{1}e^{\alpha
s}-b_{2}e^{-\alpha s}-2\nu\left(-2b_{0}+a_{1}e^{\alpha s}+a_{2}e^{\alpha
s}s+a_{3}e^{-\alpha s}\right)$ $\displaystyle=\ $
$\displaystyle\left[\bar{x}/\sigma_{S}-b_{0}+2\nu
b_{0}\right]+\left[-b_{1}+2\nu(d_{1}-a_{1})\right]e^{\alpha s}+$ (92)
$\displaystyle\left[(x_{0}-\bar{x})/\sigma_{S}-b_{2}-2\nu
a_{3}\right]e^{-\alpha s}.$ (93)

where

$\displaystyle d_{1}$
$\displaystyle=b_{1}-\frac{\sigma_{x}}{\sigma_{S}}b_{1}T+2b_{0}e^{-\alpha
T}+b_{2}e^{-2\alpha T},\quad d_{2}=\frac{\sigma_{x}}{\sigma_{S}}b_{1},$
$\displaystyle a_{1}$ $\displaystyle=d_{1}-b_{1},\quad a_{2}=d_{2},\quad
a_{3}=2b_{0}-d_{1}+b_{1}.$

Note that, compared to previous calculations there is apparently missing
factors of $\sigma_{x}/\sigma_{S}$ in the $a$-coefficients. These, however,
are being absorbed by the (also missing) factors proportional to $1/\alpha$.

As in the proof of Proposition A.4, it follows that condition (38) is
satisfied if and only if the three terms in square brackets in (92) and (93)
are zero. This in turn is equivalent to the following set of conditions,

$b_{0}=\frac{\bar{x}}{\sigma_{S}(1-2\nu)},\quad b_{1}=0,\quad 4\nu
b_{0}\left(e^{-\alpha T}-1\right)+b_{2}\left(2\nu e^{-2\alpha
T}-1\right)=-\frac{x_{0}-\bar{x}}{\sigma_{S}},$ (94)

from which the result follows. ∎

Collectively, Propositions A.4, A.5, and A.6 provide a solution to condition
(38) of Lemma 3.3 for $\nu<0$. The proofs of the propositions consist of
deriving an equivalent set of conditions characterizing the constants in (64).
Since we know that $f$ is of form (64), it might seem as if we have also shown
uniqueness. However, there is a caveat. In deriving the equivalent set of
conditions we make a priori assumptions on the value of the exponents, $c_{1}$
and $c_{2}$, and of the constant $b_{0}$ (in Proposition A.5). To prove
uniqueness we also need to check that condition (38) cannot be satisfied if
$c_{1}$ and $c_{2}$ take values initially ruled out by assumption. This
amounts to deriving alternative sets of equivalent conditions and showing that
these cannot be satisfied. In principle, this is possible, but it is rather
laborious. Fortunately, there is a shortcut to this brute force approach.

######  Proposition A.7.

Assume $\nu<0$. The solution to (38) of Lemma 3.3 provided by Propositions
A.4, A.5, and A.6 is unique.

###### Proof.

For $\nu<0$, it follows from Lemma 3.4 that $f$ is of form (64) with
$b_{0}=-D/C$. Moreover, it follows from general theory on ordinary
differential equations (ODE) that the exponents are given by $\pm\sqrt{-C/A}$,
since $-C/A>0$.

Revisiting the proof of Proposition A.4, the initial assumptions on $c_{1}$
and $c_{2}$, and the recasting of condition (38) are thus justified, not by
assumption, but as a consequence of Lemma 3.4. A fortiori, we can restrict
attention to candidate solutions of form (64) with $c_{1}=\sqrt{-C/A}$,
$c_{2}=-\sqrt{C/A}$, and $b_{0}=-D/C$, and for these candidate solutions the
derived conditions are valid. In fact, conditions (79)–(81) are automatically
satisfied, and a candidate solution is therefore an actual solution if and
only if it satisfies condition (82). Finally, since (82) has a unique
solution, the claimed uniqueness follows, in the case $\alpha\neq 0$ and
$\alpha\neq\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$. The same argument
applies to the proofs of Propositions A.5 and A.6 showing uniqueness also for
the cases $\alpha=0$, or $\alpha=\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$. ∎

##  Appendix B Overview of extremal equity strategies

To allow a full discussion of the optimization problem, we give here an
overview of the complete set of extremal equity strategies. Appendix A covers
in detail the case of prime interest, namely the strategies with positive
mean-variance trade-off. In this appendix we supplement these results with the
extremal equity strategies corresponding to a negative mean-variance trade-
off, $\nu>0$. We provide only the results and a few intermediate calculations,
which can serve as stepping stones for the dedicated reader interested in
reproducing the results. We assume throughout that $\sigma_{x}>0$ and
$\sigma_{S}>0$, while $x_{0}$ and $\bar{x}$ can have any real value.

From Lemma 3.4 we know that the extremal equity strategies satisfy the
nonhomogeneous, second-order differential equation

$Af^{\prime\prime}_{s}+Cf_{s}+D=0\quad(0\leq s\leq T),$ (95)

where

$A=1-2\nu,\quad
C=2\nu\left(\alpha-\frac{\sigma_{x}}{\sigma_{S}}\right)^{2}-\alpha^{2},\quad
D=\alpha^{2}\frac{\bar{x}}{\sigma_{S}}.$ (96)

There are three different cases, depending on the type of roots to the
characteristic polynomial, $Ar^{2}+C$, of the associated homogeneous
differential equation, $Af^{\prime\prime}_{s}+Cf_{s}=0$. The theorem below,
which we state without proof, summarizes the results.

######  Theorem B.1.

Assume independent risk factors ($\rho=0$). For given horizon $T>0$, the
extremal equity strategies are the solutions to condition (38) of Lemma 3.3.
If it exists, the extremal equity strategy, $f$, takes one of three forms.

I. Real, distinct roots

If $C<0<A$, or $A<0<C$, then

$f_{s}=b_{0}+b_{1}e^{c_{1}s}+b_{2}e^{c_{2}s}.$ (97)

For $\nu=0$, $f_{s}=\bar{x}/\sigma_{S}+e^{-\alpha
s}(x_{0}-\bar{x})/\sigma_{S}$, otherwise $b_{0}=-D/C$, $c_{1}=\sqrt{-C/A}$,
$c_{2}=-\sqrt{-C/A}$, while $b_{1}$ and $b_{2}$ are given by (44) and (45) of
Theorem 3.5.

II. Complex roots

If $C<0$ and $A<0$, or $C>0$ and $A>0$, then

$f_{s}=b_{0}+b_{1}\sin(cs)+b_{2}\cos(cs),$ (98)

where $b_{0}=-D/C$, and $c=\sqrt{C/A}$. The values of $b_{1}$ and $b_{2}$ are
given by (102) of Proposition B.2.

III. Double roots

If $C=0$ and $A\neq 0$, then

$f_{s}=b_{0}+b_{1}s+b_{2}s^{2},$ (99)

where $b_{2}=-D/(2A)$. The values of $b_{0}$ and $b_{1}$ are given by (108) of
Proposition B.3. In the special case $\nu=\alpha=0$, $f_{s}=x_{0}/\sigma_{S}$.

The theorem covers all situations, except $\nu=1/2$ ($A=0$), where generally
there is no corresponding extremal strategy, i.e., no solution to (38) of
Lemma 3.3.888For $\nu=1/2$, the solution to (95) is $f_{s}=-D/C$, i.e., a
constant not depending on $x_{0}$, nor on $T$. However, in general this
constant cannot solve (38), since the initial term of (38) is
$\xi_{s}=[\bar{x}+e^{-\alpha x}(x_{0}-\bar{x})]/\sigma_{S}$, while the
remaining terms are independent of $x_{0}$. Thus, apart from very specific
cases, e.g. $x_{0}=\bar{x}=0$, there is no solution to (38) for $\nu=1/2$. For
given model parameters, there are also other "singular" values of $\nu$ for
which there exists no corresponding extremal strategy. However, providing a
full description of necessary and sufficient conditions is outside the scope
of this paper. Except in special cases, Theorem B.1 gives the (unique)
extremal equity strategy for given model parameters and Lagrange multiplier,
$\nu$.

Noting that

$C=\left(\frac{\sigma_{x}}{\sigma_{S}}\right)^{2}\left[2\nu(\tilde{\alpha}-1)^{2}-\tilde{\alpha}^{2}\right],$
(100)

where $\tilde{\alpha}=\alpha/[\sigma_{x}/\sigma_{S}]$, we see that the sign of
$C$ depends only on $\nu$ and $\tilde{\alpha}$. This observation makes it
possible to visualize the domains corresponding to the three cases, cf. Figure
11. The solid black line is given by
$\tilde{\alpha}=[4\nu\pm\sqrt{8\nu}]/[2(2\nu-1)]$ for $\nu\geq 0$; on this
line $C=0$ and the solution is of type III. As seen previously, the strategy
is always of type I for $\nu<0$, and this is also the case for $\nu=0$ (except
for the special case $\nu=\alpha=0$). For $\nu>0$, the situation is more
complex with all three solution types occurring.

Figure 11: Plots of the parameter domains for solution types I (white area),
II (gray area), and III (solid black line). The right plot is a close-up of
the left plot. The solution type depends on the signs of $A$ and $C$, which in
turn depend on $\nu$ (Lagrange multiplier) and
$\tilde{\alpha}=\alpha/[\sigma_{x}/\sigma_{S}]$ (mean reversion ratio). The
dashed vertical line marks the excluded value $\nu=1/2$; $A$ is positive to
the left of this line and negative to the right of this line. The solid black
line corresponds to $C=0$ (solution type III), and this line forms the
boundary between solutions of type I and type II. The horizontal dashed line
marks the special case $\tilde{\alpha}=\frac{1}{2}$
($\alpha=\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$) for which the entire
family of extremal strategies is of type I.

By varying $\nu$, but keeping the model parameters fixed, we obtain the family
of all extremal strategies under this model. For $\nu<0$, the extremal
strategies are optimal with a positive mean-variance trade-off, and for
$\nu=0$ we get the global maximum. For $\nu>0$, we find both maximizing
strategies, minimizing strategies, and locally extreme strategies, cf. Section
4.2 for further discussion and examples. From Figure 11 we see that for
$\alpha=\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$ (horizontal dashed line),
all extremal strategies are of type I, and given by (45) of Theorem 3.5. This
is the only value of $\alpha$ for which all members of the family of extremal
strategies are of the same type.

###  B.1 Extremal equity strategies of type II and III

Theorem B.1 and Figure 11 give the full set of extremal strategies and the
parameter values for which the different solutions apply. In Appendix A, the
exponential solution (type I) was derived in detail for $\nu<0$. We see from
Figure 11, that the exponential solution also applies in some cases for
$\nu>0$, but we can also have a trigonometric solution (type II), or a
quadratic solution (type III). This section contains two propositions deriving
the solutions of type II and III.

The method of proof is the same as in Appendix A. Knowing that the extremal
strategies solve the ODE (95), we obtain an ansatz for the form of the
solution. Next, we rewrite condition (38) of Lemma 3.3 for this ansatz, and
this leads to a set of conditions for the constants appearing in the ansatz.
We then verify that the stated strategies satisfy these conditions. This
proves that the strategies are indeed extremal strategies. Uniqueness of the
extremal strategy, for given $\nu$, could be proved along the same lines as in
Proposition A.7, but this is left to the reader.

######  Proposition B.2.

Let $A$, $C$, and $D$ be given by (96). If $C<0$ and $A<0$, or $C>0$ and
$A>0$, then condition (38) of Lemma 3.3 is satisfied for

$f_{s}=b_{0}+b_{1}\sin(cs)+b_{2}\cos(cs),$ (101)

where $c=\sqrt{C/A}\neq 0$, $b_{0}=-D/C$, and

$\begin{pmatrix}b_{1}\\\
b_{2}\end{pmatrix}=\begin{pmatrix}k_{1}\alpha+k_{2}c&k_{2}\alpha-
k_{1}c\\\\[5.69054pt]
c&\tilde{R}\end{pmatrix}^{-1}\begin{pmatrix}\frac{\alpha\bar{x}}{\sigma_{S}}[A^{-1}+\alpha^{2}C^{-1}]\\\\[5.69054pt]
\frac{x_{0}}{\sigma_{x}}[c^{2}+\tilde{R}^{2}]+\frac{\alpha\bar{x}}{\sigma_{S}}[A^{-1}+\alpha\tilde{R}C^{-1}]\end{pmatrix},$
(102)

with $k_{1}=\sin(cT)$, $k_{2}=\cos(cT)$, and
$\tilde{R}=\frac{\sigma_{x}}{\sigma_{S}}-\alpha$.

###### Proof.

We will only give a sketch of the proof. Note that by assumption $C$ and $A$
are both non-zero and of equal sign. We are thus in the gray area of Figure
11, and we see that this implies that $\nu\neq 0$ and
$\alpha\neq\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$.

Assume first that $\alpha\neq 0$. By computations similar to those of Lemmas
A.2 and A.3 we obtain expressions for $h$ and $\int h$; the computations are
left to the reader. Using these expressions we recast the left-hand side of
condition (38) as

$\displaystyle\xi_{s}-f_{s}$ $\displaystyle+2\nu
h_{s}-2\nu\frac{\sigma_{x}}{\sigma_{S}}\int_{0}^{s}h_{u}e^{-\alpha(s-u)}du$
$\displaystyle=\ $ $\displaystyle[\bar{x}+e^{-\alpha
s}(x_{0}-\bar{x})]/\sigma_{S}-b_{0}-b_{1}\sin(cs)-b_{2}\cos(cs)+$
$\displaystyle 2\nu\left(d_{0}+d_{1}\sin(cs)+d_{2}\cos(cs)+d_{3}e^{\alpha
s}\right)-$ $\displaystyle
2\nu\left(a_{0}+a_{1}\sin(cs)+a_{2}\cos(cs)+a_{3}e^{\alpha s}+a_{4}e^{-\alpha
s}\right)$ $\displaystyle=\ $
$\displaystyle\left[\bar{x}/\sigma_{S}-b_{0}+2\nu(d_{0}-a_{0})\right]+$ (103)
$\displaystyle\left[-b_{1}+2\nu(d_{1}-a_{1})\right]\sin(cs)+\left[-b_{2}+2\nu(d_{2}-a_{2})\right]\cos(cs)+$
(104) $\displaystyle\left[2\nu(d_{3}-a_{3})\right]e^{\alpha
s}+\left[(x_{0}-\bar{x})/\sigma_{S}-2\nu a_{4}\right]e^{-\alpha s}.$ (105)

where

$\displaystyle d_{0}$
$\displaystyle=b_{0}\left[1-\frac{\sigma_{x}}{\sigma_{S}\alpha}\right],\quad
d_{1}=b_{1}-\frac{\sigma_{x}}{\sigma_{S}}\frac{\alpha
b_{1}-cb_{2}}{\alpha^{2}+c^{2}},\quad
d_{2}=b_{2}-\frac{\sigma_{x}}{\sigma_{S}}\frac{cb_{1}+\alpha
b_{2}}{\alpha^{2}+c^{2}},$ $\displaystyle d_{3}$ $\displaystyle=e^{-\alpha
T}\frac{\sigma_{x}}{\sigma_{S}}\left[\frac{b_{0}}{\alpha}+k_{1}\frac{\alpha
b_{1}-cb_{2}}{\alpha^{2}+c^{2}}+k_{2}\frac{cb_{1}+\alpha
b_{2}}{\alpha^{2}+c^{2}}\right],$ $\displaystyle a_{0}$
$\displaystyle=\frac{\sigma_{x}}{\sigma_{S}}\frac{d_{0}}{\alpha},\quad
a_{1}=\frac{\sigma_{x}}{\sigma_{S}}\frac{\alpha
d_{1}+cd_{2}}{\alpha^{2}+c^{2}},\quad
a_{2}=\frac{\sigma_{x}}{\sigma_{S}}\frac{\alpha
d_{2}-cd_{1}}{\alpha^{2}+c^{2}},\quad
a_{3}=\frac{\sigma_{x}}{\sigma_{S}}\frac{d_{3}}{2\alpha},$ $\displaystyle
a_{4}$
$\displaystyle=\frac{\sigma_{x}}{\sigma_{S}}\left[-\frac{d_{0}}{\alpha}-\frac{d_{3}}{2\alpha}+\frac{cd_{1}-\alpha
d_{2}}{\alpha^{2}+c^{2}}\right].$

As in the proof of Proposition A.4, it follows that condition (38) is
satisfied if and only if the five terms in square brackets in (103)–(105) are
zero. Note that these terms are formally equivalent to the ones of Proposition
A.4, but with different values of the constants.

The initial observation that $\nu\neq 0$ and
$\alpha\neq\frac{1}{2}\frac{\sigma_{x}}{\sigma_{S}}$ implies that the first
condition in (105), $2\nu(d_{3}-a_{3})=0$, is equivalent to $d_{3}=0$. Using
this, we get after a series of simplifications the following, equivalent set
of conditions

$\displaystyle
b_{0}=-\frac{\bar{x}}{\sigma_{S}}\left[2\nu\left(1-\frac{\sigma_{x}}{\sigma_{S}\alpha}\right)^{2}-1\right]^{-1},$
$\displaystyle
b_{1}=b_{1}2\nu\left[\left(1-\frac{\sigma_{x}}{\sigma_{S}}\frac{\alpha}{\alpha^{2}+c^{2}}\right)^{2}+\left(\frac{\sigma_{x}}{\sigma_{S}}\frac{c}{\alpha^{2}+c^{2}}\right)^{2}\right],$
$\displaystyle
b_{2}=b_{2}2\nu\left[\left(1-\frac{\sigma_{x}}{\sigma_{S}}\frac{\alpha}{\alpha^{2}+c^{2}}\right)^{2}+\left(\frac{\sigma_{x}}{\sigma_{S}}\frac{c}{\alpha^{2}+c^{2}}\right)^{2}\right],$
$\displaystyle 0=\frac{b_{0}}{\alpha}+k_{1}\frac{\alpha
b_{1}-cb_{2}}{\alpha^{2}+c^{2}}+k_{2}\frac{cb_{1}+\alpha
b_{2}}{\alpha^{2}+c^{2}},$ $\displaystyle
0=\frac{x_{0}-\bar{x}}{\sigma_{S}}+2\nu\frac{\sigma_{x}}{\sigma_{S}}\left[\frac{b_{0}}{\alpha}\left(1-\frac{\sigma_{x}}{\sigma_{S}\alpha}\right)-\frac{b_{1}c}{\alpha^{2}+c^{2}}+b_{2}\frac{\alpha-\sigma_{x}/\sigma_{S}}{\alpha^{2}+c^{2}}\right].$

In the second and third of the above conditions, the term in square brackets
is identical and we therefore in reality have only four conditions
(corresponding to the number of constants in $f$). Inserting $c^{2}=C/A$, this
term reduces to $1/(2\nu)$ and the two conditions are thereby satisfied. It is
also straightforward to verify that the remaining conditions are satisfied for
$b_{0}=-D/C$, and $(b_{1},b_{2})$ given by (102). This concludes the proof for
$\alpha\neq 0$.

For $\alpha=0$, we get by similar, but somewhat simpler calculations, that $f$
of form (101) solves (38) of Lemma 3.3 with $c=\sqrt{C/A}$, $b_{0}=0$ and

$\begin{pmatrix}b_{1}\\\
b_{2}\end{pmatrix}=\begin{pmatrix}\cos(cT)c&-\sin(cT)c\\\\[5.69054pt]
c&\sigma_{x}/\sigma_{S}\end{pmatrix}^{-1}\begin{pmatrix}0\\\\[5.69054pt]
x_{0}[c^{2}+(\sigma_{x}/\sigma_{S})^{2}]/\sigma_{x}\end{pmatrix}.$ (106)

Finally, we note that the solution for $\alpha=0$ is in fact also covered by
the general solution, i.e., it is also of the form $b_{0}=-D/C$ and
$(b_{1},b_{2})$ given by (102) evaluated at $\alpha=0$. Hence, when stating
the result we do not need to distinguish between the two cases. ∎

Finally, we give the proposition covering strategies of type III. These,
quadratic, solutions constitute the boundary between the exponential solutions
(type I) and the trigonometric solutions (type II). In the proof, we see that
we obtain the same formal set of conditions as seen before, but with different
values of the constants. It should be noted that $C=0$ implies
$\alpha\neq\sigma_{x}/\sigma_{S}$, and (108) of Proposition B.3 is therefore
well-defined.

######  Proposition B.3.

Let $A$, $C$, and $D$ be given by (96). If $C=0$ and $A\neq 0$, then condition
(38) of Lemma 3.3 is satisfied for

$f_{s}=b_{0}+b_{1}s+b_{2}s^{2},$ (107)

where $b_{2}=-D/(2A)$, and

$\begin{pmatrix}b_{0}\\\
b_{1}\end{pmatrix}=\begin{pmatrix}\alpha&T\alpha+1\\\\[5.69054pt]
\sigma_{x}/\sigma_{S}&\sigma_{x}/(\sigma_{x}-\sigma_{S}\alpha)\end{pmatrix}^{-1}\begin{pmatrix}\frac{\bar{x}\alpha}{\sigma_{S}(1-2\nu)}\left[\frac{1}{2}T^{2}\alpha^{2}+T\alpha+1\right]\\\\[5.69054pt]
\frac{x_{0}}{\sigma_{x}}\tilde{R}+\frac{\bar{x}\alpha}{\sigma_{S}(1-2\nu)}\frac{\sigma_{x}}{\sigma_{S}}\tilde{R}^{-1}\end{pmatrix},$
(108)

with $\tilde{R}=\frac{\sigma_{x}}{\sigma_{S}}-\alpha$.

###### Proof.

We will only give a sketch of the proof. Assume first that $\alpha\neq 0$. By