## Saturday, October 1, 2016

### Calculating Required Retirement Savings Rates: Part 1

In one of my early blog posts, written in December 2009 for recent college graduates , I wrote, "You must manage your spending so that you can save a significant portion of your income -- at least 10%, and more if possible." How do we determine if 10% is enough, or could it even be more than you really need to save? Although there are too many unknowns to answer this precisely, we can make various assumptions to calculate a range of savings rates that are likely to enable you to enjoy a financially secure retirement.

In this post I'll show how we can calculate that a savings rate of about 14% of gross (before-tax) income is required, given the following facts and assumptions:
• Current savings: \$0.
• Current age: 25.
• Retirement age: 65.
• A steady income with annual raises equal to the annual inflation rate.
• A 4% annualized real rate of return on your investments.
• Annual living expenses in retirement will be 80% of your current salary (adjusted for inflation).
• Social Security benefits will cover 35% of your living expenses.
• The remainder of your retirement living expenses will be covered by annual, inflation-adjusted withdrawals of 4% of your retirement savings.
So the 10% savings rate I mentioned in my 2009 post was too low given these facts and assumptions. I'll discuss the assumptions listed above in subsequent posts in this series, and make some different assumptions to see what other required savings rates we come up with.

Note that the figures above are stated in percentage terms, so the required 14% savings rate applies whether your annual earnings are \$50,000, \$100,000, or any other amount. For example, with annual earnings of \$100,000, we assume that your annual living expenses in retirement will be \$80,000 (80% x \$100,000), that Social Security will fund \$28,000 of that (35% x \$80,000), and we calculate that you will need to save \$14,000 per year (14% x \$100,000), which is \$1,167 per month. For an annual salary of \$50,000, just cut these numbers in half.

In working on a problem like this, we first need to figure out how to handle inflation. With an annual inflation rate of 2%, after one year it will take about \$102 to buy what you can buy with \$100 today. This reduction in purchasing power compounds over the years. At an average annual inflation rate of 2%, it will take about \$220 in 40 years to buy what you can buy today for \$100.

If your earnings keep pace with inflation, then in 40 years you will earn about \$220 for every \$100 you earn today. Since \$100 worth of goods in today's dollars will cost about \$220 in 40 years, your earnings in 40 years will have about the same purchasing power they have today. Another way to say this is that in 40 years, \$100 of earnings in today's dollars will buy \$100 of goods in today's dollars.

If we assume that earnings and the cost of goods both increase at the average inflation rate, then we can just work the problem in today's dollars. Today's dollars also are referred to as "real dollars", while future dollars are referred to as "nominal dollars". So 100 real dollars is equal to 100 nominal dollars today and 220 nominal dollars in 40 years (again, assuming an average annual inflation rate of 2%). To convert any real dollar amount to nominal dollars in 40 years at a 2% inflation rate, multiply by 2.2. For example, \$100,000 in real dollars is about \$220,000 in nominal dollars in 40 years at a 2% inflation rate, and \$50,000 in real dollars is about \$110,000 in nominal dollars.

Similarly, we can refer to nominal or real rates of return on investments and savings. The real rate of return is approximately equal to the nominal rate of return minus the inflation rate (the exact calculation is slightly more complex, but this approximation is close enough for the type of calculation we're doing here). So assuming an inflation rate of 2%, a nominal return rate of 6% is approximately equal to a real return rate of 4% (the exact value to two decimal places is 3.92%). At these rates, \$100 of investments will grow in one year to \$106 in nominal dollars and about \$104 in real dollars.

As another example using the same inflation rate assumption, a 2% nominal rate of return equals a 0% real rate of return, and with these rates of return, \$100 of investments today will be worth \$100 in real dollars at any time in the future. In other words, with a 2% inflation rate, you need to earn a nominal return of 2% just to keep up with inflation and maintain your purchasing power.

Remembering that we're working in real dollars, the assumption that your living expenses in retirement will be 80% of your current salary means that with an annual salary of \$50,000, your annual living expenses in retirement will be \$40,000 (80% x \$50,000). Our assumption is that Social Security will cover 35% of this, or \$14,000 (35% x \$40,000). This leaves \$26,000 of annual residual living expenses (RLE) to be funded with withdrawals from your retirement savings (\$40,000 - \$14,000). We can calculate that to fund the annual RLE of \$26,000 with annual inflation-adjusted withdrawals of 4% from your retirement savings requires that you accumulate \$650,000 in retirement savings during the 40 working years before you retire.

How do we calculate this required retirement savings value of \$650,000 based on the 4% withdrawal rate? Hopefully you recall from your elementary-school math classes that 4% can be written as the fraction 4/100, which can be reduced to the fraction 1/25. So a 4% withdrawal rate means that you will start by withdrawing 1/25 of your retirement savings in your first year of retirement, which means that you will need retirement savings equal to 25 times the annual withdrawal amount of \$26,000. Multiplying \$26,000 by 25 gives us the required retirement savings value of \$650,000.

Now that we've figured out how much you'll need in accumulate in retirement savings to retire in 40 years (using the listed facts and assumptions), we can use our assumed 4% real rate of return on investments (6% nominal rate of return minus 2% inflation) to calculate how much we'll need to save each year. We could do this by entering 40 rows of data into a spreadsheet, with each row calculating how much we'd have at the end of each year given a specified amount of savings along with our 4% real rate of return, and then use trial and error until we find the annual savings amount that gets us to \$650,000 after 40 years. But there's an easier way.

Spreadsheets like Excel and Google Sheets provide a "payment" function, PMT, that we can use to calculate the required annual savings amount directly. We plug into the spreadsheet PMT function our 4% assumed real rate of return, number of periods = 40 (years), a starting value of \$0, and a future value of \$650,000. The PMT function then calculates the required annual savings amount of \$6,840.

Dividing \$6,840 by our \$50,000 annual salary gives us a savings rate of 13.68%, which we can round to 14%. Again, this same savings rate applies to any annual salary given the assumptions we're using here; if we double the annual salary to \$100,000, the required savings at retirement doubles to \$1,300,000, and the annual savings amount doubles to \$13,680, which is 13.68% of \$100,000.

If you want to try this yourself, open up an Excel or Google Sheets spreadsheet, and enter this formula into any cell:

=PMT(4%, 40, 0, -650000)

Note that the cash flow convention used by these types of spreadsheet functions requires a minus sign in front of the future value of 650000 if we want to get a positive value for the annual savings amount (payment).

Using this PMT formula, you can enter different numbers to determine the required annual savings amount based on different assumptions. For example, to retire in 30 years assuming a 3% real rate of return and \$50,000 of current savings, with all other facts and assumptions the same, we would use this formula:

=PMT(3%, 30, 50000, -650000)

This returns an annual savings amount of \$11,112, which is a savings rate of about 22% (11,112 / 50,000). This is much larger than the 14% savings rate calculated based on our initial facts and assumptions, and illustrates the wide range of savings rates we can come up with depending on our assumptions.

In the next article in this series, we'll start looking at factors to consider in coming up with reasonable assumptions to use in calculating a required annual savings rate. For example:

• How to estimate your living expenses in retirement.
• How to estimate how much you'll receive in Social Security benefits.
• How to determine a reasonable assumption for the rate of return on your investments.
• How to determine a reasonable assumption for a safe withdrawal rate from your retirement savings.

#### 1 comment:

1. Great article with specific information that will help with retirement savings. I am looking forward to the series. dwickenh