This issue is as old as personal finance itself: What should an investor do with a large windfall, say, a bonus, gift, inheritance, etc.? Invest it all at once as one big lump sum or should we “ease into the market” and invest the cash in multiple installments? The latter is called Dollar Cost Averaging (DCA). This is a popular topic in the personal finance world and many of you might have read about it. JL Collins had a blog post on why he doesn’t like DCA and Vanguard has a nice study with extensive simulations showing that, on average, the lump sum investment pretty handily beats DCA. The intuition for that result is pretty straightforward: equities go up on average, so if you sit on your hands and voluntarily delay your investments you will have lower returns on average.

End of story! End of story? Not so fast! Even if you’re familiar with the subject already, please keep reading because we’ll have a **new spin** on this old topic. Spoiler alert: we propose a way dollar cost averaging will reduce risk and have the same average return as the lump-sum investment!

### Dollar Cost Averaging:

As Wikipedia explains it:

*“By dividing the total sum to be invested in the market […] into equal amounts put into the market at regular intervals […], DCA hopes to reduce the risk of incurring a substantial loss resulting from investing the entire “lump sum” just before a fall in the market.*

So, instead of investing a large lump sum, say $10,000 all in month 1, we could alternatively spread the investment over 2 months ($5,000 each), over 3 months ($3,333.33 each) or over 4 months ($2,500 each). The total investments are the same but DCA could potentially spread out the risk of investing one large amount all in one month.

* Side note 1: There is some confusion about the meaning of DCA. The regular monthly investments of a fixed dollar amount (into a 401(k) plan, for example) over many years will imply that when the market is down you buy more shares, and when the market is up you buy fewer shares. Some people call this dollar cost averaging as well. But the periodic investments into a 401(k) plan are more or less mandatory; ten years ago I couldn’t have funded my 401(k) with a lump sum of ten years’ worth of contributions. Dollar Cost Averaging in the context of today’s post means having the funds available and voluntarily *spreading

*out and thus delaying the investment.*

**Side note 2:** Speaking of 401(k) accounts, some personal finance bloggers advocate **“front loading”** your retirement account contributions, i.e., investing a big chunk or potentially all of your maximum annual 401(k) contribution in January. Physician on FIRE and MadFIentist had blog posts on this recently. The rationale is the same as with lump sum vs. DCA: Since the equity market goes up on average you want to have the maximum equity exposure as early as possible. But keep in mind that the regular periodic 401(k) contributions are not really the same as DCA because your investments occur when you get your income. In contrast, DCA means you **voluntarily** delay investing while holding on to cash, waiting to deploy the investments over time.

### Simulations

To see whether lump sum investments beat DCA, let’s run some simulations. We use monthly equity returns (S&P500 total return) since January 1871 minus a short-term, risk-free return (3-month T-Bill after 1926, 1-year T-bill before 1926, due to data availability). We calculate how a lump sum investment would have performed over 12-month windows relative to spreading out the investment. Vanguard’s study uses DCA spread out over 12 months and then also performs the same analysis over 6, 18, 24, 30 and 36 months, but we are more interested in shorter DCA horizons and use 2, 3, 4, 5, 6, 8, 10, and 12 months instead. Below are the results for the entire sample since 1871 and also for the more recent period since 1980.

We pretty much replicate the results from the Vanguard study: DCA indeed reduces risk but lags behind the lump sum investment in every other conceivable measure: returns are lower and Sharpe Ratios (excess return divided by risk, i.e., a measure of risk-adjusted mean return) are lower. Over the earlier period, there were a few DCA simulations that yielded marginally higher probabilities of positive returns, but that’s been reversed since 1980. Also, DCA’s probability of beating the lump sum investment is always below 50% and it declines the more we spread out the investments. Bummer!

So we confirmed the well-known result for all DCA horizons between 2 and 12 months: The peace of mind of investing in several installments over time comes at a high price. Some would argue that Dollar Cost Averaging is only for jittery chicken-little investors. But in the FIRE community, we are all cool and rational investors so there is no place for Dollar Cost Averaging around here, right? **Or, maybe there is?! **

### A new spin on DCA: Why wait for the windfall to arrive?

If the risk reduction in DCA is for real but the opportunity cost messes up our returns, why not simply spread out the investments to reduce risk, but start investing **before** the windfall arrives. Could that be the best of both worlds? Avoid the opportunity cost and still get the risk reduction, see below!

I try the DCA over 3 months starting 1 month before the windfall and the DCA over 5 months starting 2 months before the windfall, see the chart above. Notice the beauty of this assumption: We now invest 1/3 of the equity portfolio for 11, 12 and 13 months each. So the average length of the equity investment is indeed 12 months, the same as in the lump sum case! And analogously, invest one-fifth for 10, 11, 12, 13 and 14 months, also for an average of 12 months.

As the baseline, let’s assume that we are able to borrow the funds prior to the windfall date at the risk-free rate. That might be a bit of a stretch, but for folks who have an emergency fund, why not just “borrow” from that account for one or two months? The opportunity cost is exactly the risk-free rate! And what if a big expense shows up just during that one or two-month window? Come on, everybody, be creative. Use the credit card float or put the charge on a new card with introductory zero percent interest. I don’t have to tell you how to hack that, right?

Of course, as many of you may know, we don’t even have an emergency fund. But we could access a Home Equity Line of Credit (HELOC) to borrow the funds, though at a higher interest rate; the “Prime Rate” in our case (roughly the risk-free rate plus 3% p.a.). So, let’s assume that when borrowing the pre-windfall investment we have to pay that higher interest. But likewise, we also use the excess cash we have when the windfall arrives to pay down our existing HELOC balance (normally around $20,000-$25,000) until we’re done with the DCA. So, how would we implement a DCA of a $10,000 investment over 5 months with an assumed $20,000 initial HELOC balance?

- 2 months before windfall, borrow $2,000 from HELOC to invest. Balance $22,000
- 1 month before windfall, borrow $2,000 from HELOC to invest. Balance $24,000
- Windfall: Receive $10,000, invest $2,000 and use the remaining $8,000 to pay down the HELOC. New Balance: $16,000
- 1 month after the windfall: Take $2,000 out of HELOC and invest. Balance: $18,000
- 2 months after the windfall: Take $2,000 out of HELOC and invest. Balance: $20,000

This scheme generates an **average** HELOC balance of exactly $20,000, no different than our “normal” HELOC balance. So, don’t be surprised if even with borrowing costs the DCA results will look very similar to the baseline!

If you have no sizable emergency fund and no HELOC either you could still buy equities on margin at an extremely low cost: **equity index futures** deliver exactly what we want: the index total return (incl. dividends) minus risk-free rate. See our other posts on futures trading: general info here and the post on the Synthetic Roth IRA. Unfortunately, futures contracts are also very lumpy: One single S&P500 futures contract is worth 50x the index level, currently just under $120,000. You’d need a really nice size inheritance to make it worthwhile implementing DCA with equity index futures. Better be nice to your Aunt Betty!

In any case, here are the simulation results, see table below:

- There is essentially no difference in average returns between the lump sum investment and the DCA. That’s not a surprise because the average investment length in the equity market is now the same for both the lump sum and the DCA.
- Amazingly, there is no sizable deterioration in the average return even when factoring in borrowing costs. But again, that’s due to having an existing HELOC balance. The borrowing cost before the windfall is exactly offset by using some of the windfall money to temporarily pay down the HELOC until the conclusion of the DCA.
- The Sharpe Ratio under DCA is higher than with the lump sum. You get a higher probability of positive returns than with the lump sum investment and there is higher than 50% chance of beating the lump sum investment with DCA.
**How cool is that?!**

Of course, the DCA with borrowing will not be feasible all the time. Here are some important limitations:

**Uncertainty about the timing of the windfall:**If we don’t even know if or when the windfall arrives it might not be so prudent to put borrowed money in the stock market!**Borrowing limits:**If you have a net worth of $1,000,000 and the windfall is $30,000 you’ll likely find it easy to come up with $10,000 for one month before the windfall actually arrives. If it’s the other way around and you expect a $1,000,000 windfall and your current net worth is only $30,000 you will probably not get a $300k+ loan to invest the month before the windfall!**Legal and tax constraints:**If you want to invest a lump sum in a retirement account for a specific calendar year you can do that as early as January of that year. But you can’t invest the 2018 contribution before January 1, 2018, at least not in the retirement account. Back to the Front Loading technique mentioned above: Investing one large lump sum in January might be advantageous to the periodic investments. Investing three equal amounts in December, January and February (or five equal amounts in November-March) might be even more advantageous but it’s not allowed in the retirement account because of restrictions on how much you can invest in each calendar year.

### Is this all a fluke? Where is the **mathematical** proof?

We are getting beyond 2,000 words so I don’t want to push this too much more. But the superior performance of DCA with borrowing vis-a-vis the lump sum investing is not a fluke. Quite the opposite, in addition to the empirical evidence, I can **prove this mathematically**. It has to do with the fact that returns and risk in a multi-period investment problem are aggregated very differently. You don’t aggregate risk linearly.

Here’s a simple example **(please skip if you don’t like statistics and mathematics)**: Imagine there are three consecutive (monthly) returns X1, X2, X3. They have some joint distribution and all I require is that they have the same expected returns, the same variance/risk, but there could be some non-zero correlation between the returns.

**Lump sum investment:** If you invest a lump sum of $1 after X1 has been realized, you would be exposed to returns X2 and X3 only and the variance of the lump sum return is:

V_LumpSum = Var(X2) + Var(X3) + 2∙Cov(X2,X3)

The risk (=standard deviation) would be the square root of that.

**Dollar Cost Averaging:** If you invest $0.50 before X1 has been realized and another $0.50 before X3 has been realized, then this would be a form of DCA. You invest half the amount one month before the windfall and the other half in the month after the windfall. Slightly different from the 1/3, 1/3, 1/3 setup above but the intuition is the same. $0.50 would be exposed to X1 and X2 and $1.00 would be exposed to X3. The variance of the DCA portfolio return is:

V_DCA = 0.25∙Var(X1) + 0.25∙Var(X2) + Var(X3) + 0.5∙Cov(X1,X2) + Cov(X1,X3) + Cov(X2,X3)

Unless all returns are perfectly correlated, i.e., Cov(Xi,Xj) = Var(Xi) = Var(Xj), the DCA variance is smaller than the lump sum investment variance. That’s because with DCA you spread out the risk more evenly between X1 and X2. In the most basic case where the covariances are all zero (returns are uncorrelated, not a bad assumption for equity returns!), the lump sum investment has a variance that’s 1/3 higher than the DCA portfolio: 2.0∙Var(Xi) vs. 1.5∙Var(Xi). This translates into 15.5% higher risk in the lump sum investment! Despite having the same mean returns the two portfolios have very different risk profiles!

**Side note 1:** Yes, yes, yes, I know that returns are not additive but they compound. One would define the X=log(1+return) to fix that!

**Side note 2:** We clearly took today’s post from geeky to **super-geeky** with this, but there are some of us in the FIRE community who wouldn’t have it any other way!

### Summary

Dollar Cost Averaging is getting a bad reputation. But before you ridicule the proponents of DCA, keep in mind that, if done properly, DCA delivers exactly what it promises: Risk reduction **without** the opportunity cost. It will not beat the average return but it delivers the same expected return with less risk. Of course, it involves a little bit of financial hacking. Specifically, it requires leverage (gasp!) and not everybody has an appetite for that. But we do! We wrote a post a while ago on the seven benefits of debt and leverage. And we might mark this one as benefit number eight: Use **leverage to reduce risk.** Who would have thought?

You’ve given me nightmare flashbacks to my Economic Statistics and Corporate Finance classes as university ERN!

Interesting take on things though. I’m a fan of the judicious use of leverage to maximise investment returns, though I’m not sure I’d personally borrow in anticipation of an expected windfall… those rich elderly relatives with predictable mortality dates are somewhat scarce in my family 🙂

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Haha, thank! Yes, leverage is no dirty word in the ERN household. Sometimes it even helps to reduce volatility!

For me personally, this is mostly applicable for bonus season. A big chunk of money comes in at a predictable time early in the calendar year. I agree: it’s hard to time an inheritance. Don’t pull the plug on Aunt Betty trying to time the stock market! 🙂 But joke aside, the mortality date doesn’t have to be predictable. Only the date when the money goes through probate!

Cheers!

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Dude, I wish I had read this 6 months ago… I have a couple of windfalls coming. lol. But, this is insightful. Though, again, I had to read this 3 or 4 more times to understand it (though I still need to read more). I’ve actually employed a little bit of this strategy with some borrowing even without realizing it. I’m going down the rabbit hole and reading a little more with your lack of an emergency fund thought again as I’m 60 days out from pulling the plug, though I had pretty much settled on a strategy for myself which I will adjust as needed as time goes on. I appreciate you taking borrowing costs into your various analyses because I think that piece is largely ignored among the FI crowd.

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Awesome! I’m glad that people find this useful and used this intuitively before even reading this article. We seem to think very much alike! Cheers!

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Big ERN,

We worked vigorously to de-leverage our holdings over the past couple of years as we are getting close to RE and that has resulted decent-sized chunks o’ change showing up here and there. Struggled with lump sum vs DCA way too many times over my asset building years. Almost got sucked down the Value Averaging rat-hole recently, but after reading a few journal articles, I realized all was not milk-n-honey in VA land. In short, we have always ended up DCA’ing a lump over several months.

We still have a good portion of our portfolio to rebalance over the next few years while we work to minimize income for FAFSA purposes as FIRE-jrs wend through their college years. Since some of the rebalancing will be downsizing our home and as well as harvesting gains from rental properties, we will have a stream of lump sums to fold into more traditional assets and the approach you outline will definitely be something we factor into the planning.

Can’t wait for the next installment!

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Thanks for sharing DrFIRE! Yes, if you expect a large cash flow, see if it makes sense to invest it in 3 installments in months -1,0,+1. Let me know how it goes. Cheers!

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Apparently I came to the same conclusion without the math. Anticipating advanced pay and moving bonus from this summer’s military PCS move I borrowed from our emergency fund to invest. Thanks for providing the math to back up my logic. The inverse to this question is what should one do if they anticipate needing to push out a windfall? We have diverted some of our savings from our 401K into cash to be more liquid in case we get an opportunity to buy a piece of land this summer… not sure I want to keep all the cash, but for obvious reasons we cant put it into the 401k.

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Thanks for sharing. Wow, that’s great: intuition was exactly right and in line with the hard math! That doesn’t always happen.

You raise a very interesting issue: What happens when we need to withdraw a large lump sum at a specific date? The arithmetic should work exactly the same:

The worst alternative is to withdraw installments before the due date.

A better alternative is to simply take out the cash need all at once at the due date. Equities go up on average so it’s better to wait as long as possible.

An even better (lower risk, same expected return) alternative would be to take out multiple sums, e.g. one before, one at and one after the due date for risk reduction. If the little bit of leverage doesn’t cost you too much.:)

Cheers!

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Super-geeky, nice! Another good in depth post for sure, great to see this level of detail on occasion. We had to make a decision what to do with the funds coming out of the sale of our home back in 2015. We ended up opting for DCA (sort of), albeit we knew lump sum investing was supposed to be better from a return perspective. However, I can say with about 100% certainty, we ended up on the better end with (let’s call it modified) DCA.

We invested the funds in Canadian RRSP’s, solely in the Canadian Stocks (for tax reasons). Because of the drop in oil price at the same time, many of the shares got hammered and we had the opportunity to pick a few dividend stocks at very reasonable prices. If we would have done a lump sum when we got the money, we would have missed out on part of the dip. Now, this is obviously not pure DCA, as we varied the amount of money invested depending on the market fluctuations.

However, would a similar situation occur, I might try this strategy for the purchase of some index funds!

Thanks Dr ERN.

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Hi, Cheesy Finance! Thanks for sharing! Sometimes intuition beats math. If this had occurred in the late 1990s when there was only one direction (up!) for the stock market I’m sure you’d have opted for the lump sum. But recently, there has been more uncertainty and I wouldn’t blame folks for the DCA. Cheers!

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I’m a big proponent of doing things based on available data/statistics/science. But those bloody emotions/intuition….. 🙂

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We’re definitely more in tune with lump sum investing, IE. time in the market. The risk reduction of DCA just is not worth the cost to us. Then again we’re somewhere in the middle as your leverage approach is also not for us.

Some of that might be because I would essentially be doubling my exposure as most of my bonus is paid in .. you guessed it.. .stock. So in my case if the stock market went down I might lose twice. Once on the bonus to pay for the leverage and once on the investment. Some of it though, if I’m honest, is because my risk tolerance only supports limited leveraging.

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Thanks for sharing! Well, if your bonus is paid in stocks, you should be OK. No need to mess around with DCA because your employer does the lump sum investing for you. I receive much of bonus in cash, only very little in company stock.

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Very interesting idea, and a well done post. Thanks for sharing with us.

You said…

>we could access a Home Equity Line of Credit (HELOC) to borrow the funds, though at a higher interest rate; the “Prime Rate” in our case (roughly the risk-free rate plus 3% p.a.)

After some light googling, it looks HELOCs have rates from 3.5% to 8% (perhaps 4.8% as a national average). What rate did you use in your simulations? If my possible borrowing rate is far different from yours, I’ll be tempted to do my own calculations.

You said…

> V_DCA = 0.25∙Var(X1) + 0.25∙Var(X2) + Var(X3) + 0.5∙Cov(X1,X2) + Cov(X1,X3) + Cov(X2,X3)

Can you elaborate on how you derived V_DCA? I figure that the random variables X1, X2, and X3 are most precisely stated as the logarithm of the appropriate month’s return so we can be using linear combinations of {X1,X2,X3} to get a random variable for (the logarithm of) the multi-month return.

I get tripped up on the derivation of V_DCA unless I “cheat” and use the approximate identity of ln(1+x) = x at particular steps. I intuit that all the money experiences X3, and thus the coefficient of Var(X3) is 1^2. I intuit that half of the money experiences X1 and X2, and thus the coefficient of Var(X1) and Var(X2) is 0.5^2. But unfortunately I can’t derive the DCA variance rigorously without approximations.

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Thanks!

My HELOC rate is the Prime Rate, exactly. Currently, that’s the Fed Funds Rate + 3.00% p.a. If I use the FFR as the risk-free rate (close enough to 3M T-bill) and 3% spread I’d simulate roughly what I’m facing today. But the exact spread is mostly immaterial (unless it’s some extreme number like 50%). Because what you pay in borrowing fees before the windfall you’ll recover by paying down the HELOC between the windfall and the last DCA installment. In the case where you borrow at a higher rate than what you earn later you will have a less attractive picture for the DCA. I have to admit that! But keep in mind that you can buy the S&P500 and

effectivelyborrow at the risk-free rate when implementing this with futures.About the approximation of the distributions and the precise formulas for the Variances: Please refer to

Wikipedia for more info on how to go back and forth between Normal and log-Normal. There is an approximation used, true. But the results still hold qualitatively. See the empirical results. I also did some Monte Carlo simulations and confirmed. The unfortunate thing is that either you assume returns are Normal. Then the Variance equations are ok, but you run into the issue of compounding when calculating the expected returns. Or you use log-Normal, in which case the compounding works but the variances are subject to approximation. Pick your poison! Either way, whether DCA beats the Lump sum by reducing the risk by 15.5% or 15% or 14.5%. The result is still the same.

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Ah, great answers. Thanks so much.

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-1 For suggesting such a complex solution for an issue and using leverage, quite a deadly combination for most people. Straightforward, simple and de-levered options should be the go to option for most.

+1 For original thought and giving a traditional approach a different spin, plus including the math. Not all of the FI community come from tech and I do think us finance guys miss some more technical talk, empirical studies and math, hence why I love your blog.

Hence, overall net neutral with this post (but interesting, nonetheless! keep up the good work!).

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Thanks for stopping by and your feedback. Glad you enjoyed the mathematical derivation!

Just because debt is bad for most people doesn’t mean it’s always bad. Remember, you only invest money that already know will come for sure in about a month. I wouldn’t even call that debt, just financial engineering. 🙂

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Thanks for this interesting post. If I can trouble you with a question: wouldn’t the logical extension of this strategy be to lump-sum invest the full borrowed amount as soon as the windfall amount is known instead of DCAing after this time? Thanks in advance for the clarification. Love your blog!

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Excellent point! There are many ways to skin a cat. Your suggestion might be preferable depending on your risk preference. Going all in all at once with leverage increases your expected return but also your expected risk. The beauty of my suggestion is that it has the same expected return and less risk. That’s essentially a no-brainer. But again, if risk reduction is not a big concern and you want to maximize expected return, going all in all at once is the way to go!

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Great angle on a topic that I assumed to be settled already!

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Thanks for stopping by ATL!!!

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Just curious what your thoughts are on possibly “splitting the difference” when it comes to DCA vs lump sum investing? In other words, find the middle ground and invest half immediately, then dollar cost averaging the rest.

I would think this would alleviate some of the risks since any major losses would be buffered by the fact that you can put more in during the dips.

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You could do that and reduce the inefficiency from the DCA. Or just do the 1/3 before, 1/3on the day of the cash flow and 1/3 after the cash flow and get the best of both worlds. 🙂

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Thanks for your response. I like the idea in theory, but personally I’m uneasy about investing a loan.

Maybe I’ll change my tune after I read through a few more of your posts, it sounds like you’re on to something

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I had totally forgotten about this article when I wrote a couple of recent posts on this topic! I shall link back to this. I’m glad you did the risk numbers and sharpe ratio. I did not publish those and I was wondering whether to do another episode just for that, but it didn’t seem worth it. You have also covered the intriguing angle of pre-funding the LS – I love it!

https://www.actuaryonfire.com/lump-sum-dollar-cost-average-investing-part-1/

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Ha, glad you liked this one. Prefunding the LS is something that can”square the circle” i.e., no loss in expected return but a slight reduction in risk because the investments are spread out. Haven’ seen that approach anywhere else. CHeers!

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[…] If you want to read some other number-crunching posts why not look at Reprise! Sequence of Returns Risk or College Investing Strategies. Want something less number heavy? Then try Bitcoin: My Best and Worst Investment. My friend and fellow blogger ERN has also written about this subject here. […]

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