In a recent post, Canadian Capitalist showed that the effect of the HST on investor returns in mutual funds is small compared to the drag caused by fund MERs. I thought that while this conclusion is correct, his calculation of the MER impact was a little off. It turns out that we were both (slightly) wrong.
In the example, Investor A puts $100 to work in the equity market for 25 years at an average annual return of 8%, giving a final portfolio value of $685. Investor B makes a similar $100 investment in a mutual fund with a 2.5% MER. Reasoning that Investor B’s return dropped to an average of 5.5% per year, his final portfolio value works out to $381.
With the HST, the MER drag rises to 2.7% leaving 5.3% average return each year, and the final portfolio value is $363. So, compared to Investor A’s $685, the MER costs Investor B $304, and the HST costs him another $18. It’s clear that while the HST isn’t helping, the real problem is the high MER.
At first I thought that these calculations were a little off because the MER would knock 2.5% off the year-ending portfolio size, not the year-starting portfolio size. With this reasoning, Investor B’s return would be calculated as follows:
(1 + 0.08)*(1 – 0.025) = 1.053
This gives a 5.3% average return each year. Adding in the effect of the HST, we replace 0.025 with 0.027 and the average annual return drops to 5.084%.
It turns out that the truth is between Canadian Capitalist’s method and mine because of the way that the MER is reported. Mutual funds take the total costs of running the fund during the year and divide by the average assets under management during the year. The result is the MER percentage.
As it turns out, to calculate the effect of the MER on investor returns, you’ll need the e-to-the-power-of-x button on a scientific calculator or the EXP() function on a spreadsheet. For a detailed explanation of where the formula I’m about to use comes from, see the math section at the end of this post.
Here is how you can find the average yearly return after the MER is deducted:
(1 + 0.08)*(e^(–0.025)) = 1.05333
If we assume that the HST is deducted daily along with the MER, this changes to
(1 + 0.08)*(e^(–0.027)) = 1.05123
So, the average annual return is 5.333% without the HST and 5.123% with the HST. The portfolio ending value for Investor B is $367 without the HST and $349 with the HST. Compared to Investor A’s $685, the MER costs $318, and the HST costs an extra $18.
The conclusion hasn’t changed: The HST is still a minor concern compared to MERs. But, I’ve learned something new about MERs as a result of seeking precision in these calculations. Canadian Capitalist is working on a spreadsheet that takes into account more details of how mutual funds are actually run to see what effect MERs have on returns.
Given the daily values of an index (including dividends) for a year, we’re looking for the returns generated by a mutual fund 100% invested in this index. We’ll assume that there are no net fund inflows or outflows each day of the year.
m – MER (expressed as a fraction)
n – number of days in a trading year
A – average assets under management
C – total fund costs removed from the fund during the year
R – index return for the year
F – fund return for the year
v0, v1, ..., vn – index values over one year (v0 is starting value on the first day, v1 is the closing value on first day and the opening value on the second day, etc.)
f0, f1, ..., fn – corresponding fund values over one year
The average assets under management (based on each day’s closing value) are
A = (f1 + f2 + ... + fn)/n.
The one day share of the MER is m/n. Assuming the MER is taken out based on each day’s closing values, the MER for the first day is (m/n)f1. Continuing this way, the total costs for the year are
C = (m/n)*f1 + (m/n)*f2 + ... + (m/n)*fn = mA
Sanity check: MER = C/A = mA/A = m
On the first day the index rose by a factor of v1/v0. The same thing will happen to the fund on the first day before costs. On the first day, before costs are deducted, the fund will grow to
The costs will be m/n times this quantity. After costs, the fund will hold
f1 = f0*(v1/v0)*(1–m/n)
On the second day, the fund grows by a factor of v2/v1 and costs shrink it by a factor of 1–m/n:
f2 = f0*(v2/v0)*(1–m/n)^2
Continuing this way we get
fn = f0*(vn/v0)*(1–m/n)^n
The index return for the year is calculated from the index starting and ending values for the year:
vn/v0 = 1+R
Similarly, for the fund:
fn/f0 = 1+F
Substituting these into the previous equation gives
1+F = (1+R)*(1–m/n)^n
That last factor looks intimidating, but as n becomes large, it approaches e^(-m). In this case, the number of days in a year is large enough that we can use the close approximation
1+F = (1+R)*(e^(–m))
For n=250 and m=0.025, the estimate differs from the real value by only 0.0000012. In Canadian Capitalist’s example, the difference between using the estimate and using the real equation over the 25 years is just over one cent.
If we assume that HST is paid each day, then we can just replace m by m*1.08 in the previous equation. If the HST payments are delayed, then the analysis changes a little, but the final result won't be much different.