# Calculating Average Rate of Return

I'm trying to figure out how to calculate the average rate of return on some investments. I'm doing this because I'm trying to assess the impact of fees. In my theoretical scenario, let's say that I invested \$10,000 on January 1, 2007. Then, let's say that my account balance will be \$14,567 on December 31, 2017. Basically, the money has just sat in the account since January 1, 2007.

My question is, how do I calculate the average annual rate of per year using this information? In my brain, I keep coming back to this formula,

`((\$10,000 / (\$14,567 - \$10,000)) / 9) * 10`

9 is the number of years, since it takes a full year needs to be considered. Then the * 10 will get me the actual percentage. Is this correct, or am I off? Is there a better way to do this?

Thanks!

You've flipped the numerator and denominator around, and need to multiply by 100 to get percentage rather than 10:

``````(((\$14,567 - \$10,000) / \$10,000 ) / 9) * 100  = average 5.07%
``````

I like to use a simple example to assess reasonableness of an approach, if you had invested \$100 and after 1 year had \$150, your approach would yield:

``````((\$100 / (\$150 - \$100)) / 1) * 10 = 20%
``````

But since \$50 is half of \$100, we know the rate of return should be 50%, so we know that approach is off. But, flipping the numerator and denominator and multiplying by 100 gets us the 50% we expected:

``````50/100 = .5 * 100 = 50%
``````

Edit: Good catch by @DJohnM you've called it 9 years, but it's actually 11, so you'd want to adjust accordingly.

Firstly, it should be noted that the period of the investment is all of the years 2007 to 2017 inclusive.

This totals (temporarily removing one shoe and sock to extend counting range) 11 years! Not the 9 of the OP and the accepted answer...

Secondly, there are various definitions of average rate of return. One would be :

What constant annual rate of return, compounded annually, will yield the same result as the given investment?

Unfortunately, this results in an equation that cannot be solved by ordinary algebraic methods. If r is the desired annual rate, then the equation is:

(1+r)^11 * 10000 = 14567

Using logarithms:

log(1 + r) = log(1.4567)/11

r = 3.4789%

Nine years of investment would span Jan 1 2007 to Jan 1 2016.

Taking the following valuation dates

``````Jan 1 2007    p = \$10,000
Jan 1 2016    s = \$14,567
``````

with

``````r = annual interest rate
n = number of years = 9

s = p (1 + r)^n

∴ r = (s/p)^(1/n) - 1

∴ r = (14567/10000)^(1/9) - 1 = 0.0426829
``````

The annual effective interest rate is 4.268 %

Average rates of return usually assume compounding, so your formula would be

``````(\$14,567/\$10,000) ^ (1/10) - 1
``````

for annual compounding ,or

``````LN(\$14,567/\$10,000)/10
``````

for continuous compounding.

• Assuming 10 years, I would have said `r = (14567/10000)^(1/10) - 1` or `i = LN[(14567/10000)^(1/10) = 0.0376174`. Hence `p e^(r t) = 10000 e^(0.0376174 * 10) = 14567` Aug 25, 2017 at 15:50
• @ChrisDegnen You're right about the first equation - I've fixed that. Thanks. Aug 25, 2017 at 16:22