Dealing with Terminal Wealth Dispersion
Aim High!

Addressing terminal wealth dispersion (TWD) in your financial planning is absolutely necessary, as not doing so is hazardous to your wealth. Given the broad range in your portfolio's probable value at the end of your expected holding period, i.e., its terminal value and your terminal wealth (TW), you must align the lower end of this range as closely as is practical with the minimum amount of money you will require at the end of the holding period.

The active phrase here is "as closely as is practical." Practical refers to a few factors:

• Minimum must be a true minimum, not what you'd like it to be. For instance, if you're saving for retirement, the minimum would be the least you would actual need to live on, not the amount you would need to live the lifestyle you prefer.
• Matching the bottom of the range with the minimum amount of terminal wealth essentially adds a risk-free investment to your portfolio that will deliver the minimum terminal wealth with 100% certainty. The rest of your portfolio would carry risk and its return would be what enhances your lifestyle in retirement. This sounds good but it's usually not practical, as the amount of the investment required would be quite large. Unless you're a very long way from retirement, have a real good income and are willing to forgo a lot of consumption while you're young for the sake of ensuring your retirement is secure and comfortable, this option is not practical.
• A balance must be achieved between your level of risk aversion, the amount you can afford to invest and the degree of certainty that your terminal wealth will be at least the minimum you require. This is investing, after all, and one of the inherent risks is the risk of earning a return that is less than the risk-free rate. Being broadly diversified minimizes this risk but does not eliminate it.

So, although you need to aim high, your risk tolerance and income limit the elevation of your target. The good news is that by aiming high, you will have a high probability of having terminal wealth that is well above your required minimum, and being that the target is the mean, your terminal wealth is more likely to be close to the mean than at one of the extremes. Although it would be nice to end up at the upper extreme, that's equally as unlikely as ending up at the lower extreme.

You should recall from the discussion of terminal wealth dispersion on the Time Diversification page that the range of the value of a portfolio at the end of its holding period is described by the variance of the mean annual return. Specifically,

• The expected annual return is not expected to be the same each year.
• The expected value of the standard deviation of the annual returns is the same for all years.
• The magnitude of the annual deviations increases over time with the value of the portfolio and this increase is expected to be exponential.
• The standard deviation of the mean annual return, i.e., the average of the annual returns over the holding period, is expected to be the annual standard deviation of returns divided by the square root of the number of years in the holding period: S_p ÷ √n, where S_p is the standard deviation of annual portfolio returns and n is the number of years in the holding period.
• The mean annual return is the compound rate of return that determines the terminal value.

A Procedure for Dealing with Terminal Wealth Dispersion

1. Determine your risk tolerance and identify a portfolio that is consistent with your risk tolerance. This will give you values for the expected return and standard deviation of your portfolio. And don't forget that the broader your universe, the lower your risk for any given expected rate of return.
2. Determine the length of your holding period. This will be determined by the objective of your investing, such as saving for retirement of for your children's college education.
3. Determine the amount of money required at the end of your holding period. The minimum for an objective like retirement is somewhat subjective, but the amount required for a specific lump sum expense will be a fixed amount. In the case of a fixed lump sum payment, you may have no alternative but to match the lower end of the range of terminal wealth to that amount unless you have some other means of covering a shortfall.
4. If you have estimated the amount required in the future in current dollars, you will have to adjust that amount for inflation over your holding period. Multiply the current amount by (1 + i)ⁿ where i = the estimated annual rate of inflation and n = the length of your holding period in years.
5. Select the desired level of certainty, i.e., the probability, that your terminal wealth will be at least the minimum amount required after adjusting for inflation. This is the factor that is most likely to need adjusting to achieve the balance between risk aversion, affordability and level of comfort.
6. Using a table of normal probabilities, look up the z value that corresponds with your chosen level of certainty. Z is the number of standard deviations a point deviates from the mean. (The linked table lists the total probabilities from left to right. However, many normal probability tables only list the probability from the center of the distribution and 0.5 must be added to those probabilities to get the total probability left of z.)
7. Solve the following equation for the portfolio return that will produce the minimum terminal wealth Min R_p = E(R_p) - z (S_p ÷ √n), where Min R_p = the minimum required portfolio rate of return, E(R_p) = the expected portfolio rate of return, z = the number of standard deviations the minimum terminal wealth deviates from the expected terminal wealth, S_p = the expected annual portfolio standard deviation and n = the length of your holding period in years.
8. Using a financial calculator set to "end of period" or a spreadsheet, solve for the monthly investment required. This will be the payment (PMT). Input N = 12n, R (or I) = Min R_p ÷ 12 and FV = Min TW. If you have saved an initial investment, input that amount as the PV. (The PV should be input as a negative number and the payment will also be a negative number, as they are both investments, which are cash outflows.) Press the payment key to see the monthly investment required. (Assuming a fixed payment over the total holding period is a bit naive. By following the same logic described, you can build a spreadsheet model that factors in inflation of the payment.)
9. If the payment is more than you can afford, you will have to do some combination of the following:
   • Lower the level of certainty used to find the z value.
   • Lower the minimum terminal wealth, if that's a viable option.
   • Invest in a riskier portfolio that has a higher expected rate of return. (As you become more familiar with investing and gain a better understanding of risk, you will probably find that you are more risk-tolerant than your originally thought.)
   • Get a second job.
   • Marry someone who is wealthy.
10. Compute the expected value of your terminal wealth, E(TW), using a financial calculator set to "end of period" or a spreadsheet. Input N = 12n, R (or I) = E(R_p) ÷ 12 and payment (PMT) = the required monthly investment computed above. Press the future value key, FV, to find E(TW).

The six sigma bounds (99.72% probability) of your terminal wealth can be computed as E(R_p) +/- 3(S_p ÷ √n). For estimating purposes, 99.72% is essentially equal to 100%.

Now, that's a lot to digest. I know, because it took me quite a while to figure it out and decide how best to describe it. But doing an example should make it much easier to understand.

An Example of Dealing with Terminal Wealth Dispersion

Here's what he has to work with:

• Bill's holding period is 16 years.
• Two years ago, Bill started worrying about retirement and started investing $20,000 per year in a 401k sponsored by his employer. The balance in his account is now $44,000.
• His salary is $70,000 per year. The $20,000 401k contribution is deducted prior to any payroll deductions, which makes his effective salary $50,000.
• His after-tax income is $36,000 and he's been able to get by on that amount. He drives a practical car and doesn't spend much on luxuries like dining out and Caribbean vacations, but he's reasonably comfortable and saving for retirement gives him a nice sense of security.
• Bill figures his Social Security benefit will be the inflation-adjusted equivalent of $1500 per month ($18,000 per year). So he'll need another $18,000 after taxes to maintain his modest lifestyle.
• For planning purposes, he assumes inflation over the next 16 years will be equal to the long term average of 4% per year.
• Bill spent some time on the Internet reading about risk and completed some risk questionnaires to determine his level of risk aversion. He then selected a model portfolio with an annual standard deviation of returns of 8.00% and an expected return of 12.08% per year. (This portfolio was drawn from the same universe as the four-asset portfolio used in the example on the Efficient Frontier page of this web site.)
• Being that Bill's only source of retirement income other than his savings will be Social Security, he desires a relatively high level of confidence that his terminal wealth will be adequate to support him in retirement. So he decides to use a probability of 90% in his calculations.
• Bill has read that a diversified but relatively conservative portfolio has a high probability of lasting over a 30 year retirement if somewhere between 4% and 6% of the portfolio's value is withdrawn the first year and that amount is subsequently increased each year by an amount equal to the current inflation rate. So he decides to use 5% for planing purposes.
• Finally, Bill assumes his combined tax rate in retirement will be 28%.

• Holding period: 16 Years
• PV of investments: $44,000
• Salary net of 401k contribution: $50,000
• After-tax income: $36,000
• Anticipated Social Security income in current dollars; $18,000
• After-tax income required from investments in current dollars: $36,000 - $18,000 = $18,000
• Assumed inflation rate: 4%
• Expected annual return on investment portfolio: 12.08%
• Expected standard deviation of annual investment return: 8.00%
• Desired certainty of meeting or exceeding minimum terminal wealth: 90%
• Estimated length of retirement: 30 Years
• Initial withdrawal: 5% of terminal wealth, to be increase each year by the rate of inflation
• Assumed combined tax rate in retirement: 28%

To determine his minimum terminal wealth (Min TW), Bill divides $18,000 by (1 - 0.28 ) to determine the required pretax amount of $25,000 in current dollars. Then he multiplies that amount by (1 +0.04)¹⁶ to inflate it to $46,825, the amount he estimates he'll need during his first year of retirement 16 years from now. If this amount is 5% of his Min TW, then his Min TW must be $46,825 ÷ 0.05 = $936,500. (Gulp!) Bill heads to the refrigerator for a bottle of his favorite carbonated beverage.

Bill consults the table of normal probabilities and finds that 0.8997 is the closest number to the 90% level of certainty he chose for his Min TW and that the corresponding z value is 1.28 standard deviations. (0.8997 is at the intersection of row 1.20 and column 0.08, the sum of which is 1.28.)

Using the z value of 1.28, his expected portfolio return = E(R_p) = 12.08% and expected portfolio standard deviation = S_p = 8.00%, Bill calculates his minimum required portfolio return = Min R_p = E(R_p) - z(S_p ÷ √n) = 12.08% - 1.28(8.00% ÷ √16) = 12.08% - 1.28(2.00%) = 9.52%.

Now bill has all the information he needs to do his first TVM calculation with his trusty financial calculator to find his required monthly investment. To do this, he inputs N = 12n = 12x16 = 192, R (or I) = Min R_p ÷ 12 = 9.52% ÷ 12 = 0.7933% per month, the present value of his savings = PV = -$44,000 and future value = FV = Min TW = $936,500. (Be sure the PV is input as a negative. The payment calculated will also be negative, as it's a negative cash flow.) Pressing the PMT key, Bill finds that his required monthly investment is $1,640.27 which adds up to $19,683 per year, slightly less than the $20,000 he is now investing. Bill breathes a sigh of relief and makes another trip to the refrigerator.

Bill decides to stick with his $20,000 annual investment, figuring that will increase the probability of his TW being at least equal to his minimum terminal wealth. So, already having most of the information in the registers of his financial calculator, he divides $20,000 by 12 months, inputs the result as a payment of -$1,666.67 per month and presses the R (or I) key to find the corresponding minimum required return. It is 0.7839% per month x 12 months = 9.41% per year.

Now he solves the Min R_p formula for the new z value: Min R_p = E(R_p) - z(S_p ÷ √n) = 9.41% = 12.08% - z(2.00) and z = 1.34. He looks in the probability table and finds that a z value of 1.34 corresponds with a probability of 0.9099 = 90.99% ~ 91%. So there is a 91% probability that he will have at least $936,500 in 16 years and only a 9% probability that he will have less. Bill is very comfortable with this situation and decides to see what his actual target terminal wealth is.

Bill's expected terminal wealth, E(TW), can be found using most of the information still in his calculator. All he has to do is input the monthly equivalent of his expected return of 12.08%. He divides 12.08% by 12 months to find the mean monthly rate of return of 1.0067%, inputs 1.0067% per month as the rate of return and presses the FV key to find that his expected terminal wealth is $1,268,408. Bill knows enough about the characteristics of the normal probability density function to realize that his terminal wealth is apt to be closer to $1,268,408 than $936,500, so he's feeling pretty good and is glad he took the time to go through this exercise. Bill makes one more trip to the refrigerator and retires to the back porch.

The only thing Bill didn't do was compute the effective bounds of his terminal wealth, which define his terminal wealth dispersion. +/-3 standard deviations brackets 99.72% of the probable range, which is close enough to 100% for estimating purposes. Thus the range is defined by E(R_p) +/- 3(S_p ÷ √16) = 12.08% +/- 3(2.00) = 12.08% +/- 6.00%. So the lower end of the range is calculated with 6.08% ÷ 12 = 0.5067% per month and the upper end is calculated with 18.08% ÷ 12 = 1.5067% per month. All the other information is still in his calculator so all he has to do is input these new rates of return to find the limits are $655,234 and $2,619,882.

Terminal Wealth Dispersion & Individual Retirement Accounts

Bill's 16 year holding period leaves him very susceptible to the effects of a long bear market. Being broadly diversified mitigates this risk but does not eliminate it, as it's entirely possible for a worldwide bear market to occur during Bill's holding period. Then at the end of his holding period, he has to begin a new holding period, which will be the term of his retirement and it carries the same concerns as his holding period prior to retirement.

The other fault in individual retirement accounts is the assumption of longevity risk by individuals. The managers of large pension plans can depend on retirees living on average for only the average life expectancy of the population, so they only have to fund the plans to cover the cost of the average term of retirement. Individuals, however, don't know how long they're going to live, so they must over-save to ensure that they don't run out of money before they run out of time.

Bill based his estimate on living to the age of 96, but he may only live to be 76 or he might live to be 106. If he only lives to be 76, then he will have forgone a lot of pleasures, such as travel, fine dining and better vehicles, that he could otherwise have enjoyed.

Passing off the burden of retirement to individuals was a great deal for corporations but it's a very poor deal for most individuals. I say "most" because there will be some who's holding periods coincide with bull markets and who also live a long, healthy life. They will be the ones who benefit from over-saving and living beyond the average life expectancy.

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