The term cash-lock refers to a situation where the portfolio value of the CPPI is lower than the floor at some point in time, i.e. the cushion is negative. In a cash-lock situation, the investment strategy is to fully invest the portfolio into the riskless asset and this is where the terminology stems from. However, if the price process of the risky asset is continuous, this situation can not occur in continuous time and neither in discrete time with triggered trading dates as in the previous chapter. A cash-lock can only occur if trading takes place in discrete time with fixed trading dates as in chapter 1 or if the price process of the risky asset is not continuous. Since in our setup a cash-lock can not occur,
we resort to a slightly modified definition of cash-lock. In the following, we will call a situationε-cash-lock if the exposure ratio of the strategy falls below someε >0. Although all versions of the CPPI considered in this and the previous chapter can fully recover from anε-cash-lock situation, once the exposure ratio has become small, the expected time for a recovery is large such that a very small exposure ratio is already an unpleasant situation to be in.
Proposition 3.3.1 (ε-Cash-Lock) Let ε > 0, γ ∈(0,1) and n :=
=log γε (m−ε)(1−γ)
logku
>
. Then, with h(n, s) = h(n|u(s), d(s)) as in lemma A.1.1 and hk(n, s) =hk(n|u(s), d(s)) as in lemma A.1.5, it holds
a) the probability of the exposure ratio of the simple CPPI with F0 =γV0 falling below ε at some point in time before maturity time T is given by
L−s,T1
h(n, s) s
b) the probability of the exposure ratio of the CPPI with floor adjustment falling below ε at some point in time before maturity time T is given by
L−s,T1 1
s
h0(n, s) 1−hn+1(1, s)
c) with F0 =γV0, the probability of the exposure ratio of the capped CPPI falling below ε at some point in time before maturity time T is given by
L−s,T1 1
sh¯n−1(n, s) + 1 s
hn+1(¯n, s)d(s)d(s|a,∞, δ)h0(n+ 1, s) 1−d(s|a,∞, δ)hn+2(1, s)
for the case mC0 < V0 +Z and by L−s,T1
1 s
d(s)d(s|a,∞, δ)h0(n+ 1, s) 1−d(s|a,∞, δ)hn+2(1, s)
for the case mC0 ≥V0+Z, where n,d¯ (s) d(s) as in propositions 2.3.1, 2.3.2 and addi-tionally n :=
;log (m−1)F
0ε (m−ε)(F0+Z)
logku
<
.
Proof: From equation (3.1) we know that at some trading dateτ the value of the simple CPPI is given by V0erτ(γ+ (1−γ)kun) for some n ∈ Z. Therefore the exposure ratio at τ is given by γm+(1(1−γ−γ))kknun
u and it follows immediately that the exposure ratio is less than or equal to ε for alln ≤n. We know that the Laplace transform of the density for the first
time of having net n up-moves is given by h(n, s). Hence, with the help of proposition A.2.5,c) part a) of the proposition is immediate.
For part b) note that the value of the CPPI with floor adjustment is given byV0erτci(γ+ (1−γ)kun) at some trading date τ for some i ∈ N0 and some n ∈ Z\N. Hence, the exposure ratio is again given by m(1−γ)knu
γ+(1−γ)kun and less than or equal to ε for all n ≤n. The Laplace transform of the density for the first time of having i floor adjustments while not reaching the ε-cash-lock and then n net up-moves while not having a further floor adjustment is given by (hn+1(1, s))ih0(n, s). Since there can be possibly arbitrarily many floor adjustments, summing over i∈N0 yields the result.
For part c) note that h¯n(n, s)is the Laplace transform for reaching the ε-cash-lock before the cap becomes active, which explains the first summand in c) for the casemC0 < V0+Z. Likewisehn+1(¯n+1, s)is the Laplace transform for reaching the cap before theε-cash-lock.
The term d(s)is the Laplace transform for going down to the situation where mCtCap = VtCap+Zert, i.e. the situation where according to the trading rule of the CPPI the complete portfolio plus the maximum borrowing must be invested in the risky asset. From that point, it requires n up-moves to reach the ε-cash-lock. The term d(s|a,∞, δ)hn+2(1, s) stands for going down one level and going up to the situation of full exposure again without reaching the ε-cash-lock before. The term 1−d(s|a,∞,δ1)
hn+2(1,s) therefore accounts for the fact that there can be arbitrarily many switches between full exposure and less than full exposure. Finally, d(s|a,∞, δ)h0(n + 1, s) stands for going down to less than full exposure and reaching the ε-cash-lock while not reaching full exposure again. Hence, the case mC0 < V0+Z becomes apparent and the case mC0 < V0+Z is analogous. 2
Figures 3.8 and 3.9 show the probability of an ε-cash-lock occurring as a function of the maturity time for the simple CPPI, the capped CPPI and the CPPI with floor adjustment.
It is clear, that theε-cash-lock probability must be increasing in the maturity time for all strategies, since a longer maturity time increases the overall variance. It can be seen that the ε-cash-lock probability of the CPPI with floor adjustment converges to one in both figures, the ε-cash-lock probability of the capped CPPI converges to a value considerably below one in both figures and the ε-cash-lock probability of the simple CPPI converges to a value below one in figure 3.8 and to one in figure 3.9. These differences require some comment and we start with the capped CPPI. The capped CPPI has two basic barriers.
One lower barrier for reaching the ε-cash-lock and one upper barrier for reaching the cap.
There is always a positive probability of reaching the cap and this probability does not converge to zero if the time to maturity becomes large. However, once the cap is reached,
0 10 20 30 40 50 T
0 0.1 0.2 0.3 0.4 0.5
CashLockProbability
Cap Simple FA
Figure 3.8: Probability of anε-cash-lock of the simple CPPI, the capped CPPI and the CPPI with floor adjustment as a function of the matu-rity time. The parameters areku = 1.01, m= 4, μ= 0.15, r= 0.05,γ = 0.75,F0 =γV0, ε= 0.01 andσ= 0.20.
0 10 20 30 40 50
T 0
0.2 0.4 0.6 0.8 1
CashLockProbability
Figure 3.9: Probability of an ε-cash-lock of the simple CPPI, the capped CPPI and the CPPI with floor adjustment as a function of the matu-rity time. The parameters areku = 1.01,m = 4, μ= 0.15, r= 0.05, γ = 0.75, F0 =γV0, ε= 0.01 andσ= 0.30.
the strategy turns into a pure investment in the risky asset. We know that the value of the discounted risky asset at some time t > τ is given by St = Sτeσ(Wtδ−Wτδ). It is well known, that for Brownian motions with positive drifts, there is a positive probability of the Brownian motion never hitting lower barriers. Therefore, whenever the drift δ is positive and the strategy has reached full exposure, there is a positive probability of the strategy never reaching less than full exposure again.
For the simple CPPI the situation is somewhat different. From proposition 1.1.2 we know that the discounted cushion process of the continuous-time version is given by e−rtCtcont = C0e(m(μ−r)−12m2σ2)t+σmWt. With the same argument about Brownian motions with drift, we find here that there is positive probability for the discounted cushion process never falling below some lower barrier, if μ−r− 12mσ2 is positive. Since the exposure ratio is given by Fme−rtCtcont
0+e−rtCtcont and directly dependent on the discounted cushion process, the same is true for the ε-cash-lock probability. However, for negative drifts the probability of an ε-cash-lock occurring in infinite time is indeed 1. The parameters in figures 3.8 and 3.9 are the same as in figures 3.5 and 3.6 and therefore also yield μ−r− 12mσ2 >0 and < 0 respectively. Surely, for our discrete version of the simple CPPI, this condition is only an approximation as we know that for ku → ∞ the simple CPPI converges to a stop-loss strategy. Nevertheless, for reasonable values of the discretization parameter ku, the cash-lock probability of the discrete version of the simple CPPI is well explained by the cash-lock probability of the continuous time version.
For the CPPI with floor adjustment, note that after each floor adjustment, the fixed number of net n up-moves (sincen is negative these are actually down-moves) is required to reach the ε-cash-lock. Hence it is a simple consequence of the indefinitely increasing variance of the Brownian motion in infinite time that any fixed number of net down-moves is surpassed at some point in time. Therefore, independent of the parameters, the CPPI with floor adjustment will reach an ε-cash-lock situation with certainty within infinite time. Although, for a long-term or open ended strategy this is not a nice result, it is the trade-off for the increased portfolio insurance. Notice also that the probability of recovering from an ε-cash-lock situation is the same for all three strategies as their behavior is identical for low portfolio values close to the respective floor.
Nevertheless, as a consequence of the largeε-cash-lock probability it is an intuitive idea to further modify the CPPI with floor adjustment such as to introduce a minimum exposure ratio. Currently, CPPI products are often offered with a maximum as well as a minimum exposure ratio. While the maximum exposure ratio is a natural byproduct of the floor adjustments, the minimum exposure ratio must be modelled explicitly. This is what will be done in the next section.