## couple questions about SSA (67 vs 70) and breakeven points

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Topic Author
arcticpineapplecorp.
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### couple questions about SSA (67 vs 70) and breakeven points

I'm nowhere near retirement (age or otherwise) but recently I went to SSA.gov and it said If I retire at:

62 I'd get \$1324 a month
67 I'd get \$1898 a month
70 I'd get \$2358 a month

I know these numbers might change based on salary increases over the years, inflation or changes to SSA (please don't go there) but the questions I have are:

First, I thought if you wait until 70 it's an 8% increase per year (I thought from waiting from FRA at 67 to 70). But if I enter the following into excel:
=SUM(1898*1.08^3)

I get \$2390, not \$2358.

So, is there a reason SSA is saying the increase to wait until 70 is LESS than 8% additional per year?

Second, I thought I heard on James Lange's podcast that the breakeven point for waiting until 70 to claim is around age 82. (meaning: if you live to be at least 82 or older it would make sense to wait because you will have come out ahead in a total amount over the rest of your lifetime). I did a quick excel using the amounts above starting at those ages and just adjusted 3% COLA each year based on the starting amounts. I started each column with the total monthly amount X 12. Then for the second year I adjusted the first year's amount by 3% and added that to year one. For each subsequent year I continued to adjust the 3% additional based on the first year and then add to the previous year's balance. This is what the first four cells look like (for the \$1324 a month at age 62) and then continue onward in the same fashion (the first cell being B3):

=SUM(1324*12)
=SUM(B3*1.03)+B3
=SUM(B3*1.03^2)+B4
=SUM(B3*1.03^3)+B5

I've attached a copy of the spreadsheet below. Unless I'm doing something wrong (and that's highly likely) I come up with the break even point of 82 for starting at 67 (rather than 62) but either 84 (to come out ahead of starting at 62) or 86 (to come out ahead of starting at 67). The highlighted cells are what I'm seeing as the breakeven points. Are my numbers wrong? Any info you could provide would be helpful. Thanks very much.

edit: in the interim, I discovered this post at kices.com and am trying to make my way through it (https://www.kitces.com/blog/financial-p ... v-npv-irr/). Derek Tharp makes the case for break even after 14 years but uses IRR and discount NPV rates, which I didn't do in my spreadsheet. Is that (NPV/IRR) the only way the experts determine the break even points?
Last edited by arcticpineapplecorp. on Sat Jun 10, 2017 8:17 pm, edited 2 times in total.
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sometimesinvestor
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### Re: couple questions about SSA (67 vs 70) and breakeven points

Sorry , I was unwilling to check the math but have a few observations. Break even gets reduced because of the adjustments for inflation with the larger base arising from waiting. The decision fora single individual is mostly a matter of guessing how long you will live but unless your wife has earned a larger social security pension than you delaying is likely to help her/ I believe it is the june issue (but may be May) of Kiplinger's magazine has an excellent article on social security

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### Re: couple questions about SSA (67 vs 70) and breakeven points

arcticpineapplecorp. wrote:...First, I thought if you wait until 70 it's an 8% increase per year (I thought from waiting from FRA at 67 to 70). But if I enter the following into excel:
=SUM(1898*1.08^3)

I get \$2390, not \$2358.

So, is there a reason SSA is saying the increase to wait until 70 is LESS than 8% additional per year?
The increase is 8% / year, but only if you were born 1943 or later. See: Early or delayed retirement, from the Social Security Administration.

This is the illustrated table mentioned in the wiki: Social Security (the blue "info" box)

Would the bend points have anything to do with the calculation? Primary Insurance Amount
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2pedals
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### Re: couple questions about SSA (67 vs 70) and breakeven points

When you started SS at 62 you adjusted of inflation, you did not adjust your SS estimated payments for inflation at 67 and 70. I don't think this is right. SS estimate should also change due to inflation.
Last edited by 2pedals on Sun Jun 11, 2017 8:40 am, edited 1 time in total.

David Jay
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### Re: couple questions about SSA (67 vs 70) and breakeven points

arcticpineapplecorp. wrote:First, I thought if you wait until 70 it's an 8% increase per year (I thought from waiting from FRA at 67 to 70). But if I enter the following into excel:
=SUM(1898*1.08^3)

I get \$2390, not \$2358.

So, is there a reason SSA is saying the increase to wait until 70 is LESS than 8% additional per year?
Your formula is a compounding formula: 1.08*1.08*1.08 which yields a 25.9% gain over 3 years.
The 8% (actually 8/12 of 1% per month) for delaying benefits is NOT compounded. It is 24% to delay from 67 to 70
1898 * 1.24 = 2354, which is likely from rounding errors because of the "annual-whole-dollar" numbers in your statement. Actual SS benefit is monthly, not annual.
Last edited by David Jay on Sat Jun 10, 2017 9:54 pm, edited 1 time in total.
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### Re: couple questions about SSA (67 vs 70) and breakeven points

arcticpineapplecorp. wrote:. . .I went to SSA.gov and it said If I retire at:

62 I'd get \$1324 a month
67 I'd get \$1898 a month
70 I'd get \$2358 a month

I know these numbers might change based on salary increases over the years, inflation or changes to SSA (please don't go there) but the questions I have are:

First, I thought if you wait until 70 it's an 8% increase per year (I thought from waiting from FRA at 67 to 70). But if I enter the following into excel:
=SUM(1898*1.08^3)

I get \$2390, not \$2358.

So, is there a reason SSA is saying the increase to wait until 70 is LESS than 8% additional per year? . . .
For anyone born in 1943 or later, the Delayed Retirement Credits should be 2/3% per month. If you were born in 1960 or later, then your Full Retirement Age is 67, but if you were born from 1955 to 1959, you FRA could be 66 years 2 months to 66 years 10 months, changing the time for DRCs. Can you tell us your year of birth?
Last edited by whaleknives on Sun Jun 11, 2017 9:43 am, edited 1 time in total.
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David Jay
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### Re: couple questions about SSA (67 vs 70) and breakeven points

Another variation in your break-even formula may be that you receive the COLA while delaying.

You are not at a COLA disadvantage by claiming later than 62. For example: If someone your same age claims at 62 and reaches, say, an accumulated 21 percent COLA at age 70 then you will also get the same 21% COLA when you begin to draw at age 70 (your benefit is: PIA * 1.24 * 1.21).
Prediction is very difficult, especially about the future - Niels Bohr | To get the "risk premium", you really do have to take the risk - nisiprius

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### Re: couple questions about SSA (67 vs 70) and breakeven points

arcticpineapplecorp. in original post wrote:I did a quick excel using the amounts above starting at those ages and just adjusted 3% COLA each year based on the starting amounts. ... I come up with the break even point of 82 for starting at 67 (rather than 62) ...
There is no need to adjust SS benefits for estimated Cost of Living Adjustments (COLAs) to determine the breakeven for delaying when you claim benefits. It makes the calculation more complicated and prone to errors.

I suggest just using the figures from the SSA web page, Effect of Early or Delayed Retirement on Retirement Benefits, that correspond to your Full Retirement Age (FRA), aka Normal Retirement Age (NRA). If you're born in 1960 or later, your NRA is 67 and the figures to use are found on the bottom row. [*]

Code: Select all

``````   Col A    Col B    Col C    Col D
-- Breakeven --
Row  Age   % PIA     Years     Age
---  ---  -------   ------   ------
5   62   70.000
6   67  100.000   11.667   78.667
7   70  124.000   12.500   82.500``````
This shows the breakeven for delaying from 62 to 67 is 11.667 years or age 78.667; and the breakeven for delaying from 67 to 70 is 12.5 years or age 82.5.

Code: Select all

``````100 * 11.667 =  70 * 16.667
124 * 12.500 = 100 * 15.500``````
Here is the formula:

Code: Select all

``````Breakeven years = earlier \$ * (later age - earlier age) / (later \$ - earlier \$)
C6 = B5        * (A6        - A5         ) / (B6      - B5       )
11.667 = 70        * (67        - 62         ) / (100     - 70       )``````
I copied the formula on the 2nd line down from cell C6 to cell C7. I also copied it down further to cell 18 and filled in other values in columns A and B to determine the breakevens for other delaying options.

For example, to break even from delaying all the way from age 62 to 70, one must live to age 80.37.

Code: Select all

``````   Col A    Col B    Col C    Col D
-- Breakeven --
Row  Age   % PIA     Years     Age
---  ---  -------   ------   ------
8   62   70.000
9   70  124.000   10.370   80.370``````

Code: Select all

``124 * 10.37 = 70 * 18.37``
And here are the breakevens for each one year delay, 62 to 63, 63 to 64, ..., 69 to 70:

Code: Select all

``````   Col A    Col B    Col C    Col D
-- Breakeven --
Row  Age   % PIA     Years     Age
---  ---  -------   ------   ------
10   62   70.000
11   63   75.000   14.000   77.000
12   64   80.000   15.000   79.000
13   65   86.667   12.000   77.000
14   66   93.333   13.000   79.000
15   67  100.000   14.000   81.000
16   68  108.000   12.500   80.500
17   69  116.000   13.500   82.500
18   70  124.000   14.500   84.500``````

Code: Select all

``E.g., 75 * 14 = 70 * 15``
* These are the percents of the Primary Insurance Amount (PIA) payable if you claim at the NRA. Note: ages 68 & 69 are omitted, but they are easy to determine. 68's figure is 67's + 8, and 69's figure is 68's + 8.

marcopolo
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### Re: couple questions about SSA (67 vs 70) and breakeven points

#Cruncher wrote:
arcticpineapplecorp. in original post wrote:I did a quick excel using the amounts above starting at those ages and just adjusted 3% COLA each year based on the starting amounts. ... I come up with the break even point of 82 for starting at 67 (rather than 62) ...
There is no need to adjust SS benefits for estimated Cost of Living Adjustments (COLAs) to determine the breakeven for delaying when you claim benefits. It makes the calculation more complicated and prone to errors.

I suggest just using the figures from the SSA web page, Effect of Early or Delayed Retirement on Retirement Benefits, that correspond to your Full Retirement Age (FRA), aka Normal Retirement Age (NRA). If you're born in 1960 or later, your NRA is 67 and the figures to use are found on the bottom row. [*]

Code: Select all

``````   Col A    Col B    Col C    Col D
-- Breakeven --
Row  Age   % PIA     Years     Age
---  ---  -------   ------   ------
5   62   70.000
6   67  100.000   11.667   78.667
7   70  124.000   12.500   82.500``````
This shows the breakeven for delaying from 62 to 67 is 11.667 years or age 78.667; and the breakeven for delaying from 67 to 70 is 12.5 years or age 82.5.

Code: Select all

``````100 * 11.667 =  70 * 16.667
124 * 12.500 = 100 * 15.500``````
Here is the formula:

Code: Select all

``````Breakeven years = earlier \$ * (later age - earlier age) / (later \$ - earlier \$)
C6 = B5        * (A6        - A5         ) / (B6      - B5       )
11.667 = 70        * (67        - 62         ) / (100     - 70       )``````
I copied the formula on the 2nd line down from cell C6 to cell C7. I also copied it down further to cell 18 and filled in other values in columns A and B to determine the breakevens for other delaying options.

For example, to break even from delaying all the way from age 62 to 70, one must live to age 80.37.

Code: Select all

``````   Col A    Col B    Col C    Col D
-- Breakeven --
Row  Age   % PIA     Years     Age
---  ---  -------   ------   ------
8   62   70.000
9   70  124.000   10.370   80.370``````

Code: Select all

``124 * 10.37 = 70 * 18.37``
And here are the breakevens for each one year delay, 62 to 63, 63 to 64, ..., 69 to 70:

Code: Select all

``````   Col A    Col B    Col C    Col D
-- Breakeven --
Row  Age   % PIA     Years     Age
---  ---  -------   ------   ------
10   62   70.000
11   63   75.000   14.000   77.000
12   64   80.000   15.000   79.000
13   65   86.667   12.000   77.000
14   66   93.333   13.000   79.000
15   67  100.000   14.000   81.000
16   68  108.000   12.500   80.500
17   69  116.000   13.500   82.500
18   70  124.000   14.500   84.500``````

Code: Select all

``E.g., 75 * 14 = 70 * 15``
* These are the percents of the Primary Insurance Amount (PIA) payable if you claim at the NRA. Note: ages 68 & 69 are omitted, but they are easy to determine. 68's figure is 67's + 8, and 69's figure is 68's + 8.

I think the problem with this approach is it ignores the time value of money. It just totals up how much one has received by a certain age. It does not account for any benefit (investment gains) that can accrue from receiving some of the money sooner. I think this make the "breakeven" time somewhat longer.

I still think it is often advantageous for a single person, and the higher earner in a couple, to delay as long as possible. But, that is more based on value of longevity insurance rather than a breakeven analysis like this.
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David Jay
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### Re: couple questions about SSA (67 vs 70) and breakeven points

marcopolo wrote: But, that is more based on value of longevity insurance rather than a breakeven analysis like this.
This is why the Social Security debate will go on and on. There are 2 ways to evaluate SS - break even calculation or longevity insurance.

If you want to "get the most back" from all that withholding, an earlier start date is probably the better bet.

If you want to have the biggest benefit in the event that you outperform the actuarial numbers, a delay to 70 is the better choice.
Prediction is very difficult, especially about the future - Niels Bohr | To get the "risk premium", you really do have to take the risk - nisiprius

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### Re: couple questions about SSA (67 vs 70) and breakeven points

David Jay wrote:
marcopolo wrote: But, that is more based on value of longevity insurance rather than a breakeven analysis like this.
This is why the Social Security debate will go on and on. There are 2 ways to evaluate SS - break even calculation or longevity insurance.

If you want to "get the most back" from all that withholding, an earlier start date is probably the better bet.

If you want to have the biggest benefit in the event that you outperform the actuarial numbers, a delay to 70 is the better choice.

Is not the real trick to figure out how much you can keep not how much you can collect? That is when it gets interesting since you need to consider the entire tax picture as well as the potential spousal collections and tax situations for a single payout later on.
When playing with those issues it often favors waiting and converting some to Roth for both taxes as a couple and taxes as a single.
In some of my simulations you can collect (keep) more even though you may have (portfolio total) less.

Topic Author
arcticpineapplecorp.
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### Re: couple questions about SSA (67 vs 70) and breakeven points

2 pedals: I'm only using the amounts that SSA provided that I would receive at 62, 67 and 70. So I didn't adjust any starting amounts between those years, I was just using what SSA said I would get. Are you saying SSA is reporting payments at 67 and 70 without factoring inflation between 62 and 67/70?

Thank you David Jay. I wasn't aware the amounts aren't compounding. Or that the rounding errors effect the difference between what I got (25.9%) and actual (24% difference).

Whaleknives, let's just say I was born after 1959 and I'm using the amounts that SSA provided, that is FRA at 67. To be honest, since the FRA went up for folks depending on year of birth (used to be 65 and has been going up to 67) I'm surprised that early retirement is still 62 for me. I assumed that since FRA is 67 not 65, that early would be 64, not 62. So that surprised me.

Thank you #cruncher. I have to digest what you've written more closely. And thanks for the additional links.

Again, thanks everyone. I will be taking some time to read and learn more. If I continue to have questions after reading what you've provided I will respond to this post. If anyone else has anything to offer, I'm all ears. But for now, I think I've got some more work to do. Thanks!
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Topic Author
arcticpineapplecorp.
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### Re: couple questions about SSA (67 vs 70) and breakeven points

Sorry I have one more question about inflation. If we assume the following:

1. nothing changes with SSA
2. my income does not increase between now and retirement

Would the amounts SSA is providing \$1324 (62), \$1898 (67), \$2358 (70) stay the same? Or are these amounts to go up due to inflation between now and retirement age (regardless of changes/increases in income)?

So if inflation were 3% meaning costs are 2X as much in 23.5 years does that mean (all income stays the same between now and then) that my RSDI would be twice the amounts above (if all ages, 62, 67, 70 are at least 23.5 years from now) as in \$2648 (62), \$3796 (67), \$4716 (70)?

I guess what I'm asking is that if I look over previous statements, the reported RSDI amounts I'll qualify for keep increasing each year, but is this just because my salarly goes up each year, or is it due to inflation...or both?

Thanks.
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Topic Author
arcticpineapplecorp.
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### Re: couple questions about SSA (67 vs 70) and breakeven points

Thank you #cruncher. I see what you've done and I appreciate your explanation.

Marcopolo regarding the time value of money, did you read the article at kitces.com I referenced in my OP:

https://www.kitces.com/blog/financial-p ... v-npv-irr/

I believe he takes that into account using IRR and NPV.

Any info anyone can provide on my previous question (regarding whether the PIA will continue to rise due to inflation alone or due to wage increases or both)? Thanks.
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marcopolo
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### Re: couple questions about SSA (67 vs 70) and breakeven points

arcticpineapplecorp. wrote:Thank you #cruncher. I see what you've done and I appreciate your explanation.

Marcopolo regarding the time value of money, did you read the article at kitces.com I referenced in my OP:

https://www.kitces.com/blog/financial-p ... v-npv-irr/

I believe he takes that into account using IRR and NPV.

Any info anyone can provide on my previous question (regarding whether the PIA will continue to rise due to inflation alone or due to wage increases or both)? Thanks.

Yes. Kitces does consider TVM in his SS analysis. I was responding to a different post that was suggesting using the SSA description of the breakeven calculation (which, I believe does not take into account TVM), and yields a breakeven age of just 78. If I recall correctly, using a reasonable discount rate, the breakeven point is something closer to 82 when including TVM. But, as i mentioned earlier, i don't dwell on this much, as I view anything i would get from SS as adding to my longevity insurance in the out years.
Last edited by marcopolo on Sun Jun 11, 2017 9:48 pm, edited 1 time in total.
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### Re: couple questions about SSA (67 vs 70) and breakeven points

arcticpineapplecorp. wrote:Any info anyone can provide on my previous question (regarding whether the PIA will continue to rise due to inflation alone or due to wage increases or both)? Thanks.
Yes, your PIA is subject to cola increases. To the extent a new years annual wages is higher than the lowest of your previous 35 highest annual inflation adjusted years wages, your PIA will reflect inclusion of that increased amount.
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Topic Author
arcticpineapplecorp.
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### Re: couple questions about SSA (67 vs 70) and breakeven points

One Ping wrote:
arcticpineapplecorp. wrote:Any info anyone can provide on my previous question (regarding whether the PIA will continue to rise due to inflation alone or due to wage increases or both)? Thanks.
Yes, your PIA is subject to cola increases. To the extent a new years annual wages is higher than the lowest of your previous 35 highest annual inflation adjusted years wages, your PIA will reflect inclusion of that increased amount.
thanks but does that mean the numbers I'm being given by SSA now will also be greater just due to inflation...even if my income stays the same from here to retirement?
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### Re: couple questions about SSA (67 vs 70) and breakeven points

arcticpineapplecorp. wrote:thanks but does that mean the numbers I'm being given by SSA now will also be greater just due to inflation...even if my income stays the same from here to retirement?
Yes. SS uses a wage index (AWI) to index your past wages and determine "knee" points in the benefit calculation. So if average wages go up 3% this year (independent of the rate for "consumer" inflation) then our benefit will go up by 3%. This wage adjustment is applied until age 60.
Prediction is very difficult, especially about the future - Niels Bohr | To get the "risk premium", you really do have to take the risk - nisiprius

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### Re: couple questions about SSA (67 vs 70) and breakeven points

#Cruncher wrote:...

Code: Select all

``````   Col A    Col B    Col C    Col D
-- Breakeven --
Row  Age   % PIA     Years     Age
---  ---  -------   ------   ------
5   62   70.000
6   67  100.000   11.667   78.667
7   70  124.000   12.500   82.500``````
...
I think the problem with this approach is it ignores the time value of money. It just totals up how much one has received by a certain age. It does not account for any benefit (investment gains) that can accrue from receiving some of the money sooner. I think this make the "breakeven" time somewhat longer.
You'll get no argument from me on this point, marcopolo! (You didn't need to quote my entire post to make it. That post was demonstrating the right way to calculate the simple breakeven on the premise that "one must walk before one can run".)

To illustrate, if one assumes a 2% real return, the "breakeven" age when claiming at age 70 versus age 67 is 84.9 years instead of 82.5 when no growth is assumed. You can see in the table below that at age 84 one is still better off claiming at 67. But each month the results of claiming the higher amount at age 70 narrows the gap, until by 84 & 11 months it produces the same amount of real dollars.

Code: Select all

``````Row   Col A  Col B    Col C     Col D   Col E
24   Growth -> 2%
25   Amount ------>    100       124
26   Start at ---->     67        70
Age     Mo  Grows to  Grows to    Diff
------   --  --------   -------   ------
29   84.000    0    24,273    24,018    (255)
30   84.083    1    24,413    24,182    (232)
31   84.167    2    24,554    24,346    (208)
32   84.250    3    24,695    24,511    (184)
33   84.333    4    24,836    24,676    (161)
34   84.417    5    24,978    24,841    (137)
35   84.500    6    25,119    25,006    (113)``````

Code: Select all

`````` 36   84.583    7    25,261    25,172     (89)
37   84.667    8    25,403    25,338     (65)
38   84.750    9    25,546    25,504     (41)
39   84.833   10    25,688    25,671     (18)
40   84.917   11    25,831    25,837       6
41   85.000    0    25,974    26,004      30``````
Here is the formula in cell C29 that is copied right to column D and down to row 41:

Code: Select all

``C29: 24,273 = IF(\$A29 <= C\$26, 0, -FV(\$B\$24 / 12, 12 * (\$A29 - C\$26), C\$25, 0, 0))``
I don't know a formula for computing the exact breakeven as one can do when no growth is assumed. But it can be determined using Excel's Goal Seek tool.

Code: Select all

``````Row  Col A  Col B   Col C  Col D    Col E  Col F  Col G
53  Growth  2.0%
- Break Even -  ----- Grows to -----
Start   Amt    Years    Age   Earlier  Later  Diff
-----   ---   ------  ------  -------  ------ ----
54    62     70
55    67    100   14.076  81.076   19,490  19,490    0
56    70    124   14.894  84.894   25,793  25,793    0``````
Here are the formulas in cells E55 and F55 that are copied down to row 56.

Code: Select all

``````E55: 19,490 = -FV(\$B\$53 / 12, 12 * (\$C55 + A55 - A54), \$B54, 0, 0)
F55: 19,490 = -FV(\$B\$53 / 12, 12 *  \$C55             , \$B55, 0, 0)``````
Column G has a formula that simply subtracts column F from E. Note there are no formulas in column C; each cell in this column is solved for with Goal Seek. Here is the dialog to solve for the 84.894 when delaying from 67 to 70:

Code: Select all

``````Set cell:         G56
To value:           0
By changing cell: C56``````
Last edited by #Cruncher on Mon Jun 12, 2017 11:23 am, edited 1 time in total.

warowits
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### Re: couple questions about SSA (67 vs 70) and breakeven points

arcticpineapplecorp. wrote:I'm nowhere near retirement (age or otherwise) but recently I went to SSA.gov and it said If I retire at:

62 I'd get \$1324 a month
67 I'd get \$1898 a month
70 I'd get \$2358 a month
You hear a lot about breakeven points but to my mind the biggest issue is I could not live comfortably on \$1898 amonth, but I could on \$2358. I know it's (sort of) longevity insurance seen from a different angle, but it's still important to me.
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neurosphere
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Joined: Sun Jan 17, 2010 1:55 pm

### Re: couple questions about SSA (67 vs 70) and breakeven points

arcticpineapplecorp. wrote: thanks but does that mean the numbers I'm being given by SSA now will also be greater just due to inflation...even if my income stays the same from here to retirement?
I'm not sure your question has been answered directly by others. Or rather, they have answered the question in ways which did not necessarily put the answer in the way your question was framed.

First off, the benefits you were "quoted" assume no changes in inflation between now and the benefit date, and no changes in wage growth. This means that the numbers are in "today's" dollars. But once you accept/receive the benefit, it will be adjusted as per COLA/inflation.

If you have additional wages which increase your AIME, your benefit will go up. This is obviously independent of wage growth and CPI assumptions.

But inflation between now and when you collect SS does NOT impact your benefit. Only new (counted) wages and wage growth affect your benefit amount until you collect*. At that point, your benefit will (may) increase with inflation and (possibly) with new earnings. Note that wage growth may be considered a "type" of inflation, in that in general there is wage growth, and this will increase the impact of your previous earnings such that your benefit will increase based on the wage growth which occurs between now and age 60. But technically, no, CPI changes between now and when you collect do not affect your benefit*. Note that after age 60, wage growth will no longer affect your benefit.

So getting back to your example, the benefits you were estimated to receive at 67 and 70 do not take inflation into account (or more accurately, assume zero inflation). If zero inflation between today and age 67, or today and age 70, those numbers are accurate as today's dollars. Once you begin collecting, if there is "continued" zero inflation and no new earnings..well obviously your benefit does not change. But once you start collecting, your benefit will indeed (in general) change with CPI.

Neurosphere

* These comments make some simplifying assumptions and/or are not quite precise with the language. Some details were left out for now in order to make the answer more readable.
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