Author Topic: Bringing your retirement closer, dollar by dollar. But exactly how much closer?  (Read 6607 times)

dungoofed

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I searched for this and tried to calculate it myself but couldn't wrap my mind around it, but basically what I want to ask is:

Is there a formula that lets you quantify for each dollar saved, how much in minutes/seconds/etc it brings your retirement closer?

I think these two classic posts will be involved in the calculation:

http://www.mrmoneymustache.com/2012/01/13/the-shockingly-simple-math-behind-early-retirement/
http://www.mrmoneymustache.com/2011/05/12/the-coffee-machine-that-can-pay-for-a-university-education/

Basically I want to be able to say something like "If I don't go out tonight and instead save the $500 I was planning on spending on booze, coke and hookers, I'll have to work xx minutes less at this godforsaken place."

(actually... truth be told I want to be able to say it to my friends who complain about their jobs and then go out and spend money on... booze, coke and hookers. Conversation at the strip club would be something akin to "Oh so I see things must be working out for you in the office these days, right? That $70 lap dance allows you to extend your tenure by xxx minutes" etc)


JGB

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There are way too many variables to enable you to make a one-size fits all formula. Differences in income and required spending are a start, but there's a lot of detail that cab make either or both vary significantly even for people of the same age at the same company.

You could conceivably create such a formula for yourself, but it will be specific to your needs and income.

MDM

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Quick 'n' dirty estimate:
A = P * ert

P = current net worth, $
r = annual real interest earned, yr-1
t = number of years to retirement, yr
A = desired retirement net worth, $

t = ln(A/P) / r

dt/dP = - 1/(r*P)

Plugging in round numbers, let
A = $1,000,000
P = $100,000
r = .03

dt/dP = - 1/(.03 yr-1 * $100,000) * 365.25 day/yr * 24 hr/day = -2.9 hr/$.

So, under the above assumptions, every dollar saved decreases your time to retirement by almost 3 hours.  Again, the result depends on the assumed value for r and your current P.

A couple of observations:
  • The decreased time to retirement for every extra dollar is independent of your total retirement goal.
  • The change in time is greatest when your current net worth is lowest - so save early, when you don't have much!

Left

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my quick and dirty is every $2000 I save means 1 month less I have to work. So my goal is to save 1m but I won't need that much but I like how round a number it is. But anyways, while I'm retired, I don't expect I'll need more than 2,000 a month and over 15 years I'm hoping the returns double my investment. Then after 15 years, I'll have have at least double the amount of months saved up as I spent saving. So after 15 years, I'll have 30 years saved up, if I planned to live to 60 then that's all I'd need then I can rely on social security. But since I want to be completely on my own and not draw down principle, I save more than that by another 500-700 or so.

dungoofed

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Thanks for the responses.

MDM - thanks, have plugged a few values into that formula and it gives me a decent and convenient ball-park estimate. I think this has given me a better idea of how I should have asked the question.

I want to do the calculation based on percentage of income on which one lives. So I imagine the first step would be calculating the percentage I'm living on if I *do* spend this dollar, as opposed to if I *don't* spend it.

According to the tables at http://www.mrmoneymustache.com/2012/01/13/the-shockingly-simple-math-behind-early-retirement/, if I'm living on 50% of my salary with zero savings then I can expect to retire in 17 years.

So if my annual salary was $100, I'd want to compare the difference in years-until-retirement between living on $50 and $51 (of course the actual figures we'd be plugging in would be something like $38,500 and $38,501, so the

I think the only other factor we'd need would be Starting Amount and "one dollar as a percentage of salary". Hold annual salary constant; years to retirement, current salary, etc stops being relevant when talking in terms of what percentage of one's income one is living.

What do you think?
« Last Edit: June 01, 2014, 11:10:20 PM by dungoofed »

deborah

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On a sort-of-similar topic, there was recently a thread on how much your investments were earning per hour http://forum.mrmoneymustache.com/welcome-to-the-forum/what-are-your-soldiers-%28dollars%29-paid-per-hour/

Each $1 you have in your stash at retirement pays you 4c per year (if you are looking at a 4% SWR).

If you have 5% return on investments while you are accumulating your stash, each $1 you save is earning 5c per year of savings (and this is before you start adding compound interest). With compounding, if you are retiring in 10 years the $1 you save now will earn you 63c for your stash.

So each extra $ will earn you 6.5c a year for the rest of your life!

From the information above, on average you work 1700 hours a year. If you need $10,000 per year to live, each $1 you save gets you  6 minutes and 38 seconds closer to retirement.

dungoofed

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MDM - I changed the formula a little so that it only included time I'd be in the office. Each dollar I save now is worth 20 minutes that I don't have to spend here later on.

MDM

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MDM - I changed the formula a little so that it only included time I'd be in the office. Each dollar I save now is worth 20 minutes that I don't have to spend here later on.
Looks good.  It all depends on the assumptions one uses - as long as those are understood, all is well.

Rbuckyfuller

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Very nice formula MDM.  For my net worth it is approximately half-an-hour (total, not office hours).   That is very motivating.  I would pay $1 for half an hour of relaxation easy.

Very interesting that it is independent of your total intended retirement number -- that is wild!


ak907

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Just wanted to say I like this thread, keep it going, we should put together a speadsheet to calculate both total time and office time saved.

warfreak2

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MDM's formula only includes investment growth, and doesn't take into account your own contributions (which are likely to exceed the growth, until you are FI or near FI).

In fact we don't need to start with a formula for P(t) (your real wealth at time t) - we can write down dP/dt already with just a few pieces of information. If r is the real rate of return, and S(t) is your real savings ($/year) at time t, then:

dP/dt = S + rP

Simply because you earn money at a rate equal to your savings plus the return on your investments. (This makes it clearer why your retirement goal doesn't matter). This gives dt/dP = 1/(S + Pr). For MDM's example*, suppose also that your salary is $50k with a 50% savings rate, so you're putting $25k into savings per year. Then
P = $100,000
r = .03/year*
S = $25,000/year

dt/dP = 1 year/($25,000 + .03*$100,000) = 1 year/$28,000 = 18.8 minutes/$. That's a lot less than 3 hours! But you can still buy an extra hour of retirement for $3.19, that's a pretty good deal.

*The stock market averages something more like 7% after inflation, and at 3% there's no way you'd sustain a 4% SWR, but I thought I'd stick with the earlier numbers for a like-for-like comparison.
« Last Edit: June 03, 2014, 06:29:42 AM by warfreak2 »

furrychickens

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The problem with the shockingly simple math post is that your retirement budget != your current budget. Mine will be cheaper in some areas (kids will be out of house unless thing speed up considerably) but we also have a large travel budget.

Guesstimate a retirement budget, multiply by 25-33x depending on your preferred SWR. Then analyze savings in terms of what % of your FIRE number (including investment returns over x years) it represents.

Heart of Tin

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The problem with the shockingly simple math post is that your retirement budget != your current budget. Mine will be cheaper in some areas (kids will be out of house unless thing speed up considerably) but we also have a large travel budget.

Guesstimate a retirement budget, multiply by 25-33x depending on your preferred SWR. Then analyze savings in terms of what % of your FIRE number (including investment returns over x years) it represents.

As MDM and warfreak pointed out, the goal doesn't matter for this calculation assuming the increase in savings is a one time occurance. Since the goal is the only value affected by what your expenses in retirement will be, this concern can be overlooked.

I prefer annuity formulas for this sort of thing which clearly delineates regularly occurring savings, one-time immediate savings, and future expenses, but the fomulas aren't as pretty as the above. Using the earlier introduced variables, S for ongoing savings amount, P for current investment balance, r for constant growth rate, and t for time to retirement and adding my own variables swr for safe withdrawal rate and E for expenses in retirement, then the formula

t = ln((E / swr * r + S)/(P * r + S)) / ln(1 + r)

gives the time to retirement, t. Note that you can use any time period at all for these variables as long as you're consistant. For instance, you can express your savings, S, in amount per month as long as the rest of your variables (including interest rates, which are all effective intereset rates) are expressed per month. 3% interest per year, for instance, would be 1.03^(1/12) - 1 = .2466% per month. Also, you can use E, S, and P amounts that are nominal, like $20,000, or percents like a savings rate of 70% as long as the percents are all given with respect to the same variable such as current income.

To see the effect of saving an amount of money today, n, simply compare the value of t using P and the value of t using P + n as your current investment balances. If the savings can be repeated every period, then also subsitute S + n for S or, possibly, S + m where m is n times the number of times the savings can be repeated for the period your using (i.e. if you're using annual numbers and the savings is repeated monthly, then compare the value of t using P and S to the value of t using P + n and S + 12*n). If the savings will continue, not just during the duration of the acculation period, but also during retirement, then also substitute E - n or E - m for E also.

Using the numbers supplied above and assuming a 4% SWR, I can approximate warfreak's estimate using S = 25000, E = 40000, P = 100000, r = .03, and swr = .04 to get t = 22.84019053 years. If I increase P to P = 100001, then t = 22.84015428 years which is a difference of dt = 3.62473e-5 which translates to about 19 minutes/$. The difference is due to the difference in compounding period. If I compound my numbers daily instead of yearly, I get 18.8 minutes/$ just like warfreak.

If I assume that I can save that dollar every month, then I substitute P = 100001 and S = 25012 yeilding dt = .008702079 which is a difference of about 2.6 days per dollar saved every month until retirement.

If I further assume that I will save that dollar every month in retirement as well, then I also substitute E = 39988 to get dt = .014237285 or about 4.6 days per dollar saved every month permanently. 

Note that dt/dS and dt/dE are both dependent on your expenses in retirment, unlike dt/dP, so your goal does matter when computing time saved when reducing regularly occurring expenses.

MDM

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Good additions by Heart of Tin and warfreak2. 

One oh-by-the-way: a 3% real return is in fact not all that bad, and with it one can sustain a 4% inflation-adjusted withdrawal rate for ~45 years (assuming "constant" conditions).  As a "for example", copy the table below and paste into spreadsheet cell A1.  Then copy the last row down as far as desired....

Inflation0.02                   
Real return0.03                   
WR0.04                   
-----
YearBalanceSpendingReturns
11000000=B6*B3=(B6-C6/2)*($B$1+$B$2)
=A6+1=B6-C6+D6=C6*(1+$B$1)=(B7-C7/2)*($B$1+$B$2)
=A7+1=B7-C7+D7=C7*(1+$B$1)=(B8-C8/2)*($B$1+$B$2)
=A8+1=B8-C8+D8=C8*(1+$B$1)=(B9-C9/2)*($B$1+$B$2)

warfreak2

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One oh-by-the-way: a 3% real return is in fact not all that bad, and with it one can sustain a 4% inflation-adjusted withdrawal rate for ~45 years (assuming "constant" conditions).
Good point, that, though some of us might hope to be FI for longer than 45 years.

(BTW, if inflation is 2% and the real return is 3%, the nominal return isn't 2% + 3% = 5%, but (1 + 2%)(1 + 3%) - 1 = 5.06%, which can make a small difference when you're looking ahead 45 years.)

Ian

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I don't have anything to add to the formulas being considered, but maybe a small contribution to this part:

Basically I want to be able to say something like "If I don't go out tonight and instead save the $500 I was planning on spending on booze, coke and hookers, I'll have to work xx minutes less at this godforsaken place."

(actually... truth be told I want to be able to say it to my friends who complain about their jobs and then go out and spend money on... booze, coke and hookers. Conversation at the strip club would be something akin to "Oh so I see things must be working out for you in the office these days, right? That $70 lap dance allows you to extend your tenure by xxx minutes" etc)
If you want a quick way to reframe their choices that doesn't require breaking out a spreadsheet to explain, you could convert money spent into hours of life. "You spent two hours in the office for that lap dance" etc. Obviously this works better for low wage jobs and short expenditures.

MDM

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(BTW, if inflation is 2% and the real return is 3%, the nominal return isn't 2% + 3% = 5%, but (1 + 2%)(1 + 3%) - 1 = 5.06%, which can make a small difference when you're looking ahead 45 years.)
Absolutely correct.  Although one can also claim, assuming the 2% and 3% are good only to 1 significant figure, the nominal return is still 5%.  But then, to be consistent, all those yearly intermediate calculations would also have to be modified based on sig figs and that's just too much....

dungoofed

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Thanks guys - this thread turned out much more insightful than I had been expecting. Really appreciate you all taking the time.

Have updated my "Investing Principles" note accordingly.

arebelspy

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Thanks guys - this thread turned out much more insightful than I had been expecting. Really appreciate you all taking the time.

+1.  I may have to go create a new spreadsheet around this.  Fun!  :D
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