The effect of compound interest depends on frequency. Assume an annual interest rate of 12%. If we start the year with $100 and compound only once, at the end of the year, the principal grows to $112 ($100 x 1.12 = $112). If we instead compound 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

The semiannual yield is simply doubled. This is potentially confusing because the effective yield of a 12% bond-equivalent yield bond is 12.36% (i.e., 1.06^2 = 1.1236). Doubling the semiannual yield is just a bond naming convention. Therefore, if we read about an 8% bond compound semiannually, we assume this refers to 4% semiannual yield.

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

Given an annual market rate (r), the quarterly compound rate (rq) is given by:


So, for our example, where the annual market rate is 12%, the quarterly compound rate is 11.825%:

Figure 3

A similar logic applies to monthly compounding. The monthly compound rate (rm) is given here as the function of the annual market interest rate (r):


The daily compound rate (d) as a function of market interest rate (r) is given by:


Pushing It to the Limit with Continuous Compounding

Figure 4

If we increase the compound frequency to its limit, we are compounding continuously. While this may not be practical, the continuously compounded interest rate offers marvelously convenient properties. It turns out that the continuously compounded interest rate is given by:


Ln() is the natural log and in our example, the continuously compounded rate is therefore:


We get to the same place by taking the natural log of this ratio: the ending value divided by the starting value. Like this,


The latter is common when computing the continuously compounded return for a stock. For example, if the stock jumps from $10 one day to $11 on the next day, the continuously compounded daily return is given by:


What's so great about the continuously compounded rate (or return) that we will denote with rc? First, it's easy to scale it forward. Given a principal of (P), our final wealth over (n) years is given by:


'e' is the exponential function. For example, if we start with $100 and continuously compound at 8% over three years, the final wealth is given by:


Discounting to the present value (PV) is merely compounding in reverse, so the present value of a future value (F) compounded continuously at a rate of (rc) is given by:


For example, if you are going to receive $100 in three years under a 6% continuous rate, its present value 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%.


If we simply add these together, we get 40.55%. This is the two-period return:


Technically speaking, the continuous return is time consistent. Time consistency is a technical requirement for value at risk (VAR). This means that if a single-period return is a normally distributed random variable, we want multiple-period random variables to be normally distributed also. Furthermore, the multiple-period continuously compounded return is normally distributed (unlike, say, a simple percentage return).

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.

Related Articles
  1. Economics

    Understanding Interest Rates: Nominal, Real And Effective

    Interest rates can be broken down into several subcategories that incorporate various factors such as inflation. Smart investors know to look beyond the nominal or coupon rate of a bond or loan ...
  2. Investing Basics

    The Interest Rates: APR, APY And EAR

    When most people shop for financial products, all they focus on is the listed interest rate. Human eyes instinctively dismiss the fine print, which usually includes the terms APR (annual percentage ...
  3. Options & Futures

    Managing Interest Rate Risk

    Learn which tools you need to manage the risk that comes with changing rates.
  4. Investing Basics

    Overcoming Compounding's Dark Side

    Understanding how money is made and lost over time can help you improve your returns.
  5. Options & Futures

    Immunization Inoculates Against Interest Rate Risk

    Big-money investors can hedge against bond portfolio losses caused by rate fluctuations.
  6. Fundamental Analysis

    5 Basic Financial Ratios And What They Reveal

    Understanding financial ratios can help investors pick strong stocks and build wealth. Here are five to know.
  7. Investing

    What Investors Need to Know About Returns in 2016

    Last year wasn’t a great one for investors seeking solid returns, so here are three things we believe all investors need to know about returns in 2016.
  8. Economics

    The Basics Of Business Forecasting

    Whether business forecasts pertain to finances, growth, or raw materials, it’s important to remember that a forecast is little more than an informed guess.
  9. Economics

    Forces Behind Interest Rates

    Interest is a cost for one party, and income for another. Regardless of the perspective, interest rates are always changing.
  10. Markets

    The (Expected) Market Impact of the 2016 Election

    With primary season upon us, investor attention is beginning to turn to the upcoming U.S. presidential election.
  1. How do I use the rule of 72 to estimate compounding periods?

    The rule of 72 is best used to estimate compounding periods that are factors of two (2, 4, 12, 200 and so on). This is because ... Read Full Answer >>
  2. How do I use the rule of 72 to calculate continuous compounding?

    The rule of 72 is a mathematical shortcut used to predict when a population, investment or other growing category will double ... Read Full Answer >>
  3. What is continuously compounding interest?

    An interest contract with continuously compounding interest is designed to maximize the total possible interest accumulation ... Read Full Answer >>
  4. What is finance?

    "Finance" is a broad term that describes two related activities: the study of how money is managed and the actual process ... Read Full Answer >>
  5. What is the difference between positive and normative economics?

    Positive economics is objective and fact based, while normative economics is subjective and value based. Positive economic ... Read Full Answer >>
  6. Do plane tickets get cheaper closer to the date of departure?

    The price of flights usually increases one month prior to the date of departure. Flights are usually cheapest between three ... Read Full Answer >>
Hot Definitions
  1. Discouraged Worker

    A person who is eligible for employment and is able to work, but is currently unemployed and has not attempted to find employment ...
  2. Ponzimonium

    After Bernard Madoff's $65 billion Ponzi scheme was revealed, many new (smaller-scale) Ponzi schemers became exposed. Ponzimonium ...
  3. Quarterly Earnings Report

    A quarterly filing made by public companies to report their performance. Included in earnings reports are items such as net ...
  4. Dark Pool Liquidity

    The trading volume created by institutional orders that are unavailable to the public. The bulk of dark pool liquidity is ...
  5. Godfather Offer

    An irrefutable takeover offer made to a target company by an acquiring company. Typically, the acquisition price's premium ...
Trading Center