# Annualized return help?

Came across this question:

Over a period of 16 months, an investor has earned a return of 12%. The investor’s annualized return is closest to: 8.87%. 9.38%. 9.00%.

Was debating between 9.38% and 8.87%, I went with the wrong answer, 9.38%.

I calculated it as such: .12 / 16 = .0075 + 1 = 1.0075, I then took and raised it to the 12th power 1.0075^12 = 1.0938 = 9.38%

I then thought to myself, that really doesn’t seem right and did the following: 1.12^(12/16) = 1.0887 = 8.87%

I went with my first instict and was wrong, can someone just quickly explain why? Was my 9.38% a monthly compunded rate as opposed to 8.87% which was annualized?

Here’s to hoping Magician can swoop in and let me know if my reasoning is correct, or am I wrong about being wrong?

Thanks

Think at what you have been given:

12% or 1.12 is your dollar after 16 months of investment (16/12 is about ~1.33 years)

In otherwords, (1+x%)^(16/12)= 1.12 or 12%

Solve for x by raising 1.12 to (12/16) and subtracting 1 to give 0.0887 or 8.87%.

I think your first answer was the equivalent of saying 12 percent was a stated (non-compounded) rate (like an APR but for 16 months). However, it really was some rate, (x), compounded over 16 months to give you 12% return over the period.

Hope this helps!

I’d have done it the other way round, but it amounts to the same thing:

1.12^(12/16) = 1 + r

1.088713 = 1 + r

r = 8.8713%

When you multiplied 12% by 12/16, you were treating it as if it were a nominal rate, not an effective rate. To get from one effective rate to another, you compound.

Period.

You don’t multiply.

Ever.

Is there a reason why we implicity assume it’s an effective rate?

There are a few interest rates we know to be nominal:

• LIBOR
• BEY
• Mortgage rates

In each case, we’re told (or know by convention) how to convert them to an effective rate:

• LIBOR: the tenor of the LIBOR rate is the period over which it is effective (e.g., a 60-day LIBOR rate of 4.5% is an effective rate for 60 days: 4.5%(60/360) = 0.75% effective for 60 days
• BEY: this is twice the semiannual effective rate (e.g., a BEY of 3.8% means an effective semiannual rate of 1.9%)
• Mortgage rate: in the US, this is 12 times the monthly effective rate (e.g., a mortgage rate of 4.8% is a monthly effective rate of 0.4%)

Without being told explicitly the compounding period, or having an interest rate (such as LIBOR or BEY) with a known convention on its compounding period, we have no choice but to treat the rate as effective.

Furthermore, as a practical matter, if you were told that over a period of 16 months someone had earned 12% on a \$100 investment, wouldn’t you take that to mean that 16 month after they invested \$100 their portfolio was worth \$112? If you divide the ending balance by the beginning balance, the result is always (1 + r), where r is the effective rate for the holding period. That’s what effective rate means.