How long will it take for an initial investment to triple compounded continuously at an annual interest rate of 5%?
What Is Continuous Compounding?Continuous compounding is the mathematical limit that compound interest can reach if it's calculated and reinvested into an account's balance over a theoretically infinite number of periods. While this is not possible in practice, the concept of continuously compounded interest is important in finance. It is an extreme case of compounding, as most interest is compounded on a monthly, quarterly, or semiannual basis. Show
Formula and Calculation of Continuous CompoundingInstead of calculating interest on a finite number of periods, such as yearly or monthly, continuous compounding calculates interest assuming constant compounding over an infinite number of periods. The formula for compound interest over finite periods of time takes into account four variables:
The formula for continuous compounding is derived from the formula for the future value of an interest-bearing investment: Future Value (FV) = PV x [1 + (i / n)](n x t) Calculating the limit of this formula as n approaches infinity (per the definition of continuous compounding) results in the formula for continuously compounded interest: FV = PV x e (i x t), where e is the mathematical constant approximated as 2.7183. Key Takeaways
What Continuous Compounding Can Tell YouIn theory, continuously compounded interest means that an account balance is constantly earning interest, as well as refeeding that interest back into the balance so that it, too, earns interest. Continuous compounding calculates interest under the assumption that interest will be compounding over an infinite number of periods. Although continuous compounding is an essential concept, it's not possible in the real world to have an infinite number of periods for interest to be calculated and paid. As a result, interest is typically compounded based on a fixed term, such as monthly, quarterly, or annually. Even with very large investment amounts, the difference in the total interest earned through continuous compounding is not very high when compared to traditional compounding periods. Example of How to Use Continuous CompoundingAs an example, assume a $10,000 investment earns 15% interest over the next year. The following examples show the ending value of the investment when the interest is compounded annually, semiannually, quarterly, monthly, daily, and continuously.
With daily compounding, the total interest earned is $1,617.98, while with continuous compounding the total interest earned is $1,618.34, a marginal difference. Compound interest is interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan. 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). 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.) Continuously compounded returns compound the most frequently of all. Continuous compounding is the mathematical limit that compound interest can reach. It is an extreme case of compounding since most interest is compounded on a monthly, quarterly, or semiannual basis. Key Takeaways
Semiannual Rates of ReturnFirst, 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%. Image by Julie Bang © Investopedia 2019The 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. While it is not always practical to use continuous compound interest, the formula for growth is much simpler than compounding at discrete intervals. Quarterly, Monthly, and Daily Rates of ReturnNow, 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. Image by Julie Bang © Investopedia 2019Given an annual market rate (r), the quarterly compound rate (rq) is given by: r q = 4 [ ( r 2 + 1 ) 1 2 − 1 ] \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%: r q = 4 [ ( 1 2 % 2 + 1 ) 1 2 − 1 ] ≅ 1 1 . 8 2 5 % \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% Image by Julie Bang © Investopedia 2019A similar logic applies to monthly compounding. The monthly compound rate (rm) is given here as the function of the annual market interest rate (r): r m = 1 2 [ ( r 2 + 1 ) 1 6 − 1 ] = 1 2 [ ( 1 2 % 2 + 1 ) 1 6 − 1 ] ≅ 1 1 . 7 1 % \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: r d = 3 6 0 [ ( r 2 + 1 ) 1 1 8 0 − 1 ] = 3 6 0 [ ( 1 2 % 2 + 1 ) 1 1 8 0 − 1 ] ≅ 1 1 . 6 6 % \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% How Continuous Compounding WorksImage by Julie Bang © Investopedia 2019If 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: r c o n t i n u o u s = ln ( 1 + r ) \begin{aligned} &r_{continuous} = \ln ( 1 + r ) \\ \end{aligned} rcontinuous=ln(1+r) When interest is compounded more frequently, the amount of interest earned in each increment of time becomes smaller, but the total amount of accumulated interest grows faster. Ln() is the natural log and in our example, the continuously compounded rate is therefore: r c o n t i n u o u s = ln ( 1 + 0 . 1 2 ) = ln ( 1 . 1 2 ) ≅ 1 1 . 3 3 % \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 this ratio: the ending value divided by the starting value. r c o n t i n u o u s = ln ( Value End Value Start ) = ln ( 1 1 2 1 0 0 ) ≅ 1 1 . 3 3 % \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(ValueStartValueEnd)=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: r c o n t i n u o u s = ln ( Value End Value Start ) = ln ( $ 1 1 $ 1 0 ) ≅ 9 . 5 3 % \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(ValueStartValueEnd)=ln($10$11)≅9.53% 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: w = P e r c n \begin{aligned} &w = Pe ^ {r_c n} \\ \end{aligned} w=Percn 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: w = $ 1 0 0 e ( 0 . 0 8 ) ( 3 ) = $ 1 2 7 . 1 2 \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 (rc) is given by: PV of F received in (n) years = F e r c n = F e − r c n \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=ercnF=Fe−rcn For example, if you are going to receive $100 in three years under a 6% continuous rate, its present value is given by: PV = F e − r c n = ( $ 1 0 0 ) e − ( 0 . 0 6 ) ( 3 ) = $ 1 0 0 e − 0 . 1 8 ≅ $ 8 3 . 5 3 \begin{aligned} &\text{PV} = Fe ^ {-r_c n} = ( \$100 ) e ^ { -(0.06)(3) } = \$100 e ^ { -0.18 } \cong \$83.53 \\ \end{aligned} PV=Fe−rcn=($100)e−(0.06)(3)=$100e−0.18≅$83.53 Scaling Over Multiple PeriodsThe 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%. ln ( 1 2 0 1 0 0 ) ≅ 1 8 . 2 3 % \begin{aligned} &\ln \left ( \frac { 120 }{ 100 } \right ) \cong 18.23\% \\ \end{aligned} ln(100120)≅18.23% ln ( 1 5 0 1 2 0 ) ≅ 2 2 . 3 1 % \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%. This is the two-period return: ln ( 1 5 0 1 0 0 ) ≅ 4 0 . 5 5 % \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 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). What Does It Mean to Be Compounded Continuously?Continuous compounding means that there is no limit to how often interest can compound. Compounding continuously can occur an infinite number of times, meaning a balance is earning interest at all times. Does Compounded Continuously Mean Daily?Compounded continuously means that interest compounds every moment, at even the smallest quantifiable period of time. Therefore, compounded continuously occurs more frequently than daily. However, daily compounding is considered close enough to continuous compounding for most purposes. Why Is Continuous Compounding Used?Continuous compounding is used to show how much a balance can earn when interest is constantly accruing. This allows investors to calculate how much they expect to receive from an investment earning a continuously compounding rate of interest. What Is the Difference Between Discrete and Continuous Compounding?Discrete compounding applies interest at specific times, such as daily, monthly, quarterly, or annually. Discrete compounding explicitly defines the time when interest will be applied. Continuous compounding applies interest continuously, at every moment in time. When Do You Use Continuous Compound Interest?You are unlikely to encounter continuous compound interest in consumer financial products, due to the difficulty of calculating interest growth over every minute and second. Continuous compound interest is most relevant to financial professionals and other specialists because the calculation is much simpler than the corresponding formula for discrete compounding interest. The Bottom LineWe 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. How long would it take for an initial investment to triple at 5% compounded annually?1 Expert Answer
It will take 22.52 years to triple the investment at interest rate of 5%.
How long does it take an investment to triple compounded continuously?Answer and Explanation: Hence, it will take approximately 36.62 years to triple the investment when compounded continuously.
How long will it take for an investment to triple if it is compounded continuously at 12?Answer and Explanation: Hence, it will take 9.2 years for an investment to triple if it is compounded continuously at 12%.
At what rate compounded continuously Will Money triple its amount in 5 years?1 Answer. At a rate of interest of 3.86% compounded annually investment will be tripled.
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