Cpp i 20151130

A Performance Evaluation of Constant Proportion Portfolio Insurance: An Application of the Economic Index of Riskiness

呂瑞秋(Richard Lu)Associate professor in the Department of Risk Management and Insurance, Feng

Chia University,100 Wenhwa Rd., Taichung 40724, Taiwan, E-mail: [email protected],

Tel: 886-4-24517250 ext.4132

熊令瑜(Ling-Yu Hsiung)Ph.D. Student in the Program of Finance, Feng Chia University, 100 Wenhwa Rd.,

Taichung 40724, Taiwan, E-mail: [email protected],

0

mailto:[email protected]

mailto:[email protected]

ABSTRACT

The investment performance of constant proportion portfolio insurance (CPPI)

strategies is evaluated by using the economic performance measure. This

performance measure generalizes the Sharpe measure by replacing the standard

deviation by the economic index of riskiness proposed by Aumann and Serrano

(2008). For the performance evaluation, the return distributions are generated by

Monte Carlo simulations. The results show that whether the CPPI strategies can

outperform a buy-and-hold (BH) strategy depends on the level of multiplier, the

performance measure, and the market scenario. The multiplier is the most important

factor that determines whether the CPPI can outperform the BH. When the multiplier

is no more than three, the CPPI almost always outperforms the BH under the normal

return and volatility market. However, if the multiplier is five, which is a commonly

used value in applications, the CPPI is outperformed by the BH under all market

scenarios studied.

Keywords: constant proportion portfolio insurance, economic index of riskiness,

economic performance measurement

1

1. Introduction

Portfolio insurance (PI) is an investment strategy of eliminating downside risk or

obtaining guaranteed minimum returns while preserving some upward potential in

rising markets. This payoff pattern seems attractive for most investors. Portfolio

insurance can create various return-risk profiles by setting different minimum values

or floors for the portfolio. Setting a high floor means a high level of protection or

low risk for the portfolio, but there is also a low expected return. In contrast, a low

floor means a high-risk, but high-return profile. Portfolio insurance is a decision

making strategy involving different return-risk trade-off patterns.

To implement PI, Leland and Rubinstein (1976) first proposed an option-based

strategy by buying a put option on the underlying portfolio. If the put is not

available, it can be created synthetically. The put option, like a general insurance

contract, provides the coverage when losses occur. Those who insure their portfolios

have to pay the put price. Thus, portfolio insurance, in fact, has to sacrifice some

return for the risk reduction. Portfolio insurance is a decision making approach with

a return-risk trade-off.

To insure a portfolio, another approach is constant proportion portfolio insurance

(CPPI) introduced by Perold (1986) and Black and Jones (1987). In addition to the

floor setting, those who use CPPI strategies have to set a multiplier, which

determines how aggressively they want to participate in the up market. We have

shown that, comparing with a buy-and-hold (BH) strategy, the multiplier is the most

important factor that determines whether the CPPI can outperform the BH.

Is it better to have a portfolio insured? In a Black-Scholes economy with one

risk-free asset and one risky asset, for a hyperbolic absolute risk aversion (HARA)

utility investor, Merton (1971) and Brennan and Solanki (1981) derived the optimal

investment payoff, which consists of a floor and a power on the risky asset price. As

demonstrated by Perold and Sharpe (1988), the optimal payoff can be constructed by

CPPI strategies. Thus, theoretically, PI should outperform BH, which is no

protection at all. However, there is no clear empirical evidence to support PI. As

Dichtl and Drobetz (2011) pointed out, the standard expected utility theory cannot

provide an explanation for using PI strategies. They showed that PI is preferred by

prospect theory investors, but not expected utility theory investors. In this paper, we

reevaluate the performance of PI again according to an economic performance

measure. Based on this performance measure, we provide evidence that justifies PI

for expected utility theory investors.

Because of the downside protection, there is a cut-off point and no left tail for the

return distribution of an insured portfolio. Also, by CPPI, how the distribution skews

2

to the right depends on the floor and the multiplier chosen. Thus, the return

distribution is asymmetric and does not conform to a normal distribution. To

evaluate the performance, we need to consider the non-normality.

To evaluate relative performance, stochastic dominance rules may provide a

solution for investors with quite broad utility functions. Annaert, Van Osselaer, and

Verstraete (2009) used the stochastic dominance rules to compare PI strategies with

a BH strategy. Their historical simulated results showed that the BH strategy did not

stochastically dominate the portfolio insurance strategies, or vice versa. Thus,

stochastic dominance cannot provide a clear comparison or ranking of the return

distributions. This means stochastic dominance can provide partial rather than

complete ranking. This limitation can be attributed to the broad utility functions

setting. Leshno and Levy(2002) pointed out that the set of utility functions under the

second-order stochastic dominance includes all risk-averse utility functions, and

some of them only conform to very few individuals’ preferences.

By setting the utility functions, which can represent most individuals’

preferences rather than all individuals’, Leshno and Levy (2002) propose using

almost stochastic dominance instead of stochastic dominance to rank distributions.

This leads more closely to a complete ranking. By using the almost stochastic

dominance rules, Fu, Hsu, Huang, and Tzeng (2014) ranked the return distributions

between PI strategies and BH strategy. They found that the PI strategies were

dominated by the BH strategy in the sense of almost first-order and second-order

stochastic dominance only for long investment horizons. For a one-year investment

horizon, like the results of Annaert et al. (2009), neither one dominates the other.

Another way to compare PI with BH without the partial ranking problem is to

use some performance measures, such as the Sharpe ratio, the Sortino ratio, and the

Omega measure. However, recent studies indicate using these performance measures

do not provide any clear support that PI could outperform BH from. (Cesari and

Cermonini, 2003; Annaert et al., 2009; Dichtl and Drobetz, 2011).

When evaluating investment performance, the Sharpe ratio is the performance

measure popularly used. However, except for investors with quadratic utility

functions, the Sharpe ratio will not be an appropriate performance measure if

investment returns are not normally distributed. PI often creates asymmetric return

distributions, and therefore it is advisable to not just consider the first two moments

of the distribution, which are used in the Sharpe ratio.

To get a complete ranking and consider the non-normality, this paper uses an

economic performance measure proposed by Homm and Pigorsch (2012). This

performance measure is similar to the Sharpe ratio. However, instead of using

standard deviation as risk measure, the economic performance measure uses the

economic index of riskiness proposed by Aumann and Serrano (2008) (AS index

henceforth). Under this setting, the performance measure can account for mean,

3

variance and higher moments of return distributions. In addition, this confirms the

Sharpe ratio ranking if returns are normally distributed. By using the economic

performance measure, we provide evidence to justify portfolio insurance under the

standard expected utility theory without resorting to prospect theory.

The remainder of this paper is organized as follows. Section 2 presents the

constant proportion portfolio insurance model. In Section 3, we demonstrate the

simulation design. Then, performance evaluation is discussed in section 4. Section 5

provides the numerical results and discussion. The final section is the conclusion.

2. Constant Proportion Portfolio Insurance Strategy

There are three dynamic portfolio insurance strategies commonly used: stop-loss

strategy, synthetic put strategy, and constant proportion strategy. In this paper, we

only discuss and evaluate the CPPI strategies, because they can replicate the optimal

investment payoff for HARA utility investors. In fact, as demonstrated below, the

three dynamic strategies have quite similar structures. Thus, CPPI is a good

representative of PI.

To illustrate the implementation of CPPI strategies, we use a two–asset model: a

risky asset and a risk-free asset. In constructing a CPPI strategy, an investor has to

initially select a floor FTand a multiplier (m). The floor is the minimum below which

he does not want the portfolio value to fall at the end of the investment period. Thus,

given an initial amount of investmentW 0, FT /W 0 is defined as the level of

protection. A higher floor leads to a higher percent of principal protected.

For a typical constant proportion strategy, the risky asset positionStat time t is

determined by

St=m(W ¿¿ t−PV t (FT ))¿ if W t >PV t(FT ) (1)

where W t is the total investment value at time t, PV t(FT ) is the present value of the

floor at time t, andW t−PV t(FT ) is referred to as a cushion. The remainder of the

portfolio W t−St is invested in a risk-free asset. Thus, given a fixed amount of the

cushion, a higher multiplier leads to higher positions in the risky assets. The higher

the multiplier is, the more aggressive the CPPI strategies are. Whenever

W t ≤ PV t(FT ), only the risk-free asset is held and there will be no investment in the

risky assets for the rest period. Thus, the CPPI strategy is a dynamic strategy for

asset allocation. The investor also has to decide when to reset the asset allocation

based on the above equation. However, if m = 1 and FT=0, the constant proportion

strategy becomes a static BH strategy. It is an all risky asset investment throughout

the investment horizon. Thus, the buy-and-hold strategy can be regarded as a type of

CPPI strategy with no downside risk protection (Perold and Sharpe, 1988).

4

There are unconstrained as well as constrained CPPI strategies. When the

multiplier is high and the floor is low, it is possible that the risky asset position will

exceed the total investment. In this case, the unconstrained strategy allows short

positions in the risk-free asset or financial leverage by borrowing at the risk-free

rate. However, the constrained strategy rules out the short positions or using the

leverage. A stop-loss strategy can be regarded as the constrained strategy with a very

high multiplier (Black and Perold, 1992). Also, a BH strategy is a special

constrained CPPI strategy with no floor.

A synthetic put strategy for PI is an option-based strategy. A risky asset plus a

put option on the risky asset is a portfolio that is insured. Instead of buying the put

option, a synthetic put strategy creates the put synthetically by trading the risky asset

and risk-free asset dynamically according to the Black-Scholes option pricing

formula. Thus, in implementing this strategy, investors need additional information

about the volatility for trading assets. By the put-call parity, a synthetic put strategy

is the same as using the cushion in CPPI to buy call options on the risky asset. As the

call options can be replicated dynamically by a portfolio of long positions in the

risky asset and short positions in the risk-free asset. Thus, a synthetic put strategy

can be regarded as a variation of CPPI with a time-varying multiplier (Perold and

Sharpe, 1988).

3. Simulation Design

To generate the return distributions of the CPPI strategies for performance

evaluation, we use a Monte Carlo simulation. We design the simulation by mostly

following the work of Dichtl and Drobetz (2011). The key difference is that we

simulate the return of CPPI strategies with various levels of protection and

multiplier. The details of the setup are as follows.

The risky asset price is assumed to follow a geometric Brownian motion. This

price model can be written as:

(1)

where St is the risky asset price at time t, wt is a wiener process, andμ and σ are the

drift and volatility parameters. After stochastic integration, the discrete version

becomes

(2)

where ε is a Gaussian white noise with the standard deviation equal to 1. Based on

this equation, we simulate daily returns by setting ∆ t=1 /250, μ=9%∨11.5%, and

σ=20 %∨30 %. There are four market scenarios: low-return and normal volatility

market (whereμ=9% and σ=20%), low-return and high-volatility market (whereμ

5

=9% and σ=30%), normal-return and normal-volatility market (whereμ=11.5% and

σ=20%), and normal-return and high-volatility market (whereμ=11.5% and σ=30%).

The normal-return and normal-volatility market corresponds to the scenario of the

long-term developed stock markets, as in Dichtl and Drobetz (2011).

The risk-free asset process is assumed to follow the following equation:

(3)

where r is the risk-free interest rate, Mt is the price at t. In the simulations, the

interest rate is set to 4.5%.

Like most the other similar research works, we simulate the returns of the

constrained CPPI only. To simulate the CPPI strategies, the floor is 100%, 95%,

90%, 85%, or 0% of the initial amount invested. The no protection case (0%) is the

same as a BH strategy. The multiplier is set to five, three, two and one. With the

round-trip transaction costs 0.1%, the asset allocation is reset when the price of the

risk asset moves (up or down) over 2%. The investment horizon is one year. For

every simulation run, we calculate a log return which is based on the following

argument. To get the return distribution, we perform 100,000 simulation runs.

There are detail explanations about the above numbers chosen for simulations in

Dichtl and Drobetz (2011). Thus, we do not repeat them here again.

4. Performance Evaluation

In this section, we first discuss the economic index of riskiness, which is used for

constructing EPM. Then, we present the EPM for performance evaluation in the

following subsection.

4.1 Economic Index of Riskiness

Most performance measures, for instance the Sharpe measure, are kinds of

reward-to-risk measures. The risk measures often adopted are standard deviation,

semi-standard deviation, value at risk, expected shortfall, and so on. However, as

Aumann and Serrano (2008) pointed out, the common drawback of these risk

measure are the violation of monotonicity with respect to second-degree stochastic

dominance (SSD). Thus, even if portfolio A dominates portfolio B in terms of SSD,

and we know that all risk-averse investors prefer A to B, those risk measures may

indicate that portfolio B is less risky than Portfolio A.

The economic index of riskiness, the AS index, is axiomatically derived from the

theory of decision making under risk. The two key axioms are duality and positive

homogeneity. The duality requires the risk index that reflects how less risk-

averse individuals accept riskier assets. Thus, it satisfies the above

6

monotonicity.

Aumann and Serrano (2008) defined the economic index of riskiness for

a risky asset as the reciprocal of the positive risk aversion

parameter of an individual with constant absolute risk aversion

(CARA) who is indifferent between taking and not taking the risky

asset. Under their setup, the AS index must satisfy the following equation:

EU (W +St−S0 )=U (W ), (4)

where U is the utility function, W is the initial wealth, St is the price

of the risky asset at time t. Assuming no cash dividend, St−S0 is the

absolute return of holding the asset for the time interval. Aumann

and Serrano (2008) constructed the index of riskiness by using an exponential

utility function. Thus, the AS index of the risky asset, AS(St) is

defined implicitly as follows:

Ee−(St−S0 )/ AS(S t)=1 (5)

They proved that the AS(St) is a unique positive number, and any

index satisfying the two axioms will be a positive multiple of AS(St)

if some of the absolute return are negative, and the mean of the absolute return is

positive.

Under this above setup, the investment risk is a kind of additive

risk. However, if the individual places the initial wealth in the risky

asset, then the risk becomes a multiplicative risk. For a

multiplicative risk, similar to Aumann and Serrano’s approach (2008),

Schreiber (2014) defined an economic index of relative riskiness for

a risky asset as the reciprocal of the positive risk aversion

parameter of an individual with constant relative risk aversion

(CRRA) who is indifferent between taking and not taking the risky

asset. Under his setup, the index of relative riskiness must satisfy the

following equation:

EU ¿ (6)

Schreiber (2014) adopted a power utility and derived the index of

relative riskiness which, in fact, equals the AS index applied to the

log return instead of the absolute return. That is, the index of

relative riskiness, RS(St) is defined implicitly as follows:

7

Ee−(ln St−ln S0 )/RS (St)=1 (7)

The index of relative riskiness also satisfies positive homogeneity in

the sense of RS ¿), which states t-year investment risk equals one-year

investment risk repeated t times. And, if we want to take out the

time value involved, RS(St) is defined implicitly as

Ee−(ln St−ln S0−r f t )/RS(S t)=1 (8)

Thus, to measure the relative riskiness of a risky asset, we

should apply the excess log return to the formula.

4.2 Economic Performance Measurement

To evaluate the performance of the CPPI strategies, we use the economic

performance measure (EPM) proposed by Homm and Pigorsch (2012). This measure

can be regarded as a generalized Sharpe measure, because, under the normality, the

EPM is equivalent to the Sharpe measure ranking. In contrast to the Sharpe measure,

the EPM divides the mean excess return by the AS index instead of the standard

deviation. Both measures are a type of reward-to-risk measure. However, as a risk

measure, the AS index accounts for mean, variance and higher moments of return

distributions, but, for standard deviation, it just considers the second moment, and is

only a perfect risk measure under the normality. Because CPPI strategies have a

floor setting, and participate in the upside market, the resulting return distributions

are often asymmetric and non-normal. Thus, in evaluating the performance, we

should consider higher moments of the return distribution, rather than just the first

two moments that are used in the Sharpe measure.

In Homm and Pigorsch (2012), the EPM is defined as:

EPM=E(~r )

AS(~r), (9)

where E(~r ¿ is the expected excess return of an investment portfolio, and AS¿) is

the AS index of the random excess return. As demonstrated by Homm and Pigorsch

(2012), the EPM has the positive property of monotonicity with respect to

the first-order and second-order stochastic dominance. This property is not held by

most other performance measures (Homm and Pigorsch, 2012).

In applying the EPM, the subtle part is to calculate the AS index. According to

Aumann and Serrano (2008), we can obtain the index by solving AS(~r) in the

equation:

E (e−~r

AS(~r ))=1 (10)

8

Given an empirical distribution or a simulated distribution, without

assuming any distribution function, we can apply Equation (10) and

solve the AS index. This is the so-called nonparametric approach.

By assuming a probability density function for the excess return,

we might derive the formula of the AS index. This will be a

parametric approach if possible. For a normally distributed excess

return, the AS index is equal to the variance divided by two times of the mean

excess return. Thus, the EPM equals two times of the squared Sharpe measure. In

this situation, the two performance measures produce the same ranking.

5. Results of Simulations and Evaluation

In the following tables are the simulated results of investment performance for

the CPPI strategies. The tables are classified by the market volatility and the

multiplier. In each table, the first four central moments of the return distributions, the

AS index, the Sharpe ratio, and the EPM are listed across the specified level of

protection. Table 1 and 2 indicate the results of the CPPI with multiplier equal three

from the normal volatility market and the high volatility market, respectively. For

each table, there are two panels: one for the low-return market and the other for the

normal-return market. In the same way, Table 3 and 4 are the results of the CPPI

with a multiplier that equals five. Table 5 and 6 are for the multiplier that equals two,

and Table 7 and 8 are for the multiplier that equals one.1 For the BH, in fact, we can

regard it as a special case of the portfolio insurance with zero protection. As the

percent of the principal protected approaches zero, the CPPI will converge to the

BH. Actually, the simulated results also indicate the convergence.

We first examine the moments of the return distribution across all the tables. As

expected, the higher level of protection leads to the lower expected return and

standard deviation. Thus, CPPI reduces the volatility through eliminating downside

risk. In doing so, CPPI sacrifices some expected return. By looking at the skewness

and kurtosis, both of the BH is slightly higher than a normal distribution.2 It is

because we use a percentage return instead of a log return. For others, it is obvious

that the CPPI with a higher floor has higher skewness and kurtosis. As expected,

these return distributions are far from a normal distribution.

Then, the standard deviation and the AS index are worth comparing. Standard

deviation is a risk measure used the most commonly, while the AS index is a new

proposed risk measure. In all cases studied here, they produce the same ranking of

riskiness. They both indicate that the CPPI with a higher level of protection has a

lower risk. However, according to the numerical results, the AS index provides a

1 Table 5-8 are listed in the Appendix.2 A normal distribution has zero skewness and kurtosis equal 3.

9

different degree of riskiness from the standard deviation.

Under non-normality, the standard deviation might no longer be a perfect risk

measure. In this situation, the Sharpe ratio might be also questionable for a

performance measure. In Table 1, we can find that the Sharpe ratio and the EPM

produce very different performance ranking. According to the Sharpe ratio, the BH

has the highest rank. However, by the EPM, the BH is only better that the CPPI with

an 85% level of protection. The EPM gives the CPPI full protection at the highest

rank. Thus, here, we can find that the CPPI outperforms the BH only by using the

EPM.

The CPPI, with multiplier three, performs worse than the BH in the high

volatility market. From the Sharpe ratio or the EPM of Table 2, the BH gets the

highest rank over the CPPI with some level of protection. As Perold and Sharpe

(1988) demonstrated, higher volatility causes the CPPI to incur larger capital losses

because it is a dynamic strategy of buying when the market is high, and selling low

when the market is low.

When the multiplier equals five, the CPPI is outperformed by the BH under all

the market scenarios studied. The EPM and the Sharpe ratio produce the same

ranking. The lower level of the protection is, the better the CPPI performs. Thus,

even though five is the popularly used multiplier, this does not make the CPPI

perform better than the BH. Compared with the multiplier equaling three, the mean

return and the standard deviation are both higher when the multiplier equals five.

However, the percentage increase in the standard deviation is much more than that of

the mean return. Thus, although a high multiplier can create high upside potential in

the sense of mean return, it also raises the volatility. Overall, it worsens the

performance.

When we set the multiplier down to two, the CPPI with the various levels of

protection all outperform the BH under the normal volatility market in the sense of

the EPM. For the high volatility market where we expect the CPPI to perform less

well, the CPPI, with the level of protection over 95%, still performs better than the

BH. Based on the Sharpe ratio, the CPPI is also the better performer, but, the

evidence is weaker than that of the EPM. If the multiplier equals one, the CPPI

always outperforms the BH. Thus, the multiplier is quite important in determining

the performance of the CPPI.

Overall, to find out if the CPPI can outperform the BH under the expected utility

framework, there are three important keys, the performance measure, the market

scenario, and the multiplier. If we use the EPM instead of the Sharpe ratio, there is

more evidence in support of the CPPI. In the normal-return and normal-volatility

market, the CPPI performs better than in the low-return and high-volatility market.

This is consistent with Perold and Sharpe (1988) where they demonstrated that CPPI

would perform relatively well in up and less volatile markets. Finally, the CPPI often

10

outperform the BH when the multiplier is low. Regardless of the performance

measure or the market scenario, we find that the CPPI is always better than the BH

when the multiplier equals one. Dichtl and Drobetz (2011) used the multiplier five

and Annaert et al. (2009) used 14. The multipliers they used are too high to obtain

the good performers of the CPPI.

11

Table 1 Simulated Results under the Normal Volatility Market (m=3)

Panel A: Expected Market Return=9%

% of protection 100% 95% 90% 85% 0%(BH)

Mean 0.0524 0.0594 0.0663 0.0731 0.0943

STD 0.0344 0.0718 0.1081 0.1418 0.2214

Skewness 2.1972 2.1568 1.9700 1.7041 0.5893

Kurtosis 12.147 11.454 9.3145 7.2147 3.6419

AS index 0.0576 0.1326 0.2078 0.2800 0.4543

Sharpe ratio 0.2159 0.2006 0.1971 0.1979 0.2228

EPM 0.1290 0.1086 0.1025 0.1002 0.1086

Panel B: Expected Market Return=11.5%

% of protection 100% 95% 90% 85% 0%(BH)

Mean 0.0565 0.0679 0.0792 0.0901 0.1221

STD 0.0369 0.0769 0.1155 0.1508 0.2266

Skewness 2.1160 2.0765 1.8799 1.6074 0.5784

Kurtosis 11.141 10.548 8.5086 6.5744 3.5789

AS index 0.0371 0.0830 0.1291 0.1739 0.2911

Sharpe ratio 0.3118 0.2977 0.2958 0.2991 0.3402

EPM 0.3100 0.2755 0.2647 0.2595 0.2648

Notes: The m in the title is the multiplier for the CPPI. STD stands for the

standard deviation. The AS index is the economic index of riskiness

proposed by Aumann and Serrano (2008). EPM stands for the economic

performance measure.

Table 2 Simulated Results under the High Volatility Market (m = 3)

12


% of protection 100% 95% 90% 85% 0%(BH)

Mean 0.0521 0.0587 0.0651 0.0712 0.0944

STD 0.0575 0.1170 0.1707 0.2172 0.3358

Skewness 4.0962 3.4089 2.8055 2.3322 0.9190

Kurtosis 33.9903 21.4436 14.4482 10.4772 4.5322

AS index 0.1641 0.3758 0.5688 0.7331 1.0432

Sharpe ratio 0.1238 0.1170 0.1179 0.1208 0.1472

EPM 0.0434 0.0364 0.0354 0.0358 0.0474


% of protection 100% 95% 90% 85% 0%(BH)

Mean 0.0561 0.0670 0.0776 0.0875 0.1221

STD 0.0616 0.1248 0.1812 0.2292 0.3441

Skewness 3.9995 3.2847 2.6853 2.2235 0.9113

Kurtosis 31.7000 19.8429 13.3583 9.7192 4.4918

AS index 0.1026 0.2292 0.3474 0.4502 0.6682

Sharpe ratio 0.1808 0.1765 0.1800 0.1854 0.2240

EPM 0.1085 0.0960 0.0939 0.0944 0.1154





Table 3 Simulated Results under the Normal Volatility Market (m = 5)


13

% of protection 100% 95% 90% 85% 0%(BH)

Mean 0.0556 0.0667 0.0761 0.0837 0.0942

STD 0.0677 0.1267 0.1686 0.1962 0.2213

Skewness 3.6069 2.3588 1.6892 1.2532 0.5823

Kurtosis 22.3146 10.0556 6.2229 4.5663 3.5961

AS index 0.1399 0.2704 0.3589 0.4118 0.4555

Sharpe ratio 0.1569 0.1710 0.1846 0.1972 0.2225

EPM 0.0759 0.0801 0.0868 0.0940 0.1081


% of protection 100% 95% 90% 85% 0%(BH)

Mean 0.0628 0.0809 0.0960 0.1074 0.1221

STD 0.0751 0.1375 0.1801 0.2065 0.2266

Skewness 3.3795 2.1772 1.5466 1.1458 0.5784

Kurtosis 19.4728 8.8567 5.5932 4.2404 3.5789

AS index 0.0814 0.1622 0.2204 0.2574 0.2911

Sharpe ratio 0.2374 0.2612 0.2830 0.3019 0.3402

EPM 0.2189 0.2215 0.2313 0.2423 0.2648

Notes: The m in the title is the multiplier for the CPPI. STD stands for

the standard deviation. The AS index is the economic index of riskiness





% of protection 100% 95% 90% 85% 0%(BH)

14

Mean 0.0544 0.0635 0.0716 0.0785 0.0944

STD 0.1114 0.1898 0.2442 0.2818 0.3358

Skewness 4.8561 3.1399 2.3694 1.8847 0.9190

Kurtosis 34.8032 15.5467 9.9267 7.2838 4.5322

AS index 0.4816 0.7721 0.9295 1.0056 1.0432

Sharpe ratio 0.0840 0.0977 0.1087 0.1191 0.1472

EPM 0.0194 0.0240 0.0286 0.0334 0.0474


% of protection 100% 95% 90% 85% 0%(BH)

Mean 0.0610 0.0761 0.0891 0.0996 0.1221

STD 0.1209 0.2027 0.2586 0.2960 0.3441

Skewness 4.5479 2.9388 2.2159 1.7705 0.9113

Kurtosis 30.6556 13.9081 8.9748 6.7236 4.4918

AS index 0.2771 0.4597 0.5622 0.6226 0.6682

Sharpe ratio 0.1322 0.1535 0.1705 0.1845 0.2240

EPM 0.0577 0.0677 0.0784 0.0877 0.1154





15

6. Conclusion

In this paper, the investment performance of CPPI strategies is compared with a

buy-and-hold strategy. Because of the downside risk protection and the upside

potential of the CPPI, it generates an asymmetric and non-normal return distribution.

In addition to the Sharpe measure for evaluation, we use the economic performance

measure that generalizes the Sharpe measure to consider the non-normality. The

results show that the CPPI strategies can outperform a buy-and-hold (BH) strategy

under the expected utility theory. To reveal this evidence, the keys are the level of

multiplier, the performance measure, and the market scenario. The multiplier is the

most important factor that determines whether the CPPI can outperform the BH.

With the multiplier being no more than three, the CPPI almost always outperforms

the BH under the normal trend and volatility market. However, if the multiplier is

five, which is a commonly used value in applications, the CPPI is outperformed by

the BH under all market scenarios studied.

16

References

Annaert, J., Van Osselaer, S., and Verstraete, B., 2009. Performance evaluation of

portfolio insurance strategies using stochastic dominance criteria. Journal of

Banking and Finance, 33: 272-280.

Aumann, R. J., and Serrano, R., 2008. An economic index of riskiness. Journal of

Political Economy, 116: 810-835.

Black, F., and Scholes, M., 1973. The pricing of options and corporate liabilities.

Journal of Political Economy, 81: 637-654.

Black, F., and Jones, R., 1987. Simplifying portfolio insurance. Journal of Portfolio

Management, 14 (1): 48-51

Black, F., Perold, A.F., 1992. Theory of constant proportion portfolio insurance.

Journal of Economic Dynamics and Control, 16 (3-4): 403-426.

Brennan, M.J., Solanki, R., 1981. Optimal portfolio insurance. Journal of Financial

and Quantitative Analysis, 16 (3): 279-300.

Cesaria, R., and Cremoninib, D., 2003. Benchmarking, portfolio insurance and

technical analysis: A Monte Carlo comparison of dynamic strategies of asset

allocation. Journal of Economic Dynamics and Control, 27: 987-1011.

Coval, J. D., and Shumway, T., 2001. Expected option returns. Journal of Finance,

56: 983-1009.

Dichtl, H., Drobetz, W., 2011. Portfolio insurance and prospect theory investors:

popularity and optimal design of capital protected financial products.

Journal of Banking and Finance, 35 (7): 1683-1697.

Fu, H, Hsu,Y-C., Huang, R. J., Tzeng, L. Y. 2014. To hedge or not to hedge?

Evidence via almost stochastic dominance. Paper presented at the Taiwan

Risk and Insurance Association Conference, Taichung, Taiwan.

Homm, U., and Pigorsch, C., 2012. Beyond Sharpe ratio: An application of the

Aumann-Serrano index to performance measurement. Journal of Banking

and Finance, 36: 2274-2284.

Hsu, H., 2013. On the option expected returns and risk characteristics. Journal of

Futures and Options, 6: 1-31..

Leland, H.E., 1985. Option pricing and replication with transactions costs. Journal

of Finance, 40: 1283-1301.

Leland, H.E., and Rubinstein, M., 1976. The evolution of portfolio insurance. In:

Portfolio Insurance: A Guide to Dynamic Hedging. Hoboken, New Jersey:

Wiley.

___________________________ . 1988. The evolution of portfolio insurance. In:

Portfolio Insurance: A Guide to Dynamic Hedging. New York: John Wiley

& Sons.

Leshno, M., Levy, H. 2002. Preferred by all and preferred by most decision makers:

17

Almost stochastic dominance. Management Science, 48: 1074-1085.

Merton, R.C., 1971. Optimum consumption and portfolio rules in a continuous time

model. Journal of Economic Theory, 3: 373-413.

Perold, A.F., 1986. Constant Portfolio Insurance. Harvard Business School.

Perold, A.F., and Sharpe, W.F., 1988. Dynamic strategies for asset allocation.

Financial Analysts Journal, 44 (4): 16-27.

18

Appendix



% of protection 100% 95% 90% 85% 0%(BH)

Mean 0.0503 0.0549 0.0595 0.0641 0.0942

STD 0.0209 0.0436 0.0663 0.0890 0.2213

Skewness 1.2694 1.2694 1.2694 1.2694 0.5823

Kurtosis 5.9126 5.9126 5.9126 5.9124 3.5961

AS index 0.0333 0.0791 0.1257 0.1726 0.4555

Sharpe ratio 0.2518 0.2261 0.2181 0.2141 0.2225

EPM 0.1579 0.1246 0.1151 0.1104 0.1081


% of protection 100% 95% 90% 85% 0%(BH)

Mean 0.0528 0.0602 0.0677 0.0751 0.1220

STD 0.0220 0.0458 0.0697 0.0936 0.2269

Skewness 1.2739 1.2739 1.2739 1.2739 0.5841

Kurtosis 5.9475 5.9475 5.9475 5.9467 3.6071

AS index 0.0224 0.0514 0.0805 0.1097 0.2921

Sharpe ratio 0.3569 0.3326 0.3249 0.3212 0.3392

EPM 0.3498 0.2969 0.2814 0.2740 0.2636







19

% of protection 100% 95% 90% 85% 0%(BH)

Mean 0.0501 0.0546 0.0591 0.0635 0.0944

STD 0.0328 0.0684 0.1040 0.1395 0.3355

Skewness 2.1451 2.1455 2.1428 2.1099 0.9128

Kurtosis 11.4972 11.5044 11.4294 10.9180 4.4841

AS index 0.0831 0.1987 0.3156 0.4338 1.0415

Sharpe ratio 0.1566 0.1403 0.1352 0.1328 0.1472

EPM 0.0618 0.0483 0.0446 0.0427 0.0474


% of protection 100% 95% 90% 85% 0%(BH)

Mean 0.0527 0.0600 0.0673 0.0745 0.1221

STD 0.0345 0.0721 0.1096 0.1469 0.3445

Skewness 2.1684 2.1686 2.1618 2.1172 0.9176

Kurtosis 11.7517 11.7549 11.6126 10.9709 4.5270

AS index 0.0554 0.1272 0.1996 0.2721 0.6694

Sharpe ratio 0.2234 0.2079 0.2030 0.2009 0.2238

EPM 0.1393 0.1178 0.1115 0.1084 0.1152







% of protection 100% 95% 90% 85% 0%(BH)

20

Mean 0.0481 0.0504 0.0527 0.0550 0.0943

STD 0.0097 0.0203 0.0308 0.0414 0.2210

Skewness 0.5784 0.5784 0.5784 0.5784 0.5785

Kurtosis 3.5894 3.5894 3.5894 3.5894 3.5896

AS index 0.0133 0.0342 0.0561 0.0781 0.4546

Sharpe ratio 0.3219 0.2668 0.2494 0.2409 0.2229

EPM 0.2353 0.1581 0.1372 0.1277 0.1083


% of protection 100% 95% 90% 85% 0%(BH)

Mean 0.0494 0.0530 0.0566 0.0602 0.1221

STD 0.0100 0.0208 0.0317 0.0425 0.2271

Skewness 0.5857 0.5857 0.5857 0.5857 0.5858

Kurtosis 3.6172 3.6172 3.6172 3.6172 3.6175

AS index 0.0096 0.0234 0.0374 0.0516 0.2918

Sharpe ratio 0.4359 0.3823 0.3654 0.3572 0.3396

EPM 0.4531 0.3409 0.3093 0.2944 0.2643







% of protection 100% 95% 90% 85% 0%(BH)

Mean 0.0481 0.0504 0.0526 0.0549 0.0944

21

STD 0.0147 0.0307 0.0468 0.0628 0.3353

Skewness 0.9077 0.9077 0.9077 0.9077 0.9078

Kurtosis 4.4549 4.4549 4.4549 4.4549 4.4552

AS index 0.0307 0.0792 0.1299 0.1811 1.0424

Sharpe ratio 0.2105 0.1742 0.1627 0.1571 0.1472

EPM 0.1009 0.0676 0.0586 0.0545 0.0473


% of protection 100% 95% 90% 85% 0%(BH)

Mean 0.0493 0.0529 0.0565 0.0601 0.1223

STD 0.0151 0.0316 0.0480 0.0645 0.3445

Skewness 0.9196 0.9196 0.9196 0.9196 0.9197

Kurtosis 4.5346 4.5346 4.5346 4.5346 4.5351

AS index 0.0221 0.0539 0.0864 0.1190 0.6673

Sharpe ratio 0.2859 0.2505 0.2394 0.2339 0.2243

EPM 0.1956 0.1468 0.1331 0.1268 0.1158





22

Documents

Cpp i 20151130