This worksheet has been designed and tested using Netscape 3.0. It should work with later versions of Netscape's browsers but will not work with other browsers. It uses only JavaScript for computations and should cause no problems when used with the appropriate browsers. The operative word here is "should" -- the author cannot guarantee fault-free operation.

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accompanying instruction file (wi_***.htm) on your disk using the
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At a later time you may retrieve the file using the browser's **F**ile
**O**pen file command; you may then use the page as
you would if you were on the network. Note, however, that the
file saved will have all the initial settings rather than those
you have changed.

When using the worksheet, you may change any inputs. To do so, click inside the appropriate box, then make your changes or select the desired radio button. When finished, click any area outside the boxes on the form.

This worksheet is designed to find the range of likely post-retirement standards of living, based on specified savings and investment assumptions. It uses a very simple model of investment returns and hence should be considered only illustrative of more sophisticated methods that can be employed for this task.

The **worker** (principal) is assumed to purchase
a **real annuity** at retirement. Such an annuity is
guaranteed to provide a nominal amount that varies with inflation
so as to provide a specified real amount, based on its terms. The
three blocks in the first row of the input table provide the
information on the nature of the desired annuity. If only the
worker is to be covered in a **single annuity**,
only the sex and age of the worker need be provided. If a **joint
annuity** is to be purchased at retirement, the sex and
age of the **beneficiary **must also be given. If a
joint annuity is selected, the benefit will drop to 50% of its
initial amount on the death of either the worker or the
beneficiary.

The **Investment Real Return **block provides
inputs for the expected annual real return and the standard
deviation of the annual real return of the investments to be
purchased with current and future savings up to the date of
retirement. Note that the expected return is the expected value
of the one-year return, not the long-term or geometric mean.

The next block provides the assumed **Real Annuity
discount rate**. This is the rate that an annuity provider
is assumed to use to discount expected future cash flows to
determine the cost of an annuity. The higher this rate, the
cheaper will be a given annuity and the greater will be the
standard of living associated with a given investment account
value at retirement. The expected cash flows are derived using an
equally-weighted average of the mortality rates in the
Commissioners 1980 Standard Ordinary (1970-1975) table, designed
for estimating mortality for those who apply for individual life
insurance policies, and the 1983 Individual Annuity Table
(1971-1976, projected to 1983), designed for estimating mortality
for those who apply for individual annuity policies.

The **Worker Retirement Age** is the age at which
the worker wishes to retire. The worker is assumed to retire on
the same date of the year that the analysis is performed. Thus if
the Worker's **Nearest Age **is 60 and the
retirement age is 65, the worker is assumed to retire at the end
of 5 years.

The **Savings** block provides details concerning
current and planned future savings. The **Current Savings**
equals the total amount currently invested, divided by current
salary. This expresses savings in terms of the number of years of
current salary, putting it in a useful relative context. The **Future
Savings **is stated as a percent of future salary. This
percentage of salary will be saved each year up to retirement and
added to the investment account.

The **Real Salary Increase **is stated as a
percent of salary. The worker's real salary is assumed to
increase each year up to retirement by this percentage.

The **Analysis **block provides information
concerning the desired analysis. Two approaches are provided.

The first approach is **deterministic** -- the
investment return is assumed to be the same every year, with the
actual return equal to the** geometric mean** of the
investment distribution. This is the estimated long-run (50/50)
return for the specified expected return and standard deviation
of return.

The second approach is **stochastic**. **Monte
Carlo simulatio**n methods are used to estimate the range
of likely outcomes. In this procedure, many **cases**
are generated. Each such case involves a year-by-year projection
of results up to the retirement age followed by determination of
the resulting standard of living. Annual investment returns are
drawn randomly from a probability distribution with the specified
expected return and standard deviation. When a case has been
completed, the result is recorded and added to a list of results
-- one per case. When all the requested cases have been
completed, this list of results is sorted from best to worst.
Then the outcomes lying at specified **percentiles**
are determined. For example, if 1,000 cases are simulated, the
10'th percentile outcome is the 100'th from the bottom, the 50'th
percentile outcome is the median outcome, and the 90'th
percentile outcome is the 900'th from the bottom (and 100'th from
the top). The Analysis block allows the user to specify both the
number of cases and the percentiles for which outcomes are to be
shown.

In any given year the investment return is assumed to be earned on the initial account value plus half the annual contribution rate times the initial salary. This amount is added to the initial value of the account at year-end. Next, the salary is increased by the specified annual percentage. Finally, an amount equal to half the annual contribution rate times the ending salary is added to the account. The entire process is repeated for each year to retirement.

All results are stated in terms of the r**eplacement
ratio** at retirement. This is the ratio of the value of
the annuity purchased at retirement to the worker's salary just
prior to retirement. The greater the ratio, the higher the
associated standard of living.

For the simulation, annual returns are assumed to be drawn from a lognormal distribution where the return (not the logarithm of return) has the expected return (e) and standard deviation (sd) specified in the input section. This is accomplished by drawing a normal random deviate and using it to determine the logarithm of the value-relative (1 + return) for the year. Let:

z

_{t}= log ( 1 + r_{t})

where

r

_{t}= the (proportional) return in year t

The first and second moments (a and b) of the distribution of z are determined as follows:

b = sqrt ( log ( ( v / (u^2) ) + 1 ) )

a = 0.5 * log ( (u^2) / exp(b^2) )

where:

u = 1 + e / 100

v = ( sd / 100 )

^{2}

Once a value for z_{t} is drawn from a normal
distribution with mean **a** and standard deviation **b**,
the corresponding return is determined using:

1 + r

_{t}= exp (z_{t})

Random normal deviates are generated using the Box-Muller
method (Box, G.E.P and M.E. Muller, *A note on the generation
of random normal deviates*, Annals Math. Stat, V. 29, pp.
610-611).