Case Study
Retirement and Monte Carlo
Martin Dupras
ASA, F. Pl.
Senior Advisor, Aon Consulting
Cindy and Cedric are 72 years old. They will rely primarily on their RRIFs to fund their retirement. What
factors should be considered to establish this income? In the May 2004 edition of La Cible, Mr. Daniel
Laverdière, A.S.A., Pl. Fin.addressed return assumptions to use for the purpose of retirement projections.
In particular, he noted that when developing an asset return assumption, a margin of 0.5% to 1% is
needed to factor in the non-linearity of future returns. But what exactly does this mean?
A projection of future retirement income for an individual who will draw a portion of his income from his
own assets must consider future return assumptions. But should this return assumption be linear
(consistent) over time? Will the returns achieved be linear themselves?
At least two projection models can be used: the deterministic model and the stochastic model.
The deterministic model
A model is deterministic if it does not involve the calculation of probabilities (for example, a fixed annual
return of 6% during retirement or an annual return of 6% for the first 10 years and 5% thereafter). A
majority of the income projection systems currently available use this model. Although its simplicity is
attractive, this model does not necessarily provide an adequate representation of reality. Is it conceivable
to have the same return over five, 10, even 30 years? The limitation of the model must be addressed by
factoring in a margin, Mr. Laverdière noted. In the context of projecting RRIF income, the problem is even
more pronounced because of the annual withdrawal. In the event that lower or even negative returns are
obtained in the initial years (the period in which the capital is at its peak), the level of income that could be
generated for life would be at risk.
The Stochastic Model
A model is stochastic involves the calculation of probabilities (for example, an average annual return of
6% over 30 years but obtained as follows: 5.35% the first year, 8.22% the second year, 3.77% the third
year and so on and so forth). A projection using this type of model might better represent reality.
However, this approach is much more complicated than that using a fixed return. Another difficulty comes
from the fact that the choice of annual returns will affect the findings (the use of 5.35% the first year,
8.22% the second and 3.77% the third will produce a different result than the use of 3.77% the first year,
5.35% the second and 8.22% the third year, in particular where withdrawals are considered). Therefore, it
can be concluded that the use of the stochastic model must include a large number of simulations in
order to be credible. The use of such a number of simulations is sometimes called the “Monte Carlo
model.” Let's see how this model may apply to our case.
Cedric and Cindy
The couple has RRIFs in the amount of $100,000. Let’s assume a consistent annual return assumption of
6% and annual withdrawals of $7,500, indexed at 2% annually. The withdrawals must also meet the RRIF
minimum withdrawal criteria.
Using a deterministic (traditional) model, the calculation is quite simple. For $100,000 in RRIFs invested
at 6%, Cedric and Cindy will be able to earn an income of $7,500 the first year, $7,650 the second year
and so on and so forth. Following this model, the capital will be nearly exhausted on their 91st birthday.
We will assume that Cedric and Cindy are comfortable with this situation.
Using a stochastic model, let’s assume the following additional parameters:
 10,000 annual return simulations over 30 years (i.e. 300,000 data points);
 Future return obtained will follow a normal distribution;
 Average performance will be 6%, net of all costs;
 Standard deviation will be 3%.i A higher standard deviation could have been chosen but this
assumption will work for illustration purposes.
Here is what the simulated distribution of future returns looks like:
Distribution of simulated returns (300,000 data points)
Average returns 30 years
Number of simulations
300,000
Minimum return compounded over 30 years
Maximum return compounded over 30 years
3.88%
8.08%
Minimum one-year return
Maximum one-year return
-22.61%
22.09%
Probability observed in one standard deviation
Probability observed in two standard deviations
Probability observed in three standard deviations
68.14%
95.43%
99.73%
Here are the results of 10,000 simulations in terms of age of capital depletion:
Age of depletion
85
86
87
88
89
90
91
92
93
94
95
96
97
98
Number of
scenarios
2
30
295
986
1927
2,323
2,009
1,333
684
258
96
45
11
1
Probability
0.02%
0.30%
2.95%
9.86%
19.27%
23.23%
20.09%
13.33%
6.84%
2.58%
0.96%
0.45%
0.11%
0.01%
Some distortion comes from the fact that whole ages are used, whereas the middle point of this
distribution is around 90 years and nine months. This corresponds quite well to the age of depletion
based on our first deterministic model.
There is approximately a 32% chance that the capital will be depleted at 89 years old or before, leading to
a shortfall risk. This may be stressful for our couple, Cedric and Cindy. For information purposes, capital
depletion occurs at 89 years old by using a consistent annual return assumption of 5%,. The margin (to
factor in the non-linearity of future returns) discussed earlier is therefore relevant. It is possible that in light
of this information, the couple will be more at ease with such a return assumption (5%). Cindy and Cedric
might also want to take out smaller withdrawals. However, they must know that the RRIF withdrawal
minimums will limit this avenue.
The choice of assumptions will have a huge impact on the results (number of simulations, average
performance and standard deviation), but what must be noted here is the impact of non-linear returns on
the age of capital depletion. Other variables, such as survival or inflation, could also be used in the
simulations thus making the analysis more complicated, especially if non-registered capital are
considered. But, without necessarily preparing such simulations, the use of a margin based on the nonlinearity of future returns remains very relevant.
i
Based on these data, approximately 68% of future annual returns will fall between 3% and 9%, approximately 95%
of future annual returns will fall between 0% and 12% and approximately 99% of future annual returns will fall
between -3% and 15%.