Hanna, S. & Chen, P. (1996).
Efficient portfolios for saving for college, Financial Counseling and
Planning, 7, 115-122.
Copyright 1996 by AFCPE
Efficient Portfolios For Saving For College
This article uses 69 years of real rates of return for six types
of financial assets to find efficient portfolios for saving for college,
in terms of mean and minimum accumulations. Small stocks are in every efficient
portfolio. For 10 and 15 year time frames, the portfolio that was the safest
consisted of 89% intermediate term government bonds and 11% small stocks.
A family willing to stay 100% invested in small stock mutual funds until
each year’s college costs must be met can greatly reduce the burden of
saving for college, at relatively low risk. KEY WORDS: college,
investing, household portfolios, financial assets
In saving for college, the allocation of asset categories in the portfolio
is a crucial decision. Most people are not willing to take above average
risks to obtain above average returns on their investments (Avery &
Elliehausen, 1986). Saving for college is a difficult investment challenge,
as few families can afford to invest large lump sums early, so the most
common investment plan consists of periodic investments over a relatively
short time period. In considering risk versus return, various approaches
have been taken, including a focus on the possibility of a shortfall in
consumption or in some arbitrary goal (e.g., Leibowitz & Langetieg,
1989; Leibowitz & Kogelman, 1991; Ho, Milevsky & Robinson, 1994).
For college saving, consideration of a shortfall is complex, and must be
placed in the context of a comprehensive financial plan. The unique contributions
of this article are the discussion of saving for college in the context
of a comprehensive financial plan, the use of real rates of return, and
the calculation of efficient portfolios for college saving.
Saving for college is often presented as merely a mathematical exercise,
using future value tables and estimates of future college costs to generate
the required periodic contribution to a college fund (e.g., Leonetti &
Feldman, 1995). Ideally, though, saving for college should be considered
in the context of a comprehensive financial plan. All of the areas of financial
planning may be relevant to saving for a college fund. The usual advice
— to start contributing to a college fund as soon as a child is born –
should be modified if a family does not have adequate insurance coverage
or has outstanding balances on credit cards. The family’s values and goals,
and short term needs, such as finding quality child care for a young child,
need to be considered along with a goal of starting a college fund early.
Tax planning is a very important for deciding whose name should be listed
for a college fund. For some wealthy parents or grandparents, estate planning
may be important in considering a funding a child’s college education.
Retirement planning is important to saving for college, as the time between
paying for college and retirement of the parents may be important to the
acceptability of the parents taking out loans to cover part of the costs
of a child’s college.
What Will College Cost in the Future?
The tuition increases of the past 10 years are not sustainable — if
the price of anything increases much faster than wages, eventually the
entire national income would be devoted to that product. Frank (1994),
Pennar (1995), and Weagley (1995) all suggested that the rapid increases
in college tuition in the past 20 years might not persist in the future.
Colleges may not have unlimited power to increase tuition, and only 5%
of students in 1994 attended colleges with annual tuition over $15,000
(Topolnicki & O’Connell, 1994). In this article, the assumption in
examples given is that tuition will increase at the rate of general inflation.
However, none of the conclusions in the article are changed by changing
the assumption about how fast tuition increases. The possibility of continued
rapid increases in tuition makes the case for more aggressive investing
Why Don’t Most Parents Save for College?
The standard advice is to start saving as early as possible to take
advantage of compound interest. Relatively few parents save a substantial
amount of money for their children’s college education, however. Churaman
(1992) reported that only 14% of two-parent families and 11% of single-parent
families had saved money for their child’s college. Connolly (1989) noted
that a young family may be more concerned about car and house payments,
etc. than saving for college. Young parents may be saving to buy or furnish
a home. When children are very young, it is likely that wage income is
relatively low or there are high child care costs. Furthermore, parents
may expect their salaries will be much higher in the future than today.
For some occupations, salaries after 20 years may be 50% to 100% higher
in real terms than entry level salaries. For instance, the average full
professor at the University of Michigan makes about 60% more than the average
assistant professor (Wright & Dwyer, 1990, p. 113). This type of increase
between entry level salaries and the salaries of experienced workers is
common in many professions, although there may be low real growth for unskilled
jobs. If inflation is included, young parents could plausibly expect their
family income in nominal dollars to triple in the next 20 years.
Unless parents have a long period during which they can save money for
a variety of goals before having children, it is likely they will have
low levels of financial assets after the children are born. Therefore,
the most realistic scenario for saving for college will be a periodic saving
plan rather than investing of a large lump sum when the child is very young.
If parents expect family income to double or more by the time the child
enters college, it may be more rational to plan for equal proportions of
income saved each year, rather than equal amounts of nominal dollars saved
each year. For instance, if annual family income is expected to increase
from $30,000 at age 25 to $100,000 at age 45 (increase of 6% per year),
planning to save 5% of income each year would mean that $1500 would be
contributed to a college fund the first year and $5,000 the 20th year.
This would be a very different pattern than is usually recommended. It
would not take full advantage of the power of compound interest, because
of the lower early contributions. On the other hand, this pattern might
be more realistic in a life cycle context for many young families.
Are the Incentives for Saving Perverse?
The “curious game of financial aid” (Willis, 1989) may lead some parents
to conclude that it is not worthwhile to save for college. Case and McPherson
(1986) analyzed the incentive structure of the federal student aid system
and the Uniform Methodology used by many private colleges. There is a steep
penalty on current income above moderate income (in terms of reductions
in financial aid), 25% for the federal aid system and 47% for the Uniform
Methodology. The effective penalty on assets accumulated can be as high
as 21% for the four years of college for the Uniform Methodology (Case
& McPherson, 1986, p. 3). Nevertheless, Case and McPherson concluded
that the student aid system did not provide substantial disincentives to
saving for college. The prominent economist Martin Feldstein reached the
opposite conclusion, however (Fortune, 1992, p. 42). If the child
will attend a college that covers all financial need, there may be a disincentive
for saving resulting from the financial aid system. Many private colleges
such as Harvard provide grants and low cost loans to cover all costs beyond
what they calculate the family should be able to pay. However, many other
colleges do not cover estimated financial need. For those colleges, the
disincentives for saving listed above would not be relevant.
In the past, some experts advised parents to establish savings in the
child’s name because of the lower federal and state income tax rates the
child might pay compared to the parents’ rates. This advice is now outdated
by changes in federal income tax rules and by the rules of the federal
and private college aid systems (Weagley, 1995). If a child qualifies
for some financial aid, the amount of aid received by the child over four
years of college may be reduced by an amount equal to over 80% of the amount
of assets accumulated in the child’s name, so good advice now for most
parents is to avoid putting savings in a child’s name (Baldwin, 1991; Dolan
& Dolan, 1995b; Brouder, 1992; Quinn, 1995; Rowland, 1995; Wang, 1996;Willis,
1989). Saving in the parents’ names is subject to federal and state income
taxes unless it is put into tax sheltered retirement accounts, which have
the advantage of not being counted at all for the federal and many private
financial aid calculations. This strategy has been recommended by some
(e.g., Willis, 1989; Cohen, 1989) although Jane Byrant Quinn (1995) regarded
it as unethical. There may also be practical difficulties in terms of requirements
for repayment, etc. (Dolan & Dolan, 1995a), although about 75% of retirement
plans of large employers recently surveyed allowed loans against them (Schultz,
1995). If grandparents are in a position to help, the recommended strategy
is that they pay the tuition bill directly (Quinn, 1995).
How Many Years to Invest?
Even though some parents estimate SAT scores based on toddler behavior,
it is not always clear what the aptitudes and interests of a young child
will be 15 to 20 years later. For middle income families, saving for a
college fund may compete with funds for quality child care when a child
is very young and with funds for family vacations when a child is somewhat
older. Starting saving early has substantial advantages in terms of the
power of compound interest, but some experiences cannot be deferred. Parents
have only one year to experience taking a particular five year old to Disney
World and to do other things together as the child grows.
The Need to Try for a High Rate of Return
Despite the reasons listed above, the standard set of calculations presented
by experts assume that each year the parents would contribute the same
number of dollars (in nominal terms) to a college fund (e.g., Kobliner,
1989b). If parents put one dollar at the end of each year into a savings
account, at the end of 15 years they would have $15 plus accumulated interest.
The assumptions used by Garman and Forgue (1994, p. 445) are based on an
8% annual return and 6% annual increases in the price of college. Therefore,
an investment of one dollar per year would accumulate to $27 (Garman &
Forgue, p. A-7) but the price of college would have increased by a factor
of 2.4 (Garman & Forgue, p. A-3) so that the
investment of $15 would
only purchase the amount of college that would cost $11 today! One would
have to obtain a nominal rate of return of over 10% per year to accumulate
a real purchasing power equal to $15 today from an investment of one dollar
per year for 15 years.
Given the above calculations and the previous discussion concerning
the life cycle patterns of real incomes, the temptation to try for high
rates of return should be obvious. If one obtains a nominal rate of return
of 8% per year and the cost of college is increasing 6% per year, parents
who start investing for a college fund when the child is three would have
to invest almost $0.09 per year for every dollar that college costs today.
If we use the crude approximation that all 4 years of the college fund
would be needed in 15 years, a goal of saving for a college education that
today costs $50,000 would require that the parents set aside $4,413 this
year and then again next year, etc. If the parents could obtain a 12% annual
rate of return, only $3,215 would have to be invested each year in order
to reach the goal.
Strategies Recommended for Investing for College
The problem faced by parents in choosing investment alternatives is
that higher rates of return can be achieved only by accepting higher volatility
of returns. Some of the risk can be reduced at little sacrifice by diversification,
hence the common advice to invest in mutual funds rather than individual
stocks. This advice is not universal (c.f. diverse advice in Sullivan;
1993; Grover & Zweig, 1994; Connolly; 1989; The Outlook, 1995)
and some experts suggest both aggressive mutual funds and growth funds
for an infant’s college fund (e.g., Garman & Forgue, 1994, p. 445;
Kobliner, 1989a). Perry (1991) and Weagley (1995) suggested starting with
a no load mutual fund and gradually shifting to less risky investments
when five years away from college. Rowland (1995) suggested investing in
stocks to save for college. Despite the common advice to invest in stocks,
a Money Magazine poll found that half of families were investing
entirely in fixed-income accounts (Wang, 1996).
Risk Versus Return
How should parents resolve the issue of risk versus return in investing
for a college fund? It is well known that stocks have a higher mean rate
of return than bonds. Between the beginning of 1926 and the end of 1994,
a dollar invested in small stocks would have grown to $2843, compared to
$811 for the S&P 500, $26 for long term government bonds, $31 for intermediate
government bonds, $38 for corporate bonds and $12 for Treasury bills (Ibbotson
Associates, 1995, p.99). If the long run patterns from the past are the
best indicators of the future, an investor who wanted to maximize expected
return and had a long term perspective would have a portfolio consisting
only of small stocks.
In order to obtain higher rates of return, an investor must accept greater
risk, or at least greater volatility. However, even this supposed truism
is not true in the long run. Small stocks performed best of six investment
categories in 47 out of 50 possible 20 year periods between 1926 and 1994,
and the S&P 500 performed best in the other three 20 year periods (Ibbotson
Associates, 1995, p. 43). If all future 20 year periods resemble these
50 time periods, small stocks present the least risk to the investor. A
20 year time frame may not be appropriate for many investors, however.
The standard deviations of one year returns of the Ibbotson investment
categories range from 35% for small capitalization stocks to 3% for Treasury
bills (Ibbotson Associates, 1995, 33). How should an investor balance the
mean return and the volatility as represented by the standard deviations?
State Prepaid Tuition Programs as Insurance
How can parents reduce the risk that a college savings fund will not
be adequate to cover tuition? Some states have marketed prepaid tuition
programs as a way to guarantee that college costs will be covered. Michigan
was the first state to promote such a program. However, the treasurer of
the state of Michigan was quoted in 1991 as having doubts about the financial
soundness of the state’s Michigan Education Trust (Blumenstyk,1991). Despite
the fact that the marketing implies that the state guarantees payment of
actual tuition, there is not an absolute guarantee (Blumenstyk, 1991).
Several states have experienced problems with declining surpluses (e.g.,
Lively, 1993; Button & Koselka, 1994; Carmona, 1994). There is uncertainty
about the tax status of such programs, and even though the Michigan program
received a favorable decision against the IRS on appeal, it was a split
decision and the IRS is considering an appeal (Healy, 1994). Financial
aid programs usually count such investments as being in the child’s name,
so potential financial aid can be reduced substantially by these investments.
These programs are expensive — in order to be guaranteed payment of tuition
in 18 years at a college that has a tuition of $10,000 today, one might
have to invest more than $10,000 today. Given the uncertainties and disadvantages
of these programs, they should not considered unless parents or grandparents
want to increase the chance that funds will be used for a child’s education
rather than some other purpose (c.f., Wang, 1996).
This article focuses on efficient portfolios for saving for college
— what combinations of investments in six major financial asset categories
provide the highest rate of return for each level of risk. The measure
of risk in this article is not the commonly used variance or standard deviation
of the annual rates of return of investments. Instead, a shortfall measure
is used — based on the real rates of return during the period 1926-1994,
the minimum accumulated value of a portfolio.
The choice of a time frame for analysis is of fundamental importance
to analysis of optimal portfolios. A one year time frame is clearly not
valid for someone years away from college.
In this article, three analyses are conducted based on the Ibbotson
Associates (1995) data for six financial asset categoriesa from
January 1, 1926 to December 31, 1994. All possible portfolios of the six
Ibbotson Associates (1995) asset categories were analyzed for a 15 year
time frame and a 10 year time frame, for accumulation of a college fund.
Then, a 100% small stock portfolio was analyzed on the assumption that
the family started 15 years before the freshman year and stayed fully invested
until each year’s college costs had to be paid.
The Research Question
Given that many parents need to carefully evaluate risk and return of
periodic savings for college for their children, all possible portfolios
composed of the six Ibbotson Associates financial asset categories were
evaluated to find the portfolios that provide the highest minimum return
for each level of mean return, based on 15 and 10 years of periodic savings.
Real Rates of Return
The real rate of return is the appropriate basis for evaluating investments.
Tax considerations may make the nominal rate of return relevant. However,
in this article, tax considerations are ignored. This may be a reasonable
assumption if the portfolio is tax sheltered. The implications of this
assumption are discussed later in the article.
The nominal rates of return and the inflation rates were drawn from
the Ibbotson Associates Stock, Bonds, Bills and Inflation Yearbook,
1995. Table 1 shows the six asset categories and the annual nominal
geometric mean and standard deviations.
Annual Nominal Geometric Mean and Standard Deviation of Six Financial
Asset Categories, 1926-1994.
|Category||Mean Annual Rate of Return(%)||Standard Deviation (%)|
|Large company stocks||10.2||20.3|
|Small company stocks||12.2||34.6|
|Long-term corporate bonds||5.4||8.4|
|Long-term government bonds||4.8||8.8|
|Intermediate-term government bonds||5.1||5.7|
|U.S. Treasury bills||3.7||3.3|
Source: Ibbotson Associates, 1995, p. 33.
The real rate of return was calculated as: (1 + nominal rate)/(1
+ inflation rate) – 1.
Simulations of Periodic Saving for College
There were 55 overlapping 15 year periods in the Ibbotson Associates
(1995) dataset. Portfolios with all possible combinations (in increments
of 1%) of each of the six types of investments were evaluated. For each
15 year period, the real accumulation (end value) resulting from investing
one dollar per year was calculated.b For each portfolio, the
mean accumulation and the minimum accumulated for all 15 year periods were
calculated. The same process was used for all 10 year time periods.
15 Year Time Frame
The portfolio giving the highest mean accumulation consisted of 100%
small stocks, with a mean accumulation of $42.22 from an investment of
one dollar per year for 15 years. The minimum accumulation for this portfolio
was $11.31. The portfolio giving the best
worst case scenario consisted
of 89% intermediate government bonds and 11% small stocks, with a minimum
accumulation of $14.70 and a mean accumulation of $19.53. All of the portfolios
were sorted from highest mean return to lowest, and any portfolio with
a lower minimum return compared to the portfolio with the next highest
mean return was eliminated as inefficient. Figure 1 shows the resulting
efficient frontier. All of the efficient portfolios contain small
stocks, and all but the one with the highest mean return contain intermediate
government bonds. No other types of financial assets were in any efficient
portfolio. All other possible portfolios resulted in a lower mean accumulation
than was possible for the same worst case. Table 2 illustrates some
of the results for both 10 years and 15 years of investing one dollar per
year in constant dollars. For 15 years of investing, the superiority of
small stocks over large stocks is clear — a portfolio consisting of 100%
large stocks would have a lower mean accumulation ($30.07) and a lower
minimum accumulation ($10.94) than would a portfolio with 100% small stocks
(mean of $42.22, minimum of $11.31). For the 15 year horizon, reducing
the amount going into small stocks below 11% of the portfolio would not
increase the safety of the portfolio, as a portfolio consisting of 100%
intermediate government bonds would have a lower minimum accumulation ($11.60)
than an efficient portfolio consisting of 89% intermediate government bonds
and 11% small stocks ($14.70).
Figure 1 is seemingly similar to more traditional mean variance analyses
(e.g., Ibbotson Associates, 1995, p. 161), with the minimum or worst case
return replacing the variance or standard deviation for the risk measure.
The usual measure of risk, the standard deviation of annual returns, does
not provide intuitive information to the ordinary investor. For instance,
some investments may have high returns and high volatility, yet, if the
distribution of returns is above the distribution of returns for a less
volatile investment, the high volatility (high standard deviation) investment
may be superior to the less volatile investment even for very risk averse
Minimum Accumulation by Mean Accumulation for Efficient Portfolios (line)
and 2 Selected Inefficient Portfolios for a 15 year Time Frame, For $1
per Year Investment.
Examples Of Mean And Minimum Accumulations Of Investing $1 Per Year
For 10 Years and for 15 Years.
|10 year||15 year|
|100% Small Stocks||20.26||5.28||42.22||11.31|
|50% Small Stocks, 50% Intermediate Government Bonds||15.87||7.53||29.92||13.21|
|11% Small Stocks, 89% Intermediate Government Bonds||12.05||9.29||19.53||14.70|
|5% Small Stocks,
95% Intermediate Government Bonds
|100% Intermediate Government Bonds||11.24||7.68||17.13||11.60|
|100% Large Stocks||16.03||6.21||30.07||10.94|
For the efficient portfolios, only the most conservative one
(89% intermediate government bonds and 11% small stocks) has substantial
chance of not having at least a $15 accumulation. The most conservative
portfolio has a 9% chance of not having at least a $15 accumulation, whereas
the 100% small stock portfolio has only a 2% chance of not having at least
a $15 accumulation.
10 Year Time Frame
The results for 10 years of saving were similar to the results for 15
years of saving (Table 2). An investment of one dollar per year in a portfolio
consisting of 100% small stocks would on the average grow to $20.26 after
10 years. The worst result of a 100% small stock portfolio was a real accumulation
of only $5.28. The portfolio with the best worst case consisted of 89%
intermediate government bonds and 11% small stocks, just as with the 15
year time frame. The worst accumulation was $9.29, and the mean accumulation
was $12.05. No other types of investments were in efficient portfolios.
Because of the similarity to the 15 year results, the discussion below
is based only on the 15 year results.
Consequences of the Worst Case
The conservative portfolio is predictable but would require more sacrifice
on the average than the most aggressive portfolio (100% small stocks.)
Parents who put all their contributions to the college fund in a small
stock mutual fund have a small risk (less than 2%) of having an accumulation
of only $11.31 for 15 years of investing one dollar per year. The second
worst accumulation for a 100% small stock portfolio was $15.67, which was
better than the worst eight outcomes for the most conservative efficient
portfolio (89% intermediate government bonds, 11% small stocks.) There
would be some risk from assuming the mean rate of return for the 100% small
stock portfolio, as half of the time the accumulation would be less than
$42 for each dollar per year invested. If the parents started investing
shortly after the child was born, it would be possible to vary the timing
of the liquidation of the small stock portfolio until the market got better.
The parents could rely on educational loans (Kobliner, 1994) or a home
equity loan to help cover educational expenses until the time was better
to liquidate the portfolio.
Obviously, conservative investors would find the preceding strategies
distasteful. The research reported in this article demonstrates that any
desired level of risk reduction can be achieved simply by increasing the
proportion of the portfolio devoted to intermediate government bonds. These
types of bonds are not familiar to many investors, but there are mutual
funds available composed of these types of bonds (Meyer, 1991).
Liquidation Over Four Years
The preceding analysis is based on the assumption that the periodic
investment takes place for 15 years, then the fund is liquidated, and presumably
used for college expenses or put into very liquid, safe investments until
needed for college expenses. If, however, parents start investing 15 years
before their child will start college, take out enough for freshman expenses,
but leave the balance fully invested for the three subsequent years’ expenses,
what will be the result if all contributions are invested in small stock
funds that match the performance of the Ibbotson small stock category?
In real terms, the worst starting year (1958) would have produced $19.40
for every one dollar per year contributed ($0.25 each year for 15 years,
another $0.25 each year for 16 years, etc.). The mean accumulation was
$46.96. Figure 2 shows the pattern for starting points since January 1,
Real Accumulation, For $1 per Year Investment, Starting 15 Years Before
College, and Staying Fully Invested in a Small Stock Fund Until Each Year’s
College Expenses Paid.
Predicting future patterns may be difficult, but the worst
periods to start on an aggressive investing program were associated with
just before the Great Depression and with a period that went into the turmoil
of the 1970’s. If one is optimistic that the domestic and international
political patterns of those periods will not be repeated, then a prudent
assumption would be that each year $400 (in constant dollar terms) should
be invested for every $10,000 needed for college, if the freshman year
is 15 years away. Each year, the contribution should be increased by the
rate of increase in college costs. If somewhat different broadly based
small stock funds were held, the investor might be able to choose which
one to liquidate for each year’s college costs.
Starting More Than 15 Years Early
If the family starts more than 15 years before the funds are needed,
small stocks become even less risky. For annual investing with a 20 year
time frame, small stocks had a better worst-case outcome than any other
possible portfolio based on the Ibbotson asset categories. A family that
stayed fully invested would do much better most of the time, and somewhat
better almost all of the time, if the investment period were 18 to 21 years.
Saving for post-B.S. education would make small stocks even more advantageous.
The analyses presented in this article are based on the assumption that
real returns of financial assets have patterns similar to the patterns
since January 1, 1926, and that the future relationships among the asset
categories are similar to the pastc.
The technical results of this article are unique in presenting an efficient
set of investment portfolios based on periodic investing over a 10 or 15
year time frame. The similarity of the 10 and 15 year results suggests
that the results are fairly robust for time frames of 10 years or more.
All efficient portfolios in terms of mean and minimum real accumulation
contained small stocks, and for the 15 year time frame, the only other
type of investment in efficient portfolios was intermediate government
bonds. If a small stock portfolio were held until needed for each year
of college, there would be relatively little risk of a shortfall, if the
small stock mutual funds chosen matched the record of the Ibbotson small
stock index. For families who would otherwise have little chance of accumulating
a substantial amount of funds for college costs, the aggressive investment
strategy suggested in this article may provide the only alternative that
would come close to meeting a goal of sending one or more children to college.
Income taxes were not considered in the analyses. If the portfolio income
were taxable, the superiority of small stocks would increase, especially
if they were purchased in an index mutual fund which bought and held them,
generating low levels of realized capital gains.
Ideally, the evaluation of the investment choices for saving for college
would consider many aspects of the family’s financial situation, including
the expected income patterns over the years, the length of time between
college costs and retirement, and, for some families, estate planning factors.
Future changes in income tax regulations, for instance in the use of tax
sheltered retirement savings for educational costs, will have an impact
on optimal investment strategies. Many factors may objectively influence
the best choice of a portfolio for a particular family, in addition to
the family’s subjective level of risk aversion. What this article has shown,
however, is that of the thousands of possible portfolios for saving for
college, only a small number should be considered by any family. All of
the other portfolios would be inferior in the minimum accumulation likely
for any given level of expected accumulation.
The results of this article are applicable to any investment goal for
which someone made periodic contributions for a period of 10 to 20 years.
For any type of goal, each family would have different considerations in
evaluating the possibility of an investment falling short of the desired
The results presented also provide an additional argument for starting
to invest early. To prudently take advantage of high rates of return of
small stocks, one must start about 20 years before a goal.
a. The Ibbotson Associates (1995) categories are:
Large Company Stocks: S&P 500 composite with dividends reinvested.
(S&P 500, 1957 – present; S&P 90, 1926-1957)
Small Company Stocks: Fifth capitalization quintile of stocks on
the NYSE for 1926-1981. Performance of the Dimensional Fund Advisors (DFA)
Small Company fund 1982-Present.
Corporate Bonds: Salomon Brothers Long-term High grade Corporate
Long-Term Government Bonds: 20 year U.S. Bonds.
Intermediate-Term Government Bonds: Government Bonds with 5 year
Treasury Bills: 90 day T-bills.
b. The authors wrote a computer program to calculate the end values
of portfolios based on the Ibbotson Associates (1995) annual returns, assuming
that at the beginning of each year one dollar (in constant dollar terms)
is contributed to a fund. There were roughly 100 million calculations necessary
to find the efficient portfolios.
In the analysis presented in this paper, the historical record consists
of the Ibbotson Associates (1995) rates of return on six financial asset
categories. The simulations conducted by the authors implicitly take into
account relationships among the asset categories. Even though the historical
record is only 69 years, there is some evidence from almost two centuries
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1. Professor, Consumer Sciences Department,
The Ohio State University, 1787 Neil Ave., Columbus, OH 43210-1295. Phone:
(614) 292-4584. FAX: (614) 292-7536. E-mail:
2. Peng Chang, Ph.D. candidate, Consumer and Textile
Sciences Department, The Ohio State University, 1787 Neil Ave., Columbus,
OH 43210-1295. Phone: (614) 292-4389. Fax: (614) 292-7536. E-mail: email@example.com
The version of this article presented at the 1995 AFCPE meeting received
an Article of Excellence award from the Certified Financial Planner
Board of Standards in 1996.