Constant Proportion Portfolio Insurance in Defined Contribution Pension Plan Management under Discrete-Time Trading

Transcription

1 Constant Proportion Portfolio Insurance in Defined Contribution Pension Plan Management under Discrete-Time Trading Busra Zeynep Temocin 1, Ralf Korn 2, and A. Sevtap Selcuk-Kestel 1 1 Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey 2 Department of Mathematics, University of Kaiserslautern and Financial Mathematics, Fraunhofer ITWM, Kaiserslautern, Germany Abstract We consider the optimal portfolio problem under discrete-time trading with minimum guarantee protection in a defined-contribution (DC) pension scheme. We compare various versions of guarantee concepts in a labor income coupled CPPI-framework with random future labor income. Besides classical deterministic guarantees we also study path-dependent floors. Conducting risk analyses by computing risk measures such as expected shortfall and cash-lock probability and through sensitivity analyses we test the efficiency of constant proportion portfolio insurance strategies with various floors. 1 Introduction Defined-contribution type pension funds can be considered as a form of savings where external contribution payments constitute the principal amount of the portfolio wealth. Being a part of the mandatory pension system, they are the main source to finance retirement in many countries and are becoming more common in other countries where the participation is still on a voluntary basis. As a result of this rapid growth, pension systems have become a popular subject for researchers in recent years. For an overview on the development in the pension funds modelling, see [8]. Although pension systems are created to increase financial life quality by providing income after retirement, many risks are associated with these systems. The uncertainty linked to the retirement income and the fact that the pension participant/beneficiary is directly exposed to the btemocin@metu.edu.tr (Corresponding author) korn@mathematik.uni-kl.de skestel@metu.edu.tr 1

2 financial risk of the plan portfolio, place importance on the pension fund modelling. Especially, a downside-protection against market conditions has become an important main focus in this context. With the aim of reducing the impact of market/investment risk, several protection schemes have been designed and proposed in the literature [4, 15]. The main objective is to provide a floor to the value of savings that will be accumulated until retirement for a specific contribution stream. One popular example for a strategy with downside protection is Constant Proportion Portfolio Insurance (CPPI) which is a dynamic portfolio insurance strategy that aims to protect the investor against adverse market movements by guaranteeing at least an initially specified fixed amount of money at the investment horizon (see [1, 9, 6, 3, 16]). The optimal control problem is then formulated in a different way using the difference between this assumed floor and the portfolio value instead of a classical maximization of expected utility of terminal wealth. For an example of the classical stochastic optimal control problem under stochastic interest rates, see [7]. Applying this kind of a portfolio insurance strategy to pension funds raises further investment risks as the investment risk is usually much longer than a typical financial investment. While providing a simple and tractable guarantee scheme, continuous time CPPI is not realistic as it assumes continuous rebalancing unlike real discrete-time trading. Therefore, in this paper we study various discrete-time CPPI strategies. One main issue that arises in CPPI strategies with a fixed rate floor is a phenomenon known as cash-lock. This is the situation when the portfolio wealth ends up fully invested in the risk-free asset without the chance to recover. Since a cash-locked position prevents any participation in a market rise, it is considered as a critical risk, especially for investments with long horizons. In the classical (continuous-time) CPPI, once the exposure drops to zero it stays there until the end of the investment period. However, in a framework where floor interference is possible, this problem can be prevented by adjusting the floor downwards according to market conditions. A detailed study on the problem under dicrete-time trading is given by Balder and Mahayni in [12]. Another prominent risk called gap-risk is the probability of portfolio value falling below the floor level and failing to guarantee the desired amount. This risk is measured by the risk measure Expected Shortfall (ESF) which is computed based on the shortfall probabilities. Balder, Mahayni and Balder studies the problem for discrete-time classical CPPI with a fixed-growth floor in [11]. The literature also includes hedging strategies with artificial assets to model jumps and price gap-risk. In [14], the exotic so-called gap options are introduced to hedge against gap events, or jumps in a Lévy-type framework. An additional issue occurs in the case of a major increase in the market when the floor becomes insignificant comparing to the portfolio value (C(t) V(t)). Since this eventuality may create a high potential loss and prevent from taking advantage of the market rise, a modification is to include a ratchet mechanism into the floor. The main problem is that although portfolio value increases significantly in rising markets, due to high cushion amount, the potential loss is also high as long as floor remains low. This approach is introduced in [5] 2

3 under-discrete time trading assuming a fixed-growth floor. In this paper, we study a defined-contribution pension plan with different types of guarantee schemes to be met at the retirement date. We address all these risks that are present under discrete-time trading and study each variable-floor CPPI strategies modifying them specificially for our framework. The main assumption of our pension plan is that the guarantee at the retirement time will be equal to summation of the contribution payments made by the plan participants which are themselves driven by stochastic process. The first floor we assume the net present value (NPV) of these future consecutive payments. This NPV floor then grows at the risk-free rate whereas the portfolio is made up from the premium payments made regularly while evolving above this floor. The second floor we introduce is the so-called random floor. Starting with the initial contribution, this floor process is enchanced with each payment made at the fixed payment dates. We then analyze the prominent risks that these strategies involve by computing and comparing their risk measures; cash-lock probability, expected shortfall and shortfall probability. After examining these tractable CPPI strategies, to overcome the risks stated above, we consider variable floor strategies under the same framework. Modifying them for our random-growth floor process, we study constrained CPPI, Ratchet CPPI and Margin CPPI with Rathcet effect. Constrained CPPI involves a strict constraint on the exposure, excessive borrowing and leverage. Next, we assume the Ratchet floor which decreases the risk of overgrowing cushion. Lastly, we include margin effect to minimize the cash-lock risk by decreasing the floor at a minimum level. Since our framework does not have enough initial margin which enables us to decrease the floor without manipulating the end guarantee, we make a modification in the mechanism. With a detailed sensitivity analysis we discuss and compare the effectiveness of each strategy. The main novelty of our approach for portfolio insurance compared to existing literature, is that we formulate the problem in a defined-contribution setting with the presence of a randomly growing floor. Unlike the classical CPPI where the investor selects a certain fixed floor which he/she does not want the portfolio value fall under, we study different stochastic floor processes along with floors with path-dependent structures. Another difference between our approach and previous approaches is that we combine the discrete-time trading assumption with a stochastic floor CPPI, and carry out analyses for risks encoutered in real markets such as gap and cash-lock risk. The rest of the paper is organized as follows. The market model and notation are given in Section 2. CPPI under discrete-time trading and various CPPI strategies are then given in Section 3.1. Comparison of strategies based on the computed risk measures is presented in Section 4 with numerical results given in Section 5. Section 6 concludes. 2 Pension Fund Management Considering a defined-contribution plan, our main aim is to attain a guarantee on the pension fund. Besides a classical deterministic guarantee this can be 3

4 a random relative guarantee that we specify below. The horizon for the fund management is [,T] where time t = is the subscription date of the beneficiary and T denotes the retirement date. We consider regular contribution payments (i.e. monthly/yearly) that are defined as a proportion of the contributor s labor income. Furthermore, we model the labor income as a stochastic process also reflecting the risks of the financial market. 2.1 The market model To keep things simple, we consider a continuous-time securities market where the investor can invest into a money market account B and a stock (or a stock index) S with price dynamics given by db(t) = rb(t)dt, (1) ds(t) S(t) = µ Sdt+σ S dw(t), S() = S. (2) Here,W isabrownianmotiondefinedonthecompleteprobabilityspace(ω,f,p) endowed with the filtration {F t } t [,T]. µ s, σ s are real constants with σ s >, and the interest rate r is assumed to be constant. It is well-known that this modelling implies that the financial market is complete, i.e. sufficiently integrable F t -measurable random variables can be replicated via following a suitable self-financing trading strategy in the two securities B and S. 2.2 Defined-Contribution modelling In a defined contribution (DC) type pension scheme, each beneficiary periodically contributes a defined amount into his/her pension account. We assume that the contribution payments are a constant proportion, γ, of the beneficiary s labor income at fixed time instants t i [,T] and the pension plan involves an ongoing stream of contribution payments. The realistic modeling of labor income can be very hard as there are a lot of relevant stochastic variables such as disability, mortality and economic or political crises. Therefore, we suppose that the labor income, L(t), is a stochastic process satisfying the following stochastic differential equation dl(t) L(t) = µ Ldt+σ L dw(t), L() = L, (3) where µ L and σ L are assumed to be real constants. The labor income process at time t is thus L(t) = L e (µl σ2 L /2)t+σLWt, (4) assuming t = and t n = T. Under this setting, the discrete time defined-contribution stream γ(t i ) has the form γ(t i ) = γl(t i ), 4

5 for i =,1,2...,n with the dynamics dγ(t) = γdl(t), (5) for t [,T]. Note that because of the market completeness, these future contributions can be fully hedged. Therefore, the stream of future payments can be replicated using a combination of the assets and a unique price can be assigned to the claim of future premiums under the risk-neutral measure. We will make use of this in different ways below. 3 Constant Proportion Portfolio Insurance A Constant Proportion Portfolio Insurance (CPPI) is a popular dynamic portfolio insurance strategy that aims to protect the investor against adverse market movements by guaranteeing at least an initially specifed(typically) fixed amount of money at the investment horizon (see [1, 9, 6]). This method uses a simplified strategy to allocate assets dynamically over time based by the introduction of a surplus called cushion and denoted by C. The cushion is defined as the difference between the current portfolio value V and the present value of the guaranteed level Y, called floor. The investor starts by setting a floor equal to the lowest acceptable value of the portfolio. Then, he/she computes the cushion as the excess of the portfolio value over the floor and determines the amount allocated to the risky asset by multiplying the cushion by a predetermined multiplier, m. Both the floor at time T and the multiplier are representatives of the investor s risk tolerance and are exogenous to the model. The total amount allocated to the risky asset is known as the exposure, e, given by the following relation { mc(t) if C(t) > e(t) = if C(t). (6) The remaining funds are then invested in the risk-free asset. To accomplish the main aim of a CPPI strategy which is always evolving above the floor process, the floor must be hedgeable and the explicit form of the corresponding hedging strategy must be determined. 3.1 CPPI under Discrete-Time Trading In this section, we study a discrete-time version of the classical CPPI in the context of DC pension plans and describe the general discrete dynamics. The generalsettingissimilartothatofbalder, Brandl&Mahayniin[11], howeveras a distinction, we formulate the problem in the presence of consecutive random contribution payments into wealth at fixed times. After giving the discrete CPPI model, we further study more specific CPPI strategies and study pathdependent structures. Since instant rebalancing is not possible under discretetime trading, some major risks are present such as cash-lock and gap risk. Since 5

6 there is an inflow to the portfolio value on a regular basis, the CPPI portfolio is not self-financing. In our discrete-time framework, we assume that the trading is done immediately after the contribution payment γ(t i ) at time t i [,T] for i =,1,...,n 1. Let τ = {t,t 1,...,t n } be the set of fixed payment dates. Once the trading is done at time t i, the number of shares held in the risky asset is constant until the next trading date, that is over the time period (t i,t i+1 ]. Instead of considering invested fractions of wealth, which may change between rebalancing times, we take into account the number of assets at time t and denote it by φ = (α,β). Here, α and β are the number of units of the stock index and the bond, respectively. Definition 3.1. ([11]) A strategy φ = (α, β) is called simple discrete-time CPPI if for t (t k,t k+1 ] and k =,..,n 1 { } e(tk ) α t = max,, β t = V t k α t S tk. (7) S tk B tk It is important to note that due to the infeasibility of instant rebalancing, the cushion may become temporarily negative. Since negative asset exposure arises in such a case, a constraint is imposed on the number of risky asset as given in (7). Let the CPPI strategy held at time t k be represented by φ tk = (α tk,β tk ), then the value process of the DC pension portfolio under discrete-time is given by { α tk S t +β tk B t, t (t k,t k+1 ) V t (φ tk ) =. (8) α tk S tk+1 +β tk B tk+1 +γ(t k+1 ), t = t k+1 The premium payment γ(t k+1 ) which will be invested into assets an instant later at time t + k+1, incurs a jump in the portfolio value at t k+1. That is, we have V(t k+1 )(φ tk ) = V(t k+1 )(φ t k )+γ(t k+1 ) = V(t + k+1 )(φ t k+1 ), where φ tk+1 is the portfolio held after the rebalancing at time t k+1, for all k =,1,...,n 1. An additional assumption we make is the limited borrowing. Imposing the constraint e(t) = min{mc(t),v(t)} (9) we prevent unlimited borrowing for all of the strategies to be introduced. Another novelty of our study comes from the fact that we analyze the implementation the CPPI strategy, under various floor assumptions. Unlike the classical CPPI where the investor selects a certain fixed floor which he/she does not want the portfolio value fall under, we study different stochastic floor processes along with floors with path-dependent structures. 6

7 3.2 Various CPPI strategies under Discrete-Time Trading In this section, we present various floor variants under discrete-time trading and analyze the corresponding CPPI strategies CPPI with deterministic (NPV) floor The first floor we consider is the net present value (NPV) floor which by definition has deterministic dynamics. In the CPPI strategy with the NPV floor, CPPI NPV, we set the initial floor to be the discounted value of the future contributions instead of fixing a constant guarantee amount for time T. This initial floor, denoted by Ȳ then grows at the risk-free rate r until maturity date. To understand the dynamics of this deterministic floor, we next compute the expectation of future payments. Let Z(t) denote the market price at time t of the stream of future contributions payable between time t and T. Then have, Z(t) := E Q e r(ti t) γ(t i ) F t = γl(t)g(t), (1) i:t i t with g(t) = i:t i t e(µl r θσl)(ti t). Here, the risk-neutral dynamics of L(t) is given by dl(t) L(t) = (µ L σ L θ)dt+σ L d W(t), where θ = µs r σ S is the market price of risk and W is the Brownian motion under Q. In particular including the possible contribution exactly at time t in Z(t), we obtain the risk-neutral dynamics of Z(t) as whereas under P it satisfies dz(t) Z(t) = rdt+σ Ld W(t) dz(t) Z(t) = (r +σ Lθ)dt+σ L dw(t). (11) Next, we give a remark regarding the replication of the process Z(t). Remark. At payment times t i, the differential does not exist and the process evolution is given by Z(t i ) = Z(t i + )+γ(t i ), for i =,1,2,...,n. Z(t) is correlated with the assets in the market, it can be hedged perfectly between payment times. To show this replication, we consider a self-financing trading strategy ϕ = (ϕ B,ϕ S,ϕ Z ) and choose ϕ Z = 1. This strategy satisfies dv ϕ (t) = ϕ B (t)db(t)+ϕ S (t)ds(t) dz(t) = (ϕ B rb +ϕ S Sµ S Z(r +σ L θ))dt+(ϕ S Sσ S Zσ L )dw. 7

8 Equating the diffusion terms and the drift terms to zero, we obtain the number of assets as ϕ S (t) = Z σ L and ϕ B (t) = Z ( 1 σ ) L. S σ S B σ S Note in particular that due to the form of ϕ B (t) the pair (ϕ B,ϕ S ) is indeed self-financing. Moreover, the jumps or contribution payments occurring in Z(t) at payment dates are also hedgeable thanks to market completeness. Therefore, perfect replication is possible for the stream of future contributions Z(t). By Equation (1), we can now define (NPV) floor at time T as The dynamics is then given by Ȳ := Y()e rt. Ȳ(t) = Ȳ()ert (12) with Ȳ() = ρz(). Here, ρ represents the guaranteed fraction of the amount of total contributions. For numerical calibration, we assume.5 < ρ < 1 and test the effectiveness as well as practical convenience of the portfolio for different values of ρ. From (8), the dynamics wealth process are given as follows V(t) = { mc(tk ) St S tk +(V(t k ) mc(t k )) Bt B tk, C(t k ) > V(t k ) Bt B tk, C(t k ) (13) for t (t k,t k+1 ), and mc(t k ) St k+1 S V(t) = tk +(V(t k ) mc(t k )) Bt k+1 B tk +γ(t k+1 ), C(t k ) > V(t k ) Bt k+1 B tk +γ(t k+1 ), C(t k ) (14) for t = t k+1. Using (14) and (12), we derive the cushion at time t (t k,t k+1 ) as follows: { ) C(tk )( m St S C(t) = tk +(1 m)e r(t t k), C(t k ) > (15) C(t k )e r(t t k), C(t k ) with C(t k+1 ) = C(t k+1 )+γ(t k+1), for all k =,...,n 1. One should note from the dynamics (13) - (15) that to stay always above the floor, one has to use m 1. Otherwise, there is a non-zero probability of a temporarily negative cushion in a decreasing financial market. The fact that 8

9 the cushion process becomes negative only temporarily is one of the major differences between our framework and the classical CPPI. With the help of the contribution inflows, the portfolio has the chance to recover and attain a positive cushion again after a negative surplus CPPI with random-growth floor In our DC pension setting, the randomness that labor income process possesses plays an important role in the evolution of the portfolio. Since the contribution payments are made into the portfolio wealth at fixed times, it is in fact logical to increase the floor process also by the same increments for the sake of ensuring the gains. Therefore, the second floor that we introduce is the random-growth floor which not only grows at the risk-free rate between inter-payment times, but also makes jumps at the consecutive payment times as the value process does. For the sake of flexibility in gains, a fraction of each contribution is guaranteed, similar to what has been done in the NPV floor case. More precisely, for each payment γ(t i ) made at time t i [,T], cγ(t i ) is included in the floor for some real constant < c < 1. Therefore, jumps occurring in the floor process are also fractions of the inflows made into wealth. We define the random-growth floor (which we will shortly address as random floor from now on) to be the summation of the time-value of paid contributions. The payments are made at fixed trading dates t i [,T] and the accumulated premiums grow at the risk-free rate r between payment times. The guarantee at time t is then defined as k e r(t ti) cγ(t i ), t (t k,t k+1 ) Y(t) = i=. (16) Y(t k+1 )+cγ(t k+1), t = t k+1 Hence, between payment dates Y(t) has the dynamics and dy(t) = ry(t)dt, Y() = γ() Y(t) = Y(t )+γ(t) (17) for t = t k, k =,1,...,n. By (16) and (17), Y can be given as { e r(t tk) Y(t k ), t (t k,t k+1 ) Y(t) = Y(t k+1 )+γ(t. (18) k+1), t = t k+1 From (8), the wealth process is given as, V(t) = { mc(tk ) St S tk +(V(t k ) mc(t k )) Bt B tk, C(t k ) > V(t k ) Bt B tk, C(t k ), (19) 9

10 for t (t k,t k+1 ), and mc(t k ) St k+1 S V(t) = tk +(V(t k ) mc(t k )) Bt k+1 B tk +γ(t k+1 ), C(t k ) > V(t k ) Bt k+1 B tk +γ(t k+1 ), C(t k ). for t = t k+1. As cushion is defined by C(t) = V(t) Y(t), t, the cushion satisfies C(t k ) = V tk (φ tk ) Y(t k ) (2) = V t (φ tk )+γ(t k ) Y(t k ) cγ(t k) k = C(t k )+(1 c)γ(t k), (21) for all k =,1,...,n. Relation (21) implies that for c = 1 the cushion process has no discontinuity at payment times, i.e. C(t) = C(t ), t [,T]. (22) This is because of the fact that the floor is increased by the full contribution amount at each payment time when c = 1. From (18), we derive the cushion dynamics for t (t k,t k+1 ) as follows: { ) C(tk )( m St S C(t) = tk +(1 m)e r(t t k), C(t k ) > C(t k )e r(t t k), C(t k ) (23) with C(t k+1 ) = C(t k+1 )+(1 c)γ(t k+1), for all k =,...,n 1. Again from the dynamics (19), (2) and (23) one should note that it should be m 1, for the value process to always evolve above the floor. Otherwise, there is a nonzero probability of a temporarily negative cushion in a decreasing financial market. The CPPI strategies with the floors given so far possess some prominent weaknesses in real markets. One potential problem from the performance standpoint arises when market climbs. When the risky asset prices rise yielding an increasing portfolio value as well, the floor might become insignificant threatening the gains. Another risk emerges when the cushion becomes so small. In such a case the total wealth faces the danger of being fully invested in the riskfree asset and staying below the floor until maturity. Furthermore, due to the fact that rebalancements are done at discrete time intervals rather than continuously in practice, there is a risk of cushion becoming negative between two 1

11 rebalancement dates. To overcome these problems, some CPPI modifications are suggested by Boulier and Kanniganti in [5]. These modified strategies introduce some mechanisms which enable the floor to vary based on strong market conditions and results in a path-dependent structure. We extend these modified CPPI strategies to our DC setting with growing floor and make performance comparisons. Next, we study these path-dependent CPPI strategies in our DC pension plan setting CPPI strategies with variable floors In this section, we introduce a new structured CPPI for DC pension funds. As pension funds are long-term investments, the phenomena of floor becoming insignificant or being very close to portfolio value can both occur during the investment period. Therefore, a variant form of the classical CPPI which is able to prevent these risks and but still insure the desired guarantee is needed. The basic idea is to combine the ratchet effect and margin method in one strategy. The dynamics of the floor in the combined strategy is given as the combination of the equations (28) and (31) with the constraints (33) and (3). Now that we have the dynamics of different CPPI strategies, we can compare their performances using risk measures. For all path-dependent floor cases, the plan participant makes periodical contribution payments during the time he/she stays in the plan, as described in Section 3.1. Constrained CPPI In this strategy, there is a stronger constraint on the exposure compared to 9. The restriction is given as < e(t) < pv(t) for some real constant p > with the exposure having the following relation e(t) = min{mc(t),pv(t)} (24) where the floor process is as given in (18). Here, the asset units have to be redetermined based on the exposure at each time. Therefore, instead of using the unconstrained exposure as in (7), we have α t = max with e(t) as given by (24). { e(tk ) S tk, }, β t = V t k α t S tk B tk (25) Ratchet strategy The basic idea of the ratchet strategy is to increase the floor when market rises and floor threatens to become insignificant by adding the excess cushion. With the realistic constraint (24) on the exposure, the absolute value of excess cushion is given by mc(t) pv(t) m = C(t) p V(t). (26) m 11

12 This excess cushion is put on the floor when mc(t) > pv(t), (27) resulting the new floor and exposure ( ) mc(t) pv(t) Y new (t) = Y(t)+ (28) m ( = 1 p ) V(t) (29) m with the new exposure being the minimum given in (24). By increasing the floor and lowering the risk of a greater loss when market decreases, ratchet strategy increases the level of protection. However, the cushion might become so small in this case, causing a cash-lock position. Another mechanism can be employed for this problem, as given in the next section. Margin strategy with ratchet effect To avoid the problem where the exposure approaches to zero creating a cash-lock situation, one method is to decrease the floor when this happens. In the classical CPPI, this can be done by artifically augmenting the initial floor by some margin amount and adjusting the floor using this margin, as suggested in [5]. If it is the case that initial exposure is too high, the initial floor is then set at a higher value and the margin is used later when the floor falls too low. Thus, if e() is too high, the initial floor is augmented by margin M Y new () = Y()+M, the idea is to adjust the floor downward when exposure hits a predetermined lower bound e (t) where M grows at the risk-free rate r. For example, use a fraction of the remaining margin every time the exposure hits a fraction of the current lower bound e (t). Thus, if the new floor will be with new margin being and the new exposure e(t) < e (t) = e(t) ǫ Y new (t) = Y(t) M t ǫ M new t (3) (31) = (ǫ 1) M t (32) ǫ e new (t) = m(v(t) Y new (t)). (33) Here, it is a key issue to determine the lower bound e (t) to increases the portfolio perfomance. To be able to provide an insight to this lower bound 12

13 choice problem, we analyze the effect of ǫ with a sensitivity analysis and discuss the results in Section 5. Another weakness of the margin strategy suggested in [5] is that in a strongly declining market the margin might diminish the floor in time and cause a failure to meet the guaranteed amount. Apart from the fact that the margin mechanism described above has a risk of diminishing the floor, it can not be directly applied to our framework as the initial exposure given by e() = m(v() Y()) = m(1 c)γ() canneverbehighenoughtomakethemechanismwork. Namely, theinitialmargin suggested for our approach would be so small that it would fail to decrease the floor low enough to keep the cushion positive. An alternative approach for us would be to use an independent deterministic process as a margin process but it would also cause diminished floor. However, this approach would also include the risk of drastically decreased floor. Taking these facts into consideration, we adjust the margin approach in a way that the diminishing floor problem is solved and the margin is capable of bringing it down, effectively. We adopt the mechanism so that every time the condition (27) is satisfied and the floor is increased with the ratchet effect, the margin is reset to a certain percentage, of the exposure at that time. New exposure thereby is redefined as e new (t) = (1 h)pv(t) (34) and the new margin is M new (t) = hpv(t) (35) where < h < 1 is the percentage of exposure put in the new margin. 4 Risk Measures and Comparison of Strategies In this section, we compute and analyze some risk measures of the various CPPI strategies considered in this study. Besides making a comparison of different floor cases (deterministic, random or variable), our main result is a comparison between financial positions taken at the beginning of the pension plan. We examine the portfolio performance when a replicating portfolio for the future contributions is short sold with the case where the payments are made consecutively. In calculation of the risk measures to be presented, we encounter the probability of a sum of lognormal random variables. While closed-form expression of a lognormal sum probability density function is unknown, there exist well-known analytical approximation methods in the literature [17], [18]. We adopt the method of Fenton-Wilkinson who approximate the lognormal sum by a single lognormal random variable, and use moments to determine the parameters of the new lognormal distribution, as often done in pricing Asian and basket options. 13

14 We find exact values for the pre-event probabilities and approximate the more complicated post-event probabilities as will be explained in the subsections. We indicate the approximate values using the relevant approximately equal symbol. After obtaining the risk measures, we present a calibration example and discuss the effectiveness of the CPPI strategies with NPV and random floor. 4.1 Cash-lock Risk Our pension framework includes consecutive payments into the fund at fixed times. These inflows which change the portfolio allocation drastically in turn serves as a recoverer from a possible cash-locked position. Because of the pathdependent structure of the strategy, the cash-lock probability analysis has to be carried out for each period, i.e. for inter-payment times. Therefore, we first examine local dynamics. Next, we give definition of local cash-lock probability for the interval (t k,t k+1 ]. Definition 4.1. (Local cash-lock probability) For the period (t k,t k+1 ], the ζ-ξ cash-lock probability P CL ζ,ξ t k,t k+1 denotes the probability that the proportion of the risky asset at t k+1 is less than ξ given that it is equal to ζ at t k, i.e. ( P CL ζ,ξ mc(tk+1 ) t k,t k+1 := P ξ mc(t ) k) = ζ, (36) V(t k+1 ) V(t k ) for any k =,...,n 1. Notice that when cushion becomes negative, the fraction of the portfolio value which is invested in the risky asset also becomes negative. This violates the general continuous-time definition of a cash-lock probability where ζ usually satisfies ζ 1. The main reason of this violation is the fact that we assume discrete-time trading. The portfolio for the case ζ > includes risky asset, therefore the calculation of the probability becomes complicated and we obtain a summation of two lognormal random variables. Because of the inflow payments made into the wealth, we consider the cash-lock probability under two different cases. First, for the case where the payment at time t k+1 has not come in yet and for the case where it has been paid an instant ago. Since these incoming payments include an external randomness, the derivation of the cash-lock probability becomes complicated for the latter case. The cash-lock probability in the first case can be considered as an upper bound for the local cash-lock probablity. We define this upper-bound for the probability as P CL ζ,ξ := P t k,t k+1 ( mc(t k+1 ) V(t k+1 ) ξ mc(t k) = ζ V(t k ) ). (37) Namely, assuming that the payment at the beginnning has already been made and the one end of the period has not come in yet, we find the cash-lock probability for inter-payment times. It is clear that for ζ <, this upper bound is 1 as CPPI forbid any investment in risky asset for the specific period. Therefore, the 14

15 probability (37) provides some insight only for the case ζ >. The dynamics of the strategy, as given in (7), does not allow investment in risky asset at time t k+1 if C(t k ) <. Therefore, for this case, cash-lock happens if the incoming payment fails to make the cushion positive. In the next sections, we give the cash-lock probability of each strategy Cash-lock Risk NPV floor CPPI In this subsection, we analyse the cash-lock risk of the CPPI strategy with the NPV floor. Proposition 4.2. (Upper bound of local cash-lock probability) For ξ, an upper bound for ζ-ξ cash-lock probability in NPV floor CPPI is given by ) (µ S r σ2 S 2 ) T n P CL ζ,ξ = N t k,t k+1 ( ln ξ(m ζ) mζ(m ξ) 1 m m σ S T/n (38) for ζ > and P UB ζ,ξ = 1 for ζ. t k,t k+1 Proposition 4.3. (Local cash-lock probability) For ξ, the ζ-ξ cash-lock probability in NPV floor CPPI is given by ) (µ L r σ2 L 2 ) T n P CL ζ,ξ t k,t k+1 = N ln ( Y(tk )m(ξ ζ) γ(t k )(m ξ)(m ζ) σ L T/n (39) for ζ, and P CL ζ,ξ t k,t k+1 = N ( ln ln d h(a,b) f 2 (a,b) ( h(a,b) f 2 (a,b) ) ) (4) for ζ > where with a = Y(t k )ζm (m ξ) ( ) Y(t k )e r T n ξ ζ(m ξ)(1 m) m ζ. f(x,y) = xe 1 2 σ2 S T n +ye 1 2 σ2 L T n, h(x,y) = x 2 e 2σ2 S T n +y 2 e 2σ2 L T n +2xye 1 2 (σ2 S +σ2 L )T n m ζ e(µs σ 2 S 2 )T n, b = γ(t k )(m ξ)e (µl σ2 L 2 ) T n and d = Notice that for the case ζ >, both the cushion and the portfolio value include the risky asset return, as is given by the Equations (13), (14) and (15). Since the cash-lock probability does not have an explicit solution for this case due to existence of multiple randomnesses, we derive the upper bound probability which is given in the following proposition. 15

16 4.1.2 Cash-lock Risk random floor CPPI In the next propositions we give cash-lock probability and upper bound for the different cases of ζ in random floor CPPI. Proposition 4.4. (Upper bound of local cash-lock probability) For ξ, an upper bound for ζ-ξ cash-lock probability in random floor CPPI is given by ( ) P CL ζ,ξ t k = N ln ξ(m ζ) mζ(m ξ) 1 m m (µ S r σ2 S 2 ) T n (41),t k+1 σ S T/n for ζ > and P CL ζ,ξ t k,t k+1 = 1 for ζ. Proposition 4.5. (Local cash-lock probability) For ξ, the ζ-ξ cash-lock probability in random floor CPPI is given by ( ) P CL ζ,ξ t k,t k+1 = N ln Y(t k )m(ξ ζ) γ(t k )(m ζ)(m(1 c) ξ) (µ L r σ2 L 2 ) T n (42) σ L T/n for ζ, and P CL ζ,ξ t k,t k+1 = N ( d ln ln h(a,b) f 2 (a,b) ( h(a,b) f 2 (a,b) ) ) (43) for ζ > where the functions f and g are the same as in Proposition 4.3 with b = (1 c)b. Notice that for the case ζ >, both the cushion and the portfolio value include the risky asset return, as is given by the Equations (13), (14) and (15). Since the cash-lock probability does not have an explicit solution for this case due to existence of multiple randomnesses, we derive the upper bound probability which is given in the following proposition. 4.2 Gap Risk This section analyzes gap risk. To determine the effectiveness of each CPPI strategy discussed above, we study the risk measures shortfall probability and expected shortfall alongside expected terminal wealth and its standard deviation. Gap risk is the probability of a critical situation resulting in a dramatic fall in the value of the risky asset between two rebalancing dates. In worst possible scenario, the CPPI portfolio can fall below the floor before the manager could rebalance. In such a situation the CPPI portfolio would loose its capital protection. The higher the value of multiplier m, the higher would be the gap risk. The value 1/m is often referred to as the gap size which refers to the maximum loss that could be sustained between two rebalancing date before the portfolio value crashes through the floor. 16

17 4.2.1 Gap risk for NPV floor CPPI Shortfall Probability The shortfall probability is the probability of the final wealth being less than the guaranteed amount. Next, we define the shortfall probability. Definition 4.6. (Shortfall Probability) The shortfall probability P SF denotes the probability that the final value of the CPPI strategy is less than the guaranteed amount Y(T), i.e. P SF := P(C(T) < ). The local shortfall probability P LSF is the probability that the cushion is negative after one time step, given the cushion is non-negative before, that is P LSF := P(C(t k+1 ) < C(t k ) > ). It is important to note that by local shortfall probability for period (t k,t k+1 ), we describe the probability of cushion becoming negative between times t k and t k+1. Thus, we exclude the case where cushion becomes negative after time t k and recovers back before time t k+1 as this case does not affect the cushion dynamics that will prevail for the following period. Moreover, we carry out the analysis for two time instants; at time t k+1 when the contribution payment has been made and at an instant before time t k+1 where the payment has not come in yet. Denoting the probability in the latter case as P LSF := P(C(t k+1 ) < C(tk ) > ), we have the following proposition. Proposition 4.7. (Local shortfall probability) The local shortfall probability at time t k+1 in NPV floor CPPI is given by P LSF = N t k,t k+1 ( ( ln m 1 ) ) m (µs r σ2 S 2 ) T n. (44) σ S T/n At time t k+1, that is, after the payment has been made, the approximate probability is given as ( e ln r T n ( m 1 m ) ) h(ã, b) Pt LSF f k,t k+1 = N 2 (ã, b) ), (45) ln ( h(ã, b) f 2 (ã, b) 2 where ã = e (µs σ S 2 )T n and b = e (µl σ2 L 2 ) T γ(t k) n mc(t k ). Proposition 4.8. The shortfall probability is given as n 1 P SF = 1 (1 Pt LSF k,t k+1 ). (46) k= 17

18 Expected Shortfall The expected shortfall (ESF) measures the amount that is lost if a shortfall occurs. Next, we give the definition of ESF. Definition 4.9. (Expected Shortfall) The expected shortfall is the amount which is lost if a shortfall occurs, i.e. ( ) ESF := E C(T) C(T) <. Considering the path-dependent structure of our strategies, we define and analyze a localized version of ESF denoted by ESF L. Definition 4.1. (Local Expected Shortfall) The local expected shortfall at time t k+1 is given as ( ) ESF L = E C(t C(t t k+1 ) k+1 ) <. (47) k+1 Next, we give ESF L in NPV floor CPPI for the period (t k,t k+1 ). Proposition (Local Expected shortfall) The local expected shortfall at time t k+1 in NPV floor CPPI is given by C(t k )F 1, for C(t ESF L P = LSF k ) > t t k,t k+1 (48) k+1 C(t k )e r T n, for C(t k ) At time t k+1, that is, taking the end of period payment into account, we find the approximate ESF L is as follows ESFt L k+1 = C(t k )e r T n γ(t k)e µ L C(t k )F 2 e µ L T n N( d σ L T n ) P LSF t k,t k+1, for C(t k ) > T n N( d σl T n ) P LSF t k,t k+1, for C(t k ) (49) where and F 2 = me µs T n N( d σs T n )+(1 m)er T n P LSF d = ( ln d = C(t k) γ(t k ) ) (µ L r σ2 L 2 ) T n σ L T n ( e ln r T n ( m 1 ln m ) h(ã,ˆb) f 2 (ã,ˆb) ( h(ã,ˆb) f 2 (ã,ˆb) ) ). t k,t k+1, 18

19 4.2.2 Gap Risk for Random floor CPPI Shortfall Probability Note that, as the next payment is not included in the calculations, the local shortfall probability at time t k+1 for random floor case is the same as in NPVfloor case. Proposition (Local shortfall probability) The local shortfall probability at time t k+1 in random floor CPPI is the same as in NPV case ( ( P LSF ln m 1 ) ) t k,t = N m (µs r σ2 S 2 ) T n. (5) k+1 σ S T/n At time t k+1, that is, after the payment has been made, the approximate probability is given as ( e P LSF ln r T n ( m 1 m ) ) h(ã,ˆb) f t k,t k+1 = N 2 (ã,ˆb) ), (51) where ˆb = (1 c) b. ln ( h(ã,ˆb) f 2 (ã,ˆb) Expected Shortfall Following proposition gives ESF L in random floor case for the period (t k,t k+1 ). Proposition (Local Expected shortfall) The local expected shortfall at time t k+1 in NPV floor CPPI is given by C(t k )F 1, for C(t ESF L t = P LSF k ) > t k,t k+1 (52) k+1 C(t k )e r T n, for C(t k ) At time t k+1, that is, taking the end of period payment into account, we find the approximate ESF L is as follows ESF L t k+1 = C(t k )e r T n (1 c)γ(t k)e µ L C(t k )F 2 (1 c)e µ L T n N( d σ L T n ) P LSF t k,t k+1, for C(t k ) > T n N( d σl T n ) P LSF t k,t k+1, for C(t k ) (53) where F 2 = me µs T n N( d σs T n )+(1 m)er T n P LSF d = ( ln C(t k) γ(t k ) ) (µ L r σ2 L 2 ) T n σ L T n t k,t k+1, 19

20 and d = ( e ln r T n ( m 1 ln m ) h(ã,ˆb) f 2 (ã,ˆb) ( h(ã,ˆb) f 2 (ã,ˆb) where F 1 is the same as in Proposition ) ). As mentioned in Section 3.2, the CPPI strategies we consider are path-dependent. Since this dependent structure prevents tractability of the models, the only possible computable risk measures represent local dynamics. Therefore, the best way to study the evolution of the measures we calculated in the previous section is to illustrate them through trajectories. Figures 1-2 present realizations of cash-lock probability, shortfall probability and expected shortfall for the parameter set as given in Table 1 and for differing multiplier values NPV floor CPPI m=1 m=3 m=6 m=1 m=12 m= Random floor CPPI m=1 m=3 m=6 m=1 m=12 m= Figure 1: Local cash-lock probabilities for a given trajectory The regularly made inflows are provide a great protection against cash-lock and this effect can be clearly seen in Figure 1. Even for a very high multiplier value of 15, the maximum probability is bounded above. For the NPV floor CPPI, the probabilities are equal to 1 for the period that cushion is negative indicating a high risk of being close to floor and decrease drastically as the portfolio value recovers. 2

21 m=1 m=3 m=6 m=1 m=12 m=15 NPV floor CPPI Random floor CPPI m=1 m=3 m=6 m=1 m=12 m= Figure 2: Local shortfall probabilities for a given trajectory The effect that the shortfall probability increases for higher multiplier values is reflected by Figure 3. The displayed irregularity is again a result of the transition of the portfolio from being in loss to recovery. As shortfall probability quantifies the risk based on the distance of portfolio value and the floor through trajectory, as the portfolio touches the floor and cushion becomes zero, the probability suddenly jumps to 1. It is important to note that once the portfolio begins to evolve above the floor, the shortfall probabilities resemble to those of random floor CPPI as their dynamics are the same for C(t) > m=1 m=3 m=6 m=1 m=12 m=15 NPV floor CPPI Random floor CPPI m=1 m=3 m=6 m=1 m=12 m= Figure 3: Local expected shortfall values for a given trajectory Lastly, Figure 2 displays the direct relation between multiplier and the expected shortfall supporting the result of shortfall probability. Note that, at the time when NPV portfolio touches the floor and cushion becomes zero, the ESF also is zero. Note that for all cases presented above, the probabilites are bounded above. This happens because we prevent unlimited borrowing. Even in a bullish market as ours, the maximum exposure is limited to V which puts an upper bound on the shortfall risk. 21

22 5 Numerical Results To illustrate the behaviour of the various CPPI strategies discussed throughout the paper, we give some numerical examples in this section. The set of input parameters is given in Table 1. Throughout this section this parameter set is used in numerical analyses unless stated otherwise. Table 1: Values of model parameters. Interest rate, r.5 Stock parameters Drift, µ S.12 Volatility, σ S.2 Labor income parameters Drift, µ L.6 Volatility, σ L.9 Guarantee rate, c.8 Risk aversion parameter, η 2 Contribution rate, γ.1 Lower bound coefficient, ǫ.25 Multiplier, m 2 Margin parameter, h.5 Constraint parameter, p.5 Time Horizon of pension plan, T 2 Table 5 presents the sensitivities of the moments of terminal wealth to various parameters for different CPPI schemes. When we look at the table and compare horizontally, we see that there is rough ordering between the means of the five strategies. The random floor always gives the highest terminal wealth. This can be explained by the unconstrained structure of the strategy. Since there are no investment mechanisms that limit the invested amount, the portfolio with the random floor climbs up in rising markets and takes advantage of the increase in the stock. However, this freedom comes with a drawback and results in poor performance when the market performes badly. Therefore, as can be seen from the standard deviation values in the random floor column, this strategy also has the highest risk. The second highest wealth is provided by the Ratchet CPPI portfolio. As a nice advantage of the ratchetings performed during the investment horizon which decrease the cushion and therefore proportion of wealth at risk, the gains are protected. This is verified by the low standard deviation compared to random and NPV floor cases. 22

23 Table 2: Distributional properties of CPPI portfolio with various path-dependent floors for different parameter values while other parameters are held at those in Table Parameter Value Ratchet Margin Constrained Random NPV E(V T ) σ(v T ) E(V T ) σ(v T ) E(V T ) σ(v T ) E(V T ) σ(v T ) E(V T ) σ(v T ) σ S µ S σ L µ L c γ m ǫ h p

24 The next best strategy in terms of high terminal wealth is NPV floor CPPI. While being able to yield higher returns than margin and constrained strategies, it also possesses a very volatile profile represented by the high standard deviation. This is another expected result, as NPV floor CPPI is also does not have a control mechanism on the exposure. The reason that it is able to provide high returns comes from the fact that the initial floor in this case is the present value of the future premiums and evolves deterministically without tempering with the incoming payments. Therefore, the floor starts at a high value and thus at much higher value than the other floors which eventually pulls the portfolio above itself. A prominent risk in this scenario is when the participant decides to withdraw from the sistem. Because the strategy needs time to recover and provide gains by investing in the risky asset. Hence, pairing this fact with the risky structure of the strategy, we can conclude that NPV floor CPPI is profitable for those who wish to stay in the system for a long time and can bear the risk. When we look at the constrained CPPI, we see that it s mean wealth values are lower than the others that comes with a very low risk. The outcome of the low wealth and risk is due to the constrain mechanism which limits the exposure being invested in the risky asset and eventually prevents portfolio from taking advantage of a possible rise in the market. While being very convenient for the risk averse investors the constained CPPI is able to yield a reasonable return. The last strategy is the margin strategy with the ratchet effect. This strategy has the lowest risk because it possesses the most protective and varying floor. At times when the floor decreases to reduce the cash-lock risk, the portfolio temporarily follows a downward trend. This indeed affects the profits in the overall resulting in a relatively lower wealth. However, the ratchetings performed during the investment horizon guarantees the gains at each step decreasing the volatility and providing more protection. While being in a competitive level with the other floors, margin strategy offers a really small risk. Therefore, it is preferable for the most conservative investors. It is also interesting to see the sensitivities on a strategy basis. An increase in stock volatility decreases the terminal wealth while an increase in µ S enhances the terminal wealth for all strategies. As expected, higher σ S results in higher standard deviation for terminal wealth and incireasing µ S also increases the riskiness of the portfolio. Looking at the labor income parameters, we see that while being a small change, a rise in σ L also increases the mean and standard deviation of the terminal wealth. The labor drift µ L also has a positive effect on the moments. The guarantee parameter c is negatively proportional with the terminal wealth mean and standard deviation, as it increases the floor resulting in smaller invesment in risky asset and promising lower profits. The contribution parameter also not surprisingly has a positive effect on the moments as it increases the money inflow to the fund. The margin parameters ǫ and h only positively impact the margin strategy as it is not an input for the other strategies. Lastly, for an increase in the constraint parameter p, the moments also increase displaying a strong positive sensitivity. Overall, the hight standard deviations that Table 5 presents for each parameter 24