How long does it take for an investment to double in value if it is invested at 14
% compounded quarterly and compounded continuously?
a] At 14% compounded quarterly,
the investment doubles in how many years?
b] At 14% compounded continuously, the investment doubles in how many years?
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Expert Answer
Step 1
Compound interest:
In compound interest, interest is added back to the principal sum so that interest is earned on that added during the next compounding period. That is, compound interest will give an interest on the interest. The interest payments will change in the time period in which the initial sum of money stays in the bank or with the barrower.
The general formula for compound interest is,
A=P⋅[1+r
n]nt
Where:
A is the future value of the investment loan including the loan,
P is the principle amount,
r is the annual interest rate in
decimals,
n is the number of times interest is compounded per year,
t is the time of years the money is invested or borrowed.
Step 2
a] Find the number of years in which the investment will be doubled at 14%
interest compounded quarterly:
The aim is to double the invested or principal amount at the given interest rate.
The future value of the investment loan including the loan should be the double of principal amount at 14
% interest compounded quarterly.
Here,
Let the principal amount or the invested amount is P.
The future value of the invested amount including the amount is A=
2P
Annual interest rate is r=14%
=0.14
The number of times interest is compounded per year is quarterly. That is, n=4.
The number of years required to double the invested money is invested t.
The number of years in which the investment will be doubled at 14%
interest compounded quarterly is obtained as 5.04 years from the calculation given below:
A=P×[1+rn]nt
2P=P×[1+0.144]4t
2=[1+0.035]4t
2=[1.035]4t
Take natural logaritm on both sides
ln[2]=ln[1.0354t]
=4
tln[1.035]
t=ln[2
]4×ln[1.035]
=5.04
Step 3 Continuous compound interest:
In compound interest, interest is added back to the principal sum so that
interest is earned on that added during the next compounding period. That is, compound interest will give an interest on the interest.
In continuous compound interest, the principal amount will be constantly earning interest and the interest keeps earning on the interest earned.
The general formula for continuous compound interest is,
A=P⋅ ert
Where:
A is the future value of
the investment loan including the loan,
P is the principle amount,
r is the interest rate in decimals,
t is the time of years the money is invested or borrowed.
Step 4
b] Find the number of years in which the investment will be doubled at 14% interest compounded continuously:
The aim is to double the invested or principal amount at the given interest rate.
The future value of the investment loan including the loan should be the double of principal amount at
14% interest compounded continuously.
Here,
Let the principal amount or the invested amount is P.
The future value of the invested amount including the amount is A=2P
Annual interest rate is r=14%=0.14,
The number of years required to double the invested money is invested t.
The number of years in which the investment will be doubled at 14% interest compounded continuously is obtained as 4.95 years from the calculation given below:
A
=P× ert
2P=P× ert
2=ert
Take natural logarithm on both sides
ln[2]=ln[e
rt]
ln
[2]=rt
t=ln[2]
r
=ln[2]0.14; [
∵ r=14%=0.14]
=4.95
Step 5
Answer: a] In 5.04 years, the investment will be doubled at 14%
interest compounded quarterly.
b] In 4.95 years, the investment will be doubled at 14% interest compounded continuously.
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The investing information provided on this page is for educational purposes only. NerdWallet does not offer advisory or brokerage services, nor does it recommend or advise investors to buy or sell particular stocks, securities or other investments.
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.
This savings calculator includes a sample rate of return. 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.