dynamic asset allocation with event risk, … annual meetings...risk. we consider a portfolio...

Dynamic asset allocation with event risk, transaction costs andpredictable returns

Jean-Guy Simonato∗

September 2016

Abstract

Dynamic asset allocation problems with event risk have been examined in frictionless envi-ronments with continuous trading opportunities. In these frameworks, the events take the formof sudden unexpected jumps in returns and analytical solutions to the investor’s problem can becharacterized for portfolios with many assets. However, the typical investor faces many frictionssuch as transaction costs and trading restrictions and can condition his decisions on variableswith some predictive power about the returns. We thus examine here the interplay betweenevent risk, transaction costs and predictability on the dynamic asset allocation of the investor ina discrete time setting. The model is calibrated to the U.S. stock market and a Gauss-Hermitequadrature approach is used to solve the investor’s dynamic optimization problem. Variousnumerical scenarios are examined to show the impact of event risks on the asset allocation,no-trading regions and certainty equivalent. It is found that the hedging demands of an investortaking into account event risk are generally smaller than those of an investor who neglects thisrisk. It is also found that the trading frequency can have an impact on the optimal allocationand hedging demands. Neglecting event risk can also lead to non-negligible annualized certaintyequivalent losses, which get larger as the investment horizon of the investor increases and/or asthe rebalancing frequency is lowered.

Keywords: dynamic asset allocation; event risk; jumps; transaction costs; return predictability

∗Department of Finance, HEC Montréal, 3000 Chemin de la Côte-Sainte-Catherine, Montréal (Québec), Canada,H3T 2A7; [email protected];

1

1 Introduction

As is it is well known from the financial literature, stock returns are subject to large and abrupt

changes caused by unforeseen events. Examples of such events are the collapse of financial firms

during the subprime crisis, the September 11, 2001 attacks or the recent sudden drop in oil prices.

For a long lived investor, these events represent a major source of risk that should be accounted

for when choosing and adjusting the holdings of his stock portfolio.

In the literature, dynamic asset allocation problems in the presence of event risk has been

examined extensively. This literature can be loosely classified in two categories. In a first category,

the presence of event risk is examined for a frictionless environment without any trading restrictions.

For example Liu et al. (2002) examine how jumps in the returns combined with stochastic volatility

affect the investor’s choices for a two asset portfolio while Das and Uppal (2004) looks at the impact

of jumps in a multivariate context with systemic event risks. These studies found that event risk

triggers important changes in the optimal investment strategy when compared to a pure diffusion

case. More recently, in Jin and Zhang (2012), the optimal policy of a multi-asset jump-diffusion

with many state variables is characterized with a semi-closed form solution. They found that, in

this context, the investor may not reduce his investment in risky assets when facing more frequent

jumps, unlike the results obtained with a single risky-asset jump-diffusion model.

In a second part of this literature, frictions and/or trading restrictions are incorporated in the

investor’s environment. In Liu and Loewenstein (2007) event risk is examined in the presence of

proportional transaction costs with a jump diffusion process. They provide approximate solutions

for finite horizon cases and find that the boundaries of the no-trade regions, which are typically

found in the presence of transaction costs, are affected by jump sizes and the uncertainty about

the jump sizes. Jin and Zhang (2013) characterize the investor’s choice with a semi-closed form

solution in a multi asset setting and examine the effect of trading restrictions such as short sales

and borrowing restrictions.

In this paper, we contribute to this second part of the literature and examine the dynamic

asset allocation problem of an investor who faces event risk and transaction costs with predictable

returns and a constant relative risk aversion (CRRA) utility function. The model we examine is

casted in a discrete-time framework with an investor who can only trade at specific time points,

unlike the above literature where trading takes place continuously in time. As outlined in Brandt

(2010), continuous time portfolio policies are often inadmissible in discrete time because they cannot

guarantee nonnegative wealth. It is thus important to examine this case, which represents a realistic

trading behavior as investors typically rebalance their portfolios at discrete intervals. The model

examined here also allows to assess the impact of the trading frequency in the presence of event

risk.

We consider a portfolio consisting of one risky and one risk-free asset. The investor faces a risky

asset return dynamics exhibiting some predictability inherited from the dividend yield dynamics as

2

in Barberis (2000), Campbell and Viceira (1999), and Balduzzi and Lynch (1999). However, unlike

these studies, the risky return is subject to normally distributed jumps shocks arriving according to

a Bernoulli random variable. To our knowledge, this is the first study of dynamic asset allocation

simultaneously looking at event risk, predictability and trading frictions.

As it is well known from the incomplete market dynamic portfolio literature, there are no

available analytical solutions for problems similar to this one, even in the simplest cases. We

thus rely on a Gauss-Hermite quadrature integration approach to recursively compute points on

the value function obtained from the Bellman equation associated with the investor’s problem.

Closely related to this study is Czerwonko and Perrakis (2008) who also resort to a numerical

approach to examine jump risk in the presence of transaction costs. They however examine a case

without predictability by looking at a jump diffusion process leading to independent and identically

distributed (i.i.d.) returns. In contrast, the situation examined here contains as a special case the

i.i.d. jump return with normal shocks.

The model used here is calibrated on U.S. data from 1960 to 2013 and various examples are

examined to show how the risky asset allocation and no-trading regions change with respect to

changes in the different parameters. Our main results can be summarized as follows. First, when

returns are predictable, introducing event risk triggers lower proportions in allocations and sizes of

the no-trade regions when compared to the no-event risk case. Second, we find that the hedging

demands of an investor considering event risks in his optimal allocation are typically smaller than

those of an investor who does not consider such risks in his decisions. We also find that these hedging

demands are impacted by the trading frequencies. Finally, the costs associated to neglecting event

risks can be substantial, especially for an investor with a lower rebalancing frequency, such as

quarterly, or one with longer horizons, such as ten or twenty years.

The remainder of the paper is organized as follows: the investor’s dynamic optimization problem

is presented in Section 2; the return dynamics is described in Section 3; the numerical procedure

is briefly outlined in Section 4 while the calibration to the US stock market is in Section 5. The

main results about the allocation of the investor considering event risk are presented in Section 6.

Concluding remarks are offered in Section 7.

2 The investor’s problem

The situation examined here consists of an investor who has available a risky and a risk-free asset.

With an investment a time t, the risky-asset gives a simple return in excess of the risk-free rate of

Rt+1, while the constant gross risk-free rate (one plus the simple risk-free rate) is denoted as Rf .

The investor can trade the risky and the risk-free securities at times t, t + 1, ..., T − 1 and there

are no intermediate consumptions. Such a situation corresponds, for example, to a pension fund

managing the savings of an investor over a long horizon.

We assume that the risky asset excess return is predictable using a variable dt, available at t, and

will be subject to unexpected jumps. As in many dynamic asset allocation studies such as Barberis

3

(2000), Campbell and Viceira (1999) or Balduzzi and Lynch (1999), the predictive variable used

here is the dividend yield. Campbell (1987) and Fama and French (1989) find that this variable

(and others such as the term premium) has some predictive power which can be used to forecast

stock returns. The exact dynamics of this predictive variable and the excess return with jumps is

specified in the next section.

As in Constantinides (1986) and Balduzzi and Lynch (1999), we assume that the investor is

subject to variable transaction costs when trading the risky asset. This transaction cost is modeled

as a function of the difference between the current risky asset allocation proportion denoted as xt,

and the inherited asset allocation xt. The more the current optimal allocation is different from

the inherited one, the larger the transaction costs will be. Hence, the decision of the investor, in

addition to be conditioned on the current predictive variable value, will also be conditioned on this

inherited allocation which is the result of the previous allocation and the realization of a random

return. At each reallocation point, the investor assesses if the potential gain from a new optimal

allocation is suffi cient to justify the transaction costs associated with the new allocation. As shown

in Constantinides (1979) and (1986), such transaction costs trigger a no transaction region with

boundaries. When the risky asset proportion falls outside this region, the investor trades to the

closest boundary.

Finally, we assume that the investor must hold non-negative proportions of the risky and risk-

free assets. This typical trading restriction amounts to 0 ≤ xt ≤ 1, which is a restriction on short

sales often faced by pension funds and individual investors. With this short sales constraint, the

inherited proportion upon which the investor will condition his decision will also remains between

zero and one.

In this context, the problem facing the investor who wants to maximize the expected utility of

his end of investment horizon wealth is

Vt (Wt, dt, xt) = max0 ≤ xsT−1

s=t ≤ 1Et [u (WT )] (1)

where Wt is the wealth of the investor at time t, x is proportion of wealth invested in the risky

security and u (·) is the utility of the investor which is summarized here by the following CRRAutility function

u (WT ) = W 1−γT / (1− γ) (2)

with risk aversion coeffi cient γ. The function Vt (Wt, dt, xt) is the value function which denotes the

expected utility at time t, given the current wealth Wt, the known value of the predictive variable

dt and the inherited risky asset allocation xt. The t subscript on the expectation operator denotes

that it is conditional on the available information at time t.

With the above assumptions, the investor is subject to the following sequence of budget con-

straints:

Ws+1 = hsWs (xsRs+1 +Rf ) (3)

4

for s = t to T where hs is defined as one minus the transaction cost per dollar of wealth paid at

time s.We assume proportional transaction costs which are functions of the difference between the

asset allocation xs and the inherited allocation xs:

hs = 1− h |xs − xs|

where h ≥ 0 is the proportional transaction cost parameter. Given this model specification, as

shown in Appendix A, the maximization problem can be reexpressed as the following recursive

Bellman equation:

vt (dt, xt) = maxxt

Et

[(ht (xtRt+1 +Rf ))1−γ vt+1 (dt+1, xt+1)

](4)

with a terminal condition vT = 1/(1−γ). As it is well known from the dynamic portfolio literature,

wealth drops out from the value function of the CRRA investor since the optimization problem

is homogeneous in wealth. This greatly simplifies the optimization problem as it avoids the need

of computing the value function on a grid of wealths. However, because of the dependence of the

value function on the current dividend yield and the inherited proportion, an evaluation on a grid

for these values is required. For each point on this grid, the optimization problem will be solved

with a backward iteration approach, starting at T − 1.

This backward recursive solution strategy is possible with a description of the dynamics of the

two state variables (the dividend yield and the inherited allocation). For the dividend yield, a

Markovian dynamics is described next section. For the inherited allocation, the dynamics can be

written as the ratio of the wealth invested in the risky asset divided by the overall wealth:

xs+1 =xshsWs (Rs+1 +Rf )

hsWs (xsRs+1 +Rf )=xs (Rs+1 +Rf )

xsRs+1 +Rf

for s = t to T. The next section examines the return process postulated for the stock return and

dividend yield.

3 The return process

We assume that the log excess return rt+1 = ln (Rt+1 + 1) evolves with a dynamics given by the

following restricted vector autoregressive process

rt+1 = αr + βrdt + εr,t+1

√∆t+ Jt+1 ×Qt+1, (5)

dt+1 = αd + βddt + εd,t+1

√∆t (6)

where dt+1 is the log dividend yield and

αr =

(α− 1

2σ2εr

)∆t,

with βr = β∆t, αd = κθ∆t, βd = (1− κ∆t) where α, β, κ and θ are constant parameters expressed

on an annual basis with ∆t being the length of time (in years) between two observations. The error

5

terms εr,t+1 and εd,t+1 are zero-mean Gaussian shocks with annual standard deviations of σεr , σεdand a constant correlation ρ. Both the return and dividend yield processes are written in terms

of annual parameters scaled by ∆t in order to be able to examine how the trading frequency (the

length of ∆t) will impact the asset allocation and the no trading regions. For the dividend yield,

a mean-reverting parameterization is adopted since it is more convenient for this purpose, with κ

interpreted as the speed of mean reversion while θ is the long run mean of the dividend yield.

In addition to the normal shocks, an independent jump shock Jt+1×Qt+1 can affect the return

and is interpreted as the event risk component. This component is formed of Jt+1, a Gaussian

random variable written has

Jt+1 = αJ + εJ,t+1

√∆t

where αJ =(αJ − 1

2σ2J

)∆t with αJ the annualized mean jump shock, and εJ,t+1 a Gaussian shock

with a mean of zero and an annual standard deviation σJ . The arrival of these random jump shocks

is governed by Qt+1, a Bernoulli random variable taking a value of zero or one with a probability

q. Both Jt+1 and Qt+1 are independent between themselves and from the Gaussian shocks εd,t+1

and εr,t+1. Finally we note that a convexity adjustment(−1

2σ2J

)is present in the specification of

αJ as well as αr. These terms act as compensations for the convexity and obtain an annualized

expected holding period excess return, conditional on no jump (k = 0) or a jump (k = 1), equal to

e(α+βdt+kαJ ) − 1, as shown in Appendix C.

As discussed in Ball and Torous (1983), this Bernoulli specification can be used as an approxi-

mation for the Poisson model with log normal jumps adopted in Merton (1976). The assumption

made here is that in each time interval either only one jump occurs or no jump occurs. For the

Poisson model, more than one jump can occur over a discrete time interval. As found in Ball and

Torous (1983), the above specification provides a model that can be easily calibrated with a max-

imum likelihood approach. Additionally, this process is interesting in the present context since it

requires less computations than a Poisson jump when evaluating the expected values. Nevertheless,

as discussed below, with few modifications, the present framework can be generalized to handle

Poisson distributed jumps.

In this model, the stock return inherit the predictability through the AR(1) structure of the

dividend yield. Such a specification (without jumps) has been used in the multiperiod portfolio

choice literature in Barberis (2000), Campbell and Viceira (1999), and Lynch (2001). This dynamics

for the return induces substantial hedging demands, defined as the differences between the single

period and multiperiod portfolio choices at the initial period. As explained in Campbell and Viceira

(1996) and Brandt (2010), these differences are triggered by the interplay between the predictability

of the risky asset return with the dividend, and the negative correlation between the residuals of

the dividend yield and the return.

For the purpose of computing the optimal investment policy, the density of the process is

required in order to compute the expected values appearing in the Bellman equation. For the

dividend yield, the density function is Gaussian. For the risky asset return, the density is a

6

combination of normal densities (with and without jump) and is given by:

φ (rt+1 | dt) =1∑

k=0

qk × φN(µk,t, σ

2k∆t

), (7)

µk,t = αr + βrdt + kαJ , (8)

σ2k = σ2

εr + kσ2J (9)

with φN (·) denoting the univariate normal density and where q0 = q and q1 = 1− q.We note that the above process can be generalized easily for jumps modeled using a compound

Poisson process. More specifically, if the number of jumps nt+1 arriving between times t and t+1 is

distributed as a Poisson counting process with intensity λ, the probability of nt+1 = k jump shocks

is given by

pr (nt+1 = k) =e−kλk

k!

which yield the conditional density

φ (rt+1 | dt) =∞∑k=0

pr (nt+1 = k)× φN(µk,t, σ

2k∆t

). (10)

The above density requires computing an infinite sum, but the magnitude of the elements quickly

decay to negligible values.

4 Gauss-Hermite quadrature

Given this return distribution, we need to recursively compute the conditional expected value

defined by the Bellman equation (4) in order to compute the optimal asset allocation at each

trading time. Furthermore, at each time point, this expected value also needs to be computed

repeatedly since an optimization is performed on the portfolio weights in order to find the one

for which the expected value is the greatest. We examine the approach adopted in this study to

compute these conditional expected values.

In a first step, the restricted VAR system given by equations (5) and (6) conditional on state

k = 0 or 1 of the Bernoulli variable is rewritten in vector form as

yk,t+1 = µk,t (dt) + ek,t+1

where

yk,t+1 =

[rk,t+1

dt+1

], µk,t (dt) =

[αr + βrdt + kαJ

αd + βddt

],

ek,t+1 =

[εr,t+1 + kεJ,t+1

εd,t+1

]√∆t

and E(ek,t+1e

ᵀk,t+1

)= Σk, a 2× 2 symmetric matrix defined as

Σk =

[σ2k ρσkσεd

ρσkσεd σ2εd

]∆t

7

where σk is defined in equation (9) and ρ is the correlation between εr,t+1 and εd,t+1.

In a second step, the conditional expected value appearing in the Bellman equation (4) is

rewritten generically as:

E g (yt+1) =1∑

k=0

qk

∫R2

g (yk,t+1)× ΦN (yk,t+1)× dyk,t+1, (11)

where ΦN (yk,t+1) is the bivariate normal density associated with yk,t+1, and where

g (yk,t+1) = (ht (xt (erk,t+1 − 1) +Rf ))1−γ vt+1 (dt+1, xt+1) .

It is important to notice that the return density is formed of a sum of normal densities. As explained

in Judd (1998), Gauss-Hermite quadrature can be used to effi ciently compute the expected value

associated with such densities. Yet, in order to use this numerical integration scheme, the expected

value must be rewritten with a change of variable in order to make it compatible with the Gauss-

Hermite quadrature scheme. Substituting the expression for the bivariate normal density and using

the following change of variable

yk,t+1 =√

2Ωkz+µk,t (dt)

where z is a 2× 1 vector and Σk = ΩkΩᵀk, yields after few manipulations the following expression

for the integral appearing in the summation:

E g (yk,t+1) = 1/π

∫R2

g(√

2Ωkz+µk,t (dt))

exp (−zᵀz) dz.

Using this formulation, Appendix B shows the details of how Gauss-Hermite quadrature can be

used to recursively compute the expected value associated with the Bellman equation (4) on a grid

of dividend yields and inherited asset allocations.

5 Calibration

In this study, the risky asset is the CRSP value weighted stock index formed with stocks from the

NYSE, AMEX and NASDAQ markets. The log dividend yield is defined as ln(1 + Dt/Mt) where

Dt is the sum of the dividend levels for the 12 months prior to t and Mt is the market value of the

securities in the index. As in Lynch and Tan (2010) and Lynch and Balduzzi (2000), this dividend

yield series is standardized to have a mean of zero and a standard deviation of one. We use monthly

data from January 1960 to December 2013 for a total of 648 observations. The risk-free rates are

the yields on 30 days to maturity treasury bills from CRSP. Table 1 presents the summary statistics

about the annualized values of these monthly time series. Observed in this table, the stock returns

show a negative skewness, an indication of large negative returns that could potentially be linked

to jump shocks.

The two equations of the restricted VAR are estimated separately. For the dividend yield,

the parameter estimates are obtained using an ordinary least-square approach. For the log excess

8

return of the risky asset, a maximum likelihood approach is adopted with the following log likelihood

function, given by:

ln [φ (rt+1 | dt)] = ln

1∑k=0

qk1√

2πσ2k∆t

exp

(−1

2

(rt+1 − µk,t)2

σ2k∆t

) .The first panel of Table 2 shows the maximum likelihood parameter estimates (MLE) for the

dividend yield equation. The second panel shows the parameter estimates of the return equation

with the jumps turned off and on. For the dividend yield equation, these estimates have the typical

values reported in the literature with the coeffi cient of the first lag equal to(1− .1281× 1

12

)=

0.9893, showing the high persistence and predictability of this variable. The long run mean is not

significantly different from zero, as expected from the normalization of the dividend yield series.

For the return equation with and without jumps, the parameters of the lagged dividend yield

reveal results similar to those reported in the literature, with a positive value, but not highly

significant. It should be pointed out that such a result does not necessarily imply that returns are

not forecastable with the dividend yield. For example, as argued in Camponovo, Scaillet and Trojani

(2013) and the references therein, conventional econometric approaches can fail to detect predictive

relations holding for the vast majority of the data points, because of the excessive influence on the

results of some rare anomalous observations. With the jumps turned on, the likelihood increases

significantly. The jump probability is significant and estimated to be around 1− 0.7647 = 0.2353,

while the mean and standard deviation parameters of the jump component are also significant

with a negative jump mean. Finally, the correlation between the error terms, estimated from the

residuals of both equations, also shows the typical large negative value found in the literature.

6 Numerical results

We examine here various numerical examples showing how the optimal allocation of the investor is

modified by the presence of event risk. The first subsection looks at the case of independent and

identically distributed (i.i.d.) returns for which there is no predictability. This obtains an optimal

investment policy which is independent of the investor’s horizon. In the second subsection, the case

with predictability is examined.

6.1 Asset allocations with i.i.d returns

We examine the case for which the dependence of the return to the dividend yield is turned off.

Because the returns will be i.i.d., it is well known that the optimal investment policy will be

constant through time and independent from the horizon of the investor. Despite the simplicity of

this situation, it is nevertheless interesting to look at this case to show the sensitivity of optimal

allocation to changes in the model’s parameters.

9

For this case, the log excess return evolves as

rt+1 = αr + εr,t+1

√∆t+ Jt+1Qt+1.

From the formula developed in Appendix C, the annualized expected excess holding period return

is given byE [St+1]

St− 1 = qeα∆t + (1− q) e(α+αJ )∆t − 1. (12)

With this specification, using the estimated parameters of Table 2 and the expressions given in

appendix, the annual holding period excess return has an expected value of 0.0748 and a standard

deviation of 0.2173.

Figure 1 simultaneously looks at the sensitivity of the risky asset allocation and no-trade region

to changes in the parameters of the model. To generate these graphs, if not otherwise stated,

the following parameter values are used: γ = 8, T = 1, h = 0.001, ∆t = 1/12, an annualized

continuously compounded risk-free rate of 3%, and the parameters of the time series model fixed to

the values reported in Table 2. For computational purposes, ten Gauss-Hermite quadrature points

and a grid of 100 inherited allocation equally spaced between 0 and 1 are used.

Each graph in this figure plots the risky asset allocation as a function of the inherited allocation

grid. The portion of the lines at 45 degrees are the no-trade regions i.e. a region for which the

optimal asset allocation is exactly the one of the inherited allocation. The size of these no-trade

regions are linked to the cost of making a sub-optimal allocation, which should be compared to

the loss associated to the trading cost. If the cost of trading is larger than the cost associated to

a sub-optimal allocation, there will be no trade. When the cost of trading becomes smaller than

the gain associated with a new allocation, the investor will trade and the allocation will be on the

boundaries of the no-trading zone.

The first graph of Figure 1 examines the effect of changes in α. This parameter regulates the

expected excess return as shown in equation (12). As α increases, the allocation in the risky asset

increases because the expected holding period return is increasing. The size of the no-trade region

(the portion of the lines at 45 degrees) is however insensitive to changes in this parameter. As

shown in Figure 2, this phenomenon is related to the shape of the expected utility as a function

of the risky asset weight. The first graph in this figure plots the expected utility function for

different values of α and a maturity of one period. This graph shows that α changes the location

of the function but has little impact on its curvature. With similar shapes of the expected utility

function, the cost of a sub-optimal allocation is similar for all values of α, making the no-trade

region equivalent for different values of this parameter.

The second graph of Figure 1 looks at changes in the standard deviation of the shocks affecting

the returns (σεr). Higher values of this parameter bring smaller proportions of the risky asset since

it increases the risk of the returns. Unlike the previous case, this parameter has a large impact on

the no-trade region because it changes the shape of the expected utility (as a function of the risky

asset allocation). Looking at Figure 2 we see that σεr changes the location and curvature of the

10

function. As σεr increases, the expected utility function becomes more curved. For small values of

σεr (low risk and flatter expected value function), the cost of a sub-optimal allocation is small and

it takes a new allocation very different from the inherited one to compensate for the transaction

costs, which yield a large no-trade region. For larger values of σεr , the variance of the return is

higher and a sub-optimal allocation has a greater cost, which in turn makes the no-trade region

smaller.

The third graph examines changes in the length of the trading interval. As seen from the

formula of the expected value and variance of the holding period return, a change in this parameter

impacts both of these quantities. As a result, the middle of the no-trading zone remains roughly

constant since the increase in risk is compensated by an increase in expected return. However, the

increase in variance has a large impact on the no trading zone since it increases the curvature of

the expected utility function. Hence, an investor who trades less frequently will typically face a

smaller no-trade region i.e. will be more inclined to adjust his holdings.

The impact of the no-jump probability q is examined in the fourth graph. A small change in this

probability has a large impact on the expected return but a moderate one the standard deviation.

For example, the expected values and standard deviation pairs associated with the probabilities are

(0.0092 , 0.0394) for q = 0.9, (0.0058, 0.0438) for q = 0.8 and (0.0025, 0.0475) for q = 0.7. As q is

lowered (increase in jump risk), the expected return gets lower and the risk get higher, two adverse

effects which yields large changes in the optimal allocation. Since the variance is mildly affected,

the no-trade region is also mildly affected and gets slightly smaller as q gets smaller.

The fifth and sixth graphs look at αJ and σJ which are parameters affecting the expected value

and risk of the holding period return. As seen on the graph, it requires fairly large changes in these

parameters to trigger some changes in the allocation. For example to go to (roughly) a near zero

allocation to a 50% allocation αJ has to go from -0.6 to -0.2. A similar effect is also observed for

the standard deviation of the jump shock σJ which must increase by a large amount to change the

allocation. This low sensitivity of the allocation is expected since these parameters affect the risk

and return with a small probability of a jump. As for the other cases, the parameter affecting the

expected value (α) leaves a no-trade region size mostly constant, while the volatility of the jump

shock has a large effect on the size.

While the figure discussed above examined how sensitive the allocations are to changes in the

model’s parameter, Figure 3 examines how different the allocation of an investor would be with and

without event risk. The graphs in this figure present the allocation, as a function of the inherited

allocation, for different levels of transaction costs and risk aversion, T = 1 and ∆t = 1/4. To obtain

these allocations, we use the same set of numerical parameters as for the previous figure but with

jumps turned on and off. When the jumps are turned off the MLE parameter estimates for the case

without jumps are used. The cases for h = 0.001 show little variations in the allocations. Without

event risk, the allocations are slightly higher than those with event risk, to compensate for the

higher level of uncertainty associated with this situation. With a larger value of the transaction

11

cost parameter, the differences in allocations are larger, especially for lower values of the risk

aversion level. We notice however, that such differences are not observed with a higher rebalancing

frequency. For example, for the monthly case, the allocations show very little differences between

the two situations.

6.2 Asset allocations with predictable return

Here, we examine the case for which the returns of the risky asset are predictable by the dividend

yield. As shown in the literature, such a predictability triggers time-varying investment proportions

and obtains positive hedging demands i.e. a myopic risky-asset allocation which is smaller than the

initial risky-asset allocation for horizons longer than one period.

Table 3 reports statistics about the asset allocations obtained at t = 0 using an annualized

continuously compounded risk-free rate of 3%, h = 0.001 and the parameters of the time series

model fixed to the MLE values for the jump case. For computational purposes, we use ten Gauss-

Hermite quadrature points and grids of 20 dividend yields and 100 inherited allocations. The

risky asset allocations are reported for investors with horizons of 1, 12, 60 and 120 months, for

different levels of risk aversion and initial dividend yields values. Because of the transaction costs,

we characterize the initial investment opportunities with the average allocation taken over the

inherited allocation levels (first panel), the no-trade region size defined as the upper allocation

bound minus the lower bound of the region (second panel), and the hedging demands computed

as the difference between the average allocation for an horizon of T periods and the one for an

allocation of one period (third panel).

The average allocations for all horizons are increasing as the initial dividend yield value is

increased. Such a result is obtained because the equity risk premium increases with the dividend

yield level, as shown by the positive parameter estimate associated to the lagged dividend yield. As

expected, the allocation increases as the risk-aversion levels decrease. The no-trade region size1 is

typically larger for the one period cases than for the cases with longer maturities since these bring

more uncertainty and, for similar allocation changes, larger expected gains that compensate for the

cost of trading. For multiperiod horizons, the sizes of the no-trading regions are roughly similar

across horizons, but with upper and lower bounds that are varying across dividend yield levels and

risk-aversion coeffi cients.

When looking at the third panel, the myopic asset allocations (the 1 month case) are smaller

than those for the longer maturities, at the exception of the 12 months case with d0 = −1 and 0

which are slightly higher. Hence, the hedging demands are generally positive and increase with the

horizon and dividend yields, a result consistent with those obtained in the literature for the cases

without event risk.

The fourth, fifth and sixth panels present the case of an investor who does not consider event

1An allocation is in the no-trade region when the difference between the optimal allocation and the inheritedallocation is smaller than 0.01.

12

risk in his decisions. Using the MLE parameter estimates for the no-jump case, the optimal as-

set allocation is determined with q = 1 (jumps turned off). For most cases (except for the small

dividend yield case), the average allocation is larger without event risk. These differences in allo-

cation can also be large. For example, the case T = 60, γ = 5 and d0 = 1 yields a difference of

(0.9089− 0.6969) = 21 % in the initial asset allocation. Looking at the no-trade region sizes, they

are generally smaller with event risk than those without event risk because of the higher uncertainty

associated with the possibilities of jumps.

Finally, the hedging demands presented in the sixth panel are much larger for the no event risk

case. As outlined in the multiperiod portfolio choice literature when returns can be predicted with

dividend yields, relative to the myopic case, over-investing in stocks smooths out the intertemporal

risk regarding future investment opportunities. This is caused by the interplay among the positive

link between returns and dividend yields and the strong negative correlation between the dividend

yield and return shocks. Unlike the case without event risk, large negative shocks that are unrelated

to the dividend yield shocks can affect the stock return. These unrelated negative shocks reduce

the ability of stocks to smooth out the intertemporal risk, and reduce the capacity of stocks to act

as a hedging device.

Table 4 looks at the quarterly rebalancing case. For this case, the average allocations are

smaller across horizons, risk aversion and initial dividend yield levels. For the no event risk case,

the differences in allocations can be greater or smaller, depending on the horizon and initial dividend

yield level. The no-trade regions are smaller for the quarterly case than those of the monthly case,

a result similar to the i.i.d. case examined in the previous subsection. Finally, we notice that the

hedging demands are also lower for the quarterly case. With a quarterly rebalancing frequency,

event risk shocks can have a greater impact which lowers the capacity of stocks to smooth out the

intertemporal risk.

The above results are for initial allocations. For multiperiod investment problems, the alloca-

tions are typically changing through time in order to adjust to the changing horizon. To get an

idea how these proportions are changing, Figure 4 plots the no-trade region obtained at each time

point for ∆t = 1/12, a dividend level of 1.0, and an horizon of T = 24. The first graph in this

figure looks at the optimal allocation of an investor considering the possibility of event risks in his

decisions, while the bottom graph shows the case of the investors ignoring this possibility. Again,

the relevant set of MLE parameters is used to obtain the proportions. For all periods, ignoring

event risk yields higher allocations.

6.3 The economic significance of event risk consideration

An important issue related to the model examined here is the assessment of the economic signif-

icance of event risks. For this purpose, in this section, within the context of costly transactions

and simulated returns with event risks, we compare the annualized certainty equivalent returns

(CEQ) of the investor from our model these with two other investment behaviour: a first one where

13

the investor ignores event risk but accounts for costs in his decisions, and a second one where the

decisions are taken without considering event risk nor costs.

These two optimization problems can be solved by turning off some of the parameters of the

more general problem examined here. We define the CEQ as the annualized risk-free return making

the investor indifferent between holding the optimal portfolio and earning the certainty equivalent

rate over the investment horizon. The CEQ for these three policies are evaluated by a forward

simulation approach. Using simulated dividend and return paths with jumps until a maturity T ,

we invest in the risk-free and risky asset in proportions indicated by the investment policies obtained

by each problem with a transaction cost of h = 0.001. The utility of end of horizon wealths are

then evaluated and averaged over the paths to obtain an approximate expected utility level. These

averages are then converted in certainty equivalent return with the inverse of the utility function.

More specifically, using n = 106 paths, we compute:

CEQ =[u−1 (u)

]1/(∆t×T ) − 1,

u =1

n

n∑i=1

u(W

(i)T

),

W(i)T =

∏T−1t=0 h

∗t

(x∗tR

(i)t+1 +Rf

),

and

h∗t = 1− h |x∗t − x∗t | ,

x∗t =x∗t−1

(R

(i)t +Rf

)x∗t−1R

(i)t +Rf

with x∗0 = 0 and where the inverse of the utility function is given by

u−1 (u) = [u× (1− γ)]1/(1−γ) .

Here, x∗t is the optimal investment proportion at t obtained for one of the three cases considered,

R(i)t is a return simulated with the jump process and the parameters from Table 2. For each cases,

the investment policy consists of a set of optimal portfolio weights that are computed for each point

on the grid of dividend yield and inherited proportions, from t = 0 to T − 1. Given a simulated

dividend yield and inherited proportions at t, the optimal weight x∗t is obtained with a bilinear

interpolation. For the two cases without jumps, it is important to notice that we use the parameter

estimates for the no-jump case available in Table 2 in order to mimic the behavior of an investor

who does not consider jumps in his decisions.

Tables 5 and 6, report the CEQ for the monthly and quarterly cases. As shown in these tables,

the loss of not considering jumps are generally increasing with the horizon of the investor. For longer

horizons, the suboptimal behavior of the investor will become more costly as the expected wealth

losses compound themselves. The losses are also greater for the quarterly case since the suboptimal

investor will face greater shocks associated to the event risks than in the monthly case. Finally,

14

the losses are also increasing with the initial dividend yield level since the proportion of stocks

are typically larger for these cases for the suboptimal investor. For long horizon investments, the

magnitude of CEQ losses can be fairly large on an annual basis when compared to the level of the

CEQ. For example, the loss is around 83 basis points for a CEQ of 0.0469 for the quarterly 10 year

case with γ = 8 and an initial dividend of 1.0. When expressed as the difference in initial wealth that

will make the optimal and suboptimal investors indifferent at T , we obtain (1.0083)10 − 1 = 8.6%.

Although not reported in the table, computations for the case of 20 years with the same parameters

reveal even higher losses. For this case, the annualized CEQ loss is computed as 120 basis points

for an annualized CEQ level of 0.0456.

7 Conclusion

This paper examines the discrete-time multiperiod portfolio choices of an investor with finite horizon

who faces transaction costs and event risks with returns that are predictable by the current dividend

yield. Event risk take the form of jumps modeled by a Bernouilli variable and normal jump shocks.

Using monthly data from 1960 to 2014, we calibrate the model using stock index returns and

risk-free yields.

As in the previous literature, we find that proportional transaction costs trigger a no-transaction

region whose size varies across time and dividend yield values. We also find that small changes in

the probability associated to event risk can cause large changes in the optimal allocation. When

comparing the cases with and without event risk, the optimal allocations of a multiperiod investors

are lower in the presence of event risk, especially for high values of the dividend yield, longer

maturities and lower rebalancing frequencies.

The hedging demands for the event risk case are also typically much lower than those of the

no-event risk case. Finally, we find that it can be costly for the investor to ignore event risks and

transaction costs, especially for long horizons or for a lower rebalancing frequency.

References

[1] Balduzzi, P., Lynch, A., 1999, Transaction costs and predictability: some utility cost calcula-tions, Journal of Financial Economics 52, 47—78.

[2] Ball, C., Torous, W., 1983, A simplified jump process for common stock returns. Journal ofFinancial and Quantitative Analysis 18, 53—65.

[3] Barberis, N., 2000, Investing for the long run when returns are predictable, Journal of Finance55, 225—264.

[4] Brandt, M., 2010, Portfolio choice problems, in Y. Ait-Sahalia and L.P. Hansen, Handbook ofFinancial Econometrics, Volume 1: Tools and Techniques, North Holland, 269—336.

15

[5] Camponovo, L., Scaillet , O., Trojani, F., 2014, Predictability hidden by anomalousobservations, Swiss Finance Institute Research Paper No. 13-05, available at SSRN:http://ssrn.com/abstract=2237447.

[6] Campbell, J., 1987, Stock returns and the term structure. Journal of Financial Economics 18,373—399.

[7] Campbell, J., Viceira, L., 1999, Consumption and portfolio decisions when expected returnsare time-varying, Quarterly Journal of Economics 114, 433—495.

[8] Constantinides, G., 1979, Multiperiod consumption and investment behavior with convextransactions costs, Management Science 25, 1127—37.

[9] Constantinides, G., 1986, Capital market equilibrium with transaction costs, Journal of Polit-ical Economy 94, 842—862.

[10] Czerwonko, M., Perrakis, S., 2008, Portfolio selection with transaction costs and jump-diffusionasset dynamics, working paper available at SSRN: http://ssrn.com/abstract=1290424.

[11] Das, S., Uppal, R., 2004, International portfolio choice with systemic risk, Journal of Finance59, 2809—2834.

[12] Jin, X., Zhang, A., 2012, Decomposition of optimal portfolio weights in a jump-diffusion modeland its applications, Review of Financial Studies 25, 2877—2919.

[13] Jin, X., Zhang, A., 2013, Dynamic optimal portfolio choice in a jump-diffusion model withinvestment constraints, Journal of Banking and Finance 37, 1733—1746.

[14] Judd, K., 1998, Numerical methods in economics, MIT Press: Cambridge, MA.

[15] Liu, H., Loewenstein, M., 2002, Optimal portfolio selection with transaction costs and finitehorizons, Review of Financial Studies 15, 805—835.

[16] Liu, H. Loewenstein, M, 2007, Optimal portfolio selection with transaction costs and eventrisk, working paper available at SSRN: http://ssrn.com/abstract=965263.

[17] Liu, J., Longstaff, F., Pan, J, 2003, Dynamic asset allocation with event risk, Journal ofFinance 58, 231—259.

[18] Lynch, A., 2001, Portfolio choice and equity characteristics: characterizing the hedging de-mands induced by return predictability, Journal of Financial Economics 62, 67—130.

[19] Merton, R., 1976, Option pricing when the underlying process for stock returns is discontinu-ous, Journal of Financial Economics 3, 124—144.

16

A Bellman equation

For CRRA utility functions the Bellman equation simplifies to a value function independent from the wealthlevel. This standard result can be found for example in Brandt (2010), and is provided here for completeness.Using the law of iterated expectation and the definition of the CRRA utility function, the maximum expectedutility of wealth at T given by equation (1) can be rewritten as:

Vt (Wt, dt, xt) = maxxt

Et

[max

0 ≤ xsT−1s=t+1 ≤ 1Et+1

W 1−γT

1− γ

].

Using the budget constraint (3), the above equation can be rewritten as:

Vt (Wt, dt, xt) = maxxt

Et

(htWt (xtRt+1 +Rf ))

1−γ ×

max0 ≤ xsT−1s=t+1 ≤ 1

Et+1

(∏T−1

s=t+1hs(xsRs+1+Rf )

)1−γ1−γ

.

Because of the homotheticity in wealth of the utility function, we can fix W0 = 1 to obtain

vt (dt, xt) = maxxt

Et

[(ht (xtRt+1 +Rf ))

1−γvt+1 (dt+1, xt+1)

]and Vt (1, dt, xt) = vt (dt, xt) with vT (dT ) = 1/ (1− γ).

B Gauss-Hermite quadrature algorithm

The algorithm for computing the optimal portfolio weight with the Bellman equation (4) and the formulationof the expected value given by equation (11) is described here. Two main steps are associated with thealgorithm: a preliminary step where various quantities required later are computed; a recursive step, wherethe calculations of the optimal policies are performed.

Preliminary work:

- Compute the Nq × 1 vectors of univariate quadrature nodes and weights Z and ω.

- Form the N2q × 2 matrices of multivariate quadrature nodes and weights Z and ω with the Cartesian

products of Z with itself, and the Cartesian product of ω with itself.

- Compute the transformed quadrature nodes as Zk=√

2ZΩ>k for k = 0 to 1 and ω, a N2q × 1 vector with

elements ωi = ωi,1 × ωi,2.

- Compute a dividend yield grid:

. Determine δmin,t and δmax,t, the lower and upper values of the dividend yield grid at t. These valuesare determined from t = 1 to T with

δmin,t = E0 [dt]− 10×√V0 [dt] and δmax,t = E0 [dt] + 10×

√V0 [dt]

where E0 [dt] and V0 [dt] are, respectively, the conditional expected value and variance at t = 0

of dt.

. Compute the values on the grid for t = 1 to T with

δi,t = δmin,t + (i− 1)(δmax,t − δmin,t)

(Nd − 1)for i = 1 to Nd.

17

- Compute an inherited asset allocation grid of Nx values:

. With χmin = 0 and χmax = 1

χj = χmin + (j − 1)(χmax − χmin)

(Nx − 1)for j = 1 to Nx.

Recursive work

Using the terminal condition vT (dT , xT ) = 1/ (1− γ) , for each points δi,t, χj on the grid of dividend yieldand inherited allocation, recursively compute the optimal portfolio weights and value function levels with

vt (δi,t, χj) = max0≤x≤1

1

π

1∑k=0

qk × Ei,j,k

where

Ei,j,k =

N2q∑

l=1

(hj,t

[x(er

(l)k,t+1 − 1

)+Rf

])1−γvt+1

(d

(l)k,t+1, x

(l)k,t+1

)× ωl

with ωl the lth element of vector ω and where r(l)k,t+1 and d

(l)k,t+1 are given by[

r(l)k,t+1

d(l)k,t+1

]= µk,t (δi,t) +

[Z

(l)k,1

Z(l)k,2

],

hj,t = 1− h |x− χj | ,

and

x(l)k,t+1 =

x(er

(l)k,t+1 − 1 +Rf

)x(er

(l)k,t+1 − 1

)+Rf

.

The values vt+1

(d

(l)k,t+1, x

(l)k,t+1

)are obtained with a bilinear interpolation with the set of values

δi,t+1, χj , vt+1 (δi,t+1, χj)

for i = 1 to Nd and j = 1 to Nx available from the previous time step.

C Expected stock return

This appendix derives the expressions required to compute the expected value and variance of the holdingperiod return. For this purpose, using the excess return given by equation (5) , we write the log stock returnas:

lnSt+1

St=

(rf + α− 1

2σ2εr + βdt

)∆t+ εr,t+1

√∆t+Qt+1

((αJ −

1

2σ2J

)∆t+ εJ,t+1

√∆t

)where St+1 is the risky asset level at t + 1 and rf = ln (Rf ) /∆t is the annual continuously compoundedrisk-free rate. Using E [ex] = eµx+ 1

2σ2x for x ∼ N

(µx, σ

2x

), the expected stock price conditional on on k = 0

or 1 isEt [St+1 | k] = Ste

(rf+α+kαJ+βdt)∆t

obtaining an expected value of

Et [St+1] = St

1∑k=0

qke(rf+α+kαJ+βdt)∆t.

18

Similarly, the expected value of the squared stock price is

Et[S2t+1

]= S2

t

1∑k=0

qke2(rf+α+βdt+kαJ+ 1

2 (σ2εr+kσ2J))∆t.

Using the identity V ar [x] = E[x2]− E [x]

2, the variance of the holding period return can be computed

with the above formula as:

V art

[St+1

St− 1

]=Et[S2t+1

]S2t

− Et [St+1]2

S2t

.

19

Table 1: Summary statistics

Annualized Annual Annualized Annualizedrisk-free dividend index excessreturn yield return return

mean 0.0482 0.0287 0.1072 0.0590standard deviation 0.0305 0.0102 0.5351 0.5370skewness 0.8244 0.3174 -0.5329 -0.5431kurtosis 4.4971 2.4748 4.9336 4.9035

This table reports summary statistics on the monthly time series data used in this study. “Annualized risk-free

return”is (Rf,t − 1)× 12, the annualized yield to maturity of a 30 day Treasury Bill from CRSP; “Annual dividend

yield” is Dt/Mt, where Dt is the sum of the dividends for the 12 months prior to t, and Mt is the market value of

the securities in the CRSP value weighted index formed with stocks from the NYSE, AMEX and NASDAQ markets;

“Annualized index return”is (Rf,t − 1 +Rt)× 12, the annualized monthly holding period return of the CRSP value

weighted index; “Annualized excess return”is Rt×12, the annualized CRSP value weighted index return in deviation

of the risk-free rate

20

Table 2: Parameter estimates

log dividend yield log excess returnsno jumps jumps

κ 0.1281 α 0.0603 0.1495(0.0706) (0.0235) (0.0266)

θ -0.1838 β 0.0304 0.0263(0.8102) (0.0192) (0.0181)

σεd 0.5149 σεr 0.1562 0.1169(0.0083) (0.0031) (0.0081)

— — αJ — -0.4049— — — (0.1542)

— — σJ — 0.1909— — — (0.0184)

— — q — 0.7764— — — (0.0749)

loglik 315.3 — 1086.9 1113.9

ρrd -0.9028

This table reports the maximum likelihood parameter estimates for the time series model presented in Section 3.

Standard errors are reported in parenthesis. “no jumps”are the estimated parameters for the risky asset log excess

return without jumps; “jumps”are the parameters for the jump case; “loglik” is the log-likelihood value.

21

Table 3: Initial risky-asset allocation with monthly rebalancing

d0 = −1 d0 = 0 d0 = 1γ = 5 γ = 8 γ = 10 γ = 5 γ = 8 γ = 10 γ = 5 γ = 8 γ = 10

Average allocation with event riskT = 1 0.3209 0.2135 0.1743 0.4889 0.3276 0.2681 0.6531 0.4385 0.3591T = 12 0.3075 0.2010 0.1632 0.4992 0.3264 0.2646 0.6819 0.4467 0.3632T = 60 0.3645 0.2419 0.1971 0.5753 0.3823 0.3123 0.7744 0.5163 0.4221T = 120 0.4072 0.2749 0.2244 0.6253 0.4191 0.3437 0.8274 0.5577 0.4574

No-trade region size with event riskT = 1 0.2121 0.1313 0.1111 0.2020 0.1212 0.1010 0.1818 0.1212 0.1010T = 12 0.0909 0.0606 0.0505 0.0808 0.0606 0.0505 0.0808 0.0606 0.0505T = 60 0.0909 0.0707 0.0606 0.0909 0.0606 0.0505 0.0808 0.0606 0.0505T = 120 0.0909 0.0606 0.0606 0.0808 0.0606 0.0505 0.0909 0.0707 0.0505

Hedging demand with event riskT = 12 -0.0135 -0.0125 -0.0111 0.0104 -0.0012 -0.0035 0.0289 0.0082 0.0042T = 60 0.0436 0.0283 0.0228 0.0864 0.0547 0.0443 0.1213 0.0778 0.0631T = 120 0.0863 0.0613 0.0501 0.1364 0.0915 0.0756 0.1744 0.1191 0.0983

Average allocation without event riskT = 1 0.2949 0.1955 0.1594 0.4959 0.3327 0.2722 0.6969 0.4698 0.3849T = 12 0.2841 0.1854 0.1481 0.5227 0.3391 0.2748 0.7644 0.4983 0.4033T = 60 0.3621 0.2417 0.1963 0.6351 0.4237 0.3480 0.9089 0.6104 0.4977T = 120 0.4181 0.2868 0.2348 0.7024 0.4784 0.3915 0.9893 0.6717 0.5515

No-trade region size without event riskT = 1 0.2121 0.1313 0.1111 0.2020 0.1313 0.1111 0.2121 0.1414 0.1111T = 12 0.1010 0.0707 0.0606 0.1010 0.0707 0.0606 0.1010 0.0707 0.0505T = 60 0.1111 0.0808 0.0606 0.1111 0.0707 0.0707 0.1010 0.0707 0.0606T = 120 0.1111 0.0808 0.0606 0.1111 0.0707 0.0606 0.0202 0.0808 0.0606

Hedging demand without event riskT = 12 -0.0108 -0.0101 -0.0114 0.0269 0.0064 0.0026 0.0675 0.0285 0.0183T = 60 0.0673 0.0462 0.0368 0.1392 0.0910 0.0758 0.2120 0.1406 0.1127T = 120 0.1232 0.0913 0.0753 0.2066 0.1458 0.1193 0.2924 0.2020 0.1665

Optimal asset allocation computed with the MLE parameters estimates and ∆t = 1/12, Nq = 10, Nd = 20, Nx = 100,

Rf = e0.03×∆t.

22

Table 4: Initial risky-asset allocation with quarterly rebalancing

d0 = −1 d0 = 0 d0 = 1γ = 5 γ = 8 γ = 10 γ = 5 γ = 8 γ = 10 γ = 5 γ = 8 γ = 10

Average allocation with event riskT = 1 0.2641 0.1687 0.1360 0.4210 0.2694 0.2172 0.5663 0.3629 0.2928T = 4 0.2672 0.1718 0.1377 0.4272 0.2739 0.2205 0.5713 0.3690 0.2983T = 20 0.3088 0.2015 0.1643 0.4827 0.3140 0.2550 0.6379 0.4172 0.3388T = 40 0.3456 0.2289 0.1857 0.5234 0.3442 0.2802 0.6828 0.4501 0.3669

No-trade region size with event riskT = 1 0.0606 0.0505 0.0303 0.0606 0.0404 0.0404 0.0505 0.0303 0.0303T = 4 0.0505 0.0404 0.0303 0.0404 0.0303 0.0303 0.0404 0.0303 0.0303T = 20 0.0505 0.0404 0.0303 0.0505 0.0303 0.0303 0.0505 0.0404 0.0303T = 40 0.0505 0.0404 0.0303 0.0404 0.0303 0.0303 0.0505 0.0303 0.0303

Hedging demand with event riskT = 4 0.0031 0.0030 0.0018 0.0062 0.0045 0.0033 0.0050 0.0061 0.0056T = 20 0.0447 0.0328 0.0283 0.0617 0.0446 0.0378 0.0716 0.0543 0.0461T = 40 0.0815 0.0601 0.0498 0.1025 0.0748 0.0630 0.1165 0.0872 0.0742

Average allocation without event riskT = 1 0.2627 0.1679 0.1353 0.4975 0.3185 0.2568 0.7324 0.4692 0.3783T = 4 0.2746 0.1764 0.1416 0.5222 0.3349 0.2701 0.7723 0.4939 0.3979T = 20 0.3568 0.2357 0.1913 0.6483 0.4266 0.3480 0.9370 0.6205 0.5051T = 40 0.4349 0.2953 0.2419 0.7429 0.5034 0.4120 1.0000 0.7143 0.5837

No-trade region size without event riskT = 1 0.0707 0.0505 0.0404 0.0808 0.0505 0.0505 0.0808 0.0505 0.0404T = 4 0.0707 0.0505 0.0404 0.0606 0.0505 0.0404 0.0606 0.0404 0.0404T = 20 0.0707 0.0505 0.0404 0.0707 0.0404 0.0404 0.0606 0.0404 0.0404T = 40 0.0707 0.0505 0.0404 0.0707 0.0505 0.0505 0.0000 0.0505 0.0404

Hedging demand without event riskT = 4 0.0119 0.0085 0.0063 0.0247 0.0164 0.0133 0.0399 0.0247 0.0195T = 20 0.0941 0.0678 0.0560 0.1508 0.1081 0.0913 0.2046 0.1514 0.1267T = 40 0.1722 0.1274 0.1066 0.2454 0.1849 0.1552 0.2676 0.2451 0.2054

Optimal asset allocation computed with the MLE parameters estimates and ∆t = 1/4, Nq = 10, Nd = 20, Nx = 100,

Rf = e0.03×∆t.

23

Table 5: Annualized certainty equivalent returns with monthly rebalancing

d0 = −1 d0 = 0 d0 = 1γ = 5 γ = 8 γ = 10 γ = 5 γ = 8 γ = 10 γ = 5 γ = 8 γ = 10

T = 12jump, cost 0.0357 0.0338 0.0331 0.0459 0.0401 0.0382 0.0613 0.0498 0.0459no-jump, cost 0.0357 0.0337 0.0331 0.0458 0.0401 0.0381 0.0609 0.0495 0.0457no-jump, no-cost 0.0355 0.0336 0.0330 0.0455 0.0399 0.0380 0.0606 0.0493 0.0455

diff1 0.0001 0.0001 0.0000 0.0001 0.0000 0.0000 0.0004 0.0003 0.0002diff2 0.0003 0.0002 0.0001 0.0003 0.0002 0.0002 0.0007 0.0005 0.0004


diff1 0.0001 0.0001 0.0001 0.0003 0.0003 0.0002 0.0007 0.0008 0.0006diff2 0.0002 0.0002 0.0001 0.0004 0.0004 0.0003 0.0008 0.0010 0.0008


diff1 0.0002 0.0002 0.0002 0.0004 0.0006 0.0005 0.0005 0.0012 0.0010diff2 0.0003 0.0003 0.0003 0.0005 0.0007 0.0006 0.0005 0.0014 0.0012

“jump,cost”are annualized certainty equivalent returns (CEQ) for an investor considering event risk and transaction

costs in his investment decisions; “no-jump, cost” are CEQ for an investor ignoring event risk but allowing for

transaction costs in his investment decisions; “no-jump, no-cost”are CEQ for an investor not considering jumps nor

costs in his decisions; “diff1” are “jump, cost”-“no-jump, cost”CEQs; “diff2” are “jump, cost”-“no-jump, no-cost”

CEQs; The certainty equivalent are computed by Monte-Carlo simulation using the investment policies computed for

each case and 1,000,000 sample paths. ∆t = 1/12, Nq = 10, Nd = 20, Nx = 100, Rf = e0.03×∆t.

24

Table 6: Annualized certainty equivalent returns with quarterly rebalancing

d0 = −1 d0 = 0 d0 = 1γ = 5 γ = 8 γ = 10 γ = 5 γ = 8 γ = 10 γ = 5 γ = 8 γ = 10


diff1 0.0001 0.0000 0.0000 0.0008 0.0005 0.0004 0.0038 0.0022 0.0017diff2 0.0002 0.0001 0.0001 0.0011 0.0006 0.0005 0.0043 0.0025 0.0020


diff1 0.0007 0.0005 0.0004 0.0026 0.0018 0.0015 0.0059 0.0051 0.0041diff2 0.0008 0.0005 0.0004 0.0028 0.0020 0.0016 0.0060 0.0054 0.0043


diff1 0.0019 0.0016 0.0013 0.0041 0.0038 0.0031 0.0057 0.0079 0.0064diff2 0.0020 0.0017 0.0014 0.0042 0.0041 0.0033 0.0058 0.0083 0.0067

“jump, cost”are annualized certainty equivalent returns (CEQ) for an investor considering event risk and transaction

costs in his investment decisions; “no-jump, cost” are CEQ for an investor ignoring event risk but allowing for

transaction costs in his investment decisions; “no-jump, no-cost”are CEQ for an investor not considering jumps nor

costs in his decisions; “diff1” are “jump, cost”-“no-jump, cost”CEQs; “diff2” are “jump, cost”-“no-jump, no-cost”

CEQs; The certainty equivalent are computed by Monte-Carlo simulation using the investment policies computed for

each case and 1,000,000 sample paths. ∆t = 1/12, Nq = 10, Nd = 20, Nx = 100, Rf = e0.03×∆t.

25

Figure 1: Risky asset allocation sensitivity to changes in parameters for excess return withoutpredictability

0 0.5 10

0.2

0.4

0.6

0.8

1

Ris

ky a

sset

allo

catio

n α = 0.20

α = 0.15α = 0.10

0 0.5 10

0.2

0.4

0.6

0.8

1σε r

= 0.05

σε r = 0.12

σε r = 0.20

0 0.5 10

0.2

0.4

0.6

0.8

1

Ris

ky a

sset

allo

catio

n ∆ t = 1/50

∆ t = 1/12

0 0.5 10

0.2

0.4

0.6

0.8

1q = 0.9q = 0.8q = 0.7

0 0.5 10

0.2

0.4

0.6

0.8

1

Inherited risky asset allocation

Ris

ky a

sset

allo

catio

n αJ tilde = 0.2

αJ tilde = 0.4

αJ tilde = 0.6

0 0.5 10

0.2

0.4

0.6

0.8

1


σJ = 0.05

σJ = 0.25

σJ = 0.50

26

Figure 2: Expected utility as a function of the risky asset allocation for different values of α andσεr

1 0 1 20.2

0.19

0.18

0.17

0.16

0.15

0.14

0.13

0.12

Risky asset allocation

Exp

ecte

d ut

ility

α = 0.20

α = 0.15α = 0.05

1 0 1 20.2

0.19

0.18

0.17

0.16

0.15

0.14

0.13

0.12

Risky asset allocation

Exp

ecte

d ut

ility

σε r = 0.05

σε r = 0.12

σε r = 0.30

27

Figure 3: Risky asset allocations for i.i.d. excess returns with and without event risk

0 0.5 10

0.2

0.4

0.6

0.8

1h = 0.001,γ = 5

Ris

ky a

sset

allo

catio

n with event riskwithout event risk

0 0.5 10

0.2

0.4

0.6

0.8

1h = 0.005,γ = 5

Ris

ky a

sset

allo

catio


0 0.5 10

0.2

0.4

0.6

0.8

1h = 0.001,γ = 8

Ris

ky a

sset

allo

catio


0 0.5 10

0.2

0.4

0.6

0.8

1h = 0.005,γ = 8

Ris

ky a

sset

allo

catio


0 0.5 10

0.2

0.4

0.6

0.8

1h = 0.001,γ = 10

Ris

ky a

sset

allo

catio

n


with event riskwithout event risk

0 0.5 10

0.2

0.4

0.6

0.8

1h = 0.005,γ = 10

Ris

ky a

sset

allo

catio

n


with event riskwithout event risk

28

Figure 4: Upper and lower bounds off the no-trade region for time t allocations and δt = 1.0

0 5 10 15 200.6

0.7

0.8

0.9

1R

isky

ass

et a

lloca

tion

With event risk

Notrade zone lower bound for δt = 1.0

Notrade region upper bound for δt = 1.0

0 5 10 15 200.6

0.7

0.8

0.9

1

Time periods

Ris

ky a

sset

allo

catio

n

Without event risk

Notrade zone lower bound for δt = 1.0

Notrade region upper bound for δt = 1.0

29

dynamic asset allocation with event risk, … annual meetings...risk. we consider a portfolio...

Documents