Continuous compound interest is a formula for loan interest where the balance grows continuously over time, rather than being computed at discrete interva💟ls. This formula is simpler than other methods for compounding and it allows the amount due to grow faster than othe൩r methods of calculation.

The interest on a loan accumulates faster when interest is compounded more frequently. For example, a loan that compounds every quarter will accumulate more interest than the same interest ♛rate compounded annually. Because it is computed over the smallest possible interval, continuous compound interest has the highest returns of all.

Continuously compounding is the mathematical limit that compound interest can reach. It is an extreme case of&nbs✨p;compounding since most interest is compounded on a monthly, quarterly, or semiannual basis.

Understanding Compound Interest

First, let's take a look at a potentially confusing 澳洲幸运5官方开奖结果体彩网:convention. In the bond market, we refer to a 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网:effective yield of a 12% bond-equivalent yield bond is 12.36% (i.e., 1.06^2 = 1.123𝔍6). 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.

Quarterly, Monthly, and Daily Rates of Return

Now, let's discuss higher frequencies. We are still assuming a 12% annual 澳洲幸运5官方开奖结果体彩网:market interest rate. Under bond naming conventions, that impl🥀ies a 6% semiannual compound rate. We can now express the quarterly compound rate as a function of the mar꧙ket 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 ann🧸ual market rate is 12%, the quarterly compound rate is 11.825%:

 $\begin{aligned} &r_q = 4 \left [ \left ( \frac { 12\% }{ 2 } + 1 \right ) ^ \frac { 1 }{ 2 } - 1 \right ] \cong 11.825\% \\ \end{aligned}$​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):

 $\begin{aligned} r_m &= 12 \left [ \left ( \frac { r }{ 2 } + 1 \right ) ^ \frac { 1 }{ 6 } - 1 \right ] \\ &= 12 \left [ \left ( \frac { 12\% }{ 2 } + 1 \right ) ^ \frac { 1 }{ 6 } - 1 \right ] \\ &\cong 11.71\% \\ \end{aligned}$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:

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

Compounding Over Smaller Intervals

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 contin♛uously com꧒pounded 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:

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

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

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

The latter is co🎉mmon 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:

 $\begin{aligned} &r_{continuous} = \ln \left ( \frac { \text{Value}_\text{End} }{ \text{Value}_\text{Start} } \right ) = \ln \left ( \frac { \$11 }{ \$10 } \right ) \cong 9.53\% \\ \end{aligned}$​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 𒆙three years, the final wealth is given𓆏 by:

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

Discounting to the 澳洲幸运5官方开奖结果体彩网:present value (PV) is merely compounding in reverse, so the present value of a 澳洲幸运5官方开奖结果体彩网: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&nbs🌠p;F received in (꧑n) 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​

Scaling Over Multiple Periods

The convenient property of the continuouﷺsly compounded returns is that they scale over multiple periods. If the return for the fir♚st 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 t꧋o $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%.

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

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

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

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

Technically speaking, the continuous return is time-consistent. Time consistency is a technical 澳洲幸运5官方开奖结果体彩网:requirement for value at risk (VAR). This means that if a single-period return is a normally distributed 澳洲幸运5官方开奖结果体彩网:random variable, we 🍃want multip🔜le-period random variables to be normally distributed also.

Furthermore, thܫe multiple-period continuously compounded return is normally distributed (unlike, say, a simple percentage return).

Example of Continuous Compound Interest

Assume a loan with an annual 澳洲幸运5官方开奖结果体彩网: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 ap𓃲plied 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 𒅌end 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 澳洲幸运5官方开奖结果体彩网:financial institutions. In practice, the more freq🐽uently interest is compounded, the closer the total accumulation will be to the continuous compounding formula.

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, commonly used in finance due to its desirable properties. Compounding continuously provides a calculation that can scale easily over multiple periods and is time-consistent.