RRSP, TFSA or Unregistered placements?
A question that I always see on forum is the following: "How should I invest my money: RRSP, TFSA, mortgage/unregistered..."? Then the answers are generally tinted with either rumours or sentimental opinions. For example many people prefer paying down their house for the satisfaction to own it or some other would even try to give you finance you RRSP contribution. Of course it depends on your values, but I wanted to approach this on a pure mathematic basis.
First what is a RRSP? It stands for Registered Retirement Saving Plan and its purpose is to pay the taxes later so that you can earn interest on your gross income.
TFSA on the other side stands for Tax Free Saving Account and is a placement where you don’t have to pay taxes on its revenues but the money you invest in is already taxed.
So to solve that out we needs to define a few variables:
1-n : Is the actual year
2-N : Is the number of years for the placements
3-T(n) : Is the tax rate at the year “n”.
4-G : Is the original gross value invested at year 0.
5-R(n) : Is the value of the RRSP.
6-F(n) : Is the value of the TFSA.
7-U(n) : Is the value of the unregistered placement.
8-i : Is the interest rate in percentage.
In order to calculate this we need to recall the following formulas:
1-The compound interest: Value(n) = Value(0) * (1 + i/100) ^ n
2-The amount after taxes: Net_Value(n) = Gross_Value(n) * (1 – T(n)/100)
So the final value for U(n), R(n) and T(n) and be computed as following:
The gross value has compounded interest and than taxes is applied on both:
R(N) = G * (1 + i/100) ^ N * (1 – T(N)/100)
The gross value has the taxes applied first than compounded without taxes on the interest:
F(N) = G * (1 – T(0)/100) * (1 + i/100) ^ N
The gross value has the taxes applied first than compounded with taxes on the interest: (The interest rate is reduced by the taxes)
U(N) = G * (1 – T(0)/100) * (1 + i/100 * (1 – T(0)/100)) ^ N
Note: In this formula U(N) consider a constant tax rate of T(0) for simplification purpose.
The first observation to make is R(N) and T(N) are almost identical. Indeed due to the fact that multiplication is commutative, we can rewrite both equation as following:
R(N) = G * (1 + i/100) ^ N * (1 – T(N)/100)
F(N) = G * (1 + i/100) ^ N * (1 – T(0)/100)
The only difference between the two equations are T(N) and T(0) which are the taxes rate either at beginning or at the end of the placement. So choosing between the two saving turn out to be only a question of: “Do you think you will pay more or less taxes when you will withdraw that money?”. For someone working up to 67 years old with a private pension plan, there are chance that taxes increase in general, your tax bracket also has great chances to increase as well over time. In this case TFSA is probably a safer bet. For someone planning for early retirement and live on the RRSP for a while without other income source, there are good change T(N) will be lower than T(0) hence RRSP would be more recommendable.
In any case, TFSA is also more flexible as you can re-contribute withdrawn money next year. So if the amount is to serve as an emergency fund, TFSA is for sure a good thing.
Now let’s compare U(N) with either R(N) or F(N) as we will assume from there a flat tax rate (T(n) = Constant). From the equation below we had:
F(N) = G * (1 – T(0)/100) * (1 + i/100) ^ N
U(N) = G * (1 – T(0)/100) * (1 + i/100 * (1 – T(0)/100)) ^ N
It is obvious that for the same interest rate F(N) will be greater as the T(N)/100 make the interest less effective. So we will evaluate the required interest rate to compensate the difference by introducing two new variable:
iU: Is the interest rate of the placement U(n)
iF: Is the interest rate of the placement F(n)
So if we desire:
F(N) < U(N)
That mean:
G * (1 – T(0)/100) * (1 + iF/100) ^ N < G * (1 – T(0)/100) * (1 + iU/100 * (1 – T(0)/100)) ^ N
If we make the assumption N is positive (Which is the case in the real world) and after performing simplifications we obtains the following condition:
iU > iF / (1 – T(0)/100)
This equation is important because if you plan to pay down your mortgage instead of placing your money you should evaluate if the interest you will save are some significant than the one you would get from placing this money. If you the stock market doesn’t afraid you, there are good chances iU will be smaller than iF / (1 – T(0)/100) as the average value for iU those days is around 3~4% and iF around 5~6%! However if you are more the type of person that prefer the guaranteed placements yout iF risk to be around 1~2% so it is worth to do the maths!
As example:
If you can get 1.35% interest rate in guaranteed placement, your mortgage is 4% and your marginal tax rate is 35%
4 > 2 / (1-0.35/100)
4 > 2.077
Investing in the mortgage is a good idea in this case.
The following table gives some more examples:
To complete this article I created a hypothetic scenario to visualize what is happening with your money value for three types of placements. Of course the unregistered placement is out of the race as it uses the same interest rate as the other placement which may not be the case when paying mortgage. We can also clearly see at the end that both RRSP and TFSA result into the same final value when the same tax rate is involved.
