Retirement Portfolio Financial Analysis - Graduate Project

Industrial/Applied Math Masters Project 2009:
Successful Retirement Portfolio Allocation
Medicishi K. Taylor
February 26, 2009

Report
1 Introduction
This report will investigate several allocation schemes for retirement portfolios to determine
an optimal allocation of investment options that will ensure a successful retirement. We will
model three investment options and then use Monte Carlo simulation to simulate a 30-year
retirement using several different allocations of the three investments. At the end of each sim-
ulation run, we will determine if we have exhausted all retirement finances or if we still have
finances remaining.
2 Problem Analysis
To develop an optimal retirement plan, we must attempt to solve the problem of what arrange-
ments of investment options will yield high returns. Since the goal will be to preserve as much
retirement assets as possible, we will investigate several allocation schemes in an attempt to
maximize return. If we achieve this, we will be able to maintain a comfortable balance of re-
tirement assets throughout the retirement duration.
By maximizing the return on retirement assets, the effects of routine withdrawals are lessened.
Throughout retirement, a retiree could possibly be solely dependent upon the assets within
their portfolio. While some retirees may have other sources of income such as various IRAs
and annuities or may resume employment in a less demanding occupation, we will only in-
vestigate the situation where the retirement portfolio is the main source of income. This will
heighten the importance of asset preservation and stress the risk involved if a retirement plan
is not properly developed.
Based on what has already been stated, we can generalize the retirement allocation problem.
Since the purpose of finding a solution for choosing the most effective allocation of invest-
ment options is to preserve retirement assets, we can take an abstract approach to solving
this problem which will clarify its solution. Therefore, we will call this problem the “As-
set Degeneration-Regeneration Problem.” By degeneration we are referring to the effects of
withdrawals, taxes, inflation, and fees that will decrease or “degenerate” retirement assets. Re-
generation is describing the process of retirement assets increasing by return only and not by
any deposits. Therefore, along with any degenerative event, assets must increase or “regener-
ate” at a rate equal to or greater than degeneration in order to completely preserve assets. A
simplified example of this is a child eating cookies from a cookie jar. Every time the child
takes cookies from the jar, they will replenish the jar with the same amount of cookies taken or
more. By doing so, they will never run out of cookies.
In our case, the cookie jar is the retirement portfolio. However, when we introduce degenera-
tion and withdraw from these assets, we are relying entirely on the allocation of our investment
1

options to maximize return and regenerate our portfolio. Of course, our asset regeneration
process is not as immediate or fast as the cookie jar example. Therefore, we will degenerate
our retirement assets at a rate higher than regeneration. This will result in a steady decrease
of retirement assests but, the goal is not to preserve all assets completely. This is impossible
in our case since retirement assets will be the only source of income. Thus, success will be
achieved if at the end of retirement we have any amount of assets greater than zero.
3 Methodology and Assumptions
The approach of this study will require several steps to ensure that the behavior of each invest-
ment option is modeled accurately. Each option yields a different return and is characterized
by different volatility. A realistic approach will be to model each option separately and then
simulate them simultaneously during each Monte Carlo repetition.
3.1 Money Market
A major benefit of a money market account with a fixed rate of return is that it will never lose
its value due to market volatility. The annual return will always be the same proportional to the
principal balance. This eliminates the concern for estimating volatility in this case whereas the
models involving stocks and bond markets will be highly dependent on volatility estimation.
However, the annual yield of money market investments is relatively low compared to stocks
or bond markets. This is a trade-off with the very low risk. Nevertheless, for our purposes, an
average annual yield can be interpreted very easily for money market investments.
We will assume that there are no maintenance fees for our money market account as most
companies will waive the maintenance fee as long as the account balance is above a specified
amount. Therefore, the only factors we will consider are the account balance and the fixed
interest rate or rate of return. This model interprets the annual return R as
R = acur × r (1)
where acur is the current account balance and r is the guaranteed annual rate of return. The-
oretically, this simple model might seem sufficient for our study. However, most companies
do not compute interest on money market accounts annually but instead they compute the in-
terest of the account monthly. This difference might seem trivial but the effects of compound
interest will produce significant differences in evaluating interest earned annually compared to
monthly. To illustrate this, let’s take the example of a money market account where no with-
drawals are made for 10 years and the initial balance is $100,000 with a 4% fixed rate of return.
Using equation (1) to compute the total return will yield
($100, 000 × 0.04)10 = $40, 000.
So after 10 years we have earned $40,000 on our original $100,000.
Now let’s look at the more elegant example of accurately modeling the total return Rt utilizing
2

a monthly computation. From Dineen [1], we get the compound interest formula to be
Rt = P 1 +
r
n
nt
− Ainit, (2)
where P is the principal amount, r is the annual rate of return, n is the number of times the
return is computed annually, t is the life of the account, and Ainit is the starting account balance.
Using equation (2), we achieve
$100, 000 1 +
0.04
12
(12×10)
− $100, 000 = $49, 083.27.
The extra $9,083.27 makes this total return 22.7% larger than the $40,000 earned using equa-
tion (1). Therefore, we will compute the interest gained on a monthly basis for the money
market account. We will refer to the money market account as a cash investment since it will
always be cash equivalent and never lose value.
3.2 Stocks
Contrary to money market accounts, stocks and bond markets experience volatility. To model
these two investment options with an accuracy that will produce realistic results, we will use
the process of geometric Brownian motion illustrated in Dr. Stephen M. Alessandrini’s paper,
Industrial Mathematics: A Financial Application [2]. The derivation of this model results in
an elegant equation for the expected return and volatility of a stock and bond market option.
We begin our derivation by using an Itô Process of the form
dS = µSdt + σSdz. (3)
where a(S, t) = µS, b(S, t) = σS, µ is the expected return, and σ is the volatility. Since
volatility is the measure of the uncertainty about the returns of a stock, it will be an important
factor in estimating the future stock prices from historical data.
For discrete time, Dr. Alessandrini uses
∆S = µS∆t + σSε
√
∆t.
He then states that if the stochastic process of x is known, we can use the following lemma to
give the stochastic process G(x, t):
(Itô’s Lemma) Let the stochastic process x be given by the Itô process
dx = a dt + b dz,
Then, G(x,t) follows
dG = a
∂G
∂x
+
∂G
∂t
+
1
2
b2 ∂2
G
∂x2
dt + b
∂G
∂x
dz.
3

Thus, if dS = µS dt + σS dz and G(S, t) = ln S, we get
∂G
∂S
=
1
S
,
∂G
∂t
= 0,
∂2
G
∂S2
= −
1
S2
and a = µS and b = σS. Therefore,
dG = d(ln S) = µ −
σ2
2
dt + σ dz. (Generalized Wiener Process)
Now we can let ST be the stock price at time T and S0 the initial price. So in the discrete case,
the above yields
ln ST − ln S0 = µ −
σ2
2
T + σε
√
T,
where ε ∼ N(0, 1). This will imply that ln ST is normally distributed with a mean of
ln S0 + (µ − σ2
/2)T
along with a standard deviation of
σ
√
T.
Since ln ST is normal, we can say ST is lognormal and we can simulate this discretly by using
S∆t = S0e(µ− σ2/2)∆t+σ
√
∆t ε
where ε ∼ N(0, 1). This deﬁnes the process of geometric Brownian motion with mean
µ− σ2
/2 and volatility σ. It will produce nonnegative values and will be suitable for modeling
stock and bond market prices.
Based on what we have established above from Dr. Alessandrini’s paper, we can determine
how to calculate µ and σ from historical data. To do this, let Si be the value of the stock at time
ti and let τ = ti − ti−1 be the time interval in years. Then, let ui = ln (Si/Si−1). With this
information we obtain
ui = µ −
σ2
2
τ + σ
√
τ with E(ui) = µ −
σ2
2
τ and Var(ui) = σ2
τ.
By letting ¯u = ( n
i=1 ui)/n, we can use the strong law of large numbers to achieve ¯u −→ E(ui)
enabling us to approximate E(ui) by ui. We also approximate σ by
σ2
τ = Var(ui) = s2
=
1
n − 1
n
i−1
(ui − ¯u)2
.
As a result of the steps above from Dr. Alessandrini’s paper, we ﬁnally get
ˆσ2
=
s2
τ
and ˆµ =
¯µ
τ
+
ˆσ2
2
.
Hull [3] states that the standard error in this estimation is ˆσ/
√
2n. Since σ changes with time,
we cannot assume that a large n will minimize the error. We also do not want to use any data
4

that will be too old and unuseful in predicting the future behavior of stocks. In our study, we
will use 15 years of S&P 500 historical data giving n = 180 months of data.
Now that we have a method to determine µ and σ, we need to generate normally distributed
random numbers for our ε in each Monte Carlo run to ensure that we generate a lognormal
distribution for the stock and bond market price estimates. To perform this, we need the distri-
bution function of normal distribution:
F(x) =
x
−∞
1
√
2πσ
e−(t−µ)2/2σ2 dt
where µ is the mean of the density and σ is the variance of the density. We will use the Ziggurat
(stepped pyramid) method of Marsaglia and Tsang [4], which is the same normally distributed
random number generator that MATLAB uses. This will randomize our stock and bond market
prices during our Monte Carlo simulation.
3.3 Bond Market
For a bond market investment option, we will use the same process outlined by Dr. Alessan-
drini [2] to determine the µ and σ. The Dreyfus Bond Market Index will be the investment
option we use to supply our bond market historical data. We will use 10 years of historical data
for the bond market because historical data beyond 10 years was not readily available therefore,
giving n = 120 months of data for the bond market.
3.4 The Monte Carlo Model
The main model will be a Monte Carlo simulation to simulate each investment option simul-
taneously. Based on what we have determined so far, we can model each investment inde-
pendently which will allow a realistic simulation of each. The stock market functions on a
quarterly basis so we must incorporate this into the Monte Carlo simulation when modeling
stocks. We should also model the bond markets similarly since they will follow the same mar-
ket schedule. As previously stated, the money market account will operate on a monthly basis.
Since we will be modeling the withdrawal activity of the retirment plan, it is critical that we
develop a stable withdrawal schedule. It is unlikely that a retiree will withdraw a lump sum
annually. With this in mind, we will use a semi-annual withdrawal schedule. In Paul Lim’s
article on portfolio strategies [5], it is assumed that a retiree will possess $1 million in assets
at the start of retirement and tap about $70,000 from their account annually. We will withdraw
$60,000 in $30,000 semi- annual intervals to create a more realistic withdrawal schedule.
In our model, we also assume the retiree does not hold burdensome ﬁnancial responsibilities
such as excessive debt or family dependents. The retiree will only need to support the basic
ﬁnancial needs such as living expenses, utilities, food, and clothing. Therefore, $60,000 will
be a comfortable yearly income for our retiree. We will use $1 million, $1.5 million, and $3
million as tests for starting portfolio balances.
5

Along with our withdrawal schedule, we should be sure to maintain the current allocation
scheme that we are using. We will study 8 allocation schemes among which some are analyzed
in Lim’s article. The specific allocation schemes we will study are the following:
60% stocks/ 30% bonds/ 10% cash 75% stocks/ 10% bonds/ 15% cash
25% stocks/ 75% cash 50% stocks/ 50% bonds
50% stocks/ 50% cash 100% stocks
100% cash 100% bonds
After each withdrawal, the retirment portfolio will be reallocated to the specified allocation
scheme. This will ensure that we maintain the proper fund ratio for every investment option
throughout retirement. We will also adjust the portfolio value at the end of every year due
to inflation. According to the Congressional Budget Office [6], the annual inflation rate is
predicted to be 2.2% from 2008 to 2017 and then 3% afterwards. We will use this prediction
to calculate our inflation rate every year throughout retirement.
4 Solution
After running 500 Monte Carlo simulation runs for a 30-year retirement, we were able to gather
the vital information regarding a solution to the best retirement portfolio allocation. The re-
sults of our testing prove that a successful retirement portfolio needs to have a proper balance
of stability and growth. A dramatic imbalance will yield poor growth in general so achieving
maximum return consists of more than just choosing an investment option that is likely to give
the highest return.
For example, an investment option with a very high expected return is more likely to have a
higher risk than an investment option with a moderate or lower return. This means that while
the high return investment will give you greater returns on your money, it is also capable of
behaving very unpredictably resulting in periods of sharp decline or even negative returns.
Basically, your potential for losing money also increases proportionally to your potential for
gaining money with a higher yielding investment. This fact is the main reason we needed to
model the volatility of our bond market and stock funds so accurately. Volatility is a double-
edged sword in the market and must be evaluated carefully.
6

4.1 Analysis of Results
A direct analysis of our test results will help emphasize the importance of volatility when
choosing an allocation scheme. Let’s begin by analyzing a section of output from our retire-
ment model program. The following figure will list the results of our testing for three of the
eight allocations starting with an initial balance of $1 million:
Starting portfolio amount of $1000000.00:
***************************************************************
The 60/30/10 (stock/bond/cash) portfolio allocation scheme
has a 70.6 percent chance of success.
You ran out of funds 147 times out of 500.
Average ending balance: $869519.84
Minimum ending balance: $ -1657149.51
Maximum ending balance: $6943823.94
The standard deviation: $1369508.29
***************************************************************
has a 86.6 percent chance of success.
***************************************************************
The 25/75 (stock/cash) portfolio allocation scheme has a
72.4 percent chance of success.
The standard deviation: $ 432770.68
Notice the success percentage of each allocation scheme. In particular, observe that the 75/10/15
(stock/bond/cash) allocation scheme has the highest success percentage of the three at 86.6%.
Why is this the case? We get these results because the 75% of our portfolio assests that are
in the S&P 500 mutual fund are yielding an average return of about 10% a year. Since this
is the largest portion of the portfolio, the capability of higher returns is increased signficantly.
The other 10% in the bond market fund is only yielding about 4-5% a year so its affect on
the overall return of the portfolio is minimal along with the 15% of assets in the money mar-
ket. Nevertheless, the money market adds a significant feature of stability to the portfolio.
Remember, money is never lost due to volatility in a money market account. However, when
retirement assets are limited, it is especially important to maximize return so that your assets
7

are not exhausted.
With the above being mentioned, it is now easily understood why the 60/30/10 allocation re-
sults in a lower success rate of 70.6%. Not only is the highest yielding investment option 15%
smaller but the bond market fund is 20% larger. This means that this allocation increases the
percentage of assets affected by volatility to 90%. This choice might be beneficial in the case
where the 90% was invested in a relatively stable high return fund such as the S&P 500 but,
only 60% of the assets fit this category. The bond market assets which account for the other
30% of volatility offer a much lower return. Therefore, you increase the risk of loss with more
volatility while only increasing your return to approximately 5% for the extra 20% of assets in
the bond market. With inflation also taken into account, this return does not efficiently boost
asset growth. Based on this, we can conclude that unlike the money market, which yields a
relatively low return also, the extra bond market funds offer little for overall growth. Whereas
the money market may offer a low return but also is guaranteed to never lose value.
Let’s examine our test results of the 50/50 (stock/cash) allocation scheme to solidify our theory
above about the poor performance of the 60/30/10 allocation. We received the following output
when running our model for the 50/50 scheme:
***************************************************************
87.6 percent chance of success.
The output tells us that this scheme out performs the other three allocation schemes with a
success rate of 87.6%. Is this to be expected? Certainly, due to the fact that only 50% of your
portfolio is now experiencing volatility and the other half is yielding return on a stable con-
sistent basis. This is a perfect balance between high return and stability. Though the money
market’s annual return is significantly lower than the S&P 500 fund, its stability compliments
the performance of the S&P 500 investment for superior combined growth.
It is important that we analyze our results very carefully because they can be misleading at first
glance. This is illustrated in the case of a 100% stock allocation. The potential for growth and
high return is maximized in this case but the volatility adds a very high risk due to its affect on
all of your assets. Out of the three investment options, the S&P 500 yields the highest return
so the average ending balance and maximum ending balance will be very high. This can be
deceptive if further analysis is not completed. We obtain the following output for the 100%
stock allocation:
***************************************************************
8

The stock portfolio allocation scheme has a 86.8 percent
chance of success.
Notice the high average ending balance and the maximum ending balance. Nevertheless, we
can not base the performance on these values alone. Observe that the Monte Carlo simulation
for this run failed 66 out of 500 runs. This means that you ran out of money before the end of
the 30-year retirement for 66 runs. Re-visiting the output for the 50/50 (stock/cash) allocation
shows that we run out of money for 62 runs out of 500. This success ratio is actually the most
important piece of data we can analyze. Why? This is because the success rate is the Monte
Carlo simulation’s effective tool of computing the likelihood of an outcome. The higher success
rate will indicate a higher probability of success. This is what we must closely pay attention
to in finding a solution for the best investment option allocation. The 50/50 (stock bond),
100 (bond), and 100 (cash) allocations resulted in a 49.8%, 0%, and 0% success percentage
respectively. Therefore, since the 50/50 (stock/cash) allocation yields the highest success rate
of 87.6%, we can comfortably affirm that this is the best allocation when starting retirement
with a $1 million portfolio.
Increasing Success With a Larger Initial Balance
Now that we have analyzed our results for an intial portfolio balance of $1 million, let’s exam-
ine the benefits of having a higher initial balance at the start of retirement. It can be assumed
that a higher starting balance will improve the success rate of any allocation but by how much?
To analyze this, we will compare the average ending balance of 500 Monte Carlo runs for ev-
ery year during the 30-year retirement with intial balances of $1 million, $1.5 million, and $3
million. Look at the figure below for a starting portfolio balance of $1 million.
9

Figure 1: Average Growth for starting balance of $1 million
Based on the average ending balances plotted, we see that the 100% cash and 100% bond
allocation perform very poorly. Both of these allocations experience a negative growth and
eventually reach $0 before 30 years. However, by increasing the initial portfolio balance by
half, we get the following results:
Figure 2: Average Growth for starting balance of $1.5 million
10

With $1.5 million initially, the total bond and total cash allocations no longer experience a
large negative growth and in fact due not reach $0 before 30 years. Thus on average, none of
the allocations reach $0 before the end of retirement. The term average must be stressed here
because this does not mean that none of the allocations reach $0 during any run of the Monte
Carlo simulation. By examining the average balance, we obtain the probable growth pattern
for each allocation over a 30-year period. The higher averages indicate higher growths.
The average growth not only estimates the potential for return but it also indicates the average
portfolio balance at any given time during retirement. This can be particularly useful when
a retiree wants to estimate the amount of assets they will posess at a certain time while in
retirement. For example, let’s suppose a retiree holds a 75/10/15 (stock/bond/cash) portfolio
allocation with a starting amount of $3 million and is planning to move 10 years into retire-
ment. This event is not modeled in our simulation but we can closely estimate what type of
funds the retiree will have at this point in time. According to our average growth data, we can
estimate that their portfolio will be worth a little over $6 million at the end of year 10. The
retiree can use this information to determine if they would have sufficient funds to support the
financial costs of moving into another home. Once a decision is made, the portfolio balance
can be adjusted accordingly and the simulation continued to reveal if the financial costs of re-
locating will exhaust the retirement assets before 30 years.
Now let’s observe the average growth for an initial $3 million:
Figure 3: Average Growth for starting balance of $3 million
11

Here we see that when the initial balance is tripled from $1 million to $3 million the average
growth increases by a factor greater than 10. This clearly proves that an increase in the initial
portfolio balance results in a much higher growth performance leading to greater success rates.
For example, notice the program output for starting a 30-year retirement with $3 million:
Starting portfolio amount of $3000000.00:
***************************************************************
has a 100 percent chance of success.
Minimum ending balance: $1622651.68
***************************************************************
***************************************************************
100 percent chance of success.
***************************************************************
The 50/50 (stock/bond) portfolio allocation scheme
***************************************************************
The 50/50 (stock/cash) portfolio allocation scheme
12

***************************************************************
The stock portfolio allocation scheme has a 100 percent
chance of success.
***************************************************************
The cash portfolio allocation scheme has a 100 percent
chance of success.
You had sufficient funds.
Ending balance: $5405885.16
***************************************************************
The bond portfolio allocation scheme has a 100 percent
chance of success.
The standard deviation: $ 466747.93
We now see that every allocation scheme results in a 100% success rate. This means that with
$3 million, a retiree can choose any of our allocation schemes of the three investment options
and have a successful retirement. Though this information may make some content, we still
need to determine what allocation is best. Although the success rate is computed to be equal
for every allocation, it does not mean that the performances of each are equal. This can be eas-
ily determined by again observing the minimum ending balance and average ending balance of
each allocation.
Carefully examining the minimum balance ensures the success of a portfolio allocation in
terms of risk because it will give you the lowest account balance during retirement. The alloca-
tion with the largest minimum balance will be the allocation that ensures success the greatest.
Based on this information, it is determined that in this case, the 100% cash allocation performs
the best with an ending balance of $5,405,885.16 after one run. Remember that there is no
randomness with the money market because there is no volatility and the interest rate is ﬁxed.
13

This means that the ending balance is theoretically the average, minimum, and maximum end-
ing balance. Therefore, a retiree can estimate that their retirement assets will be close to $5
million at the end of 30 years. But does this allocation really have the best performance?
Recall that in this report, we started out stating that an optimum allocation scheme would
both maximize return and stability. Since all allocations are 100% successful with an initial
$3 million, let’s properly rank their performance by using the minimum ending balance and
average ending balance. As a basis for our theory, we will assume that the average ending
balance and the minimum ending balance are equally important. Thus, both will be equally
weighted. Evaluating these two values reveals the 50/50 (stock/cash) allocation as being the
best in performance with the third largest average ending balance of $14,797,825.19 and a
minimum ending balance of $3,720,149.66 which, is the also the third largest. Therefore, the
50/50 (stock/cash) allocation scheme would be the best solution for a retiree in this case.
5 Conclusion
After analyzing several sets of data, we proved our theory that a portfolio allocation that bal-
ances volatility and stability will perform the best during retirement. We initiated our study
by examining a 30-year retirement starting with $1 million. It was determined that the 50/50
(stock/cash) allocation performed the best in this situation. When we tripled the starting bal-
ance to $3 million, every allocation scheme became a potential success. However, we saw
that the performance behavior remained the same and when we analyzed the minimum ending
balance and average ending balance for each allocation, the 50/50 (stock/cash) allocation was
the optimum performer once again.
In conclusion, it is important to consider both volatility and stability when choosing an opti-
mum allocation balance. We have proved that a close balance of the two will yield maximum
results during retirement. Although, we have used the S&P 500, the Dreyfus Bond Market, and
a money market account to model our investment options, this principal of balancing volatility
and stability will be applicable for all types of investments to achieve an optimal allocation
strategy. Our retirement model could have been more realistic if we designed it to account for
random ﬁnancial events such as a new home purchase or a college fund for children or grand-
children. Nevertheless, it is an excellent predictor of future accessible funds during retirement.
The results from this study can be used in actual retirement planning as a guideline for the type
of allocations that should be used for a retirement portfolio.
14

Bibliography
[1] Seán Dineen. Probability Theory in Finance: A Mathematical Guide to the Black-Scholes
Formula, volume 70. American Mathematical Society, Providence, 2005.
[2] Dr. Stephen M. Alessandrini. Industrial mathematics: A financial application. Technical
report, Rutgers University, Graduate Math Department, 2007.
[3] J.C. Hull. Options, Futures, and Other Derivatives. Prentice Hall, New York, 5 edition,
2002.
[4] George Marsaglia and Wai Wan Tsang. The ziggurat method for generating random vari-
ables. Journal of Statistical Software, 5(8), October 2000. http://www.jstatsoft.
org/v05/i08.
[5] Paul J. Lim. Portfolio strategies for a shaky market: As retirement nears, avoiding major
losses is essential. U.S. News and World Report, September 2007.
[6] Cbo’s economic projections for calendar years 2007 to 2017, 2007. Congressional Budget
Office, http://www.cbo.gov/budget/data/econproj.shtml.
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