*I just found out that Chrome and Feedburner cannot handle square-root signs. I've updated the post to take them out.*

My son recently set up his first TFSA and has $450 per month flowing into it. His plan is to buy Vanguard Canada’s exchange-traded fund VCN with this money. Once his portfolio grows, he’ll consider adding other stock indexes and other asset classes. After the first deposit, he asked me a good question: “how often should I buy VCN?”

He was clever enough to figure out that making a trade every month might be too expensive, but if he waits too many months between trades, he’s giving up potential growth. There must be some optimum number of months between trades.

The following factors affect the optimum interval between trades:

*m*– yearly new savings

*r*– excess yearly return of stocks vs. cash

*c*– stock-trading commission

Bid-ask spreads are a real cost, but they don’t enter into consideration because they are the same over the course of time no matter how often you trade.

From these values we can calculate

*T*– threshold cash balance when you should trade to minimize costs

It’s time to trade when the cash balance reaches a threshold

^{1}

*T*equal to the square root of 2

*mc*/

*r*.

I’ll do a few examples below to show how to use this formula.

We’re assuming here that the opportunity cost rate

*r*is constant. In reality, the opportunity cost can turn out to be just about any value depending on how stocks perform. But we can’t know in advance how they’ll perform, so we just use some assumed average rate.

Another note about this formula is that it assumes that there will only be a single trade when the cash threshold is reached. It makes more sense to alternate among purchases of each ETF rather than to save up enough to make several purchases.

In my son’s case, we assumed that the opportunity cost is 6% per year (

*r*=0.06). He pays a $10 commission (

*c*=10). Plugging these figures into the formula simplifies the threshold where you should make a trade to approximately 18 times the square root of

*m*.

In my son’s case,

*m*= 12*450 = 5400. Plugging this into the formula we get that he should make a trade when his balance reaches $1320. His balance will be $450, then $900, and then $1350. Because $1350 is closest to the $1320 we calculated, he would buy some VCN every third month to minimize costs.

Let’s try another example. Suppose you receive $250 in dividends every quarter. Then

*m*=1000. Using

*r*=0.06 and

*c*=10, we get a threshold of about $570. Your balance will be closest to this threshold every second quarter ($500).

An aggressive saver puts away $500 on each bi-weekly pay cheque. So,

*m*=13,000. Using

*r*=0.06, and

*c*=10, we get a threshold of $2050. So, it makes sense to trade every fourth pay cheque.

Many investors will find it unsatisfying to let cash sit around while trying to optimize costs. It can certainly make sense to trade more often for emotional reasons. This is especially true if you are prone to spending the money if it’s just sitting there as cash. But for those interested in minimizing costs, this formula works well.

^{1}This formula is actually an approximation based on simple interest. If we use compound interest, we end up with equations that can only be solved numerically. It turns out that using simple interest gives a very close approximation, and there is little value in further precision. Let

*t*be the time between trades. Then assuming that the yearly cash contribution

*m*builds continuously, the cash balance will be

*mt*when we trade. The average cash balance is

*mt*/2. The foregone interest rate (using simple interest) is

*rt*. The opportunity cost is then (

*mt*/2)(

*rt*). Add to this the commission cost

*c*. We incur these costs 1/

*t*times per year. Then the total yearly cost is

*mrt*/2+

*c*/

*t*. This is a minimum when

*t*is the square root of 2

*c*/(

*mr*). The threshold cash balance for trading

*T*=

*mt*is the square root of 2

*mc*/

*r*. The only difference when new cash comes in discrete amounts instead of continuously is that we should trade when the balance is closest to the threshold value

*T*.

How about using Questrade, where all ETF purchases are free? (Selling ETF still cost $5 to $12 though)

ReplyDelete@Anonymous: I don't know much about Questrade, but I take their website at face value, the commission c for buying ETFs is zero, and my formula says the threshold for buying is thus zero. This would be different when selling.

DeleteDoes Questrade do anything to people who only buy with new money and never sell? I know these people are rare and that most ETF "investors" have high turnover. Perhaps Questrade just doesn't worry about their customers who just buy a few times per year.

They do sometimes charge ECN fees so it's not free, but much cheaper. There are quarterly fees for accounts under $5000.

DeleteIf the formula simplifies to 18m, doesn't that mean you would invest once every 18 years?

@Richard: The formula simplifies to 18 times the square root of m. Did the square root sign get lost in your viewer? If there is a problem with this in some viewers, then I'll have to find a different way to show this type of thing.

DeleteQuestrade's fees also depend on the investor's age. I believe there is no minimum account balance for investor's 25 and under. That said I personally don't use Questrade and I haven't liked some of the reviews I've read online so I don't know that I would personally recommend them.

DeleteThat explains a lot. I didn't see it at all.

Delete@Richard: So ends my foray into using slightly more exciting html. Chrome just left out the radical symbol (but left the symbols under the radical), and Feedburner just left out everything including the radical symbol and the things inside it.

DeleteJust give it another 15 years... using HTML has a lot in common with retirement planning :)

DeleteMike, ETFs are so complicated. Your son could set up a PAC with a well-managed, low cost mutual fund company like Steadyhand, Mawer, Leith Wheeler or GBC and not have to think about any formulas. Geez, ETFs are so much work.

ReplyDeleteTom Bradley

@Tom: Thanks for the chuckle. Fortunately, my son doesn't find a little math difficult.

DeleteOn a more serious note, do the firms you mention take on clients just starting out who have no investible assets (and whose parents aren't clients)? If so, then this is one of the paths for a young person to consider.

Steadyhand has a minimum of 10k per fund. Children of clients can get started with considerably less.

DeleteI thought there were barriers for those just starting out. Getting together $10k to $50k or so of savings creates some choices.

DeleteI personally solve this problem by purchasing TD e-series funds with my bi-weekly contributions. Then once a year during re-balancing I sell the mutual funds and buy respective ETFs. This way there is no idle cash and I only trade once a year (per asset class).

ReplyDeleteFor somebody just starting out ING Direct's - soon to be Tangerine - mutual funds look like a very simple and attractive way to invest. One would just automatically buy Streetwise Balanced Growth Portfolio (MER 1.07%) and not worry about it.

@IG: I prefer TD e-series over anything with an MER as high as 1%, but the fees aren't so important when the portfolio is small. The important thing is to start saving and when you're up to $50k to $100k, think about getting lower fees.

DeleteExactly. It will take over 5 years for the ETF to come out ahead on cost (assuming quarterly purchases, $9.99 commission and no growth) compared to a portfolio of 4 e-series funds (TDB900, TDB902, TDB909, TDB911) held in equal proportions (blended MER of 0.42%). All the while the e-series portfolio having a much greater diversification.

DeleteOf course, the ING's Streetwise fund will cost almost 2.5 times as much as the ETF over 5 years (more than $700).

@IG: How long it takes for ETFs to come out ahead of e-series will depend on how much you're saving, but 5 years is close to the mark for my son's saving level of $450/month. However, the cost of switching from e-series to ETFs after those 5 years can be fairly high as well with account-closing costs. Diversification is an issue if you stick with a single ETF, but there's no reason why you can't buy a different ETF each quarter to get great diversification.

DeleteOverall, I think the choice is somewhat ambiguous. I decided to advise my son to go the ETF route so that he'd get accustomed to the method that will serve him well for decades without having to switch from mutual funds to ETFs at some point.

"It’s time to trade when the cash balance reaches a threshold1 T equal to the square root of 2mc/r. Where:

ReplyDeletem – yearly new savings

r – excess yearly return of stocks vs. cash

c – stock-trading commission

"

Wow! Love the mathematical formula! :)

Most advisors would've said, "Just buy 2-3 times a year."

@Kanwal: I guess the difference is that I'm not an advisor. But maybe advisors can use the formula to get the right answer for a client and tell them how often to buy.

DeleteBrokerages would tell clients every week or every month. :)

ReplyDeleteFor some of my stocks, I haven't made a purchase in years. They are DRIPping.

I default to usually when the ETF in this case, hits near a 52-week low. If that doesn't happen during the year, I rebalance on an anniversary date.

The opportunity cost makes sense if this value is amortized over a 10-year cycle, such as you expect equities to rise 6% (or thereabouts) over a 10-year timeframe. There are years as you know better than I do, where stocks could rise 30% or be under 30%.

Your son is really saving...that's impressive. In my early 20s, is he in his early 20s or late-teens? I was busy paying lots of rent and drinking too much beer.

Mark

@Mark: I don't worry about 52-week highs or lows; I know I can't market-time.

DeleteYou have to look at opportunity cost in terms of the information you have at the time you make a decision. Just because stocks drop a few months later, you can't kick yourself for buying now. The opportunity cost will average out over time, even if it sometimes turns out to be large and sometimes negative.

I'm quite pleased that my son is saving. In fact, he saves more than I indicated because he has additional savings targeted at a car and other things.

Mmmmmmm beer.

If he opens an account with Questrade or with Virtual Brokers he doesn't have to worry about that since they do not charge anything for buying ETFs.

ReplyDelete@Anonymous: Those may be good choices, but I don't know enough about them yet.

DeleteHi Michael,

ReplyDeleteThis formula works when "m" is defined. In my case, however, I (we) may be able to save extra because either we went under budget for a category in a month or because we were able to obtain extra side-income.

In that case, what would you recommend? Obviously I don't want idle cash sitting in the chequing account but I also don't want to be losing out on potential gains. I kind of like what Mark said about buying when things are low (like right now!) but that is purely market timing, which is a big no-no.

I like how your "r" is at 6% which is more realistic than most other websites claim!

@R: My best suggestion is to guess your likely yearly savings (m). It's best not to get too caught up in optimizing everything perfectly. It's much better to focus on big issues and get them approximately right instead of over-optimizing smaller issues. I saw this issue with a friend who left a job and was trying to decide whether to take his money out of the pension plan or leave it in. An advisor (employed by the company) did an elaborate calculation and concluded he should leave it in. However, the real issue was whether the company could fund the pension. It turns out they couldn't fully fund it.

DeleteHi Mike,

ReplyDeleteThanks for the note. I guess the point is that if I have excess money that I won't be needing for at least the next 5 years, and the cost of purchasing the fund is less than 1% then to proceed with it.

I did read about the CanajunFinance and it was certainly interesting.