Portfolio insurance methods for index-funds

Transcription

1 Portfolio insurance methods for index-funds Andreas Lundvik U.U.D.M. Project Report 2005:6 Examensarbete i matematik, 20 poäng Handledare Paul Odermalm, Kaupthing Bank Examinator: Johan Tysk Maj 2005 Department of Mathematics Uppsala University

2 Abstract This thesis concerns the constant proportion portfolio insurance, a pathdependent strategy that forms an alternative to the more complex option based portfolio insurance. In common practice the constant proportion strategies are modified with leverage constraints, creating strategies that can more closely represent a real market decision. By refining these strategies with different rules we can take advantage of their path-dependent structure. Later the strategies are studied in detail and compared in different market situations. The result is that the refined strategies show an enhanced on average performance compared to the common practice of the strategies. This result is further strengthen when we apply our strategies to historic time series of three interesting market-based assets.

3 Contents 1 Introduction 4 2 Option based versus constant proportion portfolio insurance Option based portfolio insurance Constant proportion portfolio insurance Comparison between OBPI and CPPI CPPI strategies with restrictions CPPI gap risk Simulations and actual data scenarios Theoretical CPPI and the relation to volatility Path-dependence for the different CPPI strategies Floor strategies Growing floor strategy Constant floor strategy Comparison of trading frequencies Constant Floor strategies with leverage CPPI gap risk for real data A summary of methods for implementations Conclusions 39 5 Acknowledgements 40 3

4 1 Introduction In fluctuating markets, an investor may want to limit downward risk while allowing for some participation in upside performance. There are two common insurance policies that achieve this, the Option Based Portfolio Insurance (OBPI )and the Constant Proportion Portfolio Insurance (CPPI ). These strategies sell stock as the market falls and buy stocks as the market rises. An insurance guarantee associated with these policies, is usually set to recuperate at least the nominal amount initially invested. This amount is called the principal. The traditional way to protect the value of a portfolio from declining below a certain level is through the use of put options, either directly on the portfolio or the securities constituting the portfolio. The insured amount equals the capital invested in the portfolio less the cost of the put option, also referred to as the insurance premium. Limitations on this option-based model arise from the unavailability of options on the portfolio that need to be insured and the excessive cost of such options. To avoid some of these problems we can use dynamic allocation of risky and risk-free assets to synthetically replicate the protection of the portfolio. This is done using the Black and Scholes [5] pricing methods as further explained in Section 2.1. The CPPI strategy on the other hand is a relatively simple method to dynamically adjust a portfolio, consisting of a risky and a risk-free asset, over time. To implement the strategy, the investor selects a floor below which he/she does not want the portfolio value to fall. The floor grows at a constant rate and must initially be less than total assets. If we think of the difference between portfolio of assets and the floor as a cushion, then the CPPI decision rule is simply to keep the exposure to the risky asset a constant multiple of this cushion. Since Perold introduced the CPPI model, in [12] having its origins in earlier work by Merton [11], an extensive theoretical background has arisen on the subject. This is mostly interesting from an analytical point of view. The aim is to see how the model depends on the different included parameters. However, implementing the theory in the market, you have different factors to take into consideration. For example, restrictions on leverage and transaction-costs both result in pathdependencies, and as a result solving for the optimal trading strategy becomes generally very difficult. Transaction costs are assumed to be relatively small and we will only focus on the leverage constraint in this paper. The theory for the introduction of a borrowing limit in the CPPI model was first developed by Black and Perold in [4] but has been further investigated by many others. An extensive simulation study of how the CPPI strategy performs in different types of market environments (bull, bear and no trend) was developed by Cesari and Cremonini in [7]. They used three performance measures to determine which 4

5 strategy managed to succeed the best in each market environment. Their results show a better performance for the CPPI strategy over other common strategies in bear and no trend markets. The main objective of this paper is to determine which risky asset is most suitable to use in the CPPI strategy, but we will also try to refine existing path-dependent strategies in appropriate ways in order to get a reasonable portfolio performance over time. We begin by pointing out the differences in hedging perspective between the OBPI and CPPI as well as the delicate role of the volatility of the underlying risky asset. We will analyze what role the volatility for the CPPI given by Perold might have for a simulated price process. Later on we extend the use of the path-dependent Ratchet and Margin strategies, developed in [6], to better accommodate market conditions where leverage is constrained. By introducing different variations of these strategies, we are trying to take on the problem of the CPPI strategy going cash-out, i.e. the portfolio completely invested in risk-free assets with no possibility to recover despite the fact that the underlying asset does so. This scenario happened to several CPPI based funds in Europe, mostly in Italy and Ireland, around We will later on apply these refined CPPI strategies on actual time series, the LYNX hedge fund, the OMX stock index and the Russian stock index. 5

6 2 Option based versus constant proportion portfolio insurance In this section we describe the similarities and differences between the option based strategy and the constant proportion strategy. First the technique and requirements of hedging a simple European call option contract is presented. Then the CPPI strategy given by Perold will be explained. In Section 2.3 the comparison between the OBPI and CPPI is formulated. Further in Section 2.4, the different constrained CPPI strategies that form the basis of the application to actual time series is presented. Finally, in Section 2.5 the gap risk that effects the CPPI is clarified. 2.1 Option based portfolio insurance Hedging is a way to reduce risk and thereby protect investments in risky environments. Buying portfolio insurance is a way for the investor to outsource the actual hedging of this insurance in an uncertain market. Financial theory such as the risk-free pricing approach developed by Black and Scholes in 1973 is of highest interest when we want to price an option contract. The theory has practical use in financial institutions all over the world when pricing different kinds of derivatives. To hedge an option based contract, we first have to consider a financial market, including a price vector process S with a time horizon for every S t : t [0,T] and a risk-free asset, called a money market account, denoted by B. The S process can be interpreted as the price process for one of the underlying risky assets. If we now assume a contingent T -claim X, where a contingent claim is defined as a stochastic variable X of the form X =Φ(S T ). (1) In this case the contract function Φ, is set to be a European call option Φ(S T )=max[(s T K), 0] (2) with K being the strike price, that is the stated price per share for which the underlying stock may be purchased for a call or sold for a put, by the option holder upon exercise of the option contract. We assume that there are no transaction costs and it is possible to borrow money at a risk-free rate. Definition 2.1 A self-financing portfolio is a portfolio with no exogenous infusion or withdrawal of money. In other words, the purchase of a new portfolio must be financed solely by selling assets already in the portfolio. A common assumption in financial theory is that the market is arbitrage-free according to the following definition. 6

7 Definition 2.2 An arbitrage possibility on a financial market is a self-financed portfolio h such that V h (0) = 0, but V h (T ) > 0 (3) with probability one. We say that the market is arbitrage-free if there are no arbitrage possibilities. An arbitrage possibility is thus equivalent to the possibility of making money without investing anything with probability one. This arbitrage possibility shall be interpreted as a serious mis-pricing in the market, and the main assumption in arbitrage-free pricing theory is that the market is efficient in the sense that no arbitrage is possible. Now we will move on to the hedging perspective. Definition 2.3 A European call option contract X can be replicated or equivalently is hedgeable, if there exists a self-financing portfolio h such that with certainty VT h = X (4) where h is called a replicating or hedging portfolio and V t is the value process associated with the portfolio. From now on we will use the notional convention h(t) =[h 0 (t),h (t)] where h 0 (t) denotes the number of bonds in the portfolio and h (t) denotes the number of shares in the underlying asset. The market is said to be complete if every contingent claim X is reachable. Assuming that the T -claim X is replicated by the portfolio h, the following scenario can bring some light to the replicating perspective. Suppose that at a fix time t with t T, an investor possesses V h (t) SEK. This amount is then used to buy the portfolio h(t). To follow the portfolio strategy h in the interval [t, T ] will not generate any costs for the investor, assuming there are no transaction costs and the portfolio is self-financing. At time T the portfolio will now be worth V h (T ) SEK. The replicating or hedging assumption will give the value of the portfolio at time T to exactly equal X SEK, regardless of the stochastic price movements of the underlying asset over [t, T ]. This scenario implies that the value at time T of the contract is path independent. From a purely financial point of view, holding the portfolio h isthesameasholding the contract X, i.e. the European call option in this case. We will now try to prove that the contract X can be hedged in the Black-Scholes model. The model has the general form db t = rb t dt (5) ds t = S t [µdt + σ(t, S t )dw t ] where r is the risk-free rate, µ is the drift, σ is the volatility of S and W is called a Wiener process. This process is commonly used for modeling of asset prices and is further described in the following definition. 7

8 Definition 2.4 A stochastic process W is called a Wiener process if the following conditions hold. 1. W 0 =0 2. The process W has independent increments, i.e. if r<s t<uthen W u W t and W s W r are independent stochastic variables. 3. For s<tthe stochastic variable W t W s has the Gausian distribution N[0, t s]. 4. W has continuous trajectories. Showing that model (5) is complete for a general case is complex and requires deep results from probability theory and is not necessary here. This is because for a fairly simple contract, like the European call option, a proof of restricted completeness can easily be performed for the Black-Scholes model. The proof is based on risk-free pricing theory that forms the hedging portfolio for this option contract. But first we have to state the following lemma. Lemma 2.5 There exists an adapted process V and an adapted process λ = [λ 0,λ ] as one hedging strategy with λ 0 t + λ t =1 such that dv t = V t {λ 0 t r + λ t µ} dt + V tλ t σdw t V T =max[(s T K), 0] (6) Then the hedging portfolio has the form h 0 t = λ0 t V t (7) B t h t = λ t V t. (8) S t The strategy is now to look for a process V and a process λ that satisfies these equations. Let us begin by noticing that the process S is a Markov process and therefore the hedging portfoliocanbeassumedtohavetheform h t = h(t, S t ) (9) with the left side being a deterministic function. Further the value process is defined as V t = h 0 t B t + h t S t. (10) 8

9 This process also depends on time t and the price S of the underlying risky asset and can be stated in a similar manner as in the case of h, in equation (9), V t = f(t, S t ). (11) By applying the well known Itô formula to V,weget { df df dv = + µs dt ds + 1 } 2 σ2 S 2 d2 f dt + σs df dw (12) ds 2 ds and by factoring out V it follows that { df df + µs dt ds dv = V + } 1 2 σ2 S 2 d2 f ds 2 dt + V S df ds σdw. (13) V V Now equation (13) looks more like equation (6) and it is easy to see that the factors in front of the diffusion coefficients has the form λ t = S df ds f. (14) By remembering that V t = f(t, S t ) and substituting equation (14) into equation (13) gives us { df dt dv = V + } 1 2 σ2 S 2 d2 f ds 2 + λ µ dt + Vλ σd rf W. (15) This gives an explicit expression for λ 0 df λ 0 t = + 1 dt 2 σ2 S 2 d2 f ds 2 rf (16) and by applying the constraint λ 0 t + λ t the expression becomes =1in Lemma 2.5 and canceling out µ df + 1 dt 2 σ2 S 2 d2 f ds 2 rf = f S df ds. (17) f If we then multiply both sides by rf, weget df df + rs dt ds σ2 S 2 d2 f = rf, (18) ds2 which in fact is the Black-Scholes equation.we now have all that needs to formulate the following. If we have a contingent claim X =max[(s T K), 0] and 9

10 f(t, S t ) is the solution of equation (18) with terminal value given by the contract function, then the call option contract X can be hedged by the portfolio λ 0 t = f 1 ( S df ) (19) f ds ( S df ). (20) ds λ t = 1 f Further the hedging portfolio is given by h 0 t = 1 ( f S df ) B ds h t = df ds conclusively giving us the value process by the relation (21) (22) V h t = f(t, S t ). (23) We have now shown that a European call option contract can be hedged, or equivalently can be replicated. There is still another issue that needs to be addressed. To give the theory above a concrete form we have to perform a numerical estimation of the parameters S, r, t, T and σ. The first four parameters are all given and can be observed directly. However σ, representing the volatility in model (5), has to be estimated. The two main approaches to accomplish this are by using historical volatility or implied volatility. Let us assume that we want to value our European call option with six months left to maturity. One way is then to use historic data of the risky asset S in order to estimate σ. The standard practice here is to use the historic data for a period of the same length as the time to maturity of the contract Φ. In this case it means that we would take historical data for the last six months of S. One argument against the use of historical volatility is the fact that real life volatility is not constant, but changes over time. We do not want to estimate the volatility for the past but for the coming six months. This can be accomplished by using other assets which are already priced by the market, to price our option. Thereby, we take advantage of the market expectation of the volatility for the next six months to come. If there exists another market priced option (benchmark) written on the same underlying risky asset and with the same maturity date, this can be used to find the expected volatility σ. This value of σ, which the market has implicitly used for valuing the benchmark option, is also called the implied volatility and can be solved from the equation c =Φ(S t,t,t,r,σ,k) (24) where c is the market price of the benchmark option and Φ is our option contract. We can say that we price our option based on the benchmark. As seen the 10

11 volatility is not always easy to find for the asset that needs to be hedged which especially concerns stock indices and hedge funds of specific maturities. This imposes difficulties for the financial institution that wants to price the contract. They have to come up with good guesses of how the volatility would evolve over time. The convexity of option prices on convex contract functions like the European call, has been studied in, [1], [8] and [10] among others. These theories can give institutions a reason to overestimate the implied volatility so that they do not give out under priced derivatives. But the result of this scenario can be that the volatility estimate makes the option contract a bit more expensive than it would be if the institution knew the actual volatility. Knowing that is of course impossible from a scientific point of view. Now introducing the main topic of this section, following a similar work in [7]. We can form an insurance portfolio that consists of the replicated European call option and a position in the risk-free asset: V = Call(S, K)+F. (25) If K is a given floor, a portfolio consisting of a European call option with strike price K and the amount invested in the risk-free asset will guarantee the floor K plus any upside increase realized at maturity T. By solving the Black-Scholes equation (18) for a European call option, the following result follows. Theorem 2.6 A price of an European call option with strike K, S as the underlying risky asset and time to maturity T can be given by Call(S t,k)=s t N(d 1 (S t,k)) Ke r(t t) N(d 2 (S t,k)). (26) Here N is the cumulative distribution function for the N[0, 1] distribution and [ d 1 (S t,k)= ln S ] t σ2 1 +(r + (T t) K 2 σ (27) T t d 2 (S t,k)=d 1 (S t,k) σ T t. (28) As we can see a call option is now similar to, and can be hedged with a long position (the quantity N(d 1 (S t,k))) in the risky asset and a short position (the quantity KN(d 2 (S t,k))) in the interest bearing asset. If we now consider a European call and a European put both with strike price K and maturity date T we can form the following relation Put(S t,k)=ke r(t t) + Call(S t,k) S t. (29) By using this put-call parity relation the insured portfolio consisting of a European call option and a position in the risk-free asset can also be seen as a 11

12 European put plus a position in the risky asset. There are several ways to implement the option based portfolio insurance strategy so that it matches the initial investment and the guarantee. We are only going to show one of these because the OBPI is only being referred to as the benchmark against the CPPI strategies, defined further on. The method was first popularized by Perold and Sharpe, in [13], and is easy to implement. This method consists of buying q call options and investing the amount F 0 in risk-free bills, that will insure the principal investment if the call will expire out of the money. The proportion q and exercise price K is determined jointly by the following equations: qcall(s 0,K)+F 0 = V 0 (30) and qk = F T (31) where F T is the floor at expiration, and the initial floor is simply the present value of F T according to F 0 = F T e rt. (32) From equation (30) we can see that the value of purchased call options equals the difference between the floor and the portfolio value at time zero. Equation (31) indicates that the total exercise price equals the floor. Thus the portfolio will never be levered because there is always enough money to exercise the calls. From the function of the option price in Theorem (2.6) and using the put-call parity from equation (29) we obtain qs 0 N(d 1 (S 0,K)) qke rt N(d 2 (S 0,K)) = V 0 (33) so that the initial investment in the risk-free asset should be qn(d 2 (S 0,K)) and the amount in the risky asset should be qn(d 1 (S 0,K)). At any time t the portfolio value is: V t = F t + qcall(s t,k) (34) where the floor is: F t = F T e r(t t). (35) In terms of put options we can buy an equal amount q of shares on the risky asset S and of the European put option written on S according to q [S 0 + Put(S 0,K)] = V 0 (36) The values at maturity will now be V T = qs T if S T >K (37) V T = qk if S T <K This strategy, introduced by Perold and Sharpe in [13], will from now on be noted as OBPI PS. 12

13 2.2 Constant proportion portfolio insurance The constant proportion portfolio strategy is based on the rebalancing between two underlying assets, the risky asset S and the risk-free asset B. The CPPI is technically very simple to apply because the strategy only depends on the performance of the risky asset and is independent of an estimate of the implicit volatility, which is used in the option based model. In this section we will see that the CPPI model, introduced in [12] and further developed in [2] and [6], can be analyzed in a similar manner as given in Section 2.1. To be able to implement the CPPI strategy, the investor must select a certain floor F which he/she does not want the portfolio value V to fall under. Here F evolves according to the following equation df t = rf t dt. (38) The surplus, given by the difference between the value of the portfolio V CPPI and the floor, is called the cushion C. The determination of the amount allocated to the risky asset S is done by multiplying the cushion with a predetermined multiple m. Both the floor and the multiple are initially set according to the investors risk preferences. The exposure E t is set to be the value invested in the risky asset S at time t and is given by the following relation m(v t F t ) if V t >F t E t = (39) 0 if V t F t where m gives a convex payoff for constant values greater than one. A strategy with a convex payoff does best when the price of asset S makes continuing moves up or down or equally is characterized by trends. Such a procedure will sell risky assets as they fall and buy them as they rise. We will from here on assume that the price of the risky asset S follows the classic diffusion process: By solving for S t we get: ds t S t = µdt + σdw t. (40) σ2 σwt (µ+ S t = S 0 e 2 )t (41) with µ and σ as constants. The value of the CPPI portfolio is then obtained as the solution of the following stochastic differential equation dv t = E t ds t + r(v t E t )dt. (42) S t According to the CPPI model given by Perold the exposure has no upper bound, but with a lower limit of zero, see equation (39). In this model the multiple can 13

14 be chosen arbitrarily or to fit a certain initial exposure, assuming that the floor is fixed according to the following equation F 0 V 0 e rt. (43) By assuming that the portfolio is self-financing we then have ds t dv t = mc t + r(v t mc t )dt S t = C t (mµdt + mσdw t )+r(c t + F t mc t )dt = C t ((r + m(µ r))dt + mσdw t )+rf t dt = C t dz t + rf t dt (44) where dz t =(r + m(µ r))dt + mσdw t. If we then rewrite equation (44) using equation (38) we get and further we get dv t =(V t F t )dz t + df t d(v t F t )=(V t F t )dz t d(v t F t )=(V t F t )((r + m(µ r))dt + mσdw t ). By solving the differential equation we get σ2 (r+m(µ r) m2 V t F t =(V 0 F 0 )e 2 )t+mσwt and the expression for the portfolio value at time t is where σ2 (Zt m2 V t = F t +(V 0 F 0 )e 2 t) (45) Z t =(r + m(µ r))t + mσw t. By simplifying the above equation (45) and putting it in terms of the underlying risky asset S we get our CPPI portfolio value at time t V CPPI t ( ) m = F t + C 0 e bt St (46) S 0 where b = ( r m (r σ2 2 ) ) m 2 σ

15 By assuming that F 0 = V 0 e rt the portfolio at maturity equals V CPPI T = V 0 + C 0 e bt ( ST S 0 ) m (47) giving the floor value at time T to equal the initial investment in the portfolio. We will form here on call this model by Perold the theoretical constant proportion strategy (CPPI T ), to limit confusion between this and the restricted strategies that will be defined later on in Section 2.4. If we assume that the price of S at time zero is known, we can calculate the expected value of V T to be: [ ( ) m ] E 0 [V T ]=E V 0 + C 0 e bt ST S [ 0 σ2 σ2 r m(r ) m2 = E[V 0 ]+E C 0 e 2 2 T (S ] 0) m σ 2 (S 0 ) m (eσwt+(µ 2 )T ) m (48) = V 0 + C 0 e σ2 σ2 (r mr+m m2 2 2 )T e = V 0 + C 0 e (m(µ r)+r)t. σ2 (m2 2 +mµ m σ2 2 )T The line three simplification comes partly from the fact that E[e αwt ]=e α2 t 2.Now that we have the expected value of V T we can also calculate a expression for the variance according to Var 0 [V T ]=E 0 [(V T E 0 [V T ]) 2 ] = E 0 [VT 2 ] (E 0 [V T ]) 2 [ ( ) 2m ( ) ] m = E V0 2 + C 0e 2bT ST +2V 0 C 0 e bt ST S 0 S 0 V 2 0 C2 0 e2(m(µ r)+r)t 2V 0 C 0 e (m(µ r)+r)t = C 2 0e (2r 2mr+mσ2 m 2 σ 2 )T (S 0) 2m C 2 0 e(2m(µ r)+2r)t = C 2 0 e2(m(µ r)+r)t (e m2 σ 2T 1). (S 0 ) 2m e(2m2 σ2 +2mµ mσ2 )T (49) As we can see in equation (48), the expected value of the portfolio at maturity is independent of the constant volatility, only determined by the initial portfolio with a given multiple, the market parameters µ and r attime zero. The expression of the variance, seen in equation (49), will of course depend on the volatility of the underlying asset. The indication is however that the CPPI strategy is independent of an estimation of the market volatility in contrast to the option based portfolio insurance strategy. 15

16 2.3 Comparison between OBPI and CPPI To simplify the comparison between the OBPI PS and the CPPI T the proportion q is normalized to one without loss of generality by homogeneity property of the value of the portfolio. V 0 is set to be fixed and q is a decreasing function of the strike price K. See also [2] for an comparison study between the CPPI and the OBPI. The value at time t will now be calculated according to V OBPI t = S t + Put(S t,k)=ke r(t t) + Call(S t,k). (50) This strategy insures the exercise price K and gives the opportunity of any upside increase given at maturity T, where S T if S T >K VT OBPI = (51) K if S T K Now let us see about the CPPI strategy. Here K is the granted level of wealth at time T and F t is the floor value at any time in [0,T] giving the relation F T = F 0 e rt = K (52) and the floor value at time zero, insures K at time T, expressed as a present value of K by F 0 = Ke rt. (53) To compare the option based and the constant proportion strategies the initial portfolio values are assumed to be equal at time t =0, and by using equation (46) combined with equation (53) we have V CPPIM 0 = F t + C 0 e b 0 Sm 0 S m 0 = F 0 e r 0 + C 0 = Ke rt + C 0. (54) Further the value of the OBPI strategy at t =0, is calculated according to equation (50) and we get V OBPI 0 = Ke r(t 0) + Call(S 0,K) = Ke rt + Call(S 0,K). (55) This demonstrates that the initial cushion in the CPPI strategy equals the European call option price for the OBPI strategy calculated from the Black-Scholes 16

17 model at time t =0. An interesting fact with the CPPI function is that we can calculate the dependence of the actual volatility for this portfolio if we know S t. The so called vega function calculated from equation (46) with C 0 = Call(S 0,K) gives us the following expression: vega CPPI CPPI dvt = dσ ( = Call(S 0,K)((m m 2 )σt)e bt St S 0 =((m m 2 )σt)(v t F t ) =((m m 2 )σt)c t. ) m (56) As we can see the sensitivity of the CPPI portfolio value in relation to the actual volatility of the underlying asset S is negative for m > 1. This is further studied in Section CPPI strategies with restrictions A natural restriction on the CPPI given by Perold is to impose a constraint on leverage. This makes the model more realistic and interesting for use in concrete financial decisions. This restriction can for example be 0 <E t <pv t,where p>0 and the modified portfolio will have an exposure according to E t = min(mc t,pv t ). (57) The new portfolio is self-financing and the evolution of the portfolio will then be V t rdt if V t F t dv t = C t dz t + rf t dt if F t <V t < m F m p t (58) V t dx t if V t m F m p t where dx t =(p(µ r)+r)dt+pσdw t. A big difference between the option based strategy, the constant proportion strategy by Perold and the restricted strategies that we are going to define in this section is that the portfolio at any time is path-dependent. One well-known problem with the Constrained CPPI strategy is that it cannot handle strong climbs of the underlying asset S very well. When the portfolio value becomes big in relation to the floor, the Constrained CPPI strategy becomes sensitive to a possible fall in the risky asset. The method then fails to capture any earlier winnings in the portfolio. Further if the cushion gets too small, the exposure to the risky asset will be diminished and the CPPI portfolio becomes 17

18 unable to take off in case of an upturn on the market. These two problems can be handled by introducing variable floor strategies that can take advantage of the path-dependent characteristics in these constrained CPPI models. The Ratchet strategy introduced by Boulier and Kanniganti, in [6], tries to capture any increase in the underlying asset by diminishing the cushion. In this case the ratchet puts the excess cushion in the floor. The part of the cushion that is not used due to the constraint above is mc t pv t m When the following inequality = m p m V t F t. (59) if (m(v t F t ) >pv t ) (60) holds, then the floor and exposure are set according to { Ft+1 = m p V m t (61) E t+1 = pv t giving a new value to the floor and the exposure every time the exposure is greater than the most recent value of the portfolio. This strategy benefits from locking in money already earned by successively ratcheting up the floor. The effect is also that the insurance level is increased accordingly. A problem with this strategy is that it can cash-out, meaning that the CPPI portfolio is almost completely invested in the risk-free asset. As suggested in [6], this problem can be avoided by introducing an initial floor margin M in the Ratchet strategy according to F 0 = V 0 e rt + M 0. (62) This will reduce the initial cushion and also the exposure saving it for better use on a later occasion. The idea is here to try to enlarge the exposure every time it goes below a certain level. When the condition if (m(v t F t 1 ) < 1 2 E ) (63) is satisfied for some trigger value E, the following parameters are adjusted according to: F new t = F t M t M t+1 = 1M 2 t Et new = m(v t F new E = E new. t ) (64) The trigger value E in this Margin strategy can be chosen in different ways. One way to initialize it is to take the exposure determined at t =0,thusE = E 0. 18

19 The Margin strategy partly tampers with the principal guarantee because the margin can diminish the floor over time until another successive ratcheting of the floor is performed. As we have seen this strategy only deals with earned winnings and is thereby safe, except in the case of a huge downturn in the risky asset. This problem is addressed in Section 2.5. The Margin strategy will, in contrast to the strategy in [6], also take advantage of the future surplus in leverage for the model that is limited by the constraint in equation (57). Every time equation (60) holds and the Ratchet is moved upwards, the margin is renewed with a percentage of the maximum allowed exposure at that time. Equation (61) is thereby redefined according to F t+1 = m p V m t E t+1 =(1 x)pv t (65) M t+1 = xpv t where x is the percentage of exposure put in the new margin. This method, the Margin (x percent) CPPI strategy, can solve the problem of a diminishing margin that can happen over time, which is the case with the Margin strategy introduced by Boulier and Kanniganti, in [6]. Another way to increase the cushion, and thereby the exposure, in these constrained strategies is to keep the floor constant over time, only increasing by ratcheting up the floor. This can have a negative effect on performance if the ratchet never comes into play. However, that would require a falling trend, for the underlying risky asset, over the whole time period. This constant floor strategy has two purposes. One is to prevent the exposure from falling too low and the other one is to improve performance. We have introduced a couple of different path-dependent CPPI strategies and to summarize our discussion we list them here: 1. Constrained CPPI C (CPPI with E t = min(mc t,pv t )). 2. Ratchet strategy CPPI R (Constrained CPPI with Ratchet effect). 3. Constant Floor Ratchet strategy CPPI CFR (Ratchet strategy with constant floor instead of evolving according to equation (38). 4. Margin strategy CPPI M (Ratchet strategy with x percent margins). 5. Constant Floor Margin strategy CPPI CFM (Margin strategy with constant floor). 19

20 2.5 CPPI gap risk In this section we will clarify the so called gap risk. The discussion is based on two articles of current interest, see [14] and [9]. The investment guarantee for the CPPI can be jeopardized in extreme cases of market crashes, for example. Theoretically, the value of the CPPI is assured not to go under the floor if the underlying asset falls steeply before the insurance manager can have the chance to rebalance the portfolio. How steep the fall has to be depends on the multiplier m. A general condition between m and an asset fall on the market is 1/m (see calculations below). A historical fall of a larger amount than 10 percent on daily rebalanced data of index or hedge fund type is very unusual. However, if this were to happen it would take a multiple greater than ten for the model to crash through the floor. If we assume that a fall on the risky asset S is equal to 1/m and the time t +1value of the portfolio cannot be less than the floor value at time t, wehave V t+1 = E t S t + R t B t = mc t (1 1/m)+V t (1 + r) mc t (1 + r) =(V t C t )+(V t mc t )r = F t + R t r (66) where R is the amount invested in the risk-free asset, µ = 1/m is the yield and r is the risk-free rate from t to t +1. As we can see the portfolio value after the fall equals the floor value at time t plus the amount invested in the risk-free asset multiplied by the risk-free rate. This implies that the portfolio value cannot go under the floor value if the multiple m is less than 1/µ. Equation (66) also implies that the exposure becomes zero at time t +1if the fall is exactly 1/m. Even if the portfolio value manages to stay above the floor after this possible fall on the market, one problem still remains. The portfolio would nearly cash-out leaving no room for the market to recuperate the earlier losses. One way to avoid this is to choose a multiple with a clear distinction between the relation 1/m and the maximal expected fall of the risky asset. Otherwise this gap risk would have to be dealt with by the investor or by a bank that could give a hard guarantee. Imposing the risk of going under the floor to the investor, is not an appealing option because of the given risk preferences associated with these structured insurance portfolios. A better way can be that a hard guarantee is given by the bank that issues the CPPI or the institution that carries out the actual trading of the assets. This guarantee is performed using structured derivatives that can hedge the possible gap risk. 20

21 3 Simulations and actual data scenarios In this section we will analyze what the role of the volatility might be when comparing the OBPI and CPPI by Perold. We will also implement our refined strategies to get an idea of how they work for differing parameters and market conditions. These strategies are further applied to historical data of three marketbasedassetsandcomparedwithdifferenttrading frequencies. Comparison of the CPPI with increased leverage is analyzed in Section 3.4. In Section 3.5, the gap risk for some scenarios of the market-based assets are analyzed. In the final section we illustrate three different examples for implementation of a refined CPPI strategy. 3.1 Theoretical CPPI and the relation to volatility In this section we are going to analyze the vega function, defined in Section 2.3, for the theoretical CPPI T given by Perold. The expected volatility of the call option price stated at time zero determines the initial cushion vega(m=4) vega(m=5) vega(m=6) 0 20 vega(m=4) vega(m=5) vega(m=6) S t S t Figure 1: Vega with expected σ = 20%, Figure 2: Vega with expected σ = 10%, for CPPI strategy by Perold, as function for the CPPI strategy by Perold, as function of S at t of S at t =0.5. =0.5. As seen in Figure 1 and 2, the sensitivity of the CPPI value with respect to actual volatility is negative, and the higher the multiple the more negative it is. This observation gives us the understanding that with a given S t,thevalueof the CPPI at t would benefit from reaching there with as small actual volatility as possible. 3.2 Path-dependence for the different CPPI strategies We begin by examining the details of the simulations performed in this section. The time horizon of interest is set to be t [0,...,T]. The underlying risky assets 21

22 are simulated using a Monte Carlo method according to equation (40) with µ and σ as constants. In agreement with other working papers, and to get a significant leverage effect, the following values of m have been chosen: m = {4, 5, 6}. We have chosen a leverage constraint p =1to give a non leveraged portfolio. The proportion set in the margin cannot tamper with the principal guarantee too much; it is set to be 2.5 percent and 5 percent of exposure both initially and after every successful ratcheting of the floor. The initial trigger level for the margin is set to be E = E 0. The results we present in this Section are based on three sets of model parameters. They are chosen to give some implications of how the path-dependent structure of the CPPI strategies will work in different market conditions. The first set is a bullish market with high volatility { µ = 10%,σ = 20%,r =5%,T =5years,V0 =,F 0 = V 0 e rt}, (1) the second set is a bullish market with moderate volatility { µ = 10%,σ = 10%,r =5%,T =5years,V0 =,F 0 = V 0 e rt}, (2) and the third set is bearish market with high volatility { µ = 10%,σ = 20%,r=5%,T =5years,V0 =,F 0 = V 0 e rt} (3) with the exception that F 0 = V 0 e rt + M 0 for the Margin strategies. We will examine the evolution of the different dynamic floor portfolios and their composition over time, for sets 1 and 3, in the Figures Even though these scenarios in some cases are fairly unlikely, in terms of constant volatility and a continuously falling market, they can give us some useful indications about how the path-dependency can be used in appropriate ways in managing a CPPI strategy. As we can see in Figures 3-10 the portfolio grows exponentially in all cases S S OBPI Floor E(V) E(V) Time in weeks Time in weeks Figure 3: CPPI C strategy for set (1). Figure 4: OBPI strategy for set (1). and the yield is in between the risk-free rate and the yield of the risky asset. The 22

23 exposure in Figure 3 for the CPPI C is increasing over time as a function of the increasing gap between the floor and the portfolio value. It is notable that the multiple shows small impact on the portfolio value over the period. The CPPI R in Figure 5, has a decreasing exposure to the risky asset over the whole time period. Further, the multiple also has a negative relation to the performance of the portfolio value because of a diminishing cushion. There is a clear trade-off between the effect that locks in the historic winnings from strong market performance and the diminishing exposure that happens over time in upward sloping markets. By comparing CPPI R with CPPI CFR and CPPI M with CPPI CFM in Figures (5,6),(7, 8) and (9, 10), we can see that by introducing a constant floor, the exposure is stabilized and the performance is enhanced compared to the other CPPI strategies with a growing floor. We can also see that the difference in proportion set in the margin has no significant effect on portfolio performance and is from here set to be 2.5 percent giving a reasonably high initial cushion. The main observation for this scenario is that, though the Ratchet and Margin strategies shows a significantly smaller cushion than the Constrained strategy, the impact on performance between the strategies is relatively small. Thereby, we can get a similar performance with increased amount of money locked in by the ratchet effect increasing the insurance level accordingly. However, the OBPI strategy, seen in Figure 4, has shown to be a preferable strategy for this simulated asset because of the bullish market condition, without any downward trend. Strategy E(V T ) Std(V T ) Ratchet(%) Margin(%) Underlying asset OBPI PS CPPI C (m=6) CPPI R (m=6) CPPI CFR (m=6) CPPI M (m=6) CPPI CFM (m=6) CPPI T Table 1: Summary of performance and standard deviation on the best alternative for each strategy on set 1. The use of dynamic floor techniques, seen in the last two columns, is given in percent of all trading occasions. The marginal for the simulations was set at 2.5 percent. In the second scenario of the CPPI C, seen in Figure 11, the portfolio starts off being highly invested in the risky asset and thereby heavily exposed to the downward sloping yield. After a while it becomes disinvested in the risky asset and later on tangents the floor that grows at the risk-free rate, recuperating at least the nominal principal investment at maturity. The OBPI strategy, seen in Figure 12, shows to have a similar evolution as in the constrained case. If we look at Figure 13, we see that the Ratchet strategy in average locks in 23

24 S S E(V) E(V) Time in weeks Time in weeks Figure 5: CPPI R strategy for set (1). Figure 6: CPPI CFR strategy for set (1) S S E(V) E(V) Time in weeks Time in weeks Figure 7: CPPI M (2.5 percent) strategy for set (1). Figure 8: CPPI CFM (2.5 percent) strategy for set (1) S S E(V) E(V) Time in weeks Time in weeks Figure 9: CPPI M (5 percent) strategy for set (1). Figure 10: CPPI CFM (5 percent) strategy for set (1). 24

25 Strategy E(V T ) Std(V T ) Ratchet(%) Margin(%) Underlying asset OBPI PS CPPI C (m=6) CPPI R (m=6) CPPI CFR (m=6) CPPI M (m=6) CPPI CFM (m=6) CPPI T Table 2: Summary of performance and standard deviation on the best alternative for each strategy on set 2. The use of dynamic floor techniques, seen in the last two columns, is given in percent of all trading occasions. The marginal for the simulations was set at 2.5 percent. Strategy E(V T ) Std(V T ) Ratchet% Margin% Underlying asset OBPI PS CPPI C (m=6) CPPI R (m=6) CPPI CFR (m=6) CPPI M (m=6) CPPI CFM (m=6) CPPI T (m=6) Table 3: Summary of performance and standard deviation of the best alternative for each strategy on set 3. The use of dynamic floor techniques, seen in the last two columns, is given in percent of all trading occasions. The marginal for the simulations was set at 2.5 percent. 25

26 some of the upturns in the risky asset giving a higher expected performance at maturity than the Constrained strategy. Comparing CPPI R with CPPI CFR and CPPI M with CPPI CFM in Figures shows that the constant floor strategy would not fulfill the principal guarantee in the scenario of these model parameters. This is because the ratchet effect is almost put out of play when the underlying risky asset has a continuous down sloping trend. The assumption of such a trend is however very unlikely for stock index or other actively managed assets of time intervals stretching over 5 years. Nevertheless this can give us indications of how the strategy with constant floor would react in case of a shorter downward sloping trend within the time interval. A summary of performance and standard deviation for the different sets are presented in Tables 1-2, where we also have calculated the expected use-percentage for the ratchet and margins. An observation seen in these tables is that if the volatility of the underlying asset is lowered the expected performance of the strategies will be better for all CPPI strategies. Another interesting fact is that when the volatility is halved the use of the ratchet technique is almost doubled in terms of trading occasions. The theoretical CPPI without restriction on leverage will off course give a higher expected performance but also with a considerably higher risk. If we take a look at Table 3, we see that the CPPI strategies with growing floor will give the best performance for this bearish market environment. The use of the ratchet is reduced but the use of the margin is increased compared to the simulations of set (1) and set (2) in Tables E(V) 90 S E(V) 90 S OBPI Floor Time in weeks Time in weeks Figure 11: CPPI C strategy for set (3). Figure 12: OBPI strategy for set (3). 26

27 S E(V) 90 S E(V) Time in weeks Time in weeks Figure 13: CPPI R strategy for set (3). Figure 14: CPPI CFR strategy for set (3) E(V) 90 S E(V) S Time in weeks Time in weeks Figure 15: CPPI M (2.5 percent) strategy for set (3). Figure 16: CPPI CFM (2.5 percent) strategy for set (3) E(V) 90 S E(V) S Time in weeks Time in weeks Figure 17: CPPI M (5 percent) strategy for set (3). Figure 18: CPPI CFM (5 percent) strategy for set (3). 27

28 3.3 Floor strategies In this section we will compare the Ratchet strategies and the Margin strategies, defined in Section 2.4, to see which impact a constant floor will have on the performance of the portfolio based on real data. We will also examine which role the frequency of trading imposes on these models. In a realistic scenario the model specifications for the CPPI strategies must be modified. The underlying asset S from the earlier sections is replaced by real market-based asset. We have chosen to replace S with the LYNX hedge fund, from May 2000 to Feb 2005, the OMX stock index and the Russian stock index, from Feb 1998 to Jan B is the so called risk-free asset, but instead of evolving according to risk-free rate r, it depends on the development of Swedish treasury bills (OMRX) over time. The value of the portfolio is set at time t =0to be as in the theoretical case. When you trade with hedge funds or other actively managed funds, an issue will arise; that they often have very limited options market. Even though the options can be replicated, the volatility would be hard to approximate because of the lack of benchmark options. This makes the CPPI more appropriate to use in our first scenario even though the option-based solution might have given a better performance. Another interesting underlying asset would be the OMX stock index, which has shown both huge down and upturns over the past six years. This gives us a suspicion that a ratchet based CPPI strategy will work well for this asset. The Russian stock index has shown high volatility over the past six years. A danger with these kinds of high volatility assets is that the CPPI can cash-out, or nearly cash-out, making further performance of the portfolio very limited. For this reason we will apply a more defensive approach by lowering the multiple (m) in our CPPI strategies Growing floor strategy Looking at Figure 19 and 20, the choice between the Ratchet strategy and the Margin strategy seems to make little difference considering portfolio performance. This is because the LYNX hedge fund has a constant growth over the whole period, which makes the Margin strategy less suitable for this particular asset. The exposure will never go under the trigger value for the whole period. See equation (63) and (64) for a definition for the use of the trigger value on exposure. We will from here on only focus on the Ratchet strategy for the LYNX hedge fund. CPPI applied on the OMX stock index can be seen in Figures 21 and 22. The Ratchet strategy with growing floor cashes-out as a result of the fall on the 28

29 LYNX LYNX Value Value Time in days Time in days Figure 19: CPPI R strategy with LYNX hedge fund as underlying risky asset. Figure 20: CPPI M strategy with LYNX hedge fund as underlying risky asset OMX OMX Value 140 Value Time in days Time in days Figure 21: CPPI R strategy with OMX stock index as underlying risky asset. Figure 22: CPPI M strategy with OMX stock index as underlying risky asset. Swedish market during , giving no possibility to take advantage of future up-going trends. However, the Margin strategy succeeds to take advantage of more exposure by pushing the floor downwards and increasing the cushion over time. In our last scenario the Russian stock index, with fluctuations early in the timeperiod, shows similar results as the OMX based CPPI strategies did. The Ratchet and Margin strategies, seen in Figures 23 and 24, both cash-out on an early stage of the time-period and cannot account for the recovery of the index that happens later on. 29

30 Russian index CPPI(m=2) Floor(m=2) CPPI(m=3) Floor(m=3) RUSSIA Value 300 Value Time in days Time in days Figure 23: CPPI R strategy with Russian index as underlying asset. Figure 24: CPPI M strategy with Russian index as underlying asset Constant floor strategy As we can see in Figure 25, the Constant Floor Ratchet strategy, applied on the LYNX hedge fund, manages to use more exposure in up-going trends than the growing floor strategies in the last section, and thereby the overall portfolio performance is enhanced. This result can also be seen in Table 4 below. If LYNX Value Time in days Figure 25: CPPI CFR strategy with LYNX hedge fund as underlying risky asset. the constant floor technique for the CPPI is applied to the OMX stock index the results, seen in Figure 26 and 27, is satisfactory compared to the growing floor CPPI strategies. By using a constant floor strategy, the portfolio value will be kept above the floor after the steep fall that made the growing floor strategies to cash-out. The exposure to the risky asset can thereby increase portfolio performance later on when the index begins to recover. If we apply the constant floor strategies to the Russian stock index, the results are similar to the OMX based CPPI strategies. By observing Figure 28 and 29, we see that the 30

31 OMX OMX Value 140 Value Time in days Time in days Figure 26: CPPI CFR strategy with Figure 27: CPPI CFM strategy with OMX stock index as underlying risky asset. OMX stock index as underlying risky asset RUSSIA CPPI(m=2) Floor(m=2) CPPI(m=3) Floor(m=3) RUSSIA Value 300 Value Time in days Time in days Figure 28: CPPI CFR strategy with Russian index as underlying asset. Figure 29: CPPI CFM strategy with Russian index as underlying asset. early fluctuations on the Russian market will almost make the CPPI to cashout, but with time the constant floor manages to increase the cushion so that the market recovery, that happened during the last two years, can be accounted for. There is a slightly better performance for the Ratchet strategy over the Margin strategy for this asset. This can be explained by the huge volatility over the period, which makes the margin effect less suitable Comparison of trading frequencies We are now going to sum up the results from our different floor strategies from Section and 3.3.2, and perform a comparison between daily, weekly and monthly trading frequency of the underlying risky asset. föras seen in Table 4, if the trading frequency for the CPPI strategies on the LYNX hedge fund would be lowered, the portfolio would decrease in performance. 31

32 Strategy Daily V T Weekly V T Monthly V T LY NX CPPI R (m=4) CPPI CFR (m=4) CPPI R (m=5) CPPI CFR (m=5) CPPI R (m=6) CPPI CFR (m=6) Table 4: Summary of Ratchet CPPI performance on different trading frequencies for the LYNX hedge fund. This fact can be related to the longer trends of the daily traded asset compared to the weekly or monthly traded underlying asset. This is also the case with the OMX stock index as the underlying risky asset for the Ratchet and Margin strategies. If we look at the results in Table 5, we see that the weekly trading frequency makes the portfolio value go through the floor for m = 4, 5. This shows that by decreasing the trading frequency would not only diminish the performance, but the gap risk would also increase. Moreover, an interesting observation seen in Table 6 is that the Margin strategy manages to keep the portfolio value over the floor for the same set of CPPI parameters traded on a weekly basis. Performance is, however; equal for the best performing choice of multiple between the Ratchet and the Margin strategy with constant floor. Further, we can take a look at Table 7 and 8 representing the CPPI strategies with the Russian stock index as underlying risky asset. The main observation is that there is a huge difference in portfolio performance between the strategies with growing floor and the ones with constant floor. If we assume a lower trading frequency than daily for this asset the CPPI value would go through the floor for every scenario. When this happens the CPPI rebalancing strategy is put out of play and the results becomes insignificant. Even though this scenario has Strategy Daily V T Weekly V T Monthly V T OMX CPPI R (m=4) CPPI CFR (m=4) CPPI R (m=5) CPPI CFR (m=5) CPPI R (m=6) CPPI CFR (m=6) Table 5: Summary of Ratchet CPPI performance on different trading frequencies for the OMX stock index. Left out values means that the strategy has crashed through the floor. 32

33 Strategy Daily V T Weekly V T Monthly V T OMX CPPI M (m=4) CPPI CFM (m=4) CPPI M (m=5) CPPI CFM (m=5) CPPI M (m=6) CPPI CFM (m=6) Table 6: Summary of Margin CPPI performance on different trading frequencies for the OMX stock index. Strategy Daily V T Weekly V T Monthly V T Russia CPPI R (m=2) CPPI CFR (m=2) CPPI R (m=3) CPPI CFR (m=3) CPPI R (m=4) CPPI CFR (m=4) Table 7: Summary of Ratchet CPPI performance on different trading frequencies for the Russian stock index. Left out values means that the strategy has crashed through the floor. given the best performance of the three assets that were analyzed the approach to this asset had to be adjusted with a lower multiple. If the multiple would be higher than four, the strategies would all crash through the floor. The pathindependent structure given by option replication would probably end up with a higher performance than the CPPI strategy for this scenario. The option based insurance strategy would thereby be a more appropriate alternative for the Russian stock index. 33

34 Strategy Daily V T Weekly V T Monthly V T Russia CPPI M (m=2) CPPI CFM (m=2) CPPI M (m=3) CPPI CFM (m=3) CPPI M (m=4) CPPI CFM (m=4) Table 8: Summary of Margin CPPI performance on different trading frequencies for the Russian stock index. Left out values means that the strategy has crashed through the floor. 3.4 Constant Floor strategies with leverage We have chosen to loosen up the restriction that were imposed on the pathdependent CPPI strategies up until now and see what would happen if we compare the strategies from 3.3 with strategies allowing for increased leverage effect, see equation (60) with p=1.5. By comparing Figure 30 with Figure 25 from Section 3.3.2, we see that allowing for leverage in a restricted form heavily enhances the performance of the CPPI CFR portfolio compared to the unlevered ratchet based strategy. This is because the leverage manages to take better advantage of the up going trends of the underlying asset. Nevertheless, the investor would have a comfortable growth with a constantly increasing insurance level if he/she chooses the unlevered alternative. If we compare Figure 31 with the unlevered strategy in Figure 26 from Section 3.3.2, we see that the CPPI CFR with restricted access to leverage does not outperform the unlevered strategy which was the case with the hedge fund. This can be explained by the fact that the underlying asset is characterized by both downward and upward consecutive moves, which evidently makes these strategies more suitable without any need of leverage. 34

35 LYNX Value Time in days Figure 30: CPPI CFR (p=1.5) strategy with LYNX hedge fund as underlying risky asset OMX Value Time in days Figure 31: CPPI CFR (p=1.5) strategy with OMX stock index as underlying risky asset. 35

36 3.5 CPPI gap risk for real data In this section, we are going to make the connection between the gap risk, defined in Section 2.5, and our real time-series from Section 3.3. By analyzing the yield given by the price-movements on the risky assets we will calculate a multiple, according to equation (66), that would be the highest possible for the CPPI portfolio to stay above the floor. Between 30 Dec and 2 of Jan the LYNX hedge fund lost 4.3 percent, which represents a multiple of less then 23.1 to guarantee that the portfolio stays above the floor. This can be seen in Figure 32 on page 37, where we have chosen two multiples: one that exactly tangents the floor and one that illustrates the scenario when the portfolio crashes through the floor. The OMX stock index suffered a loss of 8.1 percent between 10 and 11 Sept This gives a theoretical multiple of less than 12.4 to secure the portfolio value over time. However, as we can see in Figure 33 on page 37, the OMX stock index is at the time in a downward trend with 11 consecutive falls of the asset price and the portfolio is barely exposed to the OMX index with only 2 percent weight in the risky asset. This has also impact on the gap risk and the multiple had to be adjusted to 15.5 to make the portfolio value go through the floor. As noted earlier the Russian stock index is highly volatile and will be very sensitive to changes of the multiple, and is therefore left out in this section. However, any value of the multiple above 4 will make the portfolio go trough the floor. The conclusion we can make of these results is that the LYNX hedge fund and the OMX stock index is safe to implement with reasonable choice of multiple while the Russian stock index would have problem staying above the floor and is much more sensitive when implementing our CPPI strategies. 3.6 A summary of methods for implementations The ways of implementing the constant proportion strategy is not always straightforward. With this in mind, we are going to give three examples of actual ways of initialize this strategy. All of these examples are based on the CPPI CFM strategy defined in Section 2.4, and traded on a daily basis to reduce gap risk and enhance performance. The ideas are based on the findings we have made in earlier sections and two articles of current interest, seen in [14] and [9]. 1. CPPI CFM with individual solution over an open time period. Initially the investor has to choose a risk profile and determine how much money that can be lost from the principal in a worst case scenario. This is performed through a specified multiple and floor value that fits the underlying asset. The maximum exposure is set to be the maximum value of 36

37 LYNX CPPI(m=23.1) Floor(m=23.1) CPPI(m=24.0) Floor(m=24.0) Value CPPI Crash July 2002 Oct 2002 Jan 2003 Mars 2003 June 2003 Time in days Figure 32: CPPI CFR strategy with LYNX hedge fund as risky asset. The first multiple (23.1) is set to tangent the floor and the second multiple (24.0) is set to go through the floor CPPI Crash OMX CPPI(m=12.4) Floor(m=12.4) CPPI(m=15.5) Floor(m=15.5) Nov 2000 Sept 2001 Juli 2002 May 2003 Figure 33: CPPI CFR strategy with OMX stock index as risky asset. The first multiple (12.4) is set to tangent the floor and the second multiple (15.5) is set to go through the floor. 37

38 the portfolio at that time. The time period for the CPPI is not fixed, and in this sense the strategy is more flexible then an option based alternative. If an investor wants to add or withdraw money this can be done within predetermined time windows. Withdrawals within the first five years are penalized by a decreasing percentage of the annual performance. This will give the investor a reason to keep his or her money invested for at least the first five years. 2. CPPI CFM with common solution over an open time period. In this solution an institution will initialize the portfolio and keep it running for an indefinite time period. The investor has to accept the way the portfolio is managed and is merely a bystander in this scenario. Investors can buy shares in a fund and subscribe to the yield that the fund gives. 3. CPPI CFM with common solution over a fixed time period. The design of this solution is more like the option based alternative. The fund runs for a fixed number of years. The multiple and the floor are set according to the underlying asset s risk profile. Investors will have a chance to invest in the fund at the begin of the first year. Early withdrawal is not allowed. At the end of the final year, the fund will give the investor the performance made by the portfolio strategy over the fixed time period and the fund is terminated. Alternatives to these solutions can be an initial limitation on the proportions between the exposure to the risky and the risk-free asset. Careful solution with 0-70 percent exposure to the risky asset. This can be interesting if the investor is careful and needs more safety than performance in the CPPI portfolio. This is also a good way to limit risk in an illiquid market without derivatives that can hedge the gap risk properly. More risky solution with 10- percent exposure to the risky asset. The main advantage with this alternative is that it eliminates the possibility of the portfolio going cash-out, but it also comes with a higher risk. 38

39 4 Conclusions In this thesis an alternative to portfolio insurance have been examined, namely the constant proportion portfolio insurance (CPPI). The traditional way to protect the value of a portfolio from declining below a certain level is through the use of put options. The CPPI strategy, on the other hand, shows to be a relatively simple method to adjust a portfolio, consisting of a risky and a risk-free asset, dynamically over time. To implement the strategy, the investor selects a floor below which she does not want the portfolio value to fall. The floor evolves over time according to the interest rate to assure the initial amount at the end of the time period. If we think of the difference between portfolio of assets and the floor as a cushion, then the CPPI decision rule is simply to keep the exposure to the risky asset a constant multiple of this cushion. There are mainly two benefits with this constant proportion over the traditional option based portfolio insurance (OBPI). The first one is that CPPI is less sensitive to an estimated value of the implicit market volatility. The second benefit is that the strategy does not have to be time-limited, as the traded option on the market does. These differences make the CPPI strategy less expensive in terms of volatility estimates, and more flexible in terms of composition and living up to the demands given by the investors. We extend the path-dependent Ratchet and Margin CPPI strategies by Boulier and Kanniganti to better accommodate market conditions where leverage has realistic constraints. The ratchet technique dynamically increases the floor after periods of good performance thereby, locking in the prior profits made by the portfolio. The benefit with the Margin strategy is that it saves surplus given by the ratchet technique which can be used on a later occasion to increase the cushion when the market turns up. By introducing different variations of these strategies, we solve a problem with the CPPI portfolio going cash-out, i.e. the portfolio is completely invested in the risk-free asset with no possibility to recover despite the fact that the underlying risky asset does so. Implementing the Ratchet and the Margin strategy with simulated assets, we observe that the cushion decreases over time limiting the possibility of market exposure but the strategies compensate for it with an increased level of protection. If we refine the strategies with a constant floor, instead of a floor growing with the interest rate, this will help to stabilize the evolution of the exposure over time. The only scenario for the growing floor that outperforms the constant floor strategies would be if the underlying risky asset has a continuous downward sloping trend over the whole time period. These results are further strengthen when we apply our strategies to historic time series of the LYNX hedge fund, the OMX stock index and the Russian stock index. In comparison with an option based insurance strategy, our refined CPPI strategies with constant floor has shown to be preferable in certain scenarios. This is the case especially when the underlying 39

40 risky asset is characterized by trends. By substituting the growing floor with a constant floor we tamper with the guarantee of the nominal initial investment, only to see that the overall performance is enhanced. The real assets that we analyze in this thesis would all benefit by being traded on a daily basis. This would both limit the gap risk and enhance performance through longer trends of the underlying risky asset. The gap risk can further be limited by choosing a value of the multiple with a clear distinction to the maximal historic fall of the risky asset. If restricted leverage is allowed in the constant floor CPPI strategies the performance for the LYNX hedge fund would be enhanced but that is not the case with the two other risky assets. The OMX stock index would not give a significant difference between leverage and no leverage, but in the case with the more volatile Russian stock index the portfolio crashes through the floor when leverage is allowed. While the use of a log-normal model on the price movements has given us some useful results, they can be further improved if we take into account the possibilities of steep falls between two replacements. More realistic simulation results could be achieved by using an alternative model such as a jump-diffusion process for the underlying asset price. 5 Acknowledgements I would like to thank my supervisor Dr. Paul Odermalm at Kaupthing Bank Sverige AB, for introducing me to the subject and giving me interesting ideas and guidance during this work. I would also like to express my gratitude to my examiner, Docent Johan Tysk at the Department of Mathematics at Uppsala University for discussions and helpful support. Finally, I would like to thank Alexandra, for being patient and understanding during a period when I needed it the most. 40

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