Electronic copy available at: http://ssrn.com/abstract=1513914

Constant Proportion Portfolio Insurance:

Discrete-time Trading and Gap Risk

Coverage∗

Cathrine JessenDepartment of Finance

Copenhagen Business School

e-mail: cj.fi@cbs.dk

December 2010

Abstract

A practical implementation of constant proportion portfolio insurance(CPPI) strategies must inevitably take market frictions into account. Istudy a CPPI in a setting with trading costs, fees and borrowing restric-tions, and relax the assumption of continuous portfolio rebalancing. Themain goals are to cover issuer’s gap risk and to maximize CPPI perfor-mance according to investor’s preferences over possible multipliers: theproportionality factor that determines the risky exposure of a CPPI. In-vestment objectives are described by the Sortino ratio and alternativelyby an S-shaped utility function known from behavioral finance. Investorswith either objective will choose a lower multiplier than if CPPI perfor-mance is measured by the expected return. Discrete-time trading requiresa portfolio rebalancing rule, which affects both performance and gap risk.Two commonly applied strategies, rebalancing at equidistant time stepsand rebalancing based on fixed market moves, are compared to a new rule,which takes trading costs into account. While the new and the market-based rules deliver similar CPPI performance, the new rebalancing ruleachieves this by fewer trading interventions. Issuer’s gap risk can be cov-ered by a fee charge, by hedging or by an artificial floor. A new approach todetermine the artificial floor is introduced. Even though all three methodsreduce losses from gap events effectively, the artificial floor and hedgingare less costly to the investor.

JEL classification: G11Keywords: CPPI, gap risk, discrete portfolio rebalancing, Sortino ratio, ex-pected utility maximization

∗I thank Rolf Poulsen and Peter Tankov for helpful comments.

Electronic copy available at: http://ssrn.com/abstract=1513914

1 Introduction

Products offering portfolio insurance are popular with both private and in-stitutional investors because it allows investors with a lower risk appetite totake exposures in attractive underlying assets without carrying the full risk. Awidely used portfolio insurance strategy is constant proportion portfolio insur-ance (CPPI), see Black & Perold (1992) and the references therein. A CPPIprovides a capital guarantee by dynamically allocating wealth between two as-sets: a risky asset, which gives the investor an upside potential, and a risk freeasset. The relative portfolio weight of risky assets is determined by a multiplierm > 1. If the portfolio value falls below the floor, defined as the present valueof the capital guarantee, all funds are invested in the risk free asset in order notto jeopardize the guarantee any further.

One strand of existing CPPI-literature considers the strategy under the as-sumption of continuous time trading in order to preserve analytical tractability.In an actual CPPI issuance the strategy must be altered, first of all becausemarket frictions such as trading costs prevents continuous trading. Secondly,when introducing a discrete-time trading rule the ability of the CPPI portfolioto honor the capital guarantee must be examined carefully. These are the twomain problems considered in this paper.

A number of studies – e.g. Black & Perold (1992), Boulier & Kanniganti(1995), Hamidi, Jurczenko & Maillet (2009), Balder, Brandl & Mahayni (2009),Constantinou & Khuman (2009) and Paulot & Lacroze (2009) – analyze theCPPI strategy in a setup with trading costs and discrete-time trading. However,none of these consider the choice of portfolio rebalancing rule explicitly. Yet, therebalancing strategy is important since too rare rebalancing increases issuer’sgap risk and too frequent rebalancing imposes high trading costs. The firstproblem addressed here is the selection between different rebalancing rules.I compare weekly portfolio rebalancing to two customized trading strategieswhich take movements in the underlying variables into account. One strategy issimply to rebalance when the change in the risky asset value exceeds a carefullychosen tolerance level. Next, I introduce in the CPPI context a trading ruleknown from option replication, see Whalley & Wilmott (1997). This can beapplied as a CPPI rebalancing strategy because the CPPI can be replicated bya position in the underlying asset, the risk free asset and a position in short-maturity put options. While both customized trading strategies are found tooutperform weekly rebalancing, the new trading rule has the advantages thatit requires notably fewer trading interventions and it avoids the delicate choiceof tolerance levels.

When managing a CPPI deal, the issuer is exposed to the risk of the portfoliovalue breaking the floor. In this event the issuer incurs a loss, since the capitalguarantee given in the CPPI contract must still be honored. This risk is denotedgap risk. The second problem addressed in this paper is the CPPI issuer’s riskmanagement problem of covering gap risk. Cont & Tankov (2009) study gaprisk for a CPPI strategy in continuous time Levy framework. They provideanalytical expressions for the probability of hitting the floor, the expected loss,and the distribution of losses, and they show that gap risk can be hedged by a

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Electronic copy available at: http://ssrn.com/abstract=1513914

position in short maturity put options. An alternative way to reduce gap riskis to introduce an artificial floor above the true floor, which gives the CPPImanager a buffer to absorb potential losses. A third approach to handle gaprisk, which is typically used in actual CPPI issuances, is for the issuer to chargea fee that covers potential losses. The appropriate size of the fee can be chosenbased on a risk analysis of the issuer’s exposure. The artificial floor approachis compared to coverage by hedging and charging a fee, and while all threemethods reduce losses due to gap events effectively, covering gap risk by thetwo former approaches are less costly.

Capital restrictions are important to include in a CPPI analysis, since banksare often reluctant to provide additional capital for gearing a risky position andif doing so typically charge a spread above the risk free rate. I find that bor-rowing restrictions improve CPPI performance, because trading costs and costsof capital exceed the increase in return obtained from higher risky exposure.

Once the issuing entity has covered its risks and administration costs theCPPI multiplier can be chosen to accommodate the CPPI investor’s risk/returnprofile. I consider two different approaches to measure CPPI performance. Thefirst employs a measure of the risk adjusted return. Since the general featuresof the CPPI return distribution is not properly captured by the commonly usedSharpe ratio, I will instead consider the Sortino ratio advocated by Pedersen &Satchell (2002). The Sortino ratio measures the expected excess return of aninvestment relative to its downside risk. An alternative performance measureis to assume that CPPI investors maximize expected utility over possible mul-tipliers. Findings in behavioral finance (Døskeland & Nordahl (2008)) suggestthat investments in products with a capital guarantee can be justified if agentsare equipped with an S-shaped utility function. I find that the two performancemeasures lead to similar choices of CPPI multiplier, which are lower than themultiplier maximizing expected CPPI return.

The rest of the paper is organized as follows. The next section describes theCPPI investment strategy and incorporates market frictions. Special attentionis paid to gap risk coverage and discrete time portfolio rebalancing. Section 3introduces the model for the underlying asset and describes investor’s prefer-ences. In section 4, numerical experiments are conducted to study the effects ofmarket frictions, especially the consequences of the choice of discrete portfoliorebalancing rule and of gap risk coverage approach. Section 5 sums up.

2 CPPI strategy in a market with frictions

In an actual implementation of a CPPI trading costs, fees and capital re-strictions are unavoidable. This section considers adjustments to the stylized,continuous-time strategy, which are necessary when incorporating market fric-tions.

2.1 Stylized CPPI

A CPPI is an investment strategy that guarantees a fixed amount of capitalG at expiration T . Let I denote the initial investment. The guarantee must

3

satisfy G ≤ Ip(0, T ), where p(s, t) denotes the time s price of a zero-couponbond maturing at time t. The value Gp(t, T ) is referred to as the floor and isthe smallest amount that will guarantee a portfolio value of G at expiry. Forsimplicity, assume G = I and normalize the initial investment to I = 1. Let(Vt)t∈[0,T ] denote the CPPI portfolio value process and (St)t∈[0,T ] the value ofthe underlying risky asset. At initiation t = 0, V0 = I > p(0, T ) if interest ratesare positive. As long as the portfolio value is above the floor, Vt > p(t, T ), thestylized CPPI trading strategy will maintain its exposure, e, to the risky assetequal to

et = mCt := m(Vt − p(t, T )

). (1)

C is called the cushion and m > 1 is the pre-specified, constant multiplier. Ifthe floor is broken at time τ ∈ [0, T ], Vτ ≤ p(τ, T ), the entire portfolio valuemust be invested in zero-coupon bonds in order not to jeopardize the capitalguarantee any further. This position is held until expiry: et = 0 for t ∈ (τ, T ].The stylized CPPI is self financing; any dividends or coupons are assumedreinvested in the CPPI portfolio.

2.2 Capital restrictions and fees

In case mCt > Vt the portfolio value does not cover the funds required to investin the risky asset. The issuer may be unwilling to provide additional liquidityand set a borrowing limit b, which restricts exposure to et = minmCt, bVt.Furthermore, the cost of additional capital is typically a spread δ above the riskfree rate.

The CPPI issuer may charge certain fees for managing the CPPI deal. Anupfront fee fu can be deducted from the initial investment thereby loweringthe highest possible capital guarantee to G < I(1 − fu)p(0, T ). In the follow-ing, a potential upfront fee is assumed to cover administration costs, and notcompensation for any risk of issuing the CPPI.

2.3 Trading costs

Trading costs must be considered in any practical implementation; assume aproportional cost of η per dollar risky asset traded. The magnitude of η reflectsthe liquidity of the underlying asset. When adjusting the CPPI portfolio, thenew exposure is computed according to (1) ex-post trading costs. In case theposition in risky assets is increased, et− < mCt, the portfolio value is

Vt = Vt− − η(mCt − et−) = Vt− − η(m(Vt − p(t, T ))− et−

)⇒

Vt =1

1 +mη

(Vt− +mηp(t, T ) + ηet−

).

If et− > mCt, the expression holds with opposite sign on η.

2.4 Discrete-time rebalancing strategies

The assumption of continuous portfolio rebalancing must be relaxed when in-troducing trading costs, and a rebalancing rule is required. Such rule can be

4

either time-based or based on moves in the underlying asset value. The sim-plest strategy is the time-based where the portfolio is rebalanced daily, weeklyor monthly as done in e.g. Balder et al. (2009) and Cesari & Cremonini (2003).However, CPPI performance can be improved significantly, as will be verified insection 4.2, by employing a trading strategy that takes movements in underlyingvariables into account. Two such strategies are introduced next.

Rebalancing depending on movements in the underlying asset

Black & Perold (1992) and Boulier & Kanniganti (1995) suggest rebalancing theCPPI portfolio whenever the underlying asset has moved a fixed percentage.Others, e.g. Paulot & Lacroze (2009), Hamidi et al. (2009) and MKaouar &Prigent (2007), apply a rebalancing rule specified by the divergence of actualexposure et from target exposure mCt. If ignoring interest accruals, the twoapproaches are in fact equivalent as stated by Black & Perold (1992): a relative

move of α in S is equivalent to a change of − (m−1)α1+mα in et

Ct. This statement is

proved here in order to clarify where approximations are made.Suppose St = (1 + α)Ss, where s < t is the latest rebalancing date prior to

time t, i.e. es = mCs. This implies et = (1 + α)es. The relative change in thecushion is approximately mα, since

Ct = Vt − p(t, T ) = et + (Vt − et)− p(t, T )

= (1 + α)es + (Vs − es)e∫ ts rudu − p(s, T )e

∫ ts rudu

≈ Cs + αes = (1 +mα)Cs, (2)

where (rt)t∈[0,T ] is the risk free interest rate process. Precision of the approx-imation relies on a small interest rate and a small time step t − s such thatbetween portfolio adjustments, changes in the floor, p(t, T ), and the in theholding of risk free assets, Vt−et, are small. If the position in the risk free assetis positive, interest earned by the risk free investment will to some extent cancelthe increase in the floor. The relative movement in et

Ctis then, as claimed

etCt

=(1 + α)es

(1 +mα)Cs=

(1− (m− 1)α

1 +mα

)esCs. (3)

A CPPI rebalancing rule αu, αd gives upper and lower bounds on relativechanges in the underlying asset that do not trigger an adjustment of the CPPIportfolio. The rule given by αu, αd states to adjust the risky exposure et toequal target exposure mCt if

St /∈[(1− αd)Ss, (1 + αu)Ss

].

Bertrand & Prigent (2002) show that the maximum drop in the underlyingasset value the CPPI portfolio value can sustain without hitting the floor is 1

m .Thus, for a given multiplier m the restrictions αd ∈

(0, 1

m

), αu ∈ (0,∞) are

imposed.This rebalancing rule is equivalent to a rule εu, εd that sets upper and

lower bounds on the divergence of actual exposure et from target exposure mCt

et /∈[(1− εd)mCt, (1 + εu)mCt

]. (4)

5

An application of (3) shows εd = (m−1)αu1+mαu

and εu = (m−1)αd1−mαd , and restrictions

on αd and αu translate to εd ∈(0, 1− 1

m

), εu ∈ (0,∞).

The strategy (4) will be referred to as the market-based rebalancing rule.

Rebalancing strategy depending on trading costs

The market-based strategy has the potential weakness that the tolerance levelsεd, εu do not depend on the level of trading cost. Thus, this rule does not allowto balance the cost of deviating from the stylized strategy against the cost oftrading.

This problem also arises when constructing a replicating portfolio for a plainvanilla option in a market setting including trading costs. In fact, the stylizedCPPI strategy can be replicated by a position in the underlying asset, the riskfree bank account and short-maturity put options as done in Cont & Tankov(2009), so an alternative rebalancing rule can be derived from results on hedgingplain vanilla options. Whalley & Wilmott (1997) consider replication of plainvanilla options in a Black Scholes model extended to include transaction costs. Iftrading costs are present, the ∆-hedge portfolio derived in a frictionless marketsetting is shown to be suboptimal to adjust, while the number of shares ∆ lieswithin the following no-trading band:

(∆t − ξt,∆t + ξt) for ξt := η13

(3Γ2

tSte−r∆t

2ω

) 13

, (5)

where Γt = ∂2Vt∂S2

t= ∂∆t

∂Stis the gamma of the derivative with value Vt to be

hedged. ∆t is the maturity of the put option in the replicating portfolio, so in(5) ∆t is set to be a single day. The parameter ω controls the trade-off betweenaversion to costs and to the risk of deviating from the replicating strategy inthe frictionless market. If ω < 1 the aversion to costs dominates. Whalley &Wilmott (1997) argue that if the boundary of the no-trading band is crossed,the minimum number of risky shares necessary to bring the position back tothe edge of the no-trading band should be bought or sold.

The derivation of this portfolio rebalancing rule rests on Black-Scholes as-sumptions. These affect both the expression for the bandwidth ξ and the calcu-lation of the option greeks. The rule (5) will be implemented as a rebalancingrule for the CPPI portfolio, even though the setting in the analysis to comeis not necessarily that of Black-Scholes. Thereby, it cannot be claimed that(5) is an optimal rebalancing strategy for the CPPI portfolio, but a reasonableapproximation.

∆ and Γ can be calculated explicitly for the continuously adjusted CPPIportfolio: ∆ is merely the number of shares held in the stylized version, ∆t =mCtSt

. Note that in general ∆t is different from the number of shares ϕt in adiscretely rebalanced CPPI portfolio since the process (ϕt) is piecewise constantwith jumps only at rebalancing dates. Γ is calculated as

Γt =∂∆t

∂St=St

∂(mCt)∂St

−mCt ∂St∂St

S2t

=m

St

mCtSt− mCt

S2t

=∆t(m− 1)

St.

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The trading strategy (5) is referred to as the bandwidth rebalancing rule, andstates to adjust the number of shares to equal ∆t ± ξt, if ϕt ≷ ∆t ± ξt.

The bandwidth rebalancing rule implies more frequent portfolio adjustmentswhen the cushion is small because Γ of a CPPI is an increasing function of thecushion. If the floor is stochastic (as would be the case with stochastic interestrates) having such cushion dependent rebalancing strategy appears crucial. Inconclusion, bandwidth rebalancing appears more customized than market-basedrebalancing as it depends on both level of trading costs and the cushion.

2.5 Gap risk coverage

The issuer may impose modifications of the CPPI strategy to cover or reducegap risk. Three approaches for covering losses from gap events are studied inthis section.

Covering gap risk by fee charge

As compensation for losses arising from potential gap events, the issuer maysimply charge a fee. Let the fee fmr be a yearly running fee given as a percentageof initial investment and suppose the size depends on the multiplier m. Thevalue of the issuer’s engagement FmT at expiry T is the sum of fee payments atdates t1 < t2 < · · · < tn < T as long as no gap event has occurred plus thevalue of the cushion if a gap event do occur:

FmT =n∑i=1

1ti<τfmr e

∫ Ttirudu + 1τ<TCτe

∫ Tτ rudu, τ = inf

t < T |Ct < 0

.

The upfront fee payment is excluded in this calculation since it is assumed tocover only administration costs. The determination of the fee fmr at initiationof the CPPI deal is based on some risk measure of the issuer’s engagement.Popular risk measures are expected loss given a loss occurs E[FmT |FmT < 0],probability of loss P [FmT < 0], Value at Risk (V aR) and expected shortfall(ES):

V aRα[FmT ] = supl ∈ R |P [FmT < l] < 1− α

:= q(1−α)

ESα[FmT ] =1

1− α

(E[FmT 1FmT <q(1−α)] + q(1−α)(P [FmT ≥ q(1−α)]− α)

).

q(1−α) is the lower (1 − α)’th quantile in the distribution of FmT . The runningfee fmr can be chosen as the smallest fee such that the issuer’s risk managementrequirement is satisfied.

Hedging gap risk

Gap risk can be hedged by a position in gap options (studied in Tankov (2008))or in short maturity put options as suggested by Cont & Tankov (2009). Inpractice this hedging approach requires that such derivatives on the underlyingasset are liquidly traded. Here, put options as hedge instruments are chosen,since these are more commonly available than gap options.

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Consider the hedge position entered at time t, and let ∆t denote the timeto expiry of the put options, which is also the length of the hedge intervals.Assuming the floor has not been broken, a new position in put options expiringat time t + ∆t is entered as the existing hedge portfolio expires. The strikeK must be chosen such that the put options are in the money, K > St+∆t, ifthe floor is broken, Ct+∆t < 0. When ∆t is small,1 it is reasonable to assumethat no CPPI-trading takes place between time t and t + ∆t. If the CPPIportfolio is rebalanced at the same time as new hedge-positions are entered (i.e.at equidistant time steps t + h∆t, h = 1, 2, ..), K can be found by applyingequation (2) as done in Cont & Tankov (2009). However, if the portfolio isrebalanced more rarely, one must allow for et 6= mCt when determining thestrike price:

0 > Ct+∆t = Vt+∆t − p(t+ ∆t, T ) ≈ Vt + (St+∆t − St)ϕt − p(t, T )

⇔ St+∆t <1

ϕt

(p(t, T )− Vt

)+ St.

Investing in ϕt put options with strike K = 1ϕt

(p(t, T ) − Vt

)+ St (almost)

eliminates gap risk in the period [t, t + ∆t]. Hedging costs are deducted fromthe CPPI portfolio value. A small cushion or a large position in risky assets,ϕt, increases the cost of hedging gap risk. Tankov (2008) finds that (next tocontinuous re-hedging) this hedge strategy performs best if the hedge position isunwound immediately after a gap event. Hedging cannot completely eliminategap risk because the calculations above rely on approximations and because theconstruction of the hedge does not take market frictions into account.

Artificial floor

Gap risk can also be reduced by introducing an artificial floor above the truefloor, such that the risky exposure is unwound if the artificial floor is broken.This gives the issuer a buffer to absorb potential losses. The true floor is stillapplied for determining the risky exposure. The artificial floor approach hasan indirect cost; it eliminates the chance of the CPPI recovering in scenarioswhere the artificial floor is broken but the true floor is not.

Paulot & Lacroze (2009) compare the reduction in gap risk from four ar-bitrarily chosen buffer sizes. However, the buffer needed to avoid the averageloss in a gap event can actually be calculated theoretically if the distributionof increments of the underlying asset is known. Assume that St+∆t = Ste

Xt ,where X is some stochastic process with a known distribution. For small in-crements, St+∆t = Ste

Xt ≈ St(1 + Xt). Recall that a relative change Xt inthe underlying asset translates approximately to a change of mXt in the cush-ion: Ct+∆t = Ct(1 + mXt). This means that the cushion turns negative ifXt < −1/m. The buffer placed on the true floor that can absorb the averageloss, given the floor is broken, is

1In practice, 1-day put options – as considered earlier to replicate the CPPI strategy – arenot traded liquidly, so here ∆t must be larger than one day.

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− C(

1 +mE

[X∣∣∣X < − 1

m

])(6)

For the risk management purpose considered here, the issuer should preferablycover losses arising in case of the maximum (or some higher quantile) dropbelow the floor. This may be incorporated by inserting some bound smallerthan − 1

m in the conditional expectation.

3 Model setup

3.1 The underlying risky asset

Consider an arbitrage-free model represented by a filtered probability space(Ω,F ,F, P ), where P denotes the real-world probability measure of marketscenarios. By the assumption of no arbitrage, there exists some pricing measureQ ∼ P . The risk free asset has the dynamics dRt = rtRtdt. In the following,the interest rate process r is assumed to be constant, although this is by nomeans necessary.

The underlying asset is modelled as a stochastic volatility process withjumps in asset value. This process has the geometric Brownian motion as aspecial case but allows for heavier tails in the return distribution. Under thereal-world probability measure P the dynamics of S is given by

dStSt

= µdt+√vtdW

1t + (Zt − 1)dNt (7)

dvt = κ(θ − vt)dt+ σ√vtdW

2t ,

where W 1,W 2 are Brownian motions with correlation ρ and N is a Poissonprocess with intensity λ and independent of W 1,W 2. The relative jump sizelogZt ∼ N (µJ , σ

2J) is assumed normally distributed with mean µJ and volatil-

ity σ2J . For the assessment of gap risk Cont & Tankov (2009) illustrate the

importance of including jumps in asset value, because gap events occurring be-cause of jumps cannot be eliminated even by continuous rebalancing nor doesthere exist a perfect hedge strategy.

3.2 CPPI investor

Now consider the CPPI investor’s problem of choosing an optimal multiplier m.Note that this analysis is not concerned with the investor’s portfolio selectionin the usual sense but is restricted to a search among CPPI portfolios withdifferent multipliers. Two approaches for describing the investment objectivesare considered: by means of a utility function and by a risk/return profile.

Utility function

A commonly employed utility function U describing the preferences of an agentas a function of wealth w is the constant relative risk aversion (CRRA) utility

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function

UCRRA(w) =w1−γ

1− γ, (8)

where γ is the risk aversion parameter. Black & Perold (1992) and El Karoui,Jeanblanc & Lacoste (2005) study optimal portfolio choice problems for CP-PIs and for general portfolios with a capital guarantee using a CRRA utilityfunction under a minimum consumption constraint.

The notion of risk aversion concerns any deviation – positive or negative –from the expected wealth. Work in behavioral finance suggests that standardutility functions such as (8) do not necessarily reflect agents’ investment deci-sions. In their seminal studies, Kahneman & Tversky (1992) find that investors’decisions are highly influenced by their loss aversion and the prospect of termi-nal wealth ending up lower than present wealth. I.e. investors care about theirrelative wealth not the absolute level of wealth. This behavior can be capturedby an S-shaped utility function. In the region of gains, utility is concave inwealth corresponding to a risk averse agent. The assumption of loss aversioncan be captured by a convex utility function in the region of losses. For theCPPI investment, losses are defined relative to the initial investment which isequal to the capital guarantee G. A simple S-shaped utility function is

US(w) =

(w −G)1−γ w ≥ G−λ(G− w)1−γ w < G

. (9)

The parameter γ describes the risk aversion in the region of gains, while λcontrols the risk seeking behavior in the loss region. Figure 1 illustrates theutility functions (8) and (9).

An S-shaped utility function was applied2 by Døskeland & Nordahl (2008)in their analysis of pension investment strategies with and without guarantees.They find that agents with such an S-shaped utility function will indeed preferan investment strategy incorporating a guarantee.

The objective of the CPPI investor is to maximize expected utility of ter-minal wealth over possible multipliers m > 1 given the set of conditions Φm inthe CPPI contract. The issuer sets the fees fu, f

mr , borrowing limit b, spread δ

charged for providing additional capital, rebalancing rule RR ∈ E,M,B (re-balancing at equidistant time steps, market-based or bandwidth rebalancing)and the choice of gap risk coverage GRC ∈ FC,H,AF (f ee charge, hedgingor artificial f loor). Trading costs η reflects the liquidity of the risky asset. Theinvestor’s utility maximization problem is

maxm>1

E0[U(WmT )] given Φm = RR,GRC, fu, fmr , δ, b, η,

where WmT is the wealth at time T from investing in a CPPI with multiplier m,

assuming this investment is held until expiry T .

2Also Dobeli & Vanini (2010), Jessen & Jørgensen (2009) and Maringer (2008) analyzeinvestor’s decisions based on S-shaped utility.

10

80 100 120 140

−0.014

−0.012

−0.010

−0.008

CRRA

wealth

utility

80 100 120 140

−10−5

05

S−shaped

wealth

utility

Figure 1: Left: CRRA utility with γ = 2. Right: S-shaped utility with γ = 0.5,

λ = 2.5.

Risk/return profile

An alternative to the utility maximization approach is to evaluate an invest-ment by its risk-adjusted return, i.e. by measuring expected return relative tosome risk measure. A well-known example is the Sharpe ratio, which measuresexpected excess return relative to its standard deviation. However, standarddeviation as risk measure can be misleading, since it punishes large losses andgains equally. This is avoided by the Sortino ratio, which measures expectedexcess return relative to the square root of the second lower partial moment(also known as semi-standard deviation). Denote by rCPPI the T -year logarith-mic return of a CPPI, and let the excess return be defined relative to a T -yearrisk free investment. Then the Sortino ratio SR is

SRx =E[rCPPI]− rT√

E[(rCPPI)2|rCPPI < x

] . (10)

Pedersen & Satchell (2002) gives the Sortino ratio a theoretical foundation asa performance measure by relating it to the maximum principle. The capitalguarantee of a CPPI ensures rCPPI ≥ 0, so the choice x = 0 implies SR0 =∞.Instead, setting x = rT (referred to as modified Sortino ratio by Pedersen &Satchell (2002)) reflects the natural requirement that the risky CPPI will out-perform the risk free investment with respect to expected return. By using thesemi-standard deviation as risk measure, the Sortino ratio is sensitive to asym-metric return distributions as e.g. the CPPI’s. The Sortino ratio has previouslybeen applied as CPPI performance measure by Constantinou & Khuman (2009)and Cesari & Cremonini (2003).

Like in the utility maximization approach, an investor with this risk/returnprofile seeks to maximize the Sortino ratio of the CPPI investment over possiblemultipliers m > 1.

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4 Numerical experiments

The performance of the CPPI strategy is analyzed by Monte Carlo simulations.The aim is to investigate implications of introducing market frictions into thesetup, and in particular to compare the rebalancing rules and the three ap-proaches for gap risk coverage. Expected values, risk measures, etc. reportedhere are based on 25 000 simulated scenarios.

4.1 Base case

The risky asset dynamics is simulated using the following set of parametersreported by Eraker (2004), who estimates S&P 500 index return data over a3-year period:

µ = 0.066 ρ = −0.586 λ = 0.504 µJ = −0.004 σJ = 0.066v0 = 0.042 θ = 0.042 κ = 4.788 σ = 0.512 r = 0.02

The base case is a CPPI contract with T = 5 years to expiry. An upfrontfee fu = 1% of initial investment is assumed, cost of capital is δ = 1% and aborrowing limit at b = 2 is imposed. Proportional transaction cost for tradingin the underlying asset is set to η = 0.5%. The effect of the parameter choicesδ, b and η determining the market frictions are studied in more detail in section4.4. The portfolio is evaluated once a day, but only rebalanced so that et = mCtaccording to the market-based strategy (4), for which the choices of εu and εdare discussed in section 4.2. No means to cover gap risk are taken yet.

The top panel in figure 2 shows that a pure index investment outperformsthe CPPI with respect to expected return. The CPPI investor indirectly pays acost of insurance: the cost of forfeiting higher returns from a direct investmentin the underlying index. The cost of insurance over the 5-year period rangesfrom 18–22 percentage points. As anticipated the CPPI gives a higher returnthan the risk free investment. The highest CPPI return is accomplished formultipliers m ∈ 4, 5. In contrast, in a frictionless Black-Scholes model theexpected return of a CPPI portfolio can be increased indefinitely by choosinga high enough multiplier as seen in figure 8 (also shown in Cont & Tankov(2009)).

If the investor is risk averse, he will care not only about expected returnsof an investment but also about higher moments of the return distribution.A higher CPPI multiplier implies higher risky exposure, and thereby a highervariance in returns. With the fairly low borrowing restriction imposed here,variance in CPPI returns with m = 9 is still slightly lower than the variance inpure index returns. However, without the borrowing restriction the variance ofCPPI returns (and expected return) is much higher. Furthermore, since CPPIreturns are bounded below the return distribution has a positive skew.

By applying the semi-standard deviation as risk measure, the Sortino ratio(10) is sensitive to the asymmetry in the CPPI return distribution. The bottompanel in figure 2 shows that investors with a risk/return profile described bythe Sortino ratio will prefer a CPPI with multiplier m ∈ 3, 4. Even though

12

2 3 4 5 6 7 8 9

0.10

0.15

0.20

0.25

0.30

0.35

Expected returns as function of multiplier

multiplier

retur

n

CPPIIndexBond

2 3 4 5 6 7 8 9

−2.0

−1.5

−1.0

−0.5

0.00.5

1.0

Sortino ratio as function of multiplier

multiplier

SR

CPPIIndex

Figure 2: Top: Expected returns of underlying stock index, risk free investment and

CPPI with market-based rebalancing as function of the multiplier. Bottom: Sortino

ratio of index and CPPI investments.

such investors favor the downside protection provided by a CPPI, they wouldchoose the uninsured index investment over a CPPI with m > 6.

Figure 3 illustrates the ranking of the index, bond and CPPIs with differentmultipliers when a utility maximization approach is taken. The top panel showsexpected CRRA utility (9) with risk aversion parameter γ = 2. An investor withsuch preferences would choose the CPPI over both the risk free and the pureindex investments but with the lowest possible multiplier (m = 1 if possible).The lower panel in figure 3 shows that an investor with an S-shaped utilityfunction having γ = 0.15 and λ = 3 prefers a CPPI with multiplier m = 3. Infact, such an investor rank the CPPI and pure index investments very similarto the investor selecting investment based on the Sortino ratio in figure 2.Other risk aversion parameters will alter the investor’s ranking of the threeinvestments, however the shape of the CPPI expected utility curve as a functionof multipliers will remain more or less unchanged.

A quantification of the difference in expected utility obtained from twoinvestments can be given in terms of the certainty equivalent, which is thecertain amount x that results in the same expected utility as obtained from arisky investment with outcome X:

U(x) = E[U(X)]

13

2 3 4 5 6 7 8 9

−0.00

94−0

.0092

−0.00

90

Expected CRRA utility as function of multiplier

multiplier

utility

CPPIIndexBond

2 3 4 5 6 7 8 9

7.58.0

8.59.0

9.5

Expected S−shaped utility as function of multiplier

multiplier

utility

CPPIIndexBond

Figure 3: Top: Expected CRRA utility with risk aversion parameter γ = 2 frominvesting in a CPPI, the underlying index and the risk free investment. Bottom: In-vestor’s expected S-shaped utility with γ = 0.15 and λ = 3 from investing in a CPPI,the underlying index and the risk free investment.

With S-shaped utility function the pure index investor should be compensatedby 2.1% of initial investment to be as well off as the CPPI investor with m = 3.The risk free investor should have an additional 4.3% to obtain the same utilityas the CPPI investor.

Since the CPPI investment chosen based on the Sortino ratio and S-shapedpreferences are similar, the former will be applied henceforth. Applying theutility maximization approach with an S-shaped utility would not alter theoverall conclusions.

4.2 Comparing rebalancing strategies

Now consider the two rebalancing strategies, market-based (base case) andbandwidth rebalancing, suggested in section 2.4. These are compared to weeklyrebalancing, where the CPPI portfolio is adjusted once a week irrespectively ofchanges in underlying variables. For all three strategies the risky portfolio isunwound as soon as the floor is observed broken.

An implementation of the market-based strategy (4) requires a choice oftolerance levels for the divergence of exposure, e, from its target, mC. Basedon numerical experiments seeking to maximize expected CPPI return the tol-

14

erance levels εd = 0.04 and εu = 0.5 are applied. For m = 5 this correspondsto rebalancing triggered by upward moves of αu = 1.1% and downward movesof αd = 7.2%. Since εd controlls the buy region where risky exposure is in-creased, the fact that εd < εu implies more frequent portfolio adjustments inrising markets than in falling markets. This is a natural consequence of choos-ing tolerance levels based on expected returns, with no consideration to gaprisk. For given εu, εd, both αu(·) and αd(·) are decreasing in m. Note that thecondition εd < 1− 1

m given in section 2.4 is satisfied for all m.Bandwidth rebalancing (5) is employed with ω = 0.25. This risk versus cost

aversion ω was found experimentally to give the highest expected return amongthe range of values tried. Recall that in this respect the notion of risk refersto deviation from the stylized CPPI strategy. A choice of ω < 1, which reflectsa higher aversion to costs, is therefore not surprising. At initiation of a CPPIwith m = 5, ω = 0.25 corresponds to a no-trading band of ±9.4% around ∆t.This translates to a ±9.4% move in the underlying index, although this numberwill change as the underlying variables change.

2 3 4 5 6 7 8 9

0.13

0.14

0.15

0.16

0.17

0.18

Effect of rebalancing strategy on expected returns

multiplier

retur

n

Market−based (base case)BandwidthWeekly

2 3 4 5 6 7 8 9

−2.0

−1.5

−1.0

−0.5

0.00.5

1.0

Effect of rebalancing strategy on Sortino ratio

multiplier

SR

Market−based (base case)BandwidthWeekly

Figure 4: CPPI investor’s expected return and Sortino ratio for CPPIs with portfolios

adjusted according to market-based, bandwidth and weekly rebalancing rules.

The expected return and Sortino ratio of a CPPI with weekly, market-basedand bandwidth rebalancing are shown in figure 4. Weekly rebalancing cannotcompete with the two customized strategies.3 Judged by expected return the

3Daily rebalancing results in even lower expected returns. Monthly rebalancing is a com-

15

market-based strategy performs slightly better than bandwidth rebalancing forthe low range of multipliers, while the opposite is true for multipliers m > 5.When performance is measured by the Sortino ratio bandwidth rebalancing ispreferred.

CPPI investors with S-shaped preferences would arrive at a ranking of thethree rebalancing strategies for different multipliers almost identical to that ofthe Sortino ratio. In terms of certainty equivalents, an investor in a CPPI em-ploying market-based rebalancing should be compensated by -0.4% of initialinvestment for m = 2 and by 1.6% for m = 9 to be as well off as if employ-ing bandwidth rebalancing. Correspondingly, a CPPI investment with weeklyrebalancing needs 0.8–1.8% compensation to provide the same utility as withbandwidth rebalancing.

With respect to trading interventions, the cushion dependence of bandwidthrebalancing makes this strategy more efficient than market-based rebalancing.The market-based strategy requires 16–250 trading interventions (increasing inm) over the 5-year period, whereas bandwidth rebalancing achieves comparableresults by only 13–33 trades. If trading costs were introduced as a fixed costper trade or a combination of fixed and proportional trading costs, bandwidthrebalancing would have a further advantage.

4.3 Covering gap risk

Figure 5 shows risk measures of the issuer’s engagement when managing a CPPIportfolio adjusted according to the market-based, bandwidth and weekly rebal-ancing strategies. Weekly rebalancing is not only more expensive to the investorbut also to the issuer in terms of higher losses. This result is not surprisingsince weekly rebalancing does not allow for rapid reduction of exposure dur-ing downside market moves. Not surprisingly the probability of loss (bottompanel) is higher for bandwidth rebalancing than for market-based, since thelatter dictates more frequent rebalancing. Yet, the expected shortfall producedby the two strategies are comparable (upper panel). Based on its previouslyobserved efficiency with respect to number of trading interventions, bandwidthrebalancing will be employed henceforth.

The three approaches for reducing issuer’s gap risk suggested in section2.5 are now implemented. First, the issuer may charge a semi-annual runningfee as compensation for potential losses. The fee size is here set such thatthe expected shortfall at the 95% level of the issuer’s engagement is positive:min fmr subject to ES95%[FmT ] > 0. The fee charge is determined based onnumerical experiments and the resulting fees are reported in table 1.4

Alternatively, the issuer can reduce gap risk by introducing the buffer givenin (6) above the true floor. When the underlying asset dynamics is specified asin (7), the distribution function of increments X is not known in closed form.

petitor to the two customized strategies with respect to expected CPPI returns, although atthe cost of large losses to the issuer. These alternatives are therefore disregarded.

4For such an analysis of tail events Monte Carlo simulation is not the best tool, and theresults in table 1 should only be considered a crude approximation of the optimal solution tothe minimization problem.

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2 3 4 5 6 7 8 9

−0.02

5−0

.015

−0.00

5

Effect of rebalancing strategy on exp. shortfall (95% level)

multiplier

% of

inves

tmen

t

Market−based (base case)BandwidthWeekly

2 3 4 5 6 7 8 9

05

1015

2025

3035

Effect of rebalancing strategy on probability of loss

multiplier

%

Market−based (base case)BandwidthWeekly

Figure 5: CPPI issuer’s expected shortfall and probability of loss when the CPPI

portfolio is adjusted according to market-based, bandwidth and weekly rebalancing.

m 2 3 4 5 6 7 8 9

fmr (%) 0 0 0.002 0.003 0.005 0.008 0.01 0.07

Table 1: Semi-annual running fee.

However, for a regular jump diffusion with constant volatility σ the distributionfunction of increments X over at time step ∆t is:

p∆t(x) = e−λ∆t∞∑k=0

(λ∆t)k exp− (x−µ∆t−kµJ )2

2(σ2∆t+kσ2J )

k√

2π(σ2∆t+ kσ2

J

) .

To account for the stochastic volatility in (7), I apply the jump diffusion distri-bution function with a stressed volatility parameter σ = 2θ. Even though theCPPI portfolio is not rebalanced daily, the portfolio is immediately unwoundif the floor is broken, so the time step applied in the is one day ∆t = 1

250 , butthe implementation of (6) uses the cushion observed on the latest rebalancingday. To cover more than the average loss when the floor is broken, I conditionon X < −1.3

m in (6).The third possibility introduced is to hedge gap risk using put options – here

options with two weeks to expiry are used as hedge instruments. Trading costson derivatives ηd are typically higher than on the underlying, and therefore

17

ηd = 2η is assumed.5 Implementation of the hedging strategy requires hedgeinstruments to be priced. Eraker (2004) also estimates the dynamics of theunderlying asset under the pricing measure Q used by the market, and reportsthe following Q-parameters:

κQ = 2.772 θQ = 0.2692 µQJ = −0.020 µQ = r − λµQJ = 0.050.

Figure 6 illustrates expected return and issuer’s expected shortfall of CPPI

2 3 4 5 6 7 8 9

0.14

0.15

0.16

0.17

0.18

Effect of gap risk coverage approach on expected returns

multiplier

retur

n

No coverageCoverage by artificial floorCoverage by hedgingCoverage by fee charge

2 3 4 5 6 7 8 9

−0.01

5−0

.005

0.005

0.015

Effect of gap risk coverage approach on exp. shortfall (95% level)

multiplier

% of

inves

tmen

t

No coverageCoverage by artificial floorCoverage by hedgingCoverage by fee charge

Figure 6: CPPI investor’s expected return and issuer’s expected shortfall when gap

risk is covered by fee charge, hedging and artificial floor.

investments incorporating the three approaches for gap risk reduction. The toppanel shows that investor’s expected return will be lowered only marginally, ifthe issuer chooses to reduce gap risk by hedging or an artificial floor, and onlyfor multipliers m > 6. In a market where derivatives are less liquidly tradedand trading costs are higher, the hedging approach would be more expensive.Covering potential losses by a running fee charge set to cover the expectedshortfall is seen to be both costly and ineffective for high multipliers. Choosing amilder risk management criterion may of course change this conclusion. Anotherproblem in covering gap risk by a fee charge is the cumbersome determinationof the appropriate fee size: the procedure is not automatic and the fee size mustbe re-calculated for a different CPPI issuance.

5This corresponds to a bid/ask spread of ±0.002 on implied volatility for at-the-money-options.

18

The effectiveness of the three approaches for gap risk coverage is studied inthe bottom panel in figure 6.6 Both hedging and the artificial floor approach re-duce the losses to almost zero, except for the highest multiplier included. SinceCPPI investors are found to prefer low multipliers, the problems of coveringgap risk for higher multipliers becomes less relevant. Furthermore, as the im-plementation of the artificial floor conditions on X < −1.3

m in (6), this approachcan be adjusted in the direction of higher expected returns or smaller losses ingap events by employing a factor lower or higher than the 1.3 chosen here.

In the following the bandwidth rebalancing strategy with gap risk coveredby an artificial floor is adapted as a reference case. By choosing this approach,concerns about whether the required short maturity put options for hedginggap risk are indeed available in an actual implementation can be ignored.

4.4 Effects of market frictions

This section explores the direct effects of introducing market frictions such astrading costs and capital restrictions in form of borrowing constraint and costof capital.

Capital restrictions

The effects of capital restrictions imposed by the CPPI issuer on expected returnand Sortino ratio are illustrated in figure 7. More precisely, the cases where anissuer provides no borrowing facility (b = 1), unlimited borrowing (b =∞) andimposes no capital restrictions at all (b =∞, δ = 0) are compared.

Figure 7 shows that the borrowing restrictions have a considerable effecton CPPI performance. While removing the borrowing facility completely doesnot affect expected CPPI return, investors with a risk/return profile describedby the Sortino ratio would actually prefer not to have any additional capitalavailable. The reason is that without additional capital, exposure will be kept ata moderate level even for higher multipliers. There is even a minor improvementin the Sortino ratio from investing in a CPPI with a high multiplier relative toa small, whereas the opposite is true in the reference case.

If the issuer facilitates unlimited additional capital, the CPPI strategy dic-tates high risky exposure, which in some scenarios results high returns. How-ever, when operating with high exposure only a minor drop in the underlyingasset will cause a severe loss in the CPPI portfolio value which may hit thefloor. Therefore the expected CPPI return is lower than in the reference casefor m > 3. If the issuer also removes the cost of capital, δ = 0, there is a minorpositive effect on expected CPPI return, although a poorer performance thanin the reference case is still reported.

A second consequence of the CPPI taking large positions in the risky asset is,that larger amounts are traded when rebalancing the portfolio, thereby causinghigher trading casts. The effect of trading costs is investigated next.

6The expected shortfall, when covering gap risk by a fee charge, is positive for the middlesection of multipliers and is caused by inaccuracies in the numerical approximation.

19

2 3 4 5 6 7 8 9

0.06

0.08

0.10

0.12

0.14

0.16

0.18

Effect of capital restrictions on expected returns

multiplier

retur

n Reference caseNo borrowingUnlimited borrowingNo capital restrictions

2 3 4 5 6 7 8 9

−3−2

−10

1

Effect of capital restrictions on Sortino ratio

multiplier

SR

Reference caseNo borrowingUnlimited borrowingNo capital restrictions

Figure 7: Effects of capital restrictions on expected return and Sortino ratio.

Trading costs and continuous rebalancing

The most common simplification in analyses of the CPPI strategy is to ignoretrading costs and allow for continuous rebalancing of the portfolio. With η = 0the bandwidth trading strategy automatically rebalances the portfolio as oftenas possible (daily in the present setup) which is a natural consequence of fric-tionless trading. The effects of this simplification on expected CPPI return andissuer’s expected shortfall are investigated in figure 8. Two cases are consid-ered: frictionless trading with the borrowing constraint at b = 2 reintroducedand frictionless trading combined with frictionless capital markets; b =∞ andδ = 0.

If trading in the underlying index was costless, the solid red curve in the toppanel of figure 8 shows that expected CPPI return would improve significantly(notice the scale on the ordinate axis). If further assuming frictionless capitalmarkets the CPPI would even outperform the pure index investment, whenmeasured by expected return. The bottom panel shows that daily portfoliorebalancing almost eliminates gap risk when measured by the expected shortfall,since exposure is kept closer to its target level.

20

2 3 4 5 6 7 8 9

0.15

0.25

0.35

0.45

Effect of trading costs on expected returns

multiplier

retur

n Reference caseNo trading costsNo trading costs, no capital restrictionsIndex

2 3 4 5 6 7 8 9

−8e−

04−4

e−04

0e+0

0

Effect of trading costs on exp. shortfall (95% level)

multiplier

% of

inves

tmen

t Reference caseNo trading costsNo trading costs, no capital restrictions

Figure 8: Effects of trading costs on investor’s expected return and issuer’s expected

shortfall.

4.5 Model risk

A popular choice of model setup is that of Black-Scholes. The effects of choosingthis simpler model specification for the underlying asset is studied in figure 9.The volatility parameter of the geometric Brownian motion reported by Eraker(2004), σGBM = 0.202, is employed. To properly compare potential lossesarising in gap events for the two model assumptions, gap risk has not beencovered/reduced in this section. Gap risk can be covered in a Black-Scholessetup by an artificial floor approach as presented in (6) by employing the normaldistribution function for the increments X.

The top panel shows that the CPPI returns for the two assumptions ofunderlying asset dynamics are comparable; the geometric Brownian motionproduces slightly higher returns for lower multipliers m ≤ 4, and lower returnsfor higher multipliers. More importantly, the underlying asset dynamics has acrucial effect on gap risk: with log-normally distributed returns the gap risk isessentially zero even without gap risk coverage as shown in the bottom panelof figure 9, and consequently, a CPPI analysis in a Black-Scholes setup willpossibly underestimate the true risk facing the issuer.

21

2 3 4 5 6 7 8 9

0.14

0.15

0.16

0.17

0.18

Index return distribution’s effect on expected returns

multiplier

retur

n

Jump diffusion with stochastic volatilityGeometric Brownian motion

2 3 4 5 6 7 8 9

−0.01

4−0

.010

−0.00

6−0

.002

Index return distribution’s effect on exp. shortfall (95% level)

multiplier

% of

inves

tmen

t

Jump diffusion with stochastic volatilityGeometric Brownian motion

Figure 9: Effects on expected return and issuer’s expected shortfall when modelling

the underlying asset as a geometric Brownian motion.

5 Summary

I have studied the CPPI strategy in a realistic setup including market frictions.Trading costs, fees and borrowing restrictions were introduced, and most im-portantly the assumption of continuous portfolio rebalancing was relaxed. Insuch a framework the main challenges are the choice of discrete-time tradingstrategy and CPPI issuer’s risk management problem of covering gap risk.

The choice of discrete-time rebalancing rule was shown to play an importantrole for the expected return delivered by the CPPI strategy. I found thatmarket-based and bandwidth rebalancing resulted in similar CPPI performance,although bandwidth rebalancing achieved this with fewer trading interventions.Moreover, the cushion-dependence of the bandwidth rebalancing rule may givethis a further advantage if the floor is stochastic, e.g. in a setup with stochasticinterest rates. An investigation of this is left for future research.

From the perspective of a CPPI issuer the main objective is to limit gap risk.I found that both hedging using short maturity put options and introduction ofan artificial floor reduce gap risk effectively at a small cost to the investor. Theartificial floor approach has the advantage that it does not depend on specifichedge instruments being available.

Given the contractual conditions imposed by the CPPI issuer, the investorwill choose the CPPI multiplier according to his/her investment objectives.

22

With expected return as performance measure, I found that the multipliersm ∈ 4, 5, 6 provided the best performance. Furthermore, risk averse investorswith S-shaped preferences and investors with a risk/return profile given by theSortino ratio had similar CPPI investment objectives. Both investor typeswould choose a multiplier m ∈ 3, 4. This is well below the upper bound onmultipliers m ∈ (10, 17) found by Bertrand & Prigent (2002) and similar to thetime and risk dependent multiplier found by Hamidi et al. (2009).7

In a setup that includes trading costs and cost of capital, investors wouldprefer a CPPI with no additional capital available, both when measuring per-formance by expected returns and by the Sortino ratio. Such a restriction willreduce gap risk, which makes it desirable also for the issuer.

References

Balder, S., Brandl, M. & Mahayni, A. (2009), ‘Effectiveness of CPPI strategiesunder discrete-time trading’, Journal of Economic Dynamics & Control33, 204–220.

Bertrand, P. & Prigent, J. (2002), ‘Portfolio insurance: the extreme value ap-proach to the CPPI method’, Finance 23, 69–86.

Black, F. & Perold, A. (1992), ‘Theory of constant proportion portfolio insur-ance’, The Journal of Economic Dynamics and Control 16, 403–426.

Boulier, J.-F. & Kanniganti, A. (1995), Expected performance and risks of var-ious portfolio insurance strategies. AFIR Colloquium, Brussels, Belgium.

Cesari, R. & Cremonini, D. (2003), ‘Benchmarking, portfolio insurance andtechnical analysis: a Monte Carlo comparison of dynamic strategies ofasset allocation’, Journal of Economic Dynamics & Control 27, 987–1011.

Constantinou, N. & Khuman, A. (2009), How does CPPI performagainst the simplest guarantee strategies? Working paper, URL:http://www.essex.ac.uk/ebs/research/efc/discussion papers/

dp 09-03.pdf.

Cont, R. & Tankov, P. (2009), ‘Constant Proportion Portfolio Insurance inPresence of Jumps in Asset Prices’, Mathematical Finance 19, 379–401.

Dobeli, B. & Vanini, P. (2010), ‘Stated and revealed investment decisionsconcerning retail structured products’, Journal of Banking & Finance34, 1400–1411.

Døskeland, T. M. & Nordahl, H. A. (2008), ‘Optimal Pension Insurance Design’,Journal of Banking & Finance 32, 382–392.

7To my knowledge these are the only papers considering the choice of CPPI multiplier. Forillustration purposes Balder et al. (2009), Cont & Tankov (2009), Paulot & Lacroze (2009)and Boulier & Kanniganti (1995) apply multipliers between m = 2 and m = 10.

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El Karoui, N., Jeanblanc, M. & Lacoste, V. (2005), ‘Optimal portfolio manage-ment with American capital guarantee’, Journal of Economic Dynamicsand Control 29, 449–468.

Eraker, B. (2004), ‘Do Stock Prices and Volatility Jump? Reconciling Evidencefrom Spot and Option Prices’, The Journal of Finance 59, 1367–1403.

Hamidi, B., Jurczenko, E. & Maillet, B. (2009), ‘A CAViaR Modelling for aSimple Time-Varying Proportion Portfolio Insurance Strategy’, Bankers,Markets & Investors forthcoming.

Jessen, P. & Jørgensen, P. (2009), Optimal Investment in Structured Bonds.Working paper, URL: http://www.asb.dk/staff/peje.

Kahneman, D. & Tversky, A. (1992), ‘Advances in Prospect Theory: Cumu-lative Representation of Uncertainty’, Journal of Risk and Uncertainty5, 297–323.

Maringer, D. (2008), ‘Risk Preferences and Loss Aversion in Portfolio Optimiza-tion, in E. J. Kontoghiorghes, B. Rustem & P. Winker’, ComputationalMethods in Financial Engineering Springer.

MKaouar, F. & Prigent, J.-L. (2007), Portfolio Insurance with transaction costs:The case of the CPPI method. Working paper,URL: http://affi2007.u-bordeaux4.fr/Actes/92.pdf.

Paulot, L. & Lacroze, X. (2009), Efficient Pricing ofCPPI using Markov Operators. Working paper, URL:http://www.citebase.org/abstract?id=oai:arXiv.org:0901.1218.

Pedersen, C. & Satchell, S. (2002), ‘On The Foundation of Performance Mea-sures under Asymmetric Returns’, Quantitative Finance 2, 217–223.

Tankov, P. (2008), Pricing and Hedging Gap Risk. Working paper, available atwww.ssrn.com.

Whalley, A. & Wilmott, P. (1997), ‘An asymptotic analysis of an optimal hedg-ing model for option pricing with transaction costs’, Mathematical Finance7, 307–324.

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