Docstoc

Omega Performance Measure and Po

Document Sample
Omega Performance Measure and Po Powered By Docstoc
					     Omega Performance Measure and Portfolio
                   Insurance
                         P. Bertrand∗              J.L. Prigent†
                                    March 16, 2006


                                         Abstract
          We analyze the performance of portfolio insurance methods : especially
      the OBPI and CPPI. For this, we use the Omega performance measure,
      which takes the entire return distribution into account. We show in par-
      ticular that for the Omega performance measure, the CPPI method is
      better than the OBPI one for “rational” thresholds.
          Keywords: Portfolio insurance, Performance, Omega measure.
          JEL classification: C 61, G 11.


1     Introduction
The goal of portfolio insurance is to give to the investors the ability to recover, at
maturity, a given percentage of their initial capital, in particular in falling mar-
kets. Two standard portfolio insurance methods are the Option Based Portfolio
Insurance (OBPI) and the Constant Proportion Portfolio Insurance (CPPI).
    The OBPI was introduced by Leland and Rubinstein (1976). This method
is based on a portfolio invested in a risky asset S (usually a financial index such
as the S&P) covered by a listed put written on it. Thus, whatever the value
of S at the terminal date T , the portfolio value will be always greater than the
strike K of the put.
    The CPPI was introduced by Perold (1986) (see also Perold and Sharpe
(1988)) for fixed-income instruments and Black and Jones (1987) for equity in-
struments. This method uses a simplified strategy to allocate assets dynamically
over time. The investor chooses a floor equal to the lowest acceptable value of
her portfolio. The cushion is defined as the excess of the portfolio value over the
floor. It determines the amount allocated to the risky asset: this amount, called
the exposure, is equal to the cushion multiplied by a predetermined multiple.
Both the floor and the multiple are functions of the investor’s risk tolerance and
are exogenous to the model. The remaining funds are invested in the reserve
asset, usually T-bills. As the cushion approaches zero, exposure approaches zero
    ∗ GREQAM    and University Montpellier I, France.
    † THEMA,   University Cergy-Pontoise, France.


                                              1
too. In continuous time, this keeps portfolio value from falling below the floor.
Portfolio value will fall below the floor only when there is a very sharp drop in
the market before the investor has a chance to trade.

    The properties of portfolio insurance have been previously studied for exam-
ple by Bookstaber and Langsam (2000), Black and Rouhani (1989), Black and
Perold (1992),...To summarize, OBPI performs better if the market increases
moderately. CPPI does better if the market drops or increases by a small or
large amount. Bertand and Prigent (2002, 2005) compare CPPI with OBPI by
introducing systematically the probability distributions of the two portfolio val-
ues and by comparing them by means of various criteria: the four first moments
of their returns, the cumulative distribution of their ratio and in particular some
of its quantiles... Hedging risk properties involved by these two strategies are
also studied, when the option has to be synthesized. The “greeks” of the OBPI
and the CPPI are derived and some of their probability distributions are com-
pared. The Greeks’ features show the different nature of the dynamic properties
of the two strategies, in particular if the risky asset value drops suddenly.
   Nevertheless, when comparing the performance of portfolios, usually we have
to choose a performance measure: for example, for standard asset allocation,
we can use Sharpe’s ratio, Treynor’s ratio or Jensen’s Alpha. Besides, since the
payoffs of portfolio insurance strategies are asymmetric, we must select a per-
formance measure which overcomes the inadequacy of traditional performance
measures when they are used to analyze return distributions which are not nor-
mally distributed.
    In the present paper, we use the Omega performance measure, recently intro-
duced by Shadwick and Keating (2002), to compare standard portfolio insurance
strategies.The Omega measure is based on a a gain-loss approach since it uses
the downside lower partial moment. It has been applied across a broad range of
models in financial analysis, in particular to examine hedge fund style or strat-
egy indices. This measure splits the return into two sub-parts according to a
threshold. The "good" returns are above this threshold and the "bad" returns
below. Therefore, the Omega measure is defined as the ratio of the gain with
respect to the threshold and the loss with respect to the same threshold. The
Omega function is defined by varying the threshold. As mentioned for example
in Bacmann and Scholz (2003), the main advantage of the Omega measure is
that it involves all the moments of the return distribution, including skewness
and kurtosis. Moreover, ranking is always possible, whatever the threshold in
contrast to the Sharpe ratio.
   Section 2 recalls the main properties of Omega measure. In Section 3, the
Omega measure is computed for OBPI and CPPI. Section 4 provides some
numerical comparisons of performance of both strategies. Some proofs are rel-
egated to an Appendix.




                                        2
2     The Omega performance measure
2.1    Definition and general properties
The Omega performance measure was first introduced by Keating and Shadwick
(2002, 2003). It was designed to overcome the shortcomings of performance mea-
sures based only on the mean and the variance of the distribution of the returns.
Omega measure takes into account the entire return distribution while requiring
no parametric assumption on the distribution. Moreover, it is a function of the
expected return threshold that is set according to investor preferences.
    Omega measure takes into account the returns below and above a given loss
threshold. More precisely, Omega is defined as the probability weighted ratio of
gains to losses relative to a return threshold. The exact mathematical definition
is given by:
                                     Rb
                                      L
                                          (1 − F (x)) dx
                          ΩF (L) =        RL             ,
                                           a
                                             F (x) dx
where F (.) is the cumulative distribution function of the asset returns defined
on the interval (a, b), with respect to the probabilty distribution P and L is the
return threshold selected by the investor. For any investor, returns below her
loss threshold are considered as losses and returns above as gains.
    At a given return threshold, investor should always prefer the portfolio with
the highest value of Omega.
    Omega function exhibits the following properties :

    • First, as shown for example in Kazemi et al (2004), Omega can be written
      as :                                h           i
                                        EP (X − L)+
                             ΩFX (L) =    h           i.
                                        EP (L − X)+

Remark 1 It is the ratio of the expectations of gains above the threshold L to
the expectations of the losses below the threshold L. As noted by Kazemi et
al (2004),Omega can be considered as the ratio of the prices of a call option
to a put option written on X with strike price L but both evaluated under the
historical probability P.

    • For L = EP [X], ΩFX (L) = 1,
    • ΩFX (.) is a monotone decreasing function.
    • ΩFX (.) = ΩGX (.) if and only if F = G.
    • Kazemi et al (2004) define the Sharpe Omega measure as:
                                       EP [X] − L
                   SharpeΩ (L) =        h          i = ΩFX (L) − 1.
                                                 +
                                     EP (L − X)


                                          3
Remark 2 If EP [X] < L, the Sharpe Omega will be negative otherwise it will
be positive.
    Typically, consider the payoff X of a stock S at time T which is modelled by
a geometric Brownian motion : X = S0 exp[(µ − σ 2 /2)T + σWT ], where WT has
the Gaussian distribution N (0, T ). Then, EP [X] = S0 exp[µT ] does not depend
on the volatility.
    Thus, if S0 exp[µT ] < L then the Sharpe Omega is an increasing function of
the volatility (due to the Vega of the put). If S0 exp[µT ] > L, the Sharpe Omega
is a decreasing function of the volatility.

   The level must be specified exogeneously: it varies according to investment
type, individual risk aversion. It might be for example a rate of inflation for
pension’s incomes or the rate of a benchmark index.

2.2    The Omega function for the Buy-and-Hold strategy.
The Omega function can be examined for standard univariate distributions: for
example the Normal, Logistic, Lognormal and Gamma distributions (see Cascon
et al (2003)) and the Johnson family of distributions (see Kazemi et al. (2004)).
    Consider for instance a portfolio manager who invests in two basic assets :
a money market account, denoted by B, and a financial index, denoted by S.
The period of time considered is [0, T ].
    The value of the riskless asset B evolves according to :

                                  dBt = Bt rdt,

where r is the deterministic interest rate.
   Assume that she wants a guaranteed level equal to pV0 (with p ≤ erT ) and
uses a Buy and Hold strategy. Thus, her portfolio value at maturity is given by
:
                                VT = pV0 + αST ,
with α equal to:
                                    V0 (1 − pe−rT )
                               α=                   .
                                           S0
   Then, the Omega of her portfolio is given by :
                                     h              i
                                                  +
                                  EP (ST − λS0 )
                      ΩVT (L) =      h              i,                         (1)
                                  EP (λS0 − ST )+

where λ = (L/V0 − p)/(1 − pe−rT ) and the threshold L is chosen obviously
higher than the guaranteed amount pV0 . Thus, it can be considered as the ratio
of the prices of a call option to a put option written on ST with strike price λS0
(evaluated under the historical probability P).
    Note that the ratio λ is higher than 1 (out-of- the-money Call) if and only if
the ratio L/V0 is higher than 1 + p(1 − e−rT ). Besides, the threshold L is smaller

                                         4
                                                   £                ¤
than the expectation EP [VT ] if and only if L/V0 ≤ 1 + p(1 − e−rT ) eµT . Thus,
if we want also to compare the ratio L/V0 with the riskless return erT , we can
for example consider three cases:
    1) The ratio L/V0 satisfies: p ≤ L/V0 ≤ 1 + p(1 − e−rT ),
    2) The ratio L/V0 satisfies: 1 + p(1 − e−rT ) ≤ L/V0 ≤ erT ,
    3) The ratio L/V0 satisfies: erT ≤ L/V0 ≤ 1 + p(1 − e−rT )eµT .

Proposition 3 The Omega performance measure for the previous Buy and
Hold strategy is a monotonous function of the guaranteed percentage p:
    1) If the ratio L/V0 satisfies: L/V0 < erT , then Omega is an increasing
function of the percentage p.
    2) If the ratio L/V0 satisfies: L/V0 > erT , then Omega is a decreasing
function of the percentage p.
    3) If the ratio L/V0 satisfies: L/V0 = erT , then Omega is a constant function
of the percentage p.
Proof. Since Omega is a decreasing function of the threshold, we deduce from
Relation (1) that it is sufficient to analyze the ratio λ as function of p. Recall
that λ(p) = (L/V0 − p)/(1 − pe−rT ). Therefore :
                    ∂λ(p)
                          = (L/V0 e−rT − 1)/(1 − pe−rT )2 ,
                     ∂p
from which we deduce the result.
   The intuition behind porposition 3 is as follows :
   • In part 1 of proposition 3, the low level of the threshold means that the
     investor is essentially concerned about risk control. As a result, the Omega
     becomes increasing in the insured percentage, p.
   • As the threshold is higher (part 2 of proposition 3), the investor worries
     more about the performance of her fund. As a result, the Omega becomes
     decreasing in the insured percentage, p.
   • If the threshold is just set equal to the initial capitalized portfolio value,
     L = V0 erT , the Omega is independent of the insured percentage, p.
   We consider the following numerical example :
   The dynamics of the market value of the risky asset S are given by the classic
diffusion process :

                            dSt = St [µdt + σdWt ] ,
where Wt is a standard Brownian motion.
Remark 4 ST = S0 exp[(µ − σ2 /2)T + σWT ], where WT has the Gaussian
distribution N (0, T ). Then, EP [VT ] = pV0 + α.S0 exp[µT ] does not depend
on the volatility. Since we choose the threshold L smaller than the expectation
EP [VT ], both Sharpe Omega and Omega are decreasing functions of the volatility.


                                        5
   The parameter values are :

              T = 1, µ = 10%, S0 = 100, V0 = 100, r = 3%, σ = 20%




              20




              15




              10




               5




                            101.5        102          102.5        103


                     Figure 1 : Ω as a function of L for p = 1.

          8




          7




          6




          5




                          101.5         102           102.5        103
                                                                         I
                   Figure 2 : Ω as a function of L for p = 0.95.
    The two previous figures illustrate the results of Proposition 3: for levels L
smaller than the riskless return, the Omega performance measure is an increas-
ing function of the guaranteed percentage p.




                                         6
3      The Omega measure of OBPI and CPPI
3.1      Definition of the two strategies
The portfolio manager is yet assumed to invest in two basic assets : a money
market account, denoted by B, and a portfolio of traded assets such as a com-
posite index, denoted by S. The strategies are self-financing.
   The OBPI method consists basically of purchasing q shares of the asset S
and q shares of European put options on S with maturity T and exercise price
K.
     Thus, the portfolio value V OBP I is given at the terminal date by :
                           OBP I
                                     ¡                  ¢
                          VT     = q ST + (K − ST )+ ,                              (2)
                 OBP I
which is also : VT     = q (K + (ST − K)+ ), due to the Put/Call parity. This
relation shows that the insured amount at maturity is the exercise price times
the number of shares, qK.

     The value VtOBP I of this portfolio at any time t in the period [0, T ] is :
                                             ³                          ´
         VtOBP I = q (St + P (t, St , K)) = q K.e−r(T −t) + C(t, St , K) ,

where P (t, St , K) and C(t, St , K) are the no-arbitrage values calculated under a
given risk-neutral probability Q (if coefficient functions µ, a and b are constant,
P (t, St , K) and C(t, St , K) are the usual Black-Scholes values of the European
Put and Call).
   Note that, for all dates t before T , the portfolio value is always above the
deterministic level qKe−r(T −t) .
    The investor is yet willing to recover a percentage p of her initial investment
V0 . Then, her portfolio manager has to choose the two adequate parameters, q
and K.
   First, since the insured amount is equal to qK, it is required that K satisfies
the relation1 :
                     pV0 = pq(K.e−rT + C(0, S0 , K)) = qK,
     which implies that :

                                   C(0, S0 , K)   1 − pe−rT
                                                =           .
                                       K              p

    Therefore, the strike K is a function K (p) of the percentage p, which is
increasing.


    1 This   relation can also take account of the smile effect.


                                                  7
   Second, the number of shares q is given by :
                                            V0
                              q=                          .
                                   S0 + P (0, S0 , K (p))

   Thus, for any initial investment value V0 , the number of shares q is a de-
creasing function of the percentage p.

    The CPPI method consists of managing a dynamic portfolio so that its value
is above a floor F at any time t. The value of the floor gives the dynamically
insured amount. It is assumed to evolve according to :

                                     dFt = Ft rdt.

Obviously, the initial floor F0 is less than the initial portfolio value V0CP P I . The
difference V0CP P I − F0 is called the cushion, denoted by C0 . Its value Ct at any
time t in [0, T ] is given by :

                                  Ct = VtCP P I − Ft .

   Denote by et the exposure, which is the total amount invested in the risky
asset. The standard CPPI method consists of letting et = mCt where m is a
constant called the multiple. The interesting case is when m > 1, that is, when
the payoff function is convex.
   The value of this portfolio VtCP P I at any time t in the period [0, T ] is :
                                                          m
                       VtCP P I (m, St ) = F0 .ert + αt .St ,          (3)
                   ³ ´                           ³        ¡     ¢  2
                                                                     ´
                    C0
    where αt      = S m exp [βt] and β = r − m r − 1 σ 2 − m2 σ .
                                                              2   2
                        0




    Thus, the CPPI method is parametrized by F0 and m. The OBPI has just
one parameter, the strike K of the put. In order to compare the two methods,
first the initial amounts V0OBP I and V0CP P I are assumed to be equal, secondly
the two strategies are supposed to provide the same guarantee qK = pV0 at
maturity. Hence, FT = qK and then F0 = qKe−rT . Moreover, the initial value
C0 of the cushion is equal to the call price C(0, S0 , K). Note that these two
conditions do not impose any constraint on the multiple, m. In what follows,
this leads us to consider CPPI strategies for various values of the multiple m2 .
   The portfolio payoffs for both strategies are given in the next figure:
   2 Note that the multiple must not be too high as shown for example in Prigent (2001) or

in Bertrand and Prigent (2002).




                                            8
         250



                                   OBPI
         200
                                   CPPI , m=4


                                   CPPI , m=6
         150
                                   CPPI , m=8



         100




          50




               0              50                100      150           200


                   Figure 3 : CPPI and OBPI Payoffs as functions of S
    Note that the value of the level L corresponding to the first intersection
of both graphics is about 103 which is approximately the value of the riskless
return. In what follows, we compare the CPPI and OPBI methods for “rational”
thresholds which are in fact around this value.

3.2     Comparison of portfolio insurance performance with
        Omega measure
3.2.1   Computations of Omega
As we analyze portfolio insurance, the threshold of the Omega measure must
be greater than the insured amount at maturity : L > q.K = p.V0 .

Proposition 5 For the OBPI strategy, the Omega function is defined by :
                                     h           i
                                 EP (ST − L/q)+
           ΩOBP I (L) =   h              i    h            i.
                        EP (L/q − ST )+ − EP (K − ST )+

Proof. See Appendix.

The following figure illustrates this risk reduction due to a Capped Put profile:

Remark 6 Note that the risk measure associated to the Omega performance
                                                h           i
                                                          +
measure at a given level H is the expectation EP (H − ST ) . Thus, the Omega
of the OBPI strategy can be viewed as the stock’s Omega to which one would
have removed the "risk" of falling under the level K. Note that the insured
amount at maturity is q.K. Therefore the reduction in the risk is clearly due to
portfolio insurance.




                                                9
           Payoff

           6




 L/q − K




                                                                      ST
                                                                      -
                     K              L/q
                             Figure 1: Capped Put


Proposition 7 For the CPPI strategy, the Omega function is defined by :
                                h                      i
                                              m      +
                              EP (p.V0 + αT .ST − L)
                ΩCP P I (L) =   h                      i.
                                                  m +
                              EP (L − p.V0 − αT .ST )

Proof. By using relation (3).

Remark 8 The expectation of the CPPI portfolio value is given by :
                   £ CP ¤
                EP VT P I = p.V0 + C0 e[r+m(µ−r)]T .

Note that this expression does not depend on the volatility. Thus, as for the stock
S (see Remark (2)), the Sharpe ratio SharpeCP P I (L) of the CPPI depends on
                                                Ω
the volatility, only through its denominator which is equal to a Put option. Thus
it is a decreasing function of the volatility and so does the Omega ratio for the
CPPI.




                                          10
3.2.2      Comparisons of Omega
General case Here, we compare numerically the Omega of the OBPI and of
the CPPI without any constraint on their expectations.
   The parameters values are the same as previously. In what follows, the
threshold level is chosen lower than the lowest expectations values of the insured
portfolios analyzed (CPPI and OBPI).


   Omega as function of the volatility σ : We first analyze the effect
of volatility on OBPI Omega and on CPPI Omega for different values of the
multiple, m. First, note that the CPPI curves intersect each other behind the
volatility level of 40 %.



             8




             6




             4




             2




                               0.1             0.2              0.3              0.4


                 Figure 4 : Ω as function of sigma for p = 1 and L = 102.

  (OBPI : solid line, line with large dashed : CPPI with m = 3, line with
medium dashed CPPI with m = 5, line with small dashed : CPPI with m = 7)

    For a small value of the threshold, L = 102, and an insured percentage equal
to 100%, the CPPI strategies dominate the OBPI strategy for all values of the
volatility as can be seen on figure3 4.
  3 In   all the figures, it is the logratithm of the Omega function that is computed.




                                               11
           3




         2.5




           2




         1.5




           1




         0.5




                        0.1           0.2           0.3            0.4


          Figure 5 : Ω as function of sigma for p = 1 and L = 103.5.

  The figure 5 shows that as the threshold rises, the OBPI can dominates some
CPPI strategies. This is particularly true for small values of m.

   We analyze now the effect of the insured percentage on the comparison
between OBPI and CPPI.

               6




               5




               4




               3




               2




               1




                        0.1           0.2          0.3           0.4


       Figure 6 : Ω as function sigma percentage p = 0.95 for L = 102.




                                     12
                 2




               1.5




                 1




               0.5




                                0.1             0.2                 0.3                   0.4



                             Figure 7 : p = 0.95 and L = 104.5
    As the percentage p decreases, the OBPI tends to dominates some CPPI
strategies:

   • for high volatility levels for L = 102,
   • for low volatility levels for L = 104.5,

    Omega as function of the threshold L The following figure corresponds
to p = 1 and σ = 20 %.


         2.5




           2




         1.5




           1




         0.5




                     102.2      102.4   102.6         102.8   103         103.2   103.4


                     Figure 8 : Ω as function of the threshold L.

  (OBPI : solid line, line with large dashed : CPPI with m = 3, line with
medium dashed : CPPI with m = 5, line with medium dashed : CPPI with
m = 7, line with small dashed : CPPI with m = 9)

   The following figure corresponds to p = 0.95 and σ = 20 %.



                                                13
              1.5




             1.25




                1




             0.75




              0.5




             0.25




                         102.5         103    103.5     104         104.5    105



                     Figure 9 : Ω as function of the threshold L.


    The effect of L becomes sensitive for high values of L as already seen in
figure 3. It is only for L greater than 103.5 that the ranking between OBPI and
CPPI’s are inverted.
    As soon as the threshold is small, the CPPI strategies dominates the OBPI
strategy.

    Special case We consider now the case where both OBPI and CPPI port-
folios values have the same expectation. Recall that the value of the multiple
such that the expectations of the two portfolio values are equal is given by4 :
                                µ      ¶ µ                    ¶
                    ∗               1         C(0, S0 , K, µ)
                  m (K) = 1 +           ln                      .
                                  µ−r         C(0, S0 , K, r)

               5




               4




               3




               2




               1




                                 0.1          0.2             0.3           0.4


                                 Figure 10 : p = 1 and L = 102
  4 See   Bertrand and Prigent (2005).




                                              14
      2.5




        2




      1.5




        1




      0.5




                     0.1           0.2           0.3             0.4


                    Figure 11 : p = 1 and L = 104

        4




        3




        2




        1




                     0.1          0.2           0.3             0.4


                   Figure 12 : p = 0.95 and L = 102
            2.5




              2




            1.5




              1




            0.5




                       0.1         0.2         0.3        0.4


     Figure 13 : Compensation of p et L : p = 0.95 and L = 105.
Then, we analyze the effect of the threshold.



                                  15
            1.75




             1.5




            1.25




               1




            0.75




             0.5




            0.25




                      102.5     103      103.5     104      104.5     105


Figure 14 : Ω as a function of L with equality of expectations : p = 1, σ = 20%.
   When the expectations of the two strategies are set equal, the six figures
above show that, according to the Omega performance criterion, most of the
time the CPPI dominates the OBPI.


4    Conclusion
The Omega performance measure takes potentially into account all the moments
of the returns distribution. Thus, it can be used to study asset with non-
normally distributed returns, such as hedge funds, equity in illiquid markets....
    However, as for performance measures based on downside deviations, we have
to assume that the return level of the omega function is exogenously defined but
the loss threshold may be defined by the investor’s preferences.
    The evaluation of an investment with the Omega function should be consid-
ered for thresholds between 0% (above the guarantee in this paper) and the risk
free rate. Intuitively, this type of threshold corresponds to the notion of capital
protection.
    In this paper, we have shown that, for this criteria, the CPPI method seems
better than the OBPI ’one, when assuming Lognormality of the stock price.
Further studies can extend this analysis when jumps may occur or may be
based on generalized downside risk-adjusted performance measures such as the
Kappa.


References
 [1] Bacmann, J.-F. and Scholz, S., (2003): Alternative performance measures
     for hedge funds, AIMA Journal, June.
 [2] Bertrand P. and Prigent J-L., (2002), “Portfolio insurance: the extreme
     value to the CPPI method”, Finance, 23, p. 69-86.



                                        16
 [3] Bertrand P. and Prigent J-L., (2005), “Portfolio insurance strategies: OBPI
     versus CPPI”, Finance, 26, p. 5-32.
 [4] Black, F. and Jones, R. (1987). Simplifying portfolio insurance. The Journal
     of Portfolio Management, 48-51.
 [5] Black, F., and Rouhani, R. (1989). Constant proportion portfolio insurance
     and the synthetic put option : a comparison, in Institutional Investor focus
     on Investment Management, edited by Frank J. Fabozzi. Cambridge, Mass.
     : Ballinger, pp 695-708.
 [6] Black, F. and Perold, A.R. (1992). Theory of constant proportion portfolio
     insurance. The Journal of Economics, Dynamics and Control, 16, 403-426.
 [7] Bookstaber, R. and Langsam, J.A. (2000). Portfolio insurance trading rules.
     The Journal of Futures Markets, 8, 15-31.
 [8] Cascon, A., Keating, C. and Shadwick, W.F., (2003), The Omega Function,
     The Finance Development Centre London.
 [9] Favre-Bulle, A. and Pache, S., (2003). The Omega Measure: Hedge Fund
     Portfolio Optimization, MBF Master’s Thesis, University of Lausanne.
[10] Kaplan, P. and Knowles, J. A.: Kappa, (2004): A Generalized Downside
     Risk-Adjusted Performance Measure. The Journal Performance Measure-
     ment, 8 (3), 42-54.
[11] Kazemi, H., Schneeweis, T. and R. Gupta, (2004), Omega as performance
     measure, Journal of performance measurement, Spring.
[12] Keating, C. and Shadwick, W.F., (2002), A universal Performance measure,
     The Journal of Performance Measurement, Spring, 59-84.
[13] Leland, H.E. & Rubinstein, M. (1976). The evolution of portfolio insurance,
     in: D.L. Luskin, ed., Portfolio insurance: a guide to dynamic hedging,
     Wiley.
[14] Perold, A. (1986). Constant portfolio insurance. Harvard Business School.
     Unpublished manuscript.
[15] Perold, A. & Sharpe, W. (1988). Dynamic strategies for asset allocation.
     Financial Analyst Journal, January-February, 16-27.
[16] Prigent, J-L. (2001). Assurance du portefeuille: analyse et extension de la
     méthode du coussin. Banque et Marchés, 51: 33-39.




                                       17
5     Appendix
Proof of Proposition (1): recall that for a given random variable X, the
Omega performance measure at the threshold L is given by:
                                     h          i
                                              +
                                  EP (X − L)
                      ΩFX (L) =      h          i.
                                  EP (L − X)+
    For the OBPI, the value of X is given by:

                            X = qK + q (ST − K)+ .
    Thus:
                               ³                    ´+
                                        +
                   (X − L)+ = q (ST − K) − (L/q − K) .
                                                    +
    Then, (X − L)+ 6= 0 is equivalent to (ST − K) > (L/q − K). Therefore,
since we must assume that L > qK, we deduce that (X − L)+ 6= 0 is equivalent
to ST > L/q and, in that case, (X − L)+ = q(ST − L/q).
    Consequently, we have:
                      h           i       h           i
                                +                   +
                   EP (X − L) = q EP (ST − L/q) .
Using the same arguments, we deduce that:
                             ³                      ´+
                                                  +
                (L − X)+ = q (L/q − K) − (ST − K)      .
    Therefore, we have two cases:
    1) If ST ≤ K, then:
                             (L − X)+ = L − qK.
    2) If ST > K, then:
                                                +                +
            (L − X)+ = q ((L/q − K) − (ST − K)) = q ((L/q − ST ) .
    Thus, for all cases:

                         £                                         ¤
         (L − X)+     = q (L/q − K)I[ST ≤K] + (L/q − ST )+ I[K<ST ] .
                         h                         i
                      = q (L/q − ST )+ − (K − ST )+ .
    Consequently, we have:
            h          i   h £          ¤    h          ii
         EP (L − X)+ = q EP (L/q − ST )+ − EP (K − ST )+ .
    Finally, we deduce:
                                       h           i
                                    EP (ST − L/q)+
               ΩOBP I (L) =   h             i   h          i.
                                          +              +
                            EP (L/q − ST ) − EP (K − ST )


                                      18

				
DOCUMENT INFO