1 Expert Answer
Viktor K. answered • 07/19/20
Finance Tutor in Boca Raton
1st: compounded daily
Let's assume we have $100 and an interest rate of 7%. For the $100 to quadruple it means that the future value would be $400. Thus, because we are talking about compounding daily we will set us the equation as follows:
100 * (1+1.07)x = 400
Then we will take 400 and divide it by 100 getting:
1.07X = 4
Now we have encountered a problem where we do not know exponent, so we will use logarithm to calculate such and transform our equation to: Log1.07(4)=X
Using our calculator we will find that it takes about 20.4895 days to quadruple the money invested under 7% interest rate compounded daily.
2nd: Using the same $100 but with the rate of 5.5% compounded continuously we will be using A=PERT formula
where:
P (principal) is equal to hypothetical $100
E (e) is a mathematical constant, which is approximately 2.718
R (rate) is the interest rate, in our case it is 5.5%
T (time) is the time required for money to grow
A (amount) is the final amount desired, which is 4 times larger of $100, thus $400
We have the following:
400 = 100 * e0.055t
400/100 = e0.055t
4 = e0.055t
Then we will apply natural log to both sides of the equations and get the following:
ln(4) = ln(e0.055t)
Since e is the base of ln(x) the equation simplifies to:
ln(4) = 0.055t
Using the calculator to find ln(4) we are getting:
1.38629 = 0.055t
Lastly find t
t = 1.38629/0.055
t = 25.20535202
Plug the answers back to the original equation to verify the answers.
1st part of the question answer: t = 20.4895
2nd part of the question answer: t = 25.20535202
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Your savings account balances and investments can grow more quickly over time through the magic of compounding. Use the compound interest calculator above to see how big a difference it could make for you.
Using this compound interest calculator
Try your calculations both with and without a monthly contribution — say, $50 to $200, depending on what you can afford.
To see the interest you can expect, compare rates on NerdWallet.
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Here’s a deeper look at how compounding works:
What is compound interest?
For savers, the definition of compound interest is basic: It’s the interest you earn on both your original money and on the interest you keep accumulating. Compound interest allows your savings to grow faster over time.
In an account that pays compound interest, such as a standard savings account, the return gets added to the original principal at the end of every compounding period, typically daily or monthly. Each time interest is calculated and added to the account, the larger balance earns more interest, resulting in higher yields.
For example, if you put $10,000 into a savings account with a 2% annual yield, compounded daily, you’d earn $203 in interest the first year, another $206 the second year and so on. After 10 years of compounding, you would have earned a total of $2,214 in interest.
But remember, that’s just an example. For longer-term savings, there are better places than savings accounts to store your money, including Roth or traditional IRAs and CDs.
Compounding investment returns
When you invest in the stock market, you don’t earn a set interest rate but rather a return based on the change in the value of your investment. When the value of your investment goes up, you earn a return.
If you leave your money and the returns you earn invested in the market, those returns are compounded over time in the same way that interest is compounded.
If you invested $10,000 in a mutual fund and the fund earned a 7% return for the year, you’d gain about $700, and your investment would be worth $10,700. If you got an average 7% return the following year, your investment would then be worth about $11,500.
Over the years, your investment can really grow: If you kept that money in a retirement account over 30 years and earned that average 7% return, for example, your $10,000 would grow to more than $76,000.
In reality, investment returns will vary year to year and even day to day. In the short term, riskier investments such as stocks or stock mutual funds may actually lose value. But over a long time horizon, history shows that a diversified growth portfolio can return an average of 6% to 7% annually. Investment returns are typically shown at an annual rate of return.
The average stock market return is historically 10% annually, though that rate is reduced by inflation. Investors can currently expect inflation to reduce purchasing power by 2% to 3% a year.
Compounding can help fulfill your long-term savings and investment goals, especially if you have time to let it work its magic over years or decades. You can earn far more than what you started with.
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Compounding with additional contributions
As impressive as compound interest might be, progress on savings goals also depends on making steady contributions.
Let’s go back to the savings account example above. We started with $10,000 and ended up with about $2,214 in interest after 10 years in an account with a 2% annual yield. But by depositing an additional $100 each month into your savings account, you’d end up with about $25,509 after 10 years, when compounded daily. The interest would be about $3,509 on total deposits of $22,000.