In a previous article I discussed interest rates and the effect of compounding interest – for and against us. There is another concept within finance that is often conflated with interest rates, even by professionals and big name entertainers – that is Rate of Return.

Rate of Return is the percentage change in the value of an investment over a specific period, expressed as a percentage of its initial cost.

ROR (%) = ((Final Value – Initial Cost) / Initial Cost) × 100

Interest rate, on the other hand, is the amount of interest due (or paid) per period, expressed as a proportion of the principal sum borrowed (or present value of the interest bearing account).  

Future Value = Present Value * (1 + Interest Rate) ^ Number of compounding periods.

With some algebra we can solve for interest rate instead of future value

Interest Rate = ((Future Value / Present Value) ^ (1/number of periods)) – 1

Incidentally, this is the same formula for calculating the “Compound Annual Growth Rate” (CAGR) or “Compound Rate of Return”

$100 grows to $106.09 over 2 years:

(($106.09/$100.00) ^ (1/2)) – 1 = 3% interest rate. You can confirm this by working it backwards from the original formula.

Do you notice the difference between these? Rate of return is backwards looking analysis. Interest Rate is used to predict future value.

Interest rate is calculated by a given time, represented in the formula by a number of periods. If not stated, number of periods = 1 and annual is assumed. You cannot calculate an interest rate without the knowing number of periods.

By contrast, calculating a rate of return does not require the variable of time. It is over one single period and, without knowing the length of that period, the number is meaningless. Rate of Return does not take into account the compounding effect of losses or gains.

A 20% rate of return sounds great, right! Invest $100 and get $120 back.

(120-100)/100 x 100 = 20%.

But what if that took 20 years to get that extra $20?

That 20% rate of return is equivalent to ((120/100) ^ (1/20)) – 1, or a 0.916% interest rate/CAGR.).

That’s a big difference. And will have a significant impact on your future planning.

I have seen many financial entertainers and financial professionals conflate these concepts with their clients implying that the “rule of 72” can be applied to the stocks, bonds, and mutual funds they recommend. Despite the title of this article, I’m sure it was not malicious. I want to assume angelic intentions. But accuracy in language is important.

The “rule of 72” is meant to be a way of estimating how long it takes to double your investment given a certain rate of return. According to the “rule”, if you can get a 12% rate of return, you should double your money in 6 years.

Despite its name, it is not actually a rule, it’s a mathematical shortcut for estimation. It works well enough for nonvolatile returns – like the constant 12% above. But it is dubious at best to apply it to volatile markets and investments. Consider the 2 hypothetical examples below, each 6 years, 5 positive, and each with average annual ROR of 12%, but drastically different ending values.

The first has a compound rate of return, or an effective annual interest rate, of 9.89%. Due to the losses in year 4, you would need an additional year at 12% return to have doubled your money. 

The second has a compound rate of return of 6.87%. You can double check the math using the above formulas.  That would require almost 3 additional years at 12% to make up for the losses in year 6.

Some financial entertainers, and the local providers they endorse, will tell you to “Just find a mutual fund that provides 10% rate of return.” I have heard one such entertainer instruct a caller on his show to “put that 10% return into an interest rate calculator and see how much money will you have?” in order to compare the conventional advice to more traditional methods.

Can you see the fallacies employed? Not only are they committing the gamblers fallacy and urging you to forget the disclaimer that “past performance does not guarantee future results,” but they commit the fallacy of equivocation – using the terms rate of return and interest rate interchangeably. If you could get a savings account with guaranteed 10% would you put any money in the stock market?

To further demonstrate the effect of volatility and that volume of return is more important than rate of return, consider this example.

If you have $100 and you invest it and get 100% return in you first year, how much do you have?

You guessed it, $200.  (Some professionals like to portray financial math as different and set themselves as experts that you need to depend on. Financial math is not different or harder than “regular” math.  They just don’t teach these concepts in school.)

Consider the second year is bad and you lose 50%, how much do you have?

You’re back to $100; your average rate of return is (100% + -50%)/2 years = 25% and your compound rate of return is 0%.

Continue it a third year, with +50% return this time. You now have $150. Your average rate of return for these 3 years is 33.3% ((100% + -50% + 50%)/3). Your compound return however is 14.5%.

How about a fourth year losing 25% this time. You now have $112.50 Your average rate of return is 19%. The compound rate of return is 2.99%.

Bear with me, I have to show you one more year. And you’ll see why.

What if you lose 25% again in year five. Now your average rate of return is 10% – that’s the number the financial entertainer told his caller to put in a future value calculator! How much money do you have? $84.38 for a compound rate of return of -3.34% for those 5 years.

Again, I’m going to ask- is rate of interest or volume of interest more important? Is rate of return more important or volume of return?

Humorist Will Rogers once quipped “I am not so much concerned with the return on capital as I am with the return of capital.”

You can lose a lot of money behind numbers showing positive Rates of Return. There are lies, damn lies, and then there are statistics. Rate of Return is a statistical measurement, useful for analyzing past performance. Applying it to calculate future values of volatile markets is invalid. It is only valid for prediction in assets with guarantees.

If you’re ready to take control of the banking function, or just want to learn more, click to book a free call with an advisor today.

Semper Reformanda

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