Continuous Compound Interest Explained: Benefits and Examples

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Continuous compounding is a method where interest is added at every possible moment rather than at set intervals, allowing a balance to grow faster than with annual, quarterly, or even daily compounding.

Because it represents the mathematical limit of compounding frequency, it yields the highest returns and is often used in theoretical finance models and certain interest rate calculations. Understanding continuous compounding helps investors and borrowers grasp how compounding frequency affects long-term growth and cost.

Basics of Compound Interest

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%.

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 compounded semiannually, we assume this refers to a 4% semiannual yield.

Calculating Compound Interest for Different Time Periods

Now, let’s discuss 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.

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


$begin{aligned} &r_q = 4 left [ left ( frac { r }{ 2 } + 1 right ) ^ frac { 1 }{ 2 } – 1 right ] end{aligned}$​rq​=4[(2r​+1)21​−1]​

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


$begin{aligned} &r_q = 4 left [ left ( frac { 12$​rq​=4[(212%​+1)21​−1]≅11.825%​

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


$begin{aligned} r_m &= 12 left [ left ( frac { r }{ 2 } + 1 right ) ^ frac { 1 }{ 6 } – 1 right ] &= 12 left [ left ( frac { 12$rm​​=12[(2r​+1)61​−1]=12[(212%​+1)61​−1]≅11.71%​

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


$begin{aligned} r_d &= 360 left [ left ( frac { r }{ 2 } + 1 right ) ^ frac { 1 }{ 180 } – 1 right ] &= 360 left [ left ( frac { 12$rd​​=360[(2r​+1)1801​−1]=360[(212%​+1)1801​−1]≅11.66%​

Benefits of Continuous Compounding

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:

$begin{aligned} &r_{continuous} = ln ( 1 + r ) end{aligned}$​rcontinuous​=ln(1+r)​

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


$begin{aligned} &r_{continuous} = ln ( 1 + 0.12 ) = ln (1.12) cong 11.33$​rcontinuous​=ln(1+0.12)=ln(1.12)≅11.33%​

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


$begin{aligned} &r_{continuous} = ln left ( frac { text{Value}_text{End} }{ text{Value}_text{Start} } right ) = ln left ( frac { 112 }{ 100 } right ) cong 11.33$​rcontinuous​=ln(ValueStart​ValueEnd​​)=ln(100112​)≅11.33%​

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:


$begin{aligned} &r_{continuous} = ln left ( frac { text{Value}_text{End} }{ text{Value}_text{Start} } right ) = ln left ( frac { $11 }{ $10 } right ) cong 9.53$​rcontinuous​=ln(ValueStart​ValueEnd​​)=ln($10$11​)≅9.53%​

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


$begin{aligned} &w = Pe ^ {r_c n} end{aligned}$​w=Perc​n​

Note that 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:


$begin{aligned} &w = $100e ^ {(0.08)(3)} = $127.12 end{aligned}$​w=$100e(0.08)(3)=$127.12​

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 (r_c) is given by:


$begin{aligned} &text{PV of F received in (n) years} = frac { F }{ e ^ {r_c n} } = Fe ^ {-r_c n} end{aligned}$​PV of F received in (n) years=erc​nF​=Fe−rc​n​

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


$begin{aligned} &text{PV} = Fe ^ {-r_c n} = ( $100 ) e ^ { -(0.06)(3) } = $100 e ^ { -0.18 } cong $83.53 end{aligned}$​PV=Fe−rc​n=($100)e−(0.06)(3)=$100e−0.18≅$83.53​

Growth Across Multiple Periods

The convenient property of the continuously compounded returns is that they scale 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%.

$begin{aligned} &ln left ( frac { 120 }{ 100 } right ) cong 18.23$​ln(100120​)≅18.23%​

$begin{aligned} &ln left ( frac { 150 }{ 120 } right ) cong 22.31$​ln(120150​)≅22.31%​

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

$begin{aligned} &ln left ( frac { 150 }{ 100 } right ) cong 40.55$​ln(100150​)≅40.55%​

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).

Continuous Compound Interest Example

Assume a loan with 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). Interest applied only to the principal is referred to as simple interest.

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 equals $112.68. (It’s higher because we compounded more frequently.)

Now assume interest is compounded continuously, starting immediately as the loan is signed. That means that the balance due grows by 0.0329% every day. Assuming 365 days in a year, the amount due will be $100 x 1.000328^365 by the end of the year, or $112.75.

It is possible to get the total interest even higher by compounding every hour, or even every minute, but such terms would be impractical for most financial institutions. In practice, the more frequently interest is compounded, the closer the total accumulation will be to the continuous compounding formula.

The Bottom Line

Continuous compounding represents the highest possible growth rate by adding interest an infinite number of times, exceeding quarterly, monthly, or even daily compounding. It shows how balances grow faster than with discrete intervals, which is why it’s used in certain bond yield and investment return calculations.

While most real-world accounts compound less frequently, understanding continuous compounding helps investors see the impact of compounding frequency and make more informed financial decisions.