your research department reports continuously compounded interest rates as

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Continuously Compounded Interest

The interest earned on both the initial principal invested and the accumulated interest from previous periods.

• What Is Continuously Compounded Interest?
• Understanding Continuous Compounding
• What Is Compound Interest?
• Annual, Semiannual, Quarterly, And Monthly Compounding

Continuously Compounded Interest Formula

• Example Of Continuous Compound Interest
• Uses Of Continuously Compounded Interest
• Drawbacks Of Continuous Compounding Interest

What is Continuously Compounded Interest?

Continuously compounding interest is the interest earned on both the initial  principal  invested and the accumulated interest from previous periods. When interest is said to be constantly compounded, it is compounded at every point in time. 

Continuous compounding, however, is an extreme case of compound interest as interest is being added and reinvested momentarily instead of at specific and distinct points in time. That is why it is also known as instantaneous compounding.

In reality, this idea is impossible to implement, but it is used in the finance industry as a benchmark for the most extreme case of compounding interest. 

One may think that money that is compounded continuously yields an infinite sum of money. However, a formula calculates the  future value  of a principal whose interest is being compounded instantaneously.

At the same growth rates, continuously compounding interest yields the most considerable future value of an asset compared to interest compounded yearly, semiannually, quarterly, monthly, daily, etc. That is because the asset will always earn and reinvest interest.

Key Takeaways

• Continuously compounded interest involves interest being added and reinvested at every moment, offering the most extreme case of compounding interest.
• The formula for continuous compounding is FV = PVe^it, where e is Euler's number, and it results in the highest future value compared to other compounding methods.
• Continuous compounding is theoretical but serves as a benchmark for extreme compounding scenarios in finance.
• Accounts with continuous compounding yield higher returns compared to those with discrete compounding, leading to the highest effective annual rates of return.

Understanding continuous compounding

Interest can be compounded more than once a year. It can be compounded semiannually, quarterly, monthly, and so on. However, interest can also be theoretically compounded continuously (that is to say, at every instant).

Continuous compounding differs from yearly, semiannual, and other compounding forms since interest isn't compounded discreetly, meaning that there is no particular instance that the interest gets compounded. Instead, interest is constantly compounding: at every moment in time.

With a little extra effort, we can also realize that this type of compounding yields the highest return possible on any asset or bank account compared to other discrete forms of compounding interest.

When interest is compounded instantaneously, you are theoretically earning the most compound interest possible. That is because we split our timeline into infinitely small periods, and interest accrues with each passing period.

Although it seems that continuous compounding can yield an infinite amount of compounded interest, that is not the case. There is a mathematical way of calculating the future value of an asset or account characterized by instantaneous compounding.

What is Compound Interest?

Compound interest is the interest earned on the principal amount and the interest piled up from the earlier periods. It is also considered the interest on the previously accrued interest.

An account that compounds the interest in addition to the principal will have a larger future balance than an account that only rewards interest on the principal deposit. That is because the former performance will earn interest on the accumulated interest, unlike the latter.

Simple interest is interest dedicated to the principal amount. It is the opposite of compound interest, in which the interest earned includes the previously piled-up interest in addition to the principal.

Let us see how accounts with compound interest earn more money than accounts with simple interest with an example. Assume there are two accounts you can deposit $1,000 in:

• An account that pays 10% simple interest (no interest on interest) means that each year, $1,000 * 10% = $100 will be added to the account.
• An account that pays 10% interest compounded yearly (includes interest on interest).

What would the balance of those accounts be one year after depositing $1,000?

• The first account will have a balance of 1,000 + $100 = $1,100.
• The second account will also have a balance of $1,000 * (1 + 0.1) = $1,100.

Note that both accounts will have the same balance after one year. That is because there is no interest from previous periods to be compounded so that both accounts will have the same balance at the end of the first year.

What would the balance of those accounts be two years after depositing $1,000?

• The first account will have a balance of $1,000 + $100 + $100 = $1,200.
• The second account will have a balance of $1,100 * (1 + 0.1) = $1,210.

Notice that starting from the end of year 2, the second account (the one with compound interest) will have a larger balance than the first account (the one with simple interest). Also, the difference between the balances (currently $10) will widen year by year.

What would the balance of those accounts be three years after depositing $1,000?

• The first account will have a balance of $1,000 + $100 + $100 + $100 = $1,300.
• The second account will have a balance of $1,210 * (1 + 0.1) = $1,331.

At the end of the third year, the gap between the two accounts has widened from $10 to $31. 

Annual, semiannual, quarterly, and monthly compounding

In the previous example, the interest rates are quoted as annual, meaning that interest was earned at the end of each year. Nevertheless, interest rates can also be cited as semiannual, quarterly, and monthly.

Differently quoted interest rates have interest compounded at different rates:

• Semiannual  compounding compounds the interest twice per year (once every six months),
• Quarterly  compounding compounds the interest four times a year (once every three months),
• Monthly  compounding compounds the interest 12 times a year (once a month)

The more frequently the interest is compounded, the more compound interest will be accumulated. That is to say, with same-sized deposits and equal interest rates, the account with monthly compounding will have higher future value than the accounts with annual compounding.

If an account rewards an interest rate of 10% compounded semiannually, the principal earns a 5% interest every six months. However, if the interest were compounded quarterly, the principal would earn 2.5% interest every three months.

The formula for calculating the future value of a principal with multiple rounds of compounding in a year is:

FV = PV [1 + i/n]^nt

• FV is the value of the account or the asset at a later date
• PV is the current balance of the account or the current value of the asset
• i is the interest rate or rate of growth
• n is the number of times the interest is being compounded in a year (hence n=2 for semiannual compounding, n=4 for quarterly compounding, and n=12 for monthly compounding)
• t is the number of years after which we want to calculate the future value of the account or asset

Example of Annual, Semiannual, Quarterly, And Monthly Compounding

With an example, let us see how accounts with more frequent compounding interest (larger n) earn more money than accounts with less frequent compounding interest (smaller n). 

Assume there are several accounts you can deposit $1,000 at an interest rate of 12%:

• The first account compounds interest annually (n=1)
• The second account compounds interest semiannually (n=2)
• The third account compounds interest quarterly (n=4)
• The fourth account compounds interest monthly (n =12)

The following table shows the future values of the four accounts after five years (t=5):

According to the table, the fourth account with monthly compounding interest has the highest FV after five years showing that accounts with more frequent compounding will generate more money compared to other accounts or assets with less frequent compounding.

Excel has specific functions that can automatically calculate these values with ease.

The formula for the future value of an asset or account with continuous compounding can be derived from the formula of the future value of a principal with multiple rounds of compounding in a year mentioned earlier:

When interest was compounded monthly, we replaced n by 12. When it was quarterly, we replaced n with 4. In the case of continuous compounding, n is an infinitely large number. 

Since we can't replace n because it would be mathematically impossible to calculate FV, we will have to work with limits. That is to say, n → + (tends to infinity) instead of n=+ (equals infinity).

Hence, we can start calculating the future value of an account with instantaneous compounding as follows.

FV = lim (n→+infinity) PV [1 + i/n]^nt = PVe^it

Where e is a mathematical constant known as  Euler's number , where e ≈ 2.718282.

How did Euler's number make its way into the equation? It is because of one of the mathematical formulas concerning limits, which states:

lim (n→+infinity) (1+ a/x)^x = e^a

Applying this formula to the FV for continuous compounding, we will have:

lim (n→+infinity) (1+ i/n)^n = e^i

Therefore, the formula for the FV with instantaneous compounding is:

FV = PVe^it

Example of Continuous Compound Interest

Suppose you can invest $1,000 in an account for five years, which yields an interest rate of 12% compounded continuously. We can calculate the future value of this account balance at the end of the fifth year by using the formula.

FV = PVe^it = $1,000 * 2.7182820.12*5 = $1,822.12

With the same parameters (same i and same t), the other accounts worked with previously had lower future values. Therefore, accounts with continuous compounding offer the highest return on investment compared to other accounts that offer discrete compounding.

The following table provides a summary of the returns of five accounts with different ways of compounding interest, along with their annualized ROIs (or effective annual rates).

As mentioned, the account with continuous compounding had the highest FV after five years. No other account with the same interest rate (12%) can have a higher FV after five years than the account with instantaneous compounding.

In addition, I noticed that effective annual rates of return for all accounts are greater than 12% (except for the one with yearly compounding). This is always the case with frequent compounding because it factors in the effect of compounding interest.

The account with the highest effective annual rate is the one with continuous compounding, with an annualized ROI of 12.75%; because the interest is compounded continuously, it has increased the returns on the initial principal so that the effective rate has surpassed 12%.

Uses of continuously compounded interest

In a variety of financial contexts, continuously compounded interest has several significant applications.

1. Theoretical Benchmark

In finance, continuously compounded interest is used as a theoretical benchmark to determine how much compound interest can increase in value.

Although continuous compounding might not be feasible in actual situations, it offers a theoretical structure for assessing compounding's effects throughout time.

2. Financial Modelling

Continuous compounding is frequently employed in financial modeling and analysis, particularly in quantitative finance. It allows analysts to accurately project the future value of investments, especially when dealing with intricate financial products like derivatives, options, and fixed-income securities.

3. Risk management

Continuous compounding is used to compute the continuous compounded rate of return, which offers a uniform metric for evaluating investment performance over several time periods. 

Continuous compounding aids in the efficient assessment and management of investment risks by portfolio managers and investors.

4. Comparative Analysis

Continuous compounding offers a uniform metric for assessing returns across various compounding frequencies, making it easier to compare investment possibilities. If all interest rates are converted to continuous compounding, investors will be able to allocate their money more wisely.

5. Interest Rate Modeling

Continuous compounding is a crucial component of interest rate modeling, especially when creating mathematical models like the Vasicek, Black-Scholes, and Cox-Ingersoll-Ross (CIR) models.

6. Economic Analysis

The long-term impacts of interest rates on inflation, economic growth , and other macroeconomic variables are examined using continuous compounding in economic analysis.

Economists can gain a better understanding of the dynamics of financial markets and how they affect the overall economy by introducing continuous compounding into economic models.

Drawbacks of Continuous Compounding Interest

Continuous compounding interest involves a number of practical obstacles and difficulties in addition to providing accuracy and theoretical elegance:

1. Complexity:  It is difficult to apply and comprehend without significant mathematical expertise since it incorporates difficult mathematical concepts like limits and exponential functions.

2. Data Requirements: Since continuous compounding relies on infinite compounding intervals, it may not be possible to gather precise and dependable data for computations.

3. Computational Intensity: When dealing with huge datasets or intricate financial models , the computation of constantly compounded interest necessitates intensive computational resources, which raises processing times and resource requirements.

Since financial transactions in real-world markets are often discontinuous, continuous compounding may not accurately reflect the ongoing accrual and reinvestment of interest.

4. Modeling Limitations: The quality and dependability of financial assessments may be impacted by continuous compounding models' tendency to oversimplify real-world complications, including transaction costs, market frictions, and liquidity limitations.

5. Interpretational Difficulties: Because continuous compounding is abstract, non-experts and practitioners who are not familiar with the theory may encounter difficulties in understanding its results or misinterpreting it.

Unlike discrete compounding techniques like annual, semiannual, quarterly, or monthly compounding, continuously compounded interest is a sophisticated idea in finance where interest is accumulated and reinvested constantly at every moment.

With continuously compounded interest, compounding happens instantly rather than at a set time, which makes it an extreme kind of compounding. It is a theoretical benchmark in finance to comprehend the maximum potential of compound interest, even if it is practically impossible to achieve.

Continuous compounding does not have set periods for compounding interest; rather, interest is continuously accumulated and reinvested. Compared to other compound interest methods, it produces the highest return on investment .

The future value of an asset with continuous compounding is found using the following formula: FV = PVe^it, where e is Euler's number. This formula is obtained from the general compounding formula by finding the limit as the compounding frequency approaches infinity.

In comparison to other compounding techniques, continuously compounded interest offers the best return on investment and effective yearly rates, representing the theoretical maximum potential of compounding interest.

Understanding continuous compounding aids in understanding the concepts of compound interest and its applications in finance, even though it is not practical for real-world implementation.

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• A Visual Guide to Simple, Compound and Continuous Interest Rates

Interest rates are confusing, despite their ubiquity. This post takes an in-depth look at why interest rates behave as they do.

Understanding these concepts will help understand finance (mortgages & savings rates), along with the omnipresent e and natural logarithm . Here’s our cheatsheet:

• P = principal, your initial investment (i.e., \$1,000)
• r = interest rate (i.e., 5% per year)
• n = number of time periods (i.e., 3 years)

And a quick calculator to convert APR to APY:

Why the fuss?

Interest rates are complex. Like Roman numerals and hieroglyphics, our first system “worked” but wasn’t quite ideal.

In the beginning, you might have had 100 gold coins and were paid 12% per year (percent = per cent = per hundred — those Roman numerals still show up!). It’s simple enough: we get 12 coins a year. But is it really 12?

If we break it down, it seems we earn 1 gold a month: 6 for January-June, and 6 for July-December. But wait a minute — after our June payout we’d have 106 gold in July, and yet earn only 6 during the rest of the year? Are you saying 100 and 106 earn the same amount in 6 months? By that logic, do 100 and 200 earn the same amount, too? Uh oh.

This issue didn’t seem to bother the ancient Egyptians, but did raise questions in the 1600s and led to Bernoulli’s discovery of e (sorry math fans, e wasn’t discovered via some hunch that a strange limit would have useful properties). There’s much to say about this riddle — just keep this in mind as we dissect interest rates:

• Interest rates and terminology were invented before the idea of compounding. Heck, loans were around in 1500 BC, before exponents, 0, or even the decimal point! So it’s no wonder our discussions can get confusing.
• Nature doesn’t wait for a human year before changing . Interest earnings are a type of “growth”, but natural phenomena like temperature and radioactive decay change constantly, every second and faster. This is one reason why physics equations model change with “e” and not “$(1+r)^n$”: Nature rudely ignores our calendar when making adjustments.

Learn the Lingo

As a result of these complications, we need a few terms to discuss interest rates:

• APR (annual percentage rate): The rate someone tells you (“12% per year!”). You’ll see this as “r” in the formula.
• APY (annual percentage yield): The rate you actually get after a year, after all compounding is taken into account. You can consider this “total return” in the formula. The APY is greater than or equal to the APR.

APR is what the bank tells you, the APY is what you pay (the price after taxes, shipping and handling, if you get my drift). And of course, banks advertise the rate that looks better.

Getting a credit card or car loan? They’ll show the “low APR” you’re paying, to hide the higher APY. But opening a savings account? Well, of course they’d tout the “high APY” they’re paying to look generous.

The APY (actual yield) is what you care about, and the way to compare competing offers .

Simple Interest

Let’s start on the ground floor: Simple interest pays a fixed amount over time . A few examples:

• Aesop’s fable of the golden goose: every day it laid a single golden egg. It couldn’t lay faster, and the eggs didn’t grow into golden geese of their own.
• Corporate bonds: A bond with a face value of \$1000 and 5% interest rate (coupon) pays you \$50 per year until it expires. You can’t increase the face value, so \$50/year is what you will get from the bond. (In reality, the bond would pay \$25 every 6 months).

Simple interest is the most basic type of return . Depositing \$100 into an account with 50% simple (annual) interest looks like this:

You start with a principal (aka investment) of \$100 and earn \$50 each year. I imagine the blue principal “shoveling” green money upwards every year.

However, this new, green money is stagnant — it can’t grow! With simple interest, the \$50 just sits there. Only the original \$100 can do “work” to generate money.

Simple interest has a simple formula: Every period you earn P * r (principal * interest rate). After n periods you have:

This formula works as long as “r” and “n” refer to the same time period. It could be years, months, or days — though in most cases, we’re considering annual interest. There’s no trickery because there’s no compounding — interest can’t grow.

Simple interest is useful when:

• Your interest earnings create something that cannot grow more . It’s like the golden goose creating eggs, or a corporate bond paying money that cannot be reinvested.
• You want simple, predictable, non-exponential results . Suppose you’re encouraging your kids to save. You could explain that you’ll put aside \$1/month in “fun money” for every \$20 in their piggybank. Most kids would be thrilled and buy comic books each month. If your last name is Greenspan, your kid might ask to reinvest the dividend.

In practice, simple interest is fairly rare because most types of earnings can be reinvested. There really isn’t an APR vs APY distinction, since your earnings can’t change: you always earn the same amount per year.

Really Understanding Growth

Most interest explanations stop there: here’s the formula, now get on your merry way. Not here: let’s see what’s really happening.

First, what does an interest rate mean? I think of it as a type of “speed” :

• 50 mph means you’ll travel 50 miles in the course of an hour
• r = 50% per year means you’ll earn 50% of your principal in the course of a year. If P = \$100, you’ll earn \$50/year (your “speed of money growth”).

But both types of speed have a subtlety: we don’t have to wait the full time period!

Does driving 50 mph mean you must go a full hour? No way! You can drive “only” 30 minutes and go 25 miles (50 mph * .5 hours). You could drive 15 minutes and go 12.5 miles (50 mph * .25 hours). You get the idea.

Interest rates are similar. An interest rate gives you a “trajectory” or “pace” to follow. If you have \$100 at a 50% simple interest rate, your pace is \$50/year. But you don’t need to follow that pace for a full year! If you grew for 6 months, you should be entitled to \$25. Take a look at this:

We start with \$100, in blue. Each year that blue contributes \$50 (in green) to our total amount. Of course, with simple interest our earnings are based on our original amount, not the “new total”. Connecting the dots gives us a trendline: we’re following a path of \$50/year. Our payouts look like a staircase because we’re only paid at the end of the year, but the trajectory still works.

Simple interest keeps the same trajectory: we earn “P*r” each year, no matter what (\$50/year in this case). That straight line perfectly predicts where we’ll end up.

The idea of “following a trajectory” may seem strange, but stick with it — it will really help when understanding the nature of e.

One point: the trajectory is “how fast” a bank account is growing at a certain moment. With simple interest, we’re stuck in a car going the same speed: \$50/year, or 50 mph. In other cases, our rate may change, like a skydiver: they start off slow, but each second fall faster and faster. But at any instant , there’s a single speed, a single trajectory.

(The math gurus will call this trajectory a “derivative” or “gradient”. No need to hit a mosquito with the calculus sledgehammer just yet.)

Basic Compound Interest

Simple interest should make you squirm. Why can’t our interest earn money? We should use the bond payouts (\$50/year) to buy more bonds. Heck, we should use the golden eggs to fund research into cloning golden geese!

Compound growth means your interest earns interest . Einstein called it “one of the most powerful forces in nature”, and it’s true. When you have a growing thing, which creates more growing things, which creates more growing things… your return adds up fast.

The most basic type is period-over-period return, which usually means “year over year”. Reinvesting our interest annually looks like this:

We earn \$50 from year 0 – 1, just like with simple interest. But in year 1-2, now that our total is \$150, we can earn \$75 this year (50% * 150) giving us \$225. In year 2-3 we have \$225, so we earn 50% of that, or \$112.50.

In general, we have (1 + r) times more “stuff” each year. After n years, this becomes:

Exponential growth outpaces simple, linear interest, which only had \$250 in year 3 (100 + 3*50). Compound growth is useful when:

• Interest can be reinvested , which is the case for most savings accounts.
• You want to predict a future value based on a growth trend . Most trends, like inflation, GDP growth, etc. are assumed to be “compoundable”. Yearly GDP growth of 3% over 10 years is really $(1.03)^10 = 1.344$, or a 34.4% increase over that decade.

Interest as a Factory

The typical interpretation sees money as a “blob” that grows over time. This view works, but sometimes I like to see interest earnings as a “factory” that generates more money:

Here’s what’s happening:

• Year 0: We start with \$100.
• Year 1: Our \$100 creates a \$50 “bond”.
• Year 2: The \$100 generates another \$50 bond. The \$50 generates a \$25 bond. The total is 50 + 25 = 75, which matches up.
• Year 3: Things get a bit crazy. The \$100 creates a third \$50 bond. The two existing \$50 bonds make \$25 each. And the \$25 makes a 12.50.
• Years 4 to infinity: Left as an exercise for the reader. (Don’t you love that textbook cop out?)

This is an interesting viewpoint. The \$100 just mindlessly cranks out \$50 “factories”, which start earning money independently (notice the 3 blue arrows from the blue principal to the green \$50s). These \$50 factories create \$25 factories, and so on.

The pattern seems complex, but it’s simpler in a way as well. The \$100 has no idea what those zany \$50s are up to: as far as the \$100 knows, we’re only making \$50/year.

So why’s this viewpoint useful?

• You can separate the impact of the parent (\$100) from the children. For example, at Year 3 we have \$337.50 total. The parent has earned \$150 (“3 * 50% * \$100 = \$150”, using the simple interest formula!). This means the various “children” have contributed \$337.50 – \$150 – \$100 = \$87.50, or about 1/3 the total value.
• Breaking earnings into components helps understand e. Knowing more about e is a good thing because it shows up everywhere.

And besides, seeing old ideas in a new light is always fun. For one of us, at least.

Understanding the Trajectory

Oh, we’re not done yet. One more insight — take a look at our trajectory:

With simple interest, we kept the same pace forever (\$50/year — pretty boring). With annually compounded interest, we get a new trajectory each year .

We deposit our money, go to sleep, and wake up at the end of the year:

• Year 1: “Hey, waittaminute. I’ve got \$150 bucks! I should be making \$75/year, not \$50!”. You yell at your banker, crank up the dial to \$75/year, and go to sleep again.
• Year 2: “Hey! I’ve got \$225, and should be making \$112.50 per year!”. You scream at your bank and get the rate adjusted.

This process repeats forever — we seem to never learn.

Compound Interest Revisited

Why are we waiting so long? Sure, waiting a year at a time is better than waiting “forever” (like simple interest), but I think we can do better. Let’s zoom in on a year:

Look at what’s happening. The green line represents our starting pace (\$50/year), and the solid area shows the cash in our account. After 6 months, we’ve earned \$25 but don’t see a dime! More importantly, after 6 months we have the same trajectory as when we started. The interest gap shows where we’ve earned interest, but stay on our original trajectory (based on the original principal). We’re losing out on what we should be making.

Imagine I took your money and returned it after 6 months. “Well, ya see, I didn’t use it for a full year, so I don’t really owe you any interest. After all, interest is measured per year. Per yeeeeeaaaaar. Not per 6 months.” You’d smile and send Bubba to break my legs.

Annual payouts are man-made artifacts, used to keep things simple. But in reality, money should be earned all the time. We can pay interest after 6 months to reduce the gap:

Here’s what happened:

• We start with \$100 and a trajectory of \$50/year, like normal
• After 6 months we get \$25, giving us \$125
• We head out using the new trajectory: 50% * \$125 = \$62.5/year
• After 6 months we collect 62.5/year times .5 year = 31.25. We have 125 + 31.25 = 156.25.

The key point is that our trajectory improved halfway through, and we earned 156.25, instead of the “expected” 150. Also, early payout gave us a smaller gap area (in white), since our \$25 of interest was doing work for the second half (it contributed the extra 6.25, or \$25 * 50% * .5 years).

For 1 year, the impact of rate r compounded n times is:

In our case, we had $(1 + 50\%/2)^2$. Repeating this for t years (multiplying t times) gives:

Compound interest reduces the “dead space” where our interest isn’t earning interest . The more frequently we compound, the smaller the gap between earning interest and updating the trajectory.

Continuous Growth

Clearly we want money to “come online” as fast as possible. Continuous growth is compound interest on steroids: you shrink the gap into oblivion, by dividing the year into more and more time periods:

The net effect is to make use of interest as soon as it’s created. We wait a millisecond, find our new sum, and go off in the new trajectory. Except it’s not every millisecond: it’s every nanosecond, picosecond, femtosecond, and intervals I don’t know the name for. Continuous growth keeps the trajectory perfectly in sync with your current amount.

Read the article on e for more details (e is a special number, like pi, and is roughly 2.718). If we have rate r and time t (in years), the result is:

If you have a 50% APR, it would be an APY of $e^(.50)$ = 64.9% if compounded continuously. That’s a pretty big difference! Notice that e takes care of the icky parts, like dividing by an infinite number of periods.

Why’s this useful?

• Most natural phenomena grow continuously . As mentioned earlier, physical phenomena grows on its own schedule: radioactive material doesn’t wait for the Earth to go around the Sun before deciding to decay. Any physical equation that models change is going to use $e^rt$.
• $e^rt$ is the adjustable, one-size-fits-all exponential . It sounds strange, but e can even model the jumpy, staircase-like growth we’ve seen with compound interest. We’ll get into this in a later article.

Most interest discussions leave e out, as continuous interest is not often used in financial calculations. (Daily compounding, $(1 + r/365)^365$, is generous enough for your bank account, thank you very much. But seriously, daily compounding is a pretty good approximation of continuous growth.)

The exponential e is the bridge from our jumpy “delayed” growth to the smooth changes of the natural world.

A Few Examples

Let’s try a few examples to make sure it’s sunk in. Remember: the APR is the rate they give you, the APY is what you actually earn (your true return).

• Is a 4.5 APY better than a 4.4 APR, compounded quarterly? You need to compare APY to APY. 4.4% compounded quarterly is $(1 + 4.4\%/4)^4 = 4.47% $, so the 4.5% APY is still better.
• Should I pay my mortgage at the end of the month, or the beginning? The beginning, for sure. This way you knock out a chunk of debt early, preventing that “debt factory” from earning interest for 30 days. Suppose your loan APY is 6% and your monthly payment is \$2000. By paying at the start of the month, you’d save \$2000 * 6% = \$120/year, or \$3600 throughout a 30-year mortgage. And a few grand is nothing to sneeze at.
• Should I use several small payments, or one large payment? . You want to pay debt off as early as possible. \$500/week for 4 weeks is better than \$2000 at the end of the month. Each payment stops a few weeks’ worth of interest. The math is a bit tricker, but think of it as 4 \$500 investments, each getting different return. In a month, the first payment saves 3 week’s worth of interest: $500 * (1 + daily rate)^21$. The next saves 2 weeks: $500 * (1 + daily rate)^14$. The third saves a week $500 * (1 + daily rate)^7$ and the last payment doesn’t save any interest. Regardless of the details, prepayment will save you money.

The general principle: When investing, get interest paid early, so it can compound. When borrowing, pay debt early to prevent that interest from compounding.

Onward and Upward

This is a lot for one sitting, but I hope you’ve seen the big picture:

• The interest rate (APR) is the “speed” at which money grows .
• Compounding lets you adjust your “speed” as you earn more interest . The APR is the initial speed; the APY is the actual change during the year.
• Man-made growth uses $(1+r)^n$ , or some variant. We like our loans to line up with years.
• Nature uses $e^{rt}$ . The universe doesn’t particularly care for our solar calendar.
• Interest rates are tricky. When in doubt, ask for the APY and pay debt early.

Treating interest in this funky way (trajectories and factories) will help us understand some of e’s cooler properties, which come in handy for calculus. Also, try the Rule of 72 for a quick way to compute the effect of interest rates mentally (that investment with 6% APY will double in 12 years). Happy math.

Other Posts In This Series

• An Intuitive Guide To Exponential Functions & e
• Demystifying the Natural Logarithm (ln)
• Common Definitions of e (Colorized)
• Understanding Exponents (Why does 0^0 = 1?)
• Using Logarithms in the Real World
• How To Think With Exponents And Logarithms
• Understanding Discrete vs. Continuous Growth
• What does an exponent really mean?
• Q: Why is e special? (2.718..., not 2, 3.7 or another number?)

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Compound Interest Calculator

How to use the compound interest calculator, interest rate definition, what is the compound interest definition, simple vs. compound interest, compounding frequency, compound interest formula, compound interest examples, example 1 – basic calculation of the value of an investment, example 2 – complex calculation of the value of an investment, example 3 – calculating the interest rate of an investment using the compound interest formula, example 4 – calculating the doubling time of an investment using the compound interest formula, compound interest table, additional information, behind the scenes of compound interest calculator.

This compound interest calculator is a tool to help you estimate how much money you will earn on your deposit . In order to make smart financial decisions, you need to be able to foresee the final result. That's why it's worth knowing how to calculate compound interest. The most common real-life application of the compound interest formula is a regular savings calculation.

Read on to find answers to the following questions:

• What is the interest rate definition?
• What is the compound interest definition, and what is the compound interest formula?
• What is the difference between simple and compound interest rates?
• How to calculate compound interest?
• What are the most common compounding frequencies?

Our compound interest calculator is a versatile tool that helps you forecast the growth of your investments over time. To effectively use it, follow these instructions:

Enter initial balance : Start by inputting the amount you have initially invested or saved.

Input interest rate : Type in the annual interest rate your investment will earn.

Set the term : Determine the number of years and months over which you want the investment to grow.

Select Compounding Frequency : Choose how often the interest will be compounded. Options range from annually to daily.

Additional deposits : Decide if you'll make additional deposits. If so, specify the amount, how often, whether these are at the beginning or end of the compounding period, and their annual growth rate.

Review results : The calculator will display the final balance, total compound interest, and the breakdown of interest earned on the initial balance and additional deposits. It will also show the total principal amount and the sum of additional deposits made over the term.

Visualization : You can choose to represent your balance growth visually by selecting a bar graph, pie chart, table, or a combined chart and table view.

For example, with an initial balance of $1,000 and an 8% interest rate compounded monthly over 20 years without additional deposits, the calculator shows a final balance of $4,926.80. The total compound interest earned is $3,926.80.

Whether for personal savings, retirement planning, or educational investments, this calculator offers the foresight needed to make informed financial decisions.

Read on to learn more about the magic of compound interest and how it's calculated.

In finance, the interest rate is defined as the amount charged by a lender to a borrower for the use of an asset . So, for the borrower, the interest rate is the cost of the debt, while for the lender, it is the rate of return.

Note that in the case where you make a deposit into a bank (e.g., put money in your savings account), you have, from a financial perspective, lent money to the bank. In such a case, the interest rate reflects your profit.

The interest rate is commonly expressed as a percentage of the principal amount (outstanding loan or value of deposit). Usually, it is presented on an annual basis, which is known as the annual percentage yield (APY) or effective annual rate (EAR).

Generally, compound interest is defined as interest that is earned not solely on the initial amount invested but also on any further interest . In other words, compound interest is the interest on both the initial principal and the interest that has been accumulated on this principal so far. Therefore, the fundamental characteristic of compound interest is that interest itself earns interest . This concept of adding a carrying charge makes a deposit or loan grow at a faster rate.

You can use the compound interest equation to find the value of an investment after a specified period or estimate the rate you have earned when buying and selling some investments. It also allows you to answer some other questions, such as how long it will take to double your investment.

We will answer these questions in the examples below.

You should know that simple interest is something different than the compound interest . It is calculated only on the initial sum of money. On the other hand, compound interest is the interest on the initial principal plus the interest which has been accumulated.

Most financial advisors will tell you that compound frequency is the number of compounding periods in a year. But if you are not sure what compounding is, this definition will be meaningless to you… To understand this term, you should know that compounding frequency is an answer to the question How often is the interest added to the principal each year? In other words, compounding frequency is the time period after which the interest will be calculated on top of the initial amount .

For example:

• Annual (1/Yr) compounding has a compounding frequency of one ,
• Quarterly (4/Yr) compounding has a compounding frequency of four ,
• Monthly (12/Yr) compounding has a compounding frequency of twelve .

Note that the greater the compounding frequency is, the greater the final balance. However, even when the frequency is unusually high, the final value can't rise above a particular limit.

As the main focus of the calculator is the compounding mechanism, we designed a chart where you can follow the progress of the annual interest balances visually. If you choose a higher than yearly compounding frequency, the diagram will display the resulting extra or additional part of interest gained over yearly compounding by the higher frequency . Thus, in this way, you can easily observe the real power of compounding.

The compound interest formula is an equation that lets you estimate how much you will earn with your savings account. It's quite complex because it takes into consideration not only the annual interest rate and the number of years but also the number of times the interest is compounded per year.

The formula for annual compound interest is as follows:

• F V \mathrm{FV} FV – Future value of the investment, in our calculator it is the final balance
• P P P – Initial balance (the value of the investment);
• r r r – Annual interest rate (in decimal);
• m m m – Number of times the interest is compounded per year ( compounding frequency ); and
• t t t – Numbers of years the money is invested for.

It is worth knowing that when the compounding period is one ( m = 1 m = 1 m = 1 ), then the interest rate ( r r r ) is called the CAGR (compound annual growth rate): you can learn about this quantity at our CAGR calculator .

• Do you want to understand the compound interest equation?
• Are you curious about the fine details of how to calculate the compound interest rate?
• Are you wondering how our calculator works?
• Do you need to know how to interpret the results of compound interest calculation?
• Are you interested in all possible uses of the compound interest formula?

The following examples are there to try and help you answer these questions. We believe that after studying them, you won't have any trouble with understanding and practical implementation of compound interest.

The first example is the simplest, in which we calculate the future value of an initial investment.

You invest $10,000 for 10 years at the annual interest rate of 5%. The interest rate is compounded yearly. What will be the value of your investment after 10 years?

Firstly let’s determine what values are given and what we need to find. We know that you are going to invest $ 10000 \$10000 $10000 – this is your initial balance P P P , and the number of years you are going to invest money is 10 10 10 . Moreover, the interest rate r r r is equal to 5 % 5\% 5% , and the interest is compounded on a yearly basis, so the m m m in the compound interest formula is equal to 1 1 1 .

We want to calculate the amount of money you will receive from this investment. That is, we want to find the future value F V \mathrm{FV} FV of your investment.

To count it, we need to plug in the appropriate numbers into the compound interest formula:

The value of your investment after 10 years will be $16,288.95.

Your profit will be F V − P \mathrm{FV} - P FV − P . It is $ 16288.95 − $ 10000.00 = $ 6288.95 \$16288.95 - \$10000.00 = \$6288.95 $16288.95 − $10000.00 = $6288.95 .

Note that when doing calculations, you must be very careful with your rounding. You shouldn't do too much until the very end. Otherwise, your answer may be incorrect. The accuracy is dependent on the values you are computing. For standard calculations, six digits after the decimal point should be enough.

In the second example, we calculate the future value of an initial investment in which interest is compounded monthly.

You invest $10,000 at the annual interest rate of 5%. The interest rate is compounded monthly. What will be the value of your investment after 10 years?

Like in the first example, we should determine the values first. The initial balance P P P is $ 10000 \$10000 $10000 , the number of years you are going to invest money is 10 10 10 , the interest rate r r r is equal to 5 % 5\% 5% , and the compounding frequency m m m is 12 12 12 . We need to obtain the future value F V \mathrm{FV} FV of the investment.

Let's plug in the appropriate numbers in the compound interest formula:

The value of your investment after 10 years will be $ 16470.09 \$16470.09 $16470.09 .

Your profit will be F V − P \mathrm{FV} - P FV − P . It is $ 16470.09 − $ 10000.00 = $ 6470.09 \$16470.09 - \$10000.00 = \$6470.09 $16470.09 − $10000.00 = $6470.09 .

Did you notice that this example is quite similar to the first one? Actually, the only difference is the compounding frequency. Note that only thanks to more frequent compounding this time you will earn $ 181.14 \$181.14 $181.14 more during the same period: $ 6470.09 − $ 6288.95 = $ 181.14 \$6470.09 - \$6288.95 = \$181.14 $6470.09 − $6288.95 = $181.14 .

Now, let's try a different type of question that can be answered using the compound interest formula. This time, some basic algebra transformations will be required. In this example, we will consider a situation in which we know the initial balance, final balance, number of years, and compounding frequency, but we are asked to calculate the interest rate. This type of calculation may be applied in a situation where you want to determine the rate earned when buying and selling an asset (e.g., property) that you are using as an investment.

Data and question You bought an original painting for $2,000. Six years later, you sold this painting for $3,000. Assuming that the painting is viewed as an investment, what annual rate did you earn?

Solution Firstly, let's determine the given values. The initial balance P P P is $ 2000 \$2000 $2000 and final balance F V \mathrm{FV} FV is $ 3000 \$3000 $3000 . The time horizon of the investment is 6 6 6 years, and the frequency of the computing is 1 1 1 . This time, we need to compute the interest rate r r r .

Let's try to plug these numbers into the basic compound interest formula:

We can solve this equation using the following steps: Divide both sides by 2000 2000 2000 :

Raise both sides to the 1/6 th power:

Subtract 1 1 1 from both sides:

Finally solve for r r r :

In this example you earned $1,000 out of the initial investment of $2,000 within the six years, meaning that your annual rate was equal to 6.9913%.

As you can see this time, the formula is not very simple and requires a lot of calculations. That's why it's worth testing our compound interest calculator, which solves the same equations in an instant, saving you time and effort.

Have you ever wondered how many years it will take for your investment to double its value? Besides its other capabilities, our calculator can help you to answer this question. To understand how it does it, let's take a look at the following example.

Data and question

You put $1,000 in your savings account. Assuming that the interest rate is equal to 4% and it is compounded yearly, find the number of years after which the initial balance will double.

The given values are as follows: the initial balance P P P is $ 1000 \$1000 $1000 and final balance F V \mathrm{FV} FV is 2 ⋅ $ 1000 = $ 2000 2 \cdot \$1000 = \$2000 2 ⋅ $1000 = $2000 , and the interest rate r r r is 4 % 4\% 4% . The frequency of the computing is 1 1 1 . The time horizon of the investment t t t is unknown.

Let's start with the basic compound interest equation:

Knowing that m = 1 m = 1 m = 1 , r = 4 % r = 4\% r = 4% , and F V = 2 ⋅ P \mathrm{FV} = 2 \cdot P FV = 2 ⋅ P we can write:

Which could be written as:

Divide both sides by P P P ( P P P mustn't be 0 0 0 !):

To solve for t t t , you need take the natural log ( ln ⁡ \ln ln ), of both sides:

In our example, it takes 18 years (18 is the nearest integer that is higher than 17.67) to double the initial investment.

Have you noticed that in the above solution, we didn't even need to know the initial and final balances of the investment? It is thanks to the simplification we made in the third step ( Divide both sides by P P P ). However, when using our compound interest rate calculator, you will need to provide this information in the appropriate fields. Don't worry if you just want to find the time in which the given interest rate would double your investment; just type in any numbers (for example, 1 1 1 and 2 2 2 ).

It is also worth knowing that exactly the same calculations may be used to compute when the investment would triple (or multiply by any number, in fact). All you need to do is just use a different multiple of P in the second step of the above example. You can also do it with our calculator.

Compound interest tables were used every day before the era of calculators, personal computers, spreadsheets, and unbelievable solutions provided by Omni Calculator 😂. The tables were designed to make the financial calculations simpler and faster (yes, really…). They are included in many older financial textbooks as an appendix.

Below, you can see what a compound interest table looks like.

Using the data provided in the compound interest table, you can calculate the final balance of your investment. All you need to know is that the column compound amount factor shows the value of the factor ( 1 + r ) t (1 + r)^t ( 1 + r ) t for the respective interest rate (first row) and t (first column). So to calculate the final balance of the investment, you need to multiply the initial balance by the appropriate value from the table.

Note that the values from the column Present worth factor are used to compute the present value of the investment when you know its future value.

Obviously, this is only a basic example of a compound interest table. In fact, they are usually much, much larger, as they contain more periods t t t various interest rates r r r and different compounding frequencies m m m ... You had to flip through dozens of pages to find the appropriate value of the compound amount factor or present worth factor.

With your new knowledge of how the world of financial calculations looked before Omni Calculator, do you enjoy our tool? Why not share it with your friends? Let them know about Omni! If you want to be financially smart, you can also try our other finance calculators.

Now that you know how to calculate compound interest, it's high time you found other applications to help you make the greatest profit from your investments:

To compare bank offers that have different compounding periods, we need to calculate the Annual Percentage Yield, also called Effective Annual Rate (EAR). This value tells us how much profit we will earn within a year. The most comfortable way to figure it out is using the APY calculator , which estimates the EAR from the interest rate and compounding frequency.

If you want to find out how long it would take for something to increase by n%, you can use our rule of 72 calculator . This tool enables you to check how much time you need to double your investment even quicker than the compound interest rate calculator.

You may also be interested in the credit card payoff calculator , which allows you to estimate how long it will take until you are completely debt-free.

The depreciation calculator enables you to use three different methods to estimate how fast the value of your asset decreases over time.

Tibor Pál, a PhD in Statistical Methods in Economics with a proven track record in financial analysis, has applied his extensive knowledge to develop the compound interest calculator.

Inspired by his own need to calculate long-term investment returns and simplify the process for others, Tibor created this tool. It's designed to help users plan their financial future, whether for retirement, saving for a home, or understanding the potential growth of their investments.

Tibor has extensively used this calculator in various projects, allowing him to project financial outcomes accurately and advise on investment strategies. It's become an essential tool for anyone needing to calculate the future value of their investments, considering different compounding frequencies and additional contributions.

Trust in the compound interest calculator is grounded in our rigorous standards of accuracy and reliability. Financial experts have thoroughly vetted it to ensure it meets the practical needs of both individual investors and financial professionals.

What's compound interest?

Compound interest is a type of interest that's calculated from both the initial balance and the interest accumulated from prior periods. Essentially you can see it as earning interest from interest .

What's the difference between simple and compound interest?

While simple interest only earns interest on the initial balance , compound interest earns interest on both the initial balance and the interest accumulated from previous periods .

How do I calculate compound interest?

To calculate compound interest is necessary to use the compound interest formula , which will show the FV future value of investment (or future balance):

FV = P × (1 + (r / m)) (m × t)

This formula takes into consideration the initial balance P , the annual interest rate r , the compounding frequency m , and the number of years t .

How long does it take for $1,000 to double?

With a compounding interest rate, it takes 17 years and 8 months to double (considering an annual compounding frequency and a 4% interest rate). To calculate this:

Use the compound interest formula:

Substitute the values . The future value FV is twice the initial balance P , the interest rate r = 4% , and the frequency m = 1 :

2P = P × (1 + (0.04 / 1)) (1 × t) 2 = (1.04) t

Solve for time t :

t = ln(2) / ln(1.04) t = 17.67 yrs = 17 years and 8 months

Effective corporate tax rate

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Discrete Compounding

Continuous compounding, special considerations, the bottom line.

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Discrete Compounding vs. Continuous Compounding: What's the Difference?

Discrete Compounding vs. Continuous Compounding: An Overview

People invest with the expectation of receiving more than what they invested. That added amount is commonly referred to as interest. Depending on the investment, interest can compound differently. The most common ways interest accrues is through discrete compounding and continuous compounding.

Discrete compounding and continuous compounding are closely related terms. Discretely compounded interest is calculated and added to the principal at specific intervals (e.g., annually, monthly, or weekly). Continuous compounding uses a natural log-based formula to calculate and add back accrued interest at the smallest possible intervals.

Interest can be compounded discretely at many different time intervals. Discrete compounding explicitly defines the number of and the distance between compounding periods. For example, an interest that compounds on the first day of every month is discrete.

There is only one way to perform continuous compounding—continuously. The distance between compounding periods is so small (smaller than even nanoseconds) that it is mathematically equal to zero.

Key Takeaways

• Compounding occurs when interest is paid not only on account balances but on previously-paid sums of interest.
• This "interest on interest" can lead to increasingly large returns over time, and has been heralded as the "miracle" or "magic" of compound interest.
• How often interest is paid on interest matters, as the more often it is paid, the more it will generate over time.
• Discrete compounding refers to payments made on balances at regular intervals such as weekly, monthly, or yearly.
• Continuous compounding yields the largest net return and computes (using calculus) interest paid hypothetically at every moment in time.

If the interest rate is simple (no compounding takes place), then the future value of any investment can be written as:

 F V = P ( 1 + r m ) m t where: F V = Future value P = Principal ( r / m ) = Interest rate m t = Time period \begin{aligned} &FV = P (1+ \frac{r}{m})^{mt}\\ &\textbf{where:}\\ &FV = \text{Future value}\\ &P = \text{Principal}\\ &(r/m) = \text{Interest rate}\\ &mt = \text{Time period}\\ \end{aligned} ​ F V = P ( 1 + m r ​ ) m t where: F V = Future value P = Principal ( r / m ) = Interest rate m t = Time period ​ 

Compounding interest calculates interest on the principal and accrued interest. When interest is compounded discretely, its formula is:

 FV = P ( 1 + r m ) m t where: t = The term of the contract (in years) m = The number of compounding periods per year \begin{aligned} &\text{FV} = \text{P} (1+ \frac{r}{m})^{mt}\\ &\textbf{where:}\\ &t = \text{The term of the contract (in years)}\\ &m = \text{The number of compounding periods per year}\\ \end{aligned} ​ FV = P ( 1 + m r ​ ) m t where: t = The term of the contract (in years) m = The number of compounding periods per year ​ 

Continuous compounding introduces the concept of the natural logarithm. This is the constant rate of growth for all naturally growing processes. It's a figure that developed out of physics.

The natural log is typically represented by the letter e. To calculate continuous compounding for an interest-generating contract, the formula needs to be written as:

F V = P × e r t FV=P\times e^{rt} F V = P × e r t

Many credit cards compound daily, resulting in extremely high credit card balances that are difficult to pay off. Make sure you're aware of how your credit card calculates interest and aim to pay off your balance every month to avoid increasing levels of debt.

Interest rates impact people in different ways. For example, an investor wants to earn the most interest possible, as it brings more of a return to their initial investment amount. A borrower wants the least amount of interest possible, because that makes the cost of borrowing lower. More interest on borrowed money ends up making it more expensive.

As an individual, you want to ensure that you are finding the best interest profile for yourself. In the case of an investor, they would benefit from compounding rather than simple interest , because simple interest calculates interest only on the principal amount. While this is good for borrowers, it is bad for investors.

As an investor, compounding is always the best scenario; however, if you can receive continuous compounding over discrete compounding, that is even better.

What Is an Example of Compounding Interest?

Compounding interest is interest earned on interest. For example, say you invest $5,000 that earns 5% every year. After the first year, you would have earned $250. This would be added to your initial investment of $5,000, for a new balance of $5,250. In the second year, you would earn 5% not on the $5,000, but on the $5,250. So in year two, you would've earned $262.50 in interest. Your new balance amount would be $5,512.50. In year three, interest would be calculated on the new balance of $5,512.50. You can see how with compounding interest you earn more interest over time.

What Are the Different Amounts of Time Interest Can Be Compounded?

Interest can be compounded at any time. It is dependent on who is determining the compounding intervals. Interest can be compounded daily, weekly, monthly, or annually. The more often it is compounded, the more interest is earned, and the faster your money grows.

Is Simple Interest Good?

Simple interest is good for a borrower because it calculates interest on just the principal amount, not on the interest amount that has accrued, making the cost of borrowing lower for a borrower. Simple interest is not good for an investor, as interest is only earned on the principal amount, but not on the accumulated interest, which would earn more money faster.

Compounding refers to how interest is calculated on interest on an investment. The two most common methods, discrete compounding and continuous compounding, will have different outcomes on the return of an investment.

Continuous compounding adds more interest, so it is better for investors, whereas discrete compounding adds less. However, all forms of compounding are better for investors than simple interest, which only calculates interest on the principal amount.

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Your research department reports continuously compounded interest rates as Maturity (Years) Interest Rate (%) 1.00 1.50...

Your research department reports continuously compounded interest rates as Maturity (Years) Interest Rate (%) 1.00 1.50 2.00 2.00 0.5 1.0 1.5 2.0 (a) Use these rates to compute the prices P.(0,1) and P.(0,2) of one- and two-year zero coupon bonds, and the price P.(0,2) of a two-year, 3% coupon bond. Coupons are paid semi- annually, and the face value of all bonds is 100. (b) Obtain the coupon bond's duration and convexity. (c) Suppose that the monthly changes in the interest rates have a mean of zero and a standard deviation of 0.5%. Obtain the monthly 95% Value at Risk and Expected Shortfall on the coupon bond. (d) Construct a hedge portfolio of 1 coupon bond and k one-year zero coupon bonds that has zero duration. What is the value of k? What is the convexity of the hedge portfolio? (e) Construct a hedge portfolio of 1 coupon bond, and k₁ one-year zero coupon bonds and k₂ two-year zero coupon bonds, that has zero duration and convexity. What are the values of k₁ and k₂? (f) Suppose that the yield curve shifts upward parallelly with dr = 1%. Recalculate P.(0,2), P.(0,1) and P:(0,2) and use this to calculate the change in the values of the hedge portfolios constructed in (d) and (e). Comment on the result. Your research department reports continuously compounded interest rates as Maturity (Years) Interest Rate (%) 1.00 1.50 2.00 2.00 0.5 1.0 1.5 2.0 (a) Use these rates to compute the prices P.(0,1) and P.(0,2) of one- and two-year zero coupon bonds, and the price P.(0,2) of a two-year, 3% coupon bond. Coupons are paid semi- annually, and the face value of all bonds is 100. (b) Obtain the coupon bond's duration and convexity. (c) Suppose that the monthly changes in the interest rates have a mean of zero and a standard deviation of 0.5%. Obtain the monthly 95% Value at Risk and Expected Shortfall on the coupon bond. (d) Construct a hedge portfolio of 1 coupon bond and k one-year zero coupon bonds that has zero duration. What is the value of k? What is the convexity of the hedge portfolio? (e) Construct a hedge portfolio of 1 coupon bond, and k₁ one-year zero coupon bonds and k₂ two-year zero coupon bonds, that has zero duration and convexity. What are the values of k₁ and k₂? (f) Suppose that the yield curve shifts upward parallelly with dr = 1%. Recalculate P.(0,2), P.(0,1) and P:(0,2) and use this to calculate the change in the values of the hedge portfolios constructed in (d) and (e). Comment on the result.

Expert answer:, a using the continuously compounded rates p01 e0015 0985 p02 e002 098039 pc02 ... view the full answer.

Managerial Economics

ISBN: 978-0133020267

7th edition

Authors: Paul Keat, Philip K Young, Steve Erfle

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• Fill in the blanks for the following statements: a. Some interfaces use a syntax that allows users to ask questions in their b. The step in user interface design is to to the steps as part of the...
• Match the definitions or examples in the first column with the terms in the second column: A Frequently required multiple level menu navigation B. An application's overall sequence of screens and...

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