I recently posted a numerical analysis that determined a retirement savings rate based on the goal that spendable income did not drop upon entering retirement. There were necessarily many assumptions that went in to this analysis, but a reader named RedG3 asked specifically about one of them: how do changes in income during working years effect that rate? I’ve been thinking through how to answer RedG3 for the last two days. That’s long enough that I figured it’s time to put some thoughts down and see what he, or others, might have to say back.

Before I jump into a numerical analysis, let me first point out some other ways of approaching this question. Such as simply pinpointing how conservative you want to be in figuring the whole set of equations out. This is important because, while there are absolutely no guarantees that the model I based my numerical analysis on will match reality, you CAN disregard some things simply by stating that you want to be conservative rather than aggressive. As an example of what I mean, I would say it is conservative to assume that the most you will ever be able to contribute to savings is exactly the amount you can contribute now. That is, perhaps your income will never go up after accounting for inflation? (There are lots of news/magazine articles showing that real income has actually decreased between the last couple generations.) Or if your income does rise, perhaps your expenses will also rise to meet the delta due to having a larger family, getting sick, or something else? Or perhaps you’ll choose to treat any increases in savings amounts as just a way to cancel out the risk of inflation rising faster than historical rates? Etc.

Another way to think about it is that you could set your savings rate assuming you don’t retire until a specific target date, then treat any extra income and resulting growth in savings rate as a way to move that date forward. I see this as another way of balancing conservative vs. aggressive strategies. You start out conservatively by acting to a plan that gives you a high confidence in being able to retire with a known income, and then use information about your progress over time to adjust your aggressiveness. Because you know the equations, you can recalculate your reachable retirement date and either see measurable progress in an earlier retirement date each time you get an increase in savings, or you can lower your savings rate to keep the date at the same point, or you can see that you need to increase savings to match a higher standard of living in retirement. What I can say without doing any math, yet with fairly high confidence, is that the more you can contribute to retirement savings now, the less you’ll need to contribute later to have the same income in retirement.

Now, let’s get on with extending our numerical analysis we started previously to account for income changes. We’ll again have to make some assumptions, the first one of which is around how our annual income changes. In my own experience, there are both big changes and small changes in income. The small changes are annual cost-of-living raises whereas the big ones are due to promotions or switching employeers. But the big changes only come at infrequent periods — though that may say something about my own career path rather than anything about normality. Since I can’t see safely assuming any particular schedule for large changes, and I’m not particularly well-versed in what a typical end-of-career income is for different education levels or types of careers, l’ll use a constant growth rate that doesn’t seem too far out of the realm of possibility (in my opinion) for a well-educated, hard-working, go-getter. Say 3.0% above the rate of inflation. This would mean that the annual income over the scenario we ran before would look something like the below in terms of 2008 dollars (i.e. we factor the inflation out):

- Year 1: income = $Z
- Year 2: income = prev. year’s income + 0.03* prev. year’s income = $Z + (0.03 * $Z) = $Z * (1.00 + 0.03) = $Z * 1.03
- Year 3: income = $Z * (1.03^2) = $Z * 1.061
- …
- Year 45: income = $Z * (1.03^44) = $Z * 3.671

So, according to our assumptions, if you’re income started out at $25,000 at age 21, then you could be earning the equivalent of $91,775 in 2008 dollars at age 66. (If that seems low to you, consider that this would be $647,625/year in absolute dollars using our previously assumed 4.5% constant inflation rate.) So now we’re ready to make one more assumption which should set us up for figuring out our target savings rate. That is, what income will you need in retirement?

If you are like most Americans, and you’ve been earning $90K-ish in buying power, you’re probably quite likely to want to continue that kind of life style and thus won’t settle for an income change upon entering retirement. You might think that this means your first years withdrawal of your retirement nest egg has to be $91,775 but you’d be wrong. You’ve forgotten to account for the difference between spendable income and total income as discussed in the previous post. But what is the amount of savings contribution here? We don’t know approaching from this direction so we’ll have to approach this in a slightly different way to do it right.

Let’s factor the above income growth into the equations we had for calculating the nest egg size given our assumptions about market growth and inflation. We’ll again let G by the annual growth rate of your assets after inflation and assume you plan to keep your savings rate constant while your income grows (not recommended — you should find it easy to save more — but it sure makes the equations easier!) The latter means that we can apply the same rate we used above (3.0%) for your savings contributions. Let’s call that S. We now have the series of equations:

Retirement Balance = = X*(1+G)^45 + X*(1+S)^1*(1+G)^44 + X*(1+S)^2*(1+G)^43 + .... + X*(1+S)^44*(1+G)^1 = X * [(1+G)^45 + (1+S)^1*(1+G)^44 + (1+S)^2*(1+G)^43 + .... + (1+S)^44*(1+G)^1] = X * [1.055^45 + (1.03^1)*(1.055^44) + (1.03^2)+(1.055^43) + .... + (1.03^44)*(1.055)^1] = X * 309.957

As before, we now can see that trying to initially put away $1,000 dollars (i.e. X = 1000) means that we’ll end up with a nest egg of $309,957 in 2008 dollars at the end of our 45 years of working for The Man. Continuing our assumption of a 4% annual withdrawal rate, that means our income during retirement is $12,398.28 in 2008 dollars. But now we have to be a little careful in figuring out what our income was just prior to retirement. Before, when we had no growth in contributions to savings, we could just add the $1,000 we started out contributing to retirement savings to match our assumption of no income change upon entering retirement. But here, we have to use that income growth factor we calculated above (remember we’re assuming our percentage contribution to retirement savings stays constant so growth in income means a corresponding growth in contributions to retirement savings.) The year before retirement, we contributed $1,000 * 3.671, or $3,671 which means we were earning $16,069.28 in total income. Now we back that down to what our income was in year one by dividing that total by the same 3.671 factor. That gives us an initial income of $4,377.36, from which we saved $1,000. Which means you needed to start out by contributing 22.84% of your income to savings!

That’s a significantly higher number than we found last time (11.40%). I wonder how dependent it is on the assumption that income shouldn’t drop when entering retirement? As I mentioned in my earlier post, I think that isn’t necessarily realistic because I suspect most people won’t have a mortgage, nor any job-related expenses (clothing, vehicle payments, gas, insurance, etc.) Let’s assume you only need two-thirds of your income instead. In this case, our $12,398.28 retirement income means a pre-retirement spendable income of $18,597.42. Which means a pre-savings income of $22,268.42, and that means a year one income of $6,066.04. And that means a savings rate of 16.49% is required. Interesting, but perhaps not really what RedG3 was after?

So, let’s look at it yet one more way. In this case, let’s figure out what our previous savings rate (11.40%) gets you in relation to the income change at retirement. We start with a savings of $1,000 in year one which means an income of $8,771.93 due to that 11.40% savings rate. Multiply by our income growth factor of 3.671 and we have an income immediately prior to retirement of $32,201.75. Subtract out 11.40% of that for a contribution to savings, and you have a spendable income of $28,530.75. So our $12,398.28 retirement income is a 43.46% drop in income when retiring. HOWEVER, this income is $4,628.44 (or 59.57%) higher than the retirement income you would have had without any income growth ($7,769.84 according to the previous post.)

In conclusion, if RedG3 holds his retirement savings rate constant but grows his income by 3.0% above inflation each year, he’d have a retirement income that is 59.57% higher than if his income didn’t change. On the other hand, it looks difficult to avoid a significant income change upon retiring — either he’ll have to start out saving a lot more than 11.40% of his income or else expect to make significant catch-up changes as he gets closer to retirement. A 66% drop could be met by a 16.49% savings rate, but a 0% drop would need a 22.84% savings rate.

As before, I need to stress the number of assumptions that went into these calculations. They’re all predicated on an assumed 10.0% investment growth rate on assets, a 4.5% inflation rate, a 4.0% withdrawal rate during retirement, a 45-year period until retirement, and now also a 3.0% income growth rate. Further, we assume these rates are perfectly regular and can be counted on to come around every year like clock-work. There is signifcant risk of these calculation being wrong if inflation spikes at the same time investment growth plummets. And worse, I was probably pretty stupid to write up a 10% investment growth rate. That isn’t very conservative of me!

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