On the Measurement of Hedging Effectiveness for Long-Term Investment Guarantees

Abstract

Although the finance literature has devoted a lot of research into the development of advanced models for improving the pricing and hedging performance, there has been much less emphasis on approaches to measure dynamic hedging effectiveness. This article discusses a statistical framework based on regression analysis to measure the effectiveness of dynamic hedges for long-term investment guarantees. The importance of taking model risk into account is emphasized. The difficulties in reducing hedging risk to an appropriately low level lead us to propose a new perspective on hedging, and recognize it as a tool to modify the risk–reward relationship of the unhedged position.

Full text

Citation: Augustyniak, Maciej;
Alexandru Badescu, and Mathieu
Boudreault. 2023. On the
Measurement of Hedging
Effectiveness for Long-Term
Investment Guarantees. Journal of
Risk and Financial Management 16:
112. https://doi.org/10.3390/
jrfm16020112
Academic Editors: Svetlozar (Zari)
Rachev and ¸Stefan Cristian
Gherghina
Received: 22 November 2022
Revised: 21 January 2023
Accepted: 6 February 2023
Published: 10 February 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Journal of
Risk and Financial
Management
Article
On the Measurement of Hedging Effectiveness for Long-Term
Investment Guarantees
Maciej Augustyniak 1,2,*, Alexandru Badescu 3and Mathieu Boudreault 2,4
1
Département de Mathématiques et de Statistique, Université de Montréal, P.O. Box 6128, Station Centre-Ville,
Montreal, QC H3C 3J7, Canada
2Quantact Actuarial and Financial Mathematics Laboratory, Centre de Recherches Mathématiques,
Université de Montréal, P.O. Box 6128, Station Centre-Ville, Montreal, QC H3C 3J7, Canada
3Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW,
Calgary, AB T2N 1N4, Canada
4Département de Mathématiques, Université du Québec à Montréal, P.O. Box 8888, Station Centre-Ville,
Montreal, QC H3C 3P8, Canada
*Correspondence: maciej.augustyniak@umontreal.ca; Tel.: +1-514-343-6111 (ext. 1696)
Abstract:
Although the finance literature has devoted a lot of research into the development of
advanced models for improving the pricing and hedging performance, there has been much less
emphasis on approaches to measure dynamic hedging effectiveness. This article discusses a statistical
framework based on regression analysis to measure the effectiveness of dynamic hedges for long-
term investment guarantees. The importance of taking model risk into account is emphasized.
The difficulties in reducing hedging risk to an appropriately low level lead us to propose a new
perspective on hedging, and recognize it as a tool to modify the risk–reward relationship of the
unhedged position.
Keywords:
hedging effectiveness; dynamic hedging; model risk; investment guarantee; variable
annuity
1. Introduction
Risk management practices in the financial industry were put to a test during the
financial crisis of the late 2000s. Many corporations did not measure their risk exposures
appropriately and numerous institutional failures occurred (e.g., Lehman Brothers). Some
financial risks, such as market risk, have a systematic component that cannot be diversified.
To avoid having a large exposure to these types of risks, the corporation can establish a
hedging strategy by trading in financial derivatives. To correctly evaluate the residual risk
that remains, it must assess the effectiveness of this strategy for mitigating the underlying
risk exposure. Suppose that an institution has sold a derivative and established a hedging
strategy to manage its risk. Assume that the value of this derivative and of the hedge
portfolio at some time in the future is
X
and
Y
, respectively. The hedged loss, or equivalently
the hedging error, of the institution then corresponds to
X−Y
(a positive value indicates a
loss while a negative value indicates a profit). The standard approach to measure hedging
effectiveness is to calculate a statistic on the distribution of X−Y.
While many studies have proposed to improve the pricing and hedging performance
of asset return models by incorporating stochastic volatility, jumps and multiple factors,
the measurement of hedging effectiveness has received considerably less attention. For
example, Bakshi et al. (1997) develop models admitting stochastic volatility, stochastic
interest rates, and random jumps and test empirically the features which improve dynamic
hedging performance of plain vanilla options. To measure hedging effectiveness, they
calculate the average absolute deviation from zero of hedging errors based on dynamic delta
and delta-vega strategies. These hedging errors are calculated based on a single rebalancing
J. Risk Financial Manag. 2023,16, 112. https://doi.org/10.3390/jrfm16020112 https://www.mdpi.com/journal/jrfm

J. Risk Financial Manag. 2023,16, 112 2 of 18
of the hedge (usually daily or weekly) and do not reflect the cumulative hedging error from
the sale of the option until maturity. Similar approaches are applied in many other articles
(e.g., Alexander and Kaeck 2012;Alexander et al. 2009,2012;Alexander and Nogueira 2007;
Bakshi et al. 2000;Branger et al. 2012;Dumas et al. 1998;François and Stentoft 2021;Hull
and White 2017;Kaeck 2013;Kim and Kim 2004;Nandi 1998).1
The studies mentioned in the previous paragraph have two weaknesses when it comes
to measuring hedging effectiveness. First, they ignore the relationship between
X
and
Y
and do not examine its strength. For instance, why is the standard deviation of
X−Y
high?
Is it due to a systematic deviation from the ideal hedging relationship
Y=X
, or is it the
consequence of a high residual standard error around an average relationship of
Y=X
? In
this work, we propose a simple statistical framework based on regression analysis to more
thoroughly examine the effectiveness of a dynamic hedging strategy. Statistics based on
the distribution of
X−Y
are good indicators of relative hedging performance and allow
us to discriminate between competing hedging strategies. However, they are not good
indicators of absolute hedging performance as they ignore the relationship between
X
and
Y
. Regression analysis allows us to assess hedging effectiveness in absolute terms and
understand the sources of hedging ineffectiveness. To our knowledge, the use of regression
analysis to measure the effectiveness of dynamic hedges has not been investigated in the
finance or actuarial science literature. On the other hand, the accounting literature (e.g.,
Charnes et al. 2003;Finnerty and Grant 2002;Hailer and Rump 2005) has investigated
the use of regression tools to measure the strength of a hedging relationship, but only in
the context of very simple hedges which are not dynamic and not model dependent (e.g.,
hedging a spot index with a futures contract).2
The second weakness of empirical studies measuring hedging effectiveness in the
finance literature is that they examine hedging errors calculated based on a single rebalanc-
ing of the hedge (usually daily or weekly) and do not reflect the cumulative hedging error
from the sale of the option until maturity. The rationale for following such an approach
is that traders assess their profits and gains on a daily or weekly basis and do not always
keep their positions open until maturity. Therefore, it is acceptable to measure hedging
effectiveness based on intermediate, rather than cumulative, hedging errors. However,
intermediate hedging errors are influenced by the time to maturity and the moneyness of
the option. For example, the hedge may be less dependent on the chosen model far from
maturity and when the option is deep out-the-money. Consequently, when we calculate
statistics based on intermediate hedging errors, we are aggregating observations that are
not identically distributed. To alleviate this problem, most of the studies display statistics
across different moneyness ranges, but not all of them separate their results according to
time to maturity. However, empirical studies in the finance literature aim to discriminate
among competing hedging strategies, and their main objective is, thus, the evaluation of rel-
ative hedging effectiveness. For this purpose, it is not problematic to consider intermediate
hedging errors. Nevertheless, these errors only provide partial information with respect to
the capacity of the hedge to replicate the desired payoff exactly, until maturity. Therefore,
absolute hedging effectiveness is best assessed by examining the cumulative hedging error
until maturity. We remark that recent studies examining the hedging performance of
discrete-time econometric models have followed this approach (see, e.g., Augustyniak and
Badescu 2021;Augustyniak et al. 2017,2021;Badescu et al. 2014;Lassance and Vrins 2018).
The cumulative performance of the hedge is especially important in the context of variable
annuities, which are investment products sold by life insurance companies that offer insurance
features and benefit from tax deferment (see Hardy 2003, for an overview, and Feng et al. 2022,
for a recent survey of the literature). This is in strong contrast to maturities of derivatives
considered in the finance literature which rarely are above one year. Moreover, insurers are not
traders and cannot close their positions by transferring them to a third party. In other words,
investment guarantees are illiquid securities. Accordingly, all studies on dynamic hedging
effectiveness of long-term investment guarantees in the actuarial literature have considered

J. Risk Financial Manag. 2023,16, 112 3 of 18
the cumulative hedging error until maturity (e.g., Augustyniak and Boudreault 2012,2017;
Coleman et al. 2006,2007;Hardy 2003;Kling et al. 2011).
This article discusses a statistical framework based on both, regression analysis and
the standard practice of calculating risk measures, to measure the effectiveness of dynamic
hedges on absolute as well as relative terms. This framework is presented with a case study
examining the effectiveness of the Black–Scholes delta hedging strategy under return path
scenarios generated with a regime-switching GARCH model (RS-GARCH) in a variable
annuity setting. We consider this hedging strategy for simplicity, but also because the
majority of insurers in Canada manage equity risk in variable annuities using dynamic
delta hedging strategies.
3
Although the focus here is on the measurement of hedging
effectiveness, rather than model risk, this risk is an integral part of the analysis as dynamic
hedging strategies depend on model assumptions. The performance of the hedge depends
on how well these assumptions represent reality. We consider an extended definition of
model risk and include parameter risk as a specific type of model uncertainty, see Cont
(2006, Remark 4.1) for a discussion. Model uncertainty refers to the combined effects of
model and parameter risks. The purpose is not to distinguish between these two types
of risks, but to examine the impact of particular types of model uncertainty on hedging
effectiveness. In our case study, the variable annuity product hedged is a Guaranteed
Minimum Maturity Benefit (GMMB) which ensures that the policyholder’s initial invested
capital is protected against a market downturn over a set period of time.
This article is structured as follows. Section 2describes the GMMB variable annu-
ity product, presents our modeling assumptions and defines the variables that must be
considered when evaluating hedging effectiveness. Section 3illustrates how regression tech-
niques can be used to evaluate hedging performance and discusses our results. Concluding
remarks are provided in Section 4.
2. Modeling Framework
Before describing the statistical framework for measuring the effectiveness of dynamic
hedges, we review some concepts in relation to the GMMB, introduce our assumptions
and models, and explain how to calculate the insurer’s unhedged and cumulative hedged
losses at maturity.
2.1. GMMB Variable Annuity Product
Suppose that an insured invests an amount
A0>
0 in a GMMB variable annuity
product that must be held to maturity
T
. The performance of the GMMB is assumed to
track an investment portfolio from time
t=
0 to
T
, denoted by
{St}0≤t≤T
. Without loss of
generality, let
S0=A0
. As compensation for the guarantee offered, the insurer deducts a
constant fee in proportion to the account value of the insured at a continuous annual rate
of δ. Therefore, the account value of the insured, {At}0≤t≤T, satisfies the relationship:
At=Ste−δt, 0 ≤t≤T.
At maturity
T
, the insured is entitled to
max(AT
,
G)
, where
G
denotes the amount of the
guarantee. If
AT<G
, the guarantee matures in-the-money and the insurer is responsible
for the shortfall between the guarantee and the account value, i.e., its liability is the payoff
of a put option:
max(
0,
G−AT)
. The value of this put option at any time
t
under the
Black–Scholes model, denoted by Pt(At,G,δ), is obtained from the following formula:
Pt(At,G,δ) = Ge−r(T−t)N(−d2)−Ate−δ(T−t)N(−d1),
d1=log(At/G) + (r−δ+σ2/2)(T−t)
σ√T−t,
d2=d1−σ√T−t,

J. Risk Financial Manag. 2023,16, 112 4 of 18
where
r
is the constant risk-free rate,
σ
is the constant annual volatility of the investment
portfolio returns and
N(·)
denotes the standard normal cumulative distribution function.
Moreover, the fair value of fees that will be collected by the insurer between times
t
and
T
is given by
At(1−e−δ(T−t)).
To see why, we can interpret the charge
δ
as a dividend rate. The fair value of dividends to
be received between times
t
and
T
is the difference between the fund value at time
t
,
At
,
and the prepaid forward price for a claim paying ATat time T,Ate−δ(T−t).
The net liability of the insurer at time
t
, i.e., the net obligation of the insurer towards
the policyholder, is
Lt=value of put option −fair value of fees that will be collected
=Pt(At,G,δ)−At(1−e−δ(T−t)).
A fair value for δcan be determined by setting the net liability at inception of the contract
to zero:
L0=P0(A0,G,δ)−A0(1−e−δT) = 0. (1)
Solving Equation
(1)
is not a difficult problem since the net liability is strictly decreasing
in δ:∂
∂δ hP0(A0,G,δ)−A0(1−e−δT)i=−TA0e−δTN(d1)<0.
Consequently, the solution is unique and easily computed numerically.
We can rewrite Equation (1) as
A0=S0e−δT+P0(A0,G,δ)
=S0e−δT+P0(S0,GeδT, 0)e−δT,
where
P0(S0
,
GeδT
, 0
)
is the price at
t=
0 of a put option on the investment portfolio with
maturity
T
and strike
GeδT
. This illustrates that an investment of
A0
in a GMMB product is
equivalent to buying
e−δT
shares of the investment portfolio and
e−δT
put options on that
index with strike price GeδT.
2.2. Market Dynamics
To realistically model return dynamics on the investment portfolio
{St}
, we use a
RS-GARCH model (see, e.g., Augustyniak 2014b;Augustyniak et al. 2018;Bauwens et al.
2010;Francq et al. 2001). We denote the log-return from time
t−
1 to
t
by
yt
, where
yt=log St−log St−1
. The RS-GARCH dynamics of
yt
, for
t=
1,
. . .
,
T
, are defined by the
following equations:
yt=µιt+σt(ι1:t)ηt, (2)
σ2
t(ι1:t) = ωιt+αe2
t−1(ιt−1) + βσ2
t−1(ι1:t−1), (3)
et−1(ιt−1) = yt−1−µιt−1, (4)
where
ηt
,
t=
1,
. . .
,
T
, are independent and identically distributed normal innovations with
zero mean and unit variance. The return dynamics depend on an unobserved discrete-time
process,
{ιt}
, which is an ergodic time-homogeneous Markov chain that can take integer
values from 1 to
M
. The
M×M
transition matrix of this Markov chain is defined by the
transition probabilities
{pij =Pr(ιt=j|ιt−1=i)}M
i,j=1
. The processes
{ιt}
and
{ηt}
are assumed independent. The notations
y1:t
and
ι1:t
are used to represent the vectors
(y1
,
. . .
,
yt)
and
(ι1
,
. . .
,
ιt)
, respectively. The symbols
µιt
and
σt(ι1:t)
denote measurable
functions with respect to a
σ
-field generated by the random variable
ιt
and the random
vector
(y1:t−1
,
ι1:t)
, respectively. Therefore,
µιt
symbolizes the conditional mean of the
return
yt
,
E(yt|y1:t−1
,
ι1:t) = E(yt|ιt)
, and
σ2
t(ι1:t)
represents its conditional variance

J. Risk Financial Manag. 2023,16, 112 5 of 18
Var(yt|y1:t−1
,
ι1:t)
. The vector
θ= ({µi
,
ωi}M
i=1
,
{pij }M
i,j=1
,
α
,
β)
denotes the parameters
of the model. To ensure positivity of the variance, the following constraints are required:
ωi>
0,
i=
1,
. . .
,
M
,
α≥
0 and
β≥
0. Since
∑M
j=1pij =
1 for
i=
1,
. . .
,
M
,
θ
contains
(2M+M(M−1) + 2) free parameters.
In addition to modeling volatility persistence with GARCH dynamics, this model
generates jumps in returns and in volatility through the switching mean in the return
equation and the switching intercept in the GARCH equation, respectively.
4
Therefore, it
contains the essential ingredients of a good financial model as the presence of stochastic
volatility, jumps in returns and jumps in volatility is widely documented in the finance
literature (e.g., Broadie et al. 2007;Eraker et al. 2003). Moreover, previous studies have
shown that the GARCH class of models has the ability to reproduce reasonably well the
cross-section of option prices (e.g., Babao˘glu et al. 2018;Badescu et al. 2017;Bégin et al.
2020;Christoffersen et al. 2006,2008,2010;Christoffersen and Jacobs 2004.)
The parameters of our RS-GARCH model are set based on the study of Augustyniak
(2014a), where the process is estimated by maximum likelihood with
M=
2 regimes on
daily and weekly percentage log-returns from the S&P 500 index. The use of two return
modeling frequencies allows us to evaluate the consistency of results obtained with daily
and weekly models. Table 1provides the parameter estimates for the daily and weekly
RS-GARCH models.
Table 1. Parameter estimates for the daily and weekly RS-GARCH models.
µ1µ2ω1ω2α β p11 p22
Daily RS-GARCH
0.081 −1.63 0.0058 0.544 0.042 0.936 0.980 0.339
(0.010) (0.20) (0.0013) (0.087) (0.006) (0.006) (0.004) (0.083)
Weekly RS-GARCH
0.339 −2.80 0.0431 2.527 0.041 0.905 0.948 0.316
(0.064) (0.54) (0.0185) (0.518) (0.018) (0.024) (0.022) (0.105)
Notes: Parameter estimates are based on the study of Augustyniak (2014a). Note that they are scaled for percentage
log-returns. Asymptotic standard errors are provided in parentheses below the estimates.
2.3. Hedging Strategy
To obtain an ideal hedge, the objective function that needs to be hedged is the net
liability of the insurer. We remark that Hardy (2003) suggests to hedge the GMMB by
replicating a long position in the underlying put position. However, the GMMB is not
a standard put option as no premium is paid at inception of the contract. Since there is
uncertainty in both the payoff and the premium, both components need to be hedged to
completely eliminate risk. The uncertainty with respect to the fees is even greater when the
policyholder is allowed to surrender their policy before maturity.
We assume that the insurer uses the Black–Scholes model to price the GMMB (i.e., to
determine the annual fee assumption
δ
) and to delta hedge its risk. A delta hedge under the
Black–Scholes framework can be established by holding a position of
∆t
in the underlying
investment portfolio5at time t, where
∆t=∂Lt
∂St
=∂Lt
∂At·∂At
∂St
= [−e−δ(T−t)N(−d1)−(1−e−δ(T−t))] ·e−δt
=−e−δTN(−d1)−(e−δt−e−δT)≤ −e−δTN(−d1).
This involves selling (
e−δt−e−δT
) more shares of the investment portfolio than if only the
put position was hedged as the delta of the put position is
−e−δTN(−d1)
. The intuition
behind this result is the following: If the value of the investment portfolio falls, the fair
value of fees will be reduced (since fees are charged in proportion to the account value) and

J. Risk Financial Manag. 2023,16, 112 6 of 18
the additional short position in the investment portfolio will provide a hedge against this
decrease in premium. Since the hedging strategy is not derived from the market model,
we are able to analyze the robustness of the Black–Scholes delta hedge under RS-GARCH
dynamics. The following assumptions are used:
A0=100, G=100, T=10, r=3%, σ=16.9% and δ=1.12%,
where
σ
is the volatility parameter used in the Black–Scholes model to determine the fair
value of fees and to calculate the hedge position
∆t
in the investment portfolio. The value
of
δ=
1.12% satisfies Equation
(1)
with these assumptions. The volatility assumption
σ
corresponds to the unconditional annualized volatility for the daily RS-GARCH model
when parameter risk is taken into account (see Section 2.4).6
2.4. Projection of the Insurer’s Loss
There are three variables that must be considered when evaluating hedging effectiveness:
X=Net unhedged loss at maturity,
Y=Cumulative mark-to-market gain on the hedge,
X−Y=Net hedged loss at maturity.
When the insurer does not use a hedging strategy, their net loss at maturity is
X
. When he
employs a hedging strategy, their net loss is
X−Y
. The losses are net because they take
into account the fee income received by the insurer. To assess hedging effectiveness, we are
interested in the relationship between
X
and
Y
and in the dispersion of
X−Y
around zero.
Consequently, we must project many realizations of these variables by simulating return
path scenarios.
First, we generated 100,000 scenarios of daily returns over
T=
10 years with the
Black–Scholes model. Log-returns under the Black–Scholes model are independent and
identically distributed normal variables. We used an annualized mean return of 7.2% and
an annualized volatility of
σ=
16.9%. This simulation allows us to assess the effectiveness
of the Black–Scholes delta hedge under an ideal situation where the hedging strategy is
derived from the market model. In this context, hedging ineffectiveness arises solely from
discrete rebalancing of the hedge portfolio as there is no model uncertainty. This type
of hedging ineffectiveness is referred to as the discretization error of the hedge. Results
obtained for the return projection under the Black–Scholes model serve as a benchmark
and allow us to properly evaluate the impact of model uncertainty on the Black–Scholes
delta hedging strategy.
Second, we generated 100,000 scenarios of daily and weekly returns over
T=
10 years
with the two RS-GARCH models presented in Table 1. To evaluate the impact of parameter
risk, we generated one set of scenarios with maximum likelihood estimates and another set
of scenarios by drawing parameter vectors from the asymptotic normal distribution of the
maximum likelihood estimator.
For each of the return path scenarios, the values of the investment portfolio,
{St}0≤t≤T
,
and the account value,
{At}0≤t≤T
, are easily calculated. The net unhedged loss at matu-
rity is
X=payoff to the insured −accumulated value of fees
=max(0, G−AT)−
T/h−1
∑
i=0
Aih (1−e−δh)er(T−ih),
where h=1/252 for daily scenarios and h=1/52 for weekly scenarios.

J. Risk Financial Manag. 2023,16, 112 7 of 18
To calculate the net hedged loss at maturity, the cumulative mark-to-market gain on
the hedge must be subtracted from the net unhedged loss. The mark-to-market gain at time
t+hof the delta hedge established at time tis
∆t·(St+h−Sterh),
where
h
represents the rebalancing frequency of the hedge portfolio. The cumulative
mark-to-market gain on the hedge corresponds to the accumulated values of these gains
to maturity:
Y=
T/h−1
∑
i=0
∆ih ·(S(i+1)h−Sih erh )·er(T−(i+1)h).
3. Statistical Framework for Measuring the Effectiveness of Dynamic Hedges
This section explains how regression techniques can be used to investigate hedging
effectiveness. The analysis is conducted for scenarios projected on a daily frequency and
the hedge portfolio is assumed to be rebalanced daily.
3.1. Relationship between Y and X
Figure 1illustrates the regression relationship between the cumulative mark-to-market
gain on the hedge,
Y
, (denoted by hedge performance on the vertical axis) and the net
unhedged loss at maturity,
X
, for 100,000 scenarios projected under the Black–Scholes
model, the RS-GARCH model and the RS-GARCH model with parameter risk. On each
graph, we show (1) the fitted linear regression line of the form “
y=a+bx
”, (2) the residual
standard error (σe), and (3) Pearson’s and Spearman’s correlation coefficients (ρand ρrank,
respectively). Spearman’s correlation coefficient measures the relationship between the
ranks of variables. These values are calculated based on all observations and based on only
the part of the distribution for the net unhedged loss (
X
) contained between the 50th and
95th percentiles (between red lines). The reason for considering this area of the distribution
is the following. When a put option is deep in-the-money or deep out-of-the money, its
delta is very close to
−
1 or 0, respectively, regardless of the model for the underlying. In
these situations, hedging the put option entails less model uncertainty as the delta under
the true model is closer to the Black–Scholes delta. Hence, the Black–Scholes delta hedge is
expected to be more effective in the lower half and the upper part of the distribution for
the net unhedged loss.
For scenarios projected with the Black–Scholes model, the correlation coefficients
between the hedge performance (
Y
) and the net unhedged loss (
X
) are close to one for all
observations and between the red lines. The regression relationship corresponds almost
exactly to the equation
Y=X
, which was to be expected as the hedging strategy is
constructed from the market model.
For RS-GARCH scenarios, the correlation coefficients calculated on all observations
misrepresent the strength of the relationship between the hedge performance and the net
unhedged loss as they markedly decrease when we consider the area between the red
lines. With parameter risk, Spearman’s correlation coefficient is approximately 0.85 which
is a sizable decrease relative to the value of 0.99 observed for the Black–Scholes scenarios.
Moreover, the slope of the regression line is below one which indicates that the hedge
portfolio falls short in offsetting the insurer’s loss. This is one advantage of using regression
analysis to investigate hedging effectiveness as it tells us what proportion of the risk is
not replicated on average. With parameter risk, over 2% of the risk is not offset while
in between red lines this value increases to 3%. The intercept in the regression is also
informative as it can detect a structural loss for the insurer due to hedging. This can in
turn suggest that the GMMB product is underpriced. For example, with parameter risk,
the intercept term is negative indicating a structural loss. Finally, the residual standard
error is magnified by a factor of almost six with respect to the Black–Scholes scenarios
and parameter risk accounts for approximately 25% of that increase. Therefore, model

J. Risk Financial Manag. 2023,16, 112 8 of 18
uncertainty much more significantly impacts
σe
than the average hedging relationship,
Y=X.
Figure 1 – Relationship between the hedge performance and the net unhedged loss based on
100,000 daily scenarios projected with the Black-Scholes model (top panel), the RS-GARCH
model (bottom-left panel) and the RS-GARCH model with parameter risk (bottom-right panel).
For RS-GARCH scenarios, the correlation coefficients calculated on all observations misrep-
resent the strength of the relationship between the hedge performance and the net unhedged loss
12
Figure 1.
Relationship between the hedge performance and the net unhedged loss based on
100,000 daily scenarios projected with the Black–Scholes model (
top panel
), the RS-GARCH model
(bottom-left panel) and the RS-GARCH model with parameter risk (bottom-right panel).
The signs of hedging ineffectiveness observed under model uncertainty can be im-
portant because by hedging the net liability, the insurer protects himself against losses
at the cost of limiting their upside potential. In fact, hedging can be viewed as a tool to
modify the risk–reward relationship of the product, while it is clear that hedging reduces
the overall risk, the resulting risk–reward relationship may be less attractive to the insurer
if the strategy employed has inefficiencies. Therefore, an effective hedging strategy should

J. Risk Financial Manag. 2023,16, 112 9 of 18
not only reduce the risk of the insurer’s position, but also offer a way to improve the
risk–reward tradeoff for the product. For example, suppose that instead of hedging the net
liability, the insurer only hedges its liability, i.e., the put option. Hence, the fee income is
not hedged and the Black–Scholes delta at time
t
is
−e−δTN(−d1)
. Figure 2illustrates the
hedging relationship for this situation under the RS-GARCH model with parameter risk.
We observe that in the first half of the distribution for the net unhedged loss, i.e., in 50%
of the scenarios, the insurer is able to take advantage of a sizable upside because the fee
income is not hedged. The tradeoff for this benefit is a reduced hedging effectiveness in the
upper half of the distribution. On average, the hedging strategy falls short in offsetting 17%
of the risk in between red lines and the negative intercept indicates a structural loss due to
hedging. In the right tail of the net unhedged loss, most of the scatter points fall below the
blue line,
Y=X
, implying that the net hedged loss of the insurer is positive. This example
demonstrates that the effectiveness of a hedge should also be evaluated with respect to the
risk–reward relationship it offers because a perfect hedge is infeasible in a practical context.
Figure 2 – Relationship between the hedge performance and the net unhedged loss based on
100,000 daily scenarios projected the RS-GARCH model with parameter risk. Only the under-
lying put option is delta hedged.
3.2 Relationship between X−Yand X
Having studied the relationship between Yand X, we now examine the dependence between
the net hedged loss, X−Y, and the net unhedged loss, X. These two variables inevitably exhibit
a form of dependence because, as mentioned previously, hedging effectiveness tends to be higher
in the lower half and the upper part of the distribution for X. However, they should ideally be
uncorrelated since what is desired is a hedging strategy whose effectiveness is uniform under a
wide variety of scenarios. This is simply because hedging must protect the insurer against an
unknown return path scenario which is to occur in the real-world.
Figure 3illustrates the relationship between the net hedged loss and the net unhedged loss for
100,000 scenarios projected under the Black-Scholes model, the RS-GARCH model and the RS-
GARCH model with parameter risk. On each graph, we show (1) Pearson’s correlation coefficient
between Xand X−Y(ρ), (2) the standard deviation of X−Y(σ), and (3) the standard deviation of
X−Yin between red lines (σred). As before, the area in between red lines spans the 50th to 95th
percentiles of the distribution for X.
14
Figure 2.
Relationship between the hedge performance and the net unhedged loss based on
100,000 daily scenarios projected the RS-GARCH model with parameter risk. Only the underly-
ing put option is delta hedged.
3.2. Relationship between X −Y and X
Having studied the relationship between
Y
and
X
, we now examine the dependence
between the net hedged loss,
X−Y
, and the net unhedged loss,
X
. These two variables
inevitably exhibit a form of dependence because, as mentioned previously, hedging effec-
tiveness tends to be higher in the lower half and the upper part of the distribution for
X
.
However, they should ideally be uncorrelated since what is desired is a hedging strategy
whose effectiveness is uniform under a wide variety of scenarios. This is simply because
hedging must protect the insurer against an unknown return path scenario which is to
occur in the real-world.
Figure 3illustrates the relationship between the net hedged loss and the net unhedged
loss for 100,000 scenarios projected under the Black–Scholes model, the RS-GARCH model
and the RS-GARCH model with parameter risk. On each graph, we show (1) Pearson’s
correlation coefficient between
X
and
X−Y
(
ρ
), (2) the standard deviation of
X−Y
(
σ
),
and (3) the standard deviation of
X−Y
in between red lines (
σred
). As before, the area in
between red lines spans the 50th to 95th percentiles of the distribution for X.

J. Risk Financial Manag. 2023,16, 112 10 of 18
Figure 3 – Relationship between the net hedged loss and the net unhedged loss based on 100,000
daily scenarios projected with the Black-Scholes model (top panel), the RS-GARCH model
(bottom-left panel) and the RS-GARCH model with parameter risk (bottom-right panel).
As expected, hedging errors are less dispersed in the tails of the distribution for the net un-
hedged loss and the scatter plots are organized in a diamond. Figure 3clearly illustrates the impact
15
Figure 3.
Relationship between the net hedged loss and the net unhedged loss based on 100,000 daily
scenarios projected with the Black–Scholes model (
top panel
), the RS-GARCH model (
bottom-left
panel) and the RS-GARCH model with parameter risk (bottom-right panel).
As expected, hedging errors are less dispersed in the tails of the distribution for the net
unhedged loss and the scatter plots are organized in a diamond. Figure 3clearly illustrates
the impact of model and parameter risks. Model risk has a considerable influence on
hedging effectiveness as the dispersion of scatter points under RS-GARCH scenarios is
much greater than that observed under Black–Scholes scenarios. The impact of parameter
risk is less than model risk, but it is still significant. In particular, parameter risk consider-
ably widens the right tail of the distribution for the net unhedged loss, which implies that
extreme investment losses are more likely.

J. Risk Financial Manag. 2023,16, 112 11 of 18
Furthermore, the correlation coefficient between the net hedged loss and the net un-
hedged loss based on Black–Scholes scenarios is close to zero. This was to be expected since
scenarios generating the largest hedging errors are generally those associated with a high
volatility. As there is no dependence between returns and volatility in the Black–Scholes
model, there should be no linear relationship between the net hedged loss and the net
unhedged loss. In contrast, the RS-GARCH model entails a leverage effect, i.e., a negative
relationship between returns and future volatility, since it simultaneously generates nega-
tive returns and positive jumps in volatility when it enters the crisis regime (regime two).
Consequently, we expect a positive correlation between the net hedged loss and the net
unhedged loss which is what is observed.
3.3. Distributions of X and X −Y
The left panel in Figure 4illustrates the empirical density of the net unhedged loss
(
X
) based on 100,000 scenarios projected under the RS-GARCH model with and without
parameter risk. The right panel shows the Q-Q plot of standardized quantiles between
these two distributions. This figure confirms that parameter risk has a non-negligible effect
on the tails of the net unhedged loss, and it should therefore not be ignored.
Figure 4 – Left panel: Empirical density of the net unhedged loss (X) based on 100,000 daily
scenarios projected under the RS-GARCH model with and without parameter risk. Right panel:
Q-Q plot of standardized quantiles between these two distributions.
of error can be reduced by rebalancing the hedge portfolio more frequently, say many times a day.
However, in practice, frequent rebalancing of the hedge portfolio entails large transaction costs
and sudden price jumps can occur in between rebalancing times. Hedging at very high frequencies
also implies additional model uncertainty (further deviations from the normal distribution) as the
distribution of high frequency returns is more volatile and has heavier tails than the one associated
with daily or weekly log-returns. This discussion emphasizes that the insurer should not aim
for a perfect hedge, but for one that offers a desired risk–reward tradeoff. In practice, hedging
significantly reduces the overall risk of loss, but a residual heavy-tailed hedging risk remains. For
a hedging strategy to be effective, the insurer must be rewarded for taking this risk.
17
Figure 4.
(
Left panel
): Empirical density of the net unhedged loss (
X
) based on 100,000 daily
scenarios projected under the RS-GARCH model with and without parameter risk. (
Right panel
):
Q-Q plot of standardized quantiles between these two distributions.
Figure 5repeats the previous exercise for the net hedged loss (
X−Y
). This is analogous
to examining the distribution of residuals from the perfect hedging relationship
Y=X
. As
observed previously, the dispersion of hedging errors with model risk is much greater than
without it (top-left panel), but the Q-Q plot (bottom-left panel) allows us to quantify the
magnitude of this discrepancy. Similar to the net unhedged loss, parameter risk widens the
tails of the net hedged loss (see bottom-right panel), but its impact on the right tail is even
more pronounced here.

J. Risk Financial Manag. 2023,16, 112 12 of 18
Figure 5 – Top-left panel: Empirical density of the net hedged loss (X−Y) based on 100,000
daily scenarios projected under the Black-Scholes model, the RS-GARCH model and the RS-
GARCH model with parameter risk. Other panels: Q-Q plots of standardized quantiles comparing
distributions for the net hedged loss.
18
Figure 5.
(
Top-left panel
): Empirical density of the net hedged loss (
X−Y
) based on 100,000 daily
scenarios projected under the Black–Scholes model, the RS-GARCH model and the RS-GARCH model
with parameter risk. (
Other panels
): Q-Q plots of standardized quantiles comparing distributions
for the net hedged loss.
However, the most interesting result is perhaps derived from the Q-Q plot in the top-
right panel. This plot compares standardized quantiles between the distribution of hedging
errors under the Black–Scholes model and those of the standard normal distribution,
denoted by
N(
0, 1
)
. It clearly shows that hedging risk is a heavy-tailed risk, even in the
absence of model uncertainty. In other words, the insurer is exposed to possibly large losses,
five to 10 standard deviations away from the mean loss, when he employs the ideal hedging
strategy in the context of a very simple data generating process (Black–Scholes model) that

J. Risk Financial Manag. 2023,16, 112 13 of 18
does not even allow for jumps or stochastic volatility. These extreme losses are due to the
discretization error of the hedge. In theory, this type of error can be reduced by rebalancing
the hedge portfolio more frequently, say many times a day. However, in practice, frequent
rebalancing of the hedge portfolio entails large transaction costs and sudden price jumps
can occur in between rebalancing times. Hedging at very high frequencies also implies
additional model uncertainty (further deviations from the normal distribution) as the
distribution of high frequency returns is more volatile and has heavier tails than the one
associated with daily or weekly log-returns. This discussion emphasizes that the insurer
should not aim for a perfect hedge, but for one that offers a desired risk–reward tradeoff. In
practice, hedging significantly reduces the overall risk of loss, but a residual heavy-tailed
hedging risk remains. For a hedging strategy to be effective, the insurer must be rewarded
for taking this risk.
3.4. Risk Measures
The conventional approach to measuring hedging effectiveness and discriminating
among competing strategies is based on measures of dispersion and risk measures calcu-
lated on the distribution for the net unhedged loss (
X−Y
). For example, we can consider
the mean, standard deviation (StDev), average absolute deviation from zero (AAD),
7
con-
ditional tail expectation (CTE), and value at risk (VaR). The StDev and the AAD measure
the dispersion of hedging errors around the mean and zero, respectively. The CTE and
VaR measure the heaviness of the right tail of the distribution for
X−Y
. These statis-
tics can offer a good way to summarize important elements of the distribution for
X−Y
and discriminate among competing hedging strategies. However, information about the
relationship between Xand Yis lost.
This section uses risk measures to determine how the rebalancing frequency of the
hedge portfolio impacts hedging effectiveness. We also examine the consistency of results
obtained with daily and weekly RS-GARCH models estimated in Table 1. Table 2shows the
results for the analysis of hedge rebalancing frequency based on 100,000 scenarios projected
under the Black–Scholes model and the RS-GARCH model with parameter risk. First, we
observe that increasing this frequency lowers risk measures and thus improves hedging
effectiveness of the Black–Scholes delta hedge with or without model uncertainty. However,
this improvement is not as significant in the context of model uncertainty. For example,
from a monthly to a daily rebalancing frequency, the 95% CTE is reduced by a factor of
five for the Black–Scholes model and by a factor of only 1.6 with model uncertainty. This
implies that the reduction in the discretization error is greater than the increase in model
error (at frequencies studied here), but model error accounts for a greater proportion of the
total hedging error at higher rebalancing frequencies. For instance, daily rebalancing offers
little improvement over weekly rebalancing and may even be less effective once transaction
costs are taken into account.
Table 2.
Risk measures for the insurer’s net loss at maturity based on the Black–Scholes model and
the RS-GARCH model with parameter risk.
Rebalancing Mean StDev AAD 95% CTE 99% VaR
B-S RSG B-S RSG B-S RSG B-S RSG B-S RSG
Unhedged −16.3 −14.7 13.0 17.6 19.4 20.9 27.4 42.9 37.2 53.7
Annual 1.5 2.4 5.5 5.9 4.4 4.7 14.5 17.4 16.8 20.3
Monthly 0.1 0.9 1.5 2.8 1.1 2.0 3.4 8.6 4.0 10.2
Weekly 0.0 0.6 0.7 2.2 0.5 1.6 1.7 6.8 2.0 8.0
Daily 0.0 0.2 0.3 1.9 0.2 1.3 0.7 5.2 0.9 6.3
Notes: B-S and RSG denote the Black–Scholes and RS-GARCH models, respectively. StDev: standard deviation.
AAD: average absolute deviation from zero. CTE: conditional tail expectation. VaR: value at risk.
Although daily rebalancing generates the lowest risk measures, the mean loss of the
insurer is approximately zero. This implies that he is not rewarded for assuming hedging

J. Risk Financial Manag. 2023,16, 112 14 of 18
risk. In contrast, when he does not hedge, their mean loss is negative, implying a profit,
but he is exposed to very large losses. Since market risk is not diversifiable, the insurer may
not tolerate such an exposure and prefer to hedge. However, when a classic risk–neutral
pricing approach is used to price the product, hedging risk is not considered and the
hedging strategy may result in an unattractive risk–reward relationship. To address this
issue, insurers generally perform pricing with conservative assumptions and include a
margin for profit. However, this methodology does not consider hedging risk directly. It
seems preferable to incorporate this risk into pricing and construct a hedging strategy that
offers a desirable risk–reward tradeoff.
Table 3examines the consistency of results obtained with daily and weekly RS-GARCH
models estimated in Table 1. 100,000 return path scenarios are projected for each one of
these models taking parameter risk into account. First, we observe that the 95% CTE and the
99% VaR of the net unhedged loss differ markedly, suggesting that long-term investment
risk is difficult to measure accurately. This is in part due to the difficulty in estimating
the average mean return, i.e., the drift dynamics of the underlying asset. For example,
the daily RS-GARCH model generates an average annualized mean return of 7.1%. In
contrast, we obtain an average annualized mean return of 4.6% for the weekly model. This
explains, in part, why higher risk measures are computed for the net unhedged loss based
on the weekly model. Moreover, the estimation period has an even greater influence on the
drift assumption. Since future drift dynamics are difficult to infer from historical data, it
is preferable to use a risk management strategy whose dependence on this assumption is
weak. Fortunately, hedging generally provides such an outcome.
Table 3.
Risk measures for the insurer’s net loss at maturity based on daily and weekly RS-GARCH
models with parameter risk.
Rebalancing Mean StDev B 95% CTE 99% VaR
D W D W D W D W D W
Unhedged −14.7 −7.8 17.6 23.4 20.9 21.2 42.9 62.0 53.7 71.9
Annual 2.4 1.8 5.9 6.6 4.7 4.8 17.4 19.4 20.3 23.1
Monthly 0.9 0.7 2.8 3.5 2.0 2.4 8.6 10.7 10.2 12.7
Weekly 0.6 0.1 2.2 2.9 1.6 2.0 6.8 8.3 8.0 9.8
Notes: Daily and weekly rebalancing frequencies are abbreviated D and W, , respectively. StDev: standard
deviation. AAD: average absolute deviation from zero. CTE: conditional tail expectation. VaR: value at risk.
We observe that risk measures at different rebalancing frequencies generally fall in a
comparable range for the daily and weekly models. However, they are a little higher for
the weekly model. This is mainly due to increased parameter risk at a weekly frequency
caused by a smaller historical sample. Although the choice of data frequency does influence
results, it does not do so enough to alter conclusions.
4. Concluding Remarks
We have discussed a statistical framework based on regression analysis to measure the
effectiveness of dynamic hedging strategies and evaluated how model uncertainties can
impact this effectiveness in a variable annuity setting. We found that model uncertainty
has a limited impact on the average proportion of the risk that is not replicated with the
Black–Scholes delta hedge .However, this model uncertainty considerably increases the
residual standard error of the hedging relationship, Y=X.
The difficulties in reducing hedging risk to an appropriately low level lead us to
suggest a new perspective on the role of hedging. The traditionally accepted purpose of
hedging is risk elimination, or maximum risk reduction. However, as a perfect hedge is
unattainable and the capacity of risk reduction is limited in practice, hedging should be
reinterpreted as a tool to modify the risk–reward relationship of the unhedged position.
In this framework, the role of hedging is to reduce risk to an acceptable level (not nec-
essarily to a maximum level), and offer the highest reward for bearing it. For instance,

J. Risk Financial Manag. 2023,16, 112 15 of 18
the risk–reward relationship of the insurer may improve if he decides not to hedge the
fees in certain situations or hedge only a portion of the liability. Moreover, as long-term
investment guarantees are complicated to hedge due to their exposure to a large variety of
risks, the valuation of these guarantees could be more accurate if the effectiveness of the
hedging strategy is taken into account in the pricing process. Although insurers can use
conservative assumptions and demand large risk premiums for assuming the underlying
risks when pricing investment guarantees, we find that hedging effectiveness should also
be an important factor to consider for valuation purposes. Such an approach deviates from
the traditional risk-neutral valuation framework by incorporating hedging risk directly
into the pricing methodology, as opposed to indirectly with conservative assumptions.
This new perspective on hedging constitutes an interesting avenue for future research in
actuarial applications.
Author Contributions:
Conceptualization, methodology, formal analysis, and investigation: M.A.
and M.B. Software and writing: M.A. Project administration, review and editing, and validation: A.B.
All authors have read and agreed to the published version of the manuscript.
Funding:
Financial support from the Natural Sciences and Engineering Research Council of Canada
(NSERC) is gratefully acknowledged by all authors.
Data Availability Statement: Not applicable.
Acknowledgments:
The authors thank the anonymous referees for their helpful comments which
improved the paper.
Conflicts of Interest: The authors declare no conflict of interest.
Notes
1
Branger et al. (2012) and Kaeck (2013) improved on the methodology to measure hedging effectiveness by comparing the
distributions of anticipated and realized hedging errors. Anticipated hedging errors are those obtained when the model used to
determine the hedge coincides with the data generating process. Realized hedging errors correspond to those that are observed
when the hedging strategy is implemented on empirical data or on simulated data based on a data generating process that is not
consistent with the model used for hedging. If the distributions of anticipated and realized hedging errors differ significantly, this
suggests some form of model misspecification in the hedge.
2
The interest of the accounting literature for measuring hedging effectiveness is due to Statements of Financial Accounting
Standards No. 133, Accounting for Derivative Instruments and Hedging Activities, commonly known as FAS 133. FAS 133, which
came into effect in the U.S. at the turn of the century, requires that all derivatives entered into by a corporation must be marked to
market and changes in their values reported in the income statement. This accounting treatment can create earnings volatility
when derivatives are used for risk management purposes as the timing of gains and losses on the hedged items may not be
matched with those on the corresponding hedging derivatives. To remedy this problem, FAS 133 allows corporations to match
the timing of these gains and losses, provided that they demonstrate and document that the hedge is highly effective in offsetting
changes in fair value for the risks being hedged. FAS 133 does not endorse any specific testing methodology, but recommends the
use of statistical tests. A good overview of approaches proposed in the accounting literature for measuring hedging effectiveness
is given by Charnes et al. (2003), Finnerty and Grant (2002), and Hailer and Rump (2005).
3
The “Report of the Task Force on Segregated Fund Liability and Capital Methodologies” produced by the Canadian Institute of
Actuaries states that “typically delta and rho are hedged while vega and gamma are only monitored” (see Canadian Institute of
Actuaries 2010). A survey performed by Towers Watson (2013) also indicates that U.S. insurers mainly hedge delta and rho risks
from their variable annuity portfolio. Although we did not include interest rate risk in our modeling framework for simplicity,
our approach to measure hedging effectiveness can be applied to more general modeling settings and hedging strategies.
4
The switching mean does not necessarily generate a jump in returns, but there is a high likelihood of observing a significant
negative return when the model enters the crisis regime. When jump-diffusion models are discretized, the occurrence of a
negative jump is equivalent to observing a large negative return in a given time interval.
5
In our study, we suppose for simplicity that one can buy and sell shares of the investment portfolio tracked by the variable annuity.
In practice, variable annuities invest in mutual funds which are relatively illiquid and cannot be shorted. As a result, insurers
must construct their hedging strategies using financial instruments that are both liquid and highly correlated with the mutual
fund. This exposes the hedging strategy to basis risk: the risk that price fluctuations in the underlying asset cannot be perfectly
replicated by trading in available instruments. Recent studies showed that basis risk significantly affects hedging effectiveness,
notably in the context of variable annuities (see, e.g., Ankirchner et al. 2014;Bauer 2020;Li et al. 2022;Trottier et al. 2018).

J. Risk Financial Manag. 2023,16, 112 16 of 18
6
Alternative approaches can be considered to determine the volatility parameter in the Black–Scholes model. For example, it
can be set based on forward-looking measures of volatility such as the implied volatility surface extracted from option price
data. Previous empirical studies have showed that the variance risk premium is typically negative (see e.g., Bakshi and Kapadia
2003;Carr and Wu 2006,2009;Clark and Dickson 2019;Israelov and Klein 2016), which implies that the risk–neutral volatility is
expected to be larger than the real-world volatility. Although a higher volatility leads to higher option prices, it is unclear whether
the use of a volatility assumption inferred from derivative prices is more effective in a variable annuity hedging context. For
instance, Augustyniak and Boudreault (2017) compared hedging effectiveness using a historical volatility estimate to a calibration
based on a measure of the VIX and found that “forward-looking measures of volatility may not lead to better volatility inputs
than measures based on historical data.” We refer to Section 5.8 in Augustyniak and Boudreault (2017) for a detailed discussion
on the calibration of the volatility parameter.
7Given nsampled values from a random variable Z, denoted by {zi}n
i=1, the AAD is given by
AAD =1
n
n
∑
i=1|zi|.
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