Interest rates are very much in the news, and rightly so: The interest earned on an investment, like bonds or bond funds, can help your investments generate returns. Simple interest refers to the simple accrual of interest on your principal; compound interest refers to the interest accrued on that principal plus the interest already earned.
Note that interest is different from investment gains, which depend on market returns and other factors.
Key Points
• Simple interest applies to the initial principal alone. Most bonds pay simple interest in the form of coupon payments, for example.
• Compound interest accumulates on both the principal and the previously earned interest. If the interest in a bond fund is reinvested, rather than distributed to investors, that creates compound interest.
• Over time, compound interest results in a higher total interest paid. More frequent compounding periods accelerate compound interest growth.
• Simple interest provides a steady increase in the total amount. Compound interest generates more rapid growth in the total amount over time.
• Continuous compounding calculates interest assuming compounding over an infinite number of periods.
What Is Simple Interest?
Simple interest is the amount of money you are able to earn upon your initial investment. Simple interest works by adding a percentage of the principal — the interest — to the principal, which increases the amount of your initial investment over time.
In the case of buying a bond, which is a debt instrument, the investor loans money to the bond issuer, who agrees to repay the principal amount, plus a fixed amount of interest. Most traditional bonds make periodic payments (coupon payments) of a fixed rate of interest on the original amount.
Simple Interest Formula
Calculating interest is important to figure out how much an interest-earning and compounding investment could generate.
The simple interest formula is I = Prt, where I = interest to be paid, P is the principal, r is the interest rate (as a decimal), and t is the time in years.
So if you’re investing $200 in an interest-earning security, such as a bond or bond fund, at a 10% rate over one year, then the interest earned would be 200 x .1 x 1 = $20.
How Bond Interest Works
But let’s say you want to know how much interest you could realize before a bond matures, as that’s what you’re concerned about when initially investing. Then, you would use a different version of the formula:
P + I = P(1 + rt)
Here, P + I is the principal of the investment and the interest, which is the total amount you should earn. So to figure that out you would calculate 200 x (1 + .1 x 1), which is 200 x (1 + .1), or 200 x 1.1, which equals $220.
Benefits and Drawbacks of Simple Interest
Interest is advantageous to investors and savers, as they accrue a bit of money without any effort. If there is a drawback, perhaps it’s that simple interest tends to accrue much more slowly than compound interest.
Example of Simple Interest
For example, let’s say you were to purchase $1,000 in bonds that paid out a simple interest rate of 1%. At the end of a year, without adding or taking out any additional money, your investment would grow to $1,010.00.
In other words, multiplying the principal by the interest rate gives you a simple interest payment of $10. If you had a longer time frame, say five years, then you’d have $1,050.00.
Though these interest yields are nothing to scoff at, simple interest rates are often not the best way to see your wealth accumulate over time. Since simple interest is usually paid out as it’s earned, and isn’t compounded, it’s difficult to make headway. So each year you will continue to be paid interest, but only on your principal — not on the new amount after interest has been added.
What Is Compound Interest?
Most real-life examples of wealth increasing over time, especially in investing, are more complex. In those cases, interest may be applied to the principal multiple times in a given year, and you might have investments for a number of years. That’s compound interest at play.
Compound interest means the amount of interest you gain is based on the principal plus all the interest that has accrued. This makes the math more complicated, but in that case the formula would be:
A = P x (1 + r/n)^(nt)
Where A is the final amount, P is the principal or starting amount, r is the interest rate, t is the number of time periods, and n is how many times compounding occurs in that time period.
Example of Compound Interest
Let’s assume you invested $200 in a bond fund earning 10% interest, but have it compound quarterly, or four times a year.
So we have:
200 x (1 + .1 / 4)^(4×1)
200 x (1 + .025)^4
200 x (1.025)^4
200 x 1.10381289062
The final amount is $220.76, which is modestly above the $220 we got using simple interest. The amount earned changes as the compounding period increases.
More Examples of Compound Interest
Let’s look at two other examples: compounding 12 times a year and 265 times a year.
For monthly interest we would start at:
200 x (1 + .1/12)^(12×1)
200 x (1 + 0.0083)^12
200 x 1.00833^12
200 x 1.10471306744
220.94
If we were to compound monthly, or 12 times in the one year, the final amount would be $220.94, which is greater than the $220 that came from simple interest, above, and the $220.76 that came from the compound interest every quarter.
• Simple interest: $220
• Quarterly interest: $220.76
• Monthly interest: $220.94
Notice how we get the biggest proportional jump when we go from simple interest to quarterly interest, compared to less than 20 cents when we triple the rate of interest to monthly.
Advantages of Compound Interest Over Simple Interest
The most obvious advantage of compound interest compared to simple interest is that it allows for exponential growth of the principal. Since interest compounds on the principal amount and interest previously accrued, a saver’s wealth will increase much faster than with simple interest, which only applies to the principal.
What Is Continuous Compounding?
Continuous compounding calculates interest assuming compounding over an infinite number of periods — which is not possible, but the continuous compounding formula can tell you how much an amount can grow over time at a fixed rate of growth.
Continuous Compounding Formula
Here is the continuous compounding formula:
A = P x e^rt
A is the final amount of money that combines the initial amount and the interest
P = principal, or the initial amount of money
e = the mathematical constant e, equal for the purposes of the formula to 2.71828
r = the rate of interest (if it’s 10%, r = .1; if it’s 25%, r = .25, and so on)
t = the number of years the compounding happens for, so either the term or length of the loan or the amount of time money is saved, with interest.
Example of Continuous Compounding
Let’s work with $200, gaining 10% interest over one year, and figure out how much money you would have at the end of that period.
Using the continuously compounding formula we get:
A = 200 x 2.71828^(.1 x 1)
A = 200 x 2.71828^(.1)
A = 200 x 1.10517084374
A = $221.03
In this hypothetical case, the interest accrued is $21.03, which is slightly more than 10% of $200, and shows how, over relatively short periods of time, continuously compounded interest does not lead to much greater gains than frequent, or even simple, interest.
To see significant gains, investments or savings must be held for substantially longer, like years. The rate matters as well. Higher rates substantially affect the amount of interest accrued as well as how frequently it’s compounded.
While this math is useful to do a few times to understand how continuous compounding works, it’s not always necessary. There are a variety of calculators online.
The Limits of Compound Interest
The reason simply jacking up the number of periods can’t result in substantially greater gains comes from the formula itself. Let’s go back to A = P x (1 + r/n)^(nt)
The frequency of compounding shows up twice. It is both the figure that the interest rate is divided by, and the figure — combined with the time period — that the factor that we multiply the starting amount is raised to.
So while making the exponent of a given number larger will make the resulting figure larger, at the same time the frequency of compounding will also make the number being raised to that greater power smaller.
What the continuous compounding formula shows you is the ultimate limit of compounding at a given rate of growth or interest rate. And compounding more and more frequently gets you fewer and fewer gains above simple interest. Ultimately a variety of factors besides frequency of compounding make a big difference in how much your principal might increase.
The rate of growth or interest makes a big difference. Using our original compounding example, 15% interest compounded continuously would get you to $232.37, which is 16.19% greater than $200, compared to the just over 10% greater than $200 that continuous compounding at 10% gets you. Even if you had merely simple interest, 15% growth of $200 gets you to $230 in a year.
How Continuous Compounding Impacts Long-Term Growth
Continuous compounding can have a massive effect on long-term growth. Since your principal is earning interest, and that interest (plus principal) is earning interest, it’s possible that your rate of growth could increase exponentially. But it requires time and patience, and the larger your principal, the larger your potential yield from long-term compounding.
Interest and Investments
As noted previously, interest can play a role in an investment portfolio, but it’s important to note the distinction between investing returns and interest. Interest refers to a percentage paid at regular intervals, i.e., quarterly. Investment returns depend on the market, and typically fluctuate widely.
However, if an investor’s portfolio contains holdings in investment vehicles or assets such as bonds, there may be interest payments in the mix, which can and likely will have an impact on overall investing returns.
The Takeaway
Simple interest is the money earned on a principal amount, and compound interest is interest earned on interest and the principal. Understanding the ways in which interest rates can work is important when managing an investment portfolio that may include bonds or bond funds.
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FAQ
Can simple interest ever outperform compound interest?
Simple interest can’t ever outperform compound interest, as compound interest will always result in a higher overall yield over time.
What industries commonly use simple interest?
Simple interest is commonly used by banks and financial institutions as interest paid on some accounts. But certain types of bonds also make simple interest payments, or coupon payments, to the bondholder.
What types of accounts benefit most from compound interest?
Several types of accounts can earn compound interest, including some savings accounts, money market accounts, and even products like CDs.
Are there downsides to compound interest?
Compound interest may work against you if you’re a borrower and your debt compounds. Because the amount you owe, plus interest, earns additional interest, putting you further into debt over time.
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