*each month*at 1%, we end up with more than $112 at the end of the year. That is, $100 x 1.01^12 at $112.68. It's higher because we compounded more frequently.

Continuously compounded returns compound the most frequently of all. Read on to learn how continuously compounded interest is calculated and why it is commonly used in finance.

**Semiannual Rates of Return, or Yields**

First, let's take a look at a potentially confusing convention. In the bond market, we refer to a bond-equivalent yield (or bond-equivalent basis). This means that if a bond yields 6% on a semiannual basis, its bond-equivalent yield is 12%.

Figure 1 |

**Quarterly, Monthly and Daily Rates of Return**

Now let's understand higher frequencies. We are still assuming a 12% annual market interest rate. Under bond naming conventions, that implies a 6% semiannual compound rate. We can now express the quarterly compound rate as a function of the market interest rate.

Figure 2 |

_{q}) is given by:

Figure 3 |

*r*) is given here as the function of the annual market interest rate (

_{m}*r*):

*d*) as a function of market interest rate (

*r*) is given by:

**Pushing It to the Limit with Continuous Compounding**

Figure 4 |

_{c}? First, it's easy to scale it forward. Given a principal of (P), our final wealth over (n) years is given by:

*compounding in reverse*, so the present value of a future value (F) compounded continuously at a rate of (r

_{c}) is given by:

**Scaling Over Multiple Periods**

The convenient property of the continuously compounded returns is that it scales over multiple periods. If the return for the first period is 4% and the return for the second period is 3%, then the two-period return is 7%. Consider we start the year with $100, which grows to $120 at the end of the first year, then $150 at the end of the second year. The continuously compounded returns are, respectively, 18.23% and 22.31%.

**The Bottom Line**

We can reformulate annual interest rates into semiannual, quarterly, monthly, or daily interest rates (or rates of return). The most frequent compounding is continuous compounding, which requires us to use a natural log and an exponential function, which is commonly used in finance due to its desirable properties: it scales easily over multiple periods and it is time consistent.