“Sound” Investment: Figuring Out How Much You Invested Before — By Solely Looking at the Current Interest Amount

By Math Vault | Financial Math

Need a dose of no-BS financial education? Here’s a good news for you. Not too long ago, we were chatting with our wildly imaginative friend and tutee Victor, who came up with the following investment question:

OK. Let’s say that a while ago, I set up a lump sum of investment with a total of \$10,000 CAD towards a Canada Savings Bond (CSB) and a Guaranteed Investment Certificate (GIC) from one of the Big Five, but because of my memory problem, I forgot what were the exact amounts I allocated to my CSB and GIC. However, since I am not totally out of my mind yet, I still remember that my bond earns at an annual interest rate of 2%, and that my GIC is a bit higher at 6%.  I also know that both of my investments are compounded daily, and payable at the end of each month.
But the question is this: Let’s say that at the end of 10 years of dedicated investment, I end up earning an interest totalling up to \$5000 CAD. Can I somehow use this information to help me recover how much I invested in CSB and GIC in the first place?

Now. That’s an excellent one! Talk about creativity here! So, what do you think? And if you just can’t wait for the answer, just read on!

Preliminary: Compound Interest

Before we move on by the way, we need to become acquainted with some basic financial math here. There are many ways as to how interests can grow and be calculated, but when we talk about (discrete) compound interest, we are also implying the investment is such that the money grows bigger — not continuously over time in a linear fashion — but incrementally and only after each period. A compounding period can be as long as a year or as short as a second, but either way, the periodic multiplier will tell us how much we get at the end of a period, for every dollar we invest in the beginning of the period.

For example, if the money is compound monthly with a periodic multiplier of 1.05 (i.e., a periodic interest rate of 5%), it just means that every time we invest 1 dollar, we will get 1.05 dollar in return in the next month. But then, this also means that if we invest $P$ dollars in the beginning of the month, we will get $P(1.05)$ by the end of the month.

Now, the term “compound” originates from the fact that the growth of money is determined — not by the original amount in the very beginning of the investment — but by the amount in the previous period. To illustrate, suppose that we start with $P$ dollars as usual (and that the periodic multiplier remains the same at 1.05), then by the end of the 2th period, what will happen is that the money will grow to 1.05 per whole of the amount at the end of the 1st period. That is, we actually get $[P(1.05)](1.05) = P(1.05)^2$ by the end of the 2th period.

Extrapolating further, if the interest is compounded semi-annually with a periodic multiplier of 1.08, then what that means is that an initial investment of P dollars will, by the end of the 12th half-year, grow to become $P(1.08)^{12} \approx P(2.518)$. That is, the amount at the end of the 12th half-year will be 2.518 per whole of the original amount, in which case, the interest earned during those 12 half-years would be $P(1.08)^{12} – P$.

And that’s pretty much all you need to know about compound interest really, with perhaps a catch: when the annual interest rate is given but the compounding period is not a year (e.g., compounded monthly or weekly), that interest rate is called the nominal interest rate (which is not the real interest rate), and it’s up to you to figure out the periodic interest rate and the associated periodic multiplier. For example, if a bank offers an investment option at a 3% of annual interest rate compounded monthly, what that really means is that the bank will — in effect — deliver the offer at a periodic interest rate of $\frac{0.03}{12}=0.0025$. In other words, the offer comes with a periodic multiplier of $1+0.0025=1.0025$.

Going Back In Time — Solving Victor’s Conundrum

In Victor’s case, we don’t really know how much he invested in CSB to begin with. But we can always call that amount $B$ (for bonds). In which case, the original amount he invested in GIC would just be $10000-B$.

Let’s figure out the corresponding periodic multipliers first. First, Victor’s CSB earns at 2% annually, compounded daily. This means that the periodic interest rate in this case would be $\frac{0.02}{365}$, so that the periodic multiplier for Victor’s CSB would be at $ 1 + \frac{0.06}{365} \approx 1.00005479452$. In a similar manner, the periodic multiplier for Victor’s GIC (which earns at 6%) would be $1 + \frac{0.06}{365} \approx 1.00016438356$.

So what’s the bottom line at the end of 10 years? Well, since both investments are compounded daily, 10 years would correspond to 3650 compounding periods in this case (while not exactly correct, it’s customary in finance to make a year equal to 365 days). This means that by the end of 10 years, Victor’s CSB will grow to $B(1.00005479452)^{3650}$, and his GIC to $(10000-B)(1.00016438356)^{3650}$. In other words, after 10 years, Victor will earn, on his CSB, an interest of:

\begin{align*} B(1.00005479452)^{3650}-B & = B(1.00005479452^{3650}-1) \\ & = B(0.22139606336) \end{align*}

and on his GIC, an interest of:

\begin{align*} & (10000-B)(1.00016438356)^{3650}-(10000-B) \\ & = 10000 (1.00016438356)^{3650}-B(1.00016438356)^{3650}-10000+B \\ & = 18220.2894361-B(1.82202894361)-10000+B \\ & = 8220.2894361-B(0.82202894361) \end{align*}

And since the total interest adds up to \$5000 CAD. The following equation is in order:

$\displaystyle B(0.22139606336)+8220.2894361-B(0.82202894361) = 5000$

Or equivalently,

$3220.2894361 = B(0.60063288025)$

Or…(drumroll)…

$B=5361.49375432$

That is, Victor must have originally invested around \$5361.5 CAD in CSB, and around 10000 – 5361.5 = \$4638.5 CAD in GIC!

(weird choice of allocation, by the way!)

All right. Problem Solved! Guess we can call it a day of financial number-crunching? 🙂

P.S. — We started the post by promising you some sound financial education. Well, it turns out our friend Victor’s fictitious investment strategy was pretty terrible. In particular, we are in no way encouraging people to invest in government bonds or GICs — Not government bonds because their yields are at a historic low these days, and not GICs because they can probably only guaranteed the purchases of some candies or veggies. 😉

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