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In the last page I showed how the annuity due formula allowed us to compute how much money was needed to be set aside monthly in order to reach a targeted future value. The example given was saving for college tuition. While there were some unknowns in this problem, it was fairly clear when and how much of a lump sum would be needed in the future. Here we consider two significantly harder, but very important time value of money applications: funding for retirement and funding a certain type of pension plan. Let’s begin with retirement planning.

Frida, a 67-year-old self-employed graphic artist, consults with her financial advisor on the question of if, and when, she can retire. She has accumulated $400,000 in retirement accounts and is currently earning $60,000 a year. She will be eligible for monthly social security benefits of $2,000 per month which begin at any point she decides to retire. How should her advisor respond? What questions need to be asked and answered? How will time value formulas be applied to the problem?

The first question is how much will Frida need each month to live? The next question is how long will she live? The first question is easier to answer. Most folks do not want to suffer any significant reduction in living standard when they retire. So, when someone asks if they can afford to retire, they usually mean can they retire comfortably. They are not asking about the best cat food recipes, or the most comfortable shoes when acting as a greeter at Walmart. So, assuming that Frida is living comfortably now, she can use her current living expenses as a guide for what will be needed in retirement. Suppose Frida spends about $42,000 per year on living expenses.

For purposes of simplicity assume that Frida has no children and is not concerned with leaving a legacy to a favored charity or relative when she dies. Basically, she does not want to outlive her money or have her money outlive her. Also, we will ignore the prospect of long-term nursing home care because Frida has purchased long term care insurance.

So, $42,000 per year translates to $3,500 per month in targeted living expenses. Since Frida has $2,000 per month in Social Security benefits her retirement savings will have to make up the $1,500 short fall.

But we still must answer the question of life expectancy. One way to handle this is to consult the US Life Tables. These tables are compiled from census and other data by the National Center for Health Statistics which is part of the Center for Disease Control and Prevention. The tables include life expectancy for individuals in different gender and ethnic groupings. An example is shown below. Essentially the table tells you on average how much longer you will live based on your attained age and classification. In Frida’s case the table indicates that the average additional life span for someone now aged 67 is about 20 more years.

Now keep in mind that this is an average and the odds of Frida living precisely 20 more years are not as high as the odds that she will either live longer or shorter than precisely 20 years. But for a first approximation let’s use this life expectancy to see where things stand.

Now we also must assume a rate of return on the diminishing balance of funds that will be used to fund monthly payouts. To be clear the funding must come from the $400,000 she has saved in her retirement account. Because she will have immediate need for these funds and cannot afford to absorb big losses in this account it makes sense to assume that the account will be invested in fixed income securities which as of 2020 do not earn much of a return. Currently Ten-year treasury notes yield a little under 2%. Fixed income index funds are yielding only slightly better. So, with an abundance of caution assuming a 3% rate of return would appear prudent.

Income taxes are another factor here because her retirement fund is almost 100% pretax which means that all distributions will be taxed not just on the income portion of the distributions but the principal portion as well. How do we handle this? Well, the simplest way to deal with this is to increase the monthly payout so that the net of taxes amount for living expenses is $1,500. Assuming a tax rate of 15% entails that we should gross up the required distribution to $1,765 ($1,500/(1-.15)). Let’s round this to $1,800.

And again, we have the perplexing problem of inflation. Do we factor it in, or don’t we? There are a couple of reasons in this situation not to. First, social security benefits are subject to inflation adjustments, and second if inflation increases significantly interest rates are likely to rise and with them the rate of return on fixed income securities. So, in this case we will not adjust living expenses for inflation.

The next question is what formula should we use to compute the needed amount in the retirement account? Well, what we have here is a in effect a reverse mortgage. We want to see, given a fixed interest rate, a fixed number of compounding periods (based on life expectancy), and fixed payments whether the retirement balance will be completely amortized, that is brought to zero. So, we can take our mortgage payment formula shown here and using some algebra convert it like this.

When we do this using this new variant, we see that under the assumptions given Frida will utilize about $325,000 of the $400,000 she has in her retirement account. Here are the assumptions and computations.

Now recall that we used her life expectancy according to the US life tables, 20 years. But what if she will live 30 more years instead? Well using the same formula, she would require about $425,000 in retirement savings, pretty close to the amount she has now. So, ignoring inflation, Frida should be able to retire now and live comfortably for the rest of her life.

Now in this example we were dealing with just one retiree. A parallel type of issue is faced in the funding of a certain type of pension plan, known as a defined benefit plan. In a defined benefit plan an employer promises to pay its employees when they retire a certain fixed level of payments for the rest of their lives. The challenge to employers that offer this type of pension is trying to figure out how much they must set aside each year to ensure that they will have the funds available to make the promised retirement payments.

A common benefit formula is a fixed percentage times the average of the highest three years earnings times the total years of employment. Here is an example of a plan that offers 2% per years of service times the average of highest three years earnings.

Essentially the pension fund must have accumulated and maintained enough assets today to payout the benefits of both current employees and retired employees. There are numerous forecasts that must be made in order to derive the annual contribution. Usually there is more certainty about benefits owed to currently retired employees because the specific monthly benefits are already fixed. The major unknowns are the rates of return on pension assets and the life expectancy of current retirees. Utilizing appropriate life expectancy tables, the total future payouts can be fairly precisely predicted.

There is more uncertainty about benefit levels that will have to be paid out for current employees who have yet to reach retirement age. The exact benefits that will be owed are not yet fully determinable because the highest years compensation and years of service are not known with certainty. Here actuaries must rely upon the company specific average of compensation and years of service at retirement age.

A series of expected pension benefit payouts will be projected over the next thirty to forty years depending upon the age of current retired and non-retired employees. Now one thing will be certain is that the projected benefit payouts will not be the same year to year. The next ten or so years may look like this for a particular company.

This variation in annual payout means that we cannot simply use the reverse mortgage formula we just used in our individual retiree example. Instead, the basic formula for the present value of a lump sum must be applied year by year and then added. Utilizing a standard spreadsheet formula like the NPV in Excel yields a present value in seconds.

Once the present value balance is computed it can be compared with the actual assets on hand in the pension fund. If the assets are not sufficient additional contributions must be made.

Because of the myriad uncertainties associated with computing future benefits under a defined benefit pension these plans have become less and less popular. Most companies now offer what is termed a defined contribution plan. With this type of plan the employer puts a percentage of an employee’s current compensation into an investment account each year. The company makes no promise to pay out any fixed benefits based upon a formula. Whatever balance the investment account grows to over the years of employment is the amount that the retiree will have available to fund their retirement. So, no time value or probability estimates need be made for these types of plans.

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