Exchange Traded Funds (ETFs) represent a revolution in investing, giving investors an unprecedented opportunity to add a dizzying array of asset classes to their portfolios at very low cost. However, these choices are not without risks. If an ETF is added to a portfolio in an unsystematic manner without regard to a sound strategy, it could wind up being a very costly mistake for an investor.

This was seen in 2008 where far too many had become overweight in equities much too close to retirement. Many of these investors panicked as the market crashed in the fall of 2008. The result was selling into a very depressed market. Investors that sold, and then failed to get back in as the market bottomed in March of 2009, missed the rebound in 2010, and in many cases locked these losses in permanently.

The Five Minute Retirement Plan was designed to quickly and efficiently create investment plans that balance risk and reward based on back-tested statistical models. The planner automatically takes into account time frame and risk tolerance to create a portfolio of ETFs to best meet an investor’s goals.

This article will discuss how these investment plans are created to take maximum advantage of the market’s potential upside over an investor’s time horizon, while systematically rebalancing into less risky assets over time, helping ensure that an investor’s nest egg will be available when needed.

The following steps were taken to create the system: (1) choosing asset classes and creating portfolios, (2) finding a function to model the reduction in risk of a portfolio over time, (3) creating an algorithm to allocate funds to portfolios, and (4) validating the results with back-tests.

**Choosing asset classes and creating a set of efficient portfolios**

The risk of a portfolio, according to Modern Portfolio Theory (MPT), can be measured by taking the standard deviation of returns of that portfolio over time. MPT considers a portfolio to be efficient if the combination of the underlying asset classes can be shown to have the highest return for a given standard deviation.

To construct a portfolio efficiently, asset classes with differing covariance need to be chosen to ensure the portfolios have the required diversification. This covariance allows a portfolio to be constructed with a higher return for a smaller standard deviation than would be possible by using any of the underlying assets alone. To achieve enough diversification, ETFs consisting of US equities, developed international equities, emerging international equities, long term bonds, short term bonds and inflation protected bonds were chosen.

To maintain a balance between too large a set of asset classes and too small a set, the following eleven asset classes were chosen:

- The United States, being the world’s largest economy, is allocated the majority of equity holdings in the portfolios with up to 55% of assets. These assets are further sub-divided into the three asset classes: 1. large (30%), 2. medium (10%) and 3. small (15%) capitalization stocks.
- In the past twenty five years, globalization has been a major economic trend, with the emerging market showing impressive gains. Since this trend is expected to continue, stocks of companies in the emerging markets (China, Brazil, South Korea, Taiwan & India, etc.) make up the fourth asset class, and are given a weight of up to 25%.
- The Asia-Pacific and European developed regions are represented by companies with large capitalizations and are given a weight of up to 10% and 5% respectively for the fifth and six asset classes.
- The US Treasury Inflation Protected Bonds and REITs are used to combat the risk of inflation and are given a weight of up to 20% and 5% respectively for the seventh and eighth asset classes.
- Long Term US Treasury bonds are used to combat deflation, and are given a weight of up to 20% for the ninth asset class.
- Total Bond Market and Short Term US Government Bonds are allocated up to 20% and 100% respectively and are included to get low risk returns over medium and short time frames for the tenth and eleventh asset classes.

The most risky portfolio was created by taking the equity asset classes only and assigning weights to them so that the best historical return to standard deviation ratio was achieved. The least risky portfolio was created simply by allocating 100% of the assets to Short Term US Government Bonds.

The remaining eighteen portfolios were constructed by moving 5% from one of the equity asset classes to the fixed income asset class in such a way that both the return and standard deviation were less in the resulting portfolio.

The resulting risk/return frontier is shown below [* all images*]:

**Function to model the reduction in risk of a portfolio over time**

If each portfolio’s risk is measured by the standard deviation in returns over time, the next question is: “how is the risk of the portfolio affected by the amount of time that it is held?”

Fortunately, this problem is well understood in statistics. The Central Limit Theorem states that the relationship between the standard deviation of a population and the standard deviation of a sample taken from that population is the standard deviation of the population times the square root of the sample size.

If holding a portfolio for one year is considered a sample size of one, and holding it for two years is a sample size of two, etc., the expected mean and standard deviation of the sample can be calculated to see how the risk of the portfolio will decline over time. A test was done to see how well this function could predict the reduction in risk of holding the S&P 500 over a sixty-one year period from 1950 through 2011. The function showed a 98% correlation with observed data. Below the results of the test are graphed with the model results shown in purple.

Next, to calculate an expected return for a portfolio, the concept of risk tolerance needs to be introduced. Risk tolerance can be thought of as the ability to accept a small or wide range of outcomes. Since this is the same as saying one wants to be X% sure of a given outcome, a good way to model risk tolerance is to use the concept of confidence levels from statistics. A confidence level is the measure of the probability that a sample from a population will return a value of X or better. For example, the probability that an adult male chosen at random from the North American population will taller than 5 feet can be estimated with 99% confidence.

The Central Limit Theorem states that the sampling distribution is normal even if the underlying distribution is not. This allows for the use of standard statistical functions for the normal distribution (bell curve) to calculate the expected return for a portfolio will not be below X% for a given confidence.

Since the mean and standard deviation for each portfolio can be calculated for each holding period, the expected return for each of the portfolios can be calculated with a given confidence. The portfolio with the best return for each holding period is the one that should be held by an investor that wants to be X% confident of an outcome over that time frame.

Below is the expected return using a 95% confidence level for all of the portfolios across a 1 to 20 year holding period (X axis). The best portfolio for each holding period and its expected return are shaded in green.

**Algorithm to allocate funds to portfolios**

To select the best portfolios to be held for a given number of years, the system simply calculates the expected return based on risk tolerance and number of years held from 1 to 24 years for all twenty portfolios. The portfolio with the highest expected return for each of the years held is then chosen, creating a list containing the best portfolio and its expected return for each of the years. If a time horizon of greater than 24 years is needed, the best portfolio for the 24 year holding period is used. Next, the system determines the expected return for the set of the best portfolios selected.

A good way to illustrate the method for calculating this is by using the example of funding a child’s college education. The length of time the child will be in college is four years. Assuming the child is starting college this year, the system determines the best portfolios to hold for one, two, three and four years based on risk tolerance, and puts one quarter of the college fund into each portfolio. Since the expected return for each portfolio given its sample size is known, the system can determine the expected return for each year by averaging all of the returns together, since each outcome is equally likely for any portfolio in any year. As each year passes, the calculation is redone after removing one of the years:

Year 1 Expected Return = AVERAGE(Return1, Return2, Return3, Return4)

Year 2 Expected Return = AVERAGE(Return1, Return2, Return3)

Year 3 Expected Return = AVERAGE(Return1, Return2)

Year 4 Expected Return = AVERAGE(Return1)

The next question is how to choose the portfolios for the years before the child enters college? The solution is to extrapolate the process backward. Here is an example for a college plan for a twelve year old where Return[N] is the expected return for the best portfolio to hold for N years based on the risk tolerance.

Age 12 – Expected Return = AVERAGE(Return6, Return7, Return8, Return9)

Age 13 – Expected Return = AVERAGE(Return5, Return6, Return7, Return8)

Age 14 – Expected Return = AVERAGE(Return4, Return5, Return6, Return7)

Age 15 – Expected Return = AVERAGE(Return3, Return4, Return5, Return6)

Age 16 – Expected Return = AVERAGE(Return2, Return3, Return4, Return5)

Age 17 – Expected Return = AVERAGE(Return1, Return2, Return3, Return4)

At age 12 the first quarter of the retirement fund will be held for six years, being sold after age 17, thus the return for the best portfolio with sample size of six is the appropriate return estimate for this part of the fund at this age. As each year passes, there is one less year to go before college, so these funds must be moved to the best portfolio for five years, etc. This calculation is done for each part of the fund for all the years going forward to create the expected return of the entire fund for each year before and during college.

Since the expected return for each year, how much money the plan started with and how much is added each year is known, the system can use simple arithmetic to find out how much money the fund is expected to have at any point in the future.

**Sample Portfolio Created with Model**

Below is an example of a retirement portfolio generated by the system. The hypothetical retiree has a $1,000,000 in assets and is age 65. Assuming a 95 year life expectancy with a very low risk tolerance (99% confidence level) and three percent inflation, the system predicts this retiree would be able to withdraw up to a maximum of $46,018 from their savings over a thirty year retirement period assuming they rebalanced annually into a less risky portfolio.

The system shows the ETF symbol and description along with number of shares needed and percentage of the portfolio each asset class represents.

Based on historical data, the retiree could expect the value of their holdings to drop by up to 17% in a given year. The stock allocation represents 46% of assets and the bond allocation represents 54% of assets.

**Back-testing results with Monte Carlo simulation**

The algorithm to rebalance a portfolio yearly to meet retirement goals uses a system of dividing the retirement balance into equal amounts, one for each year in retirement, and distributing them across the best portfolio based on risk tolerance and number of years held.

The number of years remaining in retirement is reduced by one as each year passes. This causes the existing funds to be reallocated across a set of portfolios with one less portfolio each year, which introduces an error as the funds sometimes cannot be held in the more risky portfolios for as long as needed.

To estimate how large this error is a Monte Carlo simulation was run using a $1,000,000 portfolio for a 65 year old retiree in their first year of retirement with a 95 year life expectancy over 99%, 95%, 90%, 85% and 80% confidence levels.

The simulation selected a random year and then used that year’s actual return for each portfolio in the group. Using the returns from a single year for each portfolio was done to account for the covariance between each of the portfolios to make sure real-world market fluctuations were accurately represented.

The simulation then calculated the return for each selected year and from that determined the maximum withdrawal that could be made from the portfolio over the thirty year period. This was repeated through 1,000 iterations to generate a set of Monte Carlo data.

The results of the Monte Carlo runs were compared to the model values for 99%, 95%, 90%, 85% & 80% confidence levels. The Monte Carlo simulation showed the probability of shortfall as 3.10%, 7.80%, 11.40%, 18.20% & 22.30% respectively, so the allocation algorithm is introducing an error of 2 to 3%, which is reasonable.

Below is a graph of the Monte Carlo run for the 99% confidence level. The model predicted that $46,018 could be withdrawn from the retirement fund yearly. The shortfall is represented by the bars at $45,000 and below.

**Conclusion**

In today’s turbulent markets it is more important than ever for ETF investors to manage their exposure to risk. In addition to maintaining sufficient diversification, investors must also make sure their portfolio risk matches their investing time horizon, and at a minimum must rebalance annually into a less risky portfolio to ensure that they are not facing large losses at exactly the wrong time. The methodology behind the Five Minute Retirement Planner ensures that all of these goals are met.

**Disclosure: **I am long SPY, VO, VB, VNQ, VWO, VPL, VGK, VGLT, BND, TIP, SHY.