2. CPPI strategy

The CPPI strategy is a self-financing strategy whose aim is to take potential gain on upward market move while guaranteeing at least an specified fixed amount of money at maturity time T. In the CPPI strategy, the investor initially sets a floor which is the lowest acceptable portfolio value. The cushion is calculated as the excess value of current wealth over the floor. The cushion multiplied by a pre-determined multiplier, defined as exposure, is allocated to the risky asset. Remaining funds are invested in the riskless asset. In the classic Black-Scholes market, where two basic assets are traded continuously during time horizon $[0,T]$, and the fund manager invests into the two assets: a riskless asset (money market account), B_t, and a risky asset (stock or stock index), S_t, the price dynamics under $ \mathbb{P}$ are given by

(1)\begin{equation} \frac{dB_{t}}{B_{t}}=rdt,\quad B_{0}=b, \end{equation}

(2)\begin{equation} \frac{dS_{t}}{S_{t}}=\mu_{s}d{t}+\sigma_{s}d{W_{t}},\quad S_{0}=s, \end{equation}

where r, b, s, µ_s, and σ_s are constant and W_t is a Brownian process defined on complete probability space $(\Omega , \mathcal{F}, \mathbb{P})$. Here, $ \mathbb{P}$ is real-world probability measure and the filtration $\mathcal{F}= (\mathcal{F}_{t})_{[t\geq0]}$ represents the history of the Brownian motion up to time t. The basic idea of the CPPI approach (Zagst and Kraus, Reference Zagst and Kraus2011) is that the terminal portfolio value, V_T, at the end of the investment horizon T stays above an investor-defined floor given as a percentage, $\varphi \geq 0$, of initial value, V ₀, i.e.

\begin{equation*} F_{T}=\varphi_{T} V_{0}. \end{equation*}

Since the market is arbitrage-free, it is impossible to find an investment that returns more than the risk-free rate of return, $\mathit{r}$, with no risk. Hence, the maximum guaranteed portfolio value at maturity time, T, is limited by

\begin{equation*} \varphi_{T} \leq e^{rT}. \end{equation*}

The floor F_t, $0\leq t \leq T$, denotes the present value of guarantee. By discounting with respect to deterministic interest rate, r, φ_t becomes

\begin{equation*} F_{t}=\varphi_{t}V_{0},\qquad \varphi_{t}=\varphi_{T}e^{-r(T-t)}. \end{equation*}

Accordingly, the floor follows the dynamics

\begin{equation*} dF_{t}=rF_{t}dt. \end{equation*}

The surplus of current portfolio value, V_t, above the floor F_t is called cushion. The price of cushion, C_t, at any time $t \in [0,T]$ is given as

(3)\begin{equation} C_{t}:=V_{t}-F_{t}. \end{equation}

At any time $t\in [0,T]$, the wealth invested into the risky asset, called exposure, is given by

\begin{equation*} e_{t}=mC_{t}, \end{equation*}

where m is a constant multiplier reflecting the risk attitude of the investor. The remaining funds are then invested in the risk-free asset leading to a portfolio value process

(4)\begin{equation} \frac{dV_t}{V_t} = e_t \frac{dS_t}{S_t} + (V_t-e_t) \frac{dB_t}{B_t}. \end{equation}

By Equations (3) and (4), the cushion C_t satisfies

(5)\begin{equation} dC_t = C_t \left( \left( m \left(\mu_S - r\right) +r \right)dt + m \sigma_S dW_t \right). \end{equation}

3. CPPI strategy in DC pension plan under discrete time trading setting

Let τ be set of n + 1 equidistant trading dates along time period $[0,T] $, such as $\tau= \{t_{0} \lt t_{1} \lt \ldots \lt t_{n-1} \lt t_{n}\} $, where $t_{0}=0$ is the inception date of DC pension plan, $t_{n}=T$ is the retirement date, and $\triangle t:= t_{k+1}-t_{k}=T/n $ for $k=0,1,..,n-1$.

As mentioned earlier, we consider two assets in the financial market for investing pension contributions: risky asset (stock or stock index) and riskless asset (money market account). Money market account, B_t, grows with constant interest rate, r, and its price dynamic with initial value $B_{0}=b$ given in Equation (1). Additionally, we assume that the price dynamics of risky asset, S_t, is expressed as in Equation (2). Since trading takes place at equidistant trading dates, the observed stock price is as follows:

\begin{equation*} S_{t_{k+1}}=S_{t_{k}}e^{(\mu_{S}-\frac{\sigma_{S}^{2}}{2})\triangle t+\sigma_{S}\triangle W_{t_{k}}},\qquad k=0,...,n-1, \end{equation*}

where $S_{t_{0}}=s$ and $\triangle W_{t_{k}}=W_{t_{k+1}}-W_{t_{k}}$ for $k=0,...,n-1$.

The DC fund modeling has two key aspects:

1. i. There is no consumption before the retirement date.

2. ii. Labor income plays a central role in the wealth-accumulated phase.

3. iii. The inheritance gains are not considered which avoids the consideration of longevity and mortality risks.

Labor income is hard to model realistically due to its complicated stochastic components such as financial and political crisis, disability, and mortality. Here, labor income, L_t, is assumed to have a stochastic process reflecting the risk of the financial market. If the participants contribute to the pension fund at each equidistant trading date as a certain proportion, γ, of their labor income, the labor income dynamics are defined as a stochastic differential equation as follows:

(6)\begin{equation} dL_{t}=L_{t}(\mu_{L}dt+\sigma_{L}dW^*_{t}),\;\;\;\; L_{0}=l, \end{equation}

with l, µ_L, and σ_L representing starting labor constant, average labor, and standard deviation, respectively, and $W^*_{t}$ is a Brownian process. In what follows, we restrict the stochastic behavior of the labor income process for the tractability of analytic results. The case we consider is a labor income process that perfectly correlates with risky asset ( $d{W_{t}} = dW^*_{t} $) following the approach of Bodie et al. Reference Bodie, Merton and Samuelson(1992).

Based on this setup, the observed labor income becomes

(7)\begin{equation} L_{t_{k+1}}=L_{t_{k}}e^{\left(\mu_{L}-\frac{\sigma_{L}^{2}}{2}\right)\triangle t+\sigma_{L}\triangle W_{t_{k}}},\qquad k=0,...,n-1,\\ \end{equation}

where $L_{t_{0}}=l$ and $\triangle W_{t_{k}}=W_{t_{k+1}}-W_{t_{k}}$ for $k=0,...,n-1$. Therefore, the DC with respect to labor process and the value of proportion, γ, is defined as

(8)\begin{equation} \gamma _{t_{k}}= \gamma L_{t_{k}}, \end{equation}

for all $k=0,1,..,n$ with dynamics

(9)\begin{equation} d\gamma _{t}= \gamma dL_{t}. \end{equation}

We assume that there is a portfolio which has the same dynamics as labor income. For this reason, the labor income can be replicated perfectly between equidistant trading dates using assets in the market. In order to prove this, we need to show the relevant replicating portfolio. Assuming the relation which expresses the equivalence of risk premium between the labor income and the risky asset

(10)\begin{equation} \mu_L = r + \sigma_L \frac{\mu_S-r}{\sigma_S}, \end{equation}

we consider a self-financing replicating strategy $\pi=(\pi_{B},\pi_{S},\pi_{L})$ and choose $\pi_{L}=-1$ as given in Temocin et al. (Reference Temocin, Korn and Selcuk-Kestel2017). Then, we have the value of the fund as

\begin{align*} dV^{\pi}_{t}&=\pi_{B}dB_{t}+\pi_{S}dS_{t}-dL_{t} \\ &=\pi_{B}(rB_{t}dt)+\pi_{S}(S_{t}\mu_{s}dt+S_{t}\sigma_{s}d{W_{t}})-L_{t}(\mu_{L}dt+\sigma_{L}dW_{t}) \\ &=(\pi_{B}rB_{t}+\pi_{S}S_{t}\mu_{s}-L_{t}\mu_{L})dt+(\pi_{S}S_{t}\sigma_{s}-L_{t}\sigma_{L})dW_{t}. \end{align*}

Equating the diffusion terms and the drift terms to zero, we obtain the number of assets required as

\begin{equation*} \pi_{S}(t)=\dfrac{L\sigma_{L}}{S\sigma_{S}} \;\;\;\;\;\; \text{and} \;\;\;\;\;\; \pi_{B}(t)=\dfrac{L}{rB\sigma_{S}}(\sigma_{S}\mu_{L}-\sigma_{L}\mu_{S}). \end{equation*}

Unlike the floor in the classical CPPI strategy, we consider a random floor which evolves not only with interest rate but also with the portion of each contribution. In this setup, floor, F_t becomes a stochastic process due to the dynamics of the labor income as described in (Temocin et al., Reference Temocin, Korn and Selcuk-Kestel2017). Hence, the floor at time t is defined as

(11)\begin{equation} F_{t}=\left\{\begin{array}{l}\sum _{i=0}^{k}e^{r(t-t_{i})}c\gamma_{t_{i}}, \quad t\in (t_{k},t_{k+1}) \\\\ F^{-}_{t_{k+1}}+c\gamma_{t_{k+1}},\qquad t=t_{k+1}\end{array}\right. \end{equation}

for $ k=0,1,...,n-1$, where $0 \lt c \lt 1$ is a constant that denotes the proportion of the amount of contribution to be added to the floor with $F_{0}=c\gamma_{t_{0}}$. The portfolio value is then given as

(12)\begin{equation} V_{t_{k+1}}= \left\{\begin{array}{ll} (V_{t_{k}}-mC_{t_{k}})e^{r\triangle t}+mC_{t_{k}}\frac{S_{t_{k+1}}}{S_{t_{k}}}+\gamma_{t_{k+1}}, & C_{t_{k}} \gt 0\\ V_{t_{k}}e^{r\triangle t}+\gamma_{t_{k+1}}, & C_{t_{k}}\leq0 \end{array} \right. \end{equation}

for $t \in (t_{k},t_{k+1})$ and

(13)\begin{equation} V_{t}= \left\{\begin{array}{ll} (V_{t_{k}}-mC_{t_{k}})\frac{B_{t}}{B_{t_{k}}}+mC_{t_{k}}\frac{S_{t}}{S_{t_{k}}}, & C_{t_{k}} \gt 0\\ V_{t_{k}}\frac{B_{t}}{B_{t_{k}}}, & C_{t_{k}}\leq0 \end{array} \right. \end{equation}

for $t=t_k$ for all $k=0,1,..,n-1$.

Then by its definition, the cushion process becomes

(14)\begin{equation} C_{t_{k+1}}= \left\{\begin{array}{ll} C_{t_{k}}(m\frac{S_{t_{k+1}}}{S_{t_{k}}}+(1-m)e^{r\triangle t})+(1-c)\gamma_{t_{k+1}}, & C_{t_{k}} \gt 0\\ C_{t_{k}}e^{r\triangle t}+(1-c)\gamma_{t_{k+1}}, & C_{t_{k}}\leq0 \end{array} \right. \end{equation}

for $t \in (t_{k},t_{k+1})$ and

\begin{equation*} C_{t}= \left\{\begin{array}{ll} C_{t_{k}}\left(m\frac{S_{t}}{S_{t_{k}}}+(1-m)\frac{B_{t}}{B_{t_{k}}}\right), & C_{t_{k}} \gt 0\\ C_{t_{k}}\frac{B_{t}}{B_{t_{k}}}, & C_{t_{k}}\leq0 \end{array} \right. \end{equation*}

for $t=t_k$ for all $k=0,1,..,n-1$.

Now that we have described the CPPI-DC design, we next introduce the implementation of the cushion insurance by employing a special exotic option, hereby called the cushion option.

3.1. Cushion option in the frame of a CPPI strategy

Due to our trading structure, the risk of cushion becoming negative between trading dates is always present as the re-balancements are done only at trading dates. Considering the cushion dynamics, we can define probability of gap risk, P_gap just before contribution payment comes into the system as follows:

\begin{align*} P_{gap}&=P\left( C^{-}_{t_{k+1}} \lt 0 |\ C_{t_{k}} \gt 0\right) \\ &=P\left( C_{t_{k}}\left( m\frac{S_{t}}{S_{t_{k}}}+(1-m)e^{r\triangle t}\right) \lt 0 |\ C_{t_{k}} \gt 0\right) \\ &=P\left( \frac{S_{t}}{S_{t_{k}}} \lt \frac{(m-1)}{m}e^{r\triangle t} \right). \end{align*}

To minimize P_gap, we propose the exotic cushion option with payoff function

(15)\begin{equation} (K-(1-c)\gamma L_{t_{k+1}})^{+}\mathbb{I}_{\left\lbrace \frac{S_{t_{k+1}}}{S_{t_{k}}} \lt (1-\frac{1}{m})e^{r\triangle t}\right\rbrace }, \end{equation}

where $K=C_{t_{k}}$. The main aim of this custom-made option is to generate an additional gain in case of a sudden decrease in the portfolio value. The proposed option is expected to benefit when the cushion becomes negative between times t_k and $t_{k+1}$. Hence, we compare the cushion amounts at these times with the purpose of activating the payoff only when there is a gap phenomenon in the given time period. Here, the strike price, K, represents the cushion with the aim of making sure that there is always sufficient capital for buying the option, so the option price will always be smaller than the cushion due to non-negative labor income. Since this is a theoretical derivative, our primary assumption is that there are parties in the market who sell these options. Then, based on the position of the pension fund, the beneficiary can buy the relevant option. By Girsanov’s Theorem (Shreve, Reference Shreve2004), we have the risk-neutral dynamics of the stock price as

\begin{equation*} S_{t_{k+1}}=S_{t_{k}}exp\left( \left(r-\frac{\sigma^{2}_{S}}{2}\right)\triangle t+\sigma_{S}(W^{\mathbb{Q}}_{t_{k+1}}-W^{\mathbb{Q}}_{t_{k}})\right), \end{equation*}

where $W^{\mathbb{Q}}$ is the Brownian motion under the unique risk-neutral measure $\mathbb{Q}$. Thus, we have

(16)\begin{equation} W^{\mathbb{Q}}_{t_{k+1}}-W^{\mathbb{Q}}_{t_{k}}=\dfrac{ln\frac{S_{t_{k+1}}}{S_{t_{k}}}-\left(r-\frac{\sigma^{2}_{S}}{2}\right)\triangle t}{\sigma_{S}}. \end{equation}

Substituting $W^{\mathbb{Q}}_{t_{k+1}}-W^{\mathbb{Q}}_{t_{k}}$ into the $\mathbb{Q}$-dynamics of $L_{t_{k+1}}$ yields

(17)\begin{align} L_{t_{k+1}}&=L_{t_{k}}exp\left( \left(\mu_{L}-\frac{\sigma^{2}_{L}}{2}\right)\triangle t+\sigma_{L}\dfrac{ln\frac{S_{t_{k+1}}}{S_{t_{k}}}-\left(r-\frac{\sigma^{2}_{S}}{2}\right)\triangle t}{\sigma_{S}}\right) \end{align}

(18)\begin{align} &=L_{t_{k}}\left( \frac{S_{t_{k+1}}}{S_{t_{k}}}\right) ^{\frac{\sigma_{L}}{\sigma_{S}}}exp\left( \left (\mu_{L}-\frac{\sigma^{2}_{L}}{2}-\frac{r\sigma_{L}}{\sigma_{S}}+\frac{\sigma_{L}\sigma_{S}}{2}\right)\triangle t\right). \end{align}

Hence, the payoff of the cushion option becomes

\begin{align*} ( C_{t_{k}}-(1-c)\gamma L_{t_{k+1}})^{+}\mathbb{1}_{\left\lbrace \frac{S_{t_{k+1}}}{S_{t_{k}}} \lt (1-\frac{1}{m})e^{r\triangle t}\right\rbrace } &= \left( C_{t_{k}}-(1-c)\gamma L_{t_{k}}\left(\frac{S_{t_{k+1}}}{S_{t_{k}}}\right)^{\frac{\sigma_{L}}{\sigma_{S}}}\right. \nonumber\\ & \left. exp\left( \left(\mu_{L}-\frac{\sigma^{2}_{L}}{2}-\frac{r\sigma_{L}}{\sigma_{S}}+\frac{\sigma_{L}\sigma_{S}}{2}\right)\triangle t\right) \right) ^{+}\\ &\qquad {} \times \mathbb{1}_{\left\lbrace \frac{S_{t_{k+1}}}{S_{t_{k}}} \lt (1-\frac{1}{m})e^{r\triangle t}\right\rbrace } \\ &=\zeta (K^{*}-S_{t_{k+1}}^\frac{\sigma_{L}}{\sigma_{S}})^{+}\mathbb{1}_{\left\{\frac{S_{t_{k+1}}}{S_{t_{k}}} \lt (1-\frac{1}{m})e^{r\triangle t}\right\}}, \end{align*}

where

\begin{equation*} \zeta=\dfrac{(1-c)\gamma L_{t_{k}}}{S_{t_{k}}^\frac{\sigma_{L}}{\sigma_{S}}} exp\left( \left( \mu_{L}-\frac{\sigma^{2}_{L}}{2}-\frac{r\sigma_{L}}{\sigma_{S}}+\frac{\sigma_{L}\sigma_{S}}{2}\right) \triangle t \right) \quad \text{and} \quad K^{*}=\frac{C_{t_{k}}}{\zeta}. \end{equation*}

The next proposition gives the price of the cushion with this payoff.

Proposition 1. The cushion option under CPPI-DC design with payoff

\begin{equation*} (C_{t_{k}}-(1-c)\gamma L_{t_{k+1}})^{+}\mathbb{I}_{\left\lbrace \frac{S_{t_{k+1}}}{S_{t_{k}}} \lt (1-\frac{1}{m})e^{r\triangle t}\right\rbrace }, \end{equation*}

has a price of $P_{t_{k}}$ given as

(19)\begin{equation} P_{t_{k}}=\zeta^{*}K^{*}\Phi\left( \min(A_{t_{k}},B_{t_{k}})\right)-\zeta^{*}S_{t_{k}}^{\frac{\sigma_{L}}{\sigma_{S}}}e^{\left[ \frac{\sigma_{L}}{\sigma_{S}}(r-\sigma_{S}^2/2)+\sigma_{L}^{2}/2\right] \triangle t}\Phi\left( \min(A_{t_{k}},B_{t_{k}})-\sigma_{L}\triangle t \right), \end{equation}

where

\begin{equation*} \zeta^{*}=e^{-r\triangle t} \zeta, \end{equation*}

with

\begin{equation*} A_{t_{k}} = \dfrac{ln\left( \frac{K^{*}}{S_{t_{k}}^{\frac{\sigma_{L}}{\sigma_{S}}}} \right) -\frac{\sigma_{L}}{\sigma_{S}}\left(r-\frac{\sigma_{S}^2}{2}\right)\triangle t}{\sigma_{L}\sqrt{\triangle t}} \;\;\;\;\;\; \text{and} \;\;\;\;\;\; B_{t_{k}} = \dfrac{ln(\frac{m-1}{m})+(\sigma_{S}^2/2)\triangle t}{\sigma_{S}\sqrt{\triangle t}}. \end{equation*}

Proof. The details of the proof is given in the Appendix.□

The cushion option is assumed to be purchased at time t_k for all $k=0,1,...,n-1$. Since $C_{t_{k}} \gt P_{t_{k}}$, new portfolio value, $V_{t_{k}}^{*}$, and cushion value, $C_{t_{k}}^{*}$, at time t_k are given by

\begin{equation*} V_{t_{k}}^{*}=V_{t_{k}}-P_{t_{k}} \end{equation*}

and

\begin{equation*} C_{t_{k}}^{*}=max\{V_{t_{k}}^{*}-F_{t_{k}},0\}, \end{equation*}

respectively. Then, for $C_{t_{k}} \gt 0$ for all $k=0,1,..,n-1$, the portfolio value is expressed as

\begin{align*} V_{t_{k+1}}=& \left[ (V^{*}_{t_{k}}-mC^{*}_{t_{k}})e^{r\triangle t}+mC_{t_{k}}^{*}\frac{S_{t_{k+1}}}{S_{t_{k}}}+\gamma L_{t_{k+1}}+(C_{t_{k}}-(1-c)\gamma L_{t_{k+1}})^{+}\right] \mathbb{I}_{\left\lbrace {\frac{S_{t_{k+1}}}{S_{t_{k}}} \leq (1-\frac{1}{m})e^{r\triangle t}}\right\rbrace} \\ &\qquad \qquad \qquad \qquad {} +\left[ (V^{*}_{t_{k}}-mC^{*}_{t_{k}})e^{r\triangle t}+mC_{t_{k}}^{*}\frac{S_{t_{k+1}}}{S_{t_{k}}}+\gamma L_{t_{k+1}}\right] \mathbb{I}_{\left\lbrace {\frac{S_{t_{k+1}}}{S_{t_{k}}} \gt (1-\frac{1}{m})e^{r\triangle t}}\right\rbrace}. \end{align*}

For $C_{t_{k}}\leq0$ for all $k=0,1,..,n-1$, the portfolio value, $V_{t_{k+1}}=V_{t_{k}}e^{r\triangle t}+ \gamma L_{t_{k+1}}$, yields the value of the pension fund at $t_{k+1}$. Henceforth, when $C_{t_{k}} \gt 0$, the wealth process between equidistant trading dates is given as

\begin{equation*} V_{t}=\left[ (V^{*}_{t_{k}}-mC^{*}_{t_{k}})\frac{B_{t}}{B_{t_{k}}}+mC_{t_{k}}^{*}\frac{S_{t}}{S_{t_{k}}}\right], \end{equation*}

whereas for $C_{t_{k}}\leq0$,

\begin{equation*} V_{t}=V_{t_{k}}\frac{B_{t}}{B_{t_{k}}}, \end{equation*}

for $t \in (t_{k},t_{k+1})$, $k=0,1,..,n-1$.

4. Implementation and numerical results

In order to indicate the influence of the cushion option on CPPI-DC plans under certain assumptions, we perform simulations because the realization of real data requires long years and a wide range of market conditions, DC plan, and country-specific characteristics may also change over the years. By means of 10000 Monte Carlo simulations, the CPPI strategy with cushion option and without cushion option is generated and the performance levels of both strategies are investigated under various market and DC plan assumptions. All analyses and runs are coded using MATLAB.

The simulation scenarios based on many factors are expected to expose sensitivity with respect to varying parameter values. As exotic options can be utilized for short-term leverage in financial markets, their contributions are evaluated in two basic time frames:

1. (i) short-term which corresponds to the generally assumed earliest time to terminate the pension plan; T = 3 years

2. (ii) long-term at which the participants fulfill the retirement conditions; T = 20 years.

The comparisons with respect to time frames and the parameters are performed based on the value of the fund with and without cushion options. The steps of the proposed approach start with setting up arbitrary fixed values for the contributing parameters. The algorithm then calculates the price of the cushion option for the strike price at each fixed trading date. The next step considers a CPPI scheme for each fixed trading date with and without cushion options for strike price and finds the wealth for each case. The process ends with a comparison of the setup with respect to the cushion option and different values of the contributing parameters.

Portfolio strategies are simulated using the set of parameters summarized in Table 1. The parameter values are chosen to be the same as in Temocin et al. (Reference Temocin, Korn and Selcuk-Kestel2017) in order to have a benchmark except for some adjustments in the interest rate, r, and market volatility, σ_S, to expose the influence of the cushion-option setup on fund’s development. The choice of 3 years for short-term is made with respect to the most commonly official withdrawal duration set by the life and pension insurance companies.

Table 1.

Parameter values under discrete-time trading setting

Table 2 shows the mean and standard deviation of the final wealth for the CPPI strategy without and with the cushion option for a period of 3 years and a period of 20 years. According to the results, we see that cushion option is a profitable choice for both periods. Mean, $E(V_{T})$, and standard deviation, $\sigma(V_{T})$, of final wealth for the CPPI strategy under the cushion option are higher than the mean and standard deviation of final wealth for the CPPI strategy. Considering that downside protection is provided by the CPPI strategy, the higher standard deviation indicates the longer right tail of terminal wealth distribution, and therefore it is considered a positive performance indicator, despite its more usual role as a negative performance indicator.

Table 2.

The final wealth of CPPI strategies for long and short terms

In Table 3, we illustrate the sensitivities with respect to the parameters for the time frames T = 20 years and T = 3 years. We see from the results of the simulations that the higher market volatility, σ_S, has a positive effect on moments of terminal wealth for without cushion option CPPI strategy in the 3-year time period. However, we cannot observe accurate information about its effect on the terminal wealth for the CPPI strategy under cushion option for the 20-year time period. An increase in the market drift, µ_S, raises the moment of the terminal wealth for CPPI strategies in both time periods. A rise on labor income drift, µ_L, as well as labor income volatility, σ_L, increases the moments of terminal wealth for CPPI strategies in both time horizon. Increasing the guarantee rate, c, resulting in the higher floor, has a negative effect on the moments of both strategies. That is, when the smaller amount of wealth is invested in the risky asset, we take fewer potential gains on the upward market move. CPPI with cushion option yields higher average return compared to no cushion when the contribution rate, γ, increases. When more amount of money is invested in the risky asset, investor can take more gains from increasing market. Higher multiplier, m, yields the moments of CPPI portfolio to be slightly higher in both time periods, but on the other hand higher m decreases moments of CPPI strategy with cushion option in both time horizon. The gap risk increases in the case of the high multiplier, m, and therefore the value of cushion option depending on gap risk raises. As expected, the high price of cushion option lowers the profits. Inversely, smaller multiplier, m, decreases the gap risk in line with the price of cushion option. For a small multiplier such as m = 2, the cushion option has no significant effect on the CPPI strategy. Note that the initial value for the portfolio is taken to be $0.1$ in Tables 2 and 3.

Table 3.

CPPI portfolio with and without cushion option under certain parameter values (T=20 years and T=3 years)

Figure 1 illustrates the portfolio trajectories of both strategies, highlighting their evolution over time. The figure compares the performance across two selected investment horizons. Taking a glance at the graphs, one can see that the portfolio value under the cushion option initially stands below the portfolio value of the CPPI strategy due to hedging cost for both horizons. When a sudden decrease occurs, portfolio value recovers quickly by means of cushion option and makes benefits from increasing in the market. However, portfolio value for the CPPI strategy needs time to recover and cannot make use of an advantage of potential gains on the upward move in the market.

Figure 1.

Illustrative trajectories of CPPI portfolio-scheme based on the parameters chosen in Table 1 in the 3-year time period (a and b) and 20-year time period (c and d).

Figures 2 and 3 illustrate the tail properties and the distributional behavior of terminal portfolio value for both strategies with respect to different time frames. As clearly seen, the CPPI strategy under cushion option has a longer right tail in the value of terminal wealth compared to the other case. This illustrative distributional behavior of both strategies supports the results depicted in Table 2.

Figure 2.

Short-term estimated kernel densities of final wealth for CPPI.

Figure 3.

Long-term estimated kernel densities of final wealth for CPPI.

5. Concluding comments

Fund management for DC pension plans is critical for both individuals and social security systems. In this context, the literature explores portfolio insurance as a means to provide downside protection for portfolio values. The CPPI strategy with the random floor applied to DC pension plan under discrete-time trading setting allows us to foresee the development of a guaranteed fund. Considering trading only takes place at predefined dates, the portfolio value can drop below the floor between trading dates. Additionally, contribution at trading dates may not be enough to push up the floor. Therefore, in this paper, we propose a cushion option to reduce the gap risk and derive the price of option as well as the wealth process accordingly. By means of Monte Carlo simulations, the effectiveness of the cushion option on the CPPI strategy is tested by comparing portfolio performance for the CPPI strategy with and without cushion option. To support the effectiveness of cushion option, kernel densities are estimated for terminal wealth of both strategies.

This paper contributes to the valuation of DC plan funds by explicitly pricing a cushion option and demonstrating its utilization through numerical analysis. Monte Carlo simulation results indicate that the CPPI strategy incorporating a cushion option outperforms the standard CPPI strategy over both short-term (e.g., 3-year) and long-term (e.g., 20-year) horizons, as evidenced by higher moments of terminal wealth.

Even though the CPPI strategy provides downside protection, a sudden decline in the market may cause the portfolio value to drop below the floor. If the remaining time before termination date is not enough to recover the portfolio value, the investor faces the risk (gap risk) that the portfolio value stays under the floor representing the acceptable minimum portfolio value. The cushion option which is introduced in this paper is shown to be a profitable choice for a DC plan participant who considers a risk of early withdrawal.

The findings show that for both short- and long-term pension periods, CPPI with cushion option increases the expected value of the wealth and reduces the standard deviation especially for the long-term period. This proposed approach can be utilized to depict the plausible investment strategies having certain guarantees in pension fund system. It should be noted that the market and income are assumed to have similar dynamics due to their close correlation in real life. The influence of market risk through inflation will have an real reduction on the value of the income. As a future avenue of research, we plan to generalize the salary model by accounting for imperfect correlation between labor income and financial markets. Another possible extension is to incorporate regime-switching dynamics to better capture the changing behavior of the economy over the long term. In addition to the generalizations in financial modeling, the stochastic modeling approach presented in this paper has potential applications beyond the domain of finance. Stochastic models, pricing approaches, and risk management techniques can be applied in various fields, such as management (resource allocation and decision-making), biology (population dynamics and genetic processes), environmental networks (sustainable resource management), and financial networks (systemic risk in interconnected institutions). Exploring these interdisciplinary connections could provide new avenues for future research.

Furthermore, our setup can be applied to the hybrid pension scheme for reducing funding shortfall which is prominent risk on calculation of reserve with respect to Solvency II framework by incorporating inflation and longevity risks.