# R programming

## Montserrat Guillén, Jens P. Nielsen & Anna M. Pérez-Marín

In this page, the simulation of the final wealth accumulated after the investment period for alternative life-cycle pension products is presented. A methodological overview can be found in:

##### SIMULATING THE FINAL WEALTH

Here we consider alternative pension strategies which are defined by a stock proportion process directly (are not path-dependent). Additionally, we also consider the saving scheme called Time Pension which is path-dependent. For all of them, the R-scripts that simulate the final wealth accumulated after the investment period are provided.

###### ALTERNATIVE PENSION STRATEGIES DEFINED BY A STOCK PROPORTION PROCESS

Our simulation study is based on the following assumptions. We consider a pension saver starting his savings at age x. Then, during the first T/2 years the policy holder invests a fixed amount per year c in a pension fund. After the retirement at age x+T/2, the policy holder receives constant annuities equal to the same amount per year c, so $$c(t)=\left\{ \begin{array}[c]{c} c \, \, \, \, \, \, t=t_{0},\dots,t_{T/2-1}\\ -c \, \, \, \, \, \, t=t_{T/2},\dots,t_{T-1} \end{array} \right.$$

with c>0, where c(t)>0 represents an endowment, otherwise consumption. Alternatively, decreasing annuities are also considered in the simulation study (for more details, see Guillen et al., 2013). The accumulted premiums are invested in a broadly diversified investment fund of stocks and bonds, while always keeping a proportion u(t) in the risky asset for the yearly intervals t=t0, . . ., tT-1. Finally, at age x+T the final amount, either positive or negative, is paid out to the pensioner.

We have implemented a simulation study with 10,000 runs by assuming c(t)=10 for t=t0, . . ., tT/2-1 and c(t)=-10 for t=tT/2, . . ., tT-1 for a contract starting when the investor is 30 years old and finishing at age 90, so T = 60. Alternative stock proportion processes u(t) are considered. We assume the Black Scholes market (see, Black and Scholes, 1973) and we use the following expression to simulate the investor's wealth process X(t) (see Guillen et al., 2013): $$X(t_{i^{-}}) =\Big(X(t_{{i-1}^{-}})+c(t_{i-1})\Big)\exp \left\{ u(t_{i-1})\left( \left( r_{0}-\dfrac{u(t_{i-1})\sigma ^{2}}{2}\right) \Delta t_{i-1}+\sigma Z_{i}\right) \right\} ,$$

where r0 is the expected stock return, σ is the volatility, Zi ~ N(0,1) and Δ ti-1=1. Further details about the assumptions in the Monte Carlo setting are given in the Appendix of Guillen et al. (2013).

###### TIME PENSION: A PATH-DEPENDENT STRATEGY

Here we present Time Pension, which is is path dependent saving scheme that includes a return smoothing mechanism (profits and losses are smoothed before they are credited to the investor's account). Every year the premiums are invested in an investment fund (in our simulation we assume to be approximately 70% in stocks and 30% in bonds).

Let At be the market value of the investment fund at time t. At the end of each period its balance is updated by the manager in two separate accounts: Dt, which is the primary/personal account, and Ut, which is the secondary/smoothing account. The personal account belongs to the pension saver while the smoothing account is owned by the fund manager. At the end of each period, the primary account is first credited with a periodic reference policy interest rate rtP, which is well-defined and discretely compounded. Afterwards, a fixed fraction α ∈ ]0,1[ (called the smoothing parameter) of the interim balance in the secondary account is transferred to the primary account. Once this transfer has been done, the pension saver adds to his account balance a new contribution c(t) , which may either be positive, when paying premiums, or negative, when receiving annuities. Each year, At is invested. The following accounting identity will hold at all updating times t = t0, t1, . . ., tT-1:

$$A_{t}=D_{t}+U_{t}.$$ The development of the primary account balance is given by: $$D_{t}=\left\{ \begin{array}{lc} c(t_{0})\text{} & t=t_{0} \\ (1+r_{t}^{P})D_{t-1}+\alpha \left\{ (1+u_{t}^{\ast })A_{t-1}-(1+r_{t}^{P})D_{t-1}\right\} +c(t) & t>t_{0} \end{array} \right.$$

where rtP is the policy interest rate, α is the smoothing parameter, ut* is the market investment return and c(t) is the new contribution in period t. On the other hand, the development of the secondary account balance is:

$$U_{t}=\left\{ \begin{array}{lc} 0\text{} & t=t_{0} \\ (1-\alpha )\left\{ (1+u_{t}^{\ast })A_{t-1}-(1+r_{t}^{P})D_{t-1}\right\} & t>t_{0} \end{array} \right.$$

Note that due to the path-dependence of the investment process, we cannot express the stock proportion process u(t) directly. In our simulation, the policy interest rate rtP will be equal to 0 and the smoothing parameter α will be equal to 0.2. More details can be found in Guillen et al. (2006).

##### MEASURING PERFORMANCE AGAINST NEGATIVE SAVINGS

Here we describe a method for measuring the performance of alternative pension schemes. Firstly, we need to introduce some definitions. We will use the yearly internal rate of return for evaluating the performance of an investment strategy. This is the rate of return that makes the final value of the income stream equal to zero. In our case, the internal interest rate rint would satisfy

$$\sum_{t=0}^{T-1}c(t)(1+r_{int})^{T-t}-M_{T}=0 \hspace{1.5cm} (1)$$

where MT is the amount in the pension account left for the pension saver at age x+T. We will estimate MT by the mean or the median of the simulated distribution of the final wealth.

We will also consider a reference product with a constant proportion l, l ∈ [0,1], which is defined by a constant stock proportion process ul(t) = l for all t>0. We consider the set of all reference products with a given constant proportion, i.e. Ul = {ul(t) = l, t > 0, l ∈ [0,1]}. We also define an equivalence measure within the set of all possible investment strategies U ⊃ Ul. We say that two investment strategies are equivalent if they have the same amount of risk under a given risk measure when looking at the simulated distribution of final wealth at age x+T. As risk measures we use the value at risk (VaR) and the conditional tail expectation (CTE).

Then, we use the following procedure for measuring the performance:

• 1. For every existing pension product with certain investment strategy, we find an equivalent product defined by a stock process within the reference set Ul. This is what we call the trivial benchmark
• 2. Then, our performance score for one particular pension product is now based on the comparison to its corresponding trivial benchmark. At time T we consider the mean and the median amount of the simulated distribution of X(T) for both products. Then we use (1) to calculate the estimated yearly internal interest rate for the product and its corresponding trivial benchmark.
• 3. Finally, the performance score is equal to the difference between the yearly internal interest rate when considering the mean (or median) of the final wealth of the product and the mean (or median) of the final wealth for the reference strategy.

More details can be found in Guillen et al. (2013).

##### PRICE OF THE GUARANTEE

The fair price of issuing a guarantee against having a negative saving at age T years provides a tool for measuring the downside risk of each strategy. The final expression for calculating the price of the guarantee is: $$\dfrac{1}{n}\sum_{m=1}^{n}\bigg(-\min (0,X_{T}^{(m)})\bigg),$$ where n is the number of simulations and XT(m) is the final wealth at age T of the m'th run under ℚ measure. Note that this expression is simply the mean loss of the provider of the guarantee. For more details see Bjork (2004, section 7) and Guillen et al. (2013).

###### REFERENCES

[1] Basak, S., and Shapiro A. (2001) Value-at-risk-based risk management: optimal policies and asset prices, The Review of Financial Studies 14, 2: 371-405.

[2] Bjök, T. (2004) Arbitrage theory in continuous time, Oxford University Press.

[3] Black, F. and Scholes, M. (1973) The pricing of options and corporate liabilities, Journal of Political Economy 81: 659-683.

[4] Breeden, D. T. (1979) An intertemporal asset pricing model with stochastic consumption and investment opportunities, Journal of Financial Econometrics 7, 3: 265-296.

[5] Chamorro, J. M. (2000) Mutual fund evaluation: a portfolio insurance approach. A heuristic application in Spain, Insurance: Mathematics and Economics 27: 83-104.

[6] Guillen, M., Jorgensen, P. L. and Nielsen, J. P. (2006) Return smoothing mechanisms in life and pension insurance, Insurance: Mathematics and Economics 38: 229-252.

[7] Haberman, S. and Vigna, E. (2001) Optimal investment strategy for defined contribution pension schemes, Insurance: Mathematics and Economics 28: 233-262.

[8] Jorgensen, P. L. and Nielsen, J. P. (2002) Time Pension, Finans/Invest 6: 16-21.

[9] McNeil, A. J., Frey R. and Embrechts, P. (2005) Quantitative Risk Management. Concepts, Techniques and Tools, Princeton University Press.

[10] Nielsen, P. H. and Steffensen, M. (2008) Optimal investment and life insurance strategies under minimum and maximum constraints, Insurance: Mathematics and Economics 43, 1, 15-28.

[11] Nordahl, H.A. (2008) Valuation of life insurance surrender and exchange options, Insurance: Mathematics and Economics 42, 3, 909-919.

[12] Siegmann, A. (2007) Optimal investment policies for defined benefit pension funds, Jour- nal of Pension Economic and Finance 6 (1): 1-20.

[13] Steffensen, M. (2002) Intervention options in life insurance, Insurance: Mathematics and Economics 31, 1, 71-85.

[14] Steffensen, M. (2006) Quadratic optimization of life insurance payment and Pension Insurance Payments, Astin Bulletin 36, 1, 245-267.

[15] Steffensen, M. and Waldström, S. (2008) A Two-Account Model for Pension Saving Contracts, Scandinavian Actuarial Journal, 1, 1-18.

• Universitat de Barcelona - Last Updated: 05-29-2014