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The retirement savings crisis in America is real. But it is also skewed by vast numbers of people who have saved next to nothing. Looking only at those who are making a serious effort to put something away reveals a more encouraging data set.

Pre-retirees working full-time and who have both a 401(k) plan and an IRA are tantalizingly close to securing sufficient retirement income—and their situation has improved in the past 12 months, a recent study by investment firm BlackRock found. These savers can likely close the gap with a few simple adjustments.

We are all familiar with the doomsday statistics about retirement savings: A third of workers have less than $1,000 in savings and investments that could be used for retirement, and roughly two-thirds have less than $25,000. So large numbers of people will be stuck working longer than they like and counting on Social Security for nearly all their retirement income.

BlackRock weeded out less serious savers by looking only at those with a balance in both a 401(k) plan and an IRA. The typical working 55-year-old meeting this criterion has $264,000 saved and earns $58,000 a year. That level of savings will produce $19,000 a year in guaranteed lifetime income at age 65, based on calculations from the firm’s CoRI index. (This benchmark estimates the amount of annuity income a pre-retiree would be able to purchase at retirement.) Coupled with $21,000 a year from Social Security, this saver is on track to a secure retirement income equal to 69% of final salary.

Most financial planners believe that replacing 70% to 80% of final household income is the mark savers need to hit. So this typical 55-year-old saver is just about there and can close the gap by saving a little more, spending a little less, or working just another year or two. And if market conditions remain favorable, the pre-retiree may get over the hump without changing a thing. A year ago, the typical 55-year-old saver was on track to replace just 64% of final earnings. But the stock market soared, giving savers additional funds to purchase guaranteed lifetime income when they retire.

Of course, what the market gives it can also take back. This is a moving target. But stocks usually rise over a 10-year period, and if interest rates rise over the next 10 years—most believe that will be the case—it will have the effect of boosting replacement income even further because products like immediate annuities will offer a higher return.

The picture is less rosy for older pre-retirees. The typical 60-year-old saver is on track to replace 64% of final earnings and the typical 64-year-old saver is on track to replace just 59% of final earnings. The poorer preparedness of these groups probably stems from their getting a later start saving in 401(k) plans and IRAs, says Chip Castille, head of the BlackRock Retirement Group. The working years of this age group overlapped the transition between defined-benefits plans, which began to disappear, and the rise of defined-contribution plans. They didn’t react right away and missed years of growth.

In general, the retirement readiness picture in the U.S. remains bleak. Even regular savers are falling well short of the more aggressive retirement income replacement goals. But clearly those who have taken action are much better positioned, and with only modest spending adjustments, they can easily hit the lower range of what planners advise.