Annualized and continuously compounded returns

How do you annualize a return?

Compound it. Raise the growth factor (1 + R) to the number of periods in a year, then subtract 1. Multiplying the period return ignores growth on earlier gains and understates the answer.

$$R_{annual} = (1 + R_{period})^{c} - 1$$

Here c is the number of periods in a year: 365 ÷ days, 52 for weeks, 12 for months, 1 ÷ years for a multi-year period. The result is the annualized return .

Annualizing a short period assumes it can repeat. A lucky month annualized looks spectacular and means little.

How does compounding frequency change a rate?

The more often interest is added, the more you earn, up to a limit. The stated annual rate is the quoted figure before compounding within the year. The effective annual rate (EAR) is what you actually earn over a year.

$$\text{EAR} = \left(1 + \frac{r_s}{m}\right)^{m} - 1$$

Here r_s is the stated rate and m is the number of compounding periods per year. With continuous compounding the formula becomes the following.

$$\text{EAR} = e^{r} - 1$$

What is a continuously compounded return?

It is the natural log of the growth factor. It answers: at what rate, compounded continuously, does the starting price become the ending price?

$$r_{cc} = \ln(1 + \text{HPR}) = \ln\left(\frac{P_1}{P_0}\right)$$

The continuously compounded return is always a little smaller than the holding period return for a gain, because it compounds at every instant and so needs a lower rate to get to the same place.

It has two useful properties.

• It adds across periods. The return over T periods is the sum of the period returns. Returns of 5% and 8% are 4.88% and 7.70% continuously compounded. The sum, 12.58%, is ln(1.134), and e^0.1258 − 1 = 13.4% brings you back to the holding period return.
• It projects forward simply. Future value is FV = PV × e^(r × T).

Exam tip ”Annualize” means compound. “Continuously compounded” means take the log.

Go deeper Why the log return is smaller than the simple return

For small returns, ln(1 + R) is very close to R. The two diverge as R grows, with ln(1 + R) ≈ R − R²/2 for gains. At 12% the gap is 0.67 points, at 50% it is almost 10 points: ln(1.5) = 40.5%. Gains cost more in log terms because the log curve flattens as growth factors get larger. For losses the order reverses: a 20% loss is a continuously compounded return of ln(0.8) = −22.3%, a larger negative number than −20%. In both cases the log return is the one that adds, and the simple return is the one a client sees on a statement.

Go deeper Stated rates on quoted instruments

Quoted rates on deposits, loans and bills are stated rates, and the compounding convention is part of the quote. Two accounts that both say 5% can pay different amounts if one compounds daily and the other annually. To compare them, convert each to an EAR. When a question gives a stated rate and a frequency, the first step is almost always the EAR formula, before any discounting or accumulation. The time value of money module builds on this when it works with future and present values at stated rates.