# Performance measurement of pension strategies -

# R programming

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

- Guillén, M.; Nielsen, J.P.; Pérez-Marín, A.M.; Petersen, K. (2013) "Performance measurement of pension strategies: a case study of Danish life cycle products" Scandinavian Actuarial Journal, 1, 49-68.

**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 \begin{equation} 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. \end{equation}

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=t _{0},
. . ., t_{T-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=t _{0},
. . ., t_{T/2-1}* and

*c(t)=-10*for

*t=t*for a contract starting when the investor is 30 years old and finishing at age 90, so

_{T/2}, . . ., t_{T-1}*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): \begin{equation} 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\} , \end{equation}

where *r _{0}* is the expected stock
return, σ is the volatility,

*Z*and

_{i}~ N(0,1)*Δ t*. Further details about the assumptions in the Monte Carlo setting are given in the Appendix of Guillen et al. (2013).

_{i-1}=1**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 *A*_{t} 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: *D*_{t}, which is the
primary/personal account, and *U*_{t},
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 *r _{t}^{P}*, 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,

*A*

_{t}is invested. The following accounting identity will hold at all updating times

*t = t*:

_{0}, t_{1}, . . ., t_{T-1}where *r _{t}^{P}* is the policy
interest rate, α is the smoothing parameter,

*u*is the market investment return and

_{t}^{*}*c(t)*is the new contribution in period

*t*. On the other hand, the development of the secondary account balance is:

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 *r _{t}^{P}*
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 *r _{int}* would satisfy

where *M _{T}* is the amount in the
pension account left for the pension saver at age

*x+T*. We will estimate

*M*by the mean or the median of the simulated distribution of the final wealth.

_{T}We will also consider a reference product with a
constant proportion *l*, *l* ∈ [0,1],
which is defined by a constant stock proportion
process *u _{l}(t) = l* for all

*t>0.*We consider the set of all reference products with a given constant proportion, i.e. U

*,*

_{l}= {u_{l}(t) = l*t*> 0, l ∈ [0,1]}. We also define an equivalence measure within the set of all possible investment strategies U ⊃ U

*. 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*

_{l}*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*U. This is what we call the_{l}*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: \begin{equation}
\dfrac{1}{n}\sum_{m=1}^{n}\bigg(-\min
(0,X_{T}^{(m)})\bigg), \end{equation} where *n*
is the number of simulations and *X _{T}^{(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).

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